far-field and near-field optical trapping...current optical trapping models based on ray optics...
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Far-field and near-field opticaltrapping
A thesis submitted for the degree of
Doctor of Philosophy
by
Djenan Ganic
Centre for Micro-Photonics
Faculty of Engineering and Industrial Sciences
Swinburne University of Technology
Melbourne, Australia
for Dalila, Denis , and Elma.
In science the credit goes to the man who convinces the world, not the man
to whom the idea first occurs.
— Sir Francis Darwin (1848 - 1925)
I was born not knowing and have had only a little time to change that here
and there.
— Richard Feynman (1918 - 1988)
Declaration
I, Djenan Ganic, declare that this thesis entitled:
“Far-field and near-field optical trapping”
is my own work and has not been submitted previously, in whole or in part,in respect of any other academic award.
Djenan Ganic
Centre for Micro-PhotonicsFaculty of Engineering and Industrial SciencesSwinburne University of TechnologyAustralia
Dated this day, February 25, 2005
Abstract
Optical trapping techniques have become an important and irreplaceable tool
in many research disciplines for reaching non-invasively into the microscopic
world and to manipulate, cut, assemble and transform micro-objects with
nanometer precision and sub-micrometer resolution. Further advances in
optical trapping techniques promise to bridge the gap and bring together the
macroscopic world and experimental techniques and applications of micro-
systems in areas of physics, chemistry and biology.
In order to understand the optical trapping process and to improve and
tailor experimental techniques and applications in a variety of scientific
disciplines, an accurate knowledge of trapping forces exerted on particles
and their dependency on environmental and morphological factors is of
crucial importance. Furthermore, the recent trend in novel laser trapping
experiments sees the use of complex laser beams in trapping arrangements for
achieving more controllable laser trapping techniques. Focusing of such beams
with a high numerical aperture (NA) objective required for efficient trapping
leads to a complicated amplitude, phase and polarisation distributions of an
electromagnetic field in the focal region. Current optical trapping models
based on ray optics theory and the Gaussian beam approximation are
inadequate to deal with such a focal complexity.
Novel applications of the laser trapping such as the particle-trapped
i
ABSTRACT
scanning near field optical microscopy (SNOM) and optical-trap nanometry
techniques are currently investigated largely in the experimental sense or with
approximated theoretical models. These applications are implemented using
the efficient laser trapping with high NA and evanescent wave illumination of
the sample for high resolution sensing. The proper study of these novel laser
trapping applications and the potential benefits of implementation of these
applications with complex laser beams requires an exact physical model for
the laser trapping process and a nanometric sensing model for detection of
evanescent wave scattering.
This thesis is concerned with comprehensive and rigorous modelling
and characterisation of the trapping process of spherical dielectric particles
implemented using far-field and near-field optical trapping modalities. Two
types of incident illuminations are considered, the plane wave illumination
and the doughnut beam illumination of various topological charges. The
doughnut beams represent one class of complex laser beams. However, our
optical trapping model presented in this thesis is in no way restricted to
this type of incident illumination, but is equally applicable to other types
of complex laser beam illuminations. Furthermore, the thesis is concerned
with development of a physical model for nanometric sensing, which is of
great importance for optical trapping systems that utilise evanescent field
illumination for achieving high resolution position monitoring and imaging.
The nanometric sensing model, describing the conversion of evanescent
photons into propagating photons, is realised using an analytical approach
to evanescent wave scattering by a microscopic particle. The effects of
an interface at which the evanescent wave is generated are included by
considering the scattered field reflection from the interface. Collection and
imaging of the resultant scattered field by a high numerical aperture objective
is described using vectorial diffraction theory. Using our sensing model, we
PhD thesis: Far-field and near-field optical trapping ii
ABSTRACT
have investigated the dependance of the scattering on the particle size and
refractive index, the effects of the interface on the scattering cross-section,
morphology dependent resonance effects associated with the scattering
process, and the effects of the incident angle of a laser beam undergoing
total internal reflection to generate an evanescent field. Furthermore, we
have studied the detectability of the scattered signal using a wide area
detector and a pinhole detector. A good agreement between our experimental
measurements of the focal intensity distribution in the back focal region of
the collecting objective and the theoretical predictions confirm the validity of
our approach.
The optical trapping model is implemented using a rigorous vectorial
diffraction theory for characterisation of the electromagnetic field distribution
in the focal region of a high NA objective. It is an exact model capable
of considering arbitrary amplitude, phase and polarisation of the incident
laser beam as well as apodisation functions of the focusing objective. The
interaction of a particle with the complex focused field is described by an
extension of the classical plane wave Lorentz-Mie theory with the expansion
of the incident field requiring numerical integration of finite surface integrals
only. The net force exerted on the particle is then determined using the
Maxwell stress tensor approach. Using the optical trapping model one can
consider the laser trapping process in the far-field of the focusing objective,
also known as the far-field trapping, and the laser trapping achieved by
focused evanescent field, i.e. near-field optical trapping.
Investigations of far-field laser trapping show that spherical aberration
plays a significant role in the trapping process if a refractive index mismatch
exists between the objective immersion and particle suspension media. An
optical trap efficiency is severely degraded under the presence of spherical
aberration. However, our study shows that the spherical aberration effect
PhD thesis: Far-field and near-field optical trapping iii
ABSTRACT
can be successfully dealt with using our optical trapping model. Theoretical
investigations of the trapping process achieved using an obstructed laser
beam indicate that the transverse trapping efficiency decreases rapidly with
increasing size of the obstruction, unlike the trend predicted using a ray
optics model. These theoretical investigations are in a good agreement with
our experimentally observed results.
Far-field optical trapping with complex doughnut laser beams leads to
reduced lifting force for small dielectric particles, compared with plane wave
illumination, while for large particles it is relatively unchanged. A slight
advantage of using a doughnut laser beam over plane wave illumination for far-
field trapping of large dielectric particles manifests in a higher forward axial
trapping efficiency, which increases for increasing doughnut beam topological
charge. It is indicated that the maximal transverse trapping efficiency
decreases for reducing particle size and that the rate of decrease is higher for
doughnut beam illumination, compared with plane wave illumination, which
has been confirmed by experimental measurements.
A near-field trapping modality is investigated by considering a central
obstruction placed before the focusing objective so that the obstruction size
corresponds to the minimum convergence angle larger than the critical angle.
This implies that the portion of the incident wave that is passed through
the high numerical aperture objective satisfies the total internal reflection
condition at the surface of the coverslip, so that only a focused evanescent
field is present in the particle suspension medium. Interaction of this focused
near-field with a dielectric micro-particle is described and investigated using
our optical trapping model with a central obstruction. Our investigation
shows that the maximal backward axial trapping efficiency or the lifting
force is comparable to that achieved by the far-field trapping under similar
conditions for either plane wave illumination or complex doughnut beam
PhD thesis: Far-field and near-field optical trapping iv
ABSTRACT
illumination. The dependence of the maximal axial trapping efficiency on the
particle size is nearly linear for near-field trapping with focused evanescent
wave illumination in the Mie size regime, unlike that achieved using the far-
field trapping technique.
PhD thesis: Far-field and near-field optical trapping v
Acknowledgements
In March 2000, after several discussions with Prof. Min Gu and Dr. Xiaosong
Gan, I undertook a PhD research project at the Centre for Micro-Photonics
(CMP), which forms an integral part of the Swinburne Optronics and Laser
Laboratories at Swinburne University of Technology in Hawthorn. These
discussions were very fruitful and resulted in a framework of a research project
dealing with optical trapping technology. I was very thrilled and excited about
this new project, the feeling which has not subsided ever since. Therefore, first
and foremost I would like to thank my principal supervisor Prof. Min Gu and
my associate supervisor Dr. Xiaosong Gan for the remarkable opportunity
to embark upon a journey of research and discovery into the wonderful world
of physics. Their relentless encouragements, advices and guidance made the
completion of my research work possible, for which I am immensely grateful.
I would also like to thank Swinburne University of Technology and the
Centre for Micro-Photonics for their financial support throughout the dura-
tion of my candidature, with a Swinburne Chancellor’s Research Scholarship
and a Micro-Photonics Postgraduate Scholarship, which made the research
work so much easier.
My sincerest gratitude goes to Dr. Daniel Day for introducing me to
the field of the experimental physics during my Honors project at Victoria
University. Thank you Dr. James Chon for valuable discussions on the
vi
ACKNOWLEDGEMENTS
vectorial diffraction theory and near-field trapping. To the other current
members of the CMP staff, Dr. Martin Straub, Dr. Charles Cranfield, Dr.
Guangyong Zhou, and Ms. Shuhui Wu, I thank you for your help and advices
during our meetings. I would also like to thank the previous members of the
CMP staff, Dr. Ross Ashman, Dr. Ming Gun Xu, Dr. Xiaoyuan Deng, and
Dr. Nguyen Le Huong, for their help and company during the first years of
my research.
Thank you Mr. Dru Morrish for helping and assisting me with the
laboratory equipment and for helpful discussions on many problems that
have risen not only throughout the candidature but since our first day
as undergraduate students. Mr. Dennis McPhail, I thank you for your
constructive advices on various aspects of experimental research, and for your
company since our undergraduate days. To the other fellow students of the
CMP, Mr. Michael James Ventura, Mr. Mujahid Ashraf, Ms. Baohua Jia,
Mrs. Smitha Kuriakose, and Ms. Fu Ling, I thank you all for the help and
discussions during our meetings and day to day business. A former CMP
student Dr. Damian Bird, I thank you for all your help, company and in
particular for convincing me to write my thesis using LaTeX, which makes
the writing and updating so much simpler. Thank you Mr. Mark Kivinen for
making the stages and optical mounts for my experimental setups. I would
also like to thank the CMP administrative assistants Ms. Benita Hutchinson-
Reade and Mrs. Anna Buzescu for their help with paperwork and general
office tasks.
Also, I would like to thank Prof. Theo Tschudi from the Institute of
Applied Physics at Darmstadt Technical University (Germany) for accepting
me to undertake a part of my research on doughnut beam generation using
a liquid crystal cell in his group. Many thanks go to his students Mr.
Svetomir Stankovic, Mr. Mathias Hain, and Mr. Somakanthan Somalingam,
PhD thesis: Far-field and near-field optical trapping vii
ACKNOWLEDGEMENTS
for teaching me how to make liquid crystal cells, and for their help during the
experimental work.
Finally, I would like to acknowledge the support and encouragement from
my family. To my father, Sefik Ganic, my mother, Mirjana Ganic, my
grandmother, Muruveta Topcagic, my sister, Djelila, her husband, Mirza,
my niece, Melina and my nephew, Tarik Zahirovic thank you. You have
been a great source of strength through the hard times and a fountain of joy
through the fun times. Last, but certainly not least, I would like to thank
the people to whom this thesis is dedicated; my fiancee Dalila Deronic, for
being an inspiration and a source of constant encouragement, support and
motivation, and to my children Denis and Elma Ganic, for their patience and
understanding.
Djenan Ganic
Melbourne, Australia
February 25, 2005
PhD thesis: Far-field and near-field optical trapping viii
Contents
Declaration
Abstract i
Acknowledgements vi
Contents ix
List of Figures xiii
List of Tables xxv
1 Introduction 1
1.1 Introduction to optical trapping . . . . . . . . . . . . . . . . . 1
1.2 Thesis objectives . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Preview of the thesis . . . . . . . . . . . . . . . . . . . . . . . 6
2 Literature review 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Far-field trapping . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Optical-trap nanometry . . . . . . . . . . . . . . . . . 11
2.2.2 Particle-trapped SNOM . . . . . . . . . . . . . . . . . 13
2.3 Trapping force calculation . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Ray optics method . . . . . . . . . . . . . . . . . . . . 16
ix
CONTENTS
2.3.2 Electromagnetic method . . . . . . . . . . . . . . . . . 20
2.4 Near-field Mie scattering . . . . . . . . . . . . . . . . . . . . . 24
2.4.1 Geometric optics method . . . . . . . . . . . . . . . . . 25
2.4.2 Generalised Mie theory . . . . . . . . . . . . . . . . . . 27
2.4.3 Discrete sources method . . . . . . . . . . . . . . . . . 31
2.5 Vectorial diffraction of light . . . . . . . . . . . . . . . . . . . 33
2.5.1 Focusing through mismatched refractive index materials 34
2.5.2 Focal spot splitting with high NA objective . . . . . . 39
2.5.3 Spectral splitting near phase singularities of focused waves 42
2.6 Near-field trapping . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6.1 Near-field trapping using a nano-aperture . . . . . . . . 45
2.6.2 Near-field trapping using a metallic tip . . . . . . . . . 46
2.6.3 Near-field trapping using an apertureless probe . . . . 48
2.6.4 Near-field trapping with evanescent field generated
under TIR illumination . . . . . . . . . . . . . . . . . . 50
2.7 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Three dimensional near-field Mie scattering by a small
particle 57
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Mathematical description of 3-D near-field Mie scattering . . . 59
3.2.1 Particles far from the interface . . . . . . . . . . . . . . 61
3.2.2 Particles near the interface . . . . . . . . . . . . . . . . 63
3.3 3-D scattered intensity distribution around dielectric particles 64
3.3.1 Dielectric particle situated far from the interface . . . . 65
3.3.2 Dielectric particle situated near the interface . . . . . . 69
3.4 Effects of the interface on the morphology dependent resonance 70
PhD thesis: Far-field and near-field optical trapping x
CONTENTS
3.5 Mechanism for conversion of evanescent photons into propa-
gating photons . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5.1 Physical model . . . . . . . . . . . . . . . . . . . . . . 76
3.5.2 Theoretical results and morphology dependent resonances 79
3.5.3 Experimental setup and results . . . . . . . . . . . . . 81
3.6 Pinhole detection of the scattered near-field signal . . . . . . . 84
3.7 Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . 87
4 Trapping force with a high numerical aperture objective 90
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.3 Vectorial diffraction - Gaussian approximation comparison . . 96
4.4 Model applicability . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5 Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . 99
5 Far-field optical trapping 101
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Trapping force with plane wave illumination . . . . . . . . . . 103
5.2.1 Force mapping . . . . . . . . . . . . . . . . . . . . . . 103
5.2.2 Spherical aberration . . . . . . . . . . . . . . . . . . . 105
5.2.3 Trapping efficiency with centrally obstructed plane wave 108
5.3 Trapping force with doughnut beam illumination . . . . . . . 114
5.3.1 Doughnut beam generation . . . . . . . . . . . . . . . 115
5.3.2 Vectorial diffraction of doughnut beam illumination . . 121
5.3.3 Trapping efficiency . . . . . . . . . . . . . . . . . . . . 128
5.4 Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . 133
PhD thesis: Far-field and near-field optical trapping xi
CONTENTS
6 Near-field optical trapping 136
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.2 Optical forces on microparticles in a wide area evanescent field 139
6.3 Near-field trapping with focused evanescent illumination -
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.4 Near-field trapping with focused evanescent illumination -
Experimental results . . . . . . . . . . . . . . . . . . . . . . . 149
6.4.1 Plane wave illumination . . . . . . . . . . . . . . . . . 150
6.4.2 Doughnut beam illumination . . . . . . . . . . . . . . . 151
6.5 Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . 152
7 Conclusion 154
7.1 Thesis conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.2 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2.1 Optical trapping of metallic particles . . . . . . . . . . 159
7.2.2 Near-field micromanipulation system . . . . . . . . . . 160
Bibliography 164
Author’s Publications 178
PhD thesis: Far-field and near-field optical trapping xii
List of Figures
1.1 Principle of optical tweezers trapping [1]. . . . . . . . . . . . . . 3
2.1 Optical trap nanometry for mechanical measurements of myosin [3]. 12
2.2 Schematic diagram of particle-trapped SNOM setup. . . . . . . . 15
2.3 Qualitative description of the trapping of dielectric spheres accord-
ing to Ashkin’s RO model [23]. The total force F, given by the
refraction of a typical pair of rays a and b of the trapping beam,
is always restoring for axial and transverse displacements of the
sphere from the trap focus. . . . . . . . . . . . . . . . . . . . . . 17
2.4 (a) Gradient (Fg) and scattering (Fs) forces for an incident ray.
(b) The total (Qtotal), scattering (Qs) and gradient (Qg) trapping
efficiencies for a single ray hitting a dielectric sphere of relative
index of refraction 1.2 at an angle θ. . . . . . . . . . . . . . . . . 18
2.5 Gaussian optical trapping model. . . . . . . . . . . . . . . . . . 22
2.6 (a) Geometrical optics model of scattering of evanescent waves. (b)
Typical ray paths through a dielectric particle. The integers p and
w denote the number of contacts made with the inner surface of
the dielectric particle and the substrate, respectively [71]. . . . . . 26
2.7 Geometry of the scattering system in discrete sources method. . . 32
xiii
LIST OF FIGURES
2.8 A schematic diagram of a light beam being refracted on an interface
between two media (n2 > n1). . . . . . . . . . . . . . . . . . . . 35
2.9 Contour plots of the intensity near the focus of a high NA objective
(NA = 1). (a) |E|2 for ε = 0.0, (b) |E|2 for ε = 0.98 [93]. . . . . . 40
2.10 Contour plots of the intensity near the focus of a high NA objective
focused on an interface between two media (n1 = 1.78 and n2 =
1.0), with NA = 1.65 and ε = 0.6 [97]. . . . . . . . . . . . . . . . 41
2.11 The normalised spectrum S[u1(ω0, ω]/S0 at the first axial zero-
intensity point (u1 = 4π) of the central frequency component ω0
for different values of NA: (a) NA = 0.025, (b) NA = 0.1, (c)
NA = 0.3, (d) NA = 0.4, (e) NA = 0.6, (f) NA = 0.9. . . . . . . 43
2.12 The normalised spectrum S(ω)/S0 at the first x and y zero-intensity
points of the central frequency component ω0 for different values of
NA: (a) NA = 0.025, (b) NA = 0.05, (c) NA = 0.1, (d) NA = 0.3,
(e) NA = 0.6, (f) NA = 0.9. Full lines show the variations in the
x direction, while dotted lines represent those in the y direction. . 44
2.13 The geometry of the near-field trapping model. Incident light
is x polarised and propagates along the z axis. (a) and (b)
Radiation force spatial distribution for a dielectric sphere near a
nano-aperture. The origin of each arrow represents the center of
the sphere and vectors represent the direction and the magnitude
of forces [14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.14 Near field of a gold tip in water illuminated by two different
monochromatic waves at λ = 810 nm, indicated by the k and E
vectors. The numbers in the figure give the scaling in the multiples
of the exciting field (E2), with a factor of 2 between the successive
lines. (a) No field enhancement; (b) Field enhancement of ≈ 3000 [13]. 47
PhD thesis: Far-field and near-field optical trapping xiv
LIST OF FIGURES
2.15 A dielectric sphere situated on a flat dielectric surface is illuminated
by near-field generated under total internal reflection. A tungsten
probe is used to create an optical trap [12]. . . . . . . . . . . . . 49
2.16 Force experienced by the sphere as a function of the lateral position
of the probe. Thick lines denote θ = 43, while thin lines denote
θ = 50. The probe tip is either 20 nm (solid lines) or 31 nm (dashed
lines) above the substrate. (a) TM polarisation z direction; (b) TE
polarisation z direction; (c) TM polarisation x direction; (d) TE
polarisation x direction. According to Ref. [12]. . . . . . . . . . . 50
2.17 Schematic diagram of particle movement in an evanescent field.
Velocity of the moving particle versus the incident angle of the
beam undergoing TIR [67]. . . . . . . . . . . . . . . . . . . . . . 51
2.18 Near-field trapping under focused evanescent illumination. Density
plots (a) and (b) represent the calculated modulus squared of the
electric field at wavelength 532 nm in the focal region of an objective
of NA = 1.65 at the interface between the cover slip (n = 1.78) and
water (n = 1.33). (a) No central obstruction, i.e. ε = 0; (b) With
central obstruction ε = 0.8, whose size satisfies the TIR condition [15]. 53
3.1 Illustration of the scattering model including the effect of an
interface at which an evanescent wave is generated. n, n′ and n1
denote refractive indices of the substrate, surrounding medium and
the particle, respectively. d is the distance from the center of a
particle to the interface. . . . . . . . . . . . . . . . . . . . . . . 59
PhD thesis: Far-field and near-field optical trapping xv
LIST OF FIGURES
3.2 Three-dimensional far-field distribution of the scattered intensity
around a 2 µm dielectric particle situated far away from the
interface (a) in the XZ-plane, (b) in the plane containing the X-
axis at 45o anti-clockwise from the XZ-plane, (c) in the XY -plane
and (d) in the Y Z-plane. The solid and dotted curves correspond
to the TE and TM polarisation states of the illumination wave,
respectively. n1=1.6, n′=1.0, n=1.51, λ=632.8 nm and α=45. . . 66
3.3 A qualitative interpretation of the confined intensity regions in the
scattering of an evanescent wave by a dielectric particle of radius 2
µm. The relative intensity of rays is denoted by the arrow length. . 67
3.4 Dependence of the scattered intensity distribution in the XZ plane
on the radius of a particle, when the particle is situated far away
from the interface: (a) a = 0.05 µm, (b) a = 0.1 µm, (c) a = 0.5 µm
and (d) a = 1 µm. The plots in the left and the right columns show
the intensity distributions scattered by an evanescent wave and a
plane wave respectively. The solid and dotted curves correspond
to the TE and TM polarisation states of the illumination wave,
respectively. n1 = 1.6, n′ = 1.0, n = 1.51, λ = 632.8 nm and α = 45. 68
3.5 Dependence of the scattered intensity distribution in the XZ plane
on the radius of a particle, when particle is situated on the interface:
(a) a = 0.05 µm, (b) a = 0.1 µm, (c) a = 0.5 µm and (d) a = 1
µm. The solid and dotted curves correspond to the TE and TM
polarisation states of the illumination wave, respectively. n1 = 1.6,
n′ = 1.0, n = 1.51, λ = 632.8 nm and α = 45. . . . . . . . . . . . 70
PhD thesis: Far-field and near-field optical trapping xvi
LIST OF FIGURES
3.6 Dependence of the half-space scattered intensity on the particle
radius for the TE (solid line) and TM (dotted line) polarisation
illumination, when the particle is situated on the interface. n′ =
1.0, n = 1.51, λ = 632.8 nm and α = 45. Top: n1 = 1.1, middle:
n1 = 1.3 and bottom: n1 = 1.6. . . . . . . . . . . . . . . . . . . 72
3.7 Dependence of the half-space scattered intensity on the particle
radius for the TE (solid line) and TM (dotted line) polarisation
illumination, when the particle is situated on the interface. n′ =
1.0, n = 1.51, n1 = 1.6 and λ = 632.8 nm. Top: α = 42, middle:
α = 43 and bottom: α = 45. . . . . . . . . . . . . . . . . . . . 73
3.8 A comparison of the calculated half space scattered intensity
with the intensity measured by a NA=1.3 objective in particle-
trapped near-field microscopy [50]. The solid and dotted curves
represent TE and TM incident polarisation states, respectively.
α is the incident angle and a polystyrene particle of radius 0.25
µm immersed in water is placed on the interface between the
surrounding medium and the substrate. . . . . . . . . . . . . . . 75
3.9 (a) Schematic of our theoretical model for evanescent photon con-
version. (b) Representation of the lens transformation process. (c)
Experimental setup for recording the FID of converted evanescent
photons, collected by a high NA objective O. . . . . . . . . . . . 77
3.10 Calculated FID in the image focal plane of a 0.8 NA objective. TE
(left colomn) and TM (right colomn) incident illumination. (a) and
(e) a = 100 nm. (b) and (f) a = 500 nm. (c) and (g) a = 1000 nm.
(d) and (h) a = 2000 nm. All figures are normalised to 100. . . . . 80
PhD thesis: Far-field and near-field optical trapping xvii
LIST OF FIGURES
3.11 Maximum intensity in the FID as a function of the particle radius
near MDR for TE (a) and TM (b) illumination. Insets show the full
FID representing off and on resonance cases. The particle refractive
index is 1.59 and the illumination wavelength is 633 nm. . . . . . 81
3.12 Calculated (top) and observed (bottom) FID in image focal plane
of a 0.8 NA objective collecting propagating photons converted by
a=240 nm polystyrene particle under TE (left column) and TM
(right column) incident illumination. . . . . . . . . . . . . . . . . 82
3.13 Calculated and observed y axis scan through x=0, in image
focal plane of a 0.8 NA objective collecting propagating photons
converted by 1000 nm (radius) polystyrene particle under TE
incident illumination. (a) Calculated results. (b) Observed results
(full line) where the dotted line represents the convolution of the
calculated results and the PSF of the imaging lens. Insets show the
calculated and observed FID. . . . . . . . . . . . . . . . . . . . . 83
3.14 (a) A schematic diagram of a pinhole detection process. Only
the rays coming from the front focal region are detected. (b)
Detected signal intensity as a function of a pinhole radius, in
optical coordinates, for uniformly illuminated objective. Assumed
objective NA = 0.8 in the front focal region, aperture size ρa = 3
mm and the back focal length of the objective f = 160 mm. . . . . 85
3.15 Detected scattered intensity as a function of the pinhole size (in an
optical coordinate) of a polystyrene particle for TE illumination
(left column) and TM illumination (right column). Assumed
objective NA = 0.8 in the front focal region, aperture size ρa = 3
mm and the back focal length of the objective f = 160 mm. (a)
and (d) Particle radius 0.1 µm. (b) and (e) Particle radius 0.5 µm.
(c) and (f) Particle radius 1.0 µm. . . . . . . . . . . . . . . . . . 86
PhD thesis: Far-field and near-field optical trapping xviii
LIST OF FIGURES
4.1 Schematic diagram of our trapping model. . . . . . . . . . . . . . 92
4.2 Intensity distributions in (a) axial and (b) transversal directions
(blue-X axis, red-Y axis) for the fifth-order Gaussian approximation
(dashed line) and the vectorial diffraction theory (solid line). . . . 97
4.3 Comparison between the fifth-order Gaussian approximation (empty
symbols) and the vectorial diffraction theory (filled symbols) for the
calculation of the maximal TTE (triangles) and the backward ATE
(circles) of polystyrene particles suspended in water. λ0 = 1.064
µm, ω0 = 0.4 µm and NA = 1.2. . . . . . . . . . . . . . . . . . . 97
5.1 Magnitude and direction of the trapping efficiency for various
geometrical focus positions around a polystyrene particle suspended
in water and illuminated by a λ0 = 1.064µm laser beam focused by
a NA = 1.25 water immersion objective. (a) particle radius of 2
µm. (b) Particle radius of 200 nm. . . . . . . . . . . . . . . . . . 104
5.2 Maximal backward ATE of glass particles suspended in water,
illuminated by a laser beam (λ0 = 1.064 µm), focused by an oil
immersion microscope objective (NA = 1.3). The effect of SA is
considered at a depth of 9 µm from the cover glass. . . . . . . . . 106
5.3 Maximal backward ATE and TTE of a particle illuminated by a
laser (λ0 = 1.064 µm) focused by an oil immersion microscope
objective (NA = 1.3) as a function of the distance from the cover
glass. (a) A glass particle of diameter D = 2.7 µm in water. (b)
A polystyrene particle of diameter D = 1.02 µm suspended in 60%
glycerol solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.4 Focus intensity distribution for a plane wave focused by a high NA
objective immersed in water (NA = 1.25). Top row - unobstructed
plane wave (ε = 0.0). Bottom row - obstructed plane wave (ε = 0.8). 109
PhD thesis: Far-field and near-field optical trapping xix
LIST OF FIGURES
5.5 Theoretical calculation (RO model and vectorial diffraction model)
of the maximal TTE as function of obstruction size for a polystyrene
particle of radius 1 µm immersed in water. NA = 1.25 and λ = 532
nm. The maximal TTE for the two models are normalised to start
from the same point (at ε = 0.0). . . . . . . . . . . . . . . . . . . 110
5.6 Focus intensity along a transverse direction for various obstruction
sizes. (a) Polarisation direction X and (b) Perpendicular to
polarisation direction Y. NA = 1.25 and λ = 532 nm. . . . . . . . 111
5.7 A schematic diagram of the experimental setup. . . . . . . . . . . 112
5.8 Experimental measurement of the maximal TTE as a function of
obstruction size for a polystyrene particle of radius 1 µm immersed
in water. Theoretical values are normalised by the experimental P
value at ε = 0.0. NA = 1.2 and λ = 532 nm. . . . . . . . . . . . . 113
5.9 Phase distribution of a doughnut beam. (a) The theoretical phase
distribution of a doughnut beam of charge 1 according to 16 phase
steps. (b) The electrode structure of the liquid crystal cell with 16
pie slices. (c) The phase wavefront of the doughnut beam of charge
1, measured using phase shifting interferometry. . . . . . . . . . . 116
5.10 Experimental setup for generation of a doughnut beam through
the liquid crystal cell and interference measurement of its phase
distribution (P: polariser; BS1 and BS2: beam splitters; LC: liquid
crystal cell; O: objective; L: lens; M1 and M2: mirrors; PH: pinhole;
S: screen). (a) The voltage variation as a function of the slice
position of the liquid crystal cell. (b) The unwrapped phase shift
of the liquid crystal cell as a function of applied voltage. . . . . . 117
PhD thesis: Far-field and near-field optical trapping xx
LIST OF FIGURES
5.11 The intensity distributions (a, b and c) of laser beams transmitted
through a liquid crystal cell and the corresponding interference
patterns (d, e and f). (a) and (d) plane wave. (b) and (e) Doughnut
beam of charge 1. (c) and (f) Doughnut beam of charge 2. . . . . 118
5.12 Variation of the voltage between the two contact points (see
Fig. 5.9(b)) as a function of the wavelength for the generation of a
doughnut beam of charge 1. . . . . . . . . . . . . . . . . . . . . 120
5.13 Doughnut beam of charge 1 generated using a computer controlled
SPM. (a) Applied phase-ramp pattern with 256 levels. (b) Intensity
profile. (c) Interference pattern. . . . . . . . . . . . . . . . . . . 121
5.14 Calculated intensity distribution in the focal region of a doughnut
beam focused by an objective with NA = 1 ((a)-(c)) and NA = 0.2
((d)-(f)): (a) and (d) Topological charge 1; (b) and (e) Topological
charge 2; (c) and (f) Topological charge 3. . . . . . . . . . . . . . 123
5.15 Contour plots of the intensity distribution in the focal region of
an objective with NA = 1, illuminated by a doughnut beam of
topological charge 1. (a) |Ex|2;(b) |Ey|2;(c) |Ez|2;(d) |E|2. . . . . . 124
5.16 Contour plots of the intensity distribution in the focal region of
an objective with NA = 1, illuminated by a doughnut beam of
topological charge 2. (a) |Ex|2;(b) |Ey|2;(c) |Ez|2;(d) |E|2. . . . . . 125
5.17 Contour plots of the intensity distribution in the focal region of
an objective with NA = 1, illuminated by a doughnut beam of
topological charge 3. (a) |Ex|2;(b) |Ey|2;(c) |Ez|2;(d) |E|2. . . . . . 126
5.18 Dependence of the peak ratio of |Ez|2/|Ex|2 on the numerical
aperture (a) and on the obstruction radius ε (b). . . . . . . . . . 127
PhD thesis: Far-field and near-field optical trapping xxi
LIST OF FIGURES
5.19 Maximal backward ATE of polystyrene particles suspended in
water and illuminated by a highly focused plane wave, doughnut
beam of topological charge 1 and obstructed plane wave with
ε = 0.8 as a function of particle size. NA = 1.2 and λ0 = 1.064 µm. 129
5.20 ATE of a polystyrene particle (a = 2 µm) suspended in water and
illuminated by a highly focused plane wave and doughnut beams of
different topological charges. NA = 1.2 and λ0 = 1.064 µm. . . . . 130
5.21 TTE in the polarisation (X) and perpendicular to the polarisation
(Y) directions of a polystyrene particle (a = 2 µm) suspended in
water and illuminated by a highly focused plane wave and doughnut
beams of different topological charges. NA = 1.2 and λ0 = 1.064 µm.131
6.1 Focused evanescent field produced at the coverslip interface by
high NA focusing of an obstructed plane wave polarised in the X
direction. Oil immersion and coverslip refractive index n1 = 1.78
(index matched), particle suspension medium n2 = 1.33, objective
NA = 1.65, obstruction size ε = 0.8, and illumination wavelength
λ0 = 1.064 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2 Non-dimensional horizontal Qy and vertical Qz near-field forces
exerted on polystyrene particle under TE and TM incident polarisa-
tion states as a function of the particle size parameter k′a = 2πa/λ.
The evanescent field is generated on a prism surface (refractive
index 1.722) by a plane wave incident at 51. . . . . . . . . . . . 140
6.3 Near-field trapping model. . . . . . . . . . . . . . . . . . . . . . 142
PhD thesis: Far-field and near-field optical trapping xxii
LIST OF FIGURES
6.4 Trapping efficiency mapping for a small and a large polystyrene
particle of radius a, scanned in the X direction (light polarisation
direction) across the focused evanescent field generated by a plane
wave (top) and doughnut beam illumination (bottom). NA=1.65,
λ=532 nm, ε=0.85, n1=1.78 and n2=1.33. . . . . . . . . . . . . . 143
6.5 Trapping efficiency mapping for a small and a large polystyrene
particle of radius a, scanned in the Y direction (perpendicular to the
polarisation direction) across the focused evanescent field generated
by a plane wave (top) and doughnut beam illumination (bottom).
NA=1.65, λ=532 nm, ε=0.85, n1=1.78 and n2=1.33. . . . . . . . 144
6.6 Theoretical calculations of the maximal TTE of a polystyrene
particle of 1 µm in radius as a function of the obstruction size
ε. The other conditions are the same as in Fig. 6.4. . . . . . . . . 145
6.7 Theoretical calculations of the maximal ATE of a polystyrene
particle of 1 µm in radius as a function of the obstruction size
ε. The other conditions are the same as in Fig. 6.4. . . . . . . . . 146
6.8 Dependence of the ATE on the virtual focus position d for a small
and large polystyrene particle (ε = 0.85). The other conditions are
the same as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . 147
6.9 Phase introduced by the Fresnel transmission coefficients as a
function of the incident angle. The refractive index of the incident
medium is 1.78, while the refractive index of the transmitting
medium is 1.33. . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.10 The maximal ATE as a function of a polystyrene particle size (ε =
0.85). The inset shows a schematic relation between the interaction
cross-section area and the particle size. The other conditions are
the same as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . 148
PhD thesis: Far-field and near-field optical trapping xxiii
LIST OF FIGURES
6.11 The magnitudes of the axial force for a plane wave of power 10
µW and the gravity force for different particle sizes. The other
conditions are the same as in Fig. 6.4. . . . . . . . . . . . . . . . 149
6.12 The measured and calculated plane wave illumination maximal
TTE as a function of obstruction size with a NA-1.65 objective
for both S and P scanning directions. The other conditions are the
same as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.13 The measured and calculated doughnut beam illumination maximal
TTE as a function of obstruction size with a NA-1.65 objective for
both S and P scanning directions. The other conditions are the
same as in Fig. 6.4. . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.1 Operational near-field trapping system. . . . . . . . . . . . . . . 161
7.2 Trapping efficiency mapping in the XY plane for a polystyrene
particle of radius a = 1 µm, (light polarisation is in the X direction)
across the focused evanescent field generated at a coverslip interface
by a plane wave (Charge 0) and doughnut beam illumination
(Charge 1, 2, and 3). NA=1.65, λ=532 nm, ε=0.85, n1=1.78 and
n2=1.33. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
PhD thesis: Far-field and near-field optical trapping xxiv
List of Tables
5.1 The maximal TTE for plane wave and doughnut beam illumi-
nation. Ch0 denotes plane wave input, while Ch1 denotes a
doughnut beam of topological charge 1. exp. - experimentally
measured result, th. - theoretically calculated result. . . . . . 132
5.2 The maximal TTE for a centrally obstructed plane wave and
a doughnut beam illumination. Ch0+ε denotes a centrally
obstructed plane wave input, while Ch1 denotes a doughnut
beam of topological charge 1. exp. - experimentally measured
result, th. - theoretically calculated result. Obstruction size
ε = 0.78. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
xxv
Chapter 1
Introduction
1.1 Introduction to optical trapping
Roots of an idea that light can be used to move matter date as far back
as 1609, the year when Johannes Kepler the father of the laws of planetary
motions, proposed an extraordinary sailing trip from the earth to the moon
on light itself [1]. This idea may belong to the realm of science fiction, but
Kepler himself came to it when he realised that the sunlight has an effect
on comets and that the ”sunlight pressure” turns the tails of comets away
from the sun. Today, nearly 400 years later, his ideas are becoming a reality
and matter is moved using light through optical trapping and manipulation
techniques, commonly known as optical tweezers.
Optical tweezers provide a tool for non-invasive manipulation of micro-
scopic matter. An optical tweezer system consists of a laser beam tightly
focused into a very small region, generating an extremely large electric field
gradient, using a microscope objective. When such a tightly focused laser
beam interacts with a mesoscopic particle, piconewton forces are exerted
on the particle and it is attracted toward the highest intensity region by
1
CHAPTER 1. Introduction
the so called gradient force, while the radiation force, also known as the
scattering force, acts in the direction of the light propagation, analogous to
the Kepler’s ”sunlight pressure”. Under the conditions when the gradient
force dominates, the particle with a refractive index larger than that of the
surrounding medium, is trapped in three-dimensions.
The origin of these forces can be perceived from Newton’s laws. A light
ray that is refracted through a dielectric particle changes its direction due
to the refraction process. Since light carries momentum, a change in light
direction implies that there must exist a force associated with that change.
The resulting force, manifested as a recoil action due to the momentum
redirection, draws mesoscopic particles toward the highest photon flux in the
focal region. This recoil is unnoticeable for refraction by macro objects such
as lenses, but it has a substantial and measurable influence on mesoscopic
refractive object such as small dielectric particles. An illustration of the
optical tweezers trapping principle is shown in Fig. 1.1.
Optical tweezers techniques are finding an increasing use in various
scientific disciplines such as a rapidly expanding field of single-molecule
biophysics [2–4], cell biology [5, 6], studying of molecular motors [7, 8],
manipulation of atoms and particles [9], and optical near-field imaging [10].
Optical tweezers are also useful purely as manipulators and positioning
devices, owing to its ability to confine, organise and assemble micro-objects.
An extensive resource paper on the versatility of use of optical tweezers
technology is presented in Ref. [11].
A new trapping modality based on a near-field trapping process [12–15]
is recently gaining momentum, due to its ability to reduce the trapping
volume and minimise the background noise affecting single-molecule dynamics
measurements. Furthermore, complex laser beams, such as Laguerre-
Gaussian laser beams, are finding its way into the novel optical tweezers
PhD thesis: Far-field and near-field optical trapping 2
CHAPTER 1. Introduction
Spherepusheslight to left
Lightpushessphere to right
Spherepusheslightdown
Lightpushessphereup
Intensity profile
Fig. 1.1: Principle of optical tweezers trapping [1].
technology [16,17], enabling additional controllability of the particle trapping.
A large number of these optical trapping techniques, such as particle-
trapped scanning near-field microscopy (SNOM) and optical-trap nanometry,
use an evanescent wave illumination of laser trapped particle or sample,
which increases the resolution of the measuring system [3, 10, 18, 19]. Higher
resolution derives from the scattered near-field signal [20,21], which otherwise
would not propagate and reach a detector. In order to study and fully
understand such trapping systems, an appropriate model for detection of
evanescent wave scattering is required.
Despite its versatility, theoretical treatment of optical tweezers has been
lagging behind the experimental work in this field. The main reason for this
lag is in complexity of the trapping process combined with a high numerical
aperture (NA) objective used for trapping. The tight focusing achieved by a
high NA objective precludes scalar diffraction theory to describe the trapping
PhD thesis: Far-field and near-field optical trapping 3
CHAPTER 1. Introduction
process, and necessitates the full vectorial diffraction approach [22]. To date,
there are a limited number of methods for description of optical trapping
covering a certain range of particle sizes. Particles in the Mie regime (a λ),
where λ is the wavelength of the incident illumination, are successfully treated
using the ray optics approach [23]. The ray optics approach indicates that the
trapping force is independent of the particle size. For particles in the Rayleigh
size regime with radius a < 0.2λ electromagnetic method for trapping force
characterisation, based on the Gaussian beam approximation [24], is generally
used. The electromagnetic model, on the other hand, specifies that particle
size has an influence on the trapping force exerted on the particle. Any
general theory of optical trapping needs to bridge these two models and
extend its validity into both of these size regimes. Additionally, it requires
to be capable of treating complex laser fields used in novel and future laser
trapping systems. Such an optical trapping theory would be extremely useful
for optical trapping community, as it would enhance our understanding of
the trapping process under various types of novel laser beams and it would
pave the way for predictions of new physical phenomena associated with such
beams.
1.2 Thesis objectives
Popularity of the optical trapping technique in various research disciplines,
briefly mentioned in the previous section, and a lack of an appropriate
theoretical model to understand the trapping process by focused plane wave
illumination or complex laser beams prompt for a necessity of a proper
physical model for optical trapping. Such a model would find a great use in
characterising trapping force for complex laser beams under realistic trapping
conditions and would give an important contribution to the optical trapping
community, while not being constrained to a certain particle range.
PhD thesis: Far-field and near-field optical trapping 4
CHAPTER 1. Introduction
The objective of our research is to develop such a physical model to
investigate optical trapping, achieved by the use of an arbitrary laser beam
focused by a high NA objective, in both far and near fields. The optical
trapping model needs to include two physical processes, vectorial diffraction
by a high NA objective and scattering by a small spherical particle to
characterise the resulting field on the particle surface. The particle and
the complex focused field interaction is described by an extension of the
classical plane wave Lorentz-Mie theory. The Maxwell stress tensor can then
be applied to determine the force exerted on the particle.
Such an optical trapping model would provide an exact electromagnetic
field distribution in the focal region of a high NA objective. Due to the
inherent nature of the vectorial diffraction process, an arbitrary wavefront,
incident at the entrance pupil of the high NA objective could be considered.
In other words, any apodisation function of the focusing objective could
be considered and its effects examined. Furthermore, an arbitrary input
phase modulation of the incident beam could be included and its trapping
properties studied, which would provide a tool for studying the trapping
efficiencies of Laguerre-Gaussian or doughnut beams for example. Effects
of spherical aberration (SA), nearly always present in trapping experiments,
could be considered using such a model by including vectorial diffraction of
focusing through an interface between two media with mismatched refractive
indices. Near-field trapping force exerted on a microscopic particle by a
focused evanescent field, recently experimentally demonstrated [15], could
be examined by considering a central obstruction placed perpendicularly to
the beam axis. The size of such an obstruction needs to be selected so that
it allows only the portion of the incident wave that satisfies the total internal
reflection condition at the surface of the coverslip to pass through the high NA
objective, thus generating a focused evanescent field in the particle suspension
medium.
PhD thesis: Far-field and near-field optical trapping 5
CHAPTER 1. Introduction
In addition to such an optical trapping model, our objective is to develop
an appropriate nanometric sensing model for detection of evanescent wave
scattering, which would enhance our understanding of dependence of the
collected signal on particle morphology in optical trapping systems that use
evanescent wave illumination for high resolution position monitoring and
imaging. Most important of such applications being the particle-trapped
SNOM and optical-trap nanometry systems. Such a nanometric sensing
model would include near-field Mie scattering process to determine the three-
dimensional scattered field distribution in the vicinity of a plane interface
and vectorial diffraction of the scattered signal for its transformation to the
detection plane.
1.3 Preview of the thesis
The research work presented in this thesis deals with characterisation of
optical trapping in far and near fields, based on our vectorial diffraction
model. Furthermore, it is concerned with evanescent wave scattering, which
is essential in order to understand novel optical trapping systems for high
resolution nanometry and imaging.
To introduce a foundation on which the research presented in this thesis is
built, a review of optical trapping modalities and evanescent wave scattering
is given in Chapter 2. A distinction between the far-field and near-field
trapping modalities and its technologies is outlined as well as different
theoretical treatments associated with a particular technique. Two most
important applications of optical trapping technology, optical-trap nanometry
and particle-trapped SNOM are briefly reviewed. The theoretical treatments
available for far-field trapping are classified into two groups based on the
physical characteristics of its approach. So, we have a ray optics and
PhD thesis: Far-field and near-field optical trapping 6
CHAPTER 1. Introduction
electromagnetic approaches for optical trapping evaluation reviewed. The
different theoretical treatments of near-field Mie scattering process, which is
fundamental for developing our nanometric sensing model are discussed. This
chapter also includes a review of vectorial diffraction, which is essential for
proper understanding of our optical trapping model. A particular attention
is paid to recent vectorial diffraction phenomena, linked to the depolarisation
effects, in focal and spectral splitting of focused waves. Near-field optical
trapping methods and its theoretical treatments are outlined on the basis of
its trapping technique.
Chapter 3 presents our nanometric sensing model based on three-
dimensional near-field Mie scattering process which includes the effects of
an interface at which an evanescent field is generated. This model provides a
basis for studying optical trapping applications such as the particle-trapped
SNOM and optical-trap nanometry for single molecule detection. Near-field
scattering properties of dielectric particles of various sizes, ranging from very
small to large, situated close and far from the interface are investigated
using our theoretical approach. Interface influence on morphology dependent
resonance effects associated with large dielectric particles are studied. A
physical model for conversion of evanescent photons into propagating photons
detectable by an imaging system is described, together with the theoretical
predictions and experimentally measured results with a wide area detector
such as a CCD camera. Detection of the near-field scattered signal using a
pinhole detector is also discussed.
Optical trapping model based on vectorial diffraction approach for
trapping force calculations with high NA objective is presented in Chapter 4.
This model forms a foundation for understanding and investigation of both
the far-field trapping and the near-field trapping with focused evanescent field
illumination. The theoretical approach, based on the extension of the classical
PhD thesis: Far-field and near-field optical trapping 7
CHAPTER 1. Introduction
plane wave Lorentz-Mie theory for the particle and the complex focused field
interaction and the Maxwell stress tensor approach for force evaluation is
described. A comparison of the vectorial diffraction model and a fifth order
Gaussian beam approximation is also given.
Far-field optical trapping is investigated in Chapter 5 using our optical
trapping model. Trapping efficiency of a focused plane wave is studied, while
a particular attention is paid to the trapping efficiency for small and large
dielectric particles and the effect of spherical aberration. Numerical and
experimental results for far-field trapping efficiencies achieved with centrally
obstructed laser beams are presented. Trapping efficiencies of complex
phase modulated doughnut beams are researched both theoretically and
experimentally. Furthermore, efficient methods for generation of doughnut
beams for experimental purposes are discussed, as well as the vectorial
diffraction effects associated with focusing of doughnut beams with a high
NA objective.
Near-field trapping, achieved with evanescent field generated under total
internal reflection condition, is studied in Chapter 6. Optical trapping with
a wide area evanescent field is discussed. The theoretical results of the near-
field optical trapping achieved with focused evanescent wave illumination are
presented, while the experimentally measured results and their comparison
with the theoretical results are also given. Both plane wave and doughnut
beam incident illuminations are considered.
Chapter 7 gives the conclusions drawn from the research work undertaken
in this thesis and includes a discussion of future work in this field.
PhD thesis: Far-field and near-field optical trapping 8
Chapter 2
Literature review
2.1 Introduction
Ever since researchers realised that light can move matter because photons
carry momentum, optical micromanipulation and assembly at the microscopic
scale was the ultimate goal. Back in 1970’s first optical traps were built
by Arthur Ashkin at AT&T Bell Labs in the US. His ”Levitation traps”
were based on the upward-pointing radiation pressure from a photon stream
to counter-affect the gravitational pulling force. Then, in 1986, Ashkin
and colleagues realised that a single tightly focused laser beam generates
a sufficient gradient force to trap small particles in three dimensions [25].
Thus a first ”optical tweezer” was born. Trapping forces in optical tweezers
arise from the optical momentum transfer to a transparent particle with a
refractive index slightly greater than that of the surrounding medium. The
net radiation force is directed towards the highest intensity region near the
beam focus.
Most common optical tweezers setups include a high numerical aperture
(NA) objective to tightly focus a laser beam. Such an optical trapping
9
CHAPTER 2. Literature review
modality is known as the far-field trapping scheme, because the trapping
is performed by the propagating far-field component of a focused laser
beam. Recently, a new trapping mechanism that utilises the evanescent wave
illumination, also called near-field illumination, has been proposed [12–15]
and demonstrated [15]. In this near-field trapping modality, trapping
is performed by the non-propagating evanescent field. In either of the
two trapping modalities, the trapping process and the forces involved are
described by the scattering of the incident illumination by a microscopic
particle.
This chapter is a review of available methods to physically model and
understand the far-field and the near-field trapping methods, as well as
the vectorial diffraction necessary for a proper modelling of either trapping
modality. The chapter is organised as follows. Section 2.2 looks at the far-field
trapping technique and its applications in microcopy, as well as the models
available to provide a physical insight into this trapping process. Different
models for optical trapping force calculations are reviewed in Section 2.3. In
Section 2.4, physical models for description of the near-field Mie scattering
are reviewed. Vectorial diffraction is examined in Section 2.5, while the last
Section 2.6 discusses the near-field trapping techniques.
2.2 Far-field trapping
Optical far-field traps have found a wide use in many disciplines including
physics, chemistry and biological sciences [26]. Experiments have been
performed in the far-field trapping of bacteria and viruses [27]; yeast, blood
and plant cells, protozoa and various algae [26]; and internal cell surgery [28].
Optical techniques have also been used for cell sorting [29].
Recently, spatial phase modulators [30, 31], and computer-generated
PhD thesis: Far-field and near-field optical trapping 10
CHAPTER 2. Literature review
holograms [32] are widely used for generating complex laser beams, such
as Laguerre-Gaussian beams, for novel laser trapping experiments [16, 31,
32]. Focusing of such beams with a high NA objective required for
efficient trapping leads to a complicated amplitude, phase and polarisation
distributions of an electromagnetic field in the focal region [33]. Interaction
of such a field with a micro-particle results in the controllable laser trapping
technique [16,17].
Beside these mainstream scientific fields, optical far-field trapping is
gaining momentum in the field of microscopy and particularly in the area
of the optical-trap nanometry [3] and near-field imaging (NFI) based on the
scanning near-field optical microscopy (SNOM) technique [34]. Applying
SNOM techniques researchers have achieved optical superresolution in the
range of 1-10 nm [35,36]. SNOM technique that utilises a trapped particle as
a near-field scatterer is termed particle-trapped SNOM [10,18,19].
2.2.1 Optical-trap nanometry
Optical-trap nanometry is a technique to measure nanometric movements of
biomolecules via optical means. An ordinary optical-trap nanometry system
consists of a one or two far-field laser traps with stably trapped microscopic
particles (Fig. 2.1). A biomolecule is attached to the trapped particles and
the whole system is usually illuminated by an evanescent field (near-field)
generated by a total internal reflection (TIR) [3]. One of the trapped particles
is used as an anchor to fix the one end of the molecule, while the other is
sensing its movements. The movements of the biomolecule are monitored by
observing the scattered evanescent field signal. The monitoring is usually
performed by a quadrant photodiode capable of determining particle position
in three dimensions. The quadrant photodiode is calibrated experimentally,
however it would be beneficial for the reliable high resolution position
PhD thesis: Far-field and near-field optical trapping 11
CHAPTER 2. Literature review
Laser
Glass slide
Myosin filament
Cy3-ATP Evanescent field Bead
Optical trap
Actin filament
Fluorescenceof Cy3-ATP
Nanometry
Fig. 2.1: Optical trap nanometry for mechanical measurements of myosin [3].
determination to be able to calibrate the diode based on a theoretical model
for the scattered evanescent field. Such a theoretical model would need to
incorporate the near-field Mie scattering and the vectorial diffraction of the
scattered field by a high NA lens. Both of these research areas are reviewed in
sections 2.4 and 2.5 in this chapter, and will be addressed in more details in
Chapters 3 and 4. On the other hand a reliable far-field trapping model would
provide a physical insight into the trapping performance and lead to the most
stable trap designs, which is addressed in details in Chapters 4 and 5.
Using this technique, researchers have achieved movement resolutions of
a kinesin molecule of sub-8 nm [37] and even smaller movements of a single
myosin molecule [38]. These simple movement measurements can further
be interpolated to study the complex mechanics of biomolecules [39]. More
intricate molecular dynamics, such as the DNA transcription processes by
RNA polymerase, the rotary motion of molecular motors or the enzymatic
reactions, have been also investigated using the optical-trap systems [3].
PhD thesis: Far-field and near-field optical trapping 12
CHAPTER 2. Literature review
2.2.2 Particle-trapped SNOM
Optical near-field imaging has become a rich research field in recent years
due to its ability to achieve optical resolution well below the classical
diffraction limit of approximately half the wavelength of illuminating light.
The principle of this imaging technique is to probe the near field of an object
under investigation, and thereby extract high-resolution information about
the object which otherwise does not reach a far-field detector. The most
widely applied near-field imaging (NFI) technique is SNOM [34]. The SNOM
technique is utilised in various high-resolution applications such as single-
particle plasmons observations [40], single molecule detection experiments [41,
42], light confinement studies [43] and trapping and manipulation of nano-
scale objects [13,14].
Near-field imaging, implemented using SNOM techniques, can generally
be classified into three categories: scanning aperture type [44,45], frustrated
total internal refection (FTIR) illumination with fibre tip collection [46] and
TIR with scattering collection [47]. The scanning aperture type NFI utilises
a subwavelength-diameter aperture as a localised evanescent field source.
The aperture is scanned along a sample in a close proximity, typically a
few nanometers, and the transmitted signal is collected by a conventional
optical system. The theoretical treatment of image formation of this type of
NFI has been well dealt with [48]. NFI implemented with FTIR illumination
and fibre tip collection probes a sample modulated by a localised evanescent
field and collects evanescent photons through the photon tunnelling process.
The physical model for describing the signal collection with this type of
NFI is analogous to the electron tunnelling process in scanning tunnelling
microscopy [49]. The third NFI category employs TIR illumination to
generate evanescent field. The field is probed using a small scatterer, such as
a microscopic metallic or dielectric particle or a metallic needle, to convert
PhD thesis: Far-field and near-field optical trapping 13
CHAPTER 2. Literature review
evanescent photons into propagating photons [18,19].
Optical imaging with laser trapped particle illuminated by a near-field
generated by a TIR offers higher resolution performance compared with the
classical imaging techniques. A laser trapped particle is used to probe the
near field of an object under investigation [10]. The sample image is built
by scanning the sample in two dimensions and collecting the signal scattered
by the trapped particle. Compared with a conventional SNOM, particle-
trapped SNOM enables non-invasive access to the sample, the axial pushing
force acting on the trapped particle eliminates the requirement of distance
control, and the probe deterioration problems are avoided since plenty of
particles are available for replacement.
Figure 2.2 shows a typical experimental setup used for particle-trapped
SNOM imaging. A particle is trapped with a laser beam focused by a high
NA microscope objective. The particle is laterally pulled toward the optical
axis of the focused beam by the transverse trapping force. At the same time
it is vertically pushed down or pulled up, depending on the focus position, by
the axial trapping force.
In addition to the trapping laser beam, another laser beam is incident
at the interface between the sample and its substrate satisfying the TIR
condition. Thus generated evanescent field illuminates the sample and is
scattered by the trapped particle. The scattered signal is collected by the
same high NA microscope objective and imaged onto a detector, which
is usually a photon-multiplier-tube (PMT) placed in the far-field region.
The image of the sample is reconstructed by scanning the particle in two
dimensions across the sample [10].
In relation to the other types of SNOM, the particle-trapped SNOM is
quite unique in that it relies on collecting evanescent waves scattered by a
PhD thesis: Far-field and near-field optical trapping 14
CHAPTER 2. Literature review
CCD
PMT
FilterPinhole
Beam splitter
Beam splitter
Trapping laser
High NA objective
SampleTrapped particle
Prism Illumination beamunder TIR
Scanning
Fig. 2.2: Schematic diagram of particle-trapped SNOM setup.
trapped particle for image formation [10,18,19]. The strength of the scattered
evanescent waves is dependent on the scattering efficiency of the trapped
particle and the illumination intensity [50]. The evanescent wave strength can
be further optimised by coating the substrate surface at which an evanescent
field is generated by a thin film stack [51]. The speed of image acquisition
and image resolution are determined by the laser trapping performance
and stability, i.e. trapping efficiency [52]. All of these issues involving
evanescent wave scattering, vectorial diffraction and trapping performance
are investigated in more details in Chapters 3, 4, and 5.
PhD thesis: Far-field and near-field optical trapping 15
CHAPTER 2. Literature review
2.3 Trapping force calculation
An accurate knowledge of the trapping force acting on a particle is essential for
describing and improving the trapping performance. This force depends on
many parameters of which most important ones are the particle morphology,
characteristics of the suspension medium, and the trapping illumination
distribution in the focal region. In this section, the theoretical approaches
for the calculation of trapping force on a dielectric particle are reviewed.
These approaches can be broadly classified into the ray optics (RO) approach
and the electromagnetic (EM) approach. Both of these models provide a
reasonable guidance for the evaluation of the far-field trapping force on small
spherical particles in their own region of applicability, except for the range
of λ-10λ, where λ is the wavelength of a trapping beam [53]. In Chapter 4
we present our model based on vectorial diffraction by a high NA lens for
trapping force evaluation, which is an exact model that does not suffer from
these limitations.
2.3.1 Ray optics method
In the RO method, applicable for trapping force evaluation of large particles,
a trapping laser beam focused by a high NA lens is simply decomposed into
individual rays and the ray density on the focusing lens is assumed to be
the same as that of the power density. In this method, the wave nature of a
trapping beam cannot be dealt with at all. The far-field optical trap action
on a dielectric particle, according to the RO model, can be described in terms
of the total force arising from a typical pair of rays a and b of a converging
beam [23]. The direction of the total force is dependent on the geometrical
focus position (Fig. 2.3). In this model the forces due to the refraction (Fa
and Fb) are pointing in the direction of the momentum change.
PhD thesis: Far-field and near-field optical trapping 16
CHAPTER 2. Literature review
Laser beam
Microscope objective
Particle
O
FFaFb
Particle
O
F
Fa Fb
Particle
O
FFa
Fb
ab
a
a
b
b
b
b
a
a
a
b
(a) (b) (c)
Fig. 2.3: Qualitative description of the trapping of dielectric spheres according toAshkin’s RO model [23]. The total force F, given by the refraction of a typical pairof rays a and b of the trapping beam, is always restoring for axial and transversedisplacements of the sphere from the trap focus.
A single ray of power P incident on a dielectric particle gives a rise to a
series of reflected and refracted rays. As a result, the particle experiences a
force due to the net change in momentum. This force can be resolved into
the scattering force Fs (parallel to the incident ray) and the gradient force Fg
(perpendicular to the incident ray) (Fig. 2.4(a)). The gradient and scattering
forces can be mathematically expressed as [23]
Fg =nP
c
(
1 +R cos 2θ − T 2[cos(2θ − 2θ′) +R cos 2θ]
1 +R2 + 2R cos 2θ′
)
,
Fs =nP
c
(
R sin 2θ − T 2[sin(2θ − 2θ′) +R sin 2θ]
1 +R2 + 2R cos 2θ′
)
, (2.1)
where R and T are the Fresnel reflectance and transmittance at the incident
angle θ [54], n is the refractive index of the particle relative to the surrounding
PhD thesis: Far-field and near-field optical trapping 17
CHAPTER 2. Literature review
0 10 20 30 40 50 60 70 80 90
q (DEGREES)
Q
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Qs
Qg
Qtotal
O
Z
f
ray
rmax
beam axis
n
v
q
f
ray
R=1
Fs
Fg
(a)
(b)
Fig. 2.4: (a) Gradient (Fg) and scattering (Fs) forces for an incident ray. (b) Thetotal (Qtotal), scattering (Qs) and gradient (Qg) trapping efficiencies for a singleray hitting a dielectric sphere of relative index of refraction 1.2 at an angle θ.
medium, c is the speed of light in vacuum and θ′ is the refractive angle of a
single ray incident on the particle.
If the light intensity distribution over the aperture of an objective lens
is denoted I(ρ), then the total trapping force on a dielctric particle can be
expressed as
Ft = FG + FS =
∫ 2π
0
∫ ρmax
0Fg · I(ρ) · ρdρdϕ
∫ 2π
0
∫ ρmax
0I(ρ) · ρdρdϕ
+
∫ 2π
0
∫ ρmax
0Fs · I(ρ) · ρdρdϕ
∫ 2π
0
∫ ρmax
0I(ρ) · ρdρdϕ
(2.2)
where ρ is the radial position over the trapping objective, ρmax is the
maximum radius of the objective aperture, ϕ is the azimuthal angle, Fg and
Fs are the vectorial gradient and scattering forces of a single ray, and FG and
FS are the vectorial gradient and scattering forces on a particle respectively.
The trapping efficiency Qj, a parameter independent of trapping power P
PhD thesis: Far-field and near-field optical trapping 18
CHAPTER 2. Literature review
for the evaluation of trapping force Fj, is defined as
Qj =Fjc
nP, j = g, s. (2.3)
Here Qg and Qs are termed the gradient and scattering efficiency of a single
ray respectively.
Figure 2.4(b) shows the values of the scattering efficiency Qs, gradient
efficiency Qg and the magnitude of the total trapping efficiency Qtotal for a
ray incident at an angle θ on a dielectric sphere of a relative refractive index
of n = 1.2. One can see that the maximum gradient trapping efficiency as
high as 0.5 is generated for rays at angles of θ ∼= 70.
The RO model can be further extended to include Mie metallic particles,
and determine the trapping force exerted on metallic particles. This approach
is termed a modified RO model [55]. For metallic particles the energy density
of the transmitted light falls to 1/e of the original value after the light travels
through a skin depth that is usually of several or tens of nanometers. The
optical force exerted on a metallic Mie particle is determined mainly by the
reflection at the surface of the particle. Thus the optical force, caused by the
multiple reflection on the inner surface of the particle, can be neglected for a
metallic Mie particle. As a result the gradient and scattering forces due to a
single ray incident on an Mie metallic particle at an angle θ can be expressed
as [55]
Fg =nPR
csin 2θ, Fs =
nP
c(1 +R cos 2θ) . (2.4)
A variation of the RO model for the evaluation of the axial force exerted
on a dielectric particle by the optical pressure of a focused Gaussian beam
is given in Ref. [56]. The axial force is obtained by numerical integration of
PhD thesis: Far-field and near-field optical trapping 19
CHAPTER 2. Literature review
the local force on the spherical surface with respect to the angle around the
center of the sphere. This approach does not require complicated equations
and one can easily calculate the axial force. The drawback, obviously, is that
this approach cannot give an indication of the transverse force acting on the
particle.
Other theoretical treatments for trapping force calculation that employ
the geometrical approximation as equivalent to the large-argument approxi-
mation in the Riccati-Bessel functions were published at around the same time
or little earlier as the Ashkin’s approach [23]. These theoretical treatments are
valid for large particles, with 2πa/λ ≥ 100, where a is the particle radius and
λ is the illumination wavelength. The theory for the interaction of the laser
beam and a dielectric particle was first derived by Roosen and co-workers [57],
while the extension of the theory to the tightly focused laser beams is given
by Gussgard et al. [58]. However, even though these treatments are fairly
complex, they lack the elegance and power of the Ashkin’s model [23].
2.3.2 Electromagnetic method
The EM field model for evaluation of the trapping force exerted on a
microscopic particle is based on the wave optics approach, i.e. based on
the wave nature of the incident illumination. In the EM method, the field
expansion coefficients are derived for an infinite representation of the electric
and magnetic fields external to the spherical particle. These expansion
coefficients describe the incident and scattered fields, which then can be used
to determine the force exerted on the particle using the Maxwell’s stress tensor
as
〈FN〉 = 〈∮
S
n · ←→T dS〉 , (2.5)
PhD thesis: Far-field and near-field optical trapping 20
CHAPTER 2. Literature review
where 〈 〉 represents a time average, FN is the net radiation force, n is an
outward normal unit vector,←→T is Maxwell’s stress tensor, and S is the surface
area of the particle. However, in order to correctly determine the trapping
force, the EM model requires an accurate expression for the radial component
of an incident field [24]. This method indicates that the magnitude of trapping
force is particle-size dependent and it is applicable for small particles (∼ λ
order). For the steady state optical condition,the Maxwell’s stress tensor in
the traditional Minkowski form is given as [24]
←→T =
1
4π(ε0EE + HH− 1
2(ε0E
2 +H2)←→I ) , (2.6)
where E and H are the electric and magnetic fields at the surface of the
particle, while ε0 is the permittivity of the particle and←→I is a unit tensor.
The trapping force can be further expressed as
〈FN〉 =1
4π
∫ 2π
0
∫ π
0
⟨(
εextErE+HrH−1
2(εextE
2+H2)r
)⟩
r2 sin θdθdφ
∣
∣
∣
∣
∣
r>a
,
(2.7)
where εext is the permittivity of the surrounding medium, Er and Hr are
the radial field components, r is the outward radial vector, a is the particle
radius, and r, θ and φ are spherical coordinates. In this method an accurate
knowledge of the EM field distribution in the focal region is of crucial
importance.
A simple Gaussian model (Fig. 2.5) for the incident field description is
not adequate due to its inability to represent a highly focused laser beam
required for efficient trapping. Therefore, researchers have developed fifth
order corrections [59] for the incident field components of a Gaussian laser
beam to deal with highly focused laser beams. However, even a fifth-order
Gaussian beam ignores the effect of diffraction by a high NA objective and
thus does not correctly represent the EM phase and polarisation distributions
PhD thesis: Far-field and near-field optical trapping 21
CHAPTER 2. Literature review
FN
Intensity profileIntensity profile
Fig. 2.5: Gaussian optical trapping model.
near the focus of a high NA objective. In addition, the fifth order Gaussian
beam method is not able to model the effect of spherical aberration (SA)
usually presents during laser trapping experiments due to the refractive index
mismatch between the immersion oil (or the coverslip) and aqueous solution
in which particles are situated. It is one of the aims of this thesis to deal with
trapping force under focused illumination by a high NA objective (Chapter 4)
in the far-field region (Chapter 5) and in the near-field region (Chapter 6).
Recently, Rohrbach et al. [60,61] have developed a method for calculation
of trapping forces of an EM wave on an arbitrary-shaped dielectric particle,
based on the extension of Rayleigh-Debye theory to include second-order
scattering, which considers a stronger interaction between the incident field
and the particle. This method does not use the Maxwell stress tensor to
determine the force exerted on a particle, but splits the total optical force into
two components. These components are commonly known as the scattering
force, which is due to the momentum transfer of photons, and the gradient
force, which draws dipoles toward the highest amplitude of an EM field. One
can obtain the total EM force F(r) by integrating over all volume elements
PhD thesis: Far-field and near-field optical trapping 22
CHAPTER 2. Literature review
dV ′ inside the scatterer:
F(r) = Fgrad(r) + Fsca(r) =
∫ ∫ ∫
V
αnm
2c∇Im(r′)dV ′ +
nm
cImCscag/kn ,
(2.8)
where α is the polarisability, nm is the refractive index of the medium in which
the particle is suspended, Im is the intensity distribution in the suspension
media, Csca is the scattering cross section and g is the transfer vector.
The gradient force Fgrad is determined by computing the gradient in x, y,
and z of the intensity distribution Im(x, y, z) for each volume element and
by summing all the gradient values. This is performed separately for the
intensities generated by the x and z components of the electric field. The
scattering force Fsca is obtained by the change of momentum that is due
to the scattering of the incident wave at the particle. The power of the
scattered light, Psca = ImCsca is defined by the intensity in the focus of the
incident beam Im = ε0c|Em(0, 0, 0)|2, while the scattering cross-section Csca is
calculated by the intensity scattered in the positive half-space and in negative
half-space.
However, the two components method [60,61] is limited to the case when
the maximum phase shift k0(ns − nm)2a produced by the particle of radius
a, is smaller than π/3. This is valid, for example, for a polystyrene particle
(ns = 1.57) in water illuminated by highly focused laser (λ0 = 1.064 µm) of
radius a = 370 nm or less, but for larger particles it would be inaccurate.
Generalised Lorenz-Mie theory (GLMT) is another approach that can be
used to determine the trapping force in the presence of spherical aberration,
but to date it has only been used to determine the on-axis trapping force [62,
63]. The GLMT formula for the on-axis trapping force is expressed as an
infinite series of partial wave contributions Σ as
Fz = (nm/c)(nmE20/µ0c)(π/n
2mk
2)Σ . (2.9)
PhD thesis: Far-field and near-field optical trapping 23
CHAPTER 2. Literature review
E0 denotes the field strength and k is the free-space wave number. The
incident illumination with this GLMT method is a tightly focused, truncated
and aberrated Gaussian beam and a plane wave [62]. The GLMT model is
used to calculate the on-axis trapping force on a spherical particle whose size
can range from the Rayleigh scattering limit to the ray theory limit [63].
2.4 Near-field Mie scattering
Mie theory and its later extensions and modifications are extensively used
to determine the size, shape, and orientation of small particles in vacuum
or in gaseous, liquid, or solid media, since its first formulation by Gustav
Mie [64] in 1908 and independently in 1909 by Debye [65]. Researchers from a
large variety of disciplines such as physics, electrical engineering, meteorology,
chemistry, biophysics, and astronomy are interested and concerned with this
theory. The basic formulation of the Mie scattering of evanescent waves, i.e.
near-field is an analytical continuation of the standard case of plane-wave
excitation.
As mentioned earlier in this chapter, the thorough understanding and an
appropriate physical model of near-field Mie scattering is required for proper
investigation of a SNOM technique which uses a laser trapped particle as a
near-field probe. Such a physical understanding is also needed in optical trap
nanometry for single molecule detection, in which case a single molecule is
attached to a laser trapped microscopic particle immersed into evanescent
field, and is monitored by measuring the scattered field [3].
While the evanescent photon conversion mechanism induced by other
SNOM techniques has been studied [48, 49, 66], the underlying physical
principle of the mechanism for the evanescent photon conversion by a
microscopic particle probe has not been dealt with. Furthermore, such
PhD thesis: Far-field and near-field optical trapping 24
CHAPTER 2. Literature review
physical model can be used to investigate the near-field force exerted on small
particles situated in the evanescent field [67–70]. Near-field Mie scattering
has been researched using a variety of theoretical methods, most popular of
which are the geometric optics method [71], generalised Mie theory [72–75],
and the discrete sources method [76,77]. A numerical model based on multiple
multipole (MMP) method [78] is generally applied for comparison of the
theoretically calculated results.
2.4.1 Geometric optics method
This method [71] assumes that the evanescent field is generated by a TIR of an
incident beam at an interface between two media. A particle is situated close
to the interface so that the evanescent wave propagating along the interface
is scattered (Fig. 2.6(a)). The intensity of the scattered light is a function of
the particle-interface separation due to the exponential nature of evanescent
illumination. Geometric or ray optics is an asymptotic solution to the light-
scattering problem, which is valid in the case when particle size is much
larger than the wavelength of incident illumination. Although not as rigorous
as proper Mie theory, geometric optics is mathematically simpler. This
technique can provide good approximations to the total scattering solution.
In geometric optics the incident EM wave is considered as a set of parallel
rays each of which can follow an independent path. When a given ray strikes
the particle it is divided into a reflected and a transmitted ray. The direction
of these rays is determined by the Snell’s law, while the amplitude and phase
are given by the Fresnel coefficients. This process is repeated when these rays
strike another surface, such as the particle or a substrate (Fig. 2.6(b)). The
objective of this method is to track the propagation path and intensity of each
ray as it travels within the particle or between the particle and the substrate.
It is necessary to follow only a small number of ray reflections due to reduction
PhD thesis: Far-field and near-field optical trapping 25
CHAPTER 2. Literature review
Glass
y
q
Water
ParticleEvanescentField
IncidentField
Incident ray ofevanescent wavew=2, p=0
w=0, p=1
w=1, p=1
w=1, p=2
Glass
(a) (b)
Fig. 2.6: (a) Geometrical optics model of scattering of evanescent waves. (b)Typical ray paths through a dielectric particle. The integers p and w denote thenumber of contacts made with the inner surface of the dielectric particle and thesubstrate, respectively [71].
of energy of a reflected of transmitted ray compared to the incident energy.
Integers w denotes the number of times a ray contacts the substrate surface,
while integer p denotes the number of internal contacts inside the sphere
(Fig. 2.6(b)). The total path of any ray can be calculated if the azimuthal
angle φinc, a contact angle τ , and integers w and p are given. Given these
variables, the scattering direction and all necessary optical calculations can
be performed on this ray. Following this methodology the amplitude Esca of
any scattered ray can be determined from [71]
Esca = EincCF
√
cos τ sin τdτdφinc
sin θscadθscadφsca
exp(iσt) . (2.10)
Here Einc is the amplitude of the incident ray striking the particle, CF denotes
the cumulative Fresnel coefficient for the scattered ray, θsca is a scattering
angle, while σt gives the total phase shift. The total phase shift constitutes of
PhD thesis: Far-field and near-field optical trapping 26
CHAPTER 2. Literature review
the phase change at reflection, the phase change caused by a different optical
path length and the phase change caused by the crossing of focal lines.
2.4.2 Generalised Mie theory
As early as 1979 Chew et al. [72] started to discuss elastic scattering of
evanescent waves by spherical particles. Their theory was recently slightly
corrected by Liu et al. [79]. Chew et al. [72] outlined the theory for scattering
of evanescent waves by spherical particles, which essentially consists in the
analytical continuation of the case of plane-wave excitation to the complex
angles of incidence. However, these authors were only interested in calculating
the differential scattering cross sections of relatively large dielectric particles.
Furthermore, their calculations involve an approximation, which is valid only
for two principal planes at a large distance from the scattering particle.
According to their approach the scattered field for an incident wave polarised
perpendicular to the plane of incidence is given by [72]
Esc(r) =∑
lm
icβE(l,m)
n′2ω∇×[h
(1)l (k′r)Yllm(r)]+βM(l,m)h
(1)l (k′r)Yllm(r)
large r −→ exp(ik′r)
k′r
∑
lm
(−i)l−1
βE(l,m)
n′r×Yllm(r)−βM(l,m)Yllm(r)
,
(2.11)
where l = 1 to∞ and m = -l to +l. n′ is the index of refraction of the medium
in which the particle is situated, k′ is the wave number in the same medium,
c is the speed of light in vacuum and ω denotes the angular frequency of the
incident light. The function Yllm is the vector spherical harmonics, while
h(1)l (k′r) is the spherical Hankel function of the first kind and the functions
βE(l,m) and βM(l,m) are the expansion coefficients of the scattered fields
PhD thesis: Far-field and near-field optical trapping 27
CHAPTER 2. Literature review
given by
βE(l,m) =ε′jl(k′a)[k1ajl(k1a)]
′ − ε1jl(k1a)[k′ajl(k
′a)]′ exp(−βd)αE(l,m)
ε1jl(k1a)[k′ah(1)l (k′a)]′ − ε′h(1)
l (k′a)[k1ajl(k1a)]′,
(2.12)
and
βM(l,m) =µ1jl(k1a)[k
′ajl(k′a)]′ − µ′jl(k
′a)[k1ajl(k1a)]′ exp(−βd)αM(l,m)
µ′h(1)l (k′a)[k1ajl(k1a)]′ − µ1jl(k1a)[k′ah
(1)l (k′a)]′
,
(2.13)
where ε′, µ′, k′ and ε1, µ1, k1 denote the dielectric constant, magnetic perme-
ability, and the wave number in the medium surrounding the sphere and inside
the sphere respectively. jl is the spherical Bessel function of the l-th order, a
is the radius of the sphere, d is the distance of the centre of the sphere to the
interface at which evanescent wave is generated, β = k′(n2 sin2 α/n′2 − 1)1/2,
where α is the angle of incidence at the interface. αE(l,m) and αM(l,m) are
the expansion coefficients of the incident fields given by
αE(l,m) = 2miln′
[
π(2l + 1)(l −m)!
l(l + 1)(l +m)!
]1/2
[Pml (cos θk′)/ sin θk′ ]E ′
0 , (2.14)
and
αM(l,m) = −2il+1
[
π(2l + 1)(l −m)!
l(l + 1)(l +m)!
]1/2[
d
dθk′
Pml (cos θk′)
]
E ′
0 , (2.15)
where Pml are Legendre functions, E ′
0 = 2/(1 + µ tanα/µ′ tanα′) is the
amplitude of the refracted wave, while the angle θk′ is related to the angle of
refraction α′ through sin θk′ = cosα′ and cos θk′ = sinα′.
In the case of the polarisation state parallel to the plane of incidence, the
scattered electric field is also given by Eq. 2.11 with expansion coefficients
βE(l,m) and βM(l,m) substituted by βE(l,m) and βM(l,m), respectively [72].
These scattered fields expansion coefficients are given by Eqs. 2.12 and 2.13
PhD thesis: Far-field and near-field optical trapping 28
CHAPTER 2. Literature review
with αE,M(l,m) replaced by αE,M(l,m) given by
αE(l,m) = 2il−1n′
[
π(2l + 1)(l −m)!
l(l + 1)(l +m)!
]1/2[
d
dθk′
Pml (cos θk′)
]
E ′
0 , (2.16)
and
αM(l,m) = −2mil
[
π(2l + 1)(l −m)!
l(l + 1)(l +m)!
]1/2
[Pml (cos θk′)/ sin θk′ ]E ′
0 , (2.17)
with E ′
0 = 2 sinα′ cosα/[sinα′ cosα′ + (µ/µ′) sinα cosα]. This method forms
the basis of the investigations of near-field Mie scattering described in this
thesis and is evaluated in more details in Chapter 3.
Quinten et al. [73–75] have extended the Chew et al. [72] method to
calculate the total cross sections for evanescent-wave excitation and discuss
their dependence on wavelength, angle of incidence, and particle sizes. In their
work they have not used the complex conjugates of the associated Legendre
polynomials but their actual values. The necessity of this correction was
already pointed out by Liu et al. [79]. Quinten et al. [73] have obtained
the total cross sections for extinction and scattering of evanescent waves by
applying Poynting’s law for the absorbed power density in the stationary case.
These cross sections for S-polarised light are given as
σsext =
2π
k′2N−1Re
∞∑
l=1
(2l + 1)(alΠl + blTl) ,
σssca =
2π
k′2N−1
∞∑
l=1
(2l + 1)(|al|2Πl + |bl|2Tl) , (2.18)
PhD thesis: Far-field and near-field optical trapping 29
CHAPTER 2. Literature review
and for P-polarised light as
σpext =
2π
k′2N−1Re
∞∑
l=1
(2l + 1)(alTl + blΠl) ,
σpsca =
2π
k′2N−1
∞∑
l=1
(2l + 1)(|al|2Tl + |bl|2Πl) . (2.19)
al are absolute values of the Mie coefficients for electric multipoles, while bl
denote the absolute values of Mie coefficients for magnetic multipoles which
are defined in [74]. The normalisation factorN is equal to one for plane waves,
but assumes different values for evanescent waves as discussed in Ref. [73].
The definitions of Πl and Tl are
Πl(θk) =2
n(n+ 1)
l∑
m=−l
(l −m)!
(l +m)!
∣
∣
∣
∣
∣
mPlm(cos θk)
sin θk
∣
∣
∣
∣
∣
2
,
Tl(θk) =2
n(n+ 1)
l∑
m=−l
(l −m)!
(l +m)!
∣
∣
∣
∣
∣
dPlm(cos θk)
dθk
∣
∣
∣
∣
∣
2
. (2.20)
θk represents the complex refraction angle of the incident wave and Plm are
the associated Legendre polynomials. Eqs. 2.18 and 2.19 are valid not only
for homogeneous, but also for coated spherical particles with an arbitrary
number of layers. Only the expressions for al and bl differ in these cases.
In the case of an incident plane wave, it can be proved that Πl = Tl = 1
for all multipolar orders l and angels θk with cos θk ≤ 1. In this case the
cross-sections do not differ for S and P polarisation and are the well known
results from the standard Mie theory.
Using this method Quinten et al. [73–75] have calculated scattering
and extinction of evanescent waves by small metallic, dielectric and coated
particles. They have found that the scattering and extinction results
differ strongly for evanescent and plane-wave excitation, with much stronger
morphology dependent resonances (MDR) in the former case. The special
PhD thesis: Far-field and near-field optical trapping 30
CHAPTER 2. Literature review
properties of the evanescent wave lead to increased contributions of multipolar
orders (l,m) with l > 1, which in turn leads to increased contributions of
orders l > 1 in the expansion of scattered wave. This results in polarisation-
dependent cross sections, which are much larger for P-polarised light than for
S-polarisation. Close proximity of the prism surface at which evanescent field
is generated reduce the resonant cross sections significantly. In addition, a
broadening and a redshift occur, when the particle is in contact with the prism
surface. However, the effect of multiple reflections caused by the interface has
not been dealt with. We will present a physical model to investigate the effect
of interface reflections in Chapter 3.
2.4.3 Discrete sources method
Geometry of the scattering problem considered with the discrete sources
method (DSM) is depicted in Fig. 2.7 [77]. An axisymmetric particle with
smooth boundary S and interior domain Di is situated on a plane surface. Its
symmetry axis coincides with the normal to the plane surface. The ambient
domain half space is denoted by D0, while the bottom half space consisting of
a glass prism is D1. The basis of discrete source method is in approximately
representing the electromagnetic fields as a finite linear combination of fields
of multipoles [76, 77]. Their treatment of evanescent wave scattering is an
extension of their earlier work on plane wave scattering by a small particle
near a surface [80]. Maxwell equations are satisfied in this approximation
in domains D0, D1 and Di, the radiation condition is satisfied in D0 and
D1, and transmission condition at the plane interface. Thus essentially, the
scattering problem is simplified to the approximation problem of the external
excitation on the particle surface. In this section we will not go into details
of the mathematical treatment of the DSM but instead will summarise the
main points of this technique:
PhD thesis: Far-field and near-field optical trapping 31
CHAPTER 2. Literature review
Surface
q
qk
Z
X
D0
D1
Di
O
S
Fig. 2.7: Geometry of the scattering system in discrete sources method.
• The scattered field representation is based on Green’s tensor for the half
space. As a consequence the vector potentials of the multipole fields are
expressed as Weyl-Sommerfield integrals providing the continuity of the
tangential component of the scattered fields at the plane interface [81].
• Vector potentials of regular multipole fields are used for the internal
field representation.
• The multipoles are distributed along the axis od symmetry inside
the domain Di or in the image domain of the complex plane. The
approximate solution is represented as a linear combination of Fourier
harmonics with respect to the azimuthal angle. Thus the computational
effort is significantly reduced by simplifying the approximation problem
of the external excitation to a problem of a sequence of one dimensional
approximation along a scatterer generator.
• The convergence of the approximate solution to the exact solution in
closed domain D0 is guaranteed by the completeness of the system of
PhD thesis: Far-field and near-field optical trapping 32
CHAPTER 2. Literature review
distributed multipoles [82].
The transmission condition at the particle surface is used to determine the
amplitudes of discrete sources. After the amplitudes of discrete sources have
been determined, one can calculate the far-field pattern of the scattered field
by using the asymptotic representation for the Sommerfield integrals [82].
Using DSM, Doicu et al. [77] have investigated evanescent wave differential
scattering cross sections for dielectric and metallic spherical particles. They
compare their results with the results of Liu et al. [79] which does not
include the interface effects and conclude that the effects of the interface
are significant, in particular for particles with high index of refraction.
These effects need to be taken into consideration to adequately describe the
scattering of evanescent waves by small particles near an interface.
2.5 Vectorial diffraction of light
Evaluation of trapping force on a small particle illuminated by a focused laser
beam depends, as we have pointed out in Section 2.2, on an accurate modelling
of the field distribution in the focused region. An exact model for the trapping
force calculation needs to be able to determine the field distribution of the
focused laser beam by a high NA objective precisely. Such an exact model
needs to include the vectorial diffraction of light by a high NA objective,
determining the focal distribution which interacts with a particle. As one
of the aims of this thesis, the vectorial diffraction model for laser trapping
is presented in greater details in Chapter 4. In addition to the particle
trapping discipline, focusing of light by a high NA system is attracting an ever-
increasing interest because of increasing demand of microscopy applications
in biological and material sciences [83]. Better knowledge of the diffracted-
PhD thesis: Far-field and near-field optical trapping 33
CHAPTER 2. Literature review
light distribution by a lens has also helped in the design of better objective
lenses, especially those with high NA.
The structure of the focused EM distribution has been studied by a
number of authors. The high NA focusing of EM waves in a single
homogeneous material is dealt with by E. Wolf in an early paper in 1959 [84].
The staring point of his approach was the representation of the angular
spectrum of plane waves, from which an integral representation of the image
field, similar to the Debye [85] integral were obtained. An aplanatic system
was dealt with in a subsequent paper [86]. Wolf and Li [87] later showed that
the approach based on the Debye integral is valid for systems that satisfy the
high aperture condition.
In practice, however, the focusing is performed through an interface
between two materials, such as through a coverslip. If these two materials
have different optical properties, i.e. refractive index, then the spherical
aberration occur. Microscope objectives are usually manufactured with
aberration correction for certain but fixed penetration depths. Practically, for
biological applications, this depth is compensated by an appropriate coverslip.
2.5.1 Focusing through mismatched refractive index
materials
The first work on the focusing of EM waves through mismatched refractive
index materials is that of Gasper et al. [88], who consider an arbitrary
EM wave traversing a planar interface. However, due to the complexity of
their approach, the use of their formulas for the calculation of the EM field
distribution near focus was not practical in most cases.
The most straight forward approach is that of Gu [89] and Torok et al. [90].
PhD thesis: Far-field and near-field optical trapping 34
CHAPTER 2. Literature review
Their approach is based on the extension of the Wolf’s [84] treatment of the
diffraction problem when the light is focused through a single homogeneous
material, to the case when light is focused by a high NA lens into a medium
of different refractive index to that of the medium of propagation and
introduces a considerable amount of spherical aberration. The difference in
the two treatments is that Torok et al. [90] use the matrix formalism in their
derivations.
Fig. 2.8: A schematic diagram of a light beam being refracted on an interfacebetween two media (n2 > n1).
A schematic diagram showing a linearly polarised plane wave focused by
a high NA lens into two media separated by a planar interface is shown in
Fig. 2.8. When a spherical wave is focused through an interface between
media of mismatched refractive indices, its wave front becomes distorted.
This distortion can be described by a phase function, called the aberration
function, which depends on the focus depth, the refractive indices n1 and n2,
PhD thesis: Far-field and near-field optical trapping 35
CHAPTER 2. Literature review
and the azimuthal angle [90–92]. The aberration function can be written as
Ψ(φ1, φ2,−d) = −d(n1 cosφ1 − n2 cosφ2) , (2.21)
were d denotes the focus depth, φ1 is the angle of incidence on the interface,
φ2 is the angle of refraction, which are linked by the Snell’s law. Thus the
defined aberration function has a significant effect on the energy distribution
of light focused into the second medium, especially at deep depths. As either
the NA of the focusing objective, the focusing depth, or the refractive index
of the second material (refractive index n2) increases, the main peak of the
focus distribution shifts and the energy distributions become asymmetrical
and less concentrated [91].
The electric and magnetic fields components that describe the field near
the focal region in the second material (n2), at an arbitrary point P (rp, φp, θp)
(Fig. 2.8.), can be expressed as [90]
E2x =iK
2π
∫ α
0
∫ 2π
0
(cosφ1)1/2(sinφ1)[(τp cosφ2 + τs) + (cos 2θ)
×(τp cosφ2 − τs)] expik0[rpκ+ Ψ(φ1, φ2,−d)]dφ1dθ ,
E2y =iK
2π
∫ α
0
∫ 2π
0
(cosφ1)1/2(sinφ1)(sin 2θ)(τp cosφ2 − τs)
× expik0[rpκ+ Ψ(φ1, φ2,−d)]dφ1dθ ,
E2z = − iKπ
∫ α
0
∫ 2π
0
(cosφ1)1/2(sinφ1)τp sinφ2 cos θ
× expik0[rpκ+ Ψ(φ1, φ2,−d)]dφ1dθ , (2.22)
PhD thesis: Far-field and near-field optical trapping 36
CHAPTER 2. Literature review
and
H2x =iKn2
2π
∫ α
0
∫ 2π
0
(cosφ1)1/2(sinφ1)(sin 2θ)(τs cosφ2 − τp)
× expik0[rpκ+ Ψ(φ1, φ2,−d)]dφ1dθ ,
H2y =iKn2
2π
∫ α
0
∫ 2π
0
(cosφ1)1/2(sinφ1)[(τs cosφ2 + τp) + (cos 2θ)
×(τp − τs cosφ2)] expik0[rpκ+ Ψ(φ1, φ2,−d)]dφ1dθ ,
H2z = − iKn2
π
∫ α
0
∫ 2π
0
(cosφ1)1/2(sinφ1)τs sinφ2 sin θ
× expik0[rpκ+ Ψ(φ1, φ2,−d)]dφ1dθ , (2.23)
where κ and the constant K are defined as
K =k1flo
2, (2.24)
and
κ = n1 sinφ1 sinφp cos(θ − θp) + n2 cosφ2 cosφp . (2.25)
Here θp is the spherical coordinate of the point P , k0 indicates the wave
number in vacuo, k1 is the wave number in the first medium, f is the focal
length of the lens in vacuo and l0 is an amplitude factor.
If the incident field E(0) is independent on angle θ, such as the case for a
plane wave for example, the integration in Eqs. 2.22 and 2.23 can be expressed
as the combination of two sets of three integrals, I0, I1, and I2 as derived by
Torok et al. [90] as
E2x = −iK[I(e)0 + I
(e)2 cos(2θp)] ,
E2y = −iKI(e)2 sin(2θp) ,
E2z = −2KI(e)1 cos(θp) , (2.26)
PhD thesis: Far-field and near-field optical trapping 37
CHAPTER 2. Literature review
and
H2x = −iKn2I(h)2 sin(2θp) ,
H2y = −iKn2[I(h)0 − I
(h)2 cos(2θp)] ,
H2z = −2Kn2I(h)1 sin(θp) . (2.27)
After we substitute the normalised optical coordinates
v = k1rp sinφp sinα ,
u = k2rp cosφp sin2 α (2.28)
the integrals I0, I1, and I2 are given by
I(e)0 =
∫ α
0
(cosφ1)1/2(sinφ1) exp[ik0Ψ(φ1, φ2,−d)](τs + τp cosφ2)
×J0
(
v sinφ1
sinα
)
exp
(
iu cosφ2
sin2 α
)
dφ1 ,
I(e)1 =
∫ α
0
(cosφ1)1/2(sinφ1) exp[ik0Ψ(φ1, φ2,−d)]τp sinφ2
×J1
(
v sinφ1
sinα
)
exp
(
iu cosφ2
sin2 α
)
dφ1 ,
I(e)2 =
∫ α
0
(cosφ1)1/2(sinφ1) exp[ik0Ψ(φ1, φ2,−d)](τs − τp cosφ2)
×J2
(
v sinφ1
sinα
)
exp
(
iu cosφ2
sin2 α
)
dφ1 , (2.29)
PhD thesis: Far-field and near-field optical trapping 38
CHAPTER 2. Literature review
and
I(h)0 =
∫ α
0
(cosφ1)1/2(sinφ1) exp[ik0Ψ(φ1, φ2,−d)](τp + τs cosφ2)
×J0
(
v sinφ1
sinα
)
exp
(
iu cosφ2
sin2 α
)
dφ1 ,
I(h)1 =
∫ α
0
(cosφ1)1/2(sinφ1) exp[ik0Ψ(φ1, φ2,−d)]τs sinφ2
×J1
(
v sinφ1
sinα
)
exp
(
iu cosφ2
sin2 α
)
dφ1 ,
I(h)2 =
∫ α
0
(cosφ1)1/2(sinφ1) exp[ik0Ψ(φ1, φ2,−d)](τp − τs cosφ2)
×J2
(
v sinφ1
sinα
)
exp
(
iu cosφ2
sin2 α
)
dφ1 . (2.30)
Here α denotes the maximum convergence angle, Jn is the Bessel function of
the first kind, of order n. τs and τp are the Fresnel coefficients given by [54]
τs =2 sinφ2 cosφ1
sin(φ1 + φ2),
τp =2 sinφ2 cosφ1
sin(φ1 + φ2) cos(φ1 − φ2). (2.31)
Using the vectorial diffraction approach given in Eqs. 2.22, 2.23, 2.26,
and 2.27 the EM field distribution in the region of the focus of a high NA
microscope objective, including the effects of spherical aberration, phase
modulation, and apodisation function can be determined, which would
provide a powerfull tool for the trapping force evaluation under a complex
beam illumination.
2.5.2 Focal spot splitting with high NA objective
Vectorial diffraction effects can lead to the focal splitting for objectives of
high NA. When the NA of an objective becomes large, a linearly polarised
PhD thesis: Far-field and near-field optical trapping 39
CHAPTER 2. Literature review
beam becomes depolarised in the focal region after the refraction by the
objective. In other words, if the incident electric field is polarised along the
5
0.5
0.5
0.5
0.5
1
1
1
2.5
2.5
2.5
10
203550658095
1
2
3
4
5
6
7
-7
-6
-5
-4
-3
-2
-1
0vy
1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0
vx
1
2
3
4
5
6
7
-7
-6
-5
-4
-3
-2
-1
0vy
1 2 3 4 5 6 7-7 -6 -5 -4 -3 -2 -1 0
vx
(a) (b)
Fig. 2.9: Contour plots of the intensity near the focus of a high NA objective(NA = 1). (a) |E|2 for ε = 0.0, (b) |E|2 for ε = 0.98 [93].
x direction, the diffracted field contains an x component as well as y and
z components (Eqs. 2.26 and 2.27). The effect of depolarisation becomes
stronger for increasing convergence angle of the refracted wave. As a result
of depolarisation, the focal spot of high NA objective becomes elongated
(elliptical) along the polarisation direction (Fig. 2.9(a)) [89,94–96].
Illuminated by an annular beam (also known as a ring beam), an objective
exhibits a complicated nature in the focal region. For low NA objectives the
use of an annular beam produces a reduced but circular focal spot due to
the suppression of the waves of small convergence angles [89,94–96], which is
utilised in superresolution imaging [98]. However, for objectives with high NA
and a large inner radius of an annular beam, the waves of high convergence
angles become dominant. As a consequence the diffracted pattern may not
necessarily give a single focal spot (Fig. 2.9(b)) [93,99]. Chon et al. [93] have
PhD thesis: Far-field and near-field optical trapping 40
CHAPTER 2. Literature review
Fig. 2.10: Contour plots of the intensity near the focus of a high NA objectivefocused on an interface between two media (n1 = 1.78 and n2 = 1.0), with NA =1.65 and ε = 0.6 [97].
shown that when the NA of an objective is between 0.9 and 1 in free space, the
longitudinal component of the electric field Ez at the focus becomes relatively
stronger as the obstruction of the annular beam becomes larger and results
in a split two-peak focus. The obstruction size ε is defined as the ratio of the
inner radius to the outer radius of an annular beam. The focal spot splitting
occurs only if the size of the central obstruction reaches a threshold depending
on the numerical aperture of an objective. The creation of the two-peak focus
by an objective illuminated by an annular beam may prove advantageous for
producing controllable torque [100] in laser trapping [101] systems.
Practically, the focal splitting is also observable with very high NA
objectives focused through an interface between two media, with moderate
obstruction size (Fig. 2.10) [97]. The focus-splitting effect is stronger for the
interface between the two materials with a smaller index difference, provided
the obstruction radius is chosen correctly.
PhD thesis: Far-field and near-field optical trapping 41
CHAPTER 2. Literature review
2.5.3 Spectral splitting near phase singularities of fo-
cused waves
When the incident field is spatially fully coherent, but is polychromatic
rather than monochromatic, as with ultrashort-pulsed laser illumination, the
spectrum at the zero-intensity points, called phase singularities, exhibits the
anomalous behavior that causes the splitting of the spectrum [102,103]. This
spectral splitting occurs when the focusing is performed by a lens with an
aperture that has a small maximum angle of convergence, which is called the
paraxial approximation. Such a diffraction system corresponds to a low NA
objective.
However, when the NA becomes large, the focusing process involves
depolarisation. In other words, a linearly polarised incident beam Ex exhibits
two extra components, one in the orthogonal direction, Ey , and the other
one in the longitudinal direction, Ez (Eqs. 2.26 and 2.27). As a result the
spectral splitting phenomenon may not necessarily appear in the focal region
of a high-NA objective.
The anomalous behavior of the spectral intensity, calculated at the axial
zero-intensity points under the paraxial approximation, occurs because of
the condition that the spectral intensity at frequency ω0 has a zero value
when un(ω0) = 4πn is valid for low NA lenses. These zero-intensity points
disappear for lenses with high NA as a result of the contribution of the
depolarised longitudinal component Ez [103]. The normalised spectrum at
the first axial minimum intensity point of the central frequency component
ω0 for different values of the focusing objective NA is shown in Fig. 2.11.
The normalised spectrum is split into two lines of equal intensity for a
low NA lens (Fig. 2.11(a)) for which the paraxial approximation applies.
However, when the NA of the focusing lens increases, the spectrum at the
PhD thesis: Far-field and near-field optical trapping 42
CHAPTER 2. Literature review
Fig. 2.11: The normalised spectrum S[u1(ω0, ω]/S0 at the first axial zero-intensitypoint (u1 = 4π) of the central frequency component ω0 for different values of NA:(a) NA = 0.025, (b) NA = 0.1, (c) NA = 0.3, (d) NA = 0.4, (e) NA = 0.6, (f)NA = 0.9.
first axial minimum intensity point undergoes a noticeable change. When the
NA increases (Fig. 2.11(b) and 2.11(c)), the spectrum is still split into two
lines, but with a shallow dip in the center. When the NA is larger than 0.4
(Fig. 2.11(d)), the dip in the spectrum disappears and the spectrum does not
split. In the case of a high-NA lens, the spectrum distribution is almost the
same as the input spectral distribution (Fig. 2.11(e) and 2.11(f)).
Due to the depolarisation effect, the focal spot is elongated along the
polarisation direction, which leads to the non-zero value of the first minimum
intensity point in the x direction (polarisation direction), while the first
zero-intensity point is retained in the y direction. As a consequence, the
spectral splitting in the y direction is independent of the NA value, while
the spectral behavior in the incident polarisation direction is NA dependent
(Fig. 2.12). In the paraxial case (Fig. 2.12(a)) the spectrum is split in the
PhD thesis: Far-field and near-field optical trapping 43
CHAPTER 2. Literature review
Fig. 2.12: The normalised spectrum S(ω)/S0 at the first x and y zero-intensitypoints of the central frequency component ω0 for different values of NA: (a) NA =0.025, (b) NA = 0.05, (c) NA = 0.1, (d) NA = 0.3, (e) NA = 0.6, (f) NA = 0.9.Full lines show the variations in the x direction, while dotted lines represent thosein the y direction.
x direction but the spectrum already shows a nonzero dip at the central
frequency component ω0, while it completely disappears for NA = 0.05. The
spectrum in x direction approaches that of the incident beam quickly as the
NA value increases (Fig. 2.12(c)- 2.12(f)).
2.6 Near-field trapping
The trapping volume of the far-field laser trapping geometry is approximately
three times larger in the axial direction than that in the transverse direction.
Such trapping volume elongation leads to a significant background and
poses difficulties in the observations of the single-molecule dynamics. In
recent times, a new trapping modality based on the evanescent wave
PhD thesis: Far-field and near-field optical trapping 44
CHAPTER 2. Literature review
illumination, also called near-field illumination, has been proposed [12–15]
and demonstrated [15]. This trapping technique results in a significantly
reduced trapping volume due to the fact that the strength of an evanescent
wave decays rapidly with the distance from the place at which the field is
generated. In this section, the near-field trapping mechanism based on the
different ways to generate a localised near-field is reviewed.
2.6.1 Near-field trapping using a nano-aperture
Okamoto and Kawata [14] have proposed a near-field trapping technique that
utilises a circular nano-aperture as a localised field source. The trapping is
achieved by interaction between the aperture and the dielectric sphere via
evanescent photons. The near-field trapping model geometry is shown in
Fig. 2.13. A glass substrate is coated with a metallic layer of thickness of
λ/5, where λ is the illumination wavelength in vacuum. The metallic layer
is in contact with water. A circular aperture is made in this layer with a
diameter of λ/4, and a small dielectric particle, with a diameter of λ/2 is
situated in water near the aperture.
Okamoto and Kawata [14] have made a series of numerical calculations
of the radiation force exerted on a such particle. The particle position
near the aperture is constantly changed in this calculation, thus a spatial
distribution of the radiation force, also known as a force mapping, is
determined (Fig. 2.13(a) and (b)). The EM field distribution is calculated
using using the finite-difference time-domain (FDTD) method. From each
calculated EM field distribution, the radiation force was obtained from the
Maxwell stress tensor on the surface of the sphere.
Using this methodology, it was confirmed that optical near-field trapping
using a nano-aperture configuration can be achieved. The result indicates
PhD thesis: Far-field and near-field optical trapping 45
CHAPTER 2. Literature review
polarisation polarisation
Fig. 2.13: The geometry of the near-field trapping model. Incident light is xpolarised and propagates along the z axis. (a) and (b) Radiation force spatialdistribution for a dielectric sphere near a nano-aperture. The origin of each arrowrepresents the center of the sphere and vectors represent the direction and themagnitude of forces [14].
that a particle is attracted towards the aperture. The near-field radiation
force is found to be larger than the forces due to thermal fluctuations and
to gravity. Furthermore, they have found that if two particles are near the
aperture, the first particle is trapped and the second one is also attracted to
the first one.
2.6.2 Near-field trapping using a metallic tip
Novotny et al. [13] have investigated the possibility of trapping a nano-particle
using the field enhancements close to a laser illuminated sharply pointed metal
tip. The near-field close to the tip mainly consists of evanescent components
which decay rapidly with distance from the tip. Their analysis is based
on characterising the field enhancements near the tip using the numerical
MMP method. The effect of the particle in close proximity of the tip is
PhD thesis: Far-field and near-field optical trapping 46
CHAPTER 2. Literature review
also included. Once the field distribution on the surface of the particle is
determined, the Maxwell stress tensor approach is used to determine the
near-field force exerted on the particle [13].
Fig. 2.14: Near field of a gold tip in water illuminated by two differentmonochromatic waves at λ = 810 nm, indicated by the k and E vectors. Thenumbers in the figure give the scaling in the multiples of the exciting field (E2),with a factor of 2 between the successive lines. (a) No field enhancement; (b) Fieldenhancement of ≈ 3000 [13].
Figure 2.14 shows the three dimensional MMP calculation of a field
enhancement of a gold tip with radius of 5 nm in water for two different
monochromatic plane-wave excitations. The wavelength of the illuminating
light is λ = 810 nm, which does not match the surface plasmon resonance.
When the illumination field is incident from the bottom with the polarisation
perpendicular to the tip axis (Fig. 2.14(a)), there is no field enhancement
beneath the tip. In the case when the tip is illuminated from the side
with the polarisation parallel to the tip axis (Fig. 2.14(b)), a large intensity
enhancement at the foremost part of the tip occurs. This intensity at the
foremost part of the tip is approximately 3000 times stronger than the
illuminating intensity.
PhD thesis: Far-field and near-field optical trapping 47
CHAPTER 2. Literature review
Utilising this methodology, Novotny et al. [13] have shown that nano-
particles can be trapped using the field enhancements near a metallic tip when
the polarisation of the incident light is directed along the tip axis. Based on
their findings they propose a near-field trapping scheme in which a particle
is first trapped by a conventional far-field trapping means and then a sharp
metal tip is brought to the focus. A polarisation component along the tip
axis enables trapping of the particle to the near field of the tip. The trapped
particle can be moved within the focal region of the illuminating light by
translating the tip and can be released by turning off the laser illumination.
2.6.3 Near-field trapping using an apertureless probe
Another novel approach to trap small particles using a near-field is given by
Chaumet et al. [12]. Their approach is based on the use of a combination of
evanescent illumination and light scattering at the probe apex to shape the
optical field into a localised, three-dimensional optical trap. The schematic
representation of their configuration is shown in Fig. 2.15.
The particle under consideration has a radius of 10 nm and is situated
in air. It is illuminated (λ = 500 nm) by two by two counter-propagating
evanescent waves generated by the TIR of plane waves the substrate/air
interface. These two waves have the same polarisation and a random phase
relation This symmetric illumination ensures that the sphere is not pushed
away from the tip when the sphere is just below the tip. A tungsten probe
is placed in close proximity of the dielectric sphere. As the probe tip moves
closer to the sphere, a force is exerted on the particle.
Chaumet et al. [12] have used the coupled-dipole method and the Maxwell
stress tensor to determine the near-field force exerted on the particle. They
have calculated the force experienced by the particle as the probe is moved
PhD thesis: Far-field and near-field optical trapping 48
CHAPTER 2. Literature review
Fig. 2.15: A dielectric sphere situated on a flat dielectric surface is illuminated bynear-field generated under total internal reflection. A tungsten probe is used tocreate an optical trap [12].
laterally, above the particle. The force components in axial z and lateral x
directions acting on the sphere are plotted for two distances between the tip
and the substrate (20 and 31 nm), and for two angles of illumination (43 and
50)(Fig. 2.16). It is shown that small objects can be selectively captured
and manipulated by the near-field force generated using an apertureless
probe with the trapping configuration shown in Fig. 2.15. Furthermore, the
magnitude and direction of trapping force greatly depend on the polarisation
state of the incident illumination.
For TM illumination the axial trapping force is positive, i.e. the particle is
attracted towards the probe tip. This occurs because of a large enhancement
of the field near the apex of the probe for TM illumination. This result
PhD thesis: Far-field and near-field optical trapping 49
CHAPTER 2. Literature review
(a)
(b)
(c)
(d)
Fig. 2.16: Force experienced by the sphere as a function of the lateral position ofthe probe. Thick lines denote θ = 43, while thin lines denote θ = 50. The probetip is either 20 nm (solid lines) or 31 nm (dashed lines) above the substrate. (a)TM polarisation z direction; (b) TE polarisation z direction; (c) TM polarisationx direction; (d) TE polarisation x direction. According to Ref. [12].
indicates that by using the TM illumination with this trapping configuration
one can achieve the particle lifting. In the case of the TE illumination,
however, the axial trapping force is directed away from the probe tip
as the tip gets closer to the particle. This prevents any lifting of the
particle for this polarisation state. Using this trapping configuration and its
polarisation dependence one can selectively trap a particle using TM polarised
illumination, lift it and manipulate it by moving the probe tip, and finally
eject the particle by switching to the TE polarised illumination.
2.6.4 Near-field trapping with evanescent field gener-
ated under TIR illumination
An evanescent field generated under the condition of the TIR can also trap
and move small microparticles [15, 67]. This near-field configuration can be
divided into two categories: near-field trapping with a wide area evanescent
field and near-field trapping with a focused evanescent field.
PhD thesis: Far-field and near-field optical trapping 50
CHAPTER 2. Literature review
The wide area evanescent field is usually generated at a surface of a
prism under the TIR illumination condition. Kawata and Sugiura [67]
have demonstrated a microparticle translation using the wide area near-field
configuration. The focused evanescent field is generated using an annular
beam, produced by a high NA objective that is centrally obstructed, which
satisfies the TIR condition at the objective coverslip surface. Gu et al. [15]
have recently demonstrated this type of near-field trapping. In this section,
both categories of the near-field trapping with evanescent field are reviewed.
2.6.4.1 Near-field trapping with a wide area evanescent field
The near-field trapping geometry for the wide area evanescent field trapping
is depicted in Fig. 2.17(left). It is called the wide area because the area at
which evanescent field is generated is much larger than the microparticles.
The laser beam is incident on the prism surface at an angle larger than the
Fig. 2.17: Schematic diagram of particle movement in an evanescent field. Velocityof the moving particle versus the incident angle of the beam undergoing TIR [67].
critical angle, generating a fast decaying evanescent field near the surface.
If a small dielectric particle is in the vicinity of the surface it can convert
evanescent photons into propagating photons through a photon tunneling
PhD thesis: Far-field and near-field optical trapping 51
CHAPTER 2. Literature review
process [104, 105]. The tunneled photons scatter from the particle through
the process of multiple internal reflections inside the particle. Consequently,
a part of the momentum of photons of the incident laser beam is transferred
to the particle, driving the moving particle.
The velocity of the evanescent field driven polystyrene particles of a
diameter of 6.8 µm as a function of the incident angle of the beam undergoing
the TIR is shown in Fig. 2.17(right). Due to the evanescent field depth
decrease with increasing incident angle, the particle velocity for both P and S
incident polarisation is decreasing. It is found that the particles driven by S
polarisation moved much faster than those driven by P polarisation for each
incident angle.
The theoretical treatment of this type of near-field trapping is given by
Almaas and Brevik [68], while the effects of the prism surface are discussed
by Lester and Vesperinas [69]. Both of these approaches are using an EM
model for evanescent wave illumination and its interaction with a particle,
while the force exerted on the particle is determined using the Maxwell stress
tensor approach. Almaas and Brevik [68] are using an analytical expression
for evanescent wave generated at the prism surface and interacting with a
particle far-away from the surface, so that its effects are neglected. Lester
and Vesperinas [69], on the other hand, consider the effects of the interface
through a multiple-scattering numerical simulation. They show that when
the particle approaches the surface the axial trapping force changes from
attractive (Fz < 0) to strongly repulsive (Fz > 0).
2.6.4.2 Near-field trapping with a focused evanescent field
The geometry of the near-field trapping with a focused evanescent field is
shown in Fig. 2.18. The central obstruction has such a size that the minimum
PhD thesis: Far-field and near-field optical trapping 52
CHAPTER 2. Literature review
angle of convergence of a ray is larger than the critical angle determined by
the inference between two media. This condition ensures that each ray is
incident on the interface at an angle that satisfies the TIR condition and
thus results in a focused evanescent wave on the interface. Due to the
Fig. 2.18: Near-field trapping under focused evanescent illumination. Densityplots (a) and (b) represent the calculated modulus squared of the electric fieldat wavelength 532 nm in the focal region of an objective of NA = 1.65 at theinterface between the cover slip (n = 1.78) and water (n = 1.33). (a) No centralobstruction, i.e. ε = 0; (b) With central obstruction ε = 0.8, whose size satisfiesthe TIR condition [15].
circular symmetric nature of this illumination, the resulting evanescent wave
constructively interferes at the center of the focus, enhancing the strength of
the evanescent field and reducing the lateral trapping size.
Gu et al. [15] have shown experimentally that such focused evanescent
field can trap microscopic dielectric particles due to the fast decaying nature
of the evanescent field. Using vectorial diffraction theory [97], one can show
that the axial size of the trapping volume is reduced to approximately 60
nm, while the lateral trapping size is reduced by 10% (Fig. 2.18(a) and
(b)). Additional advantage of this method compared to the other near-field
trapping methods is that the distance between the trapping site and the
objective is sufficiently large for micro-manipulation. Furthermore, there is
PhD thesis: Far-field and near-field optical trapping 53
CHAPTER 2. Literature review
no heating problems associated with this near-field trapping method. The
strength of the evanescent field under the total internal condition can be
further increased by the use of a dielectric double-layer structure coated on
the cover slip, which can result in the enhancement of the evanescent field by
approximately three orders of magnitude [51].
As far as we are aware, there has been no theoretical treatment of
this type of near-field trapping. It is one of the aims of this thesis to
present a theoretical model for this promising near-field trapping method.
Our theoretical treatment is based on the vectorial diffraction and the
Maxwell stress tensor approach, given in greater details in Chapter 4,
while the treatment of the near-field trapping under focused evanescent field
illumination is undertaken in Chapter 6.
2.7 Chapter summary
One of the aims of this thesis is to develop an exact and rigorous model
for optical trapping by a focused laser beam based on vectorial diffraction
theory. The laser beam is focused by a high NA objective, while a dielectric
particle is suspended in a medium which in general differs from the immersion
medium of the objective. Two optical trapping modalities are considered, the
far-field laser trapping and the near-field trapping under focused evanescent
field illumination. In this chapter we have reviewed the theoretical models
currently available for determining the far-field trapping force exerted on a
small dielectric particle. These approximate theoretical models cannot deal
with focal distribution complexities of laser beams focused using a high NA
objective. The physical effects such as spherical aberration and objective
apodisation, as well as an arbitrary complex phase of the incident illumination
can not be considered using these approximate models. The different near-
PhD thesis: Far-field and near-field optical trapping 54
CHAPTER 2. Literature review
field trapping techniques are also reviewed based on the method of generating
a localised near-field. The theoretical approaches for these different near-field
trapping schemes are pointed out. However, the near-field trapping technique
based on the focused evanescent field illumination has not been theoretically
dealt with. Our optical trapping model based on vectorial diffraction by a high
NA objective can be applied to provide an appropriate theoretical model for
such a trapping modality as well. In order to gain a physical insight into our
modelling of both far-field and near-field trapping modality in later chapters,
a review of vectorial diffraction theory, extremely important when dealing
with the focusing by high NA objectives, is also included. The details of our
optical trapping model are given in Chapter 4, while its applications in the
far-field and the near-field trapping are presented in Chapter 5 and Chapter 6
respectively.
The other aim of this thesis is to develop an appropriate nanometric
sensing model based on near-field (also known as evanescent field) Mie
scattering and vectorial diffraction of the scattered field. The applications
of the far-field trapping such as optical trap nanometry and particle trapped
SNOM, are based on the near-field scattering by a small trapped particle for
high resolution imaging and position measurements. The evanescent field
is generated at an interface between two different media under the TIR
condition, while the scattering particle is situated in a close proximity of
the interface. The current theoretical models of near-field Mie scattering
are dealing with differential scattering cross-sections or the asymptotic
representation of the scattered field and are reviewed in this chapter. In order
to develop an appropriate nanometric sensing model, the three-dimensional
distribution of the scattered field is required. Such a sensing model needs
to include the effects of the interface at which the near-field is generated,
which is of great importance when the particle is situated in its vicinity. The
detectability of the collected signal is addressed by considering a wide-area
PhD thesis: Far-field and near-field optical trapping 55
CHAPTER 2. Literature review
and a pinhole detector. Our nanometric sensing model is described in greater
details in Chapter 3.
PhD thesis: Far-field and near-field optical trapping 56
Chapter 3
Three dimensional near-field
Mie scattering by a small
particle
3.1 Introduction
Far-field trapping modality is often used with high resolution imaging, based
on evanescent (near-field) illumination of the sample under investigation. Two
of those high resolution imaging techniques that utilise a laser trapped particle
(far-field trapping) are reviewed in Sections 2.2.1 and 2.2.2 of Chapter 2. It is
indicated that for a proper model of near-field Mie scattering, when a particle
is near the interface at which the near-field is generated, the interface effects
need to be included (Section 2.4). Thus, this chapter investigates near-field
Mie scattering by microscopic dielectric particles situated in close proximity
of the surface at which an evanescent wave is generated and presents our
physical model for the detection of the scattered near-field signal.
Near-field Mie scattering is a scattering process caused by interaction of a
57
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
Mie particle with an evanescent wave rather than a plane wave (Section 2.4).
An evanescent wave occurs when an electromagnetic wave is incident from a
high refractive index medium into a low refractive index medium, under total
internal reflection. The strength of an evanescent wave decays quickly in the
low refractive index media; the larger the incident angle the larger the decay
constant β. Such an evanescent wave can be converted into a propagating
wave by interaction with a small particle situated in a low refractive index
medium, provided that the distance between the particle and the interface
separating the two media, d, is within a few illumination wavelengths in
length. At first the scattering of near-field is studied by neglecting the surface
effects, which corresponds to the case when the particle is far away from the
interface. Subsequently, the surface influence is included and its effects on
the scattering properties are determined.
Once the three-dimensional (3-D) electric vector field distribution in the
far-field is known a collection objective can be introduced, using the vectorial
diffraction theory, to determine the collected signal onto an optical detector.
The scalar diffraction theory would not be appropriate for this problem, due
to the vectorial nature of the scattered field.
This chapter is structured as follows. Section 3.2 gives the mathematical
treatment of near-field Mie scattering, while the 3-D scattered intensity
distribution around dielectric particles near and far from the interface is
examined in Section 3.3. Morphology dependent resonance effects are
investigated in Section 3.4. A physical model, theoretical and experimental
results for the conversion of evanescent photons into propagating photons are
given in Section 3.5. Pinhole detection of the scattered signal is examined in
Section 3.6, while the main conclusions are drawn in Section 3.7.
PhD thesis: Far-field and near-field optical trapping 58
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
3.2 Mathematical description of 3-D near-
field Mie scattering
A schematic diagram of the near-field Mie scattering model, which includes
the effect of an interface where the evanescent wave is generated, is presented
in Fig. 3.1. The incident angle of the electromagnetic wave is represented
d
P
P’
n
n’
an
n’
surrounding
medium
substrate
z
x
Y
Fig. 3.1: Illustration of the scattering model including the effect of an interface atwhich an evanescent wave is generated. n, n′ and n1 denote refractive indices ofthe substrate, surrounding medium and the particle, respectively. d is the distancefrom the center of a particle to the interface.
by α. The refractive index of the scattering particle is denoted by n1, while
it is immersed into a medium of refractive index n′. The refractive index
of the substrate is n. Particle is situated at a distance d from the interface
PhD thesis: Far-field and near-field optical trapping 59
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
at which an evanescent wave is generated. The incident electric wave for
the polarisation state of E0 perpendicular to the plane of incidence (i.e. TE
polarisation) is given by [72]
Einc(r) =∑
lm
ic
n′2ωαE(l,m)∇× [jl(k
′r)Yllm(r)]
+ αM(l,m)jl(k′r)Yllm(r)
, (3.1)
with the magnetic wave
Hinc(r) =∑
lm
αE(l,m)jl(k′r)Yllm(r)
− ic
ωαM(l,m)∇× [jl(k
′r)Yllm(r)]
. (3.2)
l = 1 to ∞ and m = -l to +l. c is the speed of light in vacuum and ω
is the angular frequency of the incident light. The functions αE(l,m) and
αM(l,m) are the expansion coefficients for the incident illumination field
given by Eqs. 2.14 and 2.15, while Yllm is the vector spherical harmonics.
jl is the spherical Bessel function of the l-th order. In the case of the
polarisation state E0 parallel to the plane of incidence (i.e. TM polarisation),
the scattered electric and magnetic fields are also given by Eq. 3.1 and Eq. 3.2
with expansion coefficients αE(l,m) and αM(l,m) substituted by αE(l,m) and
αM(l,m), respectively [72], given by Eqs. 2.16 and 2.17.
For particles which are situated far from the interface, the interaction
between the particle and the interface can be neglected. However, if the
particle is very close to the interface, then the interaction between the particle
and the interface is not negligible and it needs to be incorporated into the
scattering model.
PhD thesis: Far-field and near-field optical trapping 60
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
3.2.1 Particles far from the interface
In the geometry shown in Fig. 3.1, when the incident angle α of the
illumination electric wave E0 is larger than the critical angle, the scattered
electric field by the Mie particle can be expressed, for the TE polarisation,
as [72]
Esc(r) =∑
lm
ic
n′2ωβE(l,m)∇× [h
(1)l (k′r)Yllm(r)]
+ βM(l,m)h(1)l (k′r)Yllm(r)
, (3.3)
while the magnetic field is given by
Hsc(r) =∑
lm
βE(l,m)h(1)l (k′r)Yllm(r)
− ic
ωβM(l,m)∇× [h
(1)l (k′r)Yllm(r)]
. (3.4)
When expressed in a spherical coordinate system, Eq. 3.3 and Eq. 3.4 can be
reduced to
Esc(r) =∑
lm
cβE(l,m)
n′2ω√
l(l + 1)
h(1)l (k′r)
r sin θ
[
∂
∂θ
(
∂Ylm sin θ
∂θ
)
+1
sin θ
∂2Ylm
∂ϕ2
]
r1
+
[
(−1)βM(l,m)h(1)l (k′r)
i sin θ√
l(l + 1)
∂Ylm
∂ϕ− cβE(l,m)
n′2ω√
l(l + 1)
1
r
∂Ylm
∂θ
∂
∂r(rh
(1)l (k′r))
]
θ1
+
[
βM(l,m)h(1)l (k′r)
i√
l(l + 1)
∂Ylm
∂θ− cβE(l,m)
n′2ω√
l(l + 1)
1
r sin θ
∂Ylm
∂ϕ
∂
∂r(rh
(1)l (k′r))
]
ϕ1
,
(3.5)
PhD thesis: Far-field and near-field optical trapping 61
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
Hsc(r) =∑
lm
−cβM(l,m)
ω√
l(l + 1)
h(1)l (k′r)
r sin θ
[
∂
∂θ
(
∂Ylm sin θ
∂θ
)
+1
sin θ
∂2Ylm
∂ϕ2
]
r1
+
[
(−1)βE(l,m)h(1)l (k′r)
i sin θ√
l(l + 1)
∂Ylm
∂ϕ+
cβM(l,m)
ω√
l(l + 1)
1
r
∂Ylm
∂θ
∂
∂r(rh
(1)l (k′r))
]
θ1
+
[
βE(l,m)h(1)l (k′r)
i√
l(l + 1)
∂Ylm
∂θ+
cβM(l,m)
ω√
l(l + 1)
1
r sin θ
∂Ylm
∂ϕ
∂
∂r(rh
(1)l (k′r))
]
ϕ1
,
(3.6)
where r1, θ1 and ϕ1 are the unit vectors in the spherical coordinates. θ and
ϕ are the variables of the scalar spherical harmonics Ylm.
In the case of the polarisation state E0 parallel to the plane of incidence
(i.e. TM polarisation), the scattered electric and magnetic fields are also
given by Eq. 3.5 and Eq. 3.6 with expansion coefficients βE(l,m) and βM(l,m)
(Eqs. 2.12 and 2.13) substituted by βE(l,m) and βM(l,m), respectively [72]
as discussed in Chapter 2 Section 2.4.2.
Eq. 3.5 gives the electric field resulting from scattering of an evanescent
wave by a Mie particle without using any approximation. In the previous
calculation [72], an approximated form of Eq. 3.5, which is valid only in two
principal planes, was used. In this thesis, Eq. 3.5 is used to calculate the 3-D
distribution of the scattered field around a particle. The 3-D distribution of
the scattered field is dependent on the effective refractive index nef = n1/n′
and the size parameter q = k′a, where a is the radius of a scattering Mie
particle and k′ is the propagation constant of the electromagnetic wave in the
medium surrounding the particle. Another factor affecting the strength and
the distribution of the scattered field is the decay constant β of the evanescent
wave, defined as [72]
PhD thesis: Far-field and near-field optical trapping 62
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
β = k′
√
n2 sin2 α
n′2− 1 . (3.7)
The larger the decay constant of the near-field wave, the faster the
evanescent wave decays in the depth direction (i.e. in the X-direction).
Depending on the particle size and the decay constant, a scattering particle
may be immersed into an evanescent wave completely or partially, which
greatly affects the scattered field distribution around the particle.
3.2.2 Particles near the interface
When the scattering particle is situated close to the interface on which
an evanescent wave is generated, interaction between the particle and the
interface needs to be taken into account. Although the effect of the
interface is discussed by other researchers, only the scattering cross-sections
of evanescent wave scattering by small particles are calculated [73–75, 79].
Prieve and Walz [71] have considered interface effects but only for large
particles using a ray optics approach. Doicu et al. [76, 77] have investigated
the differential scattering cross section and the integral response of evanescent
wave scattering by small particles and a sensor tip of up to 100 nm in radius.
These numerical calculations include the interface effects in the context of
the discrete sources method and the T-matrix method, and the far-field is
determined by extrapolation of the scattered field as r →∞. However, these
calculations have not discussed scattering properties of larger particles, most
commonly used in particle trapped SNOM, whose scattering characteristics
are markedly different from scattering properties of small particles due to the
MDR effects [79,106–111].
To include the effects of the interface, the following approach is used.
The scattered electric field Etotal at an arbitrary point P in the surrounding
PhD thesis: Far-field and near-field optical trapping 63
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
medium can be expressed as a sum of the field that would be scattered
into point P if the interface between two media was not present, and the
contribution that is reflected into point P by the interface. Mathematically
it can be expressed as
Etotal(r) = Eupper(r) + rfEbottom(r) , (3.8)
where Eupper is scattered field into point P (upper space) and Ebottom is
scattered field into point P ′ (bottom space) without considering the interface,
while rf is the Fresnel amplitude reflection coefficient for a given incident
polarisation state [54]. P ′ is the mirror-reflection point of the point P .
The amount of the scattered field at point P ′ that is reflected to point P
is determined by the Fresnel amplitude reflection coefficients under given
conditions. Eupper(r) and Ebottom(r) can be determined from Eq. 3.5 without
considering the effect of the interface.
3.3 3-D scattered intensity distribution around
dielectric particles
According to the mathematical model described by Eq. 3.5, a 3-D intensity
distribution of the scattered field can be numerically calculated at any
distance from the scattering particle. To understand the effect of the interface
on scattering distribution, let us first consider that a particle is situated far
away from the interface, and then we will consider the case when the particle
is near the interface.
PhD thesis: Far-field and near-field optical trapping 64
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
3.3.1 Dielectric particle situated far from the interface
Let us consider a dielectric particle of a radius of 2 µm immersed in air.
The scattered intensity distribution produced by such a particle is shown
in Fig. 3.2. To demonstrate the 3-D distribution of the scattered field, the
scattered intensity in the XZ plane, in the plane containing the X axis at 45
anticlockwise from the XZ plane, in the XY plane and in the YZ plane are
displayed in Fig. 3.2(a), 3.2(b), 3.2(c) and 3.2(d), respectively. The scattered
intensity distribution is asymmetric in the XZ plane and in the plane at 45
from the XZ plane because the illumination field is asymmetric in these planes
while the evanescent wave propagates in the Z direction.
Fig. 3.2 shows that the evanescent wave is most intensely scattered into
certain regions around the particle. This effect is due to the fact that the
interaction of the evanescent wave with the particle is confined to the bottom
of the particle because particle size is so large compared with the decay depth
of the evanescent wave (in this case the decay depth is 186.5 nm). This
phenomenon can be qualitatively explained using Snell’s law and the Fresnel
amplitude coefficients for reflection and transmission [54] (Fig. 3.3). Three
rays representing the evanescent wave propagating along the Z direction are
chosen for demonstration. The length of the vectors in Fig. 3.3 represents the
relative strength of the intensity. Ray 1, the highest intensity ray of the three
selected rays, interacts at the very bottom of the particle (point A) with
a large incident angle. Consequently, most of the ray intensity is reflected
rather than refracted because of the high incident angle with respect to the
normal of the particle’s surface. The refracted ray, after traversing through
the particle, interacts with the particle-medium boundary at point B and the
large amount of its intensity emerges into the medium with a small portion
reflected. This reflected portion of intensity traverses through the particle
again, interacts with the boundary (point C) and emerges into the medium
PhD thesis: Far-field and near-field optical trapping 65
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
Fig. 3.2: Three-dimensional far-field distribution of the scattered intensity around a2 µm dielectric particle situated far away from the interface (a) in the XZ-plane, (b)in the plane containing the X-axis at 45o anti-clockwise from the XZ-plane, (c) inthe XY -plane and (d) in the Y Z-plane. The solid and dotted curves correspond tothe TE and TM polarisation states of the illumination wave, respectively. n1=1.6,n′=1.0, n=1.51, λ=632.8 nm and α=45.
while a negligible amount is reflected again. Rays 2 and 3, whose intensity
is weaker and determined by the decay constant of the evanescent wave
experience a similar process. As a result, the relative intensity distribution
of the three rays after three refraction processes indicates that the scattered
intensity profile is confined to certain regions around the particle and that
the highest intensity region is located in the region bellow 0 (see Fig. 3.2(a)).
The location of these intensity regions depends on the refractive indices of
PhD thesis: Far-field and near-field optical trapping 66
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
A
A
A
B
B
C
C
B
C
1 1
1
1
2
2
2
2
3
3
3
3
n’=1.0n1=1.6
Fig. 3.3: A qualitative interpretation of the confined intensity regions in thescattering of an evanescent wave by a dielectric particle of radius 2 µm. Therelative intensity of rays is denoted by the arrow length.
the dielectric particle and its immersion medium, the particle size and the
decay constant of the near-field wave.
For particles far away from the interface the dependence of the asymmetric
scattered intensity distribution on the particle size is depicted in Fig. 3.4. For
small particle sizes (Figs. 3.4(a) and 3.4(b)), the scattered intensity profiles
are similar to those under plane wave illumination [54]. This similarity to
the plane wave Mie scattering occurs because when the particle is small,
it is completely immersed into the evanescent field and the difference of
the evanescent intensity between the top and the bottom of the particle
is negligible. When the particle size becomes large the difference of
the evanescent wave intensity between the top and the bottom of the
particle becomes pronounced, resulting in an asymmetric scattered intensity
distribution as shown in the plane of incidence (i.e., in the XZ plane).
PhD thesis: Far-field and near-field optical trapping 67
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
(d)
(c)
(b)
(a)
0.00
1.50x10 -3
3.00x10 -3
4.50x10 -3
6.00x10 -3
7.50x10 -3
9.00x10 -3
0
30
60
90
120
150
180
210
240
270
300
330
0.00
1.50x10 -3
3.00x10 -3
4.50x10 -3
6.00x10 -3
7.50x10 -3
9.00x10 -3
0.0
1.0x10 -5
2.0x10 -5
3.0x10 -5
4.0x10 -5
5.0x10 -5
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7.0x10 -5
0
30
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1.0x10 -5
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6.0x10 -5
7.0x10 -5
0.0
2.0x10 -4
4.0x10 -4
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8.0x10 -4
1.0x10 -3
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30
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2.0x10 -4
4.0x10 -4
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1.0x10 -3
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1.0x10 -5
2.0x10 -5
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5.0x10 -5
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0
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330
0.0
1.0x10 -5
2.0x10 -5
3.0x10 -5
4.0x10 -5
5.0x10 -5
6.0x10 -5
0.0
5.0x10 -8
1.0x10 -7
1.5x10 -7
2.0x10 -7
2.5x10 -7
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3.5x10 -7
4.0x10 -7
0
30
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90
120
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180
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330
0.0
5.0x10 -8
1.0x10 -7
1.5x10 -7
2.0x10 -7
2.5x10 -7
3.0x10 -7
3.5x10 -7
4.0x10 -7
0.0
2.0x10 -8
4.0x10 -8
6.0x10 -8
8.0x10 -8
1.0x10 -7
1.2x10 -7
1.4x10 -7
0
30
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270
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330
0.0
2.0x10 -8
4.0x10 -8
6.0x10 -8
8.0x10 -8
1.0x10 -7
1.2x10 -7
1.4x10 -7
0.0
1.0x10 -9
2.0x10 -9
3.0x10 -9
4.0x10 -9
5.0x10 -9
0
30
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90
120
150
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330
0.0
1.0x10 -9
2.0x10 -9
3.0x10 -9
4.0x10 -9
5.0x10 -9
0.0
5.0x10 -10
1.0x10 -9
1.5x10 -9
2.0x10 -9
2.5x10 -9
0
30
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120
150
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240
270
300
330
0.0
5.0x10 -10
1.0x10 -9
1.5x10 -9
2.0x10 -9
2.5x10 -9
Fig. 3.4: Dependence of the scattered intensity distribution in the XZ plane on theradius of a particle, when the particle is situated far away from the interface: (a)a = 0.05 µm, (b) a = 0.1 µm, (c) a = 0.5 µm and (d) a = 1 µm. The plots in the leftand the right columns show the intensity distributions scattered by an evanescentwave and a plane wave respectively. The solid and dotted curves correspond to theTE and TM polarisation states of the illumination wave, respectively. n1 = 1.6,n′ = 1.0, n = 1.51, λ = 632.8 nm and α = 45.
PhD thesis: Far-field and near-field optical trapping 68
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
Fig. 3.4(c) can also be qualitatively explained using the same method as
for Fig. 3.2(a). In this case, most of the evanescent intensity on the particle
is incident with a small incident angle. As the Fresnel reflection coefficient
on the particle surface is small for small incident angles [54], the final result
is that the refracted rays lead to an intensity distribution in the region from
0 to 60.
For the 1 µm particle (Fig. 3.4(d)), the scattered intensity distribution
for the TE polarisation illumination is mostly confined to a region below the
Z axis but that for the TM polarisation illumination is mostly above the Z
axis. According to the particle size, the major contribution to the scattered
intensity distribution is from the rays with an incident angle between 70
to 80 near the bottom of the particle. The Fresnel reflection coefficients at
these incident angles for TE polarisation illumination are larger than that for
TM polarisation illumination (for example, the former is 4 times larger for
the incidence at 75). As a result, the majority of the intensity is reflected
in the case of TE polarisation, while a majority of the intensity is refracted
and transmitted through the particle in the case of TM polarisation (see
Fig. 3.4(d)).
3.3.2 Dielectric particle situated near the interface
When a particle is brought close to the interface on which the evanescent
wave is generated, the scattered intensity distribution, given by Eq. 3.8,
is drastically changed. Fig. 3.5 shows the XZ plane scattered intensity
distribution for small and large dielectric particles situated on the boundary
between the surrounding medium and the substrate. It can be seen that
not only has the scattered intensity distribution drastically changed for both
incident polarisations, but also the scattering profile for the TM polarisation
is much stronger at smaller scattering angles (0-30 from either +Z or -Z
PhD thesis: Far-field and near-field optical trapping 69
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
directions) when compared with the TE polarisation. For larger particles,
the scattering profile for the TE polarisation state exhibits similar behavior.
0.0
0.2
0.4
0.6
0.8
1.0
0
30
60
90
120
150
180
210
240
270
300
330
0.0
0.2
0.4
0.6
0.8
1.0
(a)
(c)
0.0
0.2
0.4
0.6
0.8
1.0
0
30
60
90
120
150
180
210
240
270
300
330
0.0
0.2
0.4
0.6
0.8
1.0
(b)
0.0
0.2
0.4
0.6
0.8
1.0
0
30
60
90
120
150
180
210
240
270
300
330
0.0
0.2
0.4
0.6
0.8
1.0
(d)0.0
0.2
0.4
0.6
0.8
1.0
0
30
60
90
120
150
180
210
240
270
300
330
0.0
0.2
0.4
0.6
0.8
1.0
(c)
Fig. 3.5: Dependence of the scattered intensity distribution in the XZ plane onthe radius of a particle, when particle is situated on the interface: (a) a = 0.05µm, (b) a = 0.1 µm, (c) a = 0.5 µm and (d) a = 1 µm. The solid and dottedcurves correspond to the TE and TM polarisation states of the illumination wave,respectively. n1 = 1.6, n′ = 1.0, n = 1.51, λ = 632.8 nm and α = 45.
Such a drastic change in the scattered intensity profile caused by the
particle-interface interaction, indicates that for a proper study of near-field
Mie scattering or for the modelling of the actual experiments interface effects
need to be taken into consideration.
3.4 Effects of the interface on the morphology
dependent resonance
Morphology dependent resonance (MDR) is caused by the interference of a
light beam propagating inside a dielectric particle confined by total internal
reflection [106]. As a beam of light propagating inside the particle returns
PhD thesis: Far-field and near-field optical trapping 70
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
to its starting position in phase, the constructive interference effect leads to
a series of peaks in the scattered field for given an appropriate particle size.
Due to this process, a large energy density can be created inside a particle
[106]. MDR has been observed in plane-wave Mie scattering [107–111] and
predicted in near-field Mie scattering [73–75].
Based on our method for determination of the field scattered by a
small particle situated near an interface, the intensity integrated over the
upper half-space of the particle, can be calculated. This parameter gives
a measure for the maximum signal strength in particle-trapped SNOM
and therefore can be used to study the MDR caused by near-field Mie
scattering. Fig. 3.6 shows the intensity integrated over the upper half-space
of a particle with varying refractive index immersed in air and situated on
the boundary between the surrounding medium and the substrate for TE and
TM polarisation illumination. MDR becomes evident as the particle radius
increases; the larger the particle size the sharper the resonance peaks. This
is understandable because a scattering particle can be considered to be a
micro-cavity. For a large particle, the evanescent wave interacts significantly
with the particle near its bottom. Therefore, the beam refracted into the
particle is incident on the particle boundary with a refraction angle close to
the critical angle inside the particle, which results in a high reflectance or a
large coefficient of finesse of such a cavity. Consequently, sharper resonance
peaks emerge in the scattered field.
According to Fig. 3.6-bottom, the difference of the radius between two
resonance peaks is 71.2 ± 2.0 nm, which agrees well with the result of 72.2
nm estimated by
∆a =λ
2π
arctan(√
n21 − 1)
√
n21 − 1
, (3.9)
for plane wave Mie scattering [108]. The dependence of MDR on the effective
refractive index nef = n1/n′ is shown in Fig. 3.6. As expected from Eq. 3.9,
PhD thesis: Far-field and near-field optical trapping 71
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
( m)m
0.0 0.5 1.0 1.5 2.0 2.50.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
Inte
nsi
ty(a
.u.)
Particle radius
0.0 0.5 1.0 1.5 2.0 2.50.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
Inte
nsi
ty(a
.u.)
0.0 0.5 1.0 1.5 2.0 2.50.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
0.0040
0.0045
Inte
nsi
ty(a
.u.)
Fig. 3.6: Dependence of the half-space scattered intensity on the particle radiusfor the TE (solid line) and TM (dotted line) polarisation illumination, when theparticle is situated on the interface. n′ = 1.0, n = 1.51, λ = 632.8 nm and α = 45.Top: n1 = 1.1, middle: n1 = 1.3 and bottom: n1 = 1.6.
PhD thesis: Far-field and near-field optical trapping 72
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
( m)m
0.0 0.5 1.0 1.5 2.0 2.50.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
Inte
nsi
ty(a
.u.)
Particle radius
0.0 0.5 1.0 1.5 2.0 2.50.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
Inte
nsi
ty(a
.u.)
0.0 0.5 1.0 1.5 2.0 2.50.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
Inte
nsi
ty(a
.u.)
Fig. 3.7: Dependence of the half-space scattered intensity on the particle radiusfor the TE (solid line) and TM (dotted line) polarisation illumination, when theparticle is situated on the interface. n′ = 1.0, n = 1.51, n1 = 1.6 and λ = 632.8nm. Top: α = 42, middle: α = 43 and bottom: α = 45.
PhD thesis: Far-field and near-field optical trapping 73
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
the spacing between two adjacent peaks increases as n1 decreases. As the
effective refractive index of the particle increases, MDR effects become more
pronounced with sharper resonance peaks because the coefficient of finesse
of the cavity (i.e. the reflectance of the cavity) becomes larger. It can
be seen from the Fig. 3.7 that MDR peak positions are independent of the
incident angle α of the illumination wave. This feature indicates that MDR
is mainly caused by the wave interacting at the bottom of the particle. As
the decay constant β of the evanescent wave increases with the incident angle
α, the scattering of the evanescent wave with a particle is mainly confined
to the bottom of the particle. As a result, the coefficients of finesse of the
cavity are effectively increased, resulting in the shaper peaks (see Fig. 3.7-
bottom). The calculation step in Fig. 3.7 is chosen to be 5 nm. It is found
from our calculation that resonances are found much more precisely and
more significant in strength if smaller step is used. However, smaller step
calculation also requires significantly more computational time.
Signal strength of the light scattered by a polystyrene particle of radius
0.25 µm, and illuminated by an evanescent wave produced by a He Ne laser
incident at an angle larger than the critical angle, is shown in Fig. 3.8. The
particle is immersed in water and placed on the interface. Both calculated
and experimental results [50] agree that the scattered intensity decreases
with the incident angle α for both TE and TM polarisation states of the
incident illumination. Furthermore, it shows that the rate of decrease of
the scattered intensity under TE polarisation is slower than that under TM
polarised illumination. Our model also confirms the experimental finding
that the scattered intensity for TE and TM polarisation states of incident
illumination becomes equal at an incident angle, α, of approximately 58.
Provided that the refractive index of the particle does not change
appreciably when the illumination wavelength varies, the dependence of MDR
PhD thesis: Far-field and near-field optical trapping 74
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
Fig. 3.8: A comparison of the calculated half space scattered intensity withthe intensity measured by a NA=1.3 objective in particle-trapped near-fieldmicroscopy [50]. The solid and dotted curves represent TE and TM incidentpolarisation states, respectively. α is the incident angle and a polystyrene particleof radius 0.25 µm immersed in water is placed on the interface between thesurrounding medium and the substrate.
described in Figs. 3.6 and 3.7 changes only by a scaling factor because MDR
is dependent on the size parameter q = k′a. In other words MDR in near-
field Mie scattering can be demonstrated from the fluorescence spectrum of
a fluorescent particle excited by an evanescent wave.
3.5 Mechanism for conversion of evanescent
photons into propagating photons
In the previous section, near-field (evanescent wave) scattering is investigated
in details for cases when a dielectric particle is situated near and far from the
interface. Using this treatment one can determine the scattered field distribu-
PhD thesis: Far-field and near-field optical trapping 75
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
tion at the trapping/collecting objective entrance pupil, for a given particle
and conditions. Subsequently, the effect of the trapping/collecting objective
is included by investigating the vectorial diffraction process. Consequently
the focal intensity distribution (FID) in the image space focal region of the
collecting high NA objective can be determined. Such a physical process
describes a mechanism for conversion of evanescent photons into propagating
photons.
3.5.1 Physical model
Our physical model of the evanescent photon conversion mechanism is
based on the near-field Mie scattering enhanced by the MDR and vectorial
diffraction by a high NA lens. The analytical expression for the 3-D vectorial
field distribution around a microscopic particle immersed in an evanescent
field is given by Eq. 3.5, with the inclusion of the interface effects as described
in Eq. 3.8. Subsequently, we include the effect of the trapping/collecting
objective by investigating the vectorial diffraction process, to determine the
focal intensity distribution in the image space focal region of the collecting
high NA objective. The trapping/collecting objective is one objective which
is used for both trapping of a microscopic particle and for collecting the
scattered signal.
Let us consider a microscopic particle in a close proximity of the interface
at which an evanescent field is generated by the TIR (α > αc) under either
TE or TM incident illumination (Fig. 3.9(a)). The origin of the coordinate
system is located at the center of the particle with a coordinate systems
defined in Fig. 3.9(a). The particle is observed by a high NA objective whose
focal point coincides with the particle center. The evanescent wave generated
by TIR propagates in the Y1 direction and decays exponentially in the Z1
direction, while interacting with the microscopic particle. This interaction
PhD thesis: Far-field and near-field optical trapping 76
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
j1
j2
q1
q2
r2
r1
objective
Reference spherein object space
Particle
a
a> ca
z1
y1
x1
Reference spherein image space
O1
O2
x2
z2
y2
(a)
C1 C3C2
(b)
x1
y1
z1 z2
x2
y2
r2r1
q2q1
j2j1
Laser
L2
PHL1
L3 L4
ICCD
Prism
O
BS
M1
M2 M3
(c)
Fig. 3.9: (a) Schematic of our theoretical model for evanescent photon conversion.(b) Representation of the lens transformation process. (c) Experimental setupfor recording the FID of converted evanescent photons, collected by a high NAobjective O.
can be physically described in terms of superposition of the field scattered
by the microscopic particle into upper space (space above the prism surface)
and partial reflection of the scattered field into the bottom space (space below
the prism surface)as given by Eq. 3.8. Using this method one can calculate
the superposed scattered field on the reference sphere in the object space.
The center of this reference sphere (also known as the entrance pupil of the
collecting objective) overlaps with the particle center, i.e. the origin of the
coordinate system O1.
PhD thesis: Far-field and near-field optical trapping 77
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
The precise transformation of the field from the reference sphere in object
space to the field on the reference sphere in image space requires a detailed
model of the lens. Recent studies of imaging dipole emitters [112] and
microscopic particles [113] through a high NA objective have indicated that
we can assume that the imaging objective transforms a diverging spherical
wave with its origin O1 in the center of the particle into a converging spherical
wave whose origin O2 is in the center of the focal region in image space.
Therefore, the lens effect can physically be modelled as a retardation effect
affecting the wave traversing two different dielectric media (air and glass).
Consider the spherical wavefront C1 originating from the particle center O1
(the origin of the coordinate system X1Y1Z1), just before the collecting
lens (Fig. 3.9(b)). Its curvature corresponds exactly to the curvature of
the collecting lens in object space. After traversing the lens front surface,
the wavefront becomes the plane wavefront C2. All points on the spherical
wavefront C1 arrive at the plane wavefront C2 at the same time. The plane
wavefront is then similarly transformed to the converging spherical wavefront
C3, after traversing the lens back surface. The center of the spherical
wavefront C3 is at O2 (the origin of the coordinate system X2Y2Z2). Such
transformation further indicates that the lens imparts a scaling effect and a
vector rotation. If we consider scattered field vector components, described
by its unit vectors r1, θ1, and ϕ1 in the coordinate system X1Y1Z1, they are
transformed into −r2, θ2, and ϕ2 in the coordinate system X2Y2Z2.
Considering such a transformation process of the field from the entrance
pupil to the exit pupil, the focal field distribution in image space can be
derived by the vectorial diffraction process as given by Richards and Wolf [86],
E(r2, ψ, z2) =i
λ
∫ ∫
Ω
(−Er1r2+Eθ1θ2+Eϕ1ϕ2) exp[−ikr2 sin θ2 cos(ϕ2−ψ)]
× exp(−ikz2 cos θ2) sin θ2dθ2dϕ2 , (3.10)
PhD thesis: Far-field and near-field optical trapping 78
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
where Er1, Eθ1 and Eϕ1 are given by Eq. 3.8, while r2, ψ and z2 are cylindrical
coordinates of a point in the image space with a coordinate system shown in
Fig. 3.9(a).
3.5.2 Theoretical results and morphology dependent
resonances
Applied to evanescent photon conversion by a small dielectric particle probe
for either TE or TM incident illumination, our model leads to the FID in the
far-field of the collecting lens, similar to the one obtained for imaging a dipole
emitter (Fig. 3.10(a) and 3.10(e)). This is because the probe in this case is
much smaller than the wavelength of illuminating light, and the particle is
completely immersed into the evanescent field, so the dipole approximation
applies. However, when the particle radius approaches and exceeds the
wavelength of the illuminating light, the FID shows a complex interference-
like structure (Fig. 3.10(b) - 3.10(d) and 3.10(f) - 3.10(h)). Furthermore, our
model indicates that the conversion and collection of TE evanescent photons
is somewhat different from TM evanescent photons (Fig. 3.10). The FID in
image space of the collecting lens shows a similar interference-like structure
for the conversion of either TE or TM localised photons by large particles.
However, when the conversion is performed by a small particle, this similarity
in the FID is less pronounced. The complex interference-like pattern shown
in Fig. 3.10 arises due to the enhancement of MDR and higher multipoles,
scattering properties of large particles and the effects of the interface on which
the evanescent field is generated. One would expect to see the MDR effects
in the FID of the collecting objective, due to the increase in the collected
energy. These effects are indeed manifested in the FID of our model. Two
particular MDR for TE and TM polarised incident illumination are shown
in Fig. 3.11. However, it seems that the effect is not just a mere increase in
PhD thesis: Far-field and near-field optical trapping 79
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
Fig. 3.10: Calculated FID in the image focal plane of a 0.8 NA objective. TE (leftcolomn) and TM (right colomn) incident illumination. (a) and (e) a = 100 nm.(b) and (f) a = 500 nm. (c) and (g) a = 1000 nm. (d) and (h) a = 2000 nm. Allfigures are normalised to 100.
PhD thesis: Far-field and near-field optical trapping 80
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
1445 1450 1455 1460 1465
1464 nm 1445 nm
1454 nm
(b) TM wave
a (nm)
1400 1405 1410 1415 1420
0
10
20
30
40
50
60
1419 nm 1403 nm
(a) 1411 nm
TE wave
Inte
nsity
(arb
.uni
ts)
a (nm)
Fig. 3.11: Maximum intensity in the FID as a function of the particle radius nearMDR for TE (a) and TM (b) illumination. Insets show the full FID representing offand on resonance cases. The particle refractive index is 1.59 and the illuminationwavelength is 633 nm.
collected energy due to particular MDR, but also leads to different energy
distribution for on and off resonance positions (Fig. 3.11 insets). These
different energy distributions outline the importance of detector selection for
systems operating at MDR positions, because it would be an advantage to
operate particle trapped SNOM and optical trap nanometry systems at MDR
positions to enhance the signal-to-noise ratio.
3.5.3 Experimental setup and results
To confirm the conversion mechanism given by our model we have conducted
an experiment. The experimental setup is depicted in Fig. 3.9(c). A helium-
neon laser beam was expanded and filtered using lenses L1 (microscope
objective, NA=0.2), L2 (focal length 50 mm) and a pinhole (PH). It was then
directed onto the prism-air surface by mirror M1 to form an incident angle of
51.4 ± 0.3, which in combination with a very low divergence of the helium-
PhD thesis: Far-field and near-field optical trapping 81
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
neon laser ensured that the incident beam was well above the critical angle
(αc). The prism used in the experiment had a refractive index of 1.722 and
the particles under investigation were polystyrene particles diluted in water
and dried on the prism surface. The particles, immersed into the created
localised field, were imaged using a dry 0.8 NA objective (Olympus IC 100,
infinite tube length) and projected onto an intensified CCD camera (PicoStar
HR 12 from LaVision) via lenses L3 (focal length 70 mm) and L4 (focal length
100 mm). The TIR portion of the incident beam was re-routed via mirrors
M2 and M3 and a beam-splitter (BS) into the back aperture of the collecting
objective (O), to enable us to locate the prism surface and thus the center of
the particle under consideration.
-60 -40 -20 0 20 40 60 -60
-40
-20
0
20
40
60
y (10 -6 m)
x (1
0 -6
m)
-60 -40 -20 0 20 40 60 -60
-40
-20
0
20
40
60
y (10 -6 m)
x (1
0 -6
m)
-60 -40 -20 0 20 40 60 -60
-40
-20
0
20
40
60
y (10 -6 m)
x (1
0 -6
m)
-60 -40 -20 0 20 40 60 -60
-40
-20
0
20
40
60
y (10 -6 m)
x (1
0 -6
m)
Fig. 3.12: Calculated (top) and observed (bottom) FID in image focal plane of a 0.8NA objective collecting propagating photons converted by a=240 nm polystyreneparticle under TE (left column) and TM (right column) incident illumination.
PhD thesis: Far-field and near-field optical trapping 82
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
(a)
-60 -40 -20 0 20 40 60
0.0
0.2
0.4
0.6
0.8
1.0 In
tens
ity (a
rb.u
nits
)
y (10 -6 m)
-60 -40 -20 0 20 40 60
(b)
y (10 -6 m)
Fig. 3.13: Calculated and observed y axis scan through x=0, in image focal plane ofa 0.8 NA objective collecting propagating photons converted by 1000 nm (radius)polystyrene particle under TE incident illumination. (a) Calculated results. (b)Observed results (full line) where the dotted line represents the convolution of thecalculated results and the PSF of the imaging lens. Insets show the calculated andobserved FID.
The calculated and observed results of the FID of the collecting objective,
for the conversion of both TE and TM evanescent photons by a polystyrene
particle of 240 nm in radius, are shown in Fig. 3.12. It can be seen
from Fig. 3.12 that the FID structure predicted by our model is in good
agreement with the experimentally observed results. Furthermore, the
predicted difference in the FID structure for TE and TM evanescent photons
converted by this small dielectric probe is observable in the experiment. The
interference-like FID structure for evanescent photon conversion by a large
dielectric particle probe can also be experimentally observed. Figure 3.13(a)
shows our calculated result of the FID for evanescent photon conversion by a 1
µm (radius) polystyrene particle. The corresponding experimentally observed
result is shown in Fig. 3.13(b). The measurement was repeated 10 times
and the error was estimated to be approximately 5%. Image resolution of
the observed result is somewhat degraded due to the imaging properties of
PhD thesis: Far-field and near-field optical trapping 83
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
lens L4. The observed structure is a result of the convolution of the point-
spread function (PSF) of the imaging lens L4 and the calculated result shown
in Fig. 3.13(a). The agreement between calculated and experimental results
confirm that the conversion of evanescent photons is the result of two physical
processes, near-field Mie scattering and vectorial diffraction.
3.6 Pinhole detection of the scattered near-
field signal
In this section we will look at the detectability of the scattered near-field
signal, determined using our physical model described in previous section, by
a pinhole detector. Such a detector is utilised in trapped particle SNOM to
discriminate against the out of focus signal [18, 19]. A pinhole detector is
essentially a small circular opening of a few to several tens of micrometers, in
an otherwise opaque screen, placed perpendicularly to the optical axis at the
back focal plane of the imaging objective. The back focal plane focus coincides
with the center of the pinhole. Only the signal that can pass through this
opening is detected, and it constitutes of the signal coming from the front
focal region of the imaging objective (Fig. 3.14(a)).
Mathematically, the detected signal level η of a pinhole detector can be
expressed as
η =
∫ R
0
∫ 2π
0I(r, φ)rdrdφ
∫
∞
0
∫ 2π
0I(r, φ)rdrdφ
, (3.11)
where R denotes the pinhole radius and I(r, φ) is the intensity at a point
within the pinhole detector determined by distance r from the center of the
pinhole and an angle φ. If we express the pinhole radius R in an optical
coordinate VR defined as VR = 2πρaR/(λf), where ρa denotes objective
aperture radius and f is the back focal length of the objective, then the
PhD thesis: Far-field and near-field optical trapping 84
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
rf
Pinholedetector
Objective
Front focalregion
Back focalregion
in focus rays
out of focus rays
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
Uniformly filled aperture
Sig
nall
eve
l
Pinhole radius VR
(a)
(b)
Fig. 3.14: (a) A schematic diagram of a pinhole detection process. Only the rayscoming from the front focal region are detected. (b) Detected signal intensity asa function of a pinhole radius, in optical coordinates, for uniformly illuminatedobjective. Assumed objective NA = 0.8 in the front focal region, aperture sizeρa = 3 mm and the back focal length of the objective f = 160 mm.
detected signal level as a function of the pinhole size for a uniformly filled
aperture is shown in Fig. 3.14(b). This result is essentially the same as that
given by Born and Wolf [54] for the fraction of the total energy contained
within circles of prescribed radii (varying pinhole size), in the Fraunhofer
diffraction pattern of a circular aperture. Eq. 3.11 can be used with the
intensity I(r, φ) determined by our scattering model (Eq. 3.10) for any
point within the pinhole detector. Performing the appropriate integration
in Eq. 3.11 determined by the pinhole size, the detected signal level can be
evaluated for the two polarisation states of the incident illumination and a
range of pinhole sizes and scattering particles.
If we consider a polystyrene scattering particle, the scattered signal
intensity detected by a pinhole detector is shown in Fig. 3.15. It can be
seen that for very small particles (Figs. 3.15(a) and (d)) the signal is similar
to the signal for a uniformly filled aperture (Fig. 3.14(b)). This is because a
very small particle behaves as a point source and at a far-field distance the
PhD thesis: Far-field and near-field optical trapping 85
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
Fig. 3.15: Detected scattered intensity as a function of the pinhole size (in anoptical coordinate) of a polystyrene particle for TE illumination (left column) andTM illumination (right column). Assumed objective NA = 0.8 in the front focalregion, aperture size ρa = 3 mm and the back focal length of the objective f = 160mm. (a) and (d) Particle radius 0.1 µm. (b) and (e) Particle radius 0.5 µm. (c)and (f) Particle radius 1.0 µm.
PhD thesis: Far-field and near-field optical trapping 86
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
scattered signal fills the entrance pupil of the objective nearly uniformly. For
the wavelength-size particles (Figs. 3.15(b) and (e)) a much larger pinhole size
is required to collect the signal completely. For the typical conditions given in
Fig. 3.15 and an illumination wavelength of 633 nm (helium-neon laser), it can
be estimated that a pinhole of a radius of 80 µm is required to collect the total
signal for either TE or TM incident illumination. The scattering properties of
large particles, manifested mainly through the MDR and the interaction cross-
section effects, result in the much larger pinhole size (≈ 200 µm) required for
signal collection ( Figs. 3.15(c) and (f)). As we have seen in Figs. 3.10((d)
and (h)), the FID of large particles shows the spreading of the scattered
signal in the forward direction, which is in agreement with our qualitative
interpretation of the evanescent wave scattering by large particles (Fig. 3.3),
thus the requirement for a large pinhole size for total signal collection of the
field scattered by large dielectric particles.
3.7 Chapter conclusions
3-D scattered field distribution in near-field Mie scattering is determined by
using the theory of the elastic scattering of evanescent electromagnetic waves
with the inclusion of the effects of the interface at which the evanescent wave
is generated. It is found that the scattered intensity profile is similar to
that obtained for plane wave Mie scattering, for small dielectric particles
but becomes asymmetric when dielectric particles are larger, for particles
situated far away from the interface. For particles close to the interface
scattered intensity profiles are markedly different from those expected when
the interface is not considered for both small and large dielectric particles.
MDR is evident in the scattered field of the evanescent waves and its
period is the same as that for the plane wave scattering. The peak position,
PhD thesis: Far-field and near-field optical trapping 87
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
sharpness and separation of MDR depend on the size parameter and the
effective refractive index of the particle. As the size parameter increases
resonance peaks become more pronounced and sharper. Increasing the
effective refractive index causes sharper resonant peaks but reduces the
separation between neighboring resonance peaks. It is also shown that the
main contribution to MDR results from the wave at the bottom region of
the particle because the MDR peak positions are independent of the incident
angle α.
Based on the mathematical analysis of the 3-D near-field Mie scattering,
we have developed a physical model to describe the small particle conversion
of evanescent photons into propagating photons by a small scattering particle.
This model consists of two physical processes, near-field Mie scattering
enhanced by MDR and vectorial diffraction. As a result, the far-field intensity
pattern of the collecting objective shows an interference-like pattern for a
large dielectric particle probe, while it is similar to dipole radiation for small
dielectric particles. Due to the MDR effect the energy distribution in the
detection plane is different for on and off resonance conditions. This model
provides an understanding of the evanescent photon conversion in trapped
particle SNOM and a detailed physical picture of the energy distribution in
the far-field region of a collecting objective. The theoretical predictions of our
model have been experimentally confirmed by measuring the FID of small and
large polystyrene particles.
When the detection of the scattered signal is performed by a pinhole
detector, the detector size needs to be carefully selected depending on the
scattering particle size. A very small particles act as point sources and fill the
objective entrance pupil nearly uniformly. In that case, a very small pinhole
(≈ 20-30 µm in radius) is sufficient to collect the total signal. For large
particles, on the other hand, much larger pinholes are required to completely
PhD thesis: Far-field and near-field optical trapping 88
CHAPTER 3. Three dimensional near-field Mie scattering by a small particle
collect the scattered signal using a typical imaging lens. This effect is mainly
caused by the MDR and larger interaction cross-section of the particle and
the evanescent field, which leads to the spreading of the scattered signal in
the forward direction and thus to the spreading of the signal in the imaging
plane.
Furthermore, the model is applicable for determination of the near-field
force exerted on a small particle situated in the evanescent field, which will
be explored in more details in Chapter 6. It also provides a tool for designing
novel detection arrangements in the fields of NFI, optical trap nanometry and
near-field metrology with high accuracy and resolution.
PhD thesis: Far-field and near-field optical trapping 89
Chapter 4
Trapping force with a high
numerical aperture objective
4.1 Introduction
Trapping of small particles in the far-field of a focused laser beams has been
a rich research area ever since the first ”optical tweezer” was introduced by
Ashkin et al. [25]. As we have seen in Chapter 2 (Section 2.3), there are in
principle two methods for calculating trapping forces exerted on a spherical
micro-particle, each with certain limitations and not adequate to deal with
the focal distribution complexity. Inadequacy of these methods often lies
in the severity of the approximations used in their derivations. To gain an
accurate physical insight into the trapping force a proper physical model for
particle trapping using a focused laser beam is required.
In order to generate an efficient laser trap, incident field needs to be
tightly focused using a high numerical aperture (NA) microscope objective.
Tight focusing leads to a large field intensity gradient necessary for trapping
microscopic particles. The theoretical treatment of the focusing with high
90
CHAPTER 4. Trapping force with a high numerical aperture objective
convergence angles is provided by the vectorial diffraction theory, reviewed in
Section 2.5 of Chapter 2. Thus a proper physical model for particle trapping
using a high NA objective needs to include vectorial diffraction of the incident
light by an objective. Once the exact electromagnetic (EM) field distribution
in the focal region is known its interaction with a microscopic particle can be
calculated and the field distribution on the particle surface determined [114].
The Maxwell stress tensor can then be applied to evaluate the trapping
force [24].
The aim of this chapter is to present the implementation of our vectorial
diffraction [89, 90] approach to calculate the radiation trapping forces on
a micro-particle. Such an approach enables one to consider the vectorial
properties of the EM field distribution in the focal region of a high NA
microscope objective. Effects such as the complex phase modulation on the
entrance pupil of an objective, the refractive index mismatch, i.e. spherical
aberration (SA), the polarisation dependence, and the objective apodisation
can be considered using our model without the loss of generality. Such
an exact model can deal with complex laser beams, such as the Laguerre-
Gaussian laser beams, used in the novel laser trapping experiments [16,17].
The structure of the chapter is divided into four sections beginning with
this introductory section. Our physical model for small particle trapping using
a high NA objective, including the effect of SA, is presented in Section 4.2.
Section 4.3 discusses a comparison of the vectorial diffraction approach and
another EM model based on the fifth order Gaussian beam approximation [24,
59] which is often used to estimate the trapping force exerted on a small
particle [53]. The applicability of the optical trapping model is discussed in
Section 4.4. A chapter conclusion is drawn in Section 4.5.
PhD thesis: Far-field and near-field optical trapping 91
CHAPTER 4. Trapping force with a high numerical aperture objective
4.2 Model
A schematic diagram of our model that includes the SA effect is shown
in Fig. 4.1, where the geometrical focus position defines the origin of our
coordinate system, while the position of the particle is defined as the position
of its center with respect to the origin. Such a situation usually occurs when
a dry or an oil immersion objective is used to trap particles suspended in
water. A linearly polarised laser beam with the electric field vector E0 is
E0
E2
E1
n1 n2
x
y zo
Interface z=-d
Objective
Laser beam
Particle
Fig. 4.1: Schematic diagram of our trapping model.
focused by a high NA objective through an interface between two media with
the refractive indices n1 and n2. The incident laser beam is thus brought to
a sharp focus in the medium n2, in which a small particle of the refractive
index n3 is suspended. Due to the high intensity gradients induced by the
laser focusing, the particle is attracted towards the focus provided that its
refractive index is higher than that of the surrounding medium. If the particle
refractive index is lower than that of the surrounding medium, the particle is
repelled from the focal region.
Using the vectorial Debye theory and considering a linearly polarised
PhD thesis: Far-field and near-field optical trapping 92
CHAPTER 4. Trapping force with a high numerical aperture objective
monochromatic plane wave focused into two media separated by a planar
interface, one can express the electric and magnetic field distributions in the
focal region of a high NA objective, if the polarisation direction is along the
X direction, as [90]
E2(rp,−d) = − ik1
2π
∫ ∫
Ω1
c(φ1, φ2, θ) expik0[rpκ+ Ψ(φ1, φ2,−d)]
sinφ1dφ1dθ (4.1)
H2(rp,−d) = − ik1
2π
∫ ∫
Ω1
d(φ1, φ2, θ) expik0[rpκ+ Ψ(φ1, φ2,−d)]
sinφ1dφ1dθ . (4.2)
Eqs. 4.1 and 4.2 are given in spherical polar coordinates where indices 1
and 2 refer to the first medium (refractive index n1) and the second medium
(refractive index n2), respectively. φ1 is the angle of incidence on the planar
interface, while φ2 is the angle of refraction. rp is the position vector while k0
and k1 are wave vectors in vacuum, and the first medium, respectively. The
focus depth is denoted by d, while functions c(φ1, φ2, θ) and d(φ1, φ2, θ) are
defined in ref [90]. Function Ψ(φ1, φ2,−d) is the so called spherical aberration
function caused by the refractive index mismatching [90] and is given by
Eq. 2.21, while κ is given by Eq. 2.25. Eqs. 4.1 and 4.2 can be further
expanded to give Eqs. 2.22 and 2.23 respectively. Any other polarisation
state can be resolved into two orthogonal states each of which satisfies Eqs. 4.1
and 4.2. If the incident field E(0) is independent on angle θ, such as the case
for a plane wave illumination, the integration in Eqs. 4.1 and 4.2 can be
reduced to Eqs. 2.26 and 2.27.
PhD thesis: Far-field and near-field optical trapping 93
CHAPTER 4. Trapping force with a high numerical aperture objective
If we consider a homogeneous microsphere situated in the second medium
n2 illuminated by a monochromatic EM field described by Eqs. 4.1 and 4.2,
the net radiation force on the microsphere according to the steady-state
Maxwell stress tensor analysis is given by [24]
〈F〉 =1
4π
∫ 2π
0
∫ π
0
⟨(
ε2ErE + HrH −1
2(ε2E
2 + H2)r
)⟩
r2 sinφdφdθ ,
(4.3)
where r, φ and θ are spherical polar coordinates, Er and Hr are the radial
parts of the resulting electric and magnetic fields evaluated on the spherical
surface enclosing the particle. The net force can be further expressed as
a series over the incident and scattered field coefficients Alm, Blm, alm and
blm [24],
〈Fx〉+ i〈Fy〉a2E2
0
= +i(k2a)
2
16π
∞∑
l=1
l∑
m=−l
(√
(l +m+ 2)(l +m+ 1)
(2l + 1)(2l + 3)l(l + 2)
× (2ε2alma∗
l+1,m+1 + ε2almA∗
l+1,m+1 + ε2Alma∗
l+1,m+1 + 2blmb∗
l+1,m+1
+ blmB∗
l+1,m+1 +Blmb∗
l+1,m+1) +
√
(l −m+ 1)(l −m+ 2)
(2l + 1)(2l + 3)l(l + 2)
× (2ε2al+1,m−1a∗
lm + ε2al+1,m−1A∗
lm + ε2Al+1,m−1a∗
lm + 2bl+1,m−1b∗
lm
+ bl+1,m−1B∗
lm +Bl+1,m−1b∗
lm)−√
(l +m+ 1)(l −m)ε2(−2almb∗
l,m+1
+ 2blma∗
l,m+1 − almB∗
l,m+1 + blmA∗
l,m+1 +Blma∗
l,m+1 − Almb∗
l,m+1)
)
, (4.4)
and
PhD thesis: Far-field and near-field optical trapping 94
CHAPTER 4. Trapping force with a high numerical aperture objective
〈Fz〉a2E2
0
= −(k2a)2
8π
∞∑
l=1
l∑
m=−l
Im
(√
(l −m+ 1)(l +m+ 1)
(2l + 1)(2l + 3)l(l + 2)
× (2ε2al+1,ma∗
lm + ε2al+1,mA∗
lm + ε2Al+1,ma∗
lm + 2bl+1,mb∗
lm
+ bl+1,mB∗
lm +Bl+1,mb∗
lm) +√ε2m(2almb
∗
lm + almB∗
lm + Almb∗
lm)
)
, (4.5)
.
The incident and scattered field coefficients are defined as
Alm =1
l(l + 1)ψl(k2a)
∫ 2π
0
∫ π
0
sin θE2r(a, θ, φ)Y ∗
lm(θ, φ)dθdφ , (4.6)
Blm =1
l(l + 1)ψl(k2a)
∫ 2π
0
∫ π
0
sin θH2r(a, θ, φ)Y ∗
lm(θ, φ)dθdφ , (4.7)
alm =ψ′
l(nk2a)ψl(k2a)− nψl(nk2a)ψ′
l(k2a)
nψl(nk2a)ξ(1)′
l (k2a)− ψ′
l(nk2a)ξ(1)l (k2a)
Alm , (4.8)
and
blm =nψ′
l(nk2a)ψl(k2a)− ψl(nk2a)ψ′
l(k2a)
ψl(nk2a)ξ(1)′
l (k2a)− nψ′
l(nk2a)ξ(1)l (k2a)
Blm . (4.9)
ξ(1)l = ψl − iχl, where ψl and χl are the Riccati-Bessel functions. Ylm is the
spherical harmonic function. n is defined as n = (ε3/ε2)1/2.
All calculations are performed using standard computational methods for
integral evaluations. The trapping efficiency defined as a dimensionless factor
PhD thesis: Far-field and near-field optical trapping 95
CHAPTER 4. Trapping force with a high numerical aperture objective
Q is given by
Q =cF
n2P, (4.10)
where c denotes the speed of light in vacuum, F is the trapping force and P
is the incident laser power at the focus. When trapping efficiency is evaluated
in the transverse direction it is known as the transverse trapping efficiency
(TTE), while when evaluated in the axial direction it is known as the axial
trapping efficiency (ATE). One can distinguish between two ATE; the forward
ATE (positive value) corresponding to the inverted microscope configuration
(particle pushing) and the backward ATE (negative value) corresponding to
the upright microscope configuration (particle lifting).
4.3 Vectorial diffraction - Gaussian approxi-
mation comparison
Using the methodology described in the previous section, one can incorporate
various input field characteristics, such as the apodisation function, complex
amplitude and the aberration function into the model using vectorial
diffraction (Eqs. 4.1 and 4.2). The incident field is thus represented exactly,
without approximations, resulting in the precise calculation of the trapping
force.
Since the incident illumination on a microsphere, given in Eqs. 4.1 and 4.2,
differs from the fifth-order corrected Gaussian approximation used by Barton
et al. [24](Fig. 4.2), it can be expected that the respective trapping efficiencies
predicted by our model are different. In Fig. 4.3 a comparison between
the fifth-order Gaussian and vectorial diffraction approaches in the case of
polystyrene particles suspended in water is presented. The Gaussian beam
waist is assumed to be ω0 = 0.4 µm while the vectorial diffraction method
PhD thesis: Far-field and near-field optical trapping 96
CHAPTER 4. Trapping force with a high numerical aperture objective
(b)(a)
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0
0.0
0.2
0.4
0.6
0.8
1.0
Inte
nsi
ty(a
.u.)
Z distance (mm)
Fig. 4.2: Intensity distributions in (a) axial and (b) transversal directions (blue-Xaxis, red-Y axis) for the fifth-order Gaussian approximation (dashed line) and thevectorial diffraction theory (solid line).
Fig. 4.3: Comparison between the fifth-order Gaussian approximation (emptysymbols) and the vectorial diffraction theory (filled symbols) for the calculationof the maximal TTE (triangles) and the backward ATE (circles) of polystyreneparticles suspended in water. λ0 = 1.064 µm, ω0 = 0.4 µm and NA = 1.2.
PhD thesis: Far-field and near-field optical trapping 97
CHAPTER 4. Trapping force with a high numerical aperture objective
assumes the numerical aperture NA = 1.2, which gives approximately the
same focal spot size (Fig. 4.2). For small particles both methods give nearly
the same TTE and ATE, which shows an r3 dependence as expected for
Rayleigh sized particles [53]. When their size approaches the illumination
wavelength the two methods differ significantly. However, in the case of very
large particles (r = 100 µm,) the extrapolation of the vectorial diffraction
method (dotted lines in Fig. 4.3) approaches the RO prediction.
4.4 Model applicability
Using our optical trapping model based on the vectorial diffraction theory,
the far-field optical trapping process with a high NA objective can be
investigated without approximations. Various types of incident illumination
employed in laser trapping experiments, such as incident plane wave and
doughnut beam illuminations with and without annular masks [23] can be
considered. These types of incident illuminations cannot be considered
using the current approximate methods such as the fifth-order Gaussian
approximation method. While the RO method is able to give approximate
solutions for these types of incident illuminations for very large particles, it is
completely inadequate to consider small particles which are most commonly
used in laser trapping experiments. The investigation of the far-field optical
trapping will be presented in more details in Chapter 5.
Our optical trapping model can be applied to investigate the near-field
trapping with a focused evanescent illumination as well. This can be achieved
by considering an annular mask (also called a central obstruction) of a
sufficient size to ensure that the minimum angle of convergence of each
incident ray is larger than the critical angle determined by the total internal
reflection condition between two media (objective immersion and particle
PhD thesis: Far-field and near-field optical trapping 98
CHAPTER 4. Trapping force with a high numerical aperture objective
suspension media). This arrangement results in a focused near-field generated
at the interface between the two media [15], and our model can calculate
the force exerted on a small particle due to its interaction with such a
field. Similar to the far-field trapping, our model can consider an incident
illumination with arbitrary complex input phase, polarisation and apodisation
with this type of near-field trapping. Such a near-field trapping method has
not been considered previously using any other theoretical method and will
be dealt with in greater details in Chapter 6.
The model presented in this chapter is not limited to plane wave and
doughnut incident laser beam illuminations, considered in greater details in
this thesis. It is actually applicable for an arbitrary field incident at a high
NA objective entrance pupil.
4.5 Chapter conclusions
This chapter presents an exact method for the radiation trapping force
calculation. The EM field distribution in the focal region of a microscope
objective is determined using the vectorial diffraction theory and the optical
trapping force is evaluated using the Maxwell stress tensor approach.
The vectorial diffraction method offers a number of advantages over an
approximate method such as the fifth-order Gaussian beam incident field
approximation. Firstly, the vectorial diffraction model approaches the RO
predictions at a large particle limit, to a better degree than it can be achieved
by the approximated Gaussian beam model. Secondly, and most importantly
is that it provides the appropriate treatment of the incident illumination
phase modulation, polarisation and apodisation as well as the SA occurred
in trapping experiments for both small and large particles.
PhD thesis: Far-field and near-field optical trapping 99
CHAPTER 4. Trapping force with a high numerical aperture objective
Thus, the model enables the appropriate modelling of various far-field
trapping arrangements, focused near-field trapping or the trapping systems
implemented using the novel spatial phase modulation techniques, such as
the trapping systems with focused doughnut beam illumination.
PhD thesis: Far-field and near-field optical trapping 100
Chapter 5
Far-field optical trapping
5.1 Introduction
In the previous chapter (Chapter 4) the vectorial diffraction method is
presented for calculating the optical trapping force with a high numerical
aperture (NA) objective. We have seen that our optical trapping model can be
used to calculate the far-field trapping force exerted on small particles without
using any approximations. The model incorporates the full representation
of the field in the focused region, and can be used with any particle size,
given a sufficient computing power to calculate the scattered field. Inherently,
the model can deal with a complex phase modulation of the incident field,
apodisation of an objective and different polarisation states of incident
illumination. In this chapter, we will apply our optical trapping model
to investigate the far-field trapping process under the presence of spherical
aberration (SA). The effect of a central obstruction in the beam path, which
causes an elongation of the focused spot in the axial direction, will be studied.
Furthermore, the optical trapping efficiencies of doughnut laser beams, such
as the ones used in the novel laser trapping arrangements [16, 31, 32], are
101
CHAPTER 5. Far-field optical trapping
also investigated. The comparison between a centrally obstructed plane wave
and a doughnut beam is presented and the discrepancy between our optical
trapping model and the ray optics (RO) model is discussed.
Other theoretical models for the far-field optical trapping, reviewed in
Section 2.3 of Chapter 2, are inadequate to deal with complex laser fields,
such as the focused laser beam under the influence of spherical aberration,
phase modulation, and objective apodisation. Thus, to our knowledge, there
exist no comprehensive electromagnetic (EM) theoretical treatment of the far-
field optical trapping with doughnut beam illumination. The only theoretical
treatment of such illumination type is the one given by Ashkin et al. [23]
under the RO approximation, which is inadequate because it ignores the EM
field distribution in the focal region.
The influence of a central obstruction was previously considered using
a RO approach. However, the RO model does not consider the EM field
distribution in the focal region, i.e. the elongation of the focal spot,
which limits the RO approach severely. Dependence of trapping force on
the obstruction size of a centrally obstructed laser beam is studied both
theoretically, using our optical trapping model, and experimentally in the
case of plane wave and doughnut beam illuminations. The trapping efficiency
under such conditions is compared with the RO model, and it is found that
such a model is completely inadequate. The vectorial diffraction method, on
the other hand, agrees very well with the experimental results.
This chapter is organised as follows. Section 5.2 investigates the trapping
force with plane wave illumination focused by a high NA objective. The
effects of the refractive index mismatch between the objective immersion and
particle suspension media, i.e. SA, on the trapping force are investigated
theoretically and compared with the experimental results of Felgner et
al. [115] in Section 5.2.2. Trapping force with a centrally obstructed plane
PhD thesis: Far-field and near-field optical trapping 102
CHAPTER 5. Far-field optical trapping
wave is examined in Section 5.2.3, both numerically and experimentally.
Section 5.3 studies the trapping efficiency obtained with doughnut beam
illumination, while the efficient generation of such an illumination and
the effects of vectorial diffraction of doughnut beams are presented in
Sections 5.3.1 and 5.3.2, in order to confirm the predictions experimentally
and gain a physical insight into the trapping process with doughnut beam
incident illumination. Section 5.4 presents the chapter conclusions.
5.2 Trapping force with plane wave illumina-
tion
Plane wave illumination is a most common type of illumination used in the
optical tweezers experiments. However, only the vectorial diffraction model
can deal with this case exactly in the wave optics regime. This section deals
with the force mapping of small and large dielectric particles, the SA effects
and the influence of a central obstruction on the trapping performance.
5.2.1 Force mapping
According to our analysis of the dependence of the maximal trapping
efficiency on particle size (Fig. 4.3), both the maximal transverse trapping
efficiency (TTE) and the backward axial trapping efficiency (ATE), are
greatly reduced when a particle becomes small. However, even though such
a presentation is related to the effect that is measurable in the experiments,
it does not give a clear physical picture of how the force depends on the
relative position of the geometrical focus and the particle. Such a physical
picture was presented by Ashkin et al. in the case of very large particles
and was investigated using the RO model [23], which is not applicable for
PhD thesis: Far-field and near-field optical trapping 103
CHAPTER 5. Far-field optical trapping
particles whose size is comparable to the illumination wavelength or smaller,
and is particle size invariant. However, it could be expected that the
force dependence of the relative position of the geometrical focus and the
particle is markedly different for small and large particles due to the EM field
distribution in the focal region.
0 0.50 0.5
(a)
00 0.08
(b)
Fig. 5.1: Magnitude and direction of the trapping efficiency for various geometricalfocus positions around a polystyrene particle suspended in water and illuminatedby a λ0 = 1.064µm laser beam focused by a NA = 1.25 water immersion objective.(a) particle radius of 2 µm. (b) Particle radius of 200 nm.
Two polystyrene particle sizes (a = 2 µm and a = 200 nm) suspended in
water illuminated by a λ0 = 1.064 µm laser beam focused using NA = 1.25
water immersion objective (Fig. 5.1(a) and 5.1(b)) are considered. For large
particles, the magnitude and direction of the trapping force is similar to the
one given by the RO model and it is seen that the particle is most strongly
influenced when its boundary is situated near the geometrical focus, while
away from the boundary the trapping force falls rapidly (Fig. 5.1(a)). Such a
PhD thesis: Far-field and near-field optical trapping 104
CHAPTER 5. Far-field optical trapping
rapid decrease in the trapping force magnitude is not present when one deals
with small particles (Fig. 5.1(b)). Even at a distance of twice the particle
radius, the magnitude of the trapping efficiency is relatively unchanged. This
is because the particle is much smaller than the focal field distribution so
that even at a geometrical focus distance of 2a, the particle-field interaction
is significant.
5.2.2 Spherical aberration
SA plays an important role in laser trapping because most of the trapping
experiments are performed under conditions where the refractive index
mismatch occurs. Usually a high NA oil immersion objective is used for
trapping while microparticles are suspended in water. The refractive index
difference between the immersion and the suspending media leads to the SA
when a trapping beam is focused deep into the suspending medium, which
are manifested as focal spot distortions, and degrade the trapping efficiency
of an optical trap [116].
Using the approach given in Eqs. 4.1 and 4.2, one can incorporate the
effect of SA on the trapping force at an arbitrary depth in the suspending
medium. Without considering the effect of SA produced when a refractive
index mismatch exists, as it does in most of the experimental measurements,
one not only overestimates the value of the trapping efficiency but also lose
its physical dependence on other factors such as the particle size. As it can
be seen in Fig. 5.2, by including the effect of SA the calculated backward
ATE agrees with the experimental data given by Felgner et al. [115] under
the same conditions. Furthermore, the effect of the morphology dependent
resonance (MDR) is pronounced for the particles whose size is of the order of
wavelength in the case when SA is included. The stronger MDR effect is due
to the better coupling of the incident field into the microsphere when SA is
PhD thesis: Far-field and near-field optical trapping 105
CHAPTER 5. Far-field optical trapping
present, because the focal region is larger and the field interaction with the
edge of the sphere is more pronounced.
Fig. 5.2: Maximal backward ATE of glass particles suspended in water, illuminatedby a laser beam (λ0 = 1.064 µm), focused by an oil immersion microscope objective(NA = 1.3). The effect of SA is considered at a depth of 9 µm from the cover glass.
A dependence of the trapping efficiency on the trapping distance from the
cover glass, for both axial (Fig. 5.3(a)) and transverse (Fig. 5.3(b)) directions
are investigated and compared with the experimental results given by Felgner
et al. [115]. Note that the magnitude of the error bars of the experimental
results depends on the trapping distance from the cover glass because the
measurement of the trapping force close to the cover glass is more uncertain
than that deeper into the suspending medium. In the calculation, an oil
immersion objective with NA = 1.3 and an illumination wavelength λ0 =
1.064 µm are assumed, while the refractive indices are assumed to be 1.52
for oil and cover glass (index matched), 1.57 for polystyrene, 1.51 for glass,
1.33 for water and 1.41 for 60% glycerol. The calculated maximal ATE as
a function of the distance from the coverslip for the trapping of a spherical
glass particle of diameter D = 2.7 µm and suspended in water agrees (within
PhD thesis: Far-field and near-field optical trapping 106
CHAPTER 5. Far-field optical trapping
Fig. 5.3: Maximal backward ATE and TTE of a particle illuminated by a laser(λ0 = 1.064 µm) focused by an oil immersion microscope objective (NA = 1.3) asa function of the distance from the cover glass. (a) A glass particle of diameterD = 2.7 µm in water. (b) A polystyrene particle of diameter D = 1.02 µmsuspended in 60% glycerol solution.
its error bars) with the measured results (Fig. 5.3(a)). This agreement at a
large distance from the cover glass (deeper into the suspending medium) is
better than that near the surface.
Due to the difference in the transverse EM field distribution in the focal
region of a high NA objective (Fig. 4.2), the maximal TTE in the polarisation
direction (Qx) and in the direction perpendicular to the polarisation direction
(Qy), calculated for a polystyrene particle of D = 1.02 µm suspended in a
60% glycerol solution, is different (Fig. 5.3(b)). The TTE in the Qx direction
is generally smaller than that in the Qy direction. This difference is due to
the elongation of the focal spot in the X direction. Such a difference of the
TTE is larger near the surface. Although the absolute value of the TTE
is somewhat larger than the experimental result, which may be caused by
our assumptions about the microscope objective and the suspending medium
characteristics (60% glycerol solution for which it is difficult to estimate
the viscosity and the refractive index precisely), the maximal TTE trend is
PhD thesis: Far-field and near-field optical trapping 107
CHAPTER 5. Far-field optical trapping
consistent with the experimentally measured trend if the former is normalised
by the experimental result obtained at a deep distance.
5.2.3 Trapping efficiency with centrally obstructed plane
wave
5.2.3.1 Numerical results
It has been predicted by the early researchers in the particle trapping using a
highly focused laser beam that uses of an obstructed laser beam decreases the
transverse trapping force exerted on dielectric particles [23]. Their treatment
of this problem was based on the RO approach, and they show that the
projection of the net trapping force in the transverse direction decreases with
the angle of a ray of convergence. The obstruction (ε) is achieved by centering
an opaque disk perpendicular to the beam propagation axis and it is defined as
the ratio of the radius of the obstructed part to the radius of the unobstructed
part of the beam.
The decrease in the maximal TTE for a large polystyrene particle,
predicted by the RO model, is approximately 20% for the case when the laser
beam is not obstructed and when it is nearly completely obstructed. However,
the inherent nature of the RO approach is to treat the focal distribution of
a highly focused laser beam as a geometric point. Such an approach may be
appropriate when the particle is several times larger than the wavelength of
the incident light and the incident beam is not centrally obstructed. However,
due to the fact that the focus distribution of a highly convergent beam
is significantly affected by obstructing the low angle rays, the RO model
approach is not appropriate for a centrally obstructed incident beam. The
presence of the obstruction leads to more depolarised rays reaching the focus.
Inclusion of only the high angle rays leads to the focus distribution that has a
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CHAPTER 5. Far-field optical trapping
sharper focus but enhanced airy rings in the transverse plane and is elongated
in the axial direction (Fig. 5.4). For larger obstructions this elongation can
become significant enough to lead to the drastically incorrect TTE prediction
if one relays on the RO model.
Fig. 5.4: Focus intensity distribution for a plane wave focused by a high NAobjective immersed in water (NA = 1.25). Top row - unobstructed plane wave(ε = 0.0). Bottom row - obstructed plane wave (ε = 0.8).
Figure 5.5 shows the theoretical results of the RO model and the vectorial
diffraction model for a polystyrene particle of 2 µm diameter immersed in
water and illuminated by a highly convergent laser beam with NA of 1.25 and
the wavelength λ = 532 nm. Immediately, two features are clearly evident.
Firstly, the two models predict a very different behavior of the maximal TTE
for increasing obstruction size. Secondly, the RO model predicts a small
difference in the maximal TTE for the two polarisation states of incident
illumination, with the S polarisation resulting in a slightly larger transverse
force, while the vectorial diffraction model predicts a substantial difference for
PhD thesis: Far-field and near-field optical trapping 109
CHAPTER 5. Far-field optical trapping
the two polarisation states and predicts a considerably larger transverse force
for the P polarisation when no obstruction is present. For larger obstructions
the difference in the maximal TTE for the two polarisation states diminishes,
due to different depolarisation behaviors between the S and P directions.
Fig. 5.5: Theoretical calculation (RO model and vectorial diffraction model) of themaximal TTE as function of obstruction size for a polystyrene particle of radius 1µm immersed in water. NA = 1.25 and λ = 532 nm. The maximal TTE for thetwo models are normalised to start from the same point (at ε = 0.0).
The depolarisation feature in a focal region of an objective can be
described by the vectorial diffraction theory reviewed in Section 2.5. Fig. 5.6
shows the transverse intensity distribution in a focal plane of an objective
for different obstruction sizes. The diminishing difference in the maximal
TTE of the two polarisation states for larger obstructions is caused by two
processes. The first process that determines the trapping force is the overall
field distribution incident on a particle (Fig. 5.4). Since the incident field
in the transverse focal plane is elongated, it leads to a different maximal
TTE for S and P polarisations (or for S and P scanning directions for a
given linear polarisation). The second process is the intensity gradient along
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CHAPTER 5. Far-field optical trapping
Fig. 5.6: Focus intensity along a transverse direction for various obstruction sizes.(a) Polarisation direction X and (b) Perpendicular to polarisation direction Y.NA = 1.25 and λ = 532 nm.
a particular direction (Fig. 5.6). For large obstructions the change in the
overall focal field distribution causes the reduction in the trapping efficiency.
However, the intensity gradient along the polarisation direction (Fig. 5.6(a)) is
only slightly reduced, while in the direction perpendicular to the polarisation
direction (Fig. 5.6(b)) it increases significantly. The combined effect is that
the maximal TTE decreases for both directions, S and P, due to the change in
the overall incident field distribution. However, due to the increased intensity
gradient along the S direction the decrease in this direction is slower, which
results in the diminishing difference between the two polarisation directions
(Fig. 5.5).
5.2.3.2 Experimental results
To confirm the theoretical results predicted by the two models, we have
measured the maximal TTE of polystyrene beads of 2 µm in diameter
suspended in water (Polybead polystyrene microspheres, diameter=2.134 µm
with standard deviation δ=0.039 µm. However, from now on we will refer
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CHAPTER 5. Far-field optical trapping
CCD
StageController
La
se
r
PC
Monitor
Phasemodulator
Objectivehigh NA
Controller
Scanning stage
Immersionoil
Particlecell
Obstruction
Coverslip
Dichroicbeam-splitter
L1
L2
L3
Fig. 5.7: A schematic diagram of the experimental setup.
to these beads as 2 µm beads). A schematic diagram of the experimental
system is depicted in Fig. 5.7. The illumination light beam from a 532
nm continuous-wave laser is expanded to a parallel beam by the lenses L1
(microscope objective with NA = 0.2) and L2 (focal length 50 mm). The
beam is then reflected by a phase modulator and directed into a high NA
objective lens (NA = 1.2, water immersion objective). In the plane wave case
the phase modulator was replaced by a mirror. The incident beam on the
high NA objective is focused into a sample cell containing polystyrene micro-
spheres of 2 µm diameter, suspended in water. The sample cell is mounted
on a PC-controlled scanning stage while the trapping process is monitored
using a CCD camera. A circular obstruction disk is coaxially inserted in the
PhD thesis: Far-field and near-field optical trapping 112
CHAPTER 5. Far-field optical trapping
beam path and the maximal TTE is measured using the standard Stokes law
method.
The measurement based on the Stokes method is performed in such a
way that a particle is trapped with a laser beam of fixed power and then the
particle is transversally scanned across the sample cell. The scanning speed
is increased until the particle falls out of the trap. The maximum transverse
trapping force F on a trapped particle is then calculated using the Stokes law
F = 6πavµ, where a is the radius of the trapped particle, v is the maximum
translation speed and µ is the viscosity of the surrounding medium [117]. The
presented results are averaged over three measurements and the uncertainties
are estimated from the laser power fluctuations.
The maximal TTE for various obstruction sizes, experimentally measured
using the setup in Fig. 5.7, is shown in Fig. 5.8. It can be seen that the TTE
Fig. 5.8: Experimental measurement of the maximal TTE as a function ofobstruction size for a polystyrene particle of radius 1 µm immersed in water.Theoretical values are normalised by the experimental P value at ε = 0.0. NA = 1.2and λ = 532 nm.
PhD thesis: Far-field and near-field optical trapping 113
CHAPTER 5. Far-field optical trapping
decreases rapidly for increasing obstruction size, unlike the trend predicted by
the RO model. The experiment confirms that the vectorial diffraction model
agrees well with the measured results. Furthermore, the difference in the
maximal TTE for the two polarisation states is verified in the experiment,
with the maximal TTE for the S polarisation being approximately 85% of
that for the P polarisation state when no obstruction is present (ε = 0.0).
Similar to the theoretically predicted trend, the difference in the maximal
TTE for the two polarisation states diminishes for larger obstruction sizes.
5.3 Trapping force with doughnut beam illu-
mination
Doughnut laser beams have got their name because of their characteristic
intensity profile. The electric field of a doughnut beam can be expressed as
E = E exp(imθ), where θ is the polar coordinate in the plane perpendicular
to the beam axis and m is called the topological charge. Such beams are
also known as Laguerre-Gaussian (LG) beams and are becoming increasingly
popular in the novel laser trapping arrangements [16,31,32]. To theoretically
investigate the trapping force with doughnut beam illumination focused by a
high NA objective, vectorial diffraction of doughnut beam illumination needs
to be studied. For experimental studies, however, an efficient generation of
doughnut beams is required.
In this section, a novel method to generate doughnut beams with 100%
efficiency is described, the diffraction effects when focusing doughnut beams
by a high NA objective are discussed, as well as the trapping efficiency of
doughnut beams.
PhD thesis: Far-field and near-field optical trapping 114
CHAPTER 5. Far-field optical trapping
5.3.1 Doughnut beam generation
In this section we demonstrate a novel and convenient method for producing
a doughnut laser beam using a liquid crystal (LC) cell which is capable of
dynamically controlling the phase distribution. We also demonstrate the
use of a similar method for doughnut beam generation using a spatial phase
modulator (Hamamatsu PPM X8267 Series).
5.3.1.1 Generation of doughnut beams using a liquid crystal cell
The LC-based photonic devices are increasingly used in many applications in-
cluding light focusing [118,119], phase modulation [120], beam steering [121],
and filtering [122]. A LC based element works by applying an electric field
between two walls of a cell containing appropriately oriented liquid crystals.
The applied electric field causes LC molecules to tilt, which results in a
change in refractive index. By controlling this refractive-index change one
can provide an appropriate phase shift to the incoming wavefront in order to
produce a doughnut laser beam.
If one can make a phase mask by controlling LC molecules in such a
way to produce, for example, a gradual phase change from 0π to 2πm in
a circular fashion across the incoming beam wavefront (Fig. 5.9(a)), then
a helical wavefront of topological charge m would result. To achieve this
conversion of a plane wave (m = 0) into a doughnut beam (m 6= 0) we have
made a LC cell with an indium-tin-oxide (ITO) structure consisting of 16 pie
slices on the front side of the cell (Fig. 5.9(b)). The cell diameter was 1 cm.
The structure was made by a laser lithography process using a chrome mask.
The two contact points were connected to the first and the last pies slices,
while all the slices were connected by a narrow (≈ 10µm) strip of ITO. When
a voltage was supplied to the contact points this narrow strip of ITO had high
PhD thesis: Far-field and near-field optical trapping 115
CHAPTER 5. Far-field optical trapping
0p
(a) (b) (c)
Contact points
NarrowITO layer
Pie slice
216
15
14
12
11
109 8
7
6
5
4
3
1
+10 V0 V
Fig. 5.9: Phase distribution of a doughnut beam. (a) The theoretical phasedistribution of a doughnut beam of charge 1 according to 16 phase steps. (b)The electrode structure of the liquid crystal cell with 16 pie slices. (c) Thephase wavefront of the doughnut beam of charge 1, measured using phase shiftinginterferometry.
resistance and gave a linear voltage drop from the first to the last pie slices,
as shown in Fig. 5.10(a). On the other hand, the pie slices themselves were
much wider and offered very low resistance. The back side of the cell was
made of a uniformly coated layer of ITO on a glass substrate. This uniform
ITO layer was connected to the ground. The cell was then filled with LC
molecules and sealed.
When an appropriate voltage is applied to the connection points on the
front side of the cell, it then re-distributes across each of the pie slices. LC
molecules inside the cell tilt according to the electric field strength between
the pie slices and the grounded back wall of the cell. The amount of the
tilt depends on the voltage on each pie slice, giving rise to a corresponding
change in refractive index for the light polarised along the long axis of the
liquid crystals. For a given voltage such as 10 V , one can select a proper
PhD thesis: Far-field and near-field optical trapping 116
CHAPTER 5. Far-field optical trapping
He-Ne Laser
P
BS1 LC BS2
M1 M2
O
L
S
PH
0 2 4 6 8 10
0
1
2
3
4
5
6
Phase
shift
(p)
Voltage (V)
0 2 4 6 8 10 12 14 16 180
2
4
6
8
10
Volta
ge
(V)
Pie Slice Number
(a) (b)
Fig. 5.10: Experimental setup for generation of a doughnut beam through the liquidcrystal cell and interference measurement of its phase distribution (P: polariser;BS1 and BS2: beam splitters; LC: liquid crystal cell; O: objective; L: lens; M1 andM2: mirrors; PH: pinhole; S: screen). (a) The voltage variation as a function ofthe slice position of the liquid crystal cell. (b) The unwrapped phase shift of theliquid crystal cell as a function of applied voltage.
thickness of the LC cell to produce a phase change from 0π at the first slice
to 2π at the last slice, so that the outgoing wavefront has a helical shape with
topological charge 1.
To characterise the dependence of the phase change on the voltage, a
LC cell of thickness 9.5 µm was placed between two crossed polarisers and
the intensity variation after the analyser as a function of the applied voltage
was measured. After the phase unwrapping, this dependence gives a direct
relation between the applied voltage and the phase shift of the beam (λ =
PhD thesis: Far-field and near-field optical trapping 117
CHAPTER 5. Far-field optical trapping
632.8 nm), which is shown in Fig. 5.10(b). It is clear that the LC cell can
produce a maximum of a 4π phase shift, although there exist a saturated
response caused by the nonlinear response of the liquid crystals. To extend
the linear response region, one can increase the thickness of the cell or the
concentration of the liquid crystals.
(a) (b) (c)
(e)(d) (f)
Fig. 5.11: The intensity distributions (a, b and c) of laser beams transmittedthrough a liquid crystal cell and the corresponding interference patterns (d, e andf). (a) and (d) plane wave. (b) and (e) Doughnut beam of charge 1. (c) and (f)Doughnut beam of charge 2.
To demonstrate the dynamically switching nature of the LC cell, the
LC cell is placed in an optical setup shown in Fig. 5.10. A He-Ne laser
beam (632.8 nm) of output power 5 mW was used for illumination and was
linearly polarised by a polariser (P) in the direction of the liquid crystals. It
passed through the first beam splitter (BS1) to create a reflection arm and
a transmission arm. The transmitted part of the beam passed through the
LC cell that modified the wavefront to create a doughnut beam on a screen
(S). For the interference pattern measurement, the reflected beam at BS1
was recombined with the transmitted part at BS2 to create an interference
PhD thesis: Far-field and near-field optical trapping 118
CHAPTER 5. Far-field optical trapping
pattern. The resulting beam was filtered using the microscope objective O
(NA = 0.2) and projected onto the screen using the lens L (focal length 100
mm).
When the reflection arm is blocked, the recorded patterns are displayed in
Figs. 5.11(a), 5.11(b) and 5.11(c). If there is no voltage applied to the LC cell,
the incoming laser beam wavefront is not changed (Fig. 5.11(a)). Applying an
appropriate voltage to the cell according to Fig. 5.10(b) results in a phase shift
of 2π or 4π, so that the wavefront after the cell can be dynamically converted
into a doughnut beam of charges 1 and 2 (Figs. 5.11(b) and 5.11(c)). A slight
distortion of the circular symmetry in Fig. 5.11(c) is caused by the saturated
response in Fig. 5.10(b). It was observed that the power of the generated
doughnut beams was almost the same as that of the plane wave, which leads
to a conversion efficiency near 100%.
To confirm the helical nature of the generated wavefront, we introduced
the interference arm as shown in Fig. 5.10. The measured interference
patterns corresponding to Figs. 5.11(a), 5.11(b) and 5.11(c) are shown in
Figs. 5.11(d), 5.11(e) and 5.11(f), respectively. When the LC is switched off,
the interference pattern shows the interference fringes of an equal spacing,
resulting from two plane waves (Fig. 5.11(d)). The fringe splitting in
Figs. 5.11(e) and 5.11(f) indicates that the LC modulated beam becomes
a doughnut beam and the number of splitting fringes gives the number of
topological charges [123]. A direct wavefront test was also performed using
phase shifting interferometry method [124]. The reconstructed wavefront
of the charge-one doughnut beam (Fig. 5.9(c)) is similar to the theoretical
wavefront in Fig. 5.9(a).
Another feature of the designed LC cell is tunable over a range of
wavelengths. By simply changing the applied voltage, a doughnut beam
(m = 1) at wavelengths 488 nm and 800 nm was produced, as depicted
PhD thesis: Far-field and near-field optical trapping 119
CHAPTER 5. Far-field optical trapping
450 500 550 600 650 700 750 800
2.3
2.4
2.5
2.6
2.7
2.8
2.9
DV
(V)
l (nm)
Fig. 5.12: Variation of the voltage between the two contact points (see Fig. 5.9(b))as a function of the wavelength for the generation of a doughnut beam of charge 1.
in Fig. 5.12. The dependence of the voltage change on the illumination
wavelength shown in Fig. 5.12 indicates a reduced efficiency at wavelength
800 nm, implying that the liquid crystals used in this experiment exhibits a
certain amount of dispersion and a nonlinear response near this wavelength.
5.3.1.2 Generation of doughnut beams using a spatial phase
modulator
A phase-ramp pattern, similar to the one described in the previous subsection,
can be applied to a spatial phase modulator (SPM). A computer controlled
SPM is able to modulate the phase of an incoming plane wave, according to
the pattern loaded into its RGB port. Such device is increasingly used in
the novel laser trapping and manipulation experiments [31]. However, to our
knowledge, it was not yet used with the phase-ramp method described in the
subsection 5.3.1.1.
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CHAPTER 5. Far-field optical trapping
The advantage of using an SPM to modulate the phase according to the
phase-ramp pattern is in its ease of use and ability to use a greater number
of phase levels. With the Hamamatsu SPM we can achieve 256 phase levels
(Fig. 5.13), compared to the 16 levels with the LC cell. The LC cell is in
principle also capable to achieve a greater number of levels, but at a cost of
a much greater complexity in its design and manufacture.
(b)(a) (c)
Fig. 5.13: Doughnut beam of charge 1 generated using a computer controlledSPM. (a) Applied phase-ramp pattern with 256 levels. (b) Intensity profile. (c)Interference pattern.
Figure 5.13 shows a doughnut beam of topological charge 1, achieved
using a reflection type SPM (Hamamatsu PPM X8267 Series) with a 256
levels phase-ramp. The interference pattern of a such generated doughnut
beam and a plane wave reveals a characteristic fringe splitting confirming
that the generated beam is of topological charge 1.
5.3.2 Vectorial diffraction of doughnut beam illumina-
tion
It is well known that due to the helical phase distribution of a doughnut
beam its intensity distribution gives zero on the axis, when it is focused by
a low NA lens [89]. Physically, when a high NA lens is used the electric
field in the focal region exhibits a component along the incident polarisation
PhD thesis: Far-field and near-field optical trapping 121
CHAPTER 5. Far-field optical trapping
direction as well as an orthogonal and a longitudinal component, which is
called depolarisation.
Using the vectorial Debye theory, reviewed in Chapter 2, one can express
the electric field distribution in the focal region of a linearly polarised
monochromatic doughnut beam focused by a high NA objective satisfying
the sine condition, if the polarisation is along the x direction, as [86,89]
E2(r2, ψ, z2) =i
λ
∫ ∫
Ω
√
cosφ exp(imθ) exp[−ikr2 sinφ cos(θ − ψ)]
exp(−ikz2 cosφ)[cosφ+ sin2 θ(1− cosφ)]i + cos θ sin θ(cosφ− 1)j
+ cos θ sinφk sinφdφdθ (5.1)
where i, j, and k are unit vectors in the x, y, and z directions respectively.
Variable r2, ψ, and z2 are the cylindrical coordinates of an observation point.
Any other polarisation state can be resolved in two orthogonal directions each
of which satisfies Eq. 5.1.
The intensity is proportional to the modulus squared of Eq. 5.1 and is
shown in Fig. 5.14 for doughnut beams of different topological charges and
numerical aperture. When the numerical aperture of the focusing objective is
low, the focal intensity distribution shows a well-known doughnut shape [89],
which depends on topological charges. If, on the other hand, the numerical
aperture of the focusing objective is high, the intensity distribution in the
focal region becomes distorted and loses singularity for certain topological
charges. It can be seen from Fig. 5.14 that the intensity in the doughnut
ring increases along the direction perpendicular to the incident polarisation
state and that the focal spot is elongated along the polarisation direction.
Furthermore, the doughnut beam of topological charges 1 and 2 (Fig. 5.14(a)
and Fig. 5.14(b)) looses its zero intensity on the beam axis when focused
by an objective of NA = 1 in air. For a beam of topological charge ±1,
PhD thesis: Far-field and near-field optical trapping 122
CHAPTER 5. Far-field optical trapping
Fig. 5.14: Calculated intensity distribution in the focal region of a doughnut beamfocused by an objective with NA = 1 ((a)-(c)) and NA = 0.2 ((d)-(f)): (a) and(d) Topological charge 1; (b) and (e) Topological charge 2; (c) and (f) Topologicalcharge 3.
its intensity in the center of the focal region equals 48.8% of the maximum
intensity (Fig. 5.14(a)), while for a beam of topological charge ±2 it drops
down to 13.5% of the maximum intensity (Fig. 5.14(b)). When the topological
charge becomes −3 ≥ m ≥ 3, the intensity in the center of the focal region
regains its zero value (Fig. 5.14(c)). The zero intensity in the center of the
focal region has been observed by calculating focal intensity distributions
for higher doughnut beam topological charges (m = 4 and m = 5). This
topological charge dependence of the intensity in the central focal region can
be understood from Eq. 5.1.
Considering the point at the focus at r2 = 0 and z2 = 0, we find
that |Ex|2 = 0, |Ey|2 = 0, and |Ez|2 6= 0 when m = ±1 because∫ 2π
0cos(mθ) cos θdθ = 0 for all m, except for m = ±1. In the case of
m = ±2, the electric field components in the focal region give |Ex|2 6= 0,
PhD thesis: Far-field and near-field optical trapping 123
CHAPTER 5. Far-field optical trapping
|Ey|2 6= 0, and |Ez|2 = 0 because∫ 2π
0cos(mθ) sin2 θdθ in the Ex component
and∫ 2π
0cos θ sin θ sin(mθ)dθ in the Ey component are non-zero for m = ±2.
For −3 ≥ m ≥ 3, all integrals over θ give zero, which leads to the zero values
of all the E field components at the center of the focal region.
A more detailed picture of the intensity |E|2 normalised to 100 and its
components |Ex|2, |Ey|2 and |Ez|2 near the focal region of an objective (NA =
1 in air) illuminated by a charge 1 doughnut beam polarised in the x direction
is shown in Fig. 5.15. The contour plots are presented in terms of transverse
6 4 2 0 -2 -4 -66
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2
0
-2
-4
-6
(a)
0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
Vx
Vy
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0
-2
-4
-6
(b)
0
0.3
0.5
0.8
1.0
1.3
1.5
1.8
2.0
Vx
Vy
6 4 2 0 -2 -4 -66
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2
0
-2
-4
-6
(c)
0
6.3
12.5
18.8
25.0
31.3
37.5
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50.0
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6 4 2 0 -2 -4 -66
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-2
-4
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(d)
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50.0
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75.0
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100.0
Vx
Vy
Fig. 5.15: Contour plots of the intensity distribution in the focal region of anobjective with NA = 1, illuminated by a doughnut beam of topological charge 1.(a) |Ex|2;(b) |Ey|2;(c) |Ez|2;(d) |E|2.
optical coordinates Vx and Vy, which are defined as Vx,y = k[x, y] sinαmax,
where k is the wave number and αmax is the maximum angle of convergence. It
is evident that due to the high convergence angle, |Ex|2 and |Ez|2 components
PhD thesis: Far-field and near-field optical trapping 124
CHAPTER 5. Far-field optical trapping
play a dominant role in shaping the overall intensity |E|2. The high intensity
of |Ez|2 in the central region of the geometrical focus causes the non-zero value
in the central region of the overall intensity distribution, and together with
|Ex|2 component leads to a ”two-peak focus” in the Vy direction (Fig. 5.15(d)).
6 4 2 0 -2 -4 -66
4
2
0
-2
-4
-6
(a)
0
8.8
17.5
26.3
35.0
43.8
52.5
61.3
70.0
Vx
Vy
6 4 2 0 -2 -4 -66
4
2
0
-2
-4
-6
(b)
0
0.9
1.8
2.6
3.5
4.4
5.3
6.1
7.0
Vx
Vy
6 4 2 0 -2 -4 -66
4
2
0
-2
-4
-6
(c)
0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
Vx
Vy
6 4 2 0 -2 -4 -66
4
2
0
-2
-4
-6
(d)
0
12.5
25.0
37.5
50.0
62.5
75.0
87.5
100.0
Vx
Vy
Fig. 5.16: Contour plots of the intensity distribution in the focal region of anobjective with NA = 1, illuminated by a doughnut beam of topological charge 2.(a) |Ex|2;(b) |Ey|2;(c) |Ez|2;(d) |E|2.
Fig. 5.16 gives the similar contour plots for a doughnut beam of topological
charge 2. It is interesting to note that in this case |Ey|2 component is
comparable to |Ex|2 in the central focal region, and that these two components
result in the non-zero value in the center of the focal region. The central focal
intensity for charge 2 is much smaller than that for charge 1. This is because
the relative strength of |Ez|2, which contributes to the central intensity for
charge 1, is much larger than the relative strength of |Ey|2, which contributes
PhD thesis: Far-field and near-field optical trapping 125
CHAPTER 5. Far-field optical trapping
to the central intensity for charge 2. The combination of |Ex|2 and |Ez|2
components produces higher intensity peaks in the doughnut ring along the
Vy direction (Fig. 5.16(d)).
6 4 2 0 -2 -4 -66
4
2
0
-2
-4
-6
(a)
0
8.8
17.5
26.3
35.0
43.8
52.5
61.3
70.0
Vx
Vy
6 4 2 0 -2 -4 -66
4
2
0
-2
-4
-6
(b)
0
0.4
0.9
1.3
1.8
2.2
2.6
3.1
3.5
VxV
y
6 4 2 0 -2 -4 -66
4
2
0
-2
-4
-6
(c)
0
5.6
11.3
16.9
22.5
28.1
33.8
39.4
45.0
Vx
Vy
6 4 2 0 -2 -4 -66
4
2
0
-2
-4
-6
(d)
0
12.5
25.0
37.5
50.0
62.4
74.9
87.4
100.0
Vx
Vy
Fig. 5.17: Contour plots of the intensity distribution in the focal region of anobjective with NA = 1, illuminated by a doughnut beam of topological charge 3.(a) |Ex|2;(b) |Ey|2;(c) |Ez|2;(d) |E|2.
The electric field components |Ex|2, |Ey|2 and |Ez|2 all give zero field
intensity in the central region of a doughnut beam of charge 3 (Fig. 5.17).
Component |Ex|2 gives a ring of equal intensity around the singularity
(Fig. 5.17(a)), while component produces two high intensity peaks on
either side of the singularity in the Vy direction (Fig. 5.17(c)). These two
components are comparable in strength and are dominant in determining the
overall intensity in the focal region.
The peak intensity ratio of |Ez|2/|Ex|2, gives the relative strength of the
PhD thesis: Far-field and near-field optical trapping 126
CHAPTER 5. Far-field optical trapping
transversal |Ex|2 and the longitudinal |Ez|2 field components. The peak
intensity ratio as a function of NA for topological charges 1, 2 and 3 is
shown in Fig. 5.18(a). It is seen that the peak intensity ratio increases
Fig. 5.18: Dependence of the peak ratio of |Ez|2/|Ex|2 on the numerical aperture(a) and on the obstruction radius ε (b).
rapidly with NA. The longitudinal component |Ez|2 of the doughnut beam
attains approximately a half of the strength of the |Ex|2 component even for
NA = 0.9. The increase is more rapid for topological charge 1 than that
exhibited by the charges 2 and 3 beams of lower numerical aperture. As NA
becomes larger, the most rapid increase is manifested by charge 3.
Let us turn to the effect of a central opaque disk on the phenomenon shown
in Figs. 5.14- 5.17. Such a centrally obstructed doughnut beam is analogous
to using an axicon to extend the focal depth of the doughnut beam, which
has been used in laser trapping with a high NA objective [125]. Fig. 5.18(b)
presents the dependence of the peak intensity ratio on the central obstruction
ε for doughnut beams of charges 1, 2 and 3 for NA = 1 in air. Here, ε is
defined as the ratio of the radius of the obstructed part to the radius of
the unobstructed part of the beam. Presence of obstruction leads to more
depolarised rays reaching the focus, which enhances the contribution from
PhD thesis: Far-field and near-field optical trapping 127
CHAPTER 5. Far-field optical trapping
the |Ez|2 component to the total intensity [89]. For example, for obstructions
of ε = 0.83 and ε = 0.87 for a doughnut beam of charge 1 and charge 2,
respectively, the longitudinal contribution equals the transverse one. For the
limiting case of ε→ 1.0 the longitudinal contribution for the doughnut beam
of charges 1, 2 and 3 is approximately 1.47, 1.64 and 2.25 times larger than
that from the transverse component, respectively.
5.3.3 Trapping efficiency
The consistent comparison between theory and experiments described in
section 5.2.3 exhibits the applicability of the vectorial diffraction method
for the trapping force evaluation of the complex laser beams generated using
various spatial phase modulation techniques, by including the appropriate
phase modulation in the incident illumination. Based on the vectorial
diffraction analysis of the incident doughnut beams, the trapping efficiency
can be evaluated using the method described in Section 4.2.
A comparison of the maximal backward ATE of a highly focused doughnut
beam of charge 1 with an ordinary plane wave and a centrally obstructed plane
wave incident on polystyrene particles is shown in Fig. 5.19. The difference
between an unobstructed plane wave and a doughnut beam illumination types
becomes larger for smaller particles due to the reduced interaction of small
particles with the low-intensity central field region of the focused charge 1
doughnut beam (Fig. 5.14(a)). However, according to the RO model the
backward ATE achieved with a centrally obstructed plane wave is larger
than that achieved by either an unobstructed plane wave or a doughnut
beam of charge 1 [23]. The RO model indicates that in the case of a very
large obstruction, the backward ATE is approximately 1.6 times larger than
that achieved by an unobstructed plane wave, and 1.2 times larger than
that achieved by a doughnut beam of charge 1. The RO optics model
PhD thesis: Far-field and near-field optical trapping 128
CHAPTER 5. Far-field optical trapping
Fig. 5.19: Maximal backward ATE of polystyrene particles suspended in water andilluminated by a highly focused plane wave, doughnut beam of topological charge1 and obstructed plane wave with ε = 0.8 as a function of particle size. NA = 1.2and λ0 = 1.064 µm.
however, completely ignores the EM field distribution in the focal region
of an obstructed laser beam which leads to the focal spot elongation in the
axial direction. As it can be seen from the results obtained by our trapping
model in Fig. 5.19, the maximal backward ATE is actually reduced when a
plane wave is centrally obstructed compared to either an unobstructed plane
wave or a doughnut beam of charge 1. The reduction is due to the focal spot
elongation in the axial direction which leads to a reduced intensity gradient
in this direction.
The dependence of the ATE on the topological charge of a doughnut
beam is given in Fig. 5.20 for a polystyrene particle of radius a = 2 µm
and suspended in water. The ATE in the forward direction Qf is larger for
PhD thesis: Far-field and near-field optical trapping 129
CHAPTER 5. Far-field optical trapping
higher topological charge, while the ATE in the backward direction Qb is
relatively unchanged. This result is consistent with experimental findings of
Fig. 5.20: ATE of a polystyrene particle (a = 2 µm) suspended in water andilluminated by a highly focused plane wave and doughnut beams of differenttopological charges. NA = 1.2 and λ0 = 1.064 µm.
the backward ATE [126] and the forward ATE [127] of large microparticles.
The TTE, on the other hand, is reduced for higher topological charges in
either scanning direction (Fig. 5.21).
To investigate the validity of the theoretically determined results we have
conducted experimental measurements of the TTE. The measurement is
performed in the same manner as described in Section 5.2.3.2 without the
central obstruction. For plane wave illumination the reflection type SPM is set
to uniform phase distribution, which then operates as a simple mirror, while
for the doughnut beam illumination it is loaded with a 256 levels phase-ramp
distribution described in Fig. 5.13(a). Table 5.1 shows the experimentally
measured TTE for both S (Qs) and P (Qp) scanning directions with plane
PhD thesis: Far-field and near-field optical trapping 130
CHAPTER 5. Far-field optical trapping
Fig. 5.21: TTE in the polarisation (X) and perpendicular to the polarisation (Y)directions of a polystyrene particle (a = 2 µm) suspended in water and illuminatedby a highly focused plane wave and doughnut beams of different topological charges.NA = 1.2 and λ0 = 1.064 µm.
wave and doughnut beam illuminations for three different polystyrene particle
sizes with radii of 1 µm (Polybead polystyrene microspheres, radius=1.067
µm with standard deviation δ=0.020 µm), 0.5 µm (Polybead polystyrene
microspheres, radius=0.496 µm with standard deviation δ=0.013 µm) and
0.25 µm (Polybead polystyrene microspheres, radius=0.242 µm with standard
deviation δ=0.005 µm). The large particle has a radius of twice the
wavelength size, and the radius of the medium particle is approximately the
same as the illumination wavelength, while the smallest particle radius is half
of the illumination wavelength.
It can be seen from Table 5.1 that the maximal TTE decreases for smaller
particle sizes, with the decrease being more rapid for the doughnut beam
illumination. The ratios of the experimentally measured maximal TTE of
the plane wave and doughnut beam illuminations agree with the theoretically
calculated results within the experimental error. Furthermore the trend that
the ratio is higher for smaller particles is experimentally confirmed.
Similar to the maximal backward ATE in Fig. 5.19, the maximal TTE
PhD thesis: Far-field and near-field optical trapping 131
CHAPTER 5. Far-field optical trapping
a = 1µm a = 0.5µm a = 0.25µm
Qp(Ch0) exp. 0.091 ± 0.005 0.087 ± 0.002 0.040 ± 0.002Qp(Ch1) exp. 0.079 ± 0.005 0.048 ± 0.002 0.019 ± 0.002Qs(Ch0) exp. 0.079 ± 0.005 0.068 ± 0.002 0.040 ± 0.002Qs(Ch1) exp. 0.057 ± 0.005 0.043 ± 0.002 0.015 ± 0.002Qp(Ch0)/ Qp(Ch1) exp. 1.15 ± 0.15 1.81 ± 0.12 2.11 ± 0.36Qp(Ch0)/ Qp(Ch1) th. 1.29 1.55 2.15Qs(Ch0)/ Qs(Ch1) exp. 1.38 ± 0.23 1.58 ± 0.12 2.67 ± 0.56Qs(Ch0)/ Qs(Ch1) th. 1.25 1.50 2.30
Table 5.1: The maximal TTE for plane wave and doughnut beam illumination.Ch0 denotes plane wave input, while Ch1 denotes a doughnut beam of topologicalcharge 1. exp. - experimentally measured result, th. - theoretically calculatedresult.
a = 1µm
Qp(Ch0+ε) exp. 0.018 ± 0.002Qp(Ch1) exp. 0.079 ± 0.005Qs(Ch0+ε) exp. 0.017 ± 0.002Qs(Ch1) exp. 0.057 ± 0.005Qp(Ch0+ε)/ Qp(Ch1) exp. 0.23 ± 0.04Qp(Ch0+ε)/ Qp(Ch1) th. 0.257Qs(Ch0+ε)/ Qs(Ch1) exp. 0.30 ± 0.07Qs(Ch0+ε)/ Qs(Ch1) th. 0.260
Table 5.2: The maximal TTE for a centrally obstructed plane wave and a doughnutbeam illumination. Ch0+ε denotes a centrally obstructed plane wave input, whileCh1 denotes a doughnut beam of topological charge 1. exp. - experimentallymeasured result, th. - theoretically calculated result. Obstruction size ε = 0.78.
of an obstructed plane wave is smaller than that achieved with either an
unobstructed plane wave or a doughnut beam of charge 1. The RO model
indicates that the ratio of the maximal TTE of a highly obstructed plane
wave to the one achieved by a doughnut beam of charge 1 is approximately
0.8 [23]. Our optical trapping model, which considers the exact EM field
distribution in the focal region, gives this ratio as approximately 0.26, which
agrees well with the experimentally measured results (Table 5.2). This result
PhD thesis: Far-field and near-field optical trapping 132
CHAPTER 5. Far-field optical trapping
further indicates that if one deals with complex laser beams one needs to take
into account the complexity of the focal field distribution and its interaction
with a micro-particle to properly investigate the trapping force exerted on
such a particle, which is achieved using our optical trapping model based on
the vectorial diffraction theory.
5.4 Chapter conclusions
Far-field optical trapping with high NA objective is investigated using the
vectorial diffraction model described in the previous Chapter 4 with two types
of incident illumination; plane wave and doughnut beam illumination.
It is found that the refractive index mismatch between the objective
immersion and particle suspension media, that leads to SA, severely affects
the trapping performance of an optical trap, due to the focal distribution
distortions. SA generally leads to the degradation of the trapping efficiency
for both the ATE and the TTE. The rate of degradation depends on the
particle and the environment parameters. Our optical trapping model based
on the vectorial diffraction theory, however, can successfully deal with this
issue, which has been confirmed by comparing the theoretically calculated
results with the experimental measurements of Felgner et al. [115].
When the incident plane wave is centrally obstructed the TTE of large
dielectric particles falls rapidly with the increasing size of the obstruction.
The decrease predicted by the vectorial diffraction model is much faster than
the one given by the RO model. We have experimentally measured the
dependence of the TTE on the obstruction size for large polystyrene particle
for both S and P polarisation states of the incident illuminations. It is found
that the TTE decrease occurs at a much faster rate than the one predicted
by the RO model, for both polarisations, and that it matches well the rate
PhD thesis: Far-field and near-field optical trapping 133
CHAPTER 5. Far-field optical trapping
predicted by the vectorial diffraction model. Furthermore, the difference in
the maximal TTE for the two polarisation states predicted by the vectorial
diffraction model is verified in the experiment.
Efficient generation and vectorial diffraction of doughnut beams is also
investigated in order to study the trapping force exerted on a dielectric
particle by focused doughnut laser beams. The doughnut beam generation
efficiency of 100% can be achieved when plane wave is converted into
doughnut beam using a LC cell or the reflection mode phase modulator. It
is found that the doughnut beams focused by a high NA objective yield focal
intensity distributions markedly different from those obtained when the same
beam is focused by a low NA objective. The central zero intensity points
disappear for doughnut beams of topological charges ±1 and ±2 due to the
depolarisation effect of a high NA objective. The focused distribution of a
doughnut beam of a given charge shows the increased ring intensity along the
direction perpendicular to the incident polarisation direction and the focal
spot becomes elongated in the polarisation direction. These effects are more
pronounced when such beams are centrally obstructed and may affect the
laser trapping performance.
The maximal backward ATE (lifting force) of small particles is greatly
reduced when the trapping is performed with focused doughnut beam
illumination, compared to the plane wave illumination, due to the reduced
central intensity distribution in the focal region of such an illumination.
Furthermore, when plane wave is centrally obstructed the lifting force is
reduced even further compared to either an unobstructed plane wave or
doughnut beam illumination.
For large particles, the forward ATE is increased when doughnut beam
illumination is used, while the backward ATE is relatively unchanged. On the
other hand, the TTE, under the same conditions, is reduced when doughnut
PhD thesis: Far-field and near-field optical trapping 134
CHAPTER 5. Far-field optical trapping
beams are used for trapping. It is shown theoretically and experimentally
that the TTE decreases for reducing particle size for both a plane wave and
a doughnut beam of charge 1 incident illumination. However, the rate of
decrease is faster in the case of doughnut beam illumination. The comparison
of the maximal TTE of an obstructed plane wave with ε ≈ 0.8 and a doughnut
beam of charge 1 reveals that the efficiency available with the obstructed
beam is only approximately 0.26 of that achieved by a doughnut beam,
which has been confirmed by an experimental measurement. This result
contradicts the RO model prediction which indicates that the two efficiencies
should be relatively comparable. The RO model is inadequate to describe
such a trapping process because it considers the highly obstructed beam
by calculating the trapping force produced by highly convergent rays only
(rays close to the maximum convergent angle), while completely ignoring
the diffraction effects and its influence on the focal field distribution. The
focal field distribution of a highly obstructed beam exhibits an elongation in
the axial direction and increased rings intensities in the transverse direction,
which greatly affects the trapping performance.
PhD thesis: Far-field and near-field optical trapping 135
Chapter 6
Near-field optical trapping
6.1 Introduction
The trapping force generated by a conventional high numerical aperture
(NA) optical tweezers setup, also known as the far-field trapping discussed
in Chapter 5, acts toward the high intensity focal region due to the large
field gradients. The trapping volume of the far-field laser trapping technique
is diffraction limited, leads to a significant background signal, and poses
difficulties in the single-molecule experiments. The trapping modality based
on the particle trapping using a highly localised near-field, can overcome these
problems. A review of the near-field trapping is undertaken in Section 2.6
of Chapter 2. It can be seen from this review that all of the proposed near-
field trapping techniques have been theoretically investigated, except the one
implemented using a focused evanescent illumination. Our physical model
for the trapping force evaluation, based on vectorial diffraction introduced in
Chapter 4 and implemented with the far-field optical trapping discussed in
Chapter 5, can be applied to provide a theoretical model for the near-field
trapping with focused evanescent illumination.
136
CHAPTER 6. Near-field optical trapping
The focused evanescent field is produced by placing a central obstruction
in the laser beam path, before it enters a high NA objective. The size
of the obstruction is large enough to ensure that the minimum angle of
convergence of each incident ray is larger than the critical angle determined
by the total internal reflection (TIR) condition between two media (Fig. 6.1).
The circular symmetric nature of such an illumination enhances the strength
Fig. 6.1: Focused evanescent field produced at the coverslip interface by high NAfocusing of an obstructed plane wave polarised in the X direction. Oil immersionand coverslip refractive index n1 = 1.78 (index matched), particle suspensionmedium n2 = 1.33, objective NA = 1.65, obstruction size ε = 0.8, and illuminationwavelength λ0 = 1.064 µm.
of the evanescent field and reduces its lateral size, due to the constructive
interference of the resulting evanescent field near the center of the focus. The
focal splitting effect due to the relatively large obstruction size is evident
(Fig. 6.1 XY plane). In addition to the focused evanescent field generated
by plane wave illumination, we have considered the focused evanescent
illumination obtained by centrally obstructing a doughnut beam. This
chapter studies the distribution of the near-field trapping force with focused
evanescent illumination and investigates the dependence of the near-field force
on particle size. An important question of the capability of the near-field
trapping with focused evanescent illumination to trap small particles in three
PhD thesis: Far-field and near-field optical trapping 137
CHAPTER 6. Near-field optical trapping
dimensions is also dealt with in more details. In addition to the theoretical
study of the near-field trapping with focused evanescent illumination, we have
conducted experimental investigations of the transverse trapping efficiency
(TTE) of this type of near-field trapping. A good agreement between the
theoretical and experimental results confirms the validity of our approach.
This chapter also briefly deals with the near-field forces exerted on dielec-
tric microparticles situated in the wide area evanescent field illumination. The
wide area evanescent field is considered according to the three-dimensional
consideration of the evanescent field scattering for particles situated far from
the interface at which evanescent wave is generated, described in details
in Chapter 3. The Maxwell stress tensor approach is used for the force
calculation. This method is not suitable to deal with a focused evanescent
field because the field representation is that of a wide area plane wave incident
under the TIR condition at the prism interface.
The structure of the chapter is as follows. The near-field forces exerted
on small dielectric particles by a wide area evanescent illumination, whose
scattering properties were investigated earlier, are examined in Section 6.2.
Section 6.3 gives the numerical results of the near-field trapping with a focused
evanescent illumination under both plane wave and doughnut beam incident
illumination. The near-field force distribution for a small and a large dielectric
particle is presented as well as the trapping efficiency dependence on particle
size. Experimental results are presented in Section 6.4, for the plane wave
(Section 6.4.1) and doughnut beam (Section 6.4.2) incident illumination. The
chapter conclusions are summarised in Section 6.5.
PhD thesis: Far-field and near-field optical trapping 138
CHAPTER 6. Near-field optical trapping
6.2 Optical forces on microparticles in a wide
area evanescent field
The fast decaying wide area evanescent field, whose scattering properties
by small dielectric particles were studied in Chapter 3, exerts a force on
microparticles situated in such field, due to the high intensity gradient.
The effect of such force was investigated experimentally by Kawata and
Sugiura [67] where they have observed particle movements induced by the
evanescent field, while the theoretical treatment was described by Almaas
and Brevik [68]. Even though this configuration of the near-field trapping
has been dealt with theoretically in general, we include this section for the
sake of completeness, with some specific parameters of the evanescent field
corresponding to the ones used in studies of the evanescent field scattering in
Chapter 3. If we consider a homogeneous microsphere situated in the particle
immersion medium n′ illuminated by a monochromatic electromagnetic (EM)
field described by Eq. 3.1, the resulting scattered field by Eq. 3.3, and their
corresponding magnetic fields, the net radiation force on the microsphere can
be determined using the steady-state Maxwell stress tensor analysis given by
Eq. 2.7.
Figure 6.2 shows the vertical (Qz) and the horizontal (Qy) near-field
radiation forces exerted on a polystyrene particle situated in water and
illuminated by an evanescent field generated by a TIR of the Helium-Neon
laser beam. The TIR occurs at a prism surface. The prism refractive index is
assumed to be 1.722 and the laser beam is incident at an angle of 51. This
corresponds to the values assumed in Chapter 3, when scattering properties
of microparticles were investigated. The forces are given, similarly to Almaas
PhD thesis: Far-field and near-field optical trapping 139
CHAPTER 6. Near-field optical trapping
and Brevik [68], in their non-dimensional form, defined as
Qy =Fy
ε0E20a
2, Qz =
Fz
ε0E20a
2. (6.1)
Fig. 6.2: Non-dimensional horizontal Qy and vertical Qz near-field forces exertedon polystyrene particle under TE and TM incident polarisation states as a functionof the particle size parameter k′a = 2πa/λ. The evanescent field is generated on aprism surface (refractive index 1.722) by a plane wave incident at 51.
The results in Fig. 6.2 are similar to those of Almaas and Brevik [68],
and they indicate that the vertical force is negative, i.e. directed towards
the prism surface, while the horizontal force is always positive, indicating
particle pushing in this direction. Slight modulations on the force curves
is due to the morphology dependent resonance (MDR) effects described in
more details earlier in Chapter 3, which become more pronounced for larger
particles. The force strength can be selected and maximised by tuning the
particle size, or more practically the illumination wavelength. However, we
can conclude that the force in either direction is larger for the P polarised
incident illumination (TM illumination). For example, it can be seen from
Fig. 6.2 that a small polystyrene particle of 1 µm in diameter, and under the
PhD thesis: Far-field and near-field optical trapping 140
CHAPTER 6. Near-field optical trapping
influence of an evanescent field generated by a 150 mW Helium-Neon laser
beam, spread over an interaction cross-section area of 100 µm, experiences a
force of 0.028 pN and 0.044 pN in the horizontal direction for S (TE) and P
(TM) incident polarisation states respectively. The vertical force under the
same conditions is slightly larger; being 0.041 pN for S polarisation and 0.070
pN for P polarisation.
6.3 Near-field trapping with focused evanes-
cent illumination - Numerical results
To investigate near-field trapping implemented using a focused evanescent
field we have developed a physical model based on the theoretical treatment
described in Chapter 4 (Fig. 6.3). This model includes two physical processes,
vectorial diffraction by a high NA objective under the TIR condition and
scattering by a small particle with a focused evanescent wave. The limits
of integration in Eqs. 4.1 and 4.2 are modified to incorporate the central
obstruction. The lower limit is determined by the maximum obstruction
angle (α1), while the upper limit is given by the maximum convergence angle
(α2). Thus the achieved EM distribution on the coverslip interface represents
a focused evanescent field. Based on the scattering process of a small particle
with an evanescent focal spot, one can determine the EM field inside and
outside a particle and thus the trapping force exerted on the particle.
Now let us consider that a small particle interacts with focused evanescent
illumination generated at the coverslip interface of a centrally obstructed
high NA objective (NA=1.65). An axial force caused by the fast decaying
nature of such illumination and a transverse gradient force resulting from
the focal shape are exerted on the particle. Figs. 6.4 and 6.5 shows the
trapping efficiency mapping, when a small and large polystyrene particle
PhD thesis: Far-field and near-field optical trapping 141
CHAPTER 6. Near-field optical trapping
Laser beam
Obstruction
Objective
n1
n2
Fz
Fx Fx
a1
a2
Particle
Focusedevanescentfield
Fig. 6.3: Near-field trapping model.
is transversally scanned in the X and Y directions across the focused
evanescent field distribution, generated by placing a central obstruction
(ε=0.85) perpendicularly to the path of an incoming laser beam. The
focused evanescent field is produced for ε > 0.8 when the refractive indices of
the coverslip and immersion water are n1=1.78 and n2=1.33. The laser beam
(λ=532 nm) propagates in the Z direction and two cases are considered; the
first one being an ordinary plane wave [15], while the second one is a phase
modulated doughnut beam of charge 1. For both small and large particles
the ATE for plane wave illumination is slightly larger than that for doughnut
beam illumination. However, in the case of small particles the ATE also
decreases slightly faster with plane wave illumination when the particle is
scanned in either direction. Not only is the ATE for large particles stronger,
the force mapping structure in the case of large particles is markedly different
from the small particle case due to the larger interaction cross section of the
PhD thesis: Far-field and near-field optical trapping 142
CHAPTER 6. Near-field optical trapping
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
a=0.2
5mm
X (mm)
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035T
rappin
geffic
iency
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
a=1
mm
X (mm)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Tra
ppin
geffic
iency
Laser beam
Obstruction
Objective
n1
n2
Laser beam
Obstruction
Objective
n1
n2
Laser beam
Obstruction
Objective
n1
n2
Phase modulatorLaser beam
Obstruction
Objective
n1
n2
Phase modulator
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
a=0.2
5mm
X (mm)
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
Tra
ppin
geffic
iency
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
a=1
mm
X (mm)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Tra
ppin
geffic
iency
Fig. 6.4: Trapping efficiency mapping for a small and a large polystyrene particle ofradius a, scanned in the X direction (light polarisation direction) across the focusedevanescent field generated by a plane wave (top) and doughnut beam illumination(bottom). NA=1.65, λ=532 nm, ε=0.85, n1=1.78 and n2=1.33.
small particle with the focused evanescent field. The maximal TTE for small
particles is much larger relative to the ATE than that for large particles. The
maximal TTE in the X direction, for example, constitutes 16.8% and 14.8%
of the maximal ATE for small particles in the case of the plane wave and
doughnut beam illuminations respectively, compared to the 7.4% and 7.2% for
large particles. The trapping efficiency decrease for small and large particles is
also slightly faster in the direction perpendicular to the polarisation direction
than that in the polarisation direction. This asymmetry is caused by the
focused spot elongation due to the focusing by a high NA objective.
PhD thesis: Far-field and near-field optical trapping 143
CHAPTER 6. Near-field optical trapping
Laser beam
Obstruction
Objective
n1
n2
Laser beam
Obstruction
Objective
n1
n2
Laser beam
Obstruction
Objective
n1
n2
Phase modulatorLaser beam
Obstruction
Objective
n1
n2
Phase modulator
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Y (mm)
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
a=0.2
5mm
Tra
ppin
geffic
iency
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Y (mm)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
a=1
mm
Tra
ppin
geffic
iency
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Y (mm)
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
a=0.2
5mm
Tra
ppin
geffic
iency
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Y (mm)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
a=1
mm
Tra
ppin
geffic
iency
Fig. 6.5: Trapping efficiency mapping for a small and a large polystyrene particleof radius a, scanned in the Y direction (perpendicular to the polarisation direction)across the focused evanescent field generated by a plane wave (top) and doughnutbeam illumination (bottom). NA=1.65, λ=532 nm, ε=0.85, n1=1.78 and n2=1.33.
The dependence of the maximal TTE on the obstruction size ε is critical
to capture a particle in the evanescent focal spot and is shown in Fig. 6.6 for
both plane wave and doughnut beam illumination for polystyrene particles
of diameter 2 µm. The maximal TTE decreases with increasing the size of
the beam obstruction due to the reduced contribution of the propagating
component to the transverse force and because the high angle rays are less
efficient in the transverse trapping of a dielectric particle [52]. The maximal
TTE for the plane wave case is generally slightly larger than that obtained
with doughnut beam illumination for both P and S directions. The difference
PhD thesis: Far-field and near-field optical trapping 144
CHAPTER 6. Near-field optical trapping
Fig. 6.6: Theoretical calculations of the maximal TTE of a polystyrene particle of1 µm in radius as a function of the obstruction size ε. The other conditions are thesame as in Fig. 6.4.
in the maximal TTE for the plane wave and doughnut beam cases is the
largest when no obstruction is present (plane wave TTE being 1.4 times
larger) and becomes smaller for the increasing obstruction size. In the near-
field trapping regime (ε > 0.8) this difference nearly vanishes.
The dependence of the maximal ATE on the obstruction size ε is at first
relatively unchanged until ε ∼ 0.6, at which point the maximal ATE decreases
for increasing ε (Fig. 6.7). For certain values of ε in this range it even shows
a slightly larger ATE compared to the case without the obstruction. At the
focused evanescent field condition, i.e. when ε > 0.8, the maximal ATE is
still approximately 43% of the far-field case when no obstruction is present.
So far only the case when the obstructed beam with ε corresponding to
the minimum convergence angle larger than the critical angle is focused onto
the interface between two media is considered, i.e. the geometrical focus
position is on the interface. When this geometrical focus position is brought
into the suspension medium, the fast decaying evanescent field is slightly
defocused at the interface. Since there is no propagating component, the
PhD thesis: Far-field and near-field optical trapping 145
CHAPTER 6. Near-field optical trapping
Fig. 6.7: Theoretical calculations of the maximal ATE of a polystyrene particle of1 µm in radius as a function of the obstruction size ε. The other conditions are thesame as in Fig. 6.4.
field is localised at the interface and the geometrical focus becomes virtual.
The relationship between the depth d of such a virtual focus, which describes
the spherical aberration function Ψ (Eq. 2.21) and the ATE for both small
and large dielectric particles is revealed in Fig. 6.8. Both plane wave and
doughnut beam illumination cases follow a similar trend with plane wave
illumination showing generally a slightly higher ATE, and converge at a larger
depth. It is interesting to note, however, that the highest ATE does not
occur when the geometric focus is at the interface, but it occurs for a slightly
defocused evanescent field, focused at a virtual depth of ∼ 100 nm into the
suspension medium. The physics of this phenomenon originates from phase
shifts under TIR (Fig. 6.9) [54]. According to Eqs. 2.22 and 2.23 these phase
shifts act as an equivalent spherical aberration. Consequently, the balance
between the spherical aberration function Ψ and these phase shifts determines
the maximum field at the interface. It can be shown that for the given
PhD thesis: Far-field and near-field optical trapping 146
CHAPTER 6. Near-field optical trapping
Fig. 6.8: Dependence of the ATE on the virtual focus position d for a small andlarge polystyrene particle (ε = 0.85). The other conditions are the same as inFig. 6.4.
maximum convergence angle in our case, the total phase accumulated gives
the maximum field at the interface for the depth d = 62 nm.
The dependence of the maximal near-field ATE on the polystyrene particle
size is illustrated in Fig. 6.10. The relationship between the maximal
near-field ATE is nearly linear in nature for both types of illumination
and the particle size range considered, which is different from the far-field
force relationship that depends on the cubic of the particle size. This
linear relationship can be qualitatively explained by examining the particle
interaction cross-section area (A). If the evanescent field depth is denoted
by h, it can be shown that the interaction cross-section area is given by
A = π(4ah − h2), where a is the particle radius (Fig. 6.10 inset). It can be
estimated from Fig. 6.10 that an input laser power of 10 µW is sufficient to
overcome the gravity (including the buoyancy) force and lift a polystyrene
PhD thesis: Far-field and near-field optical trapping 147
CHAPTER 6. Near-field optical trapping
Fig. 6.9: Phase introduced by the Fresnel transmission coefficients as a function ofthe incident angle. The refractive index of the incident medium is 1.78, while therefractive index of the transmitting medium is 1.33.
a a-h
h n1
n2
0.2 0.4 0.6 0.8 1.00.00
0.02
0.04
0.06
0.08
0.10
0.12
(a)
AT
E
a (mm)
Plane waveDoughnut beam
Fig. 6.10: The maximal ATE as a function of a polystyrene particle size (ε = 0.85).The inset shows a schematic relation between the interaction cross-section area andthe particle size. The other conditions are the same as in Fig. 6.4.
PhD thesis: Far-field and near-field optical trapping 148
CHAPTER 6. Near-field optical trapping
particle of 800 nm in radius or smaller (Fig. 6.11) in an upright trapping
system.
Fig. 6.11: The magnitudes of the axial force for a plane wave of power 10 µW andthe gravity force for different particle sizes. The other conditions are the same asin Fig. 6.4.
6.4 Near-field trapping with focused evanes-
cent illumination - Experimental results
The same experimental configuration as the one used for the far-field trapping
measurements (Fig. 5.7) is used for the near-field trapping measurements.
The exception is that the water immersion high NA objective (NA = 1.2) is
replaced with the Olympus (Apo 100x oil HR) high NA objective (NA =
1.65). A special coverslip glass and the immersion oil for the NA-1.65
objective have a refractive index of 1.78. Particles are suspended in water with
refractive index of 1.33 (at the room temperature of 20 C), for which the NA-
1.65 objective generates focused evanescent field for ε > 0.8. This corresponds
to the cut-off angle of 48 for the TIR in water. The illumination source
PhD thesis: Far-field and near-field optical trapping 149
CHAPTER 6. Near-field optical trapping
was a 532 nm diode-pumped cw laser (Spectra-Physics Millenia II). The
trapping efficiency measurements were performed using the Stokes method
described in Section 5.2.3.2. The presented results represent the average of
three measurements with the uncertainties estimated from the laser power
fluctuations.
6.4.1 Plane wave illumination
The measurement of the maximal TTE with plane wave illumination as a
function of the central obstruction size is shown in Fig. 6.12. In the plane wave
case the phase modulator was replaced by a mirror. Under these experimental
Fig. 6.12: The measured and calculated plane wave illumination maximal TTE asa function of obstruction size with a NA-1.65 objective for both S and P scanningdirections. The other conditions are the same as in Fig. 6.4.
conditions the evanescent field is generated for the obstruction size ε > 0.8,
thus resulting in the near-field trapping.
First of all, it can be seen from Fig. 6.12 that our model agrees well with
PhD thesis: Far-field and near-field optical trapping 150
CHAPTER 6. Near-field optical trapping
the experimental measurements for both S and P scanning directions and that
the maximal TTE decreases with the obstruction size at the rate predicted by
our model. Furthermore, the maximal TTE when no obstruction is present
is ∼ 12% higher in the polarisation direction (P direction) compared to the
direction perpendicular to the polarisation direction (S direction). However,
in the near-field case the situation is reversed, the maximal TTE in the S
direction is higher than that in the P direction. For example in the case of
a central obstruction ε = 0.9 the maximal TTE in the S direction is ∼ 12%
higher than that in the P direction. The maximal TTE in the two directions
is nearly the same for ε ∼ 0.5. All of these findings are confirmed by the
experimental measurements (Fig. 6.12).
6.4.2 Doughnut beam illumination
The doughnut beam illumination used in the experimental measurements was
generated in the same manner as for the far-field measurements described in
the previous chapter (Chapter 5). The dependence of the maximal TTE
on the central obstruction size, for doughnut beam illumination is shown in
Fig. 6.13. Similar to the measurement with the plane wave illumination,
the maximal TTE decreases for increasing obstruction size. The maximal
TTE without the obstruction is ∼ 13% higher in the polarisation P direction
compared to the S direction. However, unlike the measurement with the plane
wave illumination, the maximal TTE for the near-field trapping case with
doughnut beam illumination is nearly the same in either scanning direction.
The near-field maximal TTE with doughnut beam illumination is nearly
the same as that for plane wave illumination for either polarisation direction.
However, in the far-field case when no obstruction is present, the maximal
TTE for doughnut beam illumination is ∼ 28% lower than that achieved with
plane wave illumination.
PhD thesis: Far-field and near-field optical trapping 151
CHAPTER 6. Near-field optical trapping
Fig. 6.13: The measured and calculated doughnut beam illumination maximal TTEas a function of obstruction size with a NA-1.65 objective for both S and P scanningdirections. The other conditions are the same as in Fig. 6.4.
6.5 Chapter conclusions
The force mapping for the near-field trapping with a focused evanescent field
in the case of large polystyrene particles shows markedly different structure
from that obtained for small particles due to the larger interaction cross-
section. The ration of the maximal TTE to the maximal ATE for small
particles is much larger than that for large particles. However, the maximal
ATE is stronger for larger particles. The trapping efficiency is slightly larger
for the focused evanescent field generated by a plane wave, compared to the
one generated by a doughnut beam.
The maximal ATE for large polystyrene particles is relatively unchanged
for obstruction sizes of ε < 0.6. For certain values of ε in this range it even
shows a slightly larger ATE compared to the case without the obstruction.
For ε > 0.6, the ATE decreases rapidly for increasing ε. However, in the near-
field trapping case it still constitutes ∼ 43% of that for the far-field case. It
PhD thesis: Far-field and near-field optical trapping 152
CHAPTER 6. Near-field optical trapping
is also determined that the highest ATE does not occur when the geometric
focus is at the interface, but it occurs for a slightly defocused evanescent
field, when the geometrical focus is at a depth of ∼ 62 nm. It is found that
physically this phenomenon originates from phase shifts under TIR.
Unlike the far-field trapping, the dependence of the maximal ATE on the
particle size under focused evanescent wave illumination is nearly linear. It
is predicted that an input laser power of 10 µW is sufficient to overcome the
gravity force acting on a polystyrene particle of 800 nm in radius or smaller,
thus enabling the particle trapping in three dimensions, i.e. the particle
lifting, in an upright system.
The calculated trapping force in an evanescent focal spot for both plane
wave and doughnut beam illumination has been demonstrated to be in a good
agreement with experimental results.
PhD thesis: Far-field and near-field optical trapping 153
Chapter 7
Conclusion
7.1 Thesis conclusion
The main research work in this thesis presents an optical trapping model
based on two physical processes, vectorial diffraction of incident laser beam by
a high numerical aperture (NA) objective and scattering by a small particle
(Chapter 4). The interaction of the small particle and the focused laser
beam is described using an extension of the classical plane wave Lorentz-
Mie theory, while the force is determined using the Maxwell stress tensor
approach. An important advantage of our approach over other optical
trapping models reviewed in Chapter 2 is that it provides an exact description
of the electromagnetic (EM) field distribution in the focal region of a high
NA objective. As opposed to the other optical trapping models, our model
is capable of considering an arbitrary wavefront incident at the objective
entrance pupil. In other words, our model is capable of including any
apodisation function of the focusing objective, phase modulation of the
incident beam, polarisation effects and the effects of spherical aberration (SA)
usually present in optical trapping experiments. As a consequence we are able
154
CHAPTER 7. Conclusion
to consider the trapping process under doughnut beam illumination, which
represents one class of phase modulated laser beams (Chapter 5). We have
shown that our optical trapping model can deal with the far-field trapping
modality, as well as with the near-field trapping modality implemented with
a focused evanescent field generated by using a central obstruction in the
incident beam path. Our optical trapping model provides a first theoretical
treatment of such a near-field trapping modality (Chapter 6).
In addition to the optical trapping model, this thesis presents a nanometric
sensing model for detection of scattered evanescent waves (Chapter 3).
The sensing model, which describes the conversion of evanescent photons
into propagating photons, is implemented by considering the near-field Mie
scattering process enhanced by morphology dependent resonance (MDR) by
a small particle in the vicinity of a plane interface. The transformation
of the three-dimensional scattered field distribution to the detector plane
is achieved using vectorial diffraction of the scattered signal. The vectorial
diffraction approach for the transformation of the scattered signal is necessary
due to the vectorial nature of the scattered field. Such a sensing model
is essential for understanding optical trapping systems that use evanescent
field illumination for high resolution position monitoring and imaging, such
as optical trap nanometry and particle-trapped scanning near-field optical
microscopy (SNOM).
The nanometric sensing model reveals that the intensity pattern generated
in the far-field region of the collecting objective shows an interference-like
pattern when large dielectric particles are employed as scatterers. For small
particles the intensity pattern is similar to that given by the dipole radiation.
Our research has shown that MDR effects are evident in near-field Mie
scattering of large dielectric particles and that the resonance peaks become
sharper and less separated with increasing the effective refractive index. The
PhD thesis: Far-field and near-field optical trapping 155
CHAPTER 7. Conclusion
MDR effects are present in the image forming field collected by a high NA
objective and it results in different energy distributions in the detection plane
for on and off resonance positions. These different energy distributions outline
the importance of detector selection for systems operating at MDR positions,
because it would be an advantage to operate particle-trapped SNOM and
optical trap nanometry systems at MDR positions to enhance signal-to-noise
ratio. If the detection of the collected scattered signal is performed with a
pinhole detector, the detector size needs to be carefully selected depending
on the scattering particle size. This is because small particles act as dipole
sources and fill the objective entrance pupil uniformly, which requires a
relatively small pinhole. For large particles much larger pinholes are required
to collect the signal completely with a typical imaging lens. This effect is
contributed to larger interaction cross-section and the MDR effect of large
particles, which result in signal spreading in forward direction in the imaging
plane. The theoretical predictions of our model have been experimentally
verified by a good agreement of predicted and measured intensity distribution
of small and large polystyrene particles in the focal plane of the collecting
objective.
Applying our optical trapping model for force determination based on
the vectorial diffraction approach for light focusing under a high convergence
angle we have shown that it is superior to the commonly used Gaussian beam
model in that it exactly represents the field in the focal region of a high NA
objective and that the results obtained by our vectorial diffraction model at
large particle limit are approaching the ray optics (RO) model. The RO model
is an approximate model valid only for very large particles, and our optical
trapping model can bridge the gap between small particle case for which the
focal field distribution is crucial and large particle case for which the RO
model gives approximate solutions. Furthermore, the vectorial diffraction
model can treat arbitrary incident wavefronts and simulate real experimental
PhD thesis: Far-field and near-field optical trapping 156
CHAPTER 7. Conclusion
conditions. Investigations of the SA effects on the far-field trapping force
show that the refractive index mismatch between the objective immersion
and particle suspension media severely affects the trapping performance of an
optical trap. The rate of degradation of the trapping performance depends
on the particle and environment parameters. Our research, however, has
demonstrated that the trapping force evaluated using our optical trapping
model agrees well with the experimental results, for both axial and transverse
trapping forces.
The maximal transverse trapping efficiency (TTE) of a centrally ob-
structed plane wave focused by a high NA objective, in the case of large
dielectric particle decreases rapidly for increasing the obstruction size, unlike
the trend predicted by the RO model. The rate of decrease and the trapping
efficiency for most obstruction sizes is higher for the incident P polarisation,
compared to the incident S polarisation. At very large obstruction sizes
(ε ≈ 0.9), however, the trapping force of the P polarisation approaches that of
the S polarisation, due to the different depolarisation behaviors between the
S and P polarisation states. These findings are confirmed by experimental
measurements.
In order to study the trapping performance under a doughnut beam
illumination, we have explored efficient methods for generating such a beam.
A near 100% efficient generation of doughnut laser beams from an incident
plane wave is achievable using a liquid crystal cell or a spatial phase
modulator. It is found that the focus field distribution of doughnut beams
focused by a high NA objective is substantially different from that given by
low NA objectives. Depolarisation effects destroy the central zero intensity
points for doughnut beams of topological charge ±1 and ±2. The focused
field distribution of doughnut beams shows an increased ring intensity along
the direction perpendicular to the polarisation direction, while the focus
PhD thesis: Far-field and near-field optical trapping 157
CHAPTER 7. Conclusion
becomes elongated in the polarisation direction. These diffraction effects
are enhanced when doughnut beams are centrally obstructed and affect the
trapping efficiency of such beams. It has been found that the lifting force of
small particles is greatly reduced when focused doughnut beam is used for
trapping, compared to the plane wave, due to the field intensity reduction in
the central focal region of such an illumination. For large particles, the axial
trapping efficiency (ATE) in forward direction is increased when doughnut
beam illumination is used, while the backward ATE is relatively unchanged.
It is shown, both theoretically and experimentally, that the maximal TTE
decreases for reducing particle size and that the rate of decrease is higher for
doughnut beam illumination, compared with the plane wave illumination.
Near-field trapping modality based on particle interaction with evanescent
field generated under the total internal reflection (TIR) has been investigated
in two different configurations: wide area evanescent field and focused
evanescent field. Our research in near-field trapping force exerted on dielectric
particles by a wide area evanescent field shows that the particle is attracted
towards the interface at which the evanescent field is generated and it is
pushed in the forward scattering direction. Both of these attractive and
pushing forces are stronger for the incident P polarised illumination.
The near-field trapping force achieved by focused evanescent field attracts
particles toward the interface at which the field is generated and confines
them in the focal region. The trapping efficiency is slightly larger for the
focused evanescent field generated by a plane wave, compared to the one
generated by a doughnut beam. The maximal ATE, for large dielectric
particles, constitutes∼43% of that achieved by the far-field trapping modality
under the same conditions. Furthermore, it is determined that the highest
ATE does not occur for the geometric focus at the interface, but it occurs for
a slightly defocused evanescent field with the geometrical focus at a depth of
PhD thesis: Far-field and near-field optical trapping 158
CHAPTER 7. Conclusion
∼62 nm. Our research indicates that unlike the far-field trapping modality,
the dependence of the maximal ATE on the particle size is nearly linear for
the near-field trapping with focused evanescent wave illumination. We predict
that an input laser power of ∼10 µW suffice to overcome the gravity force
acting on a polystyrene particle of 800 nm in radius or smaller, enabling the
particle three-dimensional trapping in an upright near-field trapping system.
7.2 Future prospects
The research investigations and methodology described in this thesis can
be further extended to include the following research key areas: trapping
force determination and study of far-field and near-field trapping of metallic
particles, investigations of torque exerted on microparticles for complete near-
field manipulation, including particle rotation, and analysis of the interface
effects on the near-field particle manipulation using a focused evanescent field.
7.2.1 Optical trapping of metallic particles
7.2.1.1 Far-field trapping
For certain experimental purposes it would be more advantageous to use
metallic particles as handle points for optical trapping manipulation. Such
experiments range from a biological cell manipulation using infused metallic
nano-particles to particle-trapped SNOM with metallic particle offering much
higher near-field scattering efficiency.
Our physical methodology described in Chapter 4 is also valid for
considering metallic particle trapping in both far and near fields. Research
into metallic particles trapping efficiency would indicate and determine an
PhD thesis: Far-field and near-field optical trapping 159
CHAPTER 7. Conclusion
optimum particle parameters, such as its size and material, for a particular
optical trapping experiment. Analogous to the MDR effects with dielectric
particles, using our methodology with optical trapping of metallic particles
would also provide a physical insight into the influence of the surface plasmon
effect, associated with small metallic particles, on the optical trapping
process.
7.2.1.2 Near-field trapping
The effects of the interface at which the focused evanescent field is generated
may play a significant role for studying the near-field trapping process with
metallic particles. Due to the relatively high reflectivity of metallic particles
it could be expected that the close proximity of a dielectric interface will
affect the trapping efficiency of such particles.
This issue can be dealt with an extension of our vectorial diffraction model,
presented in Chapter 4, to include the multiple field reflection between a
metallic particle and the interface. The multiple field reflections can be
implemented using the plane wave decompositions approach of Inami and
Kawata [128]. This approach would provide a comprehensive and rigorous
method for characterising the optical near-field trapping force for highly
reflective particles.
7.2.2 Near-field micromanipulation system
On the basis of our research into the near-field trapping process using a fo-
cused evanescent field illumination described in Chapter 6 a fully operational
near-field trapping system for single molecule detection experiments can be
envisioned.
PhD thesis: Far-field and near-field optical trapping 160
CHAPTER 7. Conclusion
(a) (b)
Fig. 7.1: Operational near-field trapping system.
A far-field optical trapping is utilised for reaching deep into the par-
ticle cell and selecting an appropriate particle, possibly a particle with a
biomolecule attached to it. The particle is lifted to the top surface using
a relatively large backward axial trapping force exerted with such a far-
trapping technique (Fig. 7.1(a)). Once the particle is brought to the surface,
the incident illumination is centrally obstructed, applying an appropriate
pattern on a spatial light modulator (SLM) operating in the intensity mode
for example, which switches the optical trapping modality to the near-field
trapping (Fig. 7.1(b)). The particle is then manipulated using the localised
near-field trapping volume.
Another interesting capability of the particle manipulation using near field
trapping with focused evanescent illumination lies in the area of particle
rotation. A particle can be induced to rotate by such a near-field using
two methods. The first method is based on rotating the polarisation state
of the incident beam, which rotates the focused field distribution and the
particle or particles trapped with it. The second method relies on using
a higher charge doughnut beam to rotate the particle. Due to the helical
PhD thesis: Far-field and near-field optical trapping 161
CHAPTER 7. Conclusion
wavefront of doughnut beam illumination, it can be expected that the near-
field distribution of a highly obstructed doughnut beam becomes asymmetric,
which can lead to an interesting force distribution (Fig. 7.2). Such a force
distribution facilitates particle rotation , provided that the trapping power
is sufficient to overcome the friction force due to the viscosity of the particle
suspension medium. Near-field particle rotation, based on highly obstructed
Fig. 7.2: Trapping efficiency mapping in the XY plane for a polystyrene particleof radius a = 1 µm, (light polarisation is in the X direction) across the focusedevanescent field generated at a coverslip interface by a plane wave (Charge 0) anddoughnut beam illumination (Charge 1, 2, and 3). NA=1.65, λ=532 nm, ε=0.85,n1=1.78 and n2=1.33.
PhD thesis: Far-field and near-field optical trapping 162
CHAPTER 7. Conclusion
doughnut beams, is certainly a very rich and broad research field which
provides immense opportunities in the future work. However, due to the
limited candidature time, we have not considered this novel particle rotation
mechanism in greater details.
PhD thesis: Far-field and near-field optical trapping 163
Chapter 7
Conclusion
7.1 Thesis conclusion
The main research work in this thesis presents an optical trapping model
based on two physical processes, vectorial diffraction of incident laser beam by
a high numerical aperture (NA) objective and scattering by a small particle
(Chapter 4). The interaction of the small particle and the focused laser
beam is described using an extension of the classical plane wave Lorentz-
Mie theory, while the force is determined using the Maxwell stress tensor
approach. An important advantage of our approach over other optical
trapping models reviewed in Chapter 2 is that it provides an exact description
of the electromagnetic (EM) field distribution in the focal region of a high
NA objective. As opposed to the other optical trapping models, our model
is capable of considering an arbitrary wavefront incident at the objective
entrance pupil. In other words, our model is capable of including any
apodisation function of the focusing objective, phase modulation of the
incident beam, polarisation effects and the effects of spherical aberration (SA)
usually present in optical trapping experiments. As a consequence we are able
154
CHAPTER 7. Conclusion
to consider the trapping process under doughnut beam illumination, which
represents one class of phase modulated laser beams (Chapter 5). We have
shown that our optical trapping model can deal with the far-field trapping
modality, as well as with the near-field trapping modality implemented with
a focused evanescent field generated by using a central obstruction in the
incident beam path. Our optical trapping model provides a first theoretical
treatment of such a near-field trapping modality (Chapter 6).
In addition to the optical trapping model, this thesis presents a nanometric
sensing model for detection of scattered evanescent waves (Chapter 3).
The sensing model, which describes the conversion of evanescent photons
into propagating photons, is implemented by considering the near-field Mie
scattering process enhanced by morphology dependent resonance (MDR) by
a small particle in the vicinity of a plane interface. The transformation
of the three-dimensional scattered field distribution to the detector plane
is achieved using vectorial diffraction of the scattered signal. The vectorial
diffraction approach for the transformation of the scattered signal is necessary
due to the vectorial nature of the scattered field. Such a sensing model
is essential for understanding optical trapping systems that use evanescent
field illumination for high resolution position monitoring and imaging, such
as optical trap nanometry and particle-trapped scanning near-field optical
microscopy (SNOM).
The nanometric sensing model reveals that the intensity pattern generated
in the far-field region of the collecting objective shows an interference-like
pattern when large dielectric particles are employed as scatterers. For small
particles the intensity pattern is similar to that given by the dipole radiation.
Our research has shown that MDR effects are evident in near-field Mie
scattering of large dielectric particles and that the resonance peaks become
sharper and less separated with increasing the effective refractive index. The
PhD thesis: Far-field and near-field optical trapping 155
CHAPTER 7. Conclusion
MDR effects are present in the image forming field collected by a high NA
objective and it results in different energy distributions in the detection plane
for on and off resonance positions. These different energy distributions outline
the importance of detector selection for systems operating at MDR positions,
because it would be an advantage to operate particle-trapped SNOM and
optical trap nanometry systems at MDR positions to enhance signal-to-noise
ratio. If the detection of the collected scattered signal is performed with a
pinhole detector, the detector size needs to be carefully selected depending
on the scattering particle size. This is because small particles act as dipole
sources and fill the objective entrance pupil uniformly, which requires a
relatively small pinhole. For large particles much larger pinholes are required
to collect the signal completely with a typical imaging lens. This effect is
contributed to larger interaction cross-section and the MDR effect of large
particles, which result in signal spreading in forward direction in the imaging
plane. The theoretical predictions of our model have been experimentally
verified by a good agreement of predicted and measured intensity distribution
of small and large polystyrene particles in the focal plane of the collecting
objective.
Applying our optical trapping model for force determination based on
the vectorial diffraction approach for light focusing under a high convergence
angle we have shown that it is superior to the commonly used Gaussian beam
model in that it exactly represents the field in the focal region of a high NA
objective and that the results obtained by our vectorial diffraction model at
large particle limit are approaching the ray optics (RO) model. The RO model
is an approximate model valid only for very large particles, and our optical
trapping model can bridge the gap between small particle case for which the
focal field distribution is crucial and large particle case for which the RO
model gives approximate solutions. Furthermore, the vectorial diffraction
model can treat arbitrary incident wavefronts and simulate real experimental
PhD thesis: Far-field and near-field optical trapping 156
CHAPTER 7. Conclusion
conditions. Investigations of the SA effects on the far-field trapping force
show that the refractive index mismatch between the objective immersion
and particle suspension media severely affects the trapping performance of an
optical trap. The rate of degradation of the trapping performance depends
on the particle and environment parameters. Our research, however, has
demonstrated that the trapping force evaluated using our optical trapping
model agrees well with the experimental results, for both axial and transverse
trapping forces.
The maximal transverse trapping efficiency (TTE) of a centrally ob-
structed plane wave focused by a high NA objective, in the case of large
dielectric particle decreases rapidly for increasing the obstruction size, unlike
the trend predicted by the RO model. The rate of decrease and the trapping
efficiency for most obstruction sizes is higher for the incident P polarisation,
compared to the incident S polarisation. At very large obstruction sizes
(ε ≈ 0.9), however, the trapping force of the P polarisation approaches that of
the S polarisation, due to the different depolarisation behaviors between the
S and P polarisation states. These findings are confirmed by experimental
measurements.
In order to study the trapping performance under a doughnut beam
illumination, we have explored efficient methods for generating such a beam.
A near 100% efficient generation of doughnut laser beams from an incident
plane wave is achievable using a liquid crystal cell or a spatial phase
modulator. It is found that the focus field distribution of doughnut beams
focused by a high NA objective is substantially different from that given by
low NA objectives. Depolarisation effects destroy the central zero intensity
points for doughnut beams of topological charge ±1 and ±2. The focused
field distribution of doughnut beams shows an increased ring intensity along
the direction perpendicular to the polarisation direction, while the focus
PhD thesis: Far-field and near-field optical trapping 157
CHAPTER 7. Conclusion
becomes elongated in the polarisation direction. These diffraction effects
are enhanced when doughnut beams are centrally obstructed and affect the
trapping efficiency of such beams. It has been found that the lifting force of
small particles is greatly reduced when focused doughnut beam is used for
trapping, compared to the plane wave, due to the field intensity reduction in
the central focal region of such an illumination. For large particles, the axial
trapping efficiency (ATE) in forward direction is increased when doughnut
beam illumination is used, while the backward ATE is relatively unchanged.
It is shown, both theoretically and experimentally, that the maximal TTE
decreases for reducing particle size and that the rate of decrease is higher for
doughnut beam illumination, compared with the plane wave illumination.
Near-field trapping modality based on particle interaction with evanescent
field generated under the total internal reflection (TIR) has been investigated
in two different configurations: wide area evanescent field and focused
evanescent field. Our research in near-field trapping force exerted on dielectric
particles by a wide area evanescent field shows that the particle is attracted
towards the interface at which the evanescent field is generated and it is
pushed in the forward scattering direction. Both of these attractive and
pushing forces are stronger for the incident P polarised illumination.
The near-field trapping force achieved by focused evanescent field attracts
particles toward the interface at which the field is generated and confines
them in the focal region. The trapping efficiency is slightly larger for the
focused evanescent field generated by a plane wave, compared to the one
generated by a doughnut beam. The maximal ATE, for large dielectric
particles, constitutes∼43% of that achieved by the far-field trapping modality
under the same conditions. Furthermore, it is determined that the highest
ATE does not occur for the geometric focus at the interface, but it occurs for
a slightly defocused evanescent field with the geometrical focus at a depth of
PhD thesis: Far-field and near-field optical trapping 158
CHAPTER 7. Conclusion
∼62 nm. Our research indicates that unlike the far-field trapping modality,
the dependence of the maximal ATE on the particle size is nearly linear for
the near-field trapping with focused evanescent wave illumination. We predict
that an input laser power of ∼10 µW suffice to overcome the gravity force
acting on a polystyrene particle of 800 nm in radius or smaller, enabling the
particle three-dimensional trapping in an upright near-field trapping system.
7.2 Future prospects
The research investigations and methodology described in this thesis can
be further extended to include the following research key areas: trapping
force determination and study of far-field and near-field trapping of metallic
particles, investigations of torque exerted on microparticles for complete near-
field manipulation, including particle rotation, and analysis of the interface
effects on the near-field particle manipulation using a focused evanescent field.
7.2.1 Optical trapping of metallic particles
7.2.1.1 Far-field trapping
For certain experimental purposes it would be more advantageous to use
metallic particles as handle points for optical trapping manipulation. Such
experiments range from a biological cell manipulation using infused metallic
nano-particles to particle-trapped SNOM with metallic particle offering much
higher near-field scattering efficiency.
Our physical methodology described in Chapter 4 is also valid for
considering metallic particle trapping in both far and near fields. Research
into metallic particles trapping efficiency would indicate and determine an
PhD thesis: Far-field and near-field optical trapping 159
CHAPTER 7. Conclusion
optimum particle parameters, such as its size and material, for a particular
optical trapping experiment. Analogous to the MDR effects with dielectric
particles, using our methodology with optical trapping of metallic particles
would also provide a physical insight into the influence of the surface plasmon
effect, associated with small metallic particles, on the optical trapping
process.
7.2.1.2 Near-field trapping
The effects of the interface at which the focused evanescent field is generated
may play a significant role for studying the near-field trapping process with
metallic particles. Due to the relatively high reflectivity of metallic particles
it could be expected that the close proximity of a dielectric interface will
affect the trapping efficiency of such particles.
This issue can be dealt with an extension of our vectorial diffraction model,
presented in Chapter 4, to include the multiple field reflection between a
metallic particle and the interface. The multiple field reflections can be
implemented using the plane wave decompositions approach of Inami and
Kawata [128]. This approach would provide a comprehensive and rigorous
method for characterising the optical near-field trapping force for highly
reflective particles.
7.2.2 Near-field micromanipulation system
On the basis of our research into the near-field trapping process using a fo-
cused evanescent field illumination described in Chapter 6 a fully operational
near-field trapping system for single molecule detection experiments can be
envisioned.
PhD thesis: Far-field and near-field optical trapping 160
CHAPTER 7. Conclusion
(a) (b)
Fig. 7.1: Operational near-field trapping system.
A far-field optical trapping is utilised for reaching deep into the par-
ticle cell and selecting an appropriate particle, possibly a particle with a
biomolecule attached to it. The particle is lifted to the top surface using
a relatively large backward axial trapping force exerted with such a far-
trapping technique (Fig. 7.1(a)). Once the particle is brought to the surface,
the incident illumination is centrally obstructed, applying an appropriate
pattern on a spatial light modulator (SLM) operating in the intensity mode
for example, which switches the optical trapping modality to the near-field
trapping (Fig. 7.1(b)). The particle is then manipulated using the localised
near-field trapping volume.
Another interesting capability of the particle manipulation using near field
trapping with focused evanescent illumination lies in the area of particle
rotation. A particle can be induced to rotate by such a near-field using
two methods. The first method is based on rotating the polarisation state
of the incident beam, which rotates the focused field distribution and the
particle or particles trapped with it. The second method relies on using
a higher charge doughnut beam to rotate the particle. Due to the helical
PhD thesis: Far-field and near-field optical trapping 161
CHAPTER 7. Conclusion
wavefront of doughnut beam illumination, it can be expected that the near-
field distribution of a highly obstructed doughnut beam becomes asymmetric,
which can lead to an interesting force distribution (Fig. 7.2). Such a force
distribution facilitates particle rotation , provided that the trapping power
is sufficient to overcome the friction force due to the viscosity of the particle
suspension medium. Near-field particle rotation, based on highly obstructed
Fig. 7.2: Trapping efficiency mapping in the XY plane for a polystyrene particleof radius a = 1 µm, (light polarisation is in the X direction) across the focusedevanescent field generated at a coverslip interface by a plane wave (Charge 0) anddoughnut beam illumination (Charge 1, 2, and 3). NA=1.65, λ=532 nm, ε=0.85,n1=1.78 and n2=1.33.
PhD thesis: Far-field and near-field optical trapping 162
CHAPTER 7. Conclusion
doughnut beams, is certainly a very rich and broad research field which
provides immense opportunities in the future work. However, due to the
limited candidature time, we have not considered this novel particle rotation
mechanism in greater details.
PhD thesis: Far-field and near-field optical trapping 163
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PhD thesis: Far-field and near-field optical trapping 177
Author’s Publications
Journal Articles
Djenan Ganic, Xiaosong Gan, and Min Gu, Reduced effects of spherical
aberration on penetration depth under two-photon excitation, Appl. Opt. 39,
3945-3947, (2000).
Djenan Ganic, Daniel Day, and Min Gu, Multi-level optical data storage in
a photobleaching polymer using two-photon excitation under continuous wave
illumination, Optics and Lasers in Engineering 38, 433-437, (2002).
Djenan Ganic, Xiaosong Gan, and Min Gu, Three-dimensional evanescent
wave scattering by dielectric particles, Optik 113, 135-141, (2002).
Djenan Ganic, Xiaosong Gan, Min Gu, Mathias Hain, Somakanthan Soma-
lingam, Svetomir Stankovic, and Theo Tschudi, Generation of doughnut laser
beams by use of a liquid-crystal cell with a conversion efficiency near 100%,
Opt. Lett. 27, 1351-1353, (2002).
Djenan Ganic, Xiaosong Gan, and Min Gu, Parametric study of three-
dimensional near-field Mie scattering by dielectric particles, Opt. Commun.
216, 1-10, (2003).
178
AUTHOR’S PUBLICATIONS
Djenan Ganic, James W. M. Chon, and Min Gu, Effect of numerical aperture
on the spectral splitting feature near phase singularities of focused waves,
Appl. Phys. Lett. 82, 1527-1528, (2003).
Djenan Ganic, Xiaosong Gan, and Min Gu, Focusing of doughnut laser beams
by a high numerical-aperture objective in free space, Opt. Express 11, 2747-
2752, (2003).
Djenan Ganic, Xiaosong Gan, and Min Gu, Exact radiation trapping force
calculation based on vectorial diffraction theory, Opt. Express 12, 2670-2675,
(2004).
Djenan Ganic, Xiaosong Gan, and Min Gu, Near-field imaging by a micro-
particle: mechanism for conversion of evanescent photons into propagating
photons, Opt. Express 12, 5325-5335, (2004).
Djenan Ganic, Xiaosong Gan, and Min Gu, Trapping force and optical lifting
under focused evanescent wave illumination, Opt. Express 12, 5533-5538,
(2004).
Djenan Ganic, Xiaosong Gan, and Min Gu, Optical trapping force with
annular and doughnut laser beams based on vectorial diffraction, Opt. Express
13, 1260-1265, (2005).
Baohua Jia, Xiaosong Gan, Djenan Ganic, and Min Gu, Rotation of a
microsphere under an anomalous behavior of a focused evanescent Laguerre-
Gaussian beam, Phys. Rev. Lett. (submitted).
PhD thesis: Far-field and near-field optical trapping 179
AUTHOR’S PUBLICATIONS
Conference Papers
Djenan Ganic, Xiaosong Gan, and Min Gu, Near-field Mie scattering in
optical-trap nanometry, Proceedings of SPIE, vol. 4434-25, 158 (2001).
Djenan Ganic, Xiaosong Gan, and Min Gu, Physical model for near-field
scattering and manipulation, Proceedings of SPIE, vol. 5514-82, S13 (2004).
Conference Presentations
Djenan Ganic, Xiaosong Gan, and Min Gu, Mie scattering of evanescent
electromagnetic waves in near-field microscopy, Australian Optical Society
Conference, Adelaide, Australia, December 10-15, 2000.
Djenan Ganic, Xiaosong Gan, and Min Gu, Near-field Mie scattering
in optical-trap nanometry, European Conferences on Biomedical Optics,
Munchen, Germany, June 18-22, 2001.
Djenan Ganic, Xiaosong Gan, and Min Gu, Three-dimensional near-field Mie
scattering by dielectric particles, Multi-dimensional Microscopy Conference,
Melbourne, Australia, November 25-28, 2001.
Min Gu, Xiaosong Gan, James W. M. Chon, Djenan Ganic, and Dru Morrish,
Scanning TIR microscopy: near-field Mie scattering and localized morphology-
dependent resonance, The 3rd Asia-Pacific Workshop on Near-Field Optics,
Melbourne, Australia, November 28-30, 2001.
Xiaosong Gan, Dru Morrish, Djenan Ganic, and Min Gu, Effect of morphology
dependent resonance in near-field scattering under two-photon excitation,
PhD thesis: Far-field and near-field optical trapping 180
AUTHOR’S PUBLICATIONS
International Conference on Confocal and Near-Field Microscopy, Kaohsiung,
Taiwan, April 7-10, 2002.
Min Gu, Djenan Ganic, James W. M. Chon, and Xiaosong Gan, New features
of light focusing by a high numerical-aperture objective, Focus on Microscopy
Conference, Genova, Italy, April 13-16, 2003.
Djenan Ganic, Xiaosong Gan, and Min Gu, Near-field scattering by a
microscopic particle probe: Imaging model, Australasian Conference on optics,
Lasers and Spectroscopy, Melbourne, Australia, December 1-4, 2003.
Djenan Ganic, Xiaosong Gan, and Min Gu, Determination of trapping force
exerted on a microparticle - vectorial diffraction approach, Australian Optical
Society Conference, Canberra, Australia, July 7-8, 2004.
Djenan Ganic, Xiaosong Gan, and Min Gu, Physical model for near-
field scattering and manipulation, International Symposium on Optical
Science and Technology - Optical Trapping and Optical Micromanipulation
Conference, Denver, Colorado, USA, August 2-6, 2004.
Xiaosong Gan, Djenan Ganic, Dru Morrish, Baohua Jia, and Min Gu, Near-
field imaging sensing and manipulation, The 8th International Conference on
Optics Within Life Sciences, Melbourne, Australia, November 28 - December
1, 2004.
PhD thesis: Far-field and near-field optical trapping 181