theoretical comparison of optical traps created by standing wave and single beam · 2007. 3....

Theoretical comparison of optical traps created bystanding wave and single beam

Pavel Zem�aaneka,*, Alexandr Jon�aa�ssa, Petr J�aakla, Jan Je�zzeka,Mojm�ıır �SSer�yyb, Miroslav Li�sskab

a Institute of Scientific Instruments, Academy of Sciences of the Czech Republic, Kr�aalovopolsk�aa 147, Brno 612 64, Czech Republicb Faculty of Mechanical Engineering, Brno University of Technology, Technick�aa 2, Brno 616 69, Czech Republic

Received 4 June 2002; received in revised form 28 February 2003; accepted 3 April 2003

Abstract

We used generalised Lorenz–Mie scattering theory (GLMT) to compare submicron-sized particle optical trapping in

a single focused beam and a standing wave. We focus especially on the study of maximal axial trapping force, minimal

laser power necessary for confinement, axial trap position, and axial trap stiffness in dependency on trapped sphere

radius, refractive index, and Gaussian beam waist size. In the single beam trap (SBT), the range of refractive indices

which enable stable trapping depends strongly on the beam waist size (it grows with decreasing waist). On the contrary

to the SBT, there are certain sphere sizes (non-trapping radii) that disable sphere confinement in standing wave trap

(SWT) for arbitrary value of refractive index. For other sphere radii we show that the SWT enables confinement of high

refractive index particle in wider laser beams and provides axial trap stiffness and maximal axial trapping force at least

by two orders and one order bigger than in SBT, respectively.

� 2003 Elsevier Science B.V. All rights reserved.

PACS: 42.25.F; 42.50.Vk

Keywords: Single beam trap; Optical trapping; Optical tweezers; Standing wave; Mie scattering; Gaussian laser beam

1. Introduction

Since 1986 optical trapping has proved to be an

invaluable method for non-contact manipulation

of nanoobjects and microobjects, measurement of

extremely weak forces, study of single molecule

properties and surface properties [1–4]. The clas-sical set-up of optical tweezers uses a single laser

beam that is tightly focused by an immersion mi-

croscope objective of high numerical aperture [5].

This classical single beam trap (SBT) has been

gradually modified by using various transversal

beam profiles [6,7], interference of co-propagating

laser beams [8], self-aligned dual beam [9], or op-

tical fibres [10]. Multiple optical traps were createdusing several laser beams [11,12], diffractive optics

[13], and time sharing of a single beam [11,14,15].

Optics Communications 220 (2003) 401–412

www.elsevier.com/locate/optcom

* Corresponding author. Tel.: +42-05-4151-4202; fax: +42-05-

4151-4402.

E-mail address: [email protected] (P. Zem�aanek).

0030-4018/03/$ - see front matter � 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0030-4018(03)01409-3

mail to: [email protected]

The common characteristic of all the methods

mentioned above is that the axial force is smaller

than the radial one. This is given by physical limits

imposed upon intensity distribution in the focal

region of lens with limited numerical apertures.

Recently, it has been shown that optical trappingcan be also achieved in a standing wave created by

interference of two counter-propagating coherent

beams [16]. In this case the axial force is stronger

than the radial one because of inhomogeneous

optical intensity distribution in a periodical struc-

ture of standing wave nodes (minimums) and

antinodes (maximums). Therefore, particle con-

finement is achieved even in weakly focused oraberrated beams. An interesting problem, which

arises here, is to study the properties of this type of

optical trap (standing wave trap (SWT)) and

compare them with the classical SBT for different

parameters of a trapped object (size, refractive

index) and a trapping beam (waist size).

2. Theoretical background

Theoretical description of the interaction be-

tween a particle and an electromagnetic wavegenerally consists of description of an incident

beam, description of a field established due to the

interaction of the incident beam and the particle,

and description of forces acting on the particle. A

single beam optical trapping uses a focused laser

beam with a spot diameter comparable to the

trapping wavelength. To achieve this, high-quality

immersion objectives are usually used. In this casethe beam passes through a number of dielectric

interfaces (optical elements inside the objective,

immersion oil layer, coverslip, and water layer)

which makes a correct description of the field in

the focal region quite complex. Despite this fact,

the simple Gaussian beam is still the most fre-

quently considered form of the incident beam even

though that it is only a paraxial solution of thewave equation and it does not treat the polarisa-

tion components in the focused beam properly.

Therefore, there have been several attempts to

‘‘improve’’ the Gaussian beam so that it better

fitted the wave equation for off-axis positions in

the focused beam [17–20]. For this purpose, field

expansion in the beam size parameter s ¼ 1=ðkw0Þwas adopted but its solutions to wave equation are

known explicitly only up to the fifth order in the

parameter s. Moreover, even though the average

error of solution to Maxwell�s equations drops for

bigger s [18], it is still not sufficient for the focusedbeam that is usually used in optical tweezers (s ’0:4, which corresponds to the diffraction limited

focus, gives the average error ’3.34% in the fifth

order correction). A theoretical treatment de-

scribing the incident beam as the Gaussian beam

corrected to the fifth order (CGB) should therefore

deal with wider beam waists w0 (s6 0:2 gives

w0 P 0:8k) to reduce the average and the maximalerror (for s6 0:2 the two errors are lower than

0.062% and 1.19%, respectively [18]).

Exact Gaussian-beam-like solutions of the wave

equation are known in the oblate spheroidal co-

ordinates or using the complex-point source

method [21–23]. However, up to our knowledge,

there is no experimental method that could deter-

mine the components of the optical field distribu-tion below the objective with sufficient precision to

determine which of the above mentioned field de-

scriptions better fit the experimental conditions.

Moreover, it seems that the above mentioned im-

provements of the beam profile are still negligible

compared to the effects caused by diffraction of the

beam on the objective back aperture and by

spherical aberration due to the refractive indexmismatch at the dielectric interfaces below the

objective [24].

Description of the field produced as the result

of the interaction of the incident beam and the

particle is a general problem studied by scattering

theory. An original description of such a field,

presented by Mie and Lorenz at the beginning of

the 20th century, assumed a spherical particleilluminated by a plane wave (Lorenz–Mie the-

ory). This theory has been gradually modified so

that it can be applied for spheres, spheroids,

multilayered spheres, infinite cylinders with cir-

cular and elliptical cross-sections, sphere with one

internal eccentrically located spherical inclusion

and aggregates particle placed into an arbitrary

field distribution [18,25–28]. It is commonly re-ferred to as the generalised Lorenz–Mie theory

(GLMT).

402 P. Zem�aanek et al. / Optics Communications 220 (2003) 401–412

Description of a force interaction between the

beam and a particle is based on the momentum

conservation principle (forces acting on any finite

volume in a material body can be expressed

through the forces applied to the surface of that

volume) and on assumptions that a fluid sur-rounding the particle is isotropic, non-magnetic,

linear in its response to the applied field, and in

hydrodynamic equilibrium. In this case the time-

averaged total electromagnetic force acting on the

particle is equal to integral of the dot product of

the outward-directed unit normal vector n and the

Maxwell�s stress tensor in Minkowski form TM

over the surface S enclosing the particle [29,30]:

hFii ¼IS

Xj

T Mij nj da

* +; ð1Þ

TMij ¼

�eEiEj þ l0HiHj �

1

2ðeE2 þ l0H

2Þdij

�; ð2Þ

where e is the permittivity of a surrounding medium,l0 is the permeability of the vacuum, Ei and Hi are

components of the vectors of electric and magnetic

intensities, respectively. The components of the

optical force in cartesian co-ordinate system can be

obtained after insertion of the total outer field

components into Eq. (1) (i.e., the sum of the incident

and scattered field obtained by the GLMT).

Theoretical methods for calculation of the opti-cal forces acting on a sphere placed into a single

focused Gaussiam beam corrected to the fifth order

and to the higher orders have been presented by

Barton et al. [29] and by Ren et al. [31], respectivelly.

For our study we have chosen the Barton�s ap-

proach applied on the CGB [25,29,32] with the

Gaussian beam waist wider than 0:75k. Under this

condition, the GLMT provides reasonable basis fora theoretical comparison of the SBT and SWT. We

assume that the standing wave is created as the in-

terference of the incident and retroreflected

Gaussian beams. If a sphere is inserted into such a

field a complex scattering event occurs (see Eq.

(A.1) in Appendix A). To simplify the calculation

we omit multiply scattered fields between the re-

flective surface and the sphere (see Eq. (A.2) inAppendix A). In this case the incident beam is equal

to the standing wave created by the interference of

two independent counter-propagating focused laser

beams. Therefore, we can easily adapt the above

mentioned GLMT formalism and instead of a single

CGB we sum field components of two counter-

propagating CGB to get the initial field components

of the standing wave. To speed up the calculation,we assume that the spherical object is located on the

beam axis and we can thus employ the radial sym-

metry of the problem [32]. We also assume that in

the case of the SWT, its beam waist is placed on the

reflective surface and, therefore, the beam is re-

flected as a plane wave with equal reflectivity all

over the beam width. Although the adopted sim-

plifications (CGB, absence of spherical aberrationsand diffraction) could seem drastic, this model at

least provides correct qualitative description of the

behaviour of dielectric spheres in the SBT and SWT

[33]. We wrote the modified code for the force

evaluation ourselves but we do not present a de-

tailed mathematical description, because the meth-

od is well described in the literature [18,25,29,32].

3. Numerical study of optical trap properties

We are primarily interested in the study of the

optical trap properties for dielectric particles

smaller than the trapping wavelength which are

placed into the single focused Gaussian beam and

into the Gaussian standing wave (GSW). Figs. 1and 2 illustrate the differences between intensities,

axial forces, and trapping potentials of both types

of traps. It is assumed that the direction of the z-

axis follows the propagation of the reflected beam

and it is antiparallel to the direction of the incident

beam. Therefore, positive axial force acts against

the incident beam propagation. These plots also

define the maximal axial trapping force Fmax, axialtrap stiffness j, and potential depth of the trap

DW . In the following sections, we will study how

the parameters of both traps depend on the par-

ticle size, refractive index, and beam waist size.

3.1. Dependence of the axial force on particle size

and position in the GSW

An example of the GLMT calculation giving

the dependence of the GSW axial force on the

P. Zem�aanek et al. / Optics Communications 220 (2003) 401–412 403

sphere radius and its position in the GSW is pre-

sented in Fig. 3 in the form of a contour plot.

Force changes with the axial position and radius of

the sphere are clearly seen. Zero value of the forceand negative slope of the force with respect to the

z-co-ordinate determine a stable equilibrium po-

sition of the sphere, which is called the trap posi-

tion. Several trap positions can be found along the

optical axis for certain particle sizes. On the other

hand, there are particular radii of the sphere

(‘‘non-trapping radii’’) for which the axial force is

close to zero regardless of the position of thesphere in the GSW and, obviously, these particles

cannot be optically confined. This is caused by a

competition of the gradient forces coming from

the neighbour GSW antinodes, which pull the

sphere in opposite directions and thus cancel each

other [34]. Consequently, the weaker gradient

force caused by the focused beam envelope domi-

Fig. 2. Illustration of the SBT intensity distribution, axial in-

tensity profile, axial force profile, and potential energy profile.

The parameters used for the comparison are defined: F SBTmax –

maximal trapping force; jSBT – trap stiffness, DW SBT – potential

depth of the trap. Parameters were: beam waist w0 ¼ 0:75k;

relative refractive index m ¼ 1:19; incoming beam power P ¼ 1

W; wavelength of the beam in vacuum kvac ¼ 1064 nm.

Fig. 3. Dependence of the axial trapping force on the particle

radius a and the position z of the sphere in the GSW. Param-

eters were: w0 ¼ 1k, m ¼ 1:19, R ¼ 1, w ¼ 3p=2, P ¼ 1 W,

kvac ¼ 1064 nm, beam waist is placed on the surface.

Fig. 1. Illustration of the SWT intensity distribution, axial in-

tensity profile, axial force profile, and potential energy profile.

The parameters used for the comparison are defined: F SWTmax –

maximal trapping force; jSWT – trap stiffness, DW SWT – energy

depth of the trap. Parameters were: beam waist w0 ¼ 0:75k;

relative refractive index m ¼ 1:19; surface reflectivity R ¼ 1;

phase shift of the reflected wave w ¼ 3p=2; incoming beam

power P ¼ 1 W; wavelength of the beam in vacuum kvac ¼ 1064

nm; beam waist is placed on the surface.


nates and accelerates the sphere towards the beam

waist placed on the reflective surface–mirror. It is

also seen that very small particles are confined

near the intensity maximums, but particles bigger

than about 0:32k are confined near the intensity

minimums. This means that the optical traps areshifted from the neighbourhood of the GSW ant-

inodes to the nodes or vice versa [34].

3.2. Comparative study of the maximal trapping

force

Let us define the maximal trapping force in the

GSW as the maximal optical force directed along

the z-axis (against the incident beam direction) for

the separation between the sphere surface and the

mirror smaller than k:

F SWTmax ¼ max F SWT

GLMTða�

6 z6 aþ kÞ�: ð3Þ

The sphere surface just touches the mirror at the

initial position. Let us define the maximal trappingforce in the SBT as:

F SBTmax ¼ max F SBT

GLMTð�

�16 z61Þ�: ð4Þ

The contour plots of the maximal trapping force

values (see Fig. 4) have been chosen as the most

comprehensive way to show the principal force

dependencies on the particle size a, relative re-

fractive index m, and beam waist size w0. The

contour labels represent levels of constant forces in

pN, the shaded regions correspond to the negative

maximal trapping force (non-trapping regions).For these parametric configurations the particle is

accelerated in the incident beam direction without

a chance of its confinement. In the GSW the width

of the non-trapping regions increases with particle

size. It is also seen that the shaded regions are

getting narrower with increasing beam waist since

the gradient of the intensity of the Gaussian en-

velope becomes smaller. The dot-and-dashed (da-shed) curves link the values of parameters m and a,

for which the maximal trapping force along the z-

axis reaches its maximum (minimum) with respect

to the sphere radius for considered value of w0.

It is interesting to compare maximal axial

trapping forces in the SWT (see Fig. 4) and in the

SBT (see Fig. 5) for the same particle sizes, relative

refractive indices, and beam waists. It is seen im-

mediately that the wider is the beam waist, the

narrower is the region of relative refractive indices

where the particle can be confined using SBT and

this region reduces to the unity value of the relative

refractive index for bigger particles. This means

Fig. 4. Contour plot of the maximal SWT axial trapping force

(in pN) with respect to the z-axis as a function of the relative

refractive index m, particle radius a, and beam waist radius w0

calculated using the GLMT. Dot-and-dashed and dashed lines

represent the curves of maximum and minimum values of

maximal trapping forces with respect to the particle size and

relative refractive index, respectively. Dotted lines denote the

borders of the trapping region, the shaded areas represent the

non-trapping regions. The following parameters were used:

P ¼ 1 W, R ¼ 1, w ¼ 3p=2, kvac ¼ 1064 nm, beam waist is

placed on the surface.


that using wider beams, only those objects can be

trapped by SBT which have the refractive index

close to the value of surrounding immersion me-

dium. If the values of maximal trapping forces for

bigger particles are compared, one finds that the

SWT provides at least one order of magnitude

stronger trapping forces. For smaller particles

and wider beam waists (outside the non-trapping

regions) this disproportion becomes even more

pronounced.

3.3. Comparative study of minimal trapping power

Due to the competition of the scattering and

gradient forces, the total axial force is the mostlimiting factor of the optical confinement. The

stability of the trap along the optical axis can be

inferred from the axial profile of the potential en-

ergy. As on-axis sphere position is assumed here,

the change of the potential energy DWGLMT due to

the motion of the sphere in the field from the

starting position z1 to the end position z2 is given by:

DWGLMT ¼ �Z z2

z1

FGLMTðzÞdz: ð5Þ

Here, FGLMTðzÞ is the axial optical force calculated

by the GLMT and acting on the sphere placed at z.The integration of the GLMT force along the z-axis gives potential energy profile whose minimum

represents the equilibrium position of the sphere in

the trap.The knowledge of the trap depth can be em-

ployed to obtain minimal trapping power that is

necessary for axial confinement of a particle. It is

generally accepted that if the trap is deeper than

10kBT , the particle can be considered confined. We

defined the trap depth as the difference between the

potential energy of the lowest trap edge and the

trap bottom (see Figs. 1 and 2). The power thatprovides the trap depth equal to 10kBT will be

called the minimal trapping power. The minimal

trapping power was calculated as a function of

sphere radius a, relative refractive index m, and

beam waist size w0 (see Figs. 6 and 7). The contour

line labels give the minimal trapping power in mW.

The shaded areas correspond again to the ‘‘non-

trapping’’ configuration of parameters. The plotsreveal that the minimal trapping power in the

SWT can be smaller than 1 mW even for

w0 ¼ 1:25k if optimal sphere size and slightly

higher relative refractive index are chosen. In

contrast, much higher trapping power must be

used to confine smaller spheres or spheres close to

Fig. 5. Contour plot of maximal SBT axial trapping force (in

pN) with respect to the z-axis as a function of the relative re-

fractive index m, particle radius a, and beam waist radius w0

calculated using the GLMT. Dot-and-dashed lines represent the

curves of maximal value of the maximal trapping forces with

respect to the particle size and relative refractive index. The

dotted lines denote the borders of the trapping regions and the

shaded areas represent the non-trapping regions. The inserted

subplots in the last two plots magnify the regions of interest

with the same horizontal scale. The following parameters were

used: P ¼ 1 W, kvac ¼ 1064 nm.


the non-trapping radii. Quantitative comparison

of the SBT and SWT shows that for the samerelative refractive indices, the SWT needs lower

trapping powers for spheres smaller than about

one half of the trapping wavelength. Bigger

spheres are influenced by neighbouring antinodes

which tend to pull the sphere to opposite sides and

decrease the total axial trapping force. Thus, the

trap gets shallower. It is also seen that the differ-

ences between the minimal trapping power needed

in the SWT and in the SBT are not as big as the

differences in maximal trapping forces. This is

caused by the fact that – unlike the trapping force

Fig. 7. Contour lines of the minimal trapping power (in mW) in

the SBT as a function of the relative refractive index m, particle

radius a, and beam waist size w0 for kvac ¼ 1064 nm. Dot-and-

dashed lines represent the curves of maximal value of the


relative refractive index. Dotted lines denote the borders of the

trapping regions and the shaded areas represent the non-trap-

ping regions. The inserted subplots in the last two plots magnify

the regions of interest with the same horizontal scale.

Fig. 6. Contour lines of the minimal trapping power (in mW) in

the SWT as a function of the relative refractive index m, particle

radius a, and beam waist size w0 for R ¼ 1, w ¼ 3p=2,

kvac ¼ 1064 nm, and beam waist placed on the surface. Dot-

and-dashed lines represent the curves of maximal value of the


relative refractive index. Dotted lines denote the borders of the

trapping regions, the shaded areas represent the non-trapping

regions.


– the trap depth does not depend on the intensity

gradient but rather on the difference between the

maximal and minimal intensity, which is compa-rable in both cases.

The knowledge of the minimal trapping power

for given sphere size, refractive index, and beam

waist enables estimation of the maximal trapping

force for this power. This is the weakest maximal

trapping force that can confine this particular

sphere. In the case of the SWT this force lies be-

tween 0.31 and 0.44 pN for all studied sphere radii,refractive indices, and beam waists except the

narrow regions close to the non-trapping radii

boundaries. Even though this value is very small, it

is still significantly higher (more than 10 times for

particles smaller than the trapping wavelength)

than the sum of gravity and buoyancy forces if

glass or polystyrene spheres are considered (see

Fig. 8). However, this conclusion is not generallyvalid if the SBT is analysed. In this case, the values

of the smallest maximal axial force lie in the range

from 0.01 to 0.07 pN and for glass spheres with

radius bigger than about half of the trapping

wavelength this force becomes comparable to the

sum of gravity and buoyancy forces.

3.4. Position of the trap in the GSW

We showed in the previous section that the

maximal SWT trapping force is at least 10 times

higher than the total force done by gravity and

buoyancy if the minimal trapping power is used.

We can therefore conclude that these forces, with

respect to the high value of the axial trap stiffness

in the SWT (see Section 3.5), do not influence

significantly the axial trap position. Since the

minimal trapping power is lower than the power

that is usually used for trapping, position of the

trap can be considered as a point where FGLMTðzÞ is

equal to zero and has a negative slope (minimum

in the potential energy profile). Using this as-sumption, we can analyse the trap position gen-

erally without a precise specification of the trapped

object material. On the contrary, this conclusion is

not generally valid if the SBT is used with low

trapping power; in this case the equilibrium posi-

tion is shifted in the direction of the gravity.

The results of the SWT equilibrium position

analysis are shown in Fig. 9 for very small and bigrefractive indices as a function of the sphere size.

For the analysis we assumed that the distance

between the sphere surface and the reflective sur-

face is smaller than one trapping wavelength.

Therefore, only two trap positions nearest to the

reflective surface could be shown here because the

trap separation is close to k=2 (the minor differ-

Fig. 8. The sum of gravity and buoyancy forces (Fgrav, Fbuoy) (in

pN) as a function of the particle radius a (in kvac ¼ 1064 nm) for

polystyrene (full line) and glass (dashed line) microspheres in

water.

Fig. 9. Quantitative comparison of the trap positions above the

mirror in the GSW for two beam waist sizes, two values of

refractive index, and for the separation between the sphere

surface and the mirror smaller than k. The beam waist is placed

on the mirror, R ¼ 1, w ¼ 3p=2, kvac ¼ 1064 nm. The dashed

lines denote the position of the trapped sphere centre where it

touches the mirror.


ences are caused by the Guoy phase shift). The size

of the trapped sphere limits the distance between

the trap and the reflective surface (dashed line in

Fig. 9, because the trap cannot be closer to the

reflective surface than the sphere radius. This is the

reason of abrupt breaks in the trap position curvesnear to the reflective surface, which are caused by

contact of the sphere with the reflective surface.

The absolute position of the trap array is strongly

affected by phase shift of the reflected wave. If the

sphere size crosses the non-trapping region, the

trap position is changed approximately by k=4. It

is immediately seen from Fig. 9 that the value of

the relative refractive index does not influencenoticeably the position of the existing optical trap

but it only influences the width of the region of the

sphere sizes where the trap exists. Trap positions

for spheres with parameters close to the non-

trapping region (see Fig. 4) could be shifted in the

direction of gravity for smaller powers because the

trapping force could be smaller than the sum of

gravity and buoyancy forces.

3.5. Trap stiffness

The knowledge of the trap position enables es-timation of the trap stiffness. It is assumed that for

small deviations of the trapped particle from its

equilibrium position the restoring force is linearly

dependent on this deviation and the proportion-

ality constant defines the trap stiffness (see Figs. 1

and 2). The stiffness was obtained as the derivative

of FGLMTðzÞ with respect to the z co-ordinate at the

trap position. The results for the SWT are sum-marised in Fig. 10. The contour labels represent

the stiffness values in nN/lm. We consider only

those trap positions that form continuous curve

which is not broken by the contact of the sphere

with the surface and by the end of the sphere po-

sition range where the calculation was performed

(see Fig. 9). Non-trapping regions are shaded,

except the upper left corner where the trap posi-tions were not analysed.

Using the same procedure, the trap stiffness in

the SBT was calculated. These results, presented in

Fig. 11, are valid only for such sphere radii and

refractive indices where the maximal trapping

force (see Fig. 5) is significantly higher that the

total force produced by gravity and buoyancy (see

Fig. 8). Having this in mind, the comparison of

Figs. 10 and 11 reveals that the stiffness produced

by the SWT is at least by two orders of magnitude

bigger (the units used for the contour labels of the

SBT stiffness in Fig. 11 are in pN/lm but those for

Fig. 10. Contour lines of the SWT stiffness (in nN/lm) for

beam waist placed on the surface, P ¼ 1 W, R ¼ 1, w ¼ 3p=2,

kvac ¼ 1064 nm. Dot-and-dashed lines represent the curves of

maximal value of the maximal trapping forces with respect to

the particle size and relative refractive index. Dotted lines de-

note the borders of the trapping regions and the shaded areas

represent the non-trapping regions.


the SWT in nN/lm). Therefore, the particles in the

SWT are more precisely confined axially.

If similar calculation is repeated for the minimal

trapping power, instead of 1 W, trap stiffness jSWT

is in the range 4.3 pN/lm < jSWT < 10:5 pN=lm

for w0 ¼ 0:75k and 5 pN=lm< jSWT < 5:2 pN=lm

for w0 ¼ 2:5k, where the lower values are valid for

smaller spheres and the bigger ones for bigger

spheres. Only a weak dependence of the stiffness

on the relative refractive index was found. The

value of the stiffness is almost constant over the

trapping regions for small sphere sizes. In the case

of the SBT similar procedure would give inaccu-rate results because the trap position is shifted in

the direction of the beam propagation due to the

gravity force. Therefore, the equilibrium position

is influenced by one more parameter – density of

the microsphere and the results cannot be gener-

alised in the way used above.

4. Conclusion

In this paper the trapping properties of a re-

flection-generated GSW and a single beam werestudied theoretically using the GLMT. The

GLMT was employed to calculate the axial optical

forces acting on dielectrics spheres smaller than the

trapping wavelength and made of different dielec-

trics with the relative refractive index in the im-

mersion (water) smaller than 1.21. In the SWT, the

beam waist was placed on 100% reflective surface

and the sphere-surface distance was smaller thanone trapping wavelength. It was proved that par-

ticles can be trapped either in the GSW antinodes

(intensity maximums) or nodes (intensity mini-

mums) depending on their size and refractive in-

dex. Spheres smaller than about 0:3k are trapped

in the antinodes, spheres bigger than about 0:3kbut smaller than about 0:55k in the nodes, etc. This

can be explained by a general tendency of thesphere with relative refractive index higher than

unity to compass the high intensity region as much

as possible. When the size of such object covers

two standing wave intensity maximums, this can

only be fulfilled if the sphere centre is placed be-

tween these intensity maximums, i.e., in the GSW

node. For some sphere radii, the optical trapping

in the GSW is disabled. The precise value of theboundary non-trapping sphere radius decreases

with increasing refractive index of the trapped

sphere.

The maximal trapping force, minimal trapping

power, trap position, and trap stiffness of the

Gaussian SWT have been studied as a function of

Fig. 11. Contour lines of the SBT stiffness (in pN/lm) for P ¼ 1

W, kvac ¼ 1064 nm. Dot-and-dashed lines represent the curves

of maximal value of the maximal trapping forces with respect to

the particle size and relative refractive index. Dotted lines de-

note the borders of the trapping regions and the shaded areas

represent the non-trapping regions. The inserted subplots in the

last two plots magnify the regions of interest with the same

horizontal scale.


the beam waist size, object size and refractive in-

dex and have been compared to the SBT param-

eters. It was found that the maximal SWT

trapping force is at least by one order of magni-

tude stronger for spheres comparable with the

trapping wavelength. For smaller spheres (outsidethe non-trapping regions) and wider beam waists,

this disproportion still increases. On the contrary

to the SBT, the SWT enables confinement of high

refractive index particles even for wider beam

waist. Comparison of minimal trapping power

revealed that a stable SWT could be obtained with

much lower trapping power than a SBT for

spheres smaller than about one half of the trappingwavelength. For bigger particles influenced by

neighbour antinodes, which tend to pull the par-

ticle to the opposite sides, the GSW effect becomes

less visible. The differences between minimal

trapping power needed in the SWT and SBT are

not as big as differences in maximal trapping forces

because of much larger axial extent of the SBT.

However, the maximal SWT optical force inducedby the minimal trapping power is still 10 times

bigger than the total force produced by the gravity

and buoyancy force. Therefore, the SWT axial

position can be considered as unaffected by these

forces, which is not the case if the SBT with min-

imal trapping power is used. Axial SWT stiffness is

at least by two orders of magnitude bigger than in

the SBT and provides much more precise axialparticle confinement.

To summarise, we proved theoretically that the

SWT enables dielectric sphere confinement in a

wider range of sphere refractive indices and beam

waists compared to the SBT. Apart from small

regions of the sphere sizes and refractive indices,

where the GSW effect is cancelled, the force, stiff-

ness, and depth of the SWT for comparable pa-rameters of the sphere and the laser beam are

generally bigger than corresponding properties of

the SBT.

Acknowledgements

The authors thank the Grant Agency of the

Czech Republic for the financial support (Grant

No: 101/00/0974).

Appendix A. Multiple scattering between the sphere

and reflective surface

Let us assume, that the GSW is created by in-

terference of an incident wave and a wave reflectedat a dielectric surface [35]. An object placed into

such a GSW scatters the field incident upon it. Due

to the presence of the dielectric surface, this scat-

tered field is retro-reflected towards the object

where it is scattered again and the process is then

repeated. These multiple reflections and scattering

make the description of the total field much more

complex. Let us divide the process into two paths.Along the first one incident field Ei is scattered by

the object giving rise to scattered field SEi, where S

is a general scattering operator. Field SEi is re-

flected by the surface (RSEi, R is a general reflec-

tion operator) and then it is scattered again by the

object (SRSEi). This process goes on in a similar

way. Along the second path the incident field is not

scattered directly by the object but it is first re-flected by the surface (REi) and then scattered

(SREi) and reflected (RSREi), etc. The resulting

field is done by the superposition of all these par-

tial fields:

E ¼ Ei þ REi þ SEi þ SREi þ RSEi þ : ðA:1ÞThe partial fields due to the multiple scattering are

divergent and can be omitted if the particle is not

in the close vicinity of the surface. The total field

outside the particle then takes the form:

E ’ Ei þ REi þ SðEi þ REiÞ: ðA:2ÞThis result can be understood as the superposition

of two fields. The first one is the standing wave

created by incident and reflected waves and the

second one arises due to the scattering of thisstanding wave by the particle.

References

[1] A. Ashkin, IEEE J. Sel. Top. Quantum Electron. 6 (2000)

841.

[2] K.-O. Greulich, Micromanipulation by Light in Biology

and Medicine, Birkhauser, Basel/Boston/Berlin, 1999.

[3] M.P. Sheetz, L. Wilson, P. Matsudaira (Eds.), Laser

Tweezers in Cell Biology, vol. 55 of Methods in Cell

Biology, Academic Press, San Diego, 1998.

[4] A.D. Mehta et al., Science 283 (1999) 1689.


[5] A. Ashkin, J.M. Dziedzic, J.E. Bj€oorkholm, S. Chu, Opt.

Lett. 11 (1986) 288.

[6] S. Sato, M. Ishigure, Electron. Lett. 27 (1991) 1831.

[7] K.T. Gahagan, J.G.A. Swartzlander, J. Opt. Soc. Am. B 16

(1999) 533.

[8] A.E. Chiou et al., Opt. Commun. 133 (1997) 7.

[9] W. Wang, A.E. Chiou, G.J. Sonek, M.W. Berns, J. Opt.

Soc. Am. B 14 (1997) 697.

[10] A. Constable, J. Kim, Opt. Lett. 18 (1993) 1867.

[11] K. Visscher, S.P. Gross, S.M. Block, IEEEJ. Sel. Top.

Quantum Electron. 2 (1996) 1066.

[12] E. F€aallman, O. Axner, Appl. Opt. 36 (1997) 2107.

[13] E.R. Dufresne, D.G. Grier, Rev. Sci. Instrum. 69 (1998)

1974.

[14] K. Sasaki et al., Opt. Lett. 16 (1991) 1463.

[15] C. Mio, T. Gong, A. Terray, D.W.M. Marr, Rev. Sci.

Instrum. 71 (2000) 2196.

[16] P. Zem�aanek, A. Jon�aa�ss, L. �SSr�aamek, M. Li�sska, Opt. Lett. 24

(1999) 1448.

[17] L. Davis, Phys. Rev. A 19 (1979) 1177.

[18] J.P. Barton, D.R. Alexander, J. Appl. Phys. 66 (1989) 2800.

[19] W.L. Erikson, S. Singh, Phys. Rev. E 49 (1994) 5778.

[20] G. Gouesbet, J. Opt. 27 (1996) 35.

[21] M. Couture, P.-A. Belanger, Phys. Rev. A 24 (1981) 355.

[22] G.A. Deschamps, Electron. Lett. 7 (1971) 684.

[23] B. Landesman, H.H. Barrett, J. Opt. Soc. Am. A 5 (1988)

1610.

[24] A. Rohrbach, E.H.K. Stelzer, J. Opt. Soc. Am. A 18 (2001)

839.

[25] J.P. Barton, D.R. Alexander, S.A. Schaub, J. Appl. Phys.

64 (1988) 1632.

[26] G. Gouesbet, B. Maheu, G. Gr�eehan, J. Opt. Soc. Am. A 5

(1988) 1427.

[27] B. Maheu, G. Gouesbet, G. Gr�eehan, J. Opt. 19 (1988) 59.

[28] G. Gouesbet, G. Grehan, Atom Sprays 10 (2000) 277.

[29] J.P. Barton, D.R. Alexander, S.A. Schaub, J. Appl. Phys.

66 (1989) 4594.

[30] F.N.H. Robinson, Phys. Rep. 16 (1975) 313.

[31] K.F. Ren, G. Grehan, G. Gouesbet, Appl. Opt. 35 (1996)

2702.

[32] S.A. Schaub, J.P. Barton, D.R. Alexander, Appl. Phys.

Lett. 55 (1989) 2709.

[33] A. Jon�aa�ss, P. Zem�aanek, E.L. Florin, Opt. Lett. 26 (2001)

1466.

[34] P. Zem�aanek, A. Jon�aa�ss, M. Li�sska, J. Opt. Soc. Am. A 19

(2002) 1025.

[35] P. Zem�aanek, A. Jon�aa�ss, L. �SSr�aamek, M. Li�sska, Opt.

Commun. 151 (1998) 273.


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