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Mueller matrix polarimetry of plasmon resonantsilver nano-rods: biomedical prospects

Sayantan Ghosh, Jalpa Soni, Sudipta K. Bera, Ayan Banerjee &Nirmalya Ghosh

Dept. of Physical Sciences,Indian Institute of Science Education and Research Kolkata,

Mohanpur 741 252, West Bengal, India

Internet Lecture

Outline

Introduction to surface plasmon resonances

Theoretical aspects

Quasi Static Approximation(QSA)Beyond QSA: Mie-theory and T-matrix methodScattering matrices and polarization aspectsQuantitative polarimetryMueller matrices: Polar Decomposition

Geometric configuration of the system and methodology

Results

Resonance bands (Qscat)Polarization properties of SPREffect of ambient medium on polarization properties of SPR

Potential applications to biosensing

Conclusion

http://www.iiserkol.ac.in/~nghosh Saratov Fall Meeting 2012

http://www.iiserkol.ac.in/~nghosh

IntroductionSurface plasmon resonances

Surface plasmons are collective oscillations of free elec-trons at the metal-dielectric interface in resonance withthe incident electromagnetic (EM) field. They can prop-agate along metal-dielectric interfaces or be localized inmetal nano-particles (LSPR). They show strongly en-hanced and highly localized electromagnetic fields andhave numerous practical applications like ultra-high sen-sitive chemical and biomedical sensing, contrast enhance-ment in optical imaging, and development of new gener-ation optical devices like plasmonic wave-guiding nano-devices for optical information processing and data stor-age.

In the context of biomedical applications, bio-conjugatednano-particles/structures: changes in optical absorp-tion/scattering properties of nano-particles due tochanges in local dielectric environment obtained in thescattering/absorption spectra can be used as contrastenhancement agent in biomedical imaging with nano-particles: high scattering cross section and control on sizeand shape of nano-particles. In the context of biomedicalapplications, Quantitative plasmon polarimetry can beused for contrast mechanism in nano-particle based imag-ing as they show highly enhanced polarization response ascompared to the dielectric biological tissues. Polarizationproperties like retardance, diattenuation and depolariza-tion can be used to eliminate background Rayliegh/Miescattering from dielectric tissue scattering structures.

x

y

z

Polaroid

~E

Polarized light

Sample

n′g

n′p

Polarized anddephased light

Polaroid



TheoryQuasi Static Approximation

Available approximations and methods for characterization of LSPR in metal nano-particles:Mie theory, discrete dipole approximation (DDA), T-matrix, finite difference time domain(FDTD), finite element (FEM) and the multipole-multipole (MM) methods to name a few.Spheres Quasi static aproximation(QSA): r << λ: acts like a sphere placed in a static electricfield. Dipolar plasmon resonance:

α = 4πa3 ε1 − εmε1 + 2εm

Csca = πa2 8

3x

4

∣∣∣∣ ε1 − εmε1 + 2εm

∣∣∣∣2Cabs = kIm{α}

Condition for dipolar resonance: ε1 = −2εm Contribution of higher multi-poles for larger

particles in the extictions. Quadrupolar polarizability: α =ε1−εm

ε1−(3/2)εmand the quadrupolar

resonance condition: ε1 = −(3/2)εm.

Non spheres

Polarizability:αj = 4πabcε1−εm

3εm+3Lj(ε1−εm)

For spheres: a = b = c = 13

For spheroids: a = b 6= cFor ellipsoids: a 6= b 6= c



TheoryBeyond QSA: Mie theory & T-matrix method

Mie scattering

For EM modes of a sphere of arbitrary sizeBoundary conditions: the tangential components of the E and H are continuous atthe interface of the sphere and the dielectric environment.Solutions take the form of a set of spherical Bessel and Hankel functions.Multi-pole expansion (EM modes) possible: electric dipole, magnetic dipole, electricquadrupole and so on.If d < λ/20, only the electric dipolar term is physical and the Mie theory reducesto the Rayleigh theory.Strong dependence on the εm.For resonance, R{ε} should be negative.

T-matrix method:

The incident and scattered E fieldsare expanded in terms of vectorspherical wave functions.Relation between the input and out-put Stoke’s vectors established bymeans of a transition (T)-matrix:So = S(θ)Si

S11 S12 S13 S14

S21 S22 S23 S24

S31 S32 S33 S34

S41 S42 S43 S44

1

IoQoUoVo

=

S11 S12 S13 S14

S21 S22 S23 S24

S31 S32 S33 S34

S41 S42 S43 S44

Ii

QiUiVi

For spheres, the T-matrix hasa block diagonal form (the el-ements inside the red box arezero)

For non-spheres, the T-matrix isfully populated.



TheoryScattering matrices and polarization aspects

Amplitude scattering matrix - Jones matrix: defined in thescattering plane

(E‖sE⊥s

)=eık(r−z)

−ıkr

(S2 S3

S4 S1

)(E‖iE⊥i

)For sphere, S3, S4 = 0, non-zero otherwise.

For non-spherical axi-symmetric particles (e.g. spheroids &cylinders)

s′

= R−1

(α)s(θ)R(α), R(α) =

(cos(α) sin(α)− sin(α) cos(α)

)α, β are Euler rotation angles.

Preferential orientation: Laboratory frame’s Z-axiscoincides with the symmetry axis of the particle:

(E‖sE⊥s

)=eık(r−z)

−ıkr

(S2 00 S1

)(E‖iE⊥i

)x

y

z

θ

θ

φ

φ

r = θ × φ

1



TheoryScattering matrices and polarization aspects

Scattering Mueller matrix

S(θ) = A · (s⊗ s∗) · A−1

where,

A =

1 0 0 11 0 0 −10 1 1 00 ı −ı 0

Spherical and preferentially oriented particles: Block diagonal structure

A =

S11 S12 0 0S12 S11 0 00 0 S33 S34

0 0 −S34 S44

Relation with the Jone’s matrix: S11 = 1

2

(|S2|2 + |S1|2

), S12 = 1

2

(|S2|2 − |S1|2

),

S33 = Re(S∗2S1) and S34 = Im(S2S∗1 )

Example, Rayleigh scattering:

S(θ) =

cos2 θ+1

2cos2 θ−1

2 0 0cos2 θ−1

2cos2 θ+1

2 0 00 0 cos θ 00 0 0 cos θ



TheoryQuantitative polarimetry

Need for Quantitative Polarimetry

S =

IQUV

=

ExE

∗x + EyE

∗y

ExE∗x − EyE

∗y

ExE∗y − EyE

∗x

ı(ExE∗y − EyE

∗x)

=

IH + IVIH + IVIP − IMIR − IL

The Stokes vector:

Measurement and unique interpretation ofall the polarization parameters is difficult.

Incomplete information contained in theStokes vector S.

The individual polarimetry characteristicsare ‘lumped’.

Depends on the incident Stokes vector(non-unique)

Mueller matrix carries information about the medium and describes the medium’spolarization properties completely.Mueller matrix can be decomposed to give the complete polarization characteristics ofthe medium

Medium polarimetry characteristics

Retardance: Phase shift between two orthogonal polar-izations of light: Linear retardance: δ between H/V andP/M . Circular retardance: δc = 2ψ between L/R. Opti-cal rotation: ψDiattenuation: Differential attenuation of orthogonal po-larizations for both linear and circular polarization states.Depolarization: Reduction of the degree of polariza-tion (DOP) of a 100% polarized (DOP=1) while passingthrough a medium.

S11 S12 S13 S14

S21 S22 S23 S24

S31 S32 S33 S34

S41 S42 S43 S44

1



TheoryMueller Matrix Decomposition

Algorithm

Measured Mueller matrix M

Polar Decomposition

M∆

Depolarization

matrix

MD

Diattenuationmatrix

MR

Retardancematrix

∆L ∆C ∆Linear Circular Net

dL dCLinear Circular

δ ΨLinear Circular

1

Lu an Chipman’s polar decomposition scheme for Muellermatrices

M = M∆ · MR · MD is one ofthe six ways of performing po-lar decomposition of the Muellermatrices.

M∆: Depolarizing matrix (scat-tering)

MR: Retardance matrix (opticalactivity and birefringence)

MD: Diattenuation matrix(dichroic absorption)



TheoryPolar decomposition of Mueller Matrix

Measured Mueller matrix contains all polarization effects.

Decomposition: M = M∆ ·MR ·MD

Extracted polarization parameters:

Diattenuation d = 1M11

√M2

12 +M213 +M2

14

Net depolarization index: ∆ = 1− |tr(M∆−1)|3 .

Linear retardance: δ = cos−1[√{M(R,22) +M(R,33)}2 + {M(R,32) +M(R,23)}2 − 1

],

where, M(R,ij) is the ijth element of MR.

Optical rotation: ψ = tan−1[MR,32−MR,23MR,22+MR,33

]Total retardance: R = cos−1

[tr(MR)

2 − 1]

= cos−1[2 cos2(ψ) cos2

(δ2

)− 1]



MethodolgyGeometric configuration for nano-rods

Equal volume sphere radius: rAspect ratio (ε): d/l; Oblate: ε > 1 andProlate: ε < 1

Preferential orientation: Can be observedin single particle imaging: Dark field, con-focal, NSOM, PS OCT

Orthogonal dipolar polarizabilitiesaligned along the direction of thepolarization vectors of scattered lightpolarized perpendicular and parallel tothe scattering plane

Varying:

Aspect ratio (ε)Orientation angle (β)Scattering angle (θ)Ambient medium refractive index(n)

Z Y

X

1

Z Y

X

1

β = 0◦ β = 90◦

Incident beam along Z direction

Random orientation ⇒ Orientation averaging ⇒ Bulk studies on colloidal suspension.

Scattering matrix decomposition (preferential and random)

Intrinsic polarization propertiesDiattenuation dLinear retardance δDepolarization ∆



ResultsOptical properties of silver nano-rods

Dipolar and quadrupolar resonance in silver nano-rods

Longitudinal dipolar at longer wavelengths.

Transverse dipolar→∼400nm-450nm.

Transverse quadrupolar→∼350nm-400nm

Transverse LongitudinalVarying ε for equal volume

sphere radius of 20nm

Diattenuation effects for preferential orientation

Varying ε for β = 90◦, θ = 45◦

Strong diattenuation effects (d ∼ 0.9) with distinctspectral characteristics

d ∼ 0.33 for similar dielectric particle

Magnitude of diattenuation peaks at the dipolar &quadrupolar resonance bands.

Sharp concavity at the overlap regions of theplasmon bands.

Gradual increase in d with increasing ε.

No depolarization ∆ ∼ 0.



ResultsDiattenuation

Reasons for diattenuation

Preferential orientation:For the longitudinal and transverse dipolar modes⇒

|S2(θ)|2 ∝ |αx|2 cos2(θ)

|S1(θ)|2 ∝ |αy|2

Large enhancement at the two resonance bandscorresponding to resonances of |αx|&|αy| atdifferent wavelengths ⇒ Strong diattenuation

Quadrupolar resonance (350-400nm) ⇒Transversal ⇒ Confirmation S12(λ) ⇒ Differentialexcitation by orthogonal polarization ⇒Significant diattenuation at the QP band.

Equalvolume sphere radius 20nm,

ε = 0.65 for differentorientations β.



ResultsRetardance and diattenuation

Retardance δ effects for preferential orientation

Very strong linear retardance δ(λ) even in forwardscattering angles (θ < 90◦)

Magnitude of δ(λ) peaks around the overlap spec-tral region of the two dipolar and quadrupolarbands

S34, S43 = ±(S∗2S1 − S∗1S2) 6= 0

Non-zero complex value for both S2 & S1 (S2 6= S1)Complex polarizabilities

Longitudinal and the transverse plasmon polarizabilities with inherent phase differencesOrientation effects on Diattenuation and Retardance:



ResultsOrientation effects

Diattenuation: Decreasing β ⇒Decreasing magnitude of d.Relative change in magnitudes at dipolar andquadrupolar resonance bandsDifferential excitation of plasmon modes

Retardance: Decreasing β ⇒ Decreasing magnitudeof δ.

Enhanced depolarization for random orientation

r=20nm, θ = 45◦, varying ε

Magnitude of depolarization ∆ peaks around theoverlap spectral region of the two dipolar plasmonbands ⇒ ∆(λ)

Incoherent addition of diattenuation retarderMueller matrices (from preferentially oriented nano-particles) having random orientation of retardationaxes leads to the Mueller matrix corresponding toa pure depolarizer.



ResultsImplications of quantitative Mueller matrix polarimetry

Quantitative differences in intrinsic polarization parameters of non-spherical metal nano-particles & background tissue (cell) dielectric structuresPolarization as additional contrast mechanism to discriminate against backgroundRayleigh/Mie scattering with optimal choice of wavelength

Distinct spectral variation of polarization parameters may be exploited for sensing

r = 20nm, ε = 0.65θ = 45◦

High diattenuation figure of merit (γ) may be exploited for bio-sensing!!!



Potential applications to biosensing

Obstacles in bio-imaging using LSPR

Scattered light from tissue microscopic dielectric structures swamps nano-particle signa-ture, limiting the detectability of nano-particles in such media.Develop novel schemes to eliminate background Rayleigh/Mie scattering from tissue di-electric structure.

Biomedical applications of LSPRQuantitative plasmon polarimetry for contrast enhancement in nano-particle based imaging.

LSPR in bio-sensing

In-vitro & in-vivo diagnosisContrast enhancement in biomedical ImagingUltra sensitive bio-sensingPhoto-thermal therapyDrug delivery

Role of plasmonic polarimetry:

Can be used as a highly sensitive contrast enhancement tool.Can used in both imaging and spectroscopy for extraction of functional information fromthe background Rayleigh like scattering.Can be used in both elastic and inelastic spectroscopy of biological samples tagged withnano-particles.

Other applications

The geometric dependence of the plasmonic polarimetric characteristics could present such

nano-rods as candidates to study spin orbit coupling.



Conclusions

Enhanced polarization characteristics of nano-rods like depolarization,retardance and diattenuation as compared to similar sized dielectricparticles.

Sharp and highly media sensitive concavity of diattenuation which canbe used as a contrast enhancer in bio-sensing.



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