INTEGRAL EQUATION FORMULATION OF

ELECTROMAGNETIC SCATTERING FROM SMALL PARTICLES

Tam Ho Yin

Background Nanophotonic – the optics for

nanoparticles (1 to 100 nm) Key: surface plasmonic resonance

Discovery of Surface Plasmon

• Anomalous spectrum by Wood (1902)– Could not be explained by old diffraction

theory• Partial explanation by Rayleigh (1907)• Detailed explanation by Fano (1940)• Ritchie predicted plasmons: collective

oscillation of electron (1950)

Localized surface plasmon resonance (LSPR)

Localized surface plasmon: Surface plasmons excited in metallic nanoparticles

Observations at resonance Strong absorption and scattering Strong enhancement near the particles (“Hot

spots”)

Conditions of LSPR Resonance condition

If we apply a field E0 to a sphere with radius , the internal field is

If is real, when

- Polarization:

gets very large strong external field just outside the surfaceWe measure the resonance by

Materials for LSPR Why metal?

Negative real part of the dielectric function Usually Gold (Au) and Silver (Ag)

Why? Resonance is in visible range.

Properties of LSPR Shape dependence

Triangles

Curved Triangle

disc

rod

Properties of LSPR Material dependence

Properties of LSPR Size dependence

Spheroid, aspect ratio = 2

Size parameter L: radius of equal-volume sphere

Applications of LSPR Strong absorption and enhancement of

field leads to some applications, e.g., 1. Light trapping on solar cells2. Photothermal therapy

Plasmonic soler cell Light trapped within the waveguide by

nanoparticle

[1] H. A. Atwater and A. Polman, "Plasmonics For Improved Photovoltaic Devices," Nature Mater. 9, 865 (2010).[2] M. A. Green and S. Pillai, "Harnessing Plasmonics For Solar Cells," Nature Photon. 6, 130 (2012).

Photothermal therapy Kill cancer cells by heat

Theoretical Methods for LSPR

Differential Equations Frequency domain Time domain (FDTD)

Integral Equations Exact methods

T-Matrix Approximation

Analytic Approximation (AA) (assume constant internal field)

Differential Equation vs Integral Equation

Problems of differential equation Scalar wave equation:

Boundary condition is needed for Particle surface At infinity (Difficult!)

Discretized equation: connects one points with the neighboring points

Internal field External near field External far field

Complicated!

Simple

Why integral equation?

Why integral equation?

No Infinity boundary condition to be imposed numerically

Bypassing the external near field

Implicit outgoing boundary condition

Goals1. Introduce integral equation2. Evaluate the features of scattering by

nanoparticles numerically by T-Matrix3. Indicate the problems of FDTD4. Develop AA

1. Check the validity2. Investigate the shape dependence

1. Integral Equation

Integral Equation for Scalar Field

Scalar wave equation:

The integral equation solution for homogeneous particle:

where

(Reduces to solution to Laplace Equation when k 0)

Integral Equation for Scalar Field

“leap-frog properties

Internal field is unknown! Two methods:

T-Matrix method AA

Need internal field

Integral Equation for Scalar Field

Integral Equation for Vector Field

Vector wave equation:

The integral equation solution is:Need internal field

2. T-Matrix Method

T-Matrix Make use of the integral equation.

Goal: obtain the internal field as an expansion of spherical Bessel functions and spherical harmonics , where

T-Matrix for scalar field

Compare the and on both sides:

where

From

T-Matrix for vector field Formalism (Vector)

For the integral equation solution:

We obtain

Results Apply to different particles• measure and the

internal field• Size of particle L: the radius of equal-

volume sphere

Results Size dependence (aspect ratio=2)

The size dependence is weak (L<1 nm does not shift) consider small particle case only for a qualitative

understanding of LSPR

Results Aspect ratio dependence

Comparison

E0

k E0

k

when aspect ratio increases:• Marked red-shift

in resonance peak position

• Drastic increases of the peak value.

when aspect ratio increases:• No shift in

resonance peak position

• Slight decreases of the peak value.

vs

Higher sensitivity vs. aspect ratio!

Results Internal field (2 nm : 1 nm particles)

E0

kE0

k (AA not accurate)

ConstantRapidly varying

Problems of T-Matrix method Complicated computation Coupled shape and frequency dependence

Separate by AA!

3. Finite-Difference-Time-Domain Method

(FDTD)

FDTD Advantages:

Analytically the same for all geometry Does not involve matrix inversion, which is

troublesome for large system Find the optical response of a range of

frequencies Commercial package available

FDTD Principle

Maxwell equations

In the time domain,

FDTD solves D(t) and H(t) for an impulse Fourier transform frequency domain

Need for all frequencies fit the data with analytic model

where

FDTD Dielectric function

<10% difference in general No difference at a

given wavelength

Broadband fitting (For finding spectrum)

Fit a single point (For finding the internal field more accurately.)

Results Compare spectrum by FDTD with T-

Matrix20% Error

Results

2 nm : 1 nm gold spheroid

100 nm : 50 nm gold spheroid

• Internal Field

20%

110%

Comparing FDTD and T-Matrix

• Computational effort

Two cases:

Small sphero

id

Small cylind

er

Comparing FDTD and T-Matrix

Convergence

FDTD1. d=0.1 nm2. d=0.05 nm3. d=0.025 nm

T-Matrix4. 1 x 15. 3 x 36. 5 x 5

FDTD vs. T-Matrix T-Matrix converges more quickly FDTD can cause significant error near

the surface of particle. T-Matrix results are used as the exact

numerical results.

But, Both are complicated!

4. Simpler approach: Analytic

Approximation (AA)

Analytic Approximation Assumption:

Small particle, quasi-static case (kL 0) constant internal field and incident field

Consequence: simple formula for internal fied: dependence on frequency and shape are separated.

Shape factorDielectric constant: implicit frequency dependence

Shape factor For rotationally symmetric particle,

Where1

2

3

Follows from

Shape factor For spheroid and cylinder

decreaseincrease

Validity

Spheroid of aspect ratio 2

• Exactly agrees with T-Matrix

Cylinder of aspect ratio 2• AA (1) does not agree

with T-Matrix (3) • Modified AA (2) better

agrees with T-Matrix (3) modified

Aspect ratio dependence Consider a specific direction,

Is large when is small

Magnitude only

From

Aspect ratio dependence For a long spheroid/ cylinder:

E0E0

smalllarge

vs.

More Sensitive!

Aspect ratio dependence Shift of resonance peak

Aspect ratio dependence For cylinder (with modified factor)

23

5

Summary T-Matrix

Size dependence of LSPR: weak Aspect ratio dependence of LSPR position: large shift for

field along the long axis FDTD

significant error near the particle surface Poor converge around the surface

Analytic Approximation exact for small spheroid not good for small cylinder, unless with modified shape

factor Explain the aspect ratio dependence

External Field

External Field

From the integral equation:

Methodsa) Direct substitutionb) Multipole expansion

Known

Result (2 nm: 1 nm gold spheroid at resonance)14-fold enhancement

E on x-z plane

External Field

Enhancement around the tips:

• The typical scale of the enhancement regions is small (~ 0.5 nm), compared with– = 551 nm– particle size (~ 1 to 2 nm).

• ``Hot spot”: Important for applications

Multipole expansion of the external field

Obtain an expansion for the scattered field as:

Advantages: Avoid the singular point at |x| = |y|. Consistent with the formulation of the T-Matrix

method. Problem

Poor convergence for the near field

Issues• Poor convergence rate for external near field

compared with:

Internal Field External far field

2 nm

1 nm

0.5 nm

Field at this point

Summary of external field Strong enhancement field is reproduced

using integral equation method. T-Matrix method:

useful for internal field Not good for external near field.

Further Investigation Extend analytic approximation to and

Check its validity for Particles of other shapes Many particles system

The solution to those problems may smoothen the process of development of applications

End