Integral Equation Formulation of Electromagnetic Scattering from Small Particles
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Transcript of Integral Equation Formulation of Electromagnetic Scattering from Small Particles
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