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Multipole Analysis of Electromagnetic Scattering

Multipole Expansion• The field scattered by a particle, regardless of the particle’s geometrical

shape or composition, can always be expressed through the multipole expansion. The expression for the E-field reads 1:

• The vector functions ∇ × ℎ𝑙(1)

𝑘𝑟 𝐗𝑙𝑚 𝜃, 𝜑 and ℎ𝑙(1)

𝑘𝑟 𝐗𝑙𝑚 𝜃, 𝜑

generate a complete basis, where each function describes the field created by a unique multipole.

𝐄rel 𝐫 = 𝐸0

𝑙=1

∞

𝑚=−𝑙

𝑙

i𝑙 𝜋 2𝑙 + 1 1/2

1

𝑘𝑎E 𝑙, 𝑚 ∇ × ℎ𝑙

(1)𝑘𝑟 𝐗𝑙𝑚 𝜃, 𝜑

+𝑎M 𝑙, 𝑚 ℎ𝑙(1)

𝑘𝑟 𝐗𝑙𝑚 𝜃, 𝜑

1 J. D. Jackson, Classical Electrodynamics, 3rd edn (New York, Wiley, 2012).

Multipole Coefficients

Example: θ-component of the E-field created by a time-harmonic (a) electric dipole, and (b) electric quadrupole. The cyan arrows depict the electric current elements composing the multipole.

• The coefficients 𝑎E 𝑙, 𝑚 and 𝑎M 𝑙, 𝑚 characterize the scatterer as they reveal the electric and magnetic excitations in it.

• The integer 𝑙 describes the order of the multipole (dipole, quadrupole, …), whereas the indices E and M distinguish between electric and magnetic multipoles. The integer 𝑚 describes the amount of the 𝑧-component of angular momentum that is carried per photon.

Multipole Coefficients• For optical scatterers, the source of the scattered field is the scattering

current density

• 𝐉sca 𝐫 = −iω 𝜀 𝐫 − 𝜀h 𝐄(𝐫)

• Multipole coefficients can be extracted from it using the equations 2:

total field

permittivity of surrounding host medium

2 P. Grahn, A. Shevchenko, and M. Kaivola, Electromagnetic multipole theory for optical nanomaterials, New Journal of Physics 14, 093033 (2012). http://dx.doi.org/10.1088/1367-2630/14/9/093033

𝑎E 𝑙,𝑚 =(−i)𝑙−1𝑘2η𝑂𝑙𝑚

𝐸0 𝜋 2𝑙 + 1 1/2 𝑒−i𝑚𝜑

ψ𝑙 𝑘𝑟 + ψ𝑙′′ 𝑘𝑟 𝑃𝑙

𝑚 cos 𝜃 𝑟 ∙ 𝐉sca 𝐫

+ψ𝑙

′ 𝑘𝑟

𝑘𝑟𝜏𝑙𝑚 𝜃 𝜃 ∙ 𝐉sca 𝐫 − i𝜋𝑙𝑚 𝜃 𝜑 ∙ 𝐉sca 𝐫

d3𝑟

𝑎M 𝑙, 𝑚 =(−i)𝑙+1𝑘2η𝑂𝑙𝑚

𝐸0 𝜋 2𝑙 + 1 1/2 𝑒−i𝑚𝜑 𝑗𝑙 𝑘𝑟 i𝜋𝑙𝑚 𝜃 𝜃 ∙ 𝐉sca 𝐫 + 𝜏𝑙𝑚 𝜃 𝜑 ∙ 𝐉sca 𝐫 d3𝑟

Mie Scattering• Scattering of a plane wave by a spherical particle.

• Analytical solution exists as an expansion, in which the coefficients are called Mie coefficients.

• MATLAB code 3 exists for computing the Mie coefficients (𝛼𝑙 and 𝛽𝑙)

• The Mie expansion is a special case of the Multipole expansion, obtained by setting all coefficients with 𝑚 ≠ ±1 to zero

𝑎E 𝑙, −1 = −𝑎E 𝑙, 1

𝑎M 𝑙, −1 = 𝑎M 𝑙, 1 .

• Connection between the remaining multipole coefficients and the Mie coefficients is 𝑎E 𝑙, 1 = −𝛼𝑙 and 𝑎M 𝑙, 1 = −𝛽𝑙.

3 C. Mätzler, “MATLAB functions for Mie scattering and absorption, version 2”, in “IAP Research Report”, (University of Bern, 2002), available at: http://www.iap.unibe.ch/publications

COMSOL Implementation• Equations for extracting

𝑎E 𝑙, 𝑚 and 𝑎M 𝑙, 𝑚 are implemented in COMSOL using functions and variables

• Implementation can be added to any scattering model

• Also applicable to periodic systems

Rodrigues’ formula for associated Legendre polynomials 𝑃𝑙

𝑚 cos 𝜃

angular functions 𝜏𝑙𝑚 𝜃 and 𝜋𝑙𝑚 𝜃

spherical Bessel functions 𝑗𝑙 𝑘𝑟𝑂𝑙𝑚

calculation of 𝑎E 𝑙, 𝑚 and 𝑎M 𝑙, 𝑚

sweep the integers 𝑙 and 𝑚, using the E-field calculated in another study

Mie Benchmark• Mie scattering as a benchmark

– 600 nm vacuum wavelength

– Au sphere of 100 nm radius

– host medium with a refractive index of 1.5

• Numerical results are compared to Mie coefficients evaluated in MATLAB

Note: COMSOL uses 𝑒+j𝜔𝑡 convention, which is handled by setting the imaginary unit i → −j in the equations of this presentation. Complex-conjugation of obtained multipole coefficients returns to 𝑒−i𝜔𝑡 convention.

= +

Benchmark Results• 𝛼1 = 0.9350 − 0.0826𝑖

𝛼2 = 0.7003 − 0.2209𝑖𝛼3 = 0.0075 − 0.0384𝑖𝛽1 = 0.0941 + 0.2671𝑖𝛽2 = 0.0038 + 0.0341𝑖𝛽3 = 0.0002 + 0.0018𝑖

Move to 𝑒−i𝜔𝑡 conventionand multiply with -1 to get Mie coefficients

Numerical multipole coefficients agree with analytical results.

Scattering & Absorption Cross Sections

𝐶sca =𝜋

𝑘2

𝑙=1

∞

𝑚=−𝑙

𝑙

2𝑙 + 1 𝑎E 𝑙, 𝑚 2 + 𝑎M 𝑙, 𝑚 2

𝐶ext = −𝜋

𝑘2

𝑙=1

∞

𝑚=−1,+1

2𝑙 + 1 Re 𝑚𝑎E 𝑙, 𝑚 + 𝑎M 𝑙, 𝑚

cross section (m2)

from multipole coefficients

from MATLAB code

Directly from solved EM-field

scattering 1.4268E-13 1.4252e-13 1.4202E-13

absorption 2.7157E-14 2.7148e-14 2.7199E-14

extinction 1.6984E-13 1.6967e-13 1.6922E-13

• Scattering cross section:

• Extinction cross section:

Modal Scattering Cross Section• Analysis of each multipole’s contribution to the scattered field

𝑚 ≠ ±1 are essentially zero

electric dipoleand quadrupole