This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
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
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.