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Capabilities of the discrete dipole approximation for large particle ...
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International Symposium on Electromagnetic Theory (EMTS 2016), Espoo, Finland, 16.08.2016
Capabilities of the discrete dipole approximation for large particle systems Maxim A. Yurkin1,2
Voevodsky Institute of Chemical Kinetics and Combustion, Novosibirsk, Russia 2 Novosibirsk State University, Novosibirsk, Russia 1
Plan
DDA basics
Computational issues
Fast-multipole method Multi-grid DDA Number of iterations Orientation averaging
Application examples
Multiple scattering and physical preconditioner
Conclusion
2
Discrete dipole approximation
Solves the integral Maxwell equation for the electric field in the frequency domain using the volume discretization. Cubical subvolumes (dipoles), have typical size d ~ λ/10. Polarizability of each dipole is determined by local (complex) refractive index. Dipoles interact with each other and the incident light ⇒ system of linear equations ⇒ solved to determine dipole polarizations. Any measurable quantity is determined from dipole polarizations.
Purcell & Pennypacker, Astrophys. J. 186, 705-714 (1973). Draine & Flatau, JOSA A 11, 1491-1499 (1994). Yurkin & Hoekstra, J. Quant. Spectrosc. Radiat. Transfer 106, 558-589 (2007).
3
Master equation E(r ) = Einc (r ) +
3 d ∫ r ′G (r − r′) χ (r′)E(r′) + M(V0 , r) − L (∂V0 , r) χ (r)E(r)
V \V0
exp(ikR) 2 Rˆ Rˆ 1 − ikR Rˆ Rˆ I − 3 2 G (R ) = k I − 2 − 2 R R R R
χ (r ) = (m 2 (r ) − 1) 4π
Volume discretization
αi−1Pi − ∑ G ij P j = Einc i j ≠i
Pi = Vi χ i Ei
4
Measurable quantities (scattering) exp(ikr ) E (r ) = F (n) − ikr F(n) = −ik 3 ( I − nˆ nˆ )∑ Pi exp(−ikri ⋅ n) sca
Distribution of EM field in space
i
(
* Cext = 4πk ∑ Im Pi ⋅ Einc i
)
i
Cabs = 4πk ∑ Im(Pi ⋅ E*i )
Integral quantities
i
Csca
1 2 = 2 ∫ dΩ F (n) = Cext − Cabs k
DDA is a “numerically exact” method! Yurkin et al., JOSA A 23, 2578–2591 (2006).
5
Computational issues
System of 3Nd linear equations (up to 109)
Matrix of the system is dense.
Regular cubical grid, G (r, r′) = G (r − r′) ⇒ matrix is multilevel block-Toeplitz Memory requirements – 𝒪(N), matrix-vector product ⇒ FFT-based convolution – 𝒪(NlogN)
