theoretical and computational aspects of the svm, ebcm, and pmm methods in light scatteringby small...

Theoretical and computational aspectsof the SVM, EBCM, and PMM methods

in light scattering by small particles

V. B. Il’in1,2,3 V. G. Farafonov2 A. A. Vinokurov2,3

1Saint-Petersburg State University, Russia2Saint-Petersburg State University of Aerospace Instrumentation, Russia

3Pulkovo Observatory, Saint-Petersburg, Russia

12th Electromagnetic & Light Scattering Conference

Il’in, Farafonov, Vinokurov (Russia) ELS-XII 1 / 46

Wave Equations and Functions

Maxwell equations

Helmholts equations for E(r), H(r)

∆E(r) + k2(r)E(r) = 0,

where k is the wavenumber

Vector wave functions Fν(r)

For time-harmonic fieldsE(r, t) = E(r) exp(−iωt)

Solutions

Wave Equations and Functions

Additional condition divE(r) = 0 leads to:

Fν(r) = Maν(r) = rot(a ψν(r)),

Fν(r) = Naν(r) = rot rot(a ψν(r))/k ,

where a is a vector, ψν(r) are solutions to

∆ψν + k2ψν = 0.

Field/Potential Expansions

It looks natural to search for unknown fields as

E(r) =∑ν

aνFν(r),

or equivalentlyU,V (r) =

aνψν(r),

where U,V are scalar potentials, e.g.

E = rot(bU) + rot rot(cV ).

In all the methods vector/scalar wave functions are represented as:in spherical coordinates (r , θ, ϕ):

ψν(r) = zn(r)Pmn (θ) exp(imϕ),

in spheroidal coordinates (ξ, η, ϕ):

ψν(r) = Rnm(c , ξ)Snm(c , η) exp(imϕ),

where c is a parameter.

So, separation of variables is actually used in all 3 methods.

In all the methods vector/scalar wave functions are represented as:in spherical coordinates (r , θ, ϕ):

ψν(r) = zn(r)Pmn (θ) exp(imϕ),

in spheroidal coordinates (ξ, η, ϕ):

ψν(r) = Rnm(c , ξ)Snm(c , η) exp(imϕ),

where c is a parameter.

So, separation of variables is actually used in all 3 methods.

Separation of Variables Method (SVM)

Field expansions are substituted in the boudary conditions

(Einc + Esca)× n = Eint × n, r ∈ ∂Γ,

where n is the outer normal to the particle surface ∂Γ.The conditions are mutiplied by the angular parts of ψν with differentindices and then are integrated over ∂Γ. This yelds the followingsystem: (

A BC D

)(xsca

)xinc,

where xinc, xsca, xint are vectors of expansion coefficients, A, . . .F —matrices of surface integrals.Generalised SVM1

1see (Kahnert, 2003)Il’in, Farafonov, Vinokurov (Russia) ELS-XII 6 / 46

Extended Boundary Condition Method (EBCM)

Field expansions are substituted in the extended boundary condition:

rot∫∂Γ

n(r)× Eint(r)G(r′, r)ds − . . . =

{−Einc(r′), r′ ∈ Γ−,

Esca(r′), r′ ∈ Γ+.

Due to linear independence of wave functions we get(0 QsI Qr

)(xsca

)xinc,

where Qs , Qr are matrices, whose elements are surface integrals.

Generalized Point Matching Method (gPMM)

Residual of the standard boundary conditions

δ =M∑

(∣∣(Einc + Esca − Eint)× n∣∣2 + . . .

), r = rs ∈ ∂Γ.

Minimizing residual in the least squares sense gives(A BC D

)(xsca

)xinc,

Sum in δ can be replaced with surface integral2

2see (Farafonov & Il’in, 2006)Il’in, Farafonov, Vinokurov (Russia) ELS-XII 8 / 46

Comparison of gPMM and Integral gPMM

1 — PMM, M = N,2 — gPMM, M = 2N,3 — gPMM, M = 4N,5 — iPMM, M = N,6 — iPMM, M = 1.5N.

Key questions

1 EBCM, SVM, and PMM use the same field expansions. Does it resultin their similar behavior? How close are they?[Yes and Generally close, but in important detail not.]

2 It is well known from numerical experiments that EBCM with aspherical basis [being a widely used approach] gives high accuracyresults for some shapes, while for some others it cannot provide anyreliable results. Why? What can be said about theoretical applicabilityof EBCM?[We have some answers, but not anything is clear.]

Key questions

Singularities of analytic continuations of the fields

Obviously, a plane wave has no such singularities.But a plane wave incident at a scatterer is known to produce scatteredfield outside it and incident field inside it.Generally, analytic continuations of both the scattered field (inside thescatterer) and of the internal field (outside it) may have singularitiesdepending on the scatterer shape.

Singularities for a spheroid and Chebyshev particle

Spheroid

d sca =√

a2 − b2,

d int =∞.

Chebyshev particler(θ, ϕ) = r0(1 + ε cos nθ)

d sca = f (r0, n, ε),

d int = g(r0, n, ε).

Near Field

1 In spherical coordinates

ψν(r , θ, ϕ) = zn(r)Pmn (θ) exp(imϕ).

2 For r → 0 or ∞: zn(r) behaves like rk .3 Series U,V (r) =

∑ν aνψν(r) can be transformed into power series.

4 Radius of convergence is determined by the nearest singularity.

For scattered field: r ∈ (max d sca,∞) .

For internal field: r ∈[0,min d int) .

5 In spheroidal coordinates, it is not that simple!

Near Field

Spheroid singularities, a/b = 1.4

Spheroid convergence in the near field, a/b = 1.4

Rayleigh hypothesis

We use a generalized and simplified definition of the term as anassumption that field expansions in terms of wave functionsconverge everywhere up to the scatterer surface.In spherical coordinates and for spherical basis:

max d sca < min r(θ, ϕ) and max r(θ, ϕ) < min d int.

As field expansions are substituted in the boundary conditions,Rayleigh hypothesis seems to be required to be valid.However, we know that the methods provide accurate results whenRayleigh hypothesis is not valid.Do we realy need Rayleigh hypothesis to be valid?

Rayleigh hypothesis

Infinite linear systems analysis

a11 a12 · · ·a21 a22 · · ·...

.... . .

b1b2...

Kantorovich & Krylov (1958)Regular systems: ρi = 1−

∑∞k=1 |aik | > 0, (i = 1, 2, . . .).

Theorem. If ∃K > 0 : |bi | < Kρi , (i = 1, 2, . . .), thenI regular system is solvable,I it has the only solution,I solutions of truncated systems converge to it.

a11 a12 · · ·a21 a22 · · ·...

.... . .

b1b2...

∑∞k=1 |aik | > 0, (i = 1, 2, . . .).

a11 a12 · · ·a21 a22 · · ·...

.... . .

b1b2...

∑∞k=1 |aik | > 0, (i = 1, 2, . . .).

Analysis of EBCM, gSVM and gPMM systems

gPMMSystem has positively determined matrix and hence has always the onlysolution.

EBCMSystem is regular and satisfies solvability condition if

max d sca < min d int.

gSVMThere is no such condition for SVM.

Chebyshev particle singularities, n = 5, ε = 0.07

Solvability condition, EBCM

Solvability condition, SVM

Convergence of results, Chebyshev particle, n = 5, ε = 0.07

Paradox of the EBCM

EBCM solutions converge even if field expansions used in theboundary conditions diverge.How is this possible?Let’s consider Pattern Equation Method3.Search for far field pattern in terms of angular parts of ψν (as r →∞)As the patterns are defined only in the far field zone, one does notneed convergence of any expansions at scatterer boundary.We found that the infinite systems arisen in EBCM coincide withthose arisen in the PEM.When Rayleigh hypothesis is not valid, EBCM is notmathematically correct, but its applicability is extended in thefar field due to lucky coincindence with PEM.

3see works by Kyurkchan and SmirnovaIl’in, Farafonov, Vinokurov (Russia) ELS-XII 32 / 46

Paradox of the EBCM

Equivalence of the EBCM and gSVM systems

It was generally shown earlier (e.g., Schmidt et al., 1998).We have strictly demonstrated that the matrix of EBCM infinitesystem can be transformed into the matrix of gSVM system and viceversa.

Qs = i[CTB− ATD

], Qr = i

[FTB− ETD

where A = ASVM , B = BSVM , . . .

If in iPMM residual ∆ = 0, then

APMM = AT∗A + CT∗A, . . .

Hence, iPMM infinite system is also equivalent to gSVM system.

Qs = i[CTB− ATD

], Qr = i

[FTB− ETD

Qs = i[CTB− ATD

], Qr = i

[FTB− ETD

Qs = i[CTB− ATD

], Qr = i

[FTB− ETD

Truncation of infinite systems

For truncated systems the proof of equivalence is not correct.For EBCM and iPMM we have regular systems.For gSVM we couldn’t prove that systems are regular.Infinite EBCM and gSVM systems are equivalent, buttruncated are not.

Numerical comparison, prolate spheroid, a/b = 1.5

Numerical comparison, Chebyshev particle, n = 5, ε = 0.07

Condition number for gSVM, EBCM, iPMM systems

System matrix elements, SVM

System matrix elements, EBCM

System matrix elements, PMM

Layered scatterers

Conclusions

1 Methods are very similar, but have key differencies.2 Methods applicability ranges are affected by singularities.3 Rayleigh hypothesis is required for near field computations.4 EBCM has solvability condition for far field.5 Infinite matrices of the methods’ systems are equivalent.6 Truncated matrices are not.7 Different methods are good for different particles.8 Systems are ill-conditioned from the beginning.9 Because of solvability condition EBCM doesn’t provide accurate

results for layered scatterers.

theoretical and computational aspects of the svm, ebcm, and pmm methods in light scatteringby small...

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