This document discusses light scattering by non-spherical particles. It compares three methods for solving this problem: the separation of variables method, the extended boundary condition method, and the generalized point matching method. All three methods use the same field expansions, but differ in important details of how the boundary conditions are applied. The accuracy of the extended boundary condition method depends on the particle shape, working well for some shapes but not others. Near-field behavior and convergence are more complex for non-spherical particles compared to spheres. The concept of the Rayleigh hypothesis is also discussed.

1. LIGHT SCATTERING BY NONSPHERICAL PARTICLES
V. G. Farafonov1 V. B. Il’in1,2,3 A. A. Vinokurov1,2
1 Saint-Petersburg State University of Aerospace Instrumentation, Russia
2 Pulkovo Observatory, Saint-Petersburg, Russia
3 Saint-Petersburg State University, Russia
Fundamentals of Laser Assisted Micro- and Nanotechnologies 2010
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 1 / 49

2. The model
z
Esca , Hsca r = r (θ)
x
Einc , Hinc
Size parameter xv, = 2πa/λ ∈ [0.1, 30]
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 2 / 49

3. Wave equations and functions
Maxwell equations
For time-harmonic fields
E(r, t) = E(r) exp(−iωt)
Helmholts equations for E(r), H(r)
∆E(r) + k 2 (r)E(r) = 0,
where k is the wavenumber
Solutions
Vector wave functions Fν (r)
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 3 / 49

4. Wave equations and functions
Additional condition div E(r) = 0 leads to:
Fν (r) = Ma (r) = rot(a ψν (r)),
ν
Fν (r) = Na (r) = rot rot(a ψν (r))/k,
ν
where a is a vector, ψν (r) are solutions to
∆ψν + k 2 ψν = 0.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 4 / 49

5. Field/potential expansions
It looks natural to search for unknown fields as
E(r) = aν Fν (r),
ν
or equivalently
U, V (r) = aν ψν (r),
ν
where U, V are scalar potentials, e.g.
E = rot(bU) + rot rot(cV ).
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 5 / 49

6. Field/potential expansions
In all the methods vector/scalar wave functions are represented as:
in spherical coordinates (r , θ, ϕ):
m
ψν (r) = zn (r )Pn (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.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 6 / 49

7. Field/potential expansions
In all the methods vector/scalar wave functions are represented as:
in spherical coordinates (r , θ, ϕ):
m
ψν (r) = zn (r )Pn (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.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 6 / 49

8. 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 different
indices and then are integrated over ∂Γ. This yelds the following
system:
A B xsca E inc
= x ,
C D xint F
where xinc , xsca , xint are vectors of expansion coefficients, A, . . . F —
matrices of surface integrals.
Generalised SVM1
1
see (Kahnert, 2003)
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 7 / 49

9. Extended Boundary Condition Method (EBCM)
Field expansions are substituted in the extended boundary condition:
−Einc (r ), r ∈ Γ− ,
rot n(r) × Eint (r)G(r , r)ds − . . . =
∂Γ Esca (r ), r ∈ Γ+ .
Due to linear independence of wave functions we get
0 Qs xsca I
= xinc ,
I Qr xint 0
where Qs , Qr are matrices, whose elements are surface integrals.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 8 / 49

10. Generalized Point Matching Method (gPMM)
Residual of the standard boundary conditions
M
2
δ= Einc + Esca − Eint × n + . . . , r = rs ∈ ∂Γ.
s=1
Minimizing residual in the least squares sense gives
A B xsca E inc
= x ,
C D xint F
Sum in δ can be replaced with surface integral2
2
see (Farafonov & Il’in, 2006)
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 9 / 49

11. 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.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 10 / 49

12. Key questions
1 EBCM, SVM, and PMM use the same field expansions. Does it result
in 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 a
spherical basis [being a widely used approach] gives high accuracy
results for some shapes, while for some others it cannot provide any
reliable results. Why? What can be said about theoretical applicability
of EBCM?
[We have some answers, but not anything is clear.]
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 11 / 49

13. Key questions
1 EBCM, SVM, and PMM use the same field expansions. Does it result
in 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 a
spherical basis [being a widely used approach] gives high accuracy
results for some shapes, while for some others it cannot provide any
reliable results. Why? What can be said about theoretical applicability
of EBCM?
[We have some answers, but not anything is clear.]
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 11 / 49

14. Key questions
1 EBCM, SVM, and PMM use the same field expansions. Does it result
in 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 a
spherical basis [being a widely used approach] gives high accuracy
results for some shapes, while for some others it cannot provide any
reliable results. Why? What can be said about theoretical applicability
of EBCM?
[We have some answers, but not anything is clear.]
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 11 / 49

15. Key questions
1 EBCM, SVM, and PMM use the same field expansions. Does it result
in 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 a
spherical basis [being a widely used approach] gives high accuracy
results for some shapes, while for some others it cannot provide any
reliable results. Why? What can be said about theoretical applicability
of EBCM?
[We have some answers, but not anything is clear.]
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 11 / 49

16. Singularities for a spheroid and Chebyshev particle
Spheroid Chebyshev particle
r (θ, ϕ) = r0 (1 + ε cos nθ)
d sca = a2 − b 2 , d sca = f (r0 , n, ε),
d int = ∞. d int = g (r0 , n, ε).
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 12 / 49

17. Near Field
1 In spherical coordinates
m
ψν (r , θ, ϕ) = zn (r )Pn (θ) exp(imϕ).
2 For r → 0 or ∞: zn (r ) behaves like r k .
3 Series U, V (r) = ν aν ψν (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!
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 13 / 49

18. Near Field
1 In spherical coordinates
m
ψν (r , θ, ϕ) = zn (r )Pn (θ) exp(imϕ).
2 For r → 0 or ∞: zn (r ) behaves like r k .
3 Series U, V (r) = ν aν ψν (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!
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 13 / 49

19. Near Field
1 In spherical coordinates
m
ψν (r , θ, ϕ) = zn (r )Pn (θ) exp(imϕ).
2 For r → 0 or ∞: zn (r ) behaves like r k .
3 Series U, V (r) = ν aν ψν (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!
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 13 / 49

20. Near Field
1 In spherical coordinates
m
ψν (r , θ, ϕ) = zn (r )Pn (θ) exp(imϕ).
2 For r → 0 or ∞: zn (r ) behaves like r k .
3 Series U, V (r) = ν aν ψν (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!
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 13 / 49

21. Near Field
1 In spherical coordinates
m
ψν (r , θ, ϕ) = zn (r )Pn (θ) exp(imϕ).
2 For r → 0 or ∞: zn (r ) behaves like r k .
3 Series U, V (r) = ν aν ψν (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!
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 13 / 49

22. Spheroid singularities, a/b = 1.4
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 14 / 49

23. Spheroid singularities, a/b = 2.0
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 15 / 49

24. Spheroid singularities, a/b = 2.5
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 16 / 49

25. Spheroid convergence in the near field, a/b = 1.4
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 17 / 49

26. Spheroid convergence in the near field, a/b = 1.8
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 18 / 49

27. Spheroid convergence in the near field, a/b = 2.5
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 19 / 49

28. Spheroid convergence in the near field, a/b = 3.5
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 20 / 49

29. Rayleigh hypothesis
We use a generalized and simplified definition of the term as an
assumption that field expansions in terms of wave functions
converge 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 when
Rayleigh hypothesis is not valid.
Do we realy need Rayleigh hypothesis to be valid?
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 21 / 49

30. Rayleigh hypothesis
We use a generalized and simplified definition of the term as an
assumption that field expansions in terms of wave functions
converge 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 when
Rayleigh hypothesis is not valid.
Do we realy need Rayleigh hypothesis to be valid?
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 21 / 49

31. Rayleigh hypothesis
We use a generalized and simplified definition of the term as an
assumption that field expansions in terms of wave functions
converge 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 when
Rayleigh hypothesis is not valid.
Do we realy need Rayleigh hypothesis to be valid?
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 21 / 49

32. Rayleigh hypothesis
We use a generalized and simplified definition of the term as an
assumption that field expansions in terms of wave functions
converge 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 when
Rayleigh hypothesis is not valid.
Do we realy need Rayleigh hypothesis to be valid?
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 21 / 49

33. Rayleigh hypothesis
We use a generalized and simplified definition of the term as an
assumption that field expansions in terms of wave functions
converge 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 when
Rayleigh hypothesis is not valid.
Do we realy need Rayleigh hypothesis to be valid?
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 21 / 49

34. Infinite linear systems analysis
    
a11 a12 · · · x1 b1
a21 a22 · · · x2  b2 
  =  
. . .. . .

.
. .
. . .
. .
.
Kantorovich & Krylov (1958)
Regular systems: ρi = 1 − ∞ |aik | > 0, (i = 1, 2, . . .).
k=1
Theorem. If ∃K > 0 : |bi | < K ρi , (i = 1, 2, . . .), then
regular system is solvable,
it has the only solution,
solutions of truncated systems converge to it.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 22 / 49

35. Infinite linear systems analysis
    
a11 a12 · · · x1 b1
a21 a22 · · · x2  b2 
  =  
. . .. . .

.
. .
. . .
. .
.
Kantorovich & Krylov (1958)
Regular systems: ρi = 1 − ∞ |aik | > 0, (i = 1, 2, . . .).
k=1
Theorem. If ∃K > 0 : |bi | < K ρi , (i = 1, 2, . . .), then
regular system is solvable,
it has the only solution,
solutions of truncated systems converge to it.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 22 / 49

36. Infinite linear systems analysis
    
a11 a12 · · · x1 b1
a21 a22 · · · x2  b2 
  =  
. . .. . .

.
. .
. . .
. .
.
Kantorovich & Krylov (1958)
Regular systems: ρi = 1 − ∞ |aik | > 0, (i = 1, 2, . . .).
k=1
Theorem. If ∃K > 0 : |bi | < K ρi , (i = 1, 2, . . .), then
regular system is solvable,
it has the only solution,
solutions of truncated systems converge to it.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 22 / 49

37. Analysis of EBCM, gSVM and gPMM systems
gPMM
System has positively determined matrix and hence has always the only
solution.
EBCM
System is regular and satisfies solvability condition if
max d sca < min d int .
gSVM
There is no such condition for SVM.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 23 / 49

38. Analysis of EBCM, gSVM and gPMM systems
gPMM
System has positively determined matrix and hence has always the only
solution.
EBCM
System is regular and satisfies solvability condition if
max d sca < min d int .
gSVM
There is no such condition for SVM.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 23 / 49

39. Analysis of EBCM, gSVM and gPMM systems
gPMM
System has positively determined matrix and hence has always the only
solution.
EBCM
System is regular and satisfies solvability condition if
max d sca < min d int .
gSVM
There is no such condition for SVM.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 23 / 49

40. Chebyshev particle singularities, n = 5, ε = 0.07
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 24 / 49

41. Chebyshev particle singularities, n = 5, ε = 0.14
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 25 / 49

42. Chebyshev particle singularities, n = 5, ε = 0.21
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 26 / 49

43. Solvability condition, EBCM
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 27 / 49

44. Solvability condition, SVM
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 28 / 49

45. Convergence of results, Chebyshev particle, n = 5, ε = 0.07
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 29 / 49

46. Convergence of results, Chebyshev particle, n = 5, ε = 0.14
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 30 / 49

47. Convergence of results, Chebyshev particle, n = 5, ε = 0.21
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 31 / 49

48. Paradox of the EBCM
EBCM solutions converge even if field expansions used in the
boundary 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 not
need convergence of any expansions at scatterer boundary.
We found that the infinite systems arisen in EBCM coincide with
those arisen in the PEM.
When Rayleigh hypothesis is not valid, EBCM is not
mathematically correct, but its applicability is extended in the
far field due to lucky coincindence with PEM.
3
see works by Kyurkchan and Smirnova
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 32 / 49

49. Paradox of the EBCM
EBCM solutions converge even if field expansions used in the
boundary 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 not
need convergence of any expansions at scatterer boundary.
We found that the infinite systems arisen in EBCM coincide with
those arisen in the PEM.
When Rayleigh hypothesis is not valid, EBCM is not
mathematically correct, but its applicability is extended in the
far field due to lucky coincindence with PEM.
3
see works by Kyurkchan and Smirnova
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 32 / 49

50. Paradox of the EBCM
EBCM solutions converge even if field expansions used in the
boundary 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 not
need convergence of any expansions at scatterer boundary.
We found that the infinite systems arisen in EBCM coincide with
those arisen in the PEM.
When Rayleigh hypothesis is not valid, EBCM is not
mathematically correct, but its applicability is extended in the
far field due to lucky coincindence with PEM.
3
see works by Kyurkchan and Smirnova
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 32 / 49

51. Paradox of the EBCM
EBCM solutions converge even if field expansions used in the
boundary 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 not
need convergence of any expansions at scatterer boundary.
We found that the infinite systems arisen in EBCM coincide with
those arisen in the PEM.
When Rayleigh hypothesis is not valid, EBCM is not
mathematically correct, but its applicability is extended in the
far field due to lucky coincindence with PEM.
3
see works by Kyurkchan and Smirnova
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 32 / 49

52. Paradox of the EBCM
EBCM solutions converge even if field expansions used in the
boundary 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 not
need convergence of any expansions at scatterer boundary.
We found that the infinite systems arisen in EBCM coincide with
those arisen in the PEM.
When Rayleigh hypothesis is not valid, EBCM is not
mathematically correct, but its applicability is extended in the
far field due to lucky coincindence with PEM.
3
see works by Kyurkchan and Smirnova
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 32 / 49

53. 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 infinite
system can be transformed into the matrix of gSVM system and vice
versa.
Qs = i CT B − AT D , Qr = i FT B − ET D ,
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.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 33 / 49

54. 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 infinite
system can be transformed into the matrix of gSVM system and vice
versa.
Qs = i CT B − AT D , Qr = i FT B − ET D ,
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.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 33 / 49

55. 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 infinite
system can be transformed into the matrix of gSVM system and vice
versa.
Qs = i CT B − AT D , Qr = i FT B − ET D ,
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.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 33 / 49

56. 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 infinite
system can be transformed into the matrix of gSVM system and vice
versa.
Qs = i CT B − AT D , Qr = i FT B − ET D ,
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.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 33 / 49

57. 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, but
truncated are not.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 34 / 49

58. 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, but
truncated are not.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 34 / 49

59. 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, but
truncated are not.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 34 / 49

60. 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, but
truncated are not.
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 34 / 49

61. Numerical comparison, prolate spheroid, a/b = 1.5
Farafonov, Il’in, Vinokurov (Russia) FLAMN-10 35 / 49

62. Numerical comparison, prolate spheroid, a/b = 2.0
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63. Numerical comparison, prolate spheroid, a/b = 2.5
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64. Numerical comparison, Chebyshev particle, n = 5, ε = 0.07
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65. Numerical comparison, Chebyshev particle, n = 5, ε = 0.14
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66. Numerical comparison, Chebyshev particle, n = 5, ε = 0.21
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67. Condition number for gSVM, EBCM, iPMM systems
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68. System matrix elements, SVM
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69. System matrix elements, EBCM
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70. System matrix elements, PMM
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71. Multilayered scatterers
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72. gSVM for multilayered particles
A particle with L layers.
The electromagnetic fields in each of the domains Γ(i) satisfy the
boundary conditions
E(i) (r) × n(i) (r) = E(i+1) (r) × n(i) (r), r ∈ ∂Γ(i) , i = 1, . . . , L,
Systems for each of the layer boundaries
(i) (i)
Pi x(i) = Pi+1 x(i+1) , i = 1, . . . , L,
Iterative scheme
L
(1) xsca (1) (i) (i)
P1 = P2 (Pi )−1 Pi+1 x(L+1) .
xinc
i=2
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73. Accuracy of gSVM for multilayered particles
10-2
10-4
10-6
Relative error
10-8
10-10
10-12
10-14
10-16 0
10 101 102 103
Number of layers
xv = 0.1 xv = 10.0
xv = 0.5 xv = 15.0
xv = 1.0 xv = 30.0
xv = 5.0
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74. Polarization and intensity of layered scatterers
Homogeneous 2 layers 4 layers
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75. Polarization and intensity of layered scatterers
Homogeneous 8 layers 16 layers
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76. Conclusions
1 Methods are very similar, but have key differencies.
2 Methods applicability ranges are defined 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 efficient for different particles.
8 Systems ill-conditionedness doesn’t correlate with bad convergence.
9 SVM is the most efficient for multilayered scatterers.
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