LIGHT SCATTERING BY NONSPHERICAL PARTICLES V. G. Farafonov 1 V. B. Il’in 1,2,3 A. A. Vinokurov 1,2 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
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LIGHT SCATTERING BY NONSPHERICAL PARTICLES
V. G. Farafonov1 V. B. Il’in1,2,3 A. A. Vinokurov1,2
1Saint-Petersburg State University of Aerospace Instrumentation, Russia2Pulkovo Observatory, Saint-Petersburg, Russia
3Saint-Petersburg State University, Russia
Fundamentals of Laser Assisted Micro- and Nanotechnologies 2010
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
xint
)=
(EF
)xinc,
where xinc, xsca, xint are vectors of expansion coefficients, A, . . .F —matrices of surface integrals.Generalised SVM1
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.]
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.]
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.]
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.]
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?
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?
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?
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?
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?
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.
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.
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.
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 SmirnovaFarafonov, Il’in, Vinokurov (Russia) FLAMN-10 32 / 49
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 SmirnovaFarafonov, Il’in, Vinokurov (Russia) FLAMN-10 32 / 49
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 SmirnovaFarafonov, Il’in, Vinokurov (Russia) FLAMN-10 32 / 49
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 SmirnovaFarafonov, Il’in, Vinokurov (Russia) FLAMN-10 32 / 49
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 SmirnovaFarafonov, Il’in, Vinokurov (Russia) FLAMN-10 32 / 49
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.
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.
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.
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.
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.
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.
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.
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.
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.