Abstract

In this paper we discuss the influence of two different sets of weighting functions on the accuracy behavior of T-matrix calculations for scalar scattering problems. The first set of weighting functions is related to one of Waterman’s original approaches. The other set results into a least-squares scheme for the transmission problem. It is shown that both sets of weighting functions produce results with a converse accuracy behavior in the near and far fields. Additional information, such as reciprocity and the fulfillment of the boundary condition, are needed to choose the set of weighting functions that is most appropriate for a certain application. The obtained criteria are applied afterward to an iterative T-matrix approach we developed to analyze scattering on regular particle geometries with an impressed but slight surface irregularity. However, its usefulness is demonstrated in this paper by analyzing the far-field scattering behavior of Chebyshev particles of higher orders.

© 2010 Optical Society of America

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References

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  1. J. A. Wiscombe and A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” NASA Reference Publication 1157 (1986).
  2. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University Press, 2002).
  3. A. G. Dallas, “On the convergence and numerical stability of the second Waterman scheme for approximation of the acoustic field scattered by a hard object,” Technical Report No. 2000-7(Department of Mathematical Sciences, University of Delaware, 2000).
  4. P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
    [CrossRef]
  5. P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
    [CrossRef]
  6. T. Rother, K. Schmidt, J. Wauer, V. Shcherbakov, and J.-F. Gayet, “Light scattering on Chebyshev particles of higher order,” Appl. Opt. 45, 6030–6037 (2006).
    [CrossRef] [PubMed]
  7. T. Rother, Electromagnetic Wave Scattering on Nonspherical Particles: Basic Methodology and Simulations (Springer, 2009).
    [CrossRef]
  8. T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s function of the Helmholtz equation in spherical coordinates,” Prog. Electromagn. Res. 38, 47–95(2002).
    [CrossRef]
  9. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
    [CrossRef]
  10. J. Wauer, K. Schmidt, T. Rother, T. Ernst, and M. Hess, “Two software tools for plane wave scattering on nonspherical particles in the German Aerospace Center’s virtual laboratory,” Appl. Opt. 43, 6371–6379 (2004).
    [CrossRef] [PubMed]

2006 (1)

2004 (1)

2002 (1)

T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s function of the Helmholtz equation in spherical coordinates,” Prog. Electromagn. Res. 38, 47–95(2002).
[CrossRef]

1969 (1)

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
[CrossRef]

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Barber, P. W.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

Dallas, A. G.

A. G. Dallas, “On the convergence and numerical stability of the second Waterman scheme for approximation of the acoustic field scattered by a hard object,” Technical Report No. 2000-7(Department of Mathematical Sciences, University of Delaware, 2000).

Doicu, A.

T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s function of the Helmholtz equation in spherical coordinates,” Prog. Electromagn. Res. 38, 47–95(2002).
[CrossRef]

Ernst, T.

Gayet, J.-F.

Hess, M.

Hill, S. C.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

Kahnert, M.

T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s function of the Helmholtz equation in spherical coordinates,” Prog. Electromagn. Res. 38, 47–95(2002).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University Press, 2002).

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University Press, 2002).

Mugnai, A.

J. A. Wiscombe and A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” NASA Reference Publication 1157 (1986).

Rother, T.

Schmidt, K.

Shcherbakov, V.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University Press, 2002).

Waterman, P. C.

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
[CrossRef]

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Wauer, J.

Wiscombe, J. A.

J. A. Wiscombe and A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” NASA Reference Publication 1157 (1986).

Appl. Opt. (2)

J. Acoust. Soc. Am. (1)

P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Soc. Am. 45, 1417–1429 (1969).
[CrossRef]

Proc. IEEE (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Prog. Electromagn. Res. (1)

T. Rother, M. Kahnert, A. Doicu, and J. Wauer, “Surface Green’s function of the Helmholtz equation in spherical coordinates,” Prog. Electromagn. Res. 38, 47–95(2002).
[CrossRef]

Other (5)

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
[CrossRef]

T. Rother, Electromagnetic Wave Scattering on Nonspherical Particles: Basic Methodology and Simulations (Springer, 2009).
[CrossRef]

J. A. Wiscombe and A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” NASA Reference Publication 1157 (1986).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles(Cambridge University Press, 2002).

A. G. Dallas, “On the convergence and numerical stability of the second Waterman scheme for approximation of the acoustic field scattered by a hard object,” Technical Report No. 2000-7(Department of Mathematical Sciences, University of Delaware, 2000).

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Figures (21)

Fig. 1
Fig. 1

Geometry of the scattering problem.

Fig. 2
Fig. 2

Two scattering configurations used in this paper to check reciprocity. α, β, and γ are the Eulerian angles of orientation; θ s is the scattering angle in the x z scattering plane.

Fig. 3
Fig. 3

Differential scattering cross section of a prolate spheroid obtained with two different T-matrix approaches. Parameters: aspect ratio ar = 1.4 , refractive index n = 1.33 , size parameter of the volume equivalent sphere k 0 r eqv ( v ) = 20 .

Fig. 4
Fig. 4

n cut dependence of the maximum relative error according to the Barber–Hill criterion in the far field for the results of the two T-matrix approaches of Fig. 3.

Fig. 5
Fig. 5

n cut dependence of the relative error according to the reciprocity criterion in the far field for the two T-matrix approaches of Fig. 3.

Fig. 6
Fig. 6

Fulfillment of the reciprocity for two different numbers of n cut . Parameters as in Fig. 3.

Fig. 7
Fig. 7

n cut dependence of the maximum error regarding the fulfillment of the boundary conditions according to the Barber–Hill criterion for the two T-matrix approaches of Fig. 3.

Fig. 8
Fig. 8

Error of the fulfillment of the boundary conditions for two different numbers of n cut . Results are obtained by using the W-TM.

Fig. 9
Fig. 9

Error of the fulfilment of the boundary conditions for two different numbers of n cut . Results are obtained by using the LS-TM.

Fig. 10
Fig. 10

(a) Differential scattering cross section of a sphere obtained with Mie theory (32 expansion terms have been used to achieve convergence). Parameters: refractive index n = 1.33 , size parameter k 0 r = 20 ; (b) relative error of the differential scattering cross section of a center-off shifted sphere if compared to the result of the Mie theory of (a). Parameters: same as in (a), but with a shift s of k 0 s = 4 along the positive z axis. Calculations have been performed by use of W-TM with n cut = 32 .

Fig. 11
Fig. 11

Error of the fulfillment of the boundary conditions for the T-matrix result of the center-off shifted sphere shown in Fig. 10.

Fig. 12
Fig. 12

(a) Differential scattering cross section of a Chebyshev particle with orientation α = β = γ = 0 ° . Calculations have been performed with the Waterman and least-squares methods. Parameters: refractive index n = 1.33 , size parameter k 0 r 0 = 30 , order n o = 45 , deformation parameter ϵ = 0.0133 ; (b) relative error between the Waterman and least-squares methods.

Fig. 13
Fig. 13

Fulfillment of the reciprocity condition for the results of both methods represented in Fig. 12.

Fig. 14
Fig. 14

Relative error of the reciprocity condition as a function of the convergence parameter n cut if applied to the least-squares method.

Fig. 15
Fig. 15

(a) Differential scattering cross section of a Chebyshev particle (parameters as in Fig. 12). Calculations have been performed with the least-squares method. (b) Relative error between the convergent Waterman and the least-squares solutions. The former solution was already shown in Fig. 12.

Fig. 16
Fig. 16

Differential scattering cross section of a dielectric Chebyshev particle obtained with the W-TM approach. Parameters: refractive index n = 1.4717 + 0.389 i , size parameter k 0 r 0 = 50 , order n o = 75 , deformation parameter ϵ = 0.008 , orientation α = β = γ = 0 ° .

Fig. 17
Fig. 17

Phase function of a dielectric Chebyshev particle obtained with the W-TM approach. Parameters: refractive index n = 1.33 , size parameter k 0 r 0 = 50 , order n o = 75 , deformation parameter ϵ = 0.014 .

Fig. 18
Fig. 18

(a) Differential scattering cross section of a dielectric Chebyshev particle obtained with the T2-approximation. Parameters as in Fig. 16. (b) Errors of the T1 approximation (dashed curve) and T2 approximation (solid curve) if compared to the result of Fig. 16.

Fig. 19
Fig. 19

Phase function of a dielectric Chebyshev particle obtained with the T2 approximation. Parameters as in Fig. 17.

Fig. 20
Fig. 20

Differential scattering cross sections of a spheroidal particle without and with an impressed Chebyshev surface. Parameters: refractive index n = 1.5 + 0.4 i , size parameter k 0 r eqv ( v ) = 10 , aspect ratio ar = 1.2 , deformation parameter ϵ = 0.01 , order n o = 15 .

Fig. 21
Fig. 21

(a) Differential scattering cross section of the spheroidal particle with an impressed Chebyshev surface of Fig. 20 obtained with the T1 approximation. (b) relative error of the T1 approximation if compared to the result of Fig. 20.

Equations (50)

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( 2 + k 0 / 1 2 ) u s / int ( r ) = 0 .
u s ( r ) + u inc ( r ) = u int ( r ) ,
u s ( r ) + u inc ( r ) = u int ( r ) .
u = u n ^ = n ^ · u ,
lim r ( r i k 0 ) u s ( r ) = 0
r = r ( θ , ϕ ) · r ^ .
n ^ = r θ × r ϕ | r θ × r ϕ | ,
r θ × r ϕ = r 2 sin θ · r ^ r sin θ · r θ r · r ϕ · ϕ ^ .
u inc ( r ) = n = 0 n cut l = n n e l , n · Ψ l , n 0 ( r , θ , ϕ ) ,
u s ( r ) = n = 0 n cut l = n n a l , n 0 · Φ l , n 0 ( r , θ , ϕ ) ,
u int ( r ) = n = 0 n cut l = n n a l , n 1 · Ψ l , n 1 ( r , θ , ϕ ) .
Ψ l , n 0 / 1 ( r , θ , ϕ ) = j n ( k 0 / 1 r ) · Y l , n ( θ , ϕ ) ,
Φ l , n 0 ( r , θ , ϕ ) = h n ( 1 ) ( k 0 r ) · Y l , n ( θ , ϕ ) .
Y l , n ( θ , ϕ ) = 2 n + 1 4 π · ( n l ) ! ( n + l ) ! · P n l ( cos θ ) · e i l ϕ ,
i = n · ( n + 1 ) + l .
a = ( a 0 a 1 ) ,
T = ( T 0 T 1 )
a = T · e
i ( a i 0 · | Φ 0 i a i 1 · | Ψ 1 i ) = i e i · | Ψ 0 i ,
i ( a i 0 · | Φ 0 i a i 1 · | Ψ 1 i ) = i e i · | Ψ 0 i .
i g | f i = Γ g i * ( r ) · f i ( r ) d S ( θ , ϕ ) ,
d S ( θ , ϕ ) = | r θ × r ϕ | d θ d ϕ .
A WM · a = B WM · e ,
A WM = ( A 11 WM A 12 WM A 21 WM A 22 WM ) = ( Ψ 1 ; * | Φ 0 , Ψ 1 ; * | Ψ 1 Ψ 1 ; * | Φ 0 , Ψ 1 ; * | Ψ 1 ) ,
B WM = ( B 1 WM B 2 WM ) = ( Ψ 1 ; * | Ψ 0 Ψ 1 ; * | Ψ 0 ) .
T WM = [ A WM ] 1 · B WM .
Γ + [ Ψ 2 Φ Φ 2 Ψ ] d V = Γ [ Ψ Φ n ^ Φ Ψ n ^ ] d S .
Γ Ψ i 1 · Ψ i 1 d S = Γ Ψ i 1 · Ψ i 1 d S ,
i Ψ 1 ; * | Ψ 1 i = i Ψ 1 ; * | Ψ 1 i .
T 0 , WM = [ Ψ 1 ; * | Φ 0 Ψ 1 ; * | Φ 0 ] 1 · [ Ψ , 1 ; * | Ψ 0 Ψ 1 ; * | Ψ 0 ] .
M N = M N ( D ) + 1 k 0 M N ( N ) ,
M N ( D ) = u s N u int N + u inc N Γ 2 = u inc N + i = 0 N ( a i 0 · Φ i 0 a i 1 · Ψ i 1 ) | u inc N + i = 0 N ( a i 0 · Φ i 0 a i 1 · Ψ i 1 ) M N ( N ) ,
M N ( N ) = u s N u int N + u inc N Γ 2 = u inc N + i = 0 N ( a i 0 · Φ i 0 a i 1 · Ψ i 1 ) | u inc N + i = 0 N ( a i · Φ i 0 b i · Ψ i 1 ) .
M N a i 0 / 1 ; * = M N D a i 0 / 1 ; * + M N N k 0 a i 0 / 1 ; * = 0 .
A lsq · a = B lsq · e ,
A lsq = A D lsq + 1 k 0 · A N lsq ,
B lsq = B D lsq + 1 k 0 · B N lsq ,
A D lsq = ( [ Φ 0 | Φ 0 ] , [ Φ 0 | Ψ 1 ] [ Ψ 1 | Φ 0 ] , [ Ψ 1 | Ψ 1 ] ) , B D lsq = ( [ Φ 0 | Ψ 0 ] [ Ψ 1 | Ψ 0 ] ) ,
A N lsq = ( [ Φ , 0 | Φ , 0 ] , [ Φ , 0 | Ψ , 1 ] [ Ψ , 1 | Φ , 0 ] , [ Ψ , 1 | Ψ , 1 ] ) , B N lsq = ( [ Φ , 0 | Ψ , 0 ] [ Ψ , 1 | Ψ , 0 ] ) .
r ( θ ) = a · [ cos 2 θ + ( a b ) 2 · sin 2 θ ] 1 / 2
a r = a b .
r ( θ ) = r 0 · [ p · cos θ + ( 1 p 2 · sin 2 θ ) 1 / 2 ] ,
p = s r 0
r ( θ ) = r 0 · ( 1 + ϵ · cos n o · θ )
T 0 = [ A 0 ] 1 · B 0 .
T = [ A ] 1 · B .
( A 0 + Δ A ) · a = B · e ,
a = [ A 0 ] 1 · ( B · e Δ A · a ) ,
a ( 1 ) = [ A 0 ] 1 · ( B Δ A · [ A 0 ] 1 · B ) · e = T ( 1 ) · e
a ( 2 ) = [ A 0 ] 1 · ( B Δ A · T ( 1 ) ) · e = T ( 2 ) · e .

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