Abstract

The invariant-imbedding T-matrix (II-TM) method is a numerical method for accurately computing the single-scattering properties of dielectric particles. Because of the linearity of Maxwell’s equations, the incident electric field and the scattered electric field can be related through a transition matrix (T-Matrix). The II-TM method computes the T-matrix through a matrix recurrence formula which stems from an electromagnetic volume integral equation. The recurrence starts with an inscribed sphere within the particle and ends with the circumscribed sphere of the particle. For each iteration, a matrix known as the U-matrix is computed with the Gauss Legendre (GL) quadrature, and matrix inversion is subsequently performed to obtain the T-matrix corresponding to the portion of the particle enclosed by the spherical shell. We modify a commonly used scheme to avoid applying the quadrature scheme to discontinuities. Moreover, we apply a new scheme to generate nodes and weights in conjunction with the GL quadrature. This leads to a considerable improvement on convergence and computational efficiency in the cases of hexagonal prisms and spheroids. The basic shapes of ice crystals in the atmosphere are hexagonal columns and plates. The single-scattering properties of hexagonal ice prisms are important to atmospheric optics, radiative transfer, and remote sensing. We demonstrate that the present approach can significantly accelerate the convergence in simulating light scattering by hexagonal ice crystals.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  11. T. Wriedt and Y. Eremin, The Generalized Multipole Technique for Light Scattering. (Springer, 2018).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]

2017 (1)

2014 (3)

A. Townsend, T. Trogdon, and S. Olver, “Fast computation of Gauss quadrature nodes and weights on the whole real line,” IMA J. Numer. Anal. 36(1), 337–358 (2014).

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

I. Bogaert, “Iteration-Free Computation of Gauss–Legendre Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 36(3), A1008–A1026 (2014).
[Crossref]

2013 (2)

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[Crossref]

N. Hale and A. Townsend, “Fast and Accurate Computation of Gauss–Legendre and Gauss–Jacobi Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 35(2), A652–A674 (2013).
[Crossref]

2011 (1)

2007 (3)

A. Glaser, X. Liu, and V. Rokhlin, “A Fast Algorithm for the Calculation of the Roots of Special Functions,” SIAM J. Sci. Comput. 29(4), 1420–1438 (2007).
[Crossref]

P. C. Waterman, “The T-matrix revisited,” J. Opt. Soc. Am. A 24(8), 2257–2267 (2007).
[Crossref] [PubMed]

T. Wriedt, “Review of the null-field method with discrete sources,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 535–545 (2007).
[Crossref]

2001 (1)

1998 (1)

M. I. Mishchenko and L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
[Crossref]

1992 (1)

1991 (1)

1988 (1)

1980 (1)

1969 (1)

G. H. Golub and J. H. Welsch, “Calculation of Gauss Quadrature Rules,” Math. Comput. 23(106), 221–230 (1969).
[Crossref]

1965 (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53(8), 805–812 (1965).
[Crossref]

Auguié, B.

Bi, L.

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[Crossref]

Bogaert, I.

I. Bogaert, “Iteration-Free Computation of Gauss–Legendre Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 36(3), A1008–A1026 (2014).
[Crossref]

Glaser, A.

A. Glaser, X. Liu, and V. Rokhlin, “A Fast Algorithm for the Calculation of the Roots of Special Functions,” SIAM J. Sci. Comput. 29(4), 1420–1438 (2007).
[Crossref]

Golub, G. H.

G. H. Golub and J. H. Welsch, “Calculation of Gauss Quadrature Rules,” Math. Comput. 23(106), 221–230 (1969).
[Crossref]

Hale, N.

N. Hale and A. Townsend, “Fast and Accurate Computation of Gauss–Legendre and Gauss–Jacobi Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 35(2), A652–A674 (2013).
[Crossref]

Johnson, B. R.

Kahnert, F. M.

Kattawar, G. W.

B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044–24060 (2017).
[Crossref] [PubMed]

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[Crossref]

Khlebtsov, N. G.

Le Ru, E. C.

Liu, X.

A. Glaser, X. Liu, and V. Rokhlin, “A Fast Algorithm for the Calculation of the Roots of Special Functions,” SIAM J. Sci. Comput. 29(4), 1420–1438 (2007).
[Crossref]

Mishchenko, M. I.

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[Crossref]

M. I. Mishchenko and L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
[Crossref]

M. I. Mishchenko, “Light scattering by randomly oriented axially symmetric particles,” J. Opt. Soc. Am. A 8(6), 871–882 (1991).
[Crossref]

Olver, S.

A. Townsend, T. Trogdon, and S. Olver, “Fast computation of Gauss quadrature nodes and weights on the whole real line,” IMA J. Numer. Anal. 36(1), 337–358 (2014).

Rokhlin, V.

A. Glaser, X. Liu, and V. Rokhlin, “A Fast Algorithm for the Calculation of the Roots of Special Functions,” SIAM J. Sci. Comput. 29(4), 1420–1438 (2007).
[Crossref]

Somerville, W. R. C.

Stamnes, J. J.

Stamnes, K.

Sun, B.

Townsend, A.

A. Townsend, T. Trogdon, and S. Olver, “Fast computation of Gauss quadrature nodes and weights on the whole real line,” IMA J. Numer. Anal. 36(1), 337–358 (2014).

N. Hale and A. Townsend, “Fast and Accurate Computation of Gauss–Legendre and Gauss–Jacobi Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 35(2), A652–A674 (2013).
[Crossref]

A. Townsend, “The race to compute high-order Gauss-Legendre quadature,” SIAM J. Sci. Comput. (2015).

Travis, L. D.

M. I. Mishchenko and L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
[Crossref]

Trogdon, T.

A. Townsend, T. Trogdon, and S. Olver, “Fast computation of Gauss quadrature nodes and weights on the whole real line,” IMA J. Numer. Anal. 36(1), 337–358 (2014).

Waterman, P. C.

P. C. Waterman, “The T-matrix revisited,” J. Opt. Soc. Am. A 24(8), 2257–2267 (2007).
[Crossref] [PubMed]

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53(8), 805–812 (1965).
[Crossref]

Welsch, J. H.

G. H. Golub and J. H. Welsch, “Calculation of Gauss Quadrature Rules,” Math. Comput. 23(106), 221–230 (1969).
[Crossref]

Wiscombe, W. J.

Wriedt, T.

T. Wriedt, “Review of the null-field method with discrete sources,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 535–545 (2007).
[Crossref]

Yang, P.

B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25(20), 24044–24060 (2017).
[Crossref] [PubMed]

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[Crossref]

Zhang, X.

Appl. Opt. (4)

IMA J. Numer. Anal. (1)

A. Townsend, T. Trogdon, and S. Olver, “Fast computation of Gauss quadrature nodes and weights on the whole real line,” IMA J. Numer. Anal. 36(1), 337–358 (2014).

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transf. (4)

T. Wriedt, “Review of the null-field method with discrete sources,” J. Quant. Spectrosc. Radiat. Transf. 106(1-3), 535–545 (2007).
[Crossref]

M. I. Mishchenko and L. D. Travis, “Capabilities and limitations of a current FORTRAN implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers,” J. Quant. Spectrosc. Radiat. Transf. 60(3), 309–324 (1998).
[Crossref]

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transf. 116, 169–183 (2013).
[Crossref]

L. Bi and P. Yang, “Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. 138, 17–35 (2014).
[Crossref]

Math. Comput. (1)

G. H. Golub and J. H. Welsch, “Calculation of Gauss Quadrature Rules,” Math. Comput. 23(106), 221–230 (1969).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Proc. IEEE (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53(8), 805–812 (1965).
[Crossref]

SIAM J. Sci. Comput. (3)

I. Bogaert, “Iteration-Free Computation of Gauss–Legendre Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 36(3), A1008–A1026 (2014).
[Crossref]

N. Hale and A. Townsend, “Fast and Accurate Computation of Gauss–Legendre and Gauss–Jacobi Quadrature Nodes and Weights,” SIAM J. Sci. Comput. 35(2), A652–A674 (2013).
[Crossref]

A. Glaser, X. Liu, and V. Rokhlin, “A Fast Algorithm for the Calculation of the Roots of Special Functions,” SIAM J. Sci. Comput. 29(4), 1420–1438 (2007).
[Crossref]

Other (5)

M. Mischenko, L. Travis, and A. Lacis, Scattering, absorption, and emission of light by small particles, (Cambridge University Press,1970).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1983).

H. C. van de Hulst, Light Scattering by Small Particles. (Dover, 1981).

T. Wriedt and Y. Eremin, The Generalized Multipole Technique for Light Scattering. (Springer, 2018).

A. Townsend, “The race to compute high-order Gauss-Legendre quadature,” SIAM J. Sci. Comput. (2015).

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Figures (10)

Fig. 1
Fig. 1 Schematic of the radial recurrence. Particle in this schematic configuration is a particle with arbitrary geometry marked in green. Coordinate origin is on the geometric center of the particle. Two spherical shells are indicated by dashed circles. Arrows indicate the two radii r p and r p+1 .
Fig. 2
Fig. 2 Procedures in the sequence of expanding spherical shells in the recurrence. In the case of the spheroid, θ 1 =0. θ 2 is the angle where the spherical shell intersects the spheroid.
Fig. 3
Fig. 3 Procedures in the sequence of expanding spherical shells in the recurrence for a hexagonal column. θ 2 is the angle where the spherical shell intersects the column.
Fig. 4
Fig. 4 An example F mm' (r,θ) as function of θ for hexagonal column.
Fig. 5
Fig. 5 f 1 (x) and f 2 (x).
Fig. 6
Fig. 6 Convergence rate of the outputs ( Q ext , asymmetry factor (g factor), P 11 ( 180 )) with respect to number of quadrature nodes N q . Six panels correspond to 6 different spheroidal shapes indicated with size parameter x and aspect ratio a/b.
Fig. 7
Fig. 7 Convergence rate of the outputs ( Q ext , asymmetry factor (g factor), P 11 ( 180 )) with respect to number of quadrature nodes N q . Six panels correspond to 6 different hexagonal column shapes indicated with size parameter x and aspect ratio a/b.
Fig. 8
Fig. 8 Convergence rate of the outputs ( Q ext , asymmetry factor (g factor), P 11 ( 180 )) with respect to number of quadrature nodes N q . Six panels correspond to 6 different hexagonal column shapes indicated with size parameter x and aspect ratio a/b. Unlike Fig. 7, the integration range is split into 2 smooth intervals so the convergence rate is accelerated (see the text).
Fig. 9
Fig. 9 Phase matrix elements for hexagonal column with size parameter x = 225 (defined with the circumscribed sphere radius). Comparison is between PGOM (blue) and II-TM (red dashed). Particle refractive index is m=1.308+i1.43× 10 9 .
Fig. 10
Fig. 10 Phase matrix elements for hexagonal column with size parameter x = 225 (defined with the circumscribed sphere radius). Comparison is between PGOM (blue) and II-TM (red dashed). Particle refractive index is m=1.276+i0.413.

Tables (3)

Tables Icon

Table 1 Convergence rate of Gaussian quadrature for 2 different integrands.

Tables Icon

Table 2 CPU time required to run the same code section with Newton and Bogaert algorithms.

Tables Icon

Table 3 Acceleration for hexagonal column and spheroids of various sizes and aspect ratios.

Equations (39)

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E inc ( r )= E 0 n=0 m=n n a mn M mn (1) (k r )+ b mn N mn (1) (k r ),
E sca ( r )= E 0 n=0 m=n n p mn M mn (3) (k r )+ q mn N mn (3) (k r ),
( p mn q mn )= E 0 n ' =0 m ' = n ' n ' ( T mn m ' n ' 11 T mn m ' n ' 12 T mn m ' n ' 21 T mn m ' n ' 22 ) ( a m ' n ' b m ' n ' ).
p=Ta.
T( r p )= Q 11 ( r p )+[ I+ Q 12 ( r p ) ] [ IT( r p1 ) Q 22 ( r p ) ] 1 · T( r p )[ I+ Q 21 ( r p ) ].
Q 11 ( r p )=ik J T ( r p )Q( r p )J( r p ),
Q 12 ( r p )=ik J T ( r p )Q( r p )H( r p ),
Q 21 ( r p )=ik H T ( r p )Q( r p )J( r p ),
Q 22 ( r p )=ik H T ( r p )Q( r p )H( r p ),
J ¯ ¯ n ( r p )=( j n (k r p ) 0 0 0 1 kr r r j n (k r p ) n(n+1) j n (k r p ) k r p ),
H ¯ ¯ n ( r p )=( h n (1) (k r p ) 0 0 0 1 kr r r h n (1) (k r p ) n(n+1) h n (1) (k r p ) k r p ),
Q( r p )=Δr [ IΔrU( r p )g( r p , r p ) ] 1 U( r p ).
U mn m ' n ' ij = A mn m ' n ' 0 π dθsinθ F m m ' (r,θ) K mn m ' n ' ij (θ),i,j=1,2,
U mn m ' n ' 33 = A ' mn m ' n ' 0 π dθsinθ F ' m m ' (r,θ) d 0m n (θ) d 0 m ' n ' (θ),
F m m ' (r,θ)= 0 2π dφ e i(m m ' )φ [ ε(r,θ,φ)1 ] ,
F ' m m ' (r,θ)= 0 2π dφ e i(m m ' )φ [ ε(r,θ,φ)1 ] ε(r,θ,φ) ,
A mn m ' n ' = (kr) 2 (1) m+ m ' [ 2n+1 4πn(n+1) ] 1/2 [ 2 n ' +1 4π n ' ( n ' +1) ] 1/2 ,
A ' mn m ' n ' = n n ' (n+1)( n ' +1) A mn m ' n ' .
K mn m ' n ' ij (θ)= ( π mn (θ) π m ' n ' (θ)+ τ mn (θ) τ m ' n ' (θ) i[ π mn (θ) τ m ' n ' (θ)+ τ mn (θ) π m ' n ' (θ) ] i[ π mn (θ) τ m ' n ' (θ)+ τ mn (θ) π m ' n ' (θ) ] π mn (θ) π m ' n ' (θ)+ τ mn (θ) τ m ' n ' (θ) ),
π mn (θ)= m sinθ d 0m n (θ),
τ mn (θ)= d dθ d 0m n (θ).
ε(r,θ)1={ 0, θ 2 <θ, ε1,0<θ< θ 2 ,
[ π mn (θ) π m n ' (θ)+ τ mn (θ) τ m n ' (θ) ]sinθ = d dθ ( τ mn' (θ) d 0m n (θ)sinθ )+n'(n'+1)sinθ d 0m n (θ) d 0m n' (θ) = d dθ ( τ mn (θ) d 0m n' (θ)sinθ )+n(n+1)sinθ d 0m n (θ) d 0m n' (θ),
[ π mn (θ) τ m n ' (θ)+ τ mn (θ) π m n ' (θ) ]sinθ=m d dθ ( d 0m n (θ) d 0m n' (θ) ).
U mnm'n' 11 = U mnm'n' 22 = A mnm'n' 2π δ mm' [ε1] [ c nn' τ nn' ( θ 2 ) d 0m n ( θ 2 )sin θ 2 +n'(n'+1) 0 θ 2 dθsinθ d 0m n (θ) d 0m n' (θ) ], c nn' =1+ (1) n+n' ,
U mnm'n' 33 =A ' mnm'n' 2π δ mm' ε1 ε c nn' 0 θ 2 dθsinθ d 0m n (θ) d 0m n' (θ),
U mnm'n' 12 = U mnm'n' 21 = A mnm'n' 2π δ mm' [ε1]c ' nn' (im)[ d 0m n ( θ 2 ) d 0m n' ( θ 2 ) d 0m n (0) d 0m n' (0) ], c ' nn' =1+ (1) n+n'+1 ,
U mnm'n' 11 = U mnm'n' 22 = A mnm'n' c nn' θ 1 θ 2 dθsinθ F mm' (r,θ) K mnm'n' ii (θ), c nn' =1+ (1) n+n' ,
U mnm'n' 33 =A ' mnm'n' c nn' θ 1 θ 2 dθsinθF ' mm' (r,θ) d 0m n (θ) d 0m' n' (θ),
U mnm'n' 12 = U mnm'n' 21 = A mnm'n' c ' nn' θ 1 θ 2 dθsinθ F mm' (r,θ) K mnm'n' ij (θ), c ' nn' =1+ (1) n+n'+1 .
0 π (dθsinθ)( F mm' (r,θ) K mnm'n' ij (θ) ) l=1 N q w l F (r, θ l ) mm' K mnm'n' ij ( θ l ) .
0 1 f 1 (x)dx n=1 N q w n f 1 ( x n ) , f 1 (x)={ 1,0x0.5, 2(1x),0.5<x1.
0 1 f 2 (x)dx n=1 N q w n f 2 ( x n ) , f 2 (x)={ 1,0x0.5, 0.5[1+cos2π(x0.5)],0.5<x1.
0 θ 2 dθsinθ d 0m n (θ) d 0m n' (θ),
N=x+4.05 x 1/3 +5,
N MIE =x+4.05 x 1/3 +2,
Δr R r 0 =0.1,
θ 1 θ 2 dθsinθ F mm' (r,θ) K mnm'n' ij (θ).
θ 1 θ 2 dθsinθ F mm' (r,θ) K mnm'n' ij (θ) = θ 1 θ jump dθsinθ F mm' (r,θ) K mnm'n' ij (θ) + θ jump θ 2 dθsinθ F mm' (r,θ) K mnm'n' ij (θ) .

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