Abstract

In this paper we propose new algorithms for solution of light scattering on non-spherical particles using one-dimensional variant of discrete dipole approximation. We discuss recent advances in algorithms for matrices with structures in context of the discrete dipole approximation and show that it is possible to apply these advances to form non-iterative solvers and improve algorithmic complexity in case of many incoming plane parallel waves.

©2004 Optical Society of America

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References

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  1. P.C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, Art. No. 165404, (2003).
    [Crossref]
  2. R. E. Cline, R. J. Plemmons, and G. Worm, “Generalized inverses of certain Toeplitz matrices,” Linear Algbra and Its Applications 8,, 25–33 (1974).
    [Crossref]
  3. B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499, 1994.
    [Crossref]
  4. B. T. Draine and P. J. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT.6.0,” http://arxiv.org/abs/astro-ph/0309069, (2003).
  5. P. J. Flatau, G. L. Stephens, and B. T. Draine, “Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the block-Toeplitz structure,” J. Opt. Soc. Am. A 7, 593–600 (1990).
    [Crossref]
  6. I. Gohberg and V. Olshevsky, “Circulants, displacements and decompositions of matrices,” Integ. Equat. Oper. Th. 15, 730–743 (1992).
    [Crossref]
  7. I. Gohberg and V. Olshevsky, “Complexity of multiplication with vectors for structured matrices,” Linear Algebra Appl. 202, 163–192, (1994).
    [Crossref]
  8. J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16, 1198–1200 (1991).
    [Crossref] [PubMed]
  9. G. C. Schatz, “Electrodynamics of nonspherical noble metal nanoparticles and nanoparticle aggregates,”J. Mol. Struct.-Theochem 573, 73–80 (2001).
    [Crossref]
  10. M. Van Barel, G. Heinig, and P. A. Kravanja, “Stabilized superfast solver for nonsymmetric Toeplitz systems,” SIAM J. Matrix Anal. A 23, 494–510 (2001).
    [Crossref]

2003 (1)

P.C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, Art. No. 165404, (2003).
[Crossref]

2001 (2)

G. C. Schatz, “Electrodynamics of nonspherical noble metal nanoparticles and nanoparticle aggregates,”J. Mol. Struct.-Theochem 573, 73–80 (2001).
[Crossref]

M. Van Barel, G. Heinig, and P. A. Kravanja, “Stabilized superfast solver for nonsymmetric Toeplitz systems,” SIAM J. Matrix Anal. A 23, 494–510 (2001).
[Crossref]

1994 (2)

I. Gohberg and V. Olshevsky, “Complexity of multiplication with vectors for structured matrices,” Linear Algebra Appl. 202, 163–192, (1994).
[Crossref]

B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499, 1994.
[Crossref]

1992 (1)

I. Gohberg and V. Olshevsky, “Circulants, displacements and decompositions of matrices,” Integ. Equat. Oper. Th. 15, 730–743 (1992).
[Crossref]

1991 (1)

1990 (1)

1974 (1)

R. E. Cline, R. J. Plemmons, and G. Worm, “Generalized inverses of certain Toeplitz matrices,” Linear Algbra and Its Applications 8,, 25–33 (1974).
[Crossref]

Bryant, G. W.

P.C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, Art. No. 165404, (2003).
[Crossref]

Chaumet, P.C.

P.C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, Art. No. 165404, (2003).
[Crossref]

Cline, R. E.

R. E. Cline, R. J. Plemmons, and G. Worm, “Generalized inverses of certain Toeplitz matrices,” Linear Algbra and Its Applications 8,, 25–33 (1974).
[Crossref]

Draine, B. T.

Flatau, P. J.

Gohberg, I.

I. Gohberg and V. Olshevsky, “Complexity of multiplication with vectors for structured matrices,” Linear Algebra Appl. 202, 163–192, (1994).
[Crossref]

I. Gohberg and V. Olshevsky, “Circulants, displacements and decompositions of matrices,” Integ. Equat. Oper. Th. 15, 730–743 (1992).
[Crossref]

Goodman, J. J.

Heinig, G.

M. Van Barel, G. Heinig, and P. A. Kravanja, “Stabilized superfast solver for nonsymmetric Toeplitz systems,” SIAM J. Matrix Anal. A 23, 494–510 (2001).
[Crossref]

Kravanja, P. A.

M. Van Barel, G. Heinig, and P. A. Kravanja, “Stabilized superfast solver for nonsymmetric Toeplitz systems,” SIAM J. Matrix Anal. A 23, 494–510 (2001).
[Crossref]

Olshevsky, V.

I. Gohberg and V. Olshevsky, “Complexity of multiplication with vectors for structured matrices,” Linear Algebra Appl. 202, 163–192, (1994).
[Crossref]

I. Gohberg and V. Olshevsky, “Circulants, displacements and decompositions of matrices,” Integ. Equat. Oper. Th. 15, 730–743 (1992).
[Crossref]

Plemmons, R. J.

R. E. Cline, R. J. Plemmons, and G. Worm, “Generalized inverses of certain Toeplitz matrices,” Linear Algbra and Its Applications 8,, 25–33 (1974).
[Crossref]

Rahmani, A.

P.C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, Art. No. 165404, (2003).
[Crossref]

Schatz, G. C.

G. C. Schatz, “Electrodynamics of nonspherical noble metal nanoparticles and nanoparticle aggregates,”J. Mol. Struct.-Theochem 573, 73–80 (2001).
[Crossref]

Stephens, G. L.

Van Barel, M.

M. Van Barel, G. Heinig, and P. A. Kravanja, “Stabilized superfast solver for nonsymmetric Toeplitz systems,” SIAM J. Matrix Anal. A 23, 494–510 (2001).
[Crossref]

Worm, G.

R. E. Cline, R. J. Plemmons, and G. Worm, “Generalized inverses of certain Toeplitz matrices,” Linear Algbra and Its Applications 8,, 25–33 (1974).
[Crossref]

Integ. Equat. Oper. Th. (1)

I. Gohberg and V. Olshevsky, “Circulants, displacements and decompositions of matrices,” Integ. Equat. Oper. Th. 15, 730–743 (1992).
[Crossref]

J. Mol. Struct.-Theochem (1)

G. C. Schatz, “Electrodynamics of nonspherical noble metal nanoparticles and nanoparticle aggregates,”J. Mol. Struct.-Theochem 573, 73–80 (2001).
[Crossref]

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

Linear Algbra and Its Applications (1)

R. E. Cline, R. J. Plemmons, and G. Worm, “Generalized inverses of certain Toeplitz matrices,” Linear Algbra and Its Applications 8,, 25–33 (1974).
[Crossref]

Linear Algebra Appl. (1)

I. Gohberg and V. Olshevsky, “Complexity of multiplication with vectors for structured matrices,” Linear Algebra Appl. 202, 163–192, (1994).
[Crossref]

Opt. Lett. (1)

Phys. Rev. B (1)

P.C. Chaumet, A. Rahmani, and G. W. Bryant, “Generalization of the coupled dipole method to periodic structures,” Phys. Rev. B 67, Art. No. 165404, (2003).
[Crossref]

SIAM J. Matrix Anal. A (1)

M. Van Barel, G. Heinig, and P. A. Kravanja, “Stabilized superfast solver for nonsymmetric Toeplitz systems,” SIAM J. Matrix Anal. A 23, 494–510 (2001).
[Crossref]

Other (1)

B. T. Draine and P. J. Flatau, “User Guide for the Discrete Dipole Approximation Code DDSCAT.6.0,” http://arxiv.org/abs/astro-ph/0309069, (2003).

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Equations (20)

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k = 1 N A jk P k = E inc , j ( j = 1 , , N )
A jk P k = exp ( i k r jk ) r jk 3 { k 2 r jk × ( r jk × P k ) + ( 1 i k r jk ) r jk [ r jk 2 P k 3 r jk ( r jk · P k ) ] } ( j k ) ,
A j j = α j 1
A jk P k = f ( r jk ) P k + g ( r jk ) n jk ( n jk · P k )
A jk P k = ( f jk + g jk 0 0 0 f jk 0 0 0 f jk ) ( p x , k p y , k p z , k )
f ( r jk ) = exp ( i k r jk ) ( k 2 r jk + ik r jk 2 1 r jk 3 )
g ( r jk ) = exp ( i k r jk ) ( k 2 r jk 3 ik r jk 2 + 3 r jk 3 )
h ( r jk ) = f ( r jk ) + g ( r jk ) = exp ( i k r jk ) ( 2 ik r jk 2 2 r jk 3 ) .
( α 1 h 12 h 13 0 0 0 0 0 0 h 21 α 1 h 23 0 0 0 0 0 0 h 31 h 32 α 1 0 0 0 0 0 0 0 0 0 α 1 f 12 f 13 0 0 0 0 0 0 f 21 α 1 f 23 0 0 0 0 0 0 f 31 f 32 α 1 0 0 0 0 0 0 0 0 0 α 1 f 12 f 13 0 0 0 0 0 0 f 21 α 1 f 23 0 0 0 0 0 0 f 31 f 32 α 1 ) ( p x , 1 p x , 2 p x , 3 p y , 1 p y , 2 p y , 3 p z , 1 p z , 2 p z , 3 )
( α 1 h 12 h 13 h 1 N h 21 α 1 h 12 h 31 h 21 h 13 h 12 h N 1 h 31 h 21 α 1 ) ( P 1 P 2 P N ) x = ( E 1 E 2 E n ) inc , x
A ( { α 1 , h 12 , h 13 , , h 1 N } ) P x = E inc , x
A ( a ) A ( a c , a r ) = ( a 0 a 1 a n + 1 a 1 a 0 a 1 a 2 a 1 a n 1 a 2 a 0 )
Circ φ ( r ) = ( r 0 φ r n 1 φ r 1 r 1 r 0 φ r n 1 r 1 r 0 φ r n 1 r n 1 r 1 r 0 )
Circ φ ( r ) = D φ 1 Λ φ ( r ) D φ
D φ = diag ( ( ξ ) i = 0 n 1 ) and Λ φ ( r ) = diag ( D φ r )
Circ ( r ) = Λ ( r )
A ( a ) = 1 2 [ A ( a ) + A ( b ) ] + 1 2 [ A ( a ) A ( b ) ] = 1 2 [ A ( a + b ) + A ( a b ) ] .
A ( a ) d = 1 2 Circ 1 ( a + b ) d + 1 2 Cric 1 ( a b ) d = 1 2 Conv 1 ( a + b , d ) + 1 2 Conv 1 ( a b , d ) ,
A x = e 0 and A y = e n 1
A 1 = 1 x 0 ( φ ψ ) [ Circ ψ ( x ) Circ ( Z φ y ) Circ ψ ( Z ψ y ) Circ φ ( x ) ]

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