Abstract

We suggest a new approach to calculate the inverse matrix in scattering calculations using the T-matrix method. Instead of inversion of the full matrix, we suggest the inversion of two matrices, each of which contains half the number of rows. This approach allows significant time savings and a noticeable increase of the precision of scattering calculations due to fewer arithmetical operations. An iterative method can be applied to matrices whose dimension is also divisible by factors of 2, which can further increase the time savings and accuracy.

© 2007 Optical Society of America

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References

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  1. P. C. Waterman, Phys. Rev. D 3, 825 (1971).
    [CrossRef]
  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. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, J. Quant. Spectrosc. Radiat. Transf. 55, 535 (1996).
    [CrossRef]
  4. M. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transf. 88, 357 (2004).
  5. D. Petrov, E. Zubko, E. Synelnyk, and Yu. Shkuratov, 'Modeling polarization properties of lunar regolith with t-matrix approach,' 42nd International Microsymposium on Comparative Planetology Moscow (CD-ROM) (V.I. Vernadsky Institute, 2005), Abstract #M42lowbar56.
  6. D. V. Petrov, E. N. Synelnyk, Yu. G. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 111 (2006).
    [CrossRef]
  7. D. A. Ligon, T. W. Chen, and J. B. Gillespie, Appl. Opt. 35, 4297 (1996).
    [CrossRef] [PubMed]
  8. D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 85 (2006).
    [CrossRef]

2006 (2)

D. V. Petrov, E. N. Synelnyk, Yu. G. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 111 (2006).
[CrossRef]

D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 85 (2006).
[CrossRef]

2004 (1)

M. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transf. 88, 357 (2004).

1996 (2)

D. A. Ligon, T. W. Chen, and J. B. Gillespie, Appl. Opt. 35, 4297 (1996).
[CrossRef] [PubMed]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, J. Quant. Spectrosc. Radiat. Transf. 55, 535 (1996).
[CrossRef]

1971 (1)

P. C. Waterman, Phys. Rev. D 3, 825 (1971).
[CrossRef]

Babenko, V. A.

M. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transf. 88, 357 (2004).

Chen, T. W.

Gillespie, J. B.

Khlebtsov, N. G.

M. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transf. 88, 357 (2004).

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).

Ligon, D. A.

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, J. Quant. Spectrosc. Radiat. Transf. 55, 535 (1996).
[CrossRef]

Mishchenko, M.

M. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transf. 88, 357 (2004).

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, J. Quant. Spectrosc. Radiat. Transf. 55, 535 (1996).
[CrossRef]

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

Petrov, D.

D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 85 (2006).
[CrossRef]

D. Petrov, E. Zubko, E. Synelnyk, and Yu. Shkuratov, 'Modeling polarization properties of lunar regolith with t-matrix approach,' 42nd International Microsymposium on Comparative Planetology Moscow (CD-ROM) (V.I. Vernadsky Institute, 2005), Abstract #M42lowbar56.

Petrov, D. V.

D. V. Petrov, E. N. Synelnyk, Yu. G. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 111 (2006).
[CrossRef]

Shkuratov, Yu.

D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 85 (2006).
[CrossRef]

D. Petrov, E. Zubko, E. Synelnyk, and Yu. Shkuratov, 'Modeling polarization properties of lunar regolith with t-matrix approach,' 42nd International Microsymposium on Comparative Planetology Moscow (CD-ROM) (V.I. Vernadsky Institute, 2005), Abstract #M42lowbar56.

Shkuratov, Yu. G.

D. V. Petrov, E. N. Synelnyk, Yu. G. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 111 (2006).
[CrossRef]

Synelnyk, E.

D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 85 (2006).
[CrossRef]

D. Petrov, E. Zubko, E. Synelnyk, and Yu. Shkuratov, 'Modeling polarization properties of lunar regolith with t-matrix approach,' 42nd International Microsymposium on Comparative Planetology Moscow (CD-ROM) (V.I. Vernadsky Institute, 2005), Abstract #M42lowbar56.

Synelnyk, E. N.

D. V. Petrov, E. N. Synelnyk, Yu. G. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 111 (2006).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, J. Quant. Spectrosc. Radiat. Transf. 55, 535 (1996).
[CrossRef]

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

Videen, G.

D. V. Petrov, E. N. Synelnyk, Yu. G. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 111 (2006).
[CrossRef]

D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 85 (2006).
[CrossRef]

M. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transf. 88, 357 (2004).

Waterman, P. C.

P. C. Waterman, Phys. Rev. D 3, 825 (1971).
[CrossRef]

Wriedt, T.

M. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transf. 88, 357 (2004).

Zubko, E.

D. Petrov, E. Zubko, E. Synelnyk, and Yu. Shkuratov, 'Modeling polarization properties of lunar regolith with t-matrix approach,' 42nd International Microsymposium on Comparative Planetology Moscow (CD-ROM) (V.I. Vernadsky Institute, 2005), Abstract #M42lowbar56.

Appl. Opt. (1)

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

D. Petrov, E. Synelnyk, Yu. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 85 (2006).
[CrossRef]

D. V. Petrov, E. N. Synelnyk, Yu. G. Shkuratov, and G. Videen, J. Quant. Spectrosc. Radiat. Transf. 102, 111 (2006).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, J. Quant. Spectrosc. Radiat. Transf. 55, 535 (1996).
[CrossRef]

M. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, J. Quant. Spectrosc. Radiat. Transf. 88, 357 (2004).

Phys. Rev. D (1)

P. C. Waterman, Phys. Rev. D 3, 825 (1971).
[CrossRef]

Other (2)

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

D. Petrov, E. Zubko, E. Synelnyk, and Yu. Shkuratov, 'Modeling polarization properties of lunar regolith with t-matrix approach,' 42nd International Microsymposium on Comparative Planetology Moscow (CD-ROM) (V.I. Vernadsky Institute, 2005), Abstract #M42lowbar56.

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Figures (3)

Fig. 1
Fig. 1

Inversion time as a function of matrix size N. Points and rhombs correspond to the Gaussian inversion of the full matrix and the optimized inversion method, respectively. Note that in the optimized method we only show points for N = N max ( N max + 2 ) , where N max is divisible by 2. In this way the iterative method is used multiple times to emphasize the time savings. An actual curve showing all N appears extremely noisy, since the iterative method cannot be used for odd N, yielding significantly lower time savings for these points.

Fig. 2
Fig. 2

Intensity and degree of linear polarization for a sphere with size parameter X = 20.0 and refractive index m 0 = 1.5 + 0.1 i calculated with the optimized inversion method (solid curve) and Mie theory (points). In this case the number of rows in the matrix N = 1350 , and the time savings with our approach is approximately 75% (see Fig. 1).

Fig. 3
Fig. 3

Relative error in intensity, ε = I Mie I T matrix I Mie , using the T-matrix method with the Gaussian inversion of the full matrix (points) and the optimized inversion method (solid curve).

Equations (24)

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E inc ( ρ , γ , ϕ ) = n = 1 m = n n [ a m n Rg M m n ( ρ , γ , ϕ ) + b m n Rg N m n ( ρ , γ , ϕ ) ] ,
E sca ( ρ , γ , ϕ ) = n = 1 m = n n [ p m n M m n ( ρ , γ , ϕ ) + q m n N m n ( ρ , γ , ϕ ) ] ,
p m n = n = 1 m = n n [ T m n m n 11 a m n + T m n m n 12 b m n ] ,
q m n = n = 1 m = n n [ T m n m n 21 a m n + T m n m n 22 b m n ] ,
T = ( T 11 T 12 T 21 T 22 ) .
T = ( Rg Q ) ( Q ) 1 .
Rg Q = ( Rg Q 11 Rg Q 12 Rg Q 21 Rg Q 22 ) ,
Q = ( Q 11 Q 12 Q 21 Q 22 ) .
( T 11 T 12 T 21 T 22 ) ( Q 11 Q 12 Q 21 Q 22 ) = ( Rg Q 11 Rg Q 12 Rg Q 21 Rg Q 22 ) .
{ T 11 Q 11 + T 12 Q 21 = Rg Q 11 T 11 Q 12 + T 12 Q 22 = Rg Q 12 } ,
{ T 21 Q 11 + T 22 Q 21 = Rg Q 21 T 21 Q 12 + T 22 Q 22 = Rg Q 22 } .
T 12 = ( Rg Q 11 ( Q 11 ) 1 Q 12 + Rg Q 12 ) ( Q 21 ( Q 11 ) 1 Q 12 Q 22 ) 1 ,
T 11 = ( Rg Q 11 + T 12 Q 21 ) ( Q 11 ) 1 ,
T 22 = ( Rg Q 21 ( Q 11 ) 1 Q 12 + Rg Q 22 ) ( Q 21 ( Q 11 ) 1 Q 12 Q 22 ) 1 ,
T 21 = ( Rg Q 21 + T 22 Q 21 ) ( Q 11 ) 1 .
Q = ( Q 11 Q 12 Q 21 Q 22 ) ,
( Q 11 ) 1 Q 11 = A F = I ,
F = ( F 11 F 12 F 21 F 22 )
A = ( A 11 A 12 A 21 A 22 ) = ( F ) 1
( A 11 A 12 A 21 A 22 ) ( F 11 F 12 F 21 F 22 ) = ( I 0 0 I ) .
A 22 = ( F 21 ( F 11 ) 1 F 12 F 22 ) 1 ,
A 12 = ( F 11 ) 1 F 12 A 22 ,
A 11 = ( F 11 ) 1 A 12 F 21 ( F 11 ) 1 ,
A 21 = A 22 F 21 ( F 11 ) 1 .

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