Abstract

General equations are derived for computing the amplitude matrix for a nonspherical particle in an arbitrary orientation and for arbitrary illumination and scattering directions with respect to the laboratory reference frame, provided that the scattering problem can be solved with respect to the particle reference frame. These equations are used along with the T-matrix method to provide benchmark results for homogeneous, dielectric, rotationally symmetric particles. The computer code is publicly available on the World-Wide Web at http://www.giss.nasa.gov/~crmim.

© 2000 Optical Society of America

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References

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  1. M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, San Diego, Calif., 1999).
  2. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).
  3. M. I. Mishchenko, “Multiple scattering of polarized light in anisotropic plane-parallel media,” Transp. Theory Stat. Phys. 19, 293–316 (1990).
    [CrossRef]
  4. J. L. Haferman, “Microwave scattering by precipitation,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 1999), pp. 481–524.
  5. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).
  6. W. J. Wiscombe, A. Mugnai, “Single scattering from nonspherical Chebyshev particles: a compendium of calculations,” NASA Ref. Publ. 1157 (1986).
  7. D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
    [CrossRef]
  8. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  9. M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
    [CrossRef]
  10. J. V. Dave, B. H. Armstrong, “Computation of high-order associated Legendre polynomials,” J. Quant. Spectrosc. Radiat. Transfer 10, 557–562 (1970).
    [CrossRef]
  11. M. I. Mishchenko, 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. Transfer 60, 309–324 (1998).
    [CrossRef]
  12. C. C. Chuang, K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
    [CrossRef]
  13. K. Aydin, “Centimeter and millimeter wave scattering from nonspherical hydrometeors,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 1999), pp. 451–479.
  14. M. I. Mishchenko, “Light scattering by randomly oriented rotationally symmetric particles,” J. Opt. Soc. Am. A 8, 871–882 (1991).
    [CrossRef]

1998 (1)

M. I. Mishchenko, 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. Transfer 60, 309–324 (1998).
[CrossRef]

1996 (1)

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

1991 (1)

1990 (2)

C. C. Chuang, K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
[CrossRef]

M. I. Mishchenko, “Multiple scattering of polarized light in anisotropic plane-parallel media,” Transp. Theory Stat. Phys. 19, 293–316 (1990).
[CrossRef]

1986 (1)

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

1971 (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

1970 (1)

J. V. Dave, B. H. Armstrong, “Computation of high-order associated Legendre polynomials,” J. Quant. Spectrosc. Radiat. Transfer 10, 557–562 (1970).
[CrossRef]

Armstrong, B. H.

J. V. Dave, B. H. Armstrong, “Computation of high-order associated Legendre polynomials,” J. Quant. Spectrosc. Radiat. Transfer 10, 557–562 (1970).
[CrossRef]

Aydin, K.

K. Aydin, “Centimeter and millimeter wave scattering from nonspherical hydrometeors,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 1999), pp. 451–479.

Barber, P. W.

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

Beard, K. V.

C. C. Chuang, K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
[CrossRef]

Chuang, C. C.

C. C. Chuang, K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
[CrossRef]

Dave, J. V.

J. V. Dave, B. H. Armstrong, “Computation of high-order associated Legendre polynomials,” J. Quant. Spectrosc. Radiat. Transfer 10, 557–562 (1970).
[CrossRef]

Haferman, J. L.

J. L. Haferman, “Microwave scattering by precipitation,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 1999), pp. 481–524.

Hill, S. C.

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

Khersonskii, V. K.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, 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. Transfer 60, 309–324 (1998).
[CrossRef]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

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

M. I. Mishchenko, “Multiple scattering of polarized light in anisotropic plane-parallel media,” Transp. Theory Stat. Phys. 19, 293–316 (1990).
[CrossRef]

Moskalev, A. N.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
[CrossRef]

Mugnai, A.

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

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Travis, L. D.

M. I. Mishchenko, 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. Transfer 60, 309–324 (1998).
[CrossRef]

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

Varshalovich, D. A.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Wiscombe, W. J.

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

J. Atmos. Sci. (1)

C. C. Chuang, K. V. Beard, “A numerical model for the equilibrium shape of electrified raindrops,” J. Atmos. Sci. 47, 1374–1389 (1990).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (3)

M. I. Mishchenko, L. D. Travis, D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

J. V. Dave, B. H. Armstrong, “Computation of high-order associated Legendre polynomials,” J. Quant. Spectrosc. Radiat. Transfer 10, 557–562 (1970).
[CrossRef]

M. I. Mishchenko, 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. Transfer 60, 309–324 (1998).
[CrossRef]

NASA Ref. Publ. (1)

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

Phys. Rev. D (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Transp. Theory Stat. Phys. (1)

M. I. Mishchenko, “Multiple scattering of polarized light in anisotropic plane-parallel media,” Transp. Theory Stat. Phys. 19, 293–316 (1990).
[CrossRef]

Other (6)

J. L. Haferman, “Microwave scattering by precipitation,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 1999), pp. 481–524.

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

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds., Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications (Academic, San Diego, Calif., 1999).

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985).

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
[CrossRef]

K. Aydin, “Centimeter and millimeter wave scattering from nonspherical hydrometeors,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 1999), pp. 451–479.

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Figures (3)

Fig. 1
Fig. 1

Spherical coordinate system used to specify the direction and the polarization state of a transverse electromagnetic wave.

Fig. 2
Fig. 2

Transformation of the laboratory reference system xyz into the particle reference frame xyz′.

Fig. 3
Fig. 3

Thick curve, the shape of the generalized Chebyshev particle used in the computation of expression (40); thin curve, the equal-volume sphere.

Equations (40)

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nˆ=ϑˆL×φˆL.
EincR=EϑLincϑˆLinc+EφLincφˆLincexpiknˆincR
EscaR=EϑLscaR,nˆscaϑˆLsca+EφLscaR,nˆscaφˆLsca,  nˆsca=RR=ϑˆLsca×φˆLsca,
nˆsca · EscaR=0,
EϑLscaEφLsca=expikRR SLnˆsca, nˆinc; α, β, γEϑLincEφLinc,
EϑPscaEφPsca=expikRRSPnˆsca, nˆincEϑPincEφPinc.
EϑPϑP, φPEφPϑP, φP=ρˆnˆ; α, β, γEϑLϑL, φLEφLϑL, φL,
SLϑLsca, φLsca; ϑLinc, φLinc; α, β,γ=ρˆ-1nˆsca; α, β, γ×SPϑPsca, φPsca; ϑPinc, φPincρˆnˆinc; α, β, γ.
cos ϑP=cos  ϑL cos β+sin ϑL sin  β cosφL-α,
cosφP=1sinϑPcos β cos  γ sin ϑL cosφL-α +sin γ sin ϑL sinφL-α-sin β cos γ cos ϑL,
sinφP=1sinϑP-cos β sin γ sin ϑL cosφL-α +cos γ sin ϑL sinφL-α+sin β sin γ cos ϑL.
ExEyEz=αˆϑ, φEϑEφ,
ExPEyPEzP=βˆα, β, γExLEyLEzL.
ρˆnˆ; α, β,γ =αˆ-1ϑP, φPβˆα, β, γαˆϑL, φL.
αˆϑ, φ=cos ϑ cos φ-sin φcos ϑ sin φcos φ-sin ϑ0,
αˆ-1ϑ, φ=cos ϑ cos φcos ϑ sin φ-sin ϑ-sin φcos φ0,
βˆα, β, γ=cos α cos β cos γ - sin α sin γsin α cos β cos γ + cos α sin γ-sin β cos γ-cos α cos β sin γ - sin α cos γ-sin α cos β sin γ + cos α cos γsin β sin γcos α sin βsin α sin βcos β.
ρˆnˆ; α=0, β=0, γ=01001,
SLϑLsca, φLsca; ϑLinc, φLinc; 0, 0, 0=SPϑPsca, φPsca; ϑPinc, φPinc.
cos φP=1sin ϑPcos β sin ϑL cosφL-α-sin β cos ϑL,
sin φP=sin ϑL sinφL-αsin ϑP,
βˆα, β, γ=0=cos α cos βsin α cos β-sin β-sin αcos α0cos α sin βsin α sin βcos β.
TmnmnklP=δmmTmnmnklP,  k, l=1, 2,
S11Pnˆsca, nˆinc=1kn=1n=1m=-maxn,nmaxn,n×αmnnTmnmn11Pπmnϑscaπmnϑinc+Tmnmn21Pτmnϑscaπmnϑinc+Tmnmn12Pπmnϑscaτmnϑinc+Tmnmn22Pτmnϑscaτmnϑinc,
S12Pnˆsca, nˆinc=-ikn=1n=1m=-maxn,nmaxn,n×αmnnTmnmn11Pπmnϑscaτmnϑinc+Tmnmn21Pτmnϑscaτmnϑinc+Tmnmn12Pπmnϑscaπmnϑinc+Tmnmn22Pτmnϑscaπmnϑinc,
S21Pnˆsca, nˆinc=ikn=1n=1m=-maxn,nmaxn,n×αmnnTmnmn11Pτmnϑscaπmnϑinc+Tmnmn21Pπmnϑscaπmnϑinc+Tmnmn12Pτmnϑscaτmnϑinc+Tmnmn22Pπmnϑscaτmnϑinc,
S22Pnˆsca, nˆinc=1kn=1n=1m=-maxn,nmaxn,n×αmnnTmnmn11Pτmnϑscaτmnϑinc+Tmnmn21Pπmnϑscaτmnϑinc+Tmnmn12Pτmnϑscaπmnϑinc+Tmnmn22Pπmnϑscaπmnϑinc,
αmnn=in-n-12n+12n+1nn+1nn+11/2×expimφsca-φinc,
πmnϑ=md0mnϑsin ϑ,  τmnϑ=dd0mnϑdϑ,
Pnmcos ϑ=-1mn+m!n-m!1/2 d0mnϑ,
n+12-m21/2d0mn+1ϑ=2n+1cos ϑd0mnϑ-n2-m2d0mn-1ϑ
d0mm-1ϑ=0,
d0mmϑ=Am1-cos2ϑm/2,
A0=1,  Am+1=Am2m+12m+11/2
τmnϑ=12n+1sin ϑ-n+1n2-m2d0mn-1ϑ+n+12-m21/2d0mn+1ϑ.
rϑ, φ=r01+ n=0N cn cosnϑ,
-5.0941+24.402i-1.9425+1.9971i-1.1521-3.0978i-6.9323+24.748i,
-1.727+19.706i-0.562+0.247i-2.013-2.398i-3.088+20.401i,
4.5123+18.092i-1.6350+3.5274i-3.0970-0.9215i3.2658+18.617i,
11.307+9.6184i-2.6519+2.3589i-4.9044-0.6241i9.9947+11.295i.

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