Abstract

A computational method, based on a moment solution to the discrete dipole approximation (DDA) interaction equations, is proposed for calculation of the T matrix of arbitrary-shaped particles. It is shown that the method will automatically provide the conservation-of-energy and origin-invariance properties required of the T matrix. Furthermore, the method is significantly faster than a T-matrix calculation by direct inversion of the DDA equations. Because the method retains the dipole lattice representation of the particle, it can be applied with relative ease to particles with irregular shapes—although in the same respect it will not automatically simplify for axisymmetric particles. Calculations of scattering matrix distributions, in fixed and random orientations, are made for tetrahedron, cylindrical, and prolate spheroid particle shapes and compared with DDA and extended boundary condition method results.

© 2002 Optical Society of America

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References

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  1. P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
    [CrossRef]
  2. M. I. Mishchenko, L. D. Travis, A. Macke, “T matrix methods and its applications,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 6.
  3. D. W. Mackowski, “An effective medium method for calculation of the t matrix of aggregated spheres,” J. Quant. Spectrosc. Radiat. Transf. 70, 441–464 (2001).
    [CrossRef]
  4. B. T. Draine, “The discrete dipole approximation,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 5.
  5. D. W. Mackowski, M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
    [CrossRef]
  6. K. A. Fuller, D. W. Mackowski, “Electromagnetic scattering by compounded spherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 8.
  7. J. J. Goodman, B. T. Draine, P. J. Flatau, “Application of fast Fourier transform techniques to the discrete-dipole approximation,” Opt. Lett. 16, 1198–1200 (1991).
    [CrossRef] [PubMed]
  8. W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
    [CrossRef]
  9. P. C. Waterman, N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609–2618 (1986).
    [CrossRef]
  10. V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiment,” Radio Sci. 18, 321–327 (1983).
    [CrossRef]
  11. B. T. Draine, J. J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
    [CrossRef]
  12. T. Wriedt, A. Doicu, “Formulations of the extended boundary condition method for three dimensional scattering using the method of discrete sources,” J. Mod. Opt. 45, 199–213 (1998).
    [CrossRef]
  13. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986).

2001 (1)

D. W. Mackowski, “An effective medium method for calculation of the t matrix of aggregated spheres,” J. Quant. Spectrosc. Radiat. Transf. 70, 441–464 (2001).
[CrossRef]

1998 (1)

T. Wriedt, A. Doicu, “Formulations of the extended boundary condition method for three dimensional scattering using the method of discrete sources,” J. Mod. Opt. 45, 199–213 (1998).
[CrossRef]

1996 (1)

1993 (1)

B. T. Draine, J. J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

1991 (1)

1986 (2)

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

P. C. Waterman, N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609–2618 (1986).
[CrossRef]

1983 (1)

V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiment,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

1971 (1)

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

Bringi, V. N.

V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiment,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

Doicu, A.

T. Wriedt, A. Doicu, “Formulations of the extended boundary condition method for three dimensional scattering using the method of discrete sources,” J. Mod. Opt. 45, 199–213 (1998).
[CrossRef]

Draine, B. T.

B. T. Draine, J. J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

J. J. Goodman, B. T. Draine, P. J. Flatau, “Application of fast Fourier transform techniques to the discrete-dipole approximation,” Opt. Lett. 16, 1198–1200 (1991).
[CrossRef] [PubMed]

B. T. Draine, “The discrete dipole approximation,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 5.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986).

Flatau, P. J.

Fuller, K. A.

K. A. Fuller, D. W. Mackowski, “Electromagnetic scattering by compounded spherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 8.

Ghoul, W. A.

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

Goodman, J. J.

B. T. Draine, J. J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

J. J. Goodman, B. T. Draine, P. J. Flatau, “Application of fast Fourier transform techniques to the discrete-dipole approximation,” Opt. Lett. 16, 1198–1200 (1991).
[CrossRef] [PubMed]

Ishimaru, A.

V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiment,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

Macke, A.

M. I. Mishchenko, L. D. Travis, A. Macke, “T matrix methods and its applications,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 6.

Mackowski, D. W.

D. W. Mackowski, “An effective medium method for calculation of the t matrix of aggregated spheres,” J. Quant. Spectrosc. Radiat. Transf. 70, 441–464 (2001).
[CrossRef]

D. W. Mackowski, M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[CrossRef]

K. A. Fuller, D. W. Mackowski, “Electromagnetic scattering by compounded spherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 8.

McClain, W. M.

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

Mishchenko, M. I.

D. W. Mackowski, M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. Macke, “T matrix methods and its applications,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 6.

Pedersen, N. E.

P. C. Waterman, N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609–2618 (1986).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986).

Travis, L. D.

M. I. Mishchenko, L. D. Travis, A. Macke, “T matrix methods and its applications,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 6.

Varadan, V. K.

V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiment,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

Varadan, V. V.

V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiment,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986).

Waterman, P. C.

P. C. Waterman, N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609–2618 (1986).
[CrossRef]

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

Wriedt, T.

T. Wriedt, A. Doicu, “Formulations of the extended boundary condition method for three dimensional scattering using the method of discrete sources,” J. Mod. Opt. 45, 199–213 (1998).
[CrossRef]

Astrophys. J. (1)

B. T. Draine, J. J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

J. Appl. Phys. (1)

P. C. Waterman, N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59, 2609–2618 (1986).
[CrossRef]

J. Chem. Phys. (1)

W. M. McClain, W. A. Ghoul, “Elastic light scattering by randomly oriented macromolecules: computation of the complete set of observables,” J. Chem. Phys. 84, 6609–6622 (1986).
[CrossRef]

J. Mod. Opt. (1)

T. Wriedt, A. Doicu, “Formulations of the extended boundary condition method for three dimensional scattering using the method of discrete sources,” J. Mod. Opt. 45, 199–213 (1998).
[CrossRef]

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

J. Quant. Spectrosc. Radiat. Transf. (1)

D. W. Mackowski, “An effective medium method for calculation of the t matrix of aggregated spheres,” J. Quant. Spectrosc. Radiat. Transf. 70, 441–464 (2001).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. D (1)

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

Radio Sci. (1)

V. K. Varadan, V. N. Bringi, V. V. Varadan, A. Ishimaru, “Multiple scattering theory for waves in discrete random media and comparison with experiment,” Radio Sci. 18, 321–327 (1983).
[CrossRef]

Other (4)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986).

M. I. Mishchenko, L. D. Travis, A. Macke, “T matrix methods and its applications,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 6.

B. T. Draine, “The discrete dipole approximation,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 5.

K. A. Fuller, D. W. Mackowski, “Electromagnetic scattering by compounded spherical particles,” in Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications, M. I. Mishchenko, J. W. Hovenier, L. D. Travis, eds. (Academic, San Diego, Calif., 2000), Chap. 8.

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Figures (12)

Fig. 1
Fig. 1

DDA, DDMM, and DDSCAT S11 predictions: tetrahedron.

Fig. 2
Fig. 2

DDMM S11 predictions for equivalent incident directions: effect of mesh size.

Fig. 3
Fig. 3

DDMM orientation-averaged S11 predictions: effect of mesh size.

Fig. 4
Fig. 4

DDMM orientation-averaged Sij/S11 predictions: effect of mesh size.

Fig. 5
Fig. 5

DDA and DDMM S11 predictions: circular cylinder.

Fig. 6
Fig. 6

DDA and DDMM S12 predictions: circular cylinder. The legend is the same as that in Fig. 5.

Fig. 7
Fig. 7

DDA and DDMM orientation-averaged S11 predictions: circular cylinder.

Fig. 8
Fig. 8

DDA and DDMM orientation-averaged Sij predictions: circular cylinder. The legend is the same as that in Fig. 7.

Fig. 9
Fig. 9

EBCM and DDMM orientation-averaged S11 predictions: prolate spheroid.

Fig. 10
Fig. 10

EBCM and DDMM orientation-averaged Sij/S11 predictions: prolate spheroid.

Fig. 11
Fig. 11

EBCM and DDMM orientation-averaged S11 predictions: cylinder.

Fig. 12
Fig. 12

EBCM and DDMM orientation-averaged Sij/S11 predictions: cylinder.

Tables (1)

Tables Icon

Table 1 Orientation-Averaged Efficiency and Asymmetry Factor Predictions

Equations (73)

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Es,i(ri)=m=-11amiNm11(3)(ri)
1αami-jiNdk=-11Hmkijakjj=1Ndk=-11Amkijakj=pmi,
Es=n=1m=-nnp=12amnpNmnp(3)(r0).
amnp=i=1Ndk=-11Jmnp k110iaki,
amnp=l=1NOk=-llq=12Tmnp klqpklq,
pmi=l=1NOk=-1lq=12Jm11 klqi0pklq.
1αTmklqi-jiNdm=-11HmmijTmklqj
m=-11j=1NdAmmijTmklqj=Jm11 klqi0.
Tmnp klq=m=-11i=1NdJmnp m110iTm klqi.
Qext=-2xV2Re l=1k=-1lq=12Tklq klq.
Tklq mnp=(-1)m+kT-mnp-klq.
(Hmkij)*=(Jm11 k11ij)*-i(Ym11 k11ij)*=Jk11 m11ji-iYk11 m11ji.
Jm11 k11ij=m,n,pJm11 mnpi0Jmnp0j.
ijim,k=-11[(Tmνi)*HmkijTkμj+Tmμj(HmkjiTkνi)*]
=i,jm,k=-11[(Tmνi)*Jm11 k11ijTkμj+Tmμj(Jm11 k11jiTkνi)*]-2im=-11(Tmνi)*Tmμi=2νTνν*Tνμ-2im=-11(Tmνi)*Tmμi.
νTνν*Tνμ-Re1α+1im=-11(Tmνi)*Tmμi
=-12(Tμν*+Tνμ).
Qsca=2xV2n,m,pl,k,q|Tmnp klq|2
Qabs=-2xV2Re1α+1m=-11l,k,qi=1Nd|Tm klqi|2.
Tνμ(0)=ν,μJνν00Tνμ(0)Jμμ00.
Eint=νdνNν(1)(mpr),
dν=μWνμpμ.
Tm klqi=l,k,qJm11 klqi0Wklq klq,
μ1αJmμi0-jik=-11HmkijJkμj0Wμμ=Jmμi0.
μAνμWμμ=Bνμ,
Aνμ=m=-11i=1Nd(Jmνi0)*1αJmμi0-jiNdk=-11HmkijJkμj0,
Bνμ=m=-11i=1Nd(Jmνi0)*Jmμi0.
Tνμ=μm=-11i=1NdJνm0iJmμi0Wμμ
=μBμν*Wμμ.
T=BA-1B,
12(Aνμ+Aμν*)=Re1α+1m=-11i=1Nd(Jmνi0)*Jmμi0-μBνμBμμ*.
Wνμ(0)=ν,μJνν00Wνμ(0)Jμμ00,
Aνμ(0)=ν,μ(Jνν00)*Aνμ(0)Jμμ00,
Bνμ(0)=ν,μ(Jνν00)*Bνμ(0)Jμμ00,
(Jmνi0)*jiNdk=-11HmkijJkμj0
=(Jmνi0)*jiNdk=-11νHmkijJkνjiJνμi0.
jiNdHmkijJkνjijiNLHmkijJkνji+1xd3V-VLHmkijJkνji dVij,
k=-11V-VLHmkijJkνji dVij=Qmνi-Fδνm,
Qmνi=1mp2-102π-11n·k=-11(JkνjiHmkij-HmkijJkνji)rij2 d(cos θij)dϕij,
F=xL2mp2-102π-11k=-11JkmjiddrHmkij-HmkijddrJkmjir=xLd(cos θij)dϕij.
BνμRg Qμν*=1mp2-102π-11n·m=-11(Jmνj0Jμm0j-Jμm0jJmνj0)r0j2 d(cos θ0j)dϕ0j,
m=-11i=1Nd(Jmνi0)*ν1αδνm-jiNLk=-11HmkijJkνji
-1xd3(Qmνi-Fδνm)μJνμi0Wμμ=1xd3Rg Qμν*.
3α-jiNLm=-11k=-11HmkijJkmji+3xd3F=0.
μAνμWμμ=-Rg Qμν*,
Tνμ=ν(Rg Qνν)Wνμ
Aνμ=i=1Ndm=-11μ(Jμνi0)*Jμμi0Qmμi.
Qνμ=m,μJνm0iQmμiJμμi0,
Tνμ=-ν(Rg Qνν)Qνμ-1.
Aνμ=Cνμ-iDνμ,
Cνμ=1α+1m=-11i=1Nd(Jmνi0)*Jmμi0-μBνμBμμ*,
Dνμ=m=-11i=1Nd(Jmνi0)*k=-11jiNdYmkijJkμj0.
1-13k=-11n=1NO,im=-nnp=12|Jk11 mnpi0|2,
Amnp klq, Tmnp klq=0,mk
Amnp klq, Tmnp klq=0,mod(m+k, 4)0.
Nmn2(3)=(2n+1)(n-m)!n(n+1)(n+m)!1/2×rumn,
Nmn1(3)=×Nmn2(3),
umn=hn(r)Pnm(cos θ)exp(imϕ).
Hmnp klqij=-(-1)min-l[(2n+1)(2l+1)]1/2×wiw(w-k+m)!(w+k-m)!1/2C-mn klwC-1n 1lw×hw(rij)Pwk-m(cos θij)exp[i(k-m)ϕij],
Cmn klw=C(n, m; l, k; w, m+k).
m=-1102π-11Hm11 m11ijJm11 m11ji d(cos θij)dϕij
=4πh0(rij)j0(mprij)+12h2(rij)j2(mprij)
4πg(rij).
F(xL)=6πmp(mp2-1)[mpξ1(xL)ψ1(mpxL)-ξ1(xL)ψ1(mpxL)],
1α=j=1LMjg(xj)-1xd3F(xV,L).
NL=1+j=1LML,
F(xb)-F(xa)=xbxag(x)x2 dx,
j=LL+1Mjg(xj)4πxd3xV,LxV,L+1g(x)x2 dx=1xd3[F(xV,L+1)-F(xV,L)].
xd3α=j=1L{xd3Mjg(xj)-[F(xV,j)-F(xV,j-1)]}-F(xV,0),
xd3Mjg(xj)-[F(xV,j)-F(xV,j-1)]
=-ixd221+mp210Mjx¯j-2π(π¯V,j2-π¯V,j-12)+xd3Mj-4π3(x¯V,j3-x¯V,j-13)+,
j=1Mjx¯j-2π(x¯V,j2-x¯V,j-12)b13.448.
xd3α=-2πimp2-2mp2+1+ib21+mp210xd2-xd3,

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