Abstract

The discrete-dipole approximation (DDA) is a powerful method for calculating absorption and scattering by targets that have sizes smaller than or comparable with the wavelength of the incident radiation. We present a new prescription—the surface-corrected-lattice-dispersion relation (SCLDR)—for assigning the dipole polarizabilities while taking into account both target geometry and finite wavelength. We test the SCLDR in DDA calculations for spherical and ellipsoidal targets and show that for a fixed number of dipoles, the SCLDR prescription results in increased accuracy in the calculated cross sections for absorption and scattering. We discuss extension of the SCLDR prescription to irregular targets.

© 2004 Optical Society of America

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References

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  1. 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 (1990).
    [CrossRef]
  2. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  3. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  4. G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1988).
    [CrossRef] [PubMed]
  5. J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
    [CrossRef]
  6. B. T. Draine, J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
    [CrossRef]
  7. A. Rahmani, P. C. Chaumet, G. W. Bryant, “Coupled dipole method with an exact long-wavelength limit and improved accuracy at finite frequencies,” Opt. Lett. 27, 2118–2120 (2002).
    [CrossRef]
  8. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  9. D. Gutkowicz-Krusin, B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” http://xxx.arxiv.org/abs/astro-ph/0403082 .
  10. B. T. Draine, P. J. Flatau, “User guide for the discrete dipole approximation code ddscat (Version 5a10),” http://xxx.arxiv.org/abs/astro-ph/0008151v3 , 1–42 (2000).
  11. B. T. Draine, P. J. Flatau, “The discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]

2002 (1)

1994 (1)

1993 (1)

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

1990 (2)

1988 (2)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

G. H. Goedecke, S. G. O’Brien, “Scattering by irregular inhomogeneous particles via the digitized Green’s function algorithm,” Appl. Opt. 27, 2431–2438 (1988).
[CrossRef] [PubMed]

1973 (1)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Bohren, C. F.

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

Bryant, G. W.

Chaumet, P. C.

Draine, B. T.

B. T. Draine, P. J. Flatau, “The discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

B. T. Draine, 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 (1990).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Flatau, P. J.

Goedecke, G. H.

Goodman, J.

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

Goodman, J. J.

Greenberg, J. M.

J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[CrossRef]

Hage, J. I.

J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[CrossRef]

Huffman, D. R.

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

O’Brien, S. G.

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Rahmani, A.

Appl. Opt. (1)

Astrophys. J. (4)

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

J. I. Hage, J. M. Greenberg, “A model for the optical properties of porous grains,” Astrophys. J. 361, 251–259 (1990).
[CrossRef]

B. T. Draine, 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. Opt. Soc. Am. A (1)

Opt. Lett. (2)

Other (3)

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

D. Gutkowicz-Krusin, B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” http://xxx.arxiv.org/abs/astro-ph/0403082 .

B. T. Draine, P. J. Flatau, “User guide for the discrete dipole approximation code ddscat (Version 5a10),” http://xxx.arxiv.org/abs/astro-ph/0008151v3 , 1–42 (2000).

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Figures (7)

Fig. 1
Fig. 1

Comparison of scattering and absorption efficiency factors Qsca and Qabs computed for a pseudosphere of N=7664 dipoles and refractive index m=1.33+0.01i, averaged over 12 orientations, and using three different polarizability prescriptions: LDR, RCB, and SCLDR. The horizontal axis shows (top) |m|kd (the phase shift in radians within one lattice spacing) and (bottom) the scattering parameter x=ka. Error bars indicate the ranges of Q values obtained for the individual orientations. The top panel shows the results of Mie theory calculations; the lower panels show the fractional error in Qsca and Qabs. The SCLDR and LDR prescriptions are clearly preferred over the RCB prescription for this case.

Fig. 2
Fig. 2

Same as Fig. 1, but for refractive index m=5+4i. The SCLDR and RCB prescriptions are clearly preferred over the LDR for this case, with the SCLDR being somewhat superior to the RCB prescription.

Fig. 3
Fig. 3

Fractional error in Qabs averaged over 12 orientations for spheres with different refractive indices as a function of N-1/3, where N is the number of dipoles in the range 624–59,278. Calculations are shown for the LDR, RCB, and SCLDR polarizability prescriptions; the symbolic scheme is the same as in Fig. 1. Error bars in the lower middle panel indicate the typical ranges of Q values obtained for the individual orientations, as in Fig. 1 (shown for SCLDR prescription only). Refractive indices m, scattering parameters x=ka, and exact values of Qabs computed from Mie theory are shown inside each panel. The scattering parameters are chosen so that |m|kd0.8 (the approximate limit of applicability of the DDA) for the smallest number (N=624) of dipoles. The convergence with increasing N is quite smooth in all regions of the complex m plane with the exception of m=3+0.01i. In almost every case shown, fractional errors <2% (and often significantly lower) can be achieved for N6000 dipoles. We find that for calculating Qabs, the SCLDR is comparable with or superior in accuracy to the LDR and RCB prescriptions throughout the region of m space shown.

Fig. 4
Fig. 4

Same as Fig. 3, except that fractional errors in Qsca are plotted. Again, the SCLDR prescription is comparable with or superior to the LDR and RCB prescriptions for most m values. The SCLDR prescription produces fractional errors <2% in all cases for N6000 dipoles. In some cases, all three prescriptions produce fractional errors <2% even for small N.

Fig. 5
Fig. 5

Same as Fig. 3, but for ellipsoids with approximately 1:2:3 axial ratios. Fractional errors have been estimated based on comparison with an extrapolation of the convergence behavior of the three polarizability prescriptions as described in Section 3. Again the SCLDR prescription is comparable with or superior to the LDR and RCB prescriptions for calculating Qabs throughout the region of the complex m plane sampled.

Fig. 6
Fig. 6

Same as Fig. 5, except that fractional errors in Qsca are plotted. The SCLDR prescription is comparable with or superior to LDR for all values of m and to RCB for values of m with small imaginary parts, while RCB is superior to SCLDR for values of m with large imaginary parts. In most cases, the SCLDR prescription produces fractional errors <2% for N6000 dipoles.

Fig. 7
Fig. 7

Comparison of RCB and CMR polarizabilities. The left panel shows the distribution of polarizability eigenvalues for discrete dipole approximations to a 1:2:3 ellipsoid with m=5+4i with N=90, 688, 5456, and 43,416 dipoles. The shaded region corresponds to a fractional difference of 20% or less; the fraction of the eigenvalues within this region varies from 53% for N=90 (3d×6d×9d axes) to 91% for N=43,416 (24d×48d×72d axes). The right panel shows the fractional difference between RCB and CMR polarizabilities versus the distance (in units of the shortest axis) from the ideal ellipsoidal surface used to define the target (all dipole locations are interior to this surface). As expected, the RCB polarizability converges to the CMR polarizability for dipoles lying more than 2d from the surface.

Equations (13)

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αCMR=3d34πm2-1m2+2.
α=α(nr)1-(2/3)i(α(nr)/d3)(kd)3,
αLDR=α(0)1+(α(0)/d3)[(b1+m2b2+m2b3S)(kd)2-(2/3)i(kd)3],
S=j(ajej)2,
Ejmac=Cj-1E0,
Pj=d3mj2-14πEjmac=d3mj2-14πCj-1E0,
Pj=αjE0-kjAjkPk,
αRCB,j=d3mj2-14πΛj-1,
ΛjCj-kjAjkmk2-14πd3Ck-1Cj
Cj=1+m2-14πL,
αSCLDR,j=αRCB,j[1+(αRCB,j/d3)B]-1,
Bij=(b1+m2b2+m2b3ai2δij)(kd)2-(2/3)i(kd)3,
Frac.Err.Q(DDA)-Q(Mie)Q(Mie).

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