Abstract

A model of light scattering from an assembly of small shape-distributed ellipsoidal particles is considered. The three principal assumptions used are the neglect of multiple scattering, the dipole polarizability, and the equiprobable distribution for the ellipsoid depolarization factors. These assumptions enabled us to find analytically the effective cross section for light scattering. {That for light absorption was found in a similar way by Goncharenko et al. [J. Opt. Soc. Am. B 13, 2392 (1996)]}. The solution obtained is analyzed for some special cases, in particular for low and no absorption.

© 1999 Optical Society of America

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References

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  1. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  2. Th. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
    [CrossRef]
  3. P. W. Barber, S. C. Hill, Light Scattering by Small Particles: Computational Methods (World Scientific, Singapore, 1990).
  4. This method became renowned since the paper by E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non-spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).It was further developed, in particular, in studies by B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988);B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).One can also find an electronic review of this method at http://atol.ucsd.edu/pflatau/scatlib/dda/paperh/paperh.html .
    [CrossRef]
  5. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).
  6. K. Hinsen, B. U. Felderhof, “Dielectric constant of a suspension of uniform spheres,” Phys. Rev. B 46, 12955–12963 (1992).
    [CrossRef]
  7. R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11, 1732–1740 (1975).
    [CrossRef]
  8. D. Bergman, “The dielectric constant of a composite material—a problem in classical physics,” Phys. Rep. 43, 377–407 (1978).
    [CrossRef]
  9. R. Stognienko, Th. Henning, V. Ossenkopf, “Optical properties of coagulated particles,” Astron. Astrophys. 296, 797–809 (1995).
  10. F. Rouleau, P. G. Martin, “A new method to calculate the extinction properties of irregularly shaped particles,” Astrophys. J. 414, 803–814 (1993).
    [CrossRef]
  11. P. Latimer, “Light scattering by ellipsoids,” J. Colloid Interface Sci. 53, 102–109 (1975).
    [CrossRef]
  12. H. C. van de Hulst. Light Scattering by Small Particles (Wiley, New York, 1957).
  13. C. F. Bohren, D. R. Huffman, “Absorption cross-section maxima and minima in IR absorption bands of small ionic ellipsoidal particles,” Appl. Opt. 20, 834–841 (1981).
    [CrossRef]
  14. L. E. Paramonov, V. N. Lopatin, F. Ya. Sid’ko, “On light scattering of “soft” spheroidal particles,” Opt. Spektrosk. 61(3), 570–576 (1986).
  15. A. V. Goncharenko, E. F. Venger, S. N. Zavadskii, “Effective absorption cross section of an assembly of small ellipsoidal particles,” J. Opt. Soc. Am. B 13, 2392–2395 (1996).
    [CrossRef]
  16. B. Michel, Th. Henning, R. Stognienko, F. Rouleau, “Extinction properties in dust grains: a new computational technique,” Astrophys. J. 468, 434–441 (1996).
    [CrossRef]
  17. D. R. Huffman, C. F. Bohren, “Infrared absorption spectra of nonspherical particles treated in the Rayleigh-ellipsoid approximation,” in Light Scattering by Irregularly Shaped Particles, D. Shuerman, ed., (Plenum, New York, 1980), pp. 103–111.
  18. The upper integration limit for the inner integral was denoted improperly in Ref. 15 (however, the calculations there were performed with correct integration limits.)
  19. It is more proper to point to SiC as a wide-gap semiconductor. However, in the frequency range considered, the SiC dielectric function manifests the same characteristic behavior as the dielectric function of a classical polar dielectric in the reststrahlen range.
  20. In our calculations we used dielectric-function parameters for the SiC cubic polytype from research by L. Patrick, W. J. Choyke, “Static dielectric constant of SiC,” Phys. Rev. B 2, 2255–2564 (1970) and for Al from Ref. 1 (Chap. 9), i.e., hωp=15 eV and hγ=0.6 eV.
    [CrossRef]
  21. The Fröhlich frequency νF satisfies the condition ∊1(νF)=-2 (here we use the spatial frequency ν=1/λ=k/2π.)
  22. G. A. Baker, P. Graves-Morris, “Padé approximants,” in Encyclopedia of Mathematics and Its Applications (Addison-Wesley, London, 1981), Vols. 13 and 14.
  23. M. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
    [CrossRef] [PubMed]

1998

Th. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
[CrossRef]

1996

B. Michel, Th. Henning, R. Stognienko, F. Rouleau, “Extinction properties in dust grains: a new computational technique,” Astrophys. J. 468, 434–441 (1996).
[CrossRef]

A. V. Goncharenko, E. F. Venger, S. N. Zavadskii, “Effective absorption cross section of an assembly of small ellipsoidal particles,” J. Opt. Soc. Am. B 13, 2392–2395 (1996).
[CrossRef]

1995

R. Stognienko, Th. Henning, V. Ossenkopf, “Optical properties of coagulated particles,” Astron. Astrophys. 296, 797–809 (1995).

1994

1993

F. Rouleau, P. G. Martin, “A new method to calculate the extinction properties of irregularly shaped particles,” Astrophys. J. 414, 803–814 (1993).
[CrossRef]

1992

K. Hinsen, B. U. Felderhof, “Dielectric constant of a suspension of uniform spheres,” Phys. Rev. B 46, 12955–12963 (1992).
[CrossRef]

1986

L. E. Paramonov, V. N. Lopatin, F. Ya. Sid’ko, “On light scattering of “soft” spheroidal particles,” Opt. Spektrosk. 61(3), 570–576 (1986).

1981

C. F. Bohren, D. R. Huffman, “Absorption cross-section maxima and minima in IR absorption bands of small ionic ellipsoidal particles,” Appl. Opt. 20, 834–841 (1981).
[CrossRef]

1978

D. Bergman, “The dielectric constant of a composite material—a problem in classical physics,” Phys. Rep. 43, 377–407 (1978).
[CrossRef]

1975

P. Latimer, “Light scattering by ellipsoids,” J. Colloid Interface Sci. 53, 102–109 (1975).
[CrossRef]

R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11, 1732–1740 (1975).
[CrossRef]

1973

This method became renowned since the paper by E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non-spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).It was further developed, in particular, in studies by B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988);B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).One can also find an electronic review of this method at http://atol.ucsd.edu/pflatau/scatlib/dda/paperh/paperh.html .
[CrossRef]

1970

In our calculations we used dielectric-function parameters for the SiC cubic polytype from research by L. Patrick, W. J. Choyke, “Static dielectric constant of SiC,” Phys. Rev. B 2, 2255–2564 (1970) and for Al from Ref. 1 (Chap. 9), i.e., hωp=15 eV and hγ=0.6 eV.
[CrossRef]

Baker, G. A.

G. A. Baker, P. Graves-Morris, “Padé approximants,” in Encyclopedia of Mathematics and Its Applications (Addison-Wesley, London, 1981), Vols. 13 and 14.

Barber, P. W.

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

Bergman, D.

D. Bergman, “The dielectric constant of a composite material—a problem in classical physics,” Phys. Rep. 43, 377–407 (1978).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, “Absorption cross-section maxima and minima in IR absorption bands of small ionic ellipsoidal particles,” Appl. Opt. 20, 834–841 (1981).
[CrossRef]

D. R. Huffman, C. F. Bohren, “Infrared absorption spectra of nonspherical particles treated in the Rayleigh-ellipsoid approximation,” in Light Scattering by Irregularly Shaped Particles, D. Shuerman, ed., (Plenum, New York, 1980), pp. 103–111.

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

Choyke, W. J.

In our calculations we used dielectric-function parameters for the SiC cubic polytype from research by L. Patrick, W. J. Choyke, “Static dielectric constant of SiC,” Phys. Rev. B 2, 2255–2564 (1970) and for Al from Ref. 1 (Chap. 9), i.e., hωp=15 eV and hγ=0.6 eV.
[CrossRef]

Felderhof, B. U.

K. Hinsen, B. U. Felderhof, “Dielectric constant of a suspension of uniform spheres,” Phys. Rev. B 46, 12955–12963 (1992).
[CrossRef]

Fuchs, R.

R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11, 1732–1740 (1975).
[CrossRef]

Goncharenko, A. V.

Graves-Morris, P.

G. A. Baker, P. Graves-Morris, “Padé approximants,” in Encyclopedia of Mathematics and Its Applications (Addison-Wesley, London, 1981), Vols. 13 and 14.

Henning, Th.

B. Michel, Th. Henning, R. Stognienko, F. Rouleau, “Extinction properties in dust grains: a new computational technique,” Astrophys. J. 468, 434–441 (1996).
[CrossRef]

R. Stognienko, Th. Henning, V. Ossenkopf, “Optical properties of coagulated particles,” Astron. Astrophys. 296, 797–809 (1995).

Hill, S. C.

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

Hinsen, K.

K. Hinsen, B. U. Felderhof, “Dielectric constant of a suspension of uniform spheres,” Phys. Rev. B 46, 12955–12963 (1992).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, “Absorption cross-section maxima and minima in IR absorption bands of small ionic ellipsoidal particles,” Appl. Opt. 20, 834–841 (1981).
[CrossRef]

D. R. Huffman, C. F. Bohren, “Infrared absorption spectra of nonspherical particles treated in the Rayleigh-ellipsoid approximation,” in Light Scattering by Irregularly Shaped Particles, D. Shuerman, ed., (Plenum, New York, 1980), pp. 103–111.

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

Latimer, P.

P. Latimer, “Light scattering by ellipsoids,” J. Colloid Interface Sci. 53, 102–109 (1975).
[CrossRef]

Lopatin, V. N.

L. E. Paramonov, V. N. Lopatin, F. Ya. Sid’ko, “On light scattering of “soft” spheroidal particles,” Opt. Spektrosk. 61(3), 570–576 (1986).

Martin, P. G.

F. Rouleau, P. G. Martin, “A new method to calculate the extinction properties of irregularly shaped particles,” Astrophys. J. 414, 803–814 (1993).
[CrossRef]

Michel, B.

B. Michel, Th. Henning, R. Stognienko, F. Rouleau, “Extinction properties in dust grains: a new computational technique,” Astrophys. J. 468, 434–441 (1996).
[CrossRef]

Mishchenko, M.

Ossenkopf, V.

R. Stognienko, Th. Henning, V. Ossenkopf, “Optical properties of coagulated particles,” Astron. Astrophys. 296, 797–809 (1995).

Paramonov, L. E.

L. E. Paramonov, V. N. Lopatin, F. Ya. Sid’ko, “On light scattering of “soft” spheroidal particles,” Opt. Spektrosk. 61(3), 570–576 (1986).

Patrick, L.

In our calculations we used dielectric-function parameters for the SiC cubic polytype from research by L. Patrick, W. J. Choyke, “Static dielectric constant of SiC,” Phys. Rev. B 2, 2255–2564 (1970) and for Al from Ref. 1 (Chap. 9), i.e., hωp=15 eV and hγ=0.6 eV.
[CrossRef]

Pennypacker, C. R.

This method became renowned since the paper by E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non-spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).It was further developed, in particular, in studies by B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988);B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).One can also find an electronic review of this method at http://atol.ucsd.edu/pflatau/scatlib/dda/paperh/paperh.html .
[CrossRef]

Purcell, E. M.

This method became renowned since the paper by E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non-spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).It was further developed, in particular, in studies by B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988);B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).One can also find an electronic review of this method at http://atol.ucsd.edu/pflatau/scatlib/dda/paperh/paperh.html .
[CrossRef]

Rouleau, F.

B. Michel, Th. Henning, R. Stognienko, F. Rouleau, “Extinction properties in dust grains: a new computational technique,” Astrophys. J. 468, 434–441 (1996).
[CrossRef]

F. Rouleau, P. G. Martin, “A new method to calculate the extinction properties of irregularly shaped particles,” Astrophys. J. 414, 803–814 (1993).
[CrossRef]

Sid’ko, F. Ya.

L. E. Paramonov, V. N. Lopatin, F. Ya. Sid’ko, “On light scattering of “soft” spheroidal particles,” Opt. Spektrosk. 61(3), 570–576 (1986).

Stognienko, R.

B. Michel, Th. Henning, R. Stognienko, F. Rouleau, “Extinction properties in dust grains: a new computational technique,” Astrophys. J. 468, 434–441 (1996).
[CrossRef]

R. Stognienko, Th. Henning, V. Ossenkopf, “Optical properties of coagulated particles,” Astron. Astrophys. 296, 797–809 (1995).

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).

Travis, L. D.

van de Hulst, H. C.

H. C. van de Hulst. Light Scattering by Small Particles (Wiley, New York, 1957).

Venger, E. F.

Wriedt, Th.

Th. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
[CrossRef]

Zavadskii, S. N.

Appl. Opt.

C. F. Bohren, D. R. Huffman, “Absorption cross-section maxima and minima in IR absorption bands of small ionic ellipsoidal particles,” Appl. Opt. 20, 834–841 (1981).
[CrossRef]

M. Mishchenko, L. D. Travis, “Light scattering by polydispersions of randomly oriented spheroids with sizes comparable to wavelengths of observation,” Appl. Opt. 33, 7206–7225 (1994).
[CrossRef] [PubMed]

Astron. Astrophys.

R. Stognienko, Th. Henning, V. Ossenkopf, “Optical properties of coagulated particles,” Astron. Astrophys. 296, 797–809 (1995).

Astrophys. J.

F. Rouleau, P. G. Martin, “A new method to calculate the extinction properties of irregularly shaped particles,” Astrophys. J. 414, 803–814 (1993).
[CrossRef]

B. Michel, Th. Henning, R. Stognienko, F. Rouleau, “Extinction properties in dust grains: a new computational technique,” Astrophys. J. 468, 434–441 (1996).
[CrossRef]

This method became renowned since the paper by E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non-spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).It was further developed, in particular, in studies by B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988);B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).One can also find an electronic review of this method at http://atol.ucsd.edu/pflatau/scatlib/dda/paperh/paperh.html .
[CrossRef]

J. Colloid Interface Sci.

P. Latimer, “Light scattering by ellipsoids,” J. Colloid Interface Sci. 53, 102–109 (1975).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Spektrosk.

L. E. Paramonov, V. N. Lopatin, F. Ya. Sid’ko, “On light scattering of “soft” spheroidal particles,” Opt. Spektrosk. 61(3), 570–576 (1986).

Part. Part. Syst. Charact.

Th. Wriedt, “A review of elastic light scattering theories,” Part. Part. Syst. Charact. 15, 67–74 (1998).
[CrossRef]

Phys. Rep.

D. Bergman, “The dielectric constant of a composite material—a problem in classical physics,” Phys. Rep. 43, 377–407 (1978).
[CrossRef]

Phys. Rev. B

K. Hinsen, B. U. Felderhof, “Dielectric constant of a suspension of uniform spheres,” Phys. Rev. B 46, 12955–12963 (1992).
[CrossRef]

R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11, 1732–1740 (1975).
[CrossRef]

In our calculations we used dielectric-function parameters for the SiC cubic polytype from research by L. Patrick, W. J. Choyke, “Static dielectric constant of SiC,” Phys. Rev. B 2, 2255–2564 (1970) and for Al from Ref. 1 (Chap. 9), i.e., hωp=15 eV and hγ=0.6 eV.
[CrossRef]

Other

The Fröhlich frequency νF satisfies the condition ∊1(νF)=-2 (here we use the spatial frequency ν=1/λ=k/2π.)

G. A. Baker, P. Graves-Morris, “Padé approximants,” in Encyclopedia of Mathematics and Its Applications (Addison-Wesley, London, 1981), Vols. 13 and 14.

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

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

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).

H. C. van de Hulst. Light Scattering by Small Particles (Wiley, New York, 1957).

D. R. Huffman, C. F. Bohren, “Infrared absorption spectra of nonspherical particles treated in the Rayleigh-ellipsoid approximation,” in Light Scattering by Irregularly Shaped Particles, D. Shuerman, ed., (Plenum, New York, 1980), pp. 103–111.

The upper integration limit for the inner integral was denoted improperly in Ref. 15 (however, the calculations there were performed with correct integration limits.)

It is more proper to point to SiC as a wide-gap semiconductor. However, in the frequency range considered, the SiC dielectric function manifests the same characteristic behavior as the dielectric function of a classical polar dielectric in the reststrahlen range.

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Figures (3)

Fig. 1
Fig. 1

Frequency dependencies of the effective scattering cross section for (a) SiC and (b) Al at various values of the parameter Δ.

Fig. 2
Fig. 2

Domain of C˜sca divergence at 2=0 (shaded region).

Fig. 3
Fig. 3

Curves corresponding to Q=C˜sca/Csca0=constant at the (1,Δ) plane for 2=0 calculated from Eq. (17).

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

Csca=Ck4|α|2.
|α|2=13(|α1|2+|α2|2+|α3|2),
α=Vα˜,
Csca=Ck4V2|α˜|2P(L1, L2)dL1dL2.
C˜sca=1CCscak4V2.
αj=-11+Lj(-1),
C˜sca=13j=13|β+Lj|-2P(L1, L2)dL1dL2,
P(L1, L2)=2Δ-2χ(L1-1/3+1/3Δ)χ(L2-1/3+1/3Δ)χ(-L1-L2+2/3+1/3Δ),
C˜sca=23Δ2j=13|β+Lj|-2dL1dL2=23Δ21/3(1-Δ)1/3(1+2Δ)dL11/3(1-Δ)1/3(2+Δ)-L1dL2×j=13|β+Lj|-2.
C˜sca=1Δ22r+[arctan(r+)-arctan(r-)]-ln1+r+21+r-2=1Δ2r+π[sgn(r+)-sgn(r-)]+2r+ arctanr+-r-1+r+r--ln 1+r+21+r-2,
r+=(η+2Δ)/θ,r-=(η-Δ)/θ, η=1+3 Re(β)=1+3(1-1)/[(1-1)2+22], θ=3 Im(β)=-32/[(1-1)2+22], =1+i2.
C˜sca=1Δ22r+ arctanr+-r-1+r+r--ln1+r+21+r-2=1Δ22 η+2Δθarctan3θΔθ2+(η-Δ)(η+2Δ)-lnη2+θ2+4Δ(η+Δ)η2+θ2-Δ(2η-Δ).
-1+2Δ31-1(1-1)2+22Δ-13,
C˜sca=1Δ22r+π+arctanr+-r-1+r+r--ln1+r+21-r-2.
C˜sca0=limΔ0C˜sca=9η2+θ2=9-1+22.
C˜sca(Δ=1)=2 12+22-12arctan12+22-12-arctan1-12-ln(12+22).
1+3Δ-111-31+2Δ.
C˜scanonabs=limθ0C˜sca=2Δ23Δη-Δ-ln η+2Δη-Δ=C˜sca0Q(Δ).
Δ=-121+21-1.
C˜scanonabs=C˜sca01+32η2Δ2-45η3Δ3+.
C˜scanonabsC˜sca0 1+815Δη1+815Δη-32Δ2η2.
21-11
|δ|=|θ/η|1
C˜scaweakabs=C˜scanonabs-Rδ2,
R=9η2(η-Δ)2(η+2Δ)2η+Δη-Δ.

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