Abstract

A solution of the electromagnetic scattering problem for confocal coated spheroids has been obtained by the method of separation of variables in a spheroidal coordinate system. The main features of the solution are (i) the incident, scattered, and internal radiation fields are divided into two parts: an axisymmetric part independent of the azimuthal angle φ and a nonaxisymmetric part that with integration over φ gives zero; the diffraction problems for each part are solved separately; (ii) the scalar potentials of the solution are chosen in a special way: Abraham's potentials (for the axisymmetric part) and a superposition of the potentials used for spheres and infinitely long cylinders (for the nonaxisymmetric part). Such a procedure has been applied to homogeneous spheroids [Differential Equations 19, 1765 (1983); Astrophys. Space Sci. 204, 19, (1993)] and allows us to solve the light scattering problem for confocal spheroids with an arbitrary refractive index, size, and shape of the core or mantle. Numerical tests are described in detail. The efficiency factors have been calculated for prolate and oblate spheroids with refractive indices of 1.5 + 0.0i, 1.5 + 0.05i for the core and refractive indices of 1.3 + 0.0i, 1.3 + 0.05i for the mantle. The effects of the core size and particle shape as well as those of absorption in the core or mantle are examined. It is found that the efficiency factors of the coated and homogeneous spheroids with the volume-averaged refractive index are similar to first maximum.

© 1996 Optical Society of America

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References

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  1. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  2. J. L. Hage, J. M. Greenberg, R. T. Wang, “Scattering from arbitrarily shaped particles: theory and experiment,” Appl. Opt. 30, 1141–1152 (1991).
    [CrossRef] [PubMed]
  3. B. T. Draine, J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarized point lattice and the discrete-dipole approximation,” Astrophys. J. 405, 685–697 (1993);P. J. Flatau, G. L. Stephens, B. T. Draine, “Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the block-Toeplitz structure,” J. Opt. Soc. Am. A 7, 593–600 (1990);K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–677 (1994).
    [CrossRef]
  4. J. I. Peltoniemi, K. Lumme, K. Muinonen, W. M. Irwine, “Scattering of light by stochastically rough particles,” Appl. Opt. 28, 4088–4095 (1989).
    [CrossRef] [PubMed]
  5. F. Rouleau, P. G. Martin, “A new method to calculate the extinction properties of irregularly shaped particles,” Astrophys. J. 414, 803–814 (1993).
    [CrossRef]
  6. G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
  7. K. S. Shifrin, “Light scattering by core-mantle particles,” Izv. Akad. Nauk SSSR Ser. Geofiz. N 2, 15–21 (1952).
  8. A. C. Lind, J. M. Greenberg, “Electromagnetic scattering by obliquely oriented cylinders,” J. Appl. Phys. 37, 3195–3203 (1966).
    [CrossRef]
  9. G. A. Shah, “Scattering of plane electromagnetic waves by infinite concentric circular cylinders at oblique incidence,” Mon. Not. R. Astron. Soc. 148, 93–102 (1972).
  10. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [PubMed]
  11. V. G. Farafonov, “The scattering of a plane electromagnetic wave by a dielectric spheroid,” Differential Equations (Sov.) 19, 1765–1777 (1983).
  12. N. V. Voshchinnikov, V. G. Farafonov, “Light scattering by dielectric spheroids. I,” Opt. Spektrosk. 58, 81–85 (1985).
  13. N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
    [CrossRef]
  14. T. Onaka, “Light scattering by spheroidal grains,” Ann. Tokyo Astron. Obs. 18, 1–54 (1980).
  15. M. F. R. Cooray, I. R. Ciric, “Scattering of electromagnetic waves by a coated dielectric spheroid,” J. Electro. Waves Appl. 6, 1491–1507 (1992).
    [CrossRef]
  16. B. P. Sinha, R. H. MacPhie, “Electromagnetic scattering by prolate spheroids for plane waves with arbitrary polarization and angle of incidence,” Radio Sci. 12, 171–184 (1977).
    [CrossRef]
  17. B. Peterson, S. Ström, “T-matrix formulation of electromagnetic scattering from multilayered scatters,” Phys. Rev. D 10, 2670–2684 (1974).
    [CrossRef]
  18. P. W. Barber, “Scattering and absorption by homogeneous and layered dielectrics,” in Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach, V. K. Varadan, V. V. Varadan, eds. (Pergamon, New York, 1980), pp. 191–209;D.-S. Wang, P. W. Barber, “Scattering by inhomogeneous nonspherical objects,” Appl. Opt. 18, 1190–1197 (1979);D.-S. Wang, C. H. Chen, P. W. Barber, P. J. Wyatt, “Light scattering by polydisperse suspensions of inhomogeneous nonspherical particles,” Appl. Opt. 18, 2672–2678 (1979).
    [CrossRef] [PubMed]
  19. V. N. Lopatin, F. Ya. Sid'ko, Introduction to Optics of Cell Suspension (Nauka, Novosibirsk, Russia, 1987).
  20. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).
  21. I. V. Komarov, L. I. Ponomarev, S. Yu. Slavyanov, Spheroidal and Coulomb Spheroidal Functions (Nauka, Moscow, 1976).
  22. J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).
  23. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  24. S. Asano, “Light scattering properties of spheroidal particles,” Appl. Opt. 18, 712–723 (1979).
    [CrossRef] [PubMed]
  25. P. W. Barber, “Resonance electromagnetic absorption by non-spherical dielectric objects,” IEEE Trans. Microwave Theory Tech. MTT-25, 373–381 (1977).
    [CrossRef]
  26. M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
    [CrossRef]
  27. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  28. N. V. Voshchinnikov, “Dust grains in reflection nebulae. Spherical core-mantle grains,” Sov. Astron. 22, 561–566 (1978).
  29. The numerical code is available on request from N. V. Voshchinnikov, e-mail: nvv@aispbu.spb.su.

1994 (1)

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

1993 (3)

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

B. T. Draine, J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarized point lattice and the discrete-dipole approximation,” Astrophys. J. 405, 685–697 (1993);P. J. Flatau, G. L. Stephens, B. T. Draine, “Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the block-Toeplitz structure,” J. Opt. Soc. Am. A 7, 593–600 (1990);K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–677 (1994).
[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]

1992 (1)

M. F. R. Cooray, I. R. Ciric, “Scattering of electromagnetic waves by a coated dielectric spheroid,” J. Electro. Waves Appl. 6, 1491–1507 (1992).
[CrossRef]

1991 (1)

1989 (1)

1988 (1)

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

1985 (1)

N. V. Voshchinnikov, V. G. Farafonov, “Light scattering by dielectric spheroids. I,” Opt. Spektrosk. 58, 81–85 (1985).

1983 (1)

V. G. Farafonov, “The scattering of a plane electromagnetic wave by a dielectric spheroid,” Differential Equations (Sov.) 19, 1765–1777 (1983).

1980 (1)

T. Onaka, “Light scattering by spheroidal grains,” Ann. Tokyo Astron. Obs. 18, 1–54 (1980).

1979 (1)

1978 (1)

N. V. Voshchinnikov, “Dust grains in reflection nebulae. Spherical core-mantle grains,” Sov. Astron. 22, 561–566 (1978).

1977 (2)

P. W. Barber, “Resonance electromagnetic absorption by non-spherical dielectric objects,” IEEE Trans. Microwave Theory Tech. MTT-25, 373–381 (1977).
[CrossRef]

B. P. Sinha, R. H. MacPhie, “Electromagnetic scattering by prolate spheroids for plane waves with arbitrary polarization and angle of incidence,” Radio Sci. 12, 171–184 (1977).
[CrossRef]

1975 (1)

1974 (1)

B. Peterson, S. Ström, “T-matrix formulation of electromagnetic scattering from multilayered scatters,” Phys. Rev. D 10, 2670–2684 (1974).
[CrossRef]

1972 (1)

G. A. Shah, “Scattering of plane electromagnetic waves by infinite concentric circular cylinders at oblique incidence,” Mon. Not. R. Astron. Soc. 148, 93–102 (1972).

1966 (1)

A. C. Lind, J. M. Greenberg, “Electromagnetic scattering by obliquely oriented cylinders,” J. Appl. Phys. 37, 3195–3203 (1966).
[CrossRef]

1952 (1)

K. S. Shifrin, “Light scattering by core-mantle particles,” Izv. Akad. Nauk SSSR Ser. Geofiz. N 2, 15–21 (1952).

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Asano, S.

Barber, P. W.

P. W. Barber, “Resonance electromagnetic absorption by non-spherical dielectric objects,” IEEE Trans. Microwave Theory Tech. MTT-25, 373–381 (1977).
[CrossRef]

P. W. Barber, “Scattering and absorption by homogeneous and layered dielectrics,” in Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach, V. K. Varadan, V. V. Varadan, eds. (Pergamon, New York, 1980), pp. 191–209;D.-S. Wang, P. W. Barber, “Scattering by inhomogeneous nonspherical objects,” Appl. Opt. 18, 1190–1197 (1979);D.-S. Wang, C. H. Chen, P. W. Barber, P. J. Wyatt, “Light scattering by polydisperse suspensions of inhomogeneous nonspherical particles,” Appl. Opt. 18, 2672–2678 (1979).
[CrossRef] [PubMed]

Bohren, C. F.

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

Bowman, J. J.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Ciric, I. R.

M. F. R. Cooray, I. R. Ciric, “Scattering of electromagnetic waves by a coated dielectric spheroid,” J. Electro. Waves Appl. 6, 1491–1507 (1992).
[CrossRef]

Cooray, M. F. R.

M. F. R. Cooray, I. R. Ciric, “Scattering of electromagnetic waves by a coated dielectric spheroid,” J. Electro. Waves Appl. 6, 1491–1507 (1992).
[CrossRef]

Draine, B. T.

B. T. Draine, J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarized point lattice and the discrete-dipole approximation,” Astrophys. J. 405, 685–697 (1993);P. J. Flatau, G. L. Stephens, B. T. Draine, “Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the block-Toeplitz structure,” J. Opt. Soc. Am. A 7, 593–600 (1990);K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–677 (1994).
[CrossRef]

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

Farafonov, V. G.

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

N. V. Voshchinnikov, V. G. Farafonov, “Light scattering by dielectric spheroids. I,” Opt. Spektrosk. 58, 81–85 (1985).

V. G. Farafonov, “The scattering of a plane electromagnetic wave by a dielectric spheroid,” Differential Equations (Sov.) 19, 1765–1777 (1983).

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

Goodman, J.

B. T. Draine, J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarized point lattice and the discrete-dipole approximation,” Astrophys. J. 405, 685–697 (1993);P. J. Flatau, G. L. Stephens, B. T. Draine, “Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the block-Toeplitz structure,” J. Opt. Soc. Am. A 7, 593–600 (1990);K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–677 (1994).
[CrossRef]

Greenberg, J. M.

J. L. Hage, J. M. Greenberg, R. T. Wang, “Scattering from arbitrarily shaped particles: theory and experiment,” Appl. Opt. 30, 1141–1152 (1991).
[CrossRef] [PubMed]

A. C. Lind, J. M. Greenberg, “Electromagnetic scattering by obliquely oriented cylinders,” J. Appl. Phys. 37, 3195–3203 (1966).
[CrossRef]

Hage, J. L.

Huffman, D. R.

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

Irwine, W. M.

Komarov, I. V.

I. V. Komarov, L. I. Ponomarev, S. Yu. Slavyanov, Spheroidal and Coulomb Spheroidal Functions (Nauka, Moscow, 1976).

Lind, A. C.

A. C. Lind, J. M. Greenberg, “Electromagnetic scattering by obliquely oriented cylinders,” J. Appl. Phys. 37, 3195–3203 (1966).
[CrossRef]

Lopatin, V. N.

V. N. Lopatin, F. Ya. Sid'ko, Introduction to Optics of Cell Suspension (Nauka, Novosibirsk, Russia, 1987).

Lumme, K.

MacPhie, R. H.

B. P. Sinha, R. H. MacPhie, “Electromagnetic scattering by prolate spheroids for plane waves with arbitrary polarization and angle of incidence,” Radio Sci. 12, 171–184 (1977).
[CrossRef]

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]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Muinonen, K.

Onaka, T.

T. Onaka, “Light scattering by spheroidal grains,” Ann. Tokyo Astron. Obs. 18, 1–54 (1980).

Peltoniemi, J. I.

Peterson, B.

B. Peterson, S. Ström, “T-matrix formulation of electromagnetic scattering from multilayered scatters,” Phys. Rev. D 10, 2670–2684 (1974).
[CrossRef]

Ponomarev, L. I.

I. V. Komarov, L. I. Ponomarev, S. Yu. Slavyanov, Spheroidal and Coulomb Spheroidal Functions (Nauka, Moscow, 1976).

Rouleau, F.

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

Senior, T. B. A.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Shah, G. A.

G. A. Shah, “Scattering of plane electromagnetic waves by infinite concentric circular cylinders at oblique incidence,” Mon. Not. R. Astron. Soc. 148, 93–102 (1972).

Shifrin, K. S.

K. S. Shifrin, “Light scattering by core-mantle particles,” Izv. Akad. Nauk SSSR Ser. Geofiz. N 2, 15–21 (1952).

Sid'ko, F. Ya.

V. N. Lopatin, F. Ya. Sid'ko, Introduction to Optics of Cell Suspension (Nauka, Novosibirsk, Russia, 1987).

Sinha, B. P.

B. P. Sinha, R. H. MacPhie, “Electromagnetic scattering by prolate spheroids for plane waves with arbitrary polarization and angle of incidence,” Radio Sci. 12, 171–184 (1977).
[CrossRef]

Slavyanov, S. Yu.

I. V. Komarov, L. I. Ponomarev, S. Yu. Slavyanov, Spheroidal and Coulomb Spheroidal Functions (Nauka, Moscow, 1976).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Ström, S.

B. Peterson, S. Ström, “T-matrix formulation of electromagnetic scattering from multilayered scatters,” Phys. Rev. D 10, 2670–2684 (1974).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Uslenghi, P. L. E.

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Voshchinnikov, N. V.

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

N. V. Voshchinnikov, V. G. Farafonov, “Light scattering by dielectric spheroids. I,” Opt. Spektrosk. 58, 81–85 (1985).

N. V. Voshchinnikov, “Dust grains in reflection nebulae. Spherical core-mantle grains,” Sov. Astron. 22, 561–566 (1978).

Wang, R. T.

Yamamoto, G.

Ann. Phys. (Leipzig) (1)

G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).

Ann. Tokyo Astron. Obs. (1)

T. Onaka, “Light scattering by spheroidal grains,” Ann. Tokyo Astron. Obs. 18, 1–54 (1980).

Appl. Opt. (4)

Astrophys. J. (3)

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

B. T. Draine, J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarized point lattice and the discrete-dipole approximation,” Astrophys. J. 405, 685–697 (1993);P. J. Flatau, G. L. Stephens, B. T. Draine, “Light scattering by rectangular solids in the discrete-dipole approximation: a new algorithm exploiting the block-Toeplitz structure,” J. Opt. Soc. Am. A 7, 593–600 (1990);K. Lumme, J. Rahola, “Light scattering by porous dust particles in the discrete-dipole approximation,” Astrophys. J. 425, 653–677 (1994).
[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]

Astrophys. Space Sci. (1)

N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
[CrossRef]

Differential Equations (Sov.) (1)

V. G. Farafonov, “The scattering of a plane electromagnetic wave by a dielectric spheroid,” Differential Equations (Sov.) 19, 1765–1777 (1983).

IEEE Trans. Microwave Theory Tech. (1)

P. W. Barber, “Resonance electromagnetic absorption by non-spherical dielectric objects,” IEEE Trans. Microwave Theory Tech. MTT-25, 373–381 (1977).
[CrossRef]

Izv. Akad. Nauk SSSR Ser. Geofiz. N (1)

K. S. Shifrin, “Light scattering by core-mantle particles,” Izv. Akad. Nauk SSSR Ser. Geofiz. N 2, 15–21 (1952).

J. Appl. Phys. (1)

A. C. Lind, J. M. Greenberg, “Electromagnetic scattering by obliquely oriented cylinders,” J. Appl. Phys. 37, 3195–3203 (1966).
[CrossRef]

J. Electro. Waves Appl. (1)

M. F. R. Cooray, I. R. Ciric, “Scattering of electromagnetic waves by a coated dielectric spheroid,” J. Electro. Waves Appl. 6, 1491–1507 (1992).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

G. A. Shah, “Scattering of plane electromagnetic waves by infinite concentric circular cylinders at oblique incidence,” Mon. Not. R. Astron. Soc. 148, 93–102 (1972).

Opt. Commun. (1)

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Opt. Spektrosk. (1)

N. V. Voshchinnikov, V. G. Farafonov, “Light scattering by dielectric spheroids. I,” Opt. Spektrosk. 58, 81–85 (1985).

Phys. Rev. D (1)

B. Peterson, S. Ström, “T-matrix formulation of electromagnetic scattering from multilayered scatters,” Phys. Rev. D 10, 2670–2684 (1974).
[CrossRef]

Radio Sci. (1)

B. P. Sinha, R. H. MacPhie, “Electromagnetic scattering by prolate spheroids for plane waves with arbitrary polarization and angle of incidence,” Radio Sci. 12, 171–184 (1977).
[CrossRef]

Sov. Astron. (1)

N. V. Voshchinnikov, “Dust grains in reflection nebulae. Spherical core-mantle grains,” Sov. Astron. 22, 561–566 (1978).

Other (8)

The numerical code is available on request from N. V. Voshchinnikov, e-mail: nvv@aispbu.spb.su.

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

P. W. Barber, “Scattering and absorption by homogeneous and layered dielectrics,” in Acoustic, Electromagnetic and Elastic Wave Scattering—Focus on the T-Matrix Approach, V. K. Varadan, V. V. Varadan, eds. (Pergamon, New York, 1980), pp. 191–209;D.-S. Wang, P. W. Barber, “Scattering by inhomogeneous nonspherical objects,” Appl. Opt. 18, 1190–1197 (1979);D.-S. Wang, C. H. Chen, P. W. Barber, P. J. Wyatt, “Light scattering by polydisperse suspensions of inhomogeneous nonspherical particles,” Appl. Opt. 18, 2672–2678 (1979).
[CrossRef] [PubMed]

V. N. Lopatin, F. Ya. Sid'ko, Introduction to Optics of Cell Suspension (Nauka, Novosibirsk, Russia, 1987).

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

I. V. Komarov, L. I. Ponomarev, S. Yu. Slavyanov, Spheroidal and Coulomb Spheroidal Functions (Nauka, Moscow, 1976).

J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

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Figures (9)

Fig. 1
Fig. 1

Scattering geometry for a prolate spheroid with a confocal core–mantle structure. The space is divided into three parts: the outer medium, the mantle, and the core. The scattered field in the far-field zone is represented in the spherical coordinate system. The origin of the coordinate system is at the center of the spheroid whereas the z axis coincides with its axis of revolution. The angle of incidence α is the angle in the xz plane between the direction of incidence and the z axis.

Fig. 2
Fig. 2

Percent difference between coated spheres and coated spheroids ∊ defined by Eq. (64): m core = 1.5 + 0.0 i, m mantle = 1.3 + 0.0 i, Vcore/Vtotal = 0.5, a1/b1 = 1.0001, ●, prolate spheroids, ○, oblate spheroids.

Fig. 3
Fig. 3

Scattering efficiency factors Qsca at α = 0° as a function of size parameter 2πa1/λ for the prolate coated spheroids (solid curves): m core = 1.5 + 0.0 i, m mantle = 1.3 + 0.0 i, a1/b1 = 2. The dashed curve represents the results for homogeneous spheroids with m = 1.4 + 0.0 i.

Fig. 4
Fig. 4

Same as Fig. 3 but for oblate spheroids.

Fig. 5
Fig. 5

Scattering efficiency factors Qsca at a α = 90° as a function of size parameter 2p πa1/λ for the prolate coated spheroids (solid curves): m core = 1.5 + 0.0 i, m mantle = 1.3 + 0.0 i, a1/b1 = 2. The dashed curve represents the results for homogeneous with m = 1.4 + 0.0 i.

Fig. 6
Fig. 6

Same as Fig. 5 but for oblate spheroids.

Fig. 7
Fig. 7

Scattering efficiency factors Qsca αt α = 0° as a function of size parameter 2πα1/λ for the coated spheroids (solid curves): m core = 1.5 + 0.0 i, m mantle = 1.3 + 0.0 i, α1/b1 = 10. The dashed curves represent the results for homogeneous spheroids with m = 1.4 + 0.0 i. The factors for oblate spheroids were multiplied by 20.

Fig. 8
Fig. 8

Efficiency factors for a, extinction Qext; b, scattering Qsca; and c, absorption Qabs at α = 0° as a function of size parameter 2πa1/λ for the prolate coated spheroids: a1/b1 = 2; 1, m core = 1.5 + 0.05 i, m mantle = 1.3 + 0.05 i; 2, m core = 1.5 + 0.05 i, m mantle = 1.3 + 0.0 i; 3, m core = 1.5 + 0.0 i, m mantle = 1.3 + 0.05 i; 4, m core = 1.5 + 0.0 i, m mantle = 1.3 + 0.0 i.

Fig. 9
Fig. 9

Same as Fig. 8 but for oblate spheroids.

Tables (2)

Tables Icon

Table 1 Efficiency Factors for Extinction Qext and Scattering Qsca for Core–Mantle Spheres and Spheroids at α = 0°a

Tables Icon

Table 2 Efficiency Factors for Scattering Qsca TM sca for Core–MantleSpheroidsa

Equations (77)

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x = d 2 ( ξ 2 1 ) 1 / 2 ( 1 η 2 ) 1 / 2 cos φ , y = d 2 ( ξ 2 1 ) 1 / 2 ( 1 η 2 ) 1 / 2 sin φ , z = d 2 ξ η ,
2 E ( i ) + k i 2 E ( i ) = 0 ,
E ( i ) = 0 ,
E τ ( 0 ) + E τ ( 1 ) = E τ ( 2 ) H τ ( 0 ) + H τ ( 1 ) = H τ ( 2 ) } ξ = ξ 1 ,
E τ ( 2 ) = E τ ( 3 ) H τ ( 2 ) = H τ ( 3 ) } ξ = ξ 2 ,
lim r r ( E ( 1 ) r i k 1 E ( 1 ) ) = 0 ,
H = 1 i μ k 0 × E , E = 1 i ɛ k 0 × H .
E ( 0 ) = i y exp [ i k 1 ( x sin α + z cos α ) ] , H ( 0 ) = ɛ 1 μ 1 ( i x cos α i z sin α ) × exp [ i k 1 ( x sin α + z cos α ) ] ;
E ( 0 ) = ( i x cos α i z sin α ) exp [ i k 1 ( x sin α + z cos α ) ] , H ( 0 ) = ɛ 1 μ 1 i y exp [ i k 1 ( x sin α + z cos α ) ] .
E ( i ) = E 1 ( i ) + E 2 ( i ) , H ( i ) = H 1 ( i ) + H 2 ( i ) , i = 0 , 1 , 2 , 3 ,
P = h φ E φ , Q = h φ H φ .
E ξ = i k 0 ɛ 1 h η h φ Q η , H ξ = i k 0 μ 1 h η h φ P η , E η = i k 0 ɛ 1 h ξ h φ Q ξ , H η = i k 0 μ 1 h ξ h φ P ξ ,
h ξ = d 2 ( ξ 2 η 2 ξ 2 1 ) 1 / 2 , h η = d 2 ( ξ 2 η 2 1 η 2 ) 1 / 2 , h φ = d 2 [ ( ξ 2 1 ) ( 1 η 2 ) ] 1 / 2
E 1 φ ( 0 ) = l = 1 a l ( 0 ) R 1 l ( 1 ) ( c 1 , ξ ) S 1 l ( c 1 , η ) , H 1 φ ( 0 ) = l = 1 b l ( 0 ) R 1 l ( 1 ) ( c 1 , ξ ) S 1 l ( c 1 , η ) ,
E 1 φ ( 0 ) = l = 1 a l ( 0 ) R 1 l ( 3 ) ( c 1 , ξ ) S 1 l ( c 1 , η ) , H 1 φ ( 0 ) = l = 1 b l ( 0 ) R 1 l ( 3 ) ( c 1 , ξ ) S 1 l ( c 1 , η ) ,
E 1 φ ( 2 ) = l = 1 [ a l ( 2 ) R 1 l ( 1 ) ( c 2 , ξ ) + c l ( 2 ) × R 1 l ( 3 ) ( c 2 , ξ ) ] S 1 l ( c 2 , η ) , H 1 φ ( 2 ) = l = 1 [ b l ( 2 ) R 1 l ( 1 ) ( c 2 , ξ ) + d l ( 2 ) R 1 l ( 3 ) × ( c 2 , ξ ) ] S 1 l ( c 2 , η ) ,
E 1 φ ( 3 ) = l = 1 a l ( 3 ) R 1 l ( 1 ) ( c 3 , ξ ) S 1 l ( c 3 , η ) , H 1 φ ( 3 ) = l = 1 b l ( 3 ) R 1 l ( 1 ) ( c 3 , ξ ) S 1 l ( c 3 , η ) ,
a l ( 0 ) = 2 i l N 1 l 2 ( c 1 ) S 1 l ( c 1 , cos α ) , b l ( 0 ) = 0 ;
a l ( 0 ) = 0 , b l ( 0 ) = 2 i l ɛ 1 μ 1 N 1 l 2 ( c 1 ) S 1 l ( c 1 , cos α ) .
P ( 0 ) + P ( 1 ) = P ( 2 ) 1 μ 1 ξ [ P ( 0 ) + P ( 1 ) ] = 1 μ 2 ξ P ( 2 ) } ξ = ξ 1 ,
P ( 2 ) = P ( 3 ) 1 μ 2 ξ P ( 2 ) = 1 μ 3 ξ P ( 3 ) } ξ = ξ 2 .
{ ( μ 2 μ 1 1 ) ξ 1 I + ( ξ 1 2 f ) [ μ 2 μ 1 3 1 ( c 2 , ξ 1 ) ] } Z ( 2 ) + { ( μ 2 μ 1 1 ) ξ 1 I + ( ξ 1 2 f ) [ μ 2 μ 1 3 3 ( c 2 , ξ 1 ) ] } Y ( 2 ) = μ 2 μ 1 Δ ( c 2 , c 1 ) ( ξ 1 2 f ) [ 3 ( c 1 , ξ 1 ) 1 ( c 1 , ξ 1 ) ] F ( 0 ) , [ ( μ 2 μ 3 1 ) ξ 2 I + ( ξ 2 2 f ) ( μ 2 μ 3 1 1 ( c 2 , ξ 2 ) ) ] P 1 Z ( 2 ) + { ( μ 2 μ 3 1 ) ξ 2 I + ( ξ 2 2 f ) [ μ 2 μ 3 3 1 ( c 2 , ξ 2 ) ] } P 3 Y ( 2 ) = 0 ,
Z ( j ) = { z l ( j ) } 1 , Y ( 2 ) = { y l ( 2 ) } 1 , F ( 0 ) = { f l ( 0 ) } 1 , z l ( 1 ) = a l ( 1 ) R 1 l ( 3 ) ( c 1 , ξ 1 ) N 1 l ( c 1 ) , z l ( 2 ) = a l ( 2 ) R 1 l ( 1 ) ( c 2 , ξ 1 ) N 1 l ( c 2 ) , y l ( 2 ) = c l ( 2 ) R 1 l ( 3 ) ( c 2 , ξ 1 ) N 1 l ( c 2 ) , f l ( 0 ) = a l ( 0 ) R 1 l ( 1 ) ( c 1 , ξ 1 ) N 1 l ( c 1 ) , Δ ( c i , c j ) = { δ n l ( m ) ( c i , c j ) } m , j ( c i , ξ ) = { R m l ( j ) ( c i , ξ ) / R m l ( j ) ( c i , ξ ) δ n l } m , P j = P j ( c 2 , ξ 1 , ξ 2 ) = { R m l ( j ) ( c 2 , ξ 2 ) / R m l ( j ) × ( c 2 , ξ 1 ) δ n l } m , 1 = Δ ( c 2 , c 3 ) 1 ( c 3 , ξ 2 ) Δ ( c 3 , c 2 ) , 3 = Δ ( c 2 , c 1 ) 3 ( c 1 , ξ 1 ) Δ ( c 1 , c 2 ) , I = { δ n l } m is the unit matrix .
Z ( 1 ) = Δ ( c 1 , c 2 ) ( Z ( 2 ) + Y ( 2 ) ) F ( 0 ) .
E 2 ( i ) = × [ U ( i ) i z + V ( i ) r ] , H 2 ( i ) = 1 i μ i k 0 × × [ U ( i ) i z + V ( i ) r ] ;
E 2 ( i ) = 1 i ɛ i k 0 × × [ U ( i ) i z + V ( i ) r ] , H 2 ( i ) = × [ U ( i ) i z + V ( i ) r ] .
U ( 0 ) = m = 1 l = m a m l ( 0 ) R m l ( 1 ) ( c 1 , ξ ) S m l ( c 1 , η ) cos m φ , V ( 0 ) = m = 1 l = m b m l ( 0 ) R m l ( 1 ) ( c 1 , ξ ) S m l ( c 1 , η ) cos m φ ,
U ( 1 ) = m = 1 l = m a m l ( 1 ) R m l ( 3 ) ( c 1 , ξ ) S m l ( c 1 , η ) cos m φ , V ( 1 ) = m = 1 l = m b m l ( 1 ) R m l ( 3 ) ( c 1 , ξ ) S m l ( c 1 , η ) cos m φ ,
U ( 2 ) = m = 1 l = m [ a m l ( 2 ) R m l ( 1 ) ( c 2 , ξ ) + c m l ( 2 ) R m l ( 3 ) ( c 2 , ξ ) ] S m l ( c 2 , η ) cos m φ , V ( 2 ) = m = 1 l = m [ b m l ( 2 ) R m l ( 1 ) ( c 2 , ξ ) + d m l ( 2 ) R m l ( 3 ) ( c 2 , ξ ) ] S m l ( c 2 , η ) cos m φ ,
U ( 3 ) = m = 1 l = m a m l ( 3 ) R m l ( 1 ) ( c 3 , ξ ) S m l ( c 3 , η ) cos m φ , V ( 3 ) = m = 1 l = m b m l ( 3 ) R m l ( 1 ) ( c 3 , ξ ) S m l ( c 3 , η ) cos m φ .
a m l ( 0 ) = 4 i l 1 k 1 N m l 2 ( c 1 ) S m l ( c 1 , cos α ) sin α , b m l ( 0 ) = 0 .
M m l a = × ( ψ m l a ) , N m l a = 1 k × M m l a ,
η [ U ( 0 ) + U ( 1 ) ] + d 2 ξ V ( 1 ) = η U ( 2 ) + d 2 ξ V ( 2 ) ξ { ξ [ U ( 0 ) + U ( 1 ) ] + f d 2 η V ( 1 ) } = ξ [ ξ U ( 2 ) + f d 2 η V ( 2 ) ] ɛ 1 { ξ [ U ( 0 ) + U ( 1 ) ] + f d 2 η V ( 1 ) } = ɛ 2 [ ξ U ( 2 ) + f d 2 η V ( 2 ) ] 1 μ 1 ξ { η [ U ( 0 ) + U ( 1 ) ] + d 2 ξ V ( 1 ) } = 1 μ 2 { ξ [ η U ( 2 ) + d 2 ξ V ( 2 ) ] + ( 1 c 2 2 c 1 2 ) 1 η 2 ξ 2 f η [ ξ U ( 2 ) + f d 2 η V ( 2 ) ] } } ξ = ξ 1 ,
η U ( 2 ) + d 2 ξ V ( 2 ) = η U ( 3 ) + d 2 ξ V ( 3 ) ξ [ ξ U ( 2 ) + f d 2 η V ( 2 ) ] = ξ [ ξ U ( 3 ) + f d 2 η V ( 3 ) ] ɛ 2 [ ξ U ( 2 ) + f d 2 η V ( 2 ) ] = ɛ 3 [ ξ U ( 3 ) + f d 2 η V ( 3 ) ] 1 μ 2 { ξ [ η U ( 2 ) + d 2 ξ V ( 2 ) ] + ( 1 c 2 2 c 3 2 ) 1 η 2 ξ 2 f η [ ξ U ( 2 ) + f d 2 η V ( 2 ) ] } = 1 μ 3 ξ [ η U ( 3 ) + d 2 ξ V ( 3 ) ] } ξ = ξ 2 .
U ( 0 ) + U ( 1 ) = U ( 2 ) ξ { ξ [ U ( 0 ) + U ( 1 ) ] + f η d 2 V ( 1 ) } = ξ [ ξ U ( 2 ) + f η d 2 V ( 2 ) ] V ( 1 ) = V ( 2 ) 1 ɛ 1 ξ { η [ U ( 0 ) + U ( 1 ) ] + d 2 ξ V ( 1 ) } = 1 ɛ 2 { ξ [ η U ( 2 ) + d 2 ξ V ( 2 ) ] + ( 1 c 2 2 c 1 2 ) 1 η 2 ξ 2 f η [ ξ U ( 2 ) + f d 2 η V ( 2 ) ] } } ξ = ξ 1 ,
U ( 2 ) = U ( 3 ) ξ [ ξ U ( 2 ) + f ξ d 2 V ( 2 ) ] = ξ [ ξ U ( 3 ) + f ξ d 2 V ( 3 ) ] V ( 2 ) = V ( 3 ) 1 ɛ 2 { ξ [ η U ( 2 ) + d 2 ξ V ( 2 ) ] + ( 1 c 2 2 c 3 2 ) 1 η 2 ξ 2 f η [ ξ U ( 2 ) + f d 2 η V ( 2 ) ] } = 1 ɛ 3 ξ [ η U ( 3 ) + d 2 ξ V ( 3 ) ] } ξ = ξ 2 .
Z j = { z j , l ( m ) } m , Y j = { y j , l ( m ) } m , X j = { x j , l ( m ) } m , F m = { f l ( m ) } m , z 1 , l ( m ) = κ 1 a m l ( 1 ) R m l ( 3 ) ( c 1 , ξ 1 ) N m l ( c 1 ) , z 2 , l ( m ) = c 1 b m l ( 1 ) R m l ( 3 ) ( c 1 , ξ 1 ) N m l ( c 1 ) , y 1 , l ( m ) = κ 1 c m l ( 2 ) R m l ( 3 ) ( c 2 , ξ 1 ) N m l ( c 2 ) , y 2 , l ( m ) = c 1 d m l ( 2 ) R m l ( 3 ) ( c 2 , ξ 1 ) N m l ( c 2 ) , x 1 , l ( m ) = κ 1 a m l ( 2 ) R m l ( 1 ) ( c 2 , ξ 1 ) N m l ( c 2 ) , x 2 , l ( m ) = c 1 b m l ( 2 ) R m l ( 1 ) ( c 2 , ξ 1 ) N m l ( c 2 ) , f l ( m ) = κ 1 a m l ( 0 ) R m l ( 1 ) ( c 1 , ξ 1 ) N m l ( c 1 ) , = 4 i l 1 N m l 1 ( c 1 ) S m l ( c 1 , cos α ) sin α R m l ( 1 ) ( c 1 , ξ 1 ) , Γ ( c i , c j ) = { γ n l ( m ) ( c i , c j ) } m , K ( c i , c j ) = { κ n l ( m ) ( c i , c j ) } m , Σ ( c i , c j ) = { σ n l ( m ) ( c i , c j ) } m .
{ ξ 1 [ 3 1 ( c 2 , ξ 1 ) ] + ξ 1 ( ɛ 2 ɛ 1 1 ) A 3 } X 1 + f [ Γ [ 3 1 ( c 2 , ξ 1 ) ] + ( ɛ 2 ɛ 1 1 ) A 3 Γ ] X 2 + { ξ 1 [ 3 3 ( c 2 , ξ 1 ) ] + ξ 1 ( ɛ 2 ɛ 1 1 ) A 3 } Y 1 + f { Γ [ 3 3 ( c 2 , ξ 1 ) ] + ( ɛ 2 ɛ 1 1 ) A 3 Γ } Y 2 = ξ 1 Δ ( c 2 , c 1 ) [ 3 ( c 1 , ξ 1 ) 1 ( c 1 , ξ 1 ) ] F m , { Γ [ 3 1 ( c 2 , ξ 1 ) ] + ξ 1 ( ɛ 2 ɛ 1 1 ) 3 } X 1 + { ξ 1 [ 3 1 ( c 2 , ξ 1 ) ] + f ( ɛ 2 ɛ 1 1 ) 3 Γ } X 2 + { Γ [ 3 3 ( c 2 , ξ 1 ) ] + ξ 1 ( ɛ 2 ɛ 1 1 ) 3 } Y 1 + { ξ 1 [ 3 3 ( c 2 , ξ 1 ) ] + f ( ɛ 2 ɛ 1 1 ) 3 Γ } Y 2 = Γ ( c 2 , c 1 ) [ 3 ( c 1 , ξ 1 ) 1 ( c 1 , ξ 1 ) ] F m , { ξ 2 [ 1 1 ( c 2 , ξ 2 ) ] + ξ 2 ( ɛ 2 ɛ 3 1 ) A 1 } P 1 X 1 + f { Γ [ 1 1 ( c 2 , ξ 2 ) ] + ( ɛ 2 ɛ 3 1 ) A 1 Γ } P 1 X 2 + { ξ 2 [ 1 3 ( c 2 , ξ 2 ) ] + ξ 2 ( ɛ 2 ɛ 3 1 ) A 1 } P 3 Y 1 + f { Γ [ 1 3 ( c 2 , ξ 2 ) ] + ( ɛ 2 ɛ 3 1 ) A 1 Γ } P 3 Y 2 = 0 , { Γ [ 1 1 ( c 2 , ξ 2 ) ] + ξ 2 ( ɛ 2 ɛ 3 1 ) 1 } P 1 X 1 + { ξ 2 [ 1 1 ( c 2 , ξ 2 ) ] + f ( ɛ 2 ɛ 3 1 ) 1 Γ } P 1 X 2 + { Γ [ 1 3 ( c 2 , ξ 2 ) ] + ξ 2 ( ɛ 2 ɛ 3 1 ) 1 } P 3 Y 1 + { ξ 2 [ 1 3 ( c 2 , ξ 2 ) ] + f ( ɛ 2 ɛ 3 1 ) 1 Γ } P 3 Y 2 = 0 ,
A 1 = [ ξ 2 ( I + ξ 2 1 ) f Γ 1 Γ ] Q ( c 2 , ξ 2 ) , A 3 = [ ξ 1 ( I + ξ 1 3 ) f Γ 3 Γ ] Q ( c 2 , ξ 1 ) , 1 = [ ξ 2 Γ 1 ( I + ξ 2 1 ) Γ ] Q ( c 2 , ξ 2 ) + 1 ξ 2 2 f K ( c 2 , c 2 ) , 3 = [ ξ 1 Γ 3 ( I + ξ 1 3 ) Γ ] Q ( c 2 , ξ 1 ) + 1 ξ 1 2 f K ( c 2 , c 2 ) , Γ = Γ ( c 2 , c 2 ) , Q ( c , ξ ) = [ ξ 2 I f Γ 2 ( c , c ) ] 1 ;
ξ 1 [ 3 1 ( c 2 , ξ 1 ) ] X 1 + f Γ [ 3 1 ( c 2 , ξ 1 ) ] X 2 + ξ 1 [ 3 3 ( c 2 , ξ 1 ) ] Y 1 + f Γ [ 3 3 ( c 2 , ξ 1 ) ] Y 2 = ξ 1 Δ ( c 2 , c 1 ) [ 3 ( c 1 , ξ 1 ) 1 ( c 1 , ξ 1 ) ] F m , { Γ [ 3 1 ( c 2 , ξ 1 ) ] + ( ɛ 2 ɛ 1 1 ) [ Γ 3 + ξ 1 ξ 1 2 f K ( c 2 , c 2 ) ] } X 1 + { ξ 1 [ 3 1 ( c 2 , ξ 1 ) ] + ( ɛ 2 ɛ 1 1 ) [ I + ξ 1 3 + f ξ 1 2 f ( c 2 , c 2 ) ] } X 2 + { Γ [ 3 3 ( c 2 , ξ 1 ) ] + ( ɛ 2 ɛ 1 1 ) [ Γ 3 + f ξ 1 2 f K ( c 2 , c 2 ) ] } Y 1 + { ξ 1 [ 3 3 ( c 2 , ξ 1 ) ] + ( ɛ 2 ɛ 1 1 ) [ I + ξ 1 3 + f ξ 1 2 f ( c 2 , c 2 ) ] } Y 2 = ɛ 2 ɛ 1 Γ ( c 2 , c 1 ) [ 3 ( c 1 , ξ 1 ) 1 ( c 1 , ξ 1 ) ] F m , ξ 2 [ 1 1 ( c 2 , ξ 2 ) ] P 1 X 1 + f Γ [ 1 1 ( c 2 , ξ 2 ) ] P 1 X 2 + ξ 2 [ 1 3 ( c 2 , ξ 2 ) ] P 3 Y 1 + f Γ [ 1 3 ( c 2 , ξ 2 ) ] P 3 Y 2 = 0 , { Γ [ 1 1 ( c 2 , ξ 2 ) ] + ( ɛ 2 ɛ 3 1 ) [ Γ 1 + ξ 2 ξ 2 2 f K ( c 2 , c 2 ) ] } P 1 X 1 + { ξ 2 [ 1 1 ( c 2 , ξ 2 ) ] + ( ɛ 2 ɛ 3 1 ) [ I + ξ 2 1 + f ξ 2 2 f ( c 2 , c 2 ) ] } P 1 X 2 + { Γ [ 1 3 ( c 2 , ξ 2 ) ] + ( ɛ 2 ɛ 3 1 ) [ Γ 1 + ξ 2 ξ 2 2 f K ( c 2 , c 2 ) ] } P 3 Y 1 + { ξ 2 [ 1 3 ( c 2 , ξ 2 ) ] + ( ɛ 2 ɛ 3 1 ) [ I + ξ 2 1 + f ξ 2 2 f ( c 2 , c 2 ) ] } P 3 Y 2 = 0
Z 1 = Δ ( c 1 , c 2 ) { [ I + ( ɛ 2 ɛ 1 1 ) ξ 1 2 Q ( c 2 , ξ 1 ) ] ( X 1 + Y 1 ) + f ( ɛ 2 ɛ 1 1 ) ξ 1 Γ Q ( c 2 , ξ 1 ) ( X 2 + Y 2 ) } F m , Z 2 = Δ ( c 1 , c 2 ) { [ ɛ 2 ɛ 1 I + ( ɛ 2 ɛ 1 1 ) ξ 1 2 Q ( c 2 , ξ 1 ) ] ( X 2 + Y 2 ) ( ɛ 2 ɛ 1 1 ) ξ 1 Γ Q ( c 2 , ξ 1 ) ( X 1 + Y 1 ) } ;
Z 1 = Δ ( c 1 , c 2 ) ( X 1 + Y 1 ) F m , Z 2 = Δ ( c 1 , c 2 ) ( X 2 + Y 2 ) .
z n = β 1 z n + 1 + β 2 z n 1 + i = 1 g n i z i + f n ,
| β 1 | + | β 2 | p < 1 , i = 1 | g n i | const / n , n = 1 | f n | 2 < .
| β 1 | + | β 2 | + i = 1 | g n i | p < 1 .
( E ( 1 ) E ( 1 ) ) = 1 i k 1 r exp [ i ( k 1 r k 1 r ) ] ( T 22 T 12 T 21 T 11 ) ( E ( 0 ) E ( 0 ) ) ,
T 11 = l = 1 i l a l ( 1 ) S 1 l ( c 1 , cos ϑ ) m = 1 l = m i ( l 1 ) × [ k 1 a m l ( 1 ) S m l ( c 1 , cos ϑ ) + i b m l ( 1 ) S m l ( c 1 , cos ϑ ) ] sin ϑ cos m φ ,
T 12 = m = 1 l = m i l b m l ( 1 ) m S m l ( c 1 , cos ϑ ) sin ϑ sin m φ ,
T 21 = m = 1 l = m i l b m l ( 1 ) m S m l ( c 1 , cos ϑ ) sin ϑ sin m φ ,
T 22 = l = 1 i l b l ( 1 ) S 1 l ( c 1 , cos ϑ ) + m = 1 l = m i ( l 1 ) × [ k 1 a m l ( 1 ) S m l ( c 1 , cos ϑ ) + i b m l ( 1 ) S m l ( c 1 , cos ϑ ) ] sin ϑ cos m φ .
C ext = 4 π k 1 2 Re [ A ( 1 ) , i ( 0 ) ] Θ = 0 ° ,
C sca = 1 k 1 2 4 π | A ( 1 ) | 2 d Ω ,
C abs = C ext C sca ,
G ( α ) = π b 1 ( a 1 2 sin 2 α + b 1 2 cos 2 α ) 1 / 2 for a prolate spheroid ,
G ( α ) = π a 1 ( a 1 2 cos 2 α + b 1 2 sin 2 α ) 1 / 2 for an oblate spheroid ,
Q ext = 4 c 1 2 [ ( ξ 1 2 f ) ( ξ 1 2 f cos 2 α ) ] 1 / 2 × Re { l = 1 i l a l ( 1 ) S 1 l ( c 1 , cos α ) + m = 1 l = m i ( l 1 ) [ k 1 a m l ( 1 ) S m l ( c 1 , cos α ) + i b m l ( 1 ) S m l ( c 1 , cos α ) ] sin α } ,
Q sca = 1 c 1 2 [ ( ξ 1 2 f ) ( ξ 1 2 f cos 2 α ) ] 1 / 2 × { 2 l = 1 | a l ( 1 ) | 2 N 1 l 2 ( c 1 ) + Re m = 1 l = m n = m i n l × { k 1 2 a m l 1 a m n ( 1 ) * ω l n ( m ) ( c 1 , c 1 ) + i k 1 [ b m l ( 1 ) a m n ( 1 ) * κ l n ( m ) ( c 1 , c 1 ) a m l ( 1 ) b m n ( 1 ) * κ n l ( m ) ( c 1 , c 1 ) ] + b m l ( 1 ) b m n ( 1 ) * τ l n ( m ) ( c 1 , c 1 ) } N m l ( c 1 ) N m n ( c 1 ) } ,
C π r υ 1 2 = ( ξ 1 2 f cos 2 α ) 1 / 2 [ ξ 1 4 ( ξ 1 2 f ) ] 1 / 6 Q .
r υ 1 , 2 3 = a 1 , 2 b 1 , 2 2 for prolate spheroids ,
r υ 1 , 2 3 = a 1 , 2 2 b 1 , 2 for oblate spheroids ,
ξ 1 , 2 = a 1 , 2 b 1 , 2 [ ( a 1 , 2 b 1 , 2 ) 2 1 ] 1 / 2 for prolate spheroid ,
ξ 1 , 2 = [ ( a 1 , 2 b 1 , 2 ) 2 1 ] 1 / 2 for an oblate spheroid .
2 π a 1 λ = c 1 ξ 1 for prolate spheroid ,
2 π a 1 λ = c 1 ( ξ 1 2 + 1 ) 1 / 2 for an oblate spheroid .
V core V total = ξ 2 ( ξ 2 2 f ) ξ 1 ( ξ 1 2 f ) .
( ξ 1 , 2 ) ( n ) = [ ( ξ 1 , 2 ) ( n 1 ) + V total , core V core , total ξ 2 , 1 ( ξ 2 , 1 2 1 ) ] 1 / 3 ,
( ξ 1 , 2 ) ( n ) = 2 [ ( ξ 1 , 2 ) ( n 1 ) ] 3 + V total , core V core , total ξ 2 , 1 ( ξ 2 , 1 2 + 1 ) ] 3 [ ( ξ 1 , 2 ) ( n 1 ) 2 + 1 ,
τ = a 1 [ ( a 1 b 1 ) 2 sin 2 α + cos 2 α ] 1 / 2 a 2 [ ( a 2 b 2 ) 2 sin 2 α + cos 2 α ] 1 / 2
τ = a 1 [ ( a 1 b 1 ) 2 cos 2 α + sin 2 α ] 1 / 2 a 2 [ ( a 2 b 2 ) 2 cos 2 α + sin 2 α ] 1 / 2
= Q sca ( sphere ) C sca ( spheroid ) / π r υ 2 Q sca ( sphere ) 100 % .
δ n l ( m ) ( c i , c j ) = 1 1 S m n ( c i , η ) S m l ( c j , η ) d η = N m n 1 ( c i ) N m l 1 ( c j ) r = 0 , 1 d r m n ( c i ) d r m l ( c j ) × 2 2 r + 2 m + 1 ( r + 2 m ) ! r ! ,
γ n l ( m ) ( c i , c j ) = 1 1 S m n ( c i , η ) S m l ( c j , η ) η d η = N m n 1 ( c i ) N m l 1 ( c j ) r = 0 , 1 d r m n ( c i ) × [ d r 1 m l ( c j ) r 2 r + 2 m 1 + d r + 1 m l ( c j ) r + 2 m + 1 2 r + 2 m + 3 ] × 2 2 r + 2 m + 1 ( r + 2 m ) ! r ! ,
κ n l ( m ) ( c i , c j ) = 1 1 S m n ( c i , η ) S m l ( c j , η ) ( 1 η 2 ) d η = N m n 1 ( c i ) N m l 1 ( c j ) r = 0 , 1 d r m n ( c i ) × [ d r 1 m l ( c j ) r ( r + 2 m + 1 ) 2 r + 2 m 1 d r + 1 m l ( c j ) ( r + m ) ( r + 2 m + 1 ) 2 r + 2 m + 3 ] × 2 2 r + 2 m + 1 ( r + 2 m ) ! r ! ,
σ n l ( m ) ( c i , c j ) = 1 1 [ η S m n ( c i , η ) ] S m l ( c j , η ) ( 1 η 2 ) d η = N m n 1 ( c i ) N m l 1 ( c j ) r = 0 , 1 d r m n ( c i ) × [ d r 2 m l ( c j ) ( r 1 ) r ( r + m ) ( 2 r + 2 m 3 ) ( 2 r + 2 m 1 ) + d r m l ( c j ) 3 ( r + m ) ( r + m + 1 ) m 2 2 ( 2 r + 2 m 1 ) ( 2 r + 2 m + 3 ) d r + 2 m l ( c j ) × ( r + m + 1 ) ( r + 2 m + 1 ) ( r + 2 m + 2 ) ( 2 r + 2 m + 3 ) ( 2 r + 2 m + 5 ) ] × 2 2 r + 2 m + 1 ( r + 2 m ) ! r ! ,
ω n l ( m ) ( c i , c j ) = 1 1 S m n ( c i , η ) S m l ( c j , η ) ( 1 η 2 ) d η = N m n 1 ( c i ) N m l 1 ( c j ) r = 0 , 1 d r m n ( c i ) × [ d r 2 m l ( c j ) r ( r 1 ) ( 2 r + 2 m 3 ) ( 2 r + 2 m 1 ) d r m l ( c j ) 2 [ ( r + m ) ( r + m + 1 ) + m 2 1 ] ( 2 r + 2 m 1 ) ( 2 r + 2 m + 3 ) + d r + 2 m l ( c j ) ( r + 2 m + 1 ) ( r + 2 m + 2 ) ( 2 r + 2 m + 3 ) ( 2 r + 2 m + 5 ) ] × 2 2 r + 2 m + 1 ( r + 2 m ) ! r ! ,
τ n l ( m ) ( c i , c j ) = 1 1 [ S m n ( c i , η ) S m l ( c j , η ) ( 1 η 2 ) + m 2 S m n ( c i , η ) S m l ( c j , η ) 1 η 2 ] d η = N m n 1 ( c i ) N m l 1 ( c j ) r = 0 , 1 d r m n ( c i ) d r m l ( c j ) × 2 ( r + m ) ( r + m + 1 ) 2 r + 2 m + 1 × ( r + 2 m ) ! r ! .
κ n l ( m ) ( c i , c j ) + κ ln ( m ) ( c j , c i ) = 2 γ n l ( m ) ( c i , c j ) , σ n l ( m ) ( c i , c j ) + σ ln ( m ) ( c j , c i ) = 2 δ n l ( m ) ( c i , c j ) ω n l ( m ) ( c i , c j ) .

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