Abstract

The light scattering problem for a confocal multilayered spheroid has been solved by the extended boundary condition method with a corresponding spheroidal basis. The solution preserves the advantages of the approach applied previously to homogeneous and core-mantle spheroids, i.e., the separation of the radiation fields into two parts and a special choice of scalar potentials for each of the parts. The method is known to be useful in a wide range of the particle parameters. It is particularly efficient for strongly prolate and oblate spheroids. Numerical tests are described. Illustrative calculations have shown that the extinction factors converge to average values with a growing number of layers and how the extinction varies with a growth of particle porosity.

© 2012 Optical Society of America

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  1. V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered nonspherical particles: a set of methods,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 599–626 (2003).
    [CrossRef]
  2. F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
    [CrossRef]
  3. A. Vinokurov, V. Farafonov, and V. Il’in, “Separation of variables method for multilayered nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1356–1368 (2009).
    [CrossRef]
  4. V. G. Farafonov, “A unified approach, using spheroidal functions, for solving the problem of light scattering by axisymmetric particles,” J. Math. Sci. 175, 698–723 (2011).
    [CrossRef]
  5. I. Gurwich, M. Kleiman, N. Shiloah, and A. Cohen, “Scattering of electromagnetic radiation by multilayered spheroidal particles: recursive procedure,” Appl. Opt. 39, 470–477 (2000).
    [CrossRef]
  6. V. G. Farafonov, “New recursive solution to the problem of scattering of electromagnetic radiation by multilayered spheroidal particles,” Opt. Spectrosc. 90, 743–752 (2001).
    [CrossRef]
  7. I. Gurwich, M. Kleiman, N. Shiloah, and D. Oaknin, “Scattering by an arbitrary multi-layered spheroid: theory and numerical results,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 649–660 (2003).
    [CrossRef]
  8. Y. Han, H. Zhang, and X. Sun, “Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries,” Appl. Phys. B 84, 485–492 (2006).
    [CrossRef]
  9. D. S. Wang and P. W. Barber, “Scattering by inhomogeneous nonspherical objects,” Appl. Opt. 18, 1190–1197 (1979).
    [CrossRef]
  10. D. Petrov, Y. Shkuratov, E. Zubko, and G. Videen, “Sh-matrices method as applied to scattering by particles with layered structure,” J. Quant. Spectrosc. Radiat. Transfer 106, 437–454(2007).
    [CrossRef]
  11. A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).
  12. N. V. Voshchinnikov, V. G. Farafonov, G. Videen, and L. S. Ivlev, “Development of the separation of variables method for multi-layered spheroids,” in Proceedings of the 9th Conference on Electromagnetic and Light Scattering by Nonspherical Particles, N. V. Voshchinnikov, ed. (VVM Com. Ltd., 2006), pp. 271–274.
  13. V. G. Farafonov, N. V. Voshchinnikov, and V. V. Somsikov, “Light scattering by a core-mantle spheroidal particle,” Appl. Opt. 35, 5412–5426 (1996).
    [CrossRef]
  14. C. Flammer, Spheroidal Wave Functions (Stanford University, 1957).
  15. I. V. Komarov, L. I. Ponomarev, and S. Yu. Slavyanov, Spheroidal and Coulomb Spheroidal Functions (Nauka, 1976).
  16. V. G. Farafonov and V. B. Il’in, “Single light scattering: computational methods,” in Light Scattering Reviews, A. A. Kokhanovsky, ed. (Springer, 2006), Vol. 1, pp. 125–177.
  17. N. V. Voshchinnikov and V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
    [CrossRef]
  18. N. V. Voshchinnikov and V. G. Farafonov, “Numerical treatment of spheroidal wave functions,” in International Conference on Electromagnetic and Light Scattering by Nonspherical Particles, B. Å. S. Gustafson, L. Kolokolova, and G. Videen, eds. (Adelphi Army Research Laboratory, 2002), pp. 325–328.
  19. N. V. Voshchinnikov and V. G. Farafonov, “Computation of radial prolate spheroidal wave functions using Jáffe’s series expansions,” J. Comp. Math. Math. Phys. 43, 1299–1309 (2003).
  20. N. V. Voshchinnikov and J. S. Mathis, “Calculating cross sections of composite interstellar grains,” Astrophys. J. 526, 257–264 (1999).
    [CrossRef]
  21. B. Posselt, V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered ellipsoidal particles in the quasistatic approximation,” Meas. Sci. Technol. 13, 256–262 (2002).
    [CrossRef]
  22. N. V. Voshchinnikov, V. B. Il’in, and Th. Henning, “Modelling the optical properties of composite and porous interstellar grains,” Astron. Astrophys. 429, 371–381 (2005).
    [CrossRef]
  23. E. Krügel and R. Siebenmorgen, “Dust in protostellar cores and stellar disks,” Astron. Astrophys. 288, 929–941 (1994).

2011

V. G. Farafonov, “A unified approach, using spheroidal functions, for solving the problem of light scattering by axisymmetric particles,” J. Math. Sci. 175, 698–723 (2011).
[CrossRef]

2009

A. Vinokurov, V. Farafonov, and V. Il’in, “Separation of variables method for multilayered nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1356–1368 (2009).
[CrossRef]

2007

D. Petrov, Y. Shkuratov, E. Zubko, and G. Videen, “Sh-matrices method as applied to scattering by particles with layered structure,” J. Quant. Spectrosc. Radiat. Transfer 106, 437–454(2007).
[CrossRef]

2006

Y. Han, H. Zhang, and X. Sun, “Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries,” Appl. Phys. B 84, 485–492 (2006).
[CrossRef]

2005

N. V. Voshchinnikov, V. B. Il’in, and Th. Henning, “Modelling the optical properties of composite and porous interstellar grains,” Astron. Astrophys. 429, 371–381 (2005).
[CrossRef]

2003

I. Gurwich, M. Kleiman, N. Shiloah, and D. Oaknin, “Scattering by an arbitrary multi-layered spheroid: theory and numerical results,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 649–660 (2003).
[CrossRef]

N. V. Voshchinnikov and V. G. Farafonov, “Computation of radial prolate spheroidal wave functions using Jáffe’s series expansions,” J. Comp. Math. Math. Phys. 43, 1299–1309 (2003).

V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered nonspherical particles: a set of methods,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 599–626 (2003).
[CrossRef]

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
[CrossRef]

2002

B. Posselt, V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered ellipsoidal particles in the quasistatic approximation,” Meas. Sci. Technol. 13, 256–262 (2002).
[CrossRef]

2001

V. G. Farafonov, “New recursive solution to the problem of scattering of electromagnetic radiation by multilayered spheroidal particles,” Opt. Spectrosc. 90, 743–752 (2001).
[CrossRef]

2000

1999

N. V. Voshchinnikov and J. S. Mathis, “Calculating cross sections of composite interstellar grains,” Astrophys. J. 526, 257–264 (1999).
[CrossRef]

1996

1994

E. Krügel and R. Siebenmorgen, “Dust in protostellar cores and stellar disks,” Astron. Astrophys. 288, 929–941 (1994).

1993

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

1979

Barber, P. W.

Cohen, A.

Doicu, A.

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Eremin, Y.

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Farafonov, V.

A. Vinokurov, V. Farafonov, and V. Il’in, “Separation of variables method for multilayered nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1356–1368 (2009).
[CrossRef]

Farafonov, V. G.

V. G. Farafonov, “A unified approach, using spheroidal functions, for solving the problem of light scattering by axisymmetric particles,” J. Math. Sci. 175, 698–723 (2011).
[CrossRef]

V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered nonspherical particles: a set of methods,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 599–626 (2003).
[CrossRef]

N. V. Voshchinnikov and V. G. Farafonov, “Computation of radial prolate spheroidal wave functions using Jáffe’s series expansions,” J. Comp. Math. Math. Phys. 43, 1299–1309 (2003).

B. Posselt, V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered ellipsoidal particles in the quasistatic approximation,” Meas. Sci. Technol. 13, 256–262 (2002).
[CrossRef]

V. G. Farafonov, “New recursive solution to the problem of scattering of electromagnetic radiation by multilayered spheroidal particles,” Opt. Spectrosc. 90, 743–752 (2001).
[CrossRef]

V. G. Farafonov, N. V. Voshchinnikov, and V. V. Somsikov, “Light scattering by a core-mantle spheroidal particle,” Appl. Opt. 35, 5412–5426 (1996).
[CrossRef]

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

N. V. Voshchinnikov and V. G. Farafonov, “Numerical treatment of spheroidal wave functions,” in International Conference on Electromagnetic and Light Scattering by Nonspherical Particles, B. Å. S. Gustafson, L. Kolokolova, and G. Videen, eds. (Adelphi Army Research Laboratory, 2002), pp. 325–328.

V. G. Farafonov and V. B. Il’in, “Single light scattering: computational methods,” in Light Scattering Reviews, A. A. Kokhanovsky, ed. (Springer, 2006), Vol. 1, pp. 125–177.

N. V. Voshchinnikov, V. G. Farafonov, G. Videen, and L. S. Ivlev, “Development of the separation of variables method for multi-layered spheroids,” in Proceedings of the 9th Conference on Electromagnetic and Light Scattering by Nonspherical Particles, N. V. Voshchinnikov, ed. (VVM Com. Ltd., 2006), pp. 271–274.

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford University, 1957).

Gurwich, I.

I. Gurwich, M. Kleiman, N. Shiloah, and D. Oaknin, “Scattering by an arbitrary multi-layered spheroid: theory and numerical results,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 649–660 (2003).
[CrossRef]

I. Gurwich, M. Kleiman, N. Shiloah, and A. Cohen, “Scattering of electromagnetic radiation by multilayered spheroidal particles: recursive procedure,” Appl. Opt. 39, 470–477 (2000).
[CrossRef]

Han, Y.

Y. Han, H. Zhang, and X. Sun, “Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries,” Appl. Phys. B 84, 485–492 (2006).
[CrossRef]

Henning, Th.

N. V. Voshchinnikov, V. B. Il’in, and Th. Henning, “Modelling the optical properties of composite and porous interstellar grains,” Astron. Astrophys. 429, 371–381 (2005).
[CrossRef]

Il’in, V.

A. Vinokurov, V. Farafonov, and V. Il’in, “Separation of variables method for multilayered nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1356–1368 (2009).
[CrossRef]

Il’in, V. B.

N. V. Voshchinnikov, V. B. Il’in, and Th. Henning, “Modelling the optical properties of composite and porous interstellar grains,” Astron. Astrophys. 429, 371–381 (2005).
[CrossRef]

V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered nonspherical particles: a set of methods,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 599–626 (2003).
[CrossRef]

B. Posselt, V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered ellipsoidal particles in the quasistatic approximation,” Meas. Sci. Technol. 13, 256–262 (2002).
[CrossRef]

V. G. Farafonov and V. B. Il’in, “Single light scattering: computational methods,” in Light Scattering Reviews, A. A. Kokhanovsky, ed. (Springer, 2006), Vol. 1, pp. 125–177.

Ivlev, L. S.

N. V. Voshchinnikov, V. G. Farafonov, G. Videen, and L. S. Ivlev, “Development of the separation of variables method for multi-layered spheroids,” in Proceedings of the 9th Conference on Electromagnetic and Light Scattering by Nonspherical Particles, N. V. Voshchinnikov, ed. (VVM Com. Ltd., 2006), pp. 271–274.

Kahnert, F. M.

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
[CrossRef]

Kleiman, M.

I. Gurwich, M. Kleiman, N. Shiloah, and D. Oaknin, “Scattering by an arbitrary multi-layered spheroid: theory and numerical results,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 649–660 (2003).
[CrossRef]

I. Gurwich, M. Kleiman, N. Shiloah, and A. Cohen, “Scattering of electromagnetic radiation by multilayered spheroidal particles: recursive procedure,” Appl. Opt. 39, 470–477 (2000).
[CrossRef]

Komarov, I. V.

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

Krügel, E.

E. Krügel and R. Siebenmorgen, “Dust in protostellar cores and stellar disks,” Astron. Astrophys. 288, 929–941 (1994).

Mathis, J. S.

N. V. Voshchinnikov and J. S. Mathis, “Calculating cross sections of composite interstellar grains,” Astrophys. J. 526, 257–264 (1999).
[CrossRef]

Oaknin, D.

I. Gurwich, M. Kleiman, N. Shiloah, and D. Oaknin, “Scattering by an arbitrary multi-layered spheroid: theory and numerical results,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 649–660 (2003).
[CrossRef]

Petrov, D.

D. Petrov, Y. Shkuratov, E. Zubko, and G. Videen, “Sh-matrices method as applied to scattering by particles with layered structure,” J. Quant. Spectrosc. Radiat. Transfer 106, 437–454(2007).
[CrossRef]

Ponomarev, L. I.

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

Posselt, B.

B. Posselt, V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered ellipsoidal particles in the quasistatic approximation,” Meas. Sci. Technol. 13, 256–262 (2002).
[CrossRef]

Prokopjeva, M. S.

V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered nonspherical particles: a set of methods,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 599–626 (2003).
[CrossRef]

B. Posselt, V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered ellipsoidal particles in the quasistatic approximation,” Meas. Sci. Technol. 13, 256–262 (2002).
[CrossRef]

Shiloah, N.

I. Gurwich, M. Kleiman, N. Shiloah, and D. Oaknin, “Scattering by an arbitrary multi-layered spheroid: theory and numerical results,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 649–660 (2003).
[CrossRef]

I. Gurwich, M. Kleiman, N. Shiloah, and A. Cohen, “Scattering of electromagnetic radiation by multilayered spheroidal particles: recursive procedure,” Appl. Opt. 39, 470–477 (2000).
[CrossRef]

Shkuratov, Y.

D. Petrov, Y. Shkuratov, E. Zubko, and G. Videen, “Sh-matrices method as applied to scattering by particles with layered structure,” J. Quant. Spectrosc. Radiat. Transfer 106, 437–454(2007).
[CrossRef]

Siebenmorgen, R.

E. Krügel and R. Siebenmorgen, “Dust in protostellar cores and stellar disks,” Astron. Astrophys. 288, 929–941 (1994).

Slavyanov, S. Yu.

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

Somsikov, V. V.

Sun, X.

Y. Han, H. Zhang, and X. Sun, “Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries,” Appl. Phys. B 84, 485–492 (2006).
[CrossRef]

Videen, G.

D. Petrov, Y. Shkuratov, E. Zubko, and G. Videen, “Sh-matrices method as applied to scattering by particles with layered structure,” J. Quant. Spectrosc. Radiat. Transfer 106, 437–454(2007).
[CrossRef]

N. V. Voshchinnikov, V. G. Farafonov, G. Videen, and L. S. Ivlev, “Development of the separation of variables method for multi-layered spheroids,” in Proceedings of the 9th Conference on Electromagnetic and Light Scattering by Nonspherical Particles, N. V. Voshchinnikov, ed. (VVM Com. Ltd., 2006), pp. 271–274.

Vinokurov, A.

A. Vinokurov, V. Farafonov, and V. Il’in, “Separation of variables method for multilayered nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1356–1368 (2009).
[CrossRef]

Voshchinnikov, N. V.

N. V. Voshchinnikov, V. B. Il’in, and Th. Henning, “Modelling the optical properties of composite and porous interstellar grains,” Astron. Astrophys. 429, 371–381 (2005).
[CrossRef]

N. V. Voshchinnikov and V. G. Farafonov, “Computation of radial prolate spheroidal wave functions using Jáffe’s series expansions,” J. Comp. Math. Math. Phys. 43, 1299–1309 (2003).

N. V. Voshchinnikov and J. S. Mathis, “Calculating cross sections of composite interstellar grains,” Astrophys. J. 526, 257–264 (1999).
[CrossRef]

V. G. Farafonov, N. V. Voshchinnikov, and V. V. Somsikov, “Light scattering by a core-mantle spheroidal particle,” Appl. Opt. 35, 5412–5426 (1996).
[CrossRef]

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

N. V. Voshchinnikov and V. G. Farafonov, “Numerical treatment of spheroidal wave functions,” in International Conference on Electromagnetic and Light Scattering by Nonspherical Particles, B. Å. S. Gustafson, L. Kolokolova, and G. Videen, eds. (Adelphi Army Research Laboratory, 2002), pp. 325–328.

N. V. Voshchinnikov, V. G. Farafonov, G. Videen, and L. S. Ivlev, “Development of the separation of variables method for multi-layered spheroids,” in Proceedings of the 9th Conference on Electromagnetic and Light Scattering by Nonspherical Particles, N. V. Voshchinnikov, ed. (VVM Com. Ltd., 2006), pp. 271–274.

Wang, D. S.

Wriedt, T.

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

Zhang, H.

Y. Han, H. Zhang, and X. Sun, “Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries,” Appl. Phys. B 84, 485–492 (2006).
[CrossRef]

Zubko, E.

D. Petrov, Y. Shkuratov, E. Zubko, and G. Videen, “Sh-matrices method as applied to scattering by particles with layered structure,” J. Quant. Spectrosc. Radiat. Transfer 106, 437–454(2007).
[CrossRef]

Appl. Opt.

Appl. Phys. B

Y. Han, H. Zhang, and X. Sun, “Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries,” Appl. Phys. B 84, 485–492 (2006).
[CrossRef]

Astron. Astrophys.

N. V. Voshchinnikov, V. B. Il’in, and Th. Henning, “Modelling the optical properties of composite and porous interstellar grains,” Astron. Astrophys. 429, 371–381 (2005).
[CrossRef]

E. Krügel and R. Siebenmorgen, “Dust in protostellar cores and stellar disks,” Astron. Astrophys. 288, 929–941 (1994).

Astrophys. J.

N. V. Voshchinnikov and J. S. Mathis, “Calculating cross sections of composite interstellar grains,” Astrophys. J. 526, 257–264 (1999).
[CrossRef]

Astrophys. Space Sci.

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

J. Quant. Spectrosc. Radiat. Transfer

I. Gurwich, M. Kleiman, N. Shiloah, and D. Oaknin, “Scattering by an arbitrary multi-layered spheroid: theory and numerical results,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 649–660 (2003).
[CrossRef]

V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered nonspherical particles: a set of methods,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 599–626 (2003).
[CrossRef]

A. Vinokurov, V. Farafonov, and V. Il’in, “Separation of variables method for multilayered nonspherical particles,” J. Quant. Spectrosc. Radiat. Transfer 110, 1356–1368 (2009).
[CrossRef]

J. Comp. Math. Math. Phys.

N. V. Voshchinnikov and V. G. Farafonov, “Computation of radial prolate spheroidal wave functions using Jáffe’s series expansions,” J. Comp. Math. Math. Phys. 43, 1299–1309 (2003).

J. Math. Sci.

V. G. Farafonov, “A unified approach, using spheroidal functions, for solving the problem of light scattering by axisymmetric particles,” J. Math. Sci. 175, 698–723 (2011).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transfer 79–80, 775–824 (2003).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

D. Petrov, Y. Shkuratov, E. Zubko, and G. Videen, “Sh-matrices method as applied to scattering by particles with layered structure,” J. Quant. Spectrosc. Radiat. Transfer 106, 437–454(2007).
[CrossRef]

Meas. Sci. Technol.

B. Posselt, V. G. Farafonov, V. B. Il’in, and M. S. Prokopjeva, “Light scattering by multilayered ellipsoidal particles in the quasistatic approximation,” Meas. Sci. Technol. 13, 256–262 (2002).
[CrossRef]

Opt. Spectrosc.

V. G. Farafonov, “New recursive solution to the problem of scattering of electromagnetic radiation by multilayered spheroidal particles,” Opt. Spectrosc. 90, 743–752 (2001).
[CrossRef]

Other

A. Doicu, T. Wriedt, and Y. Eremin, Light Scattering by Systems of Particles (Springer, 2006).

N. V. Voshchinnikov, V. G. Farafonov, G. Videen, and L. S. Ivlev, “Development of the separation of variables method for multi-layered spheroids,” in Proceedings of the 9th Conference on Electromagnetic and Light Scattering by Nonspherical Particles, N. V. Voshchinnikov, ed. (VVM Com. Ltd., 2006), pp. 271–274.

C. Flammer, Spheroidal Wave Functions (Stanford University, 1957).

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

V. G. Farafonov and V. B. Il’in, “Single light scattering: computational methods,” in Light Scattering Reviews, A. A. Kokhanovsky, ed. (Springer, 2006), Vol. 1, pp. 125–177.

N. V. Voshchinnikov and V. G. Farafonov, “Numerical treatment of spheroidal wave functions,” in International Conference on Electromagnetic and Light Scattering by Nonspherical Particles, B. Å. S. Gustafson, L. Kolokolova, and G. Videen, eds. (Adelphi Army Research Laboratory, 2002), pp. 325–328.

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Figures (6)

Fig. 1.
Fig. 1.

Scattering geometry for a prolate spheroid with the confocal layered structure and a 1 / b 1 = 2 . The space is divided into N + 1 parts: the outer medium (1), the outermost layer (2), ⋯, the core ( N + 1 ). The scattered field in the far-field zone is represented in the spherical coordinate system ( r , ϑ , φ ). Θ is the scattering angle. The origin of the Cartesian coordinate system is at the center of the spheroid while the z axis coincides with its axis of revolution. The angle of incidence α is the angle between the direction of incidence and the z axis in the x y plane.

Fig. 2.
Fig. 2.

Percent difference between three-layered spheres and three-layered spheroids ϵ defined by Eq. (68): m 3 = 1.7 + 0.0 i , m 2 = 1.5 + 0.0 i , m 1 = 1.3 + 0.0 i , V j / V t o t a l = 0.33 , a 1 / b 1 = 1.0001 , α = 0 ° , (•)–prolate spheroids, (o)–oblate spheroids.

Fig. 3.
Fig. 3.

Size dependence of the extinction efficiency factors for layered prolate spheroids with a 1 / b 1 = 3 . Each particle contains an equal fraction of carbon, silicate, and vacuum (the porosity P = 1 / 3 ) separated in equivolume confocal layers. The cyclic order of the different material layers is indicated (starting from the core). The effect of the increase of the number of layers is illustrated.

Fig. 4.
Fig. 4.

Size dependence of the normalized extinction cross sections for 18-layered prolate and oblate spheroids with a 1 / b 1 = 3 . Particles contain an equal fraction of carbon and silicate without vacuum (the porosity P = 0.0 ) or 50% of vacuum (the porosity P = 0.50 ). For a given value of the size parameter, the compact and porous particles have the same mass. The cyclic order of the different material layers is: carbon/vacuum/silicate (starting from the core). The effect of the increase of particle porosity and oblique incidence is illustrated.

Fig. 5.
Fig. 5.

The normalized extinction cross sections [see Eq. (70)] for layered prolate and oblate spheroids with a 1 / b 1 = 3 . For α = 90 ° , the curves are plotted for the sum of the TM and TE modes. The effect of variation of particle type and orientation is illustrated.

Fig. 6.
Fig. 6.

The normalized extinction cross sections [see Eq. (70)] for layered oblate spheroids. The effect of variation of particle shape is illustrated.

Tables (1)

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Table 1. Efficiency Factors for Extinction Q ext and Scattering Q sca for Prolate and Oblate Multilayered Spheroids at α = 0 ° a

Equations (74)

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x = d 2 ( ξ 2 f ˜ ) 1 / 2 ( 1 η 2 ) 1 / 2 cos φ , y = d 2 ( ξ 2 f ˜ ) 1 / 2 ( 1 η 2 ) 1 / 2 sin φ , z = d 2 ξ η ,
ξ = ξ j ,
a 1 2 b 1 2 = a 2 2 b 2 2 = = a N 2 b N 2 = ( d 2 ) 2 .
E⃗ ( 0 ) = i⃗ y exp [ i k 1 ( x sin α + z cos α ) ] ;
E⃗ ( 0 ) = ( i⃗ x cos α i⃗ z sin α ) exp [ i k 1 ( x sin α + z cos α ) ] .
E⃗ ( i ) = E⃗ ( i ) + E⃗ 2 ( i ) , H⃗ ( i ) = H⃗ ( i ) + H⃗ 2 ( i ) , i = 0 , 1 , 2 , , N + 1 ,
P ( i ) = E 1 φ ( i ) cos φ , Q ( i ) = H 1 φ ( i ) cos φ ,
Δ P ( i ) + k i 2 P ( i ) = 0 , Δ Q ( i ) + k i 2 Q ( i ) = 0.
P ( 0 ) + P ( 1 ) = P ( 2 ) , [ ξ 2 f ˜ ( P ( 0 ) + P ( 1 ) ) ] ξ = μ 1 μ 2 [ ξ 2 f ˜ P ( 2 ) ] ξ , } ξ = ξ 1
P ( j ) = P ( j + 1 ) , [ ξ 2 f ˜ P ( j ) ] ξ = μ j μ j + 1 [ ξ 2 f ˜ P ( j + 1 ) ] ξ , } ξ = ξ j
P ( j ) = P A ( j ) + P B ( j ) ,
d 2 ( ξ 1 2 f ˜ ) 0 2 π 0 π { P ( 2 ) ( r⃗ ) G 1 ξ [ μ 1 μ 2 P ( 2 ) ( r⃗ ) ξ + ( μ 1 μ 2 1 ) ξ 1 ξ 1 2 f ˜ P ( 2 ) ( r⃗ ) ] G 1 } d η d φ = { P ( 0 ) ( r⃗ ) , r⃗ D 1 , P ( 1 ) ( r⃗ ) , r⃗ R 3 1 ,
d 2 ( ξ j 2 f ˜ ) 0 2 π 0 π { P ( j + 1 ) ( r⃗ ) G j ξ [ μ j μ j + 1 P ( j + 1 ) ( r⃗ ) ξ + ( μ j μ j + 1 1 ) ξ j ξ j 2 f ˜ P ( j + 1 ) ( r⃗ ] G j } d η d φ = { P A ( j ) ( r⃗ ) , r⃗ D j , P B ( j ) ( r⃗ ) , r⃗ R 3 j ,
G j = G ( k j , r⃗ , r⃗ ) = exp i k j | r⃗ r⃗ | 4 π | r⃗ r⃗ |
P ( 0 ) Q ( 0 ) = l = 1 a l ( 0 ) b l ( 0 ) R 1 l ( 1 ) ( c 1 , ξ ) S 1 l ( c 1 , η ) cos φ ,
P ( 1 ) Q ( 1 ) = l = 1 a l ( 1 ) b l ( 1 ) R 1 l ( 3 ) ( c 1 , ξ ) S 1 l ( c 1 , η ) cos φ ,
P A ( j ) Q A ( j ) = l = 1 a l ( j ) b l ( j ) R 1 l ( 1 ) ( c j , ξ ) S 1 l ( c j , η ) cos φ ,
P B ( j ) Q B ( j ) = l = 1 c l ( j ) d l ( j ) R 1 l ( 3 ) ( c j , ξ ) S 1 l ( c j , η ) cos φ ,
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 ε 1 μ 1 N 1 l 2 ( c 1 ) S 1 l ( c 1 , cos α ) .
G ( k j , r⃗ , r⃗ ) = i k j 2 π m = 0 l = m ( 2 δ 0 m ) × N m l 2 ( c j ) R m l ( 1 ) ( c j , ξ < ) R m l ( 3 ) ( c j , ξ > ) × S m l ( c j , η ) S m l ( c j , η ) cos m ( φ φ ) ,
δ 0 m = { 1 , m = 0 , 0 , m 0 ,
( z⃗ A ( j ) z⃗ B ( j ) ) = ( A 31 ( j ) A 33 ( j ) A 11 ( j ) A 13 ( j ) ) ( z⃗ A ( j + 1 ) z⃗ B ( j + 1 ) ) ,
A 31 ( j ) = W j [ R [ 3 ] ( c j , ξ j ) Δ ( 1 ) ( c j , c j + 1 ) μ j μ j + 1 Δ ( 1 ) ( c j , c j + 1 ) R [ 1 ] ( c j + 1 , ξ j ) ( μ j μ j + 1 1 ) ξ j ξ j 2 f ˜ Δ ( 1 ) ( c j , c j + 1 ) ] P [ 1 ] ( c j + 1 , ξ j , ξ j + 1 ) ,
A 33 ( j ) = W j [ R [ 3 ] ( c j , ξ j ) Δ ( 1 ) ( c j , c j + 1 ) μ j μ j + 1 Δ ( 1 ) ( c j , c j + 1 ) R [ 3 ] ( c j + 1 , ξ j ) ( μ j μ j + 1 1 ) ξ j ξ j 2 f ˜ Δ ( 1 ) ( c j , c j + 1 ) ] P [ 3 ] ( c j + 1 , ξ j , ξ j + 1 ) ,
A 11 ( j ) = W j [ R [ 1 ] ( c j , ξ j ) Δ ( 1 ) ( c j , c j + 1 ) μ j μ j + 1 Δ ( 1 ) ( c j , c j + 1 ) R [ 1 ] ( c j + 1 , ξ j ) ( μ j μ j + 1 1 ) ξ j ξ j 2 f ˜ Δ ( 1 ) ( c j , c j + 1 ) ] P [ 1 ] ( c j + 1 , ξ j , ξ j + 1 ) ,
A 13 ( j ) = W j [ R [ 1 ] ( c j , ξ j ) Δ ( 1 ) ( c j , c j + 1 ) μ j μ j + 1 Δ ( 1 ) ( c j , c j + 1 ) R [ 3 ] ( c j + 1 , ξ j ) ( μ j μ j + 1 1 ) ξ j ξ j 2 f ˜ Δ ( 1 ) ( c j , c j + 1 ) ] P [ 3 ] ( c j + 1 , ξ j , ξ j + 1 ) .
( z⃗ A ( 1 ) z⃗ B ( 1 ) ) = ( { a l ( 0 ) R 1 l ( 1 ) ( c 1 , ξ 1 ) N 1 l ( c 1 ) } 1 { a l ( 1 ) R 1 l ( 3 ) ( c 1 , ξ 1 ) N 1 l ( c 1 ) } 1 ) ,
( z⃗ A ( j ) z⃗ B ( j ) ) = ( { a l ( j ) R 1 l ( 1 ) ( c j , ξ j ) N 1 l ( c j ) } 1 { c l ( j ) R 1 l ( 3 ) ( c j , ξ j ) N 1 l ( c j ) } 1 ) ,
R [ i ] ( c j , ξ j ) = { R m l ( i ) ( c j , ξ j ) / R m l ( i ) ( c j , ξ j ) δ n l } m ,
W j = [ R [ 3 ] ( c j , ξ j ) R [ 1 ] ( c j , ξ j ) ] 1 ,
P [ i ] ( c j , ξ j 1 , ξ j ) = { R m l ( i ) ( c j , ξ j 1 ) / R m l ( i ) ( c j , ξ j ) δ n l } m .
z⃗ B ( 1 ) = A 2 A 1 ( 1 ) z⃗ A ( 1 ) ,
( A 1 A ) 2 = ( A 31 ( 1 ) A 33 ( 1 ) A 11 ( 1 ) A 13 ( 1 ) ) ( A 31 ( j ) A 33 ( j ) A 11 ( j ) A 13 ( j ) ) ( A 31 ( N 1 ) A 33 ( N 1 ) A 11 ( N 1 ) A 13 ( N 1 ) ) ( A 31 ( N ) A 11 ( N ) ) .
A 31 ( j ) = W j [ R [ 3 ] ( i c j , i ξ j ) Δ ( 1 ) ( i c j , i c j + 1 ) ε j ε j + 1 Δ ( 1 ) ( i c j , i c j + 1 ) R [ 1 ] ( i c j + 1 , i ξ j ) ( ε j ε j + 1 1 ) i ξ j ( i ξ ) j 2 f ˜ Δ ( 1 ) ( i c j , i c j + 1 ) ] P [ 1 ] ( i c j + 1 , i ξ j , i ξ j + 1 ) .
E⃗ 2 ( i ) = × ( U ( i ) i z + V ( i ) r ) , H⃗ 2 ( i ) = 1 i μ i k 0 ⃗∇ × ⃗∇ × ( U ( i ) i⃗ z + V ( i ) r⃗ ) ;
E⃗ 2 ( i ) = 1 i ε i k 0 ⃗∇ × ⃗∇ × ( U ( i ) i⃗ z + V ( i ) r⃗ ) , H⃗ 2 ( i ) = ⃗∇ × ( U ( i ) i⃗ z + V ( i ) r⃗ ) ,
η U ( j ) + d 2 ξ V ( j ) = η U ( j + 1 ) + d 2 ξ V ( j + 1 ) , ξ ( ξ U ( j ) + f ˜ d 2 η V ( j ) ) = ξ ( ξ U ( j + 1 ) + f ˜ d 2 η V ( j + 1 ) ) , ε j ( ξ U ( j ) + f ˜ d 2 η V ( j ) ) = ε j + 1 ( ξ U ( j + 1 ) + f ˜ d 2 η V ( j + 1 ) ) , 1 μ j ξ ( η U ( j ) + d 2 ξ V ( j ) ) = 1 μ j + 1 [ ξ ( η U ( j + 1 ) + d 2 ξ V ( j + 1 ) ) + ( 1 c j + 1 2 c j 2 ) 1 η 2 ξ 2 f ˜ η ( ξ U ( j + 1 ) + f ˜ d 2 η V ( j + 1 ) ) ] , } ξ = ξ j
d 2 ( ξ j 2 f ˜ ) 0 2 π 0 π { U ( j + 1 ) G j ξ μ j μ j + 1 U ( j + 1 ) ξ G j + ( ε j + 1 ε j 1 ) [ ξ j 2 ξ j 2 f ˜ η 2 U ( j + 1 ) + f ˜ ξ j η ξ j 2 f ˜ η 2 d 2 V ( j + 1 ) ] G j ξ + ( μ j μ j + 1 1 ) [ ξ j 2 ξ j 2 f ˜ η 2 U ( j + 1 ) ξ + f ˜ ξ j η ξ j 2 f ˜ η 2 d 2 V ( j + 1 ) ξ ] G j + ( ε j + 1 ε j 1 ) ξ j ξ j 2 f ˜ η 2 [ U ( j + 1 ) + 2 ξ j 2 ξ j 2 f ˜ η 2 U ( j + 1 ) + 2 f ˜ ξ j η ξ j 2 f ˜ η 2 d 2 V ( j + 1 ) ] G j ( ε j + 1 ε j μ j μ j + 1 ) f ˜ η ξ j 2 f ˜ η 2 [ 1 η 2 ξ j 2 f ˜ η ( ξ j U ( j + 1 ) + f ˜ η d 2 V ( j + 1 ) ) + d 2 V ( j + 1 ) ] G j } d η d φ = { U A ( j ) ( r⃗ ) , r⃗ D j , U B ( j ) ( r⃗ ) , r⃗ R 3 j ,
d 2 ( ξ j 2 f ˜ ) 0 2 π 0 π { ε j + 1 ε j d 2 V ( j + 1 ) G j ξ d 2 V ( j + 1 ) ξ G j ( ε j + 1 ε j 1 ) [ ξ j η ξ j 2 f ˜ η 2 U ( j + 1 ) + ξ j 2 ξ j 2 f ˜ η 2 d 2 V ( j + 1 ) ] G j ξ ( μ j μ j + 1 1 ) [ ξ j η ξ j 2 f ˜ η 2 U ( j + 1 ) ξ + ξ j 2 ξ j 2 f ˜ η 2 d 2 V ( j + 1 ) ξ ] G j ( ε j + 1 ε j 1 ) ξ j ξ j 2 f ˜ η 2 [ d 2 V ( j + 1 ) + 2 ξ j η ξ j 2 f ˜ η 2 U ( j + 1 ) + 2 ξ j 2 ξ j 2 f ˜ η 2 d 2 V ( j + 1 ) ] G j + ( ε j + 1 ε j μ j μ j + 1 ) ξ j ξ j 2 f ˜ η 2 [ 1 η 2 ξ j 2 f ˜ η ( ξ j U ( j + 1 ) + f ˜ η d 2 V ( j + 1 ) ) + d 2 V ( j + 1 ) ] G j } d η d φ = { V A ( j ) ( r⃗ ) , r⃗ D j , V B ( j ) ( r⃗ ) , r⃗ R 3 j ,
U ( 0 ) V ( 0 ) = m = 1 l = m a m l ( 0 ) b m l ( 0 ) R m l ( 1 ) ( c 1 , ξ ) S m l ( c 1 , η ) cos m φ ,
U ( 1 ) V ( 1 ) = m = 1 l = m a m l ( 1 ) b m l ( 1 ) R m l ( 3 ) ( c 1 , ξ ) S m l ( c 1 , η ) cos m φ ,
U A ( j ) V A ( j ) = m = 1 l = m a m l ( j ) b m l ( j ) R m l ( 1 ) ( c j , ξ ) S m l ( c j , η ) cos m φ ,
U B ( j ) V B ( j ) = m = 1 l = m c m l ( j ) d m l ( j ) R m l ( 3 ) ( c j , ξ ) S m l ( c j , 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 .
( Z⃗ A ( j ) Z⃗ B ( j ) ) = ( A 31 ( j ) A 33 ( j ) A 11 ( j ) A 13 ( j ) ) ( Z⃗ A ( j + 1 ) Z⃗ B ( j + 1 ) ) ,
( Z⃗ A ( j ) Z⃗ B ( j ) ) = ( { k 1 a m l ( j ) R m l ( 3 ) ( c j , ξ j ) N m l ( c j ) } m { c 1 b m l ( j ) R m l ( 3 ) ( c j , ξ j ) N m l ( c j ) } m { k 1 c m l ( j ) R m l ( 3 ) ( c j , ξ j ) N m l ( c j ) } m { c 1 d m l ( j ) R m l ( 3 ) ( c j , ξ j ) N m l ( c j ) } m ) ,
A i k ( j ) = ( A i k , A ( j ) B i k , A ( j ) A i k , B ( j ) B i k , B ( j ) ) ,
A 31 , A ( j ) = W j { R [ 3 ] ( c j , ξ j ) Δ ( m ) ( c j , c j + 1 ) μ j μ j + 1 Δ ( m ) ( c j , c j + 1 ) R [ 1 ] ( c j + 1 , ξ j ) + ( ε j + 1 ε j 1 ) ξ j [ ξ j R [ 3 ] ( c j , ξ j ) Q ( m ) ( c j , c j + 1 , ξ j ) Q ( m ) ( c j , c j + 1 , ξ j ) ( I 2 ξ j 2 Q ( m ) ( c j + 1 , c j + 1 , ξ j ) ) ] + ( μ j μ j + 1 1 ) ξ j 2 Q ( m ) ( c j , c j + 1 , ξ j ) R [ 1 ] ( c j + 1 , ξ j ) ( ε j + 1 ε j μ j μ j + 1 ) f ˜ ξ j ξ j 2 f ˜ Q ( m ) ( c j , c j + 1 , ξ j ) E ( m ) ( c j + 1 , c j + 1 ) } P [ 1 ] ( c j + 1 , ξ j , ξ j + 1 ) ,
B 31 , A ( j ) = W j { ( ε j + 1 ε j 1 ) f ˜ ξ j [ R [ 3 ] ( c j , ξ j ) Q ( m ) ( c j , c j + 1 , ξ j ) + 2 ξ j Q ( m ) ( c j , c j + 1 , ξ j ) Q ( m ) ( c j + 1 , c j + 1 , ξ j ) ] Γ ( m ) ( c j + 1 , c j + 1 ) + ( μ j μ j + 1 1 ) f ˜ ξ j Q ( m ) ( c j , c j + 1 , ξ j ) Γ ( m ) ( c j + 1 , c j + 1 ) R [ 1 ] ( c j + 1 , ξ j ) ( ε j + 1 ε j μ j μ j + 1 ) f ˜ ξ j 2 f ˜ [ ( ξ j 2 Q ( m ) ( c j , c j + 1 , ξ j ) Δ ( m ) ( c j , c j + 1 ) ) K ( m ) ( c j + 1 , c j + 1 ) + Γ ( m ) ( c j , c j + 1 ) ] } × P [ 1 ] ( c j + 1 , ξ j , ξ j + 1 ) ,
A 31 , B ( j ) = W j { ( ε j + 1 ε j 1 ) ξ j [ R [ 3 ] ( c j , ξ j ) Q ( m ) ( c j , c j + 1 , ξ j ) + 2 ξ j Q ( m ) ( c j , c j + 1 , ξ j ) Q ( m ) ( c j + 1 , c j + 1 , ξ j ) ] Γ ( m ) ( c j + 1 , c j + 1 ) ( μ j μ j + 1 1 ) ξ j Q ( m ) ( c j , c j + 1 , ξ j ) Γ ( m ) ( c j + 1 , c j + 1 ) R [ 1 ] ( c j + 1 , ξ j ) + ( ε j + 1 ε j μ j μ j + 1 ) × ξ j 2 ξ j 2 f ˜ Q ( m ) ( c j , c j + 1 , ξ j ) K ( m ) ( c j + 1 , c j + 1 ) } P [ 1 ] ( c j + 1 , ξ j , ξ j + 1 ) ,
B 31 , B ( j ) = W j { ε j + 1 ε j R [ 3 ] ( c j , ξ j ) Δ ( m ) ( c j , c j + 1 ) Δ ( m ) ( c j , c j + 1 ) R [ 1 ] ( c j + 1 , ξ j ) ( ε j + 1 ε j 1 ) ξ j [ ξ j R [ 3 ] ( c j , ξ j ) Q ( m ) ( c j , c j + 1 , ξ j ) Q ( m ) ( c j , c j + 1 , ξ j ) ( I 2 ξ j 2 Q ( m ) ( c j + 1 , c j + 1 , ξ j ) ) ] ( μ j μ j + 1 1 ) ξ j 2 Q ( m ) ( c j , c j + 1 , ξ j ) R [ 1 ] ( c j + 1 , ξ j ) + ( ε j + 1 ε j μ j μ j + 1 ) ξ j ξ j 2 f ˜ [ f ˜ Q ( m ) ( c j , c j + 1 , ξ j ) E ( m ) ( c j + 1 , c j + 1 ) + Δ ( m ) ( c j , c j + 1 ) ] } P [ 1 ] ( c j + 1 , ξ j , ξ j + 1 ) ,
Q ( m ) ( c j + 1 , c j + 1 , ξ j ) = { ξ j 2 I f ˜ [ Γ ( m ) ( c j + 1 , c j + 1 ) ] 2 } 1 ,
Q ( m ) ( c j , c j + 1 , ξ j ) = Δ ( m ) ( c j , c j + 1 ) Q ( m ) ( c j + 1 , c j + 1 , ξ j )
Z⃗ B ( 1 ) = A 2 A 1 ( 1 ) Z⃗ A ( 1 ) ,
Z⃗ B ( 1 ) = ( { k 1 a m l ( 1 ) R m l ( 3 ) ( c 1 , ξ 1 ) N m l ( c 1 ) } m { c 1 b m l ( 1 ) R m l ( 3 ) ( c 1 , ξ 1 ) N m l ( c 1 ) } m ) ,
Z⃗ A ( 1 ) = ( { k 1 a m l ( 0 ) R m l ( 1 ) ( c 1 , ξ 1 ) N m l ( c 1 ) } m 0 ) ,
C = G Q ,
G ( α ) = π b 1 ( a 1 2 sin 2 α + b 1 2 cos 2 α ) 1 / 2 for prolate spheroids,
G ( α ) = π a 1 ( a 1 2 cos 2 α + b 1 2 sin 2 α ) 1 / 2 for oblate spheroids,
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 ] ,
C π r V 2 = [ ( a 1 / b 1 ) 2 sin 2 α + cos 2 α ] 1 / 2 ( a 1 / b 1 ) 2 / 3 Q for prolate spheroids,
C π r V 2 = [ ( a 1 / b 1 ) 2 cos 2 α + sin 2 α ] 1 / 2 ( a 1 / b 1 ) 1 / 3 Q for oblate spheroids.
r V 3 = a 1 b 1 2 for prolate spheroids,
r V 3 = a 1 2 b 1 for oblate spheroids.
x V = 2 π r V / λ ,
ξ j = ( a j b j ) ( 1 + f ˜ ) / 2 [ ( a j b j ) 2 1 ] 1 / 2 .
2 π a 1 λ = ( a 1 b 1 ) ( 1 f ˜ ) / 2 c 1 ξ 1 = x V ( a 1 b 1 ) ( 3 + f ˜ ) / 6 .
l = j N f l = l = j N V l V total = ξ j ( ξ j 2 f ˜ ) ξ 1 ( ξ 1 2 f ˜ ) .
( ξ j ) ( n ) = ( ξ j ) ( n 1 ) + l = j N f l [ ξ 1 ( ξ 1 2 1 ) ] 3 ,
( ξ j ) ( n ) = 2 [ ( ξ j ) ( n 1 ) ] 3 + l = j N f l [ ξ 1 ( ξ 1 2 + 1 ) ] 3 [ ( ξ j ) ( n 1 ) ] 2 + 1 ,
ϵ = Q ext ( sphere ) C ext ( spheroid ) / π r V 2 Q ext ( sphere ) 100 %
x porous = x compact ( 1 P ) 1 / 3 = x compact ( V solid / V total ) 1 / 3 ,
C ( n ) = C ( porous grain ) C ( compact grain of same mass and shape ) = ( 1 P ) 2 / 3 Q ( porous grain ) Q ( compact grain of same mass and shape ) .

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