Abstract

We introduce the concept of the equiphase sphere for light scattering by nonspherical dielectric particles. This concept facilitates the derivation of a simple analytical expression for the total scattering cross section of such particles. We tested this concept for spheroidal particles and obtained a bound on the minor-to-major axis ratio for the valid application of this technique. We show that this technique yields results that agree well with the rigorous numerical solution of Maxwell’s equations obtained with the finite-difference time-domain method. The new technique has the potential to be extended to the study of light scattering by arbitrarily shaped convex dielectric particles.

© 2004 Optical Society of America

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  1. M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, San Diego, Calif., 2000).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  3. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  4. Z. Chen, A. Taflove, V. Backman, “Equivalent volume-averaged light scattering behavior of randomly inhomogeneous dielectric spheres in the resonant range,” Opt. Lett. 28, 765–767 (2003).
    [CrossRef] [PubMed]
  5. S. Asano, G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [CrossRef] [PubMed]
  6. N. V. Voshchinnikov, V. G. Farafonov, “Optical properties of spheroidal particles,” Astrophys. Space Sci. 204, 19–86 (1993).
    [CrossRef]
  7. N. V. Voshchinnikov, V. G. Farafonov, “Light scattering by an elongated particle: spheroid versus infinite cylinder,” Meas. Sci. Technol. 13, 249–255 (2002).
    [CrossRef]
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  9. J. D. Klett, R. A. Sutherland, “Approximate methods for modeling the scattering properties of non-spherical particles: evaluation of the Wentzel–Kramers–Brillouin method,” Appl. Opt. 31, 373–386 (1992).
    [CrossRef] [PubMed]
  10. S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
    [CrossRef]
  11. A. A. Kokhanovsy, A. Macke, “Integral light scattering and absorption characteristics of large nonspherical particles,” Appl. Opt. 36, 8785–8790 (1997).
    [CrossRef]
  12. H. Nussenzveig, W. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
    [CrossRef]
  13. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 176.
  14. A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, Boston, Mass., 2000).
  15. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  16. V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
    [CrossRef] [PubMed]

2003 (1)

2002 (1)

N. V. Voshchinnikov, V. G. Farafonov, “Light scattering by an elongated particle: spheroid versus infinite cylinder,” Meas. Sci. Technol. 13, 249–255 (2002).
[CrossRef]

2000 (1)

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

1997 (1)

1994 (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

1993 (1)

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

1992 (1)

1987 (1)

S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
[CrossRef]

1980 (1)

H. Nussenzveig, W. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

1975 (1)

Ackerman, S. A.

S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
[CrossRef]

Arendt, J. T.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Asano, S.

Backman, V.

Z. Chen, A. Taflove, V. Backman, “Equivalent volume-averaged light scattering behavior of randomly inhomogeneous dielectric spheres in the resonant range,” Opt. Lett. 28, 765–767 (2003).
[CrossRef] [PubMed]

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Badizadegan, K.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Chen, Z.

Crawford, J. M.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Dasari, R. R.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Farafonov, V. G.

N. V. Voshchinnikov, V. G. Farafonov, “Light scattering by an elongated particle: spheroid versus infinite cylinder,” Meas. Sci. Technol. 13, 249–255 (2002).
[CrossRef]

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

Feld, M. S.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Fitzmaurice, M.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Gurjar, R.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Hagness, S.

A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, Boston, Mass., 2000).

Hovenier, J. W.

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, San Diego, Calif., 2000).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Itzkan, I.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Kabani, S.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Klett, J. D.

Kline, E.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Kokhanovsy, A. A.

Levin, H. S.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Macke, A.

McGillican, T.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Mishchenko, M. I.

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, San Diego, Calif., 2000).

Muller, M. G.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Nussenzveig, H.

H. Nussenzveig, W. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Perelman, L. T.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Seiler, M.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Stephens, G. L.

S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
[CrossRef]

Sutherland, R. A.

Taflove, A.

Travis, L. D.

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, San Diego, Calif., 2000).

Valdez, T.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Van Dam, J.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 176.

Voshchinnikov, N. V.

N. V. Voshchinnikov, V. G. Farafonov, “Light scattering by an elongated particle: spheroid versus infinite cylinder,” Meas. Sci. Technol. 13, 249–255 (2002).
[CrossRef]

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

Wallace, M.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Wiscombe, W.

H. Nussenzveig, W. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Yamamoto, G.

Zhang, Q.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Zonios, G.

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Appl. Opt. (3)

Astrophys. Space Sci. (1)

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

J. Atmos. Sci. (1)

S. A. Ackerman, G. L. Stephens, “The absorption of solar radiation by cloud droplets: an application of anomalous diffraction theory,” J. Atmos. Sci. 44, 1574–1588 (1987).
[CrossRef]

J. Comput. Phys. (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Meas. Sci. Technol. (1)

N. V. Voshchinnikov, V. G. Farafonov, “Light scattering by an elongated particle: spheroid versus infinite cylinder,” Meas. Sci. Technol. 13, 249–255 (2002).
[CrossRef]

Nature (1)

V. Backman, L. T. Perelman, J. T. Arendt, R. Gurjar, M. G. Muller, Q. Zhang, G. Zonios, E. Kline, T. McGillican, T. Valdez, J. Van Dam, M. Wallace, K. Badizadegan, J. M. Crawford, M. Fitzmaurice, S. Kabani, H. S. Levin, M. Seiler, R. R. Dasari, I. Itzkan, M. S. Feld, “Detection of preinvasive cancer cells in situ,” Nature 406, 35–36 (2000).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

H. Nussenzveig, W. Wiscombe, “Efficiency factors in Mie scattering,” Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Other (6)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), p. 176.

A. Taflove, S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, Boston, Mass., 2000).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

M. I. Mishchenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, San Diego, Calif., 2000).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

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Figures (7)

Fig. 1
Fig. 1

Schematic of different combinations of incidence and polarization state of light. (a) TM mode, (b) TE mode, (c) TEM mode.

Fig. 2
Fig. 2

Condition parameter β versus b/a ratios of spheroids for λ=500 nm. (a) TE and TM modes, (b) TEM modes, (c) expanded view of Fig. 2(b).

Fig. 3
Fig. 3

(a) Total scattering cross section of (a=5 µm, b=4.5 µm) spheroid versus wavelength. The FDTD results for the spheroid are represented by solid dots (TE mode) and open circles (TM mode) and the Mie results for the corresponding equiphase sphere by a solid curve. (b) Corresponding results for the TEM mode.

Fig. 4
Fig. 4

(a) Total scattering cross section of (a=5 µm, b=4.5 µm) spheroid versus wavelength for TE and TM modes. The FDTD results are represented by solid dots (TE mode) and open circles (TM mode) and the results of the approximate analysis by a solid curve. (b) Corresponding results for (a=5 µm, b=2.5 µm) spheroid.

Fig. 5
Fig. 5

(a) Total scattering cross section of (a=5 µm, b=1 µm) spheroid versus wavelength for TE and TM modes. The FDTD results are represented by solid dots (TE mode) and open circles (TM mode) and the results of the approximate analysis by a solid curve. (b) Corresponding results for (a=5 µm, b=0.5 µm) spheroid.

Fig. 6
Fig. 6

Total scattering cross section of spheroids for the TM mode versus wavelength for b/a=0.2, 0.7, and 0.9, corresponding to the three b/a ratios designated by circles in Fig. 2(a). (a) b/a=0.9, (b) b/a=0.7, (c) b/a=0.2.

Fig. 7
Fig. 7

Total scattering cross section of spheroids for the TEM mode versus wavelength for b/a=0.5, 0.7, and 0.9, corresponding to the three b/a ratios designated by circles in Figs. 2(b) and 2(c). (a) b/a=0.9, (b) b/a=0.7, (c) b/a=0.5.

Equations (16)

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

σs=σs(v)+σs(s),
σs(s)2Sm(kd/2)-2/3[1+χ(kd, n)],
σs(s)2Sm[k(abc)1/3/2]-2/3,
σs(v)=2 ReS{1-exp[iξ(r)]}d2r,
ξ(r)=L(r)k[n(r)-1]dl,
Lb{1-2[|r|/R(r)]2}1/2 cos[γ(r)-γ(r)+ϕ(r)],
ξ(r)kb(n-1){1-1/n2+[|r|/R(r)]2/n2}1/2[1+δL(r)],
δL=[L(r)-bη(r)]/[bη(r)],
η(r)={1-1/n2+[|r|/R(r)]2/n2}1/2.
σs=σs(s)+2S[1-2n sin ρ/ρ+4n sin2(ρ/2)/ρ2],
σs=(πab/2)[k(ab2)1/3/2]-2/3+(πab/2)[1-2n sin ρ/ρ+4n sin2(ρ/2)/ρ2].
σs=(πab/2)[k(ab2)1/3/2]-2/3+(πb2/2)[1-2n sin ρa/ρa+4n sin2(ρa/2)/ρa2],
σs=σs(v)+σs(s)=(πab/2)[k(ab2)1/3/2]-2/3+2S[1-2n sin ρ/ρ+4n sin2(ρ/2)/ρ2].
σs=2S[1-2 sin ρ/ρ+4 sin2(ρ/2)/ρ2].
β4(n-1)bδL/λ(16/π2)(b/λ)×[(n-1)2/n](1-b/a)(1+a2/b2)-1<1.
β4(n-1)aδL/λ(16/π2)(a/λ)×[(n-1)2/n](1-a/b)(1+b2/a2)-1<1,

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