Abstract

The parallel iteration procedure for computing scattering by a multilayer sphere is described. The procedure uses a successive doubling strategy applied to four sets of multiple-scattering amplitudes, which is reminiscent of the fast Fourier transform (FFT) algorithm. The procedure is then used to calculate scattering of a plane wave by a modified Luneburg lens. The evolution of the transmission rainbow for the Luneburg lens parameter f>1 into an orbiting ray for f=1 and into a series of morphology-dependent resonances for f<1 is studied, and various features of the scattered intensity as a function of scattering angle are commented on. It is found that some resonances are formed without the presence of an exterior centrifugal barrier to confine them.

© 2008 Optical Society of America

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  3. O. B. Toon and T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657-3660 (1981).
  4. R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960-1967 (1985).
  5. D. W. Mackowski, R. A. Altenkirch, and M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551-1559 (1990).
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    [CrossRef]
  7. L. Kai and P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501-511 (1994).
  8. B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286-3296 (1996).
  9. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Grehan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36, 5188-5198 (1997).
  10. J. A. Lock, “Debye series analysis of scattering of a plane wave by a spherical Bragg grating,” Appl. Opt. 44, 5594-5603 (2005).
    [CrossRef]
  11. R. C. Gonzalez, Digital Image Processing, 3rd ed. (Pearson/Prentice Hall, 2008), pp. 299-302.
  12. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 124-125.
  13. M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 365, Eq. (9.3.1); p. 437, Eq. (10.1.1).
  14. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505-1509 (1980).
  15. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 127, 128, and 478.
  16. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82-124 (1969).
    [CrossRef]
  17. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 176-178.
  18. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 209-210.
  19. P. Chylek, “Partial-wave resonances and the ripple structure in the Mie normalized extinction cross section,” J. Opt. Soc. Am. 66, 285-287 (1976).
  20. P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229-2233 (1978).
    [CrossRef]
  21. C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585-1592 (1992).
  22. P. Chylek, J. D. Pendleton, and R. G. Pinnick, “Internal and near-surface scattered field of a spherical particle at resonant conditions,” Appl. Opt. 24, 3940-3942 (1985).
  23. M. Schneider and E. D. Hirleman, “Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry,” Appl. Opt. 33, 2379-2388 (1994).
  24. J. P. A. J. van Beeck and M. L. Reithmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259-2266 (1996).
  25. P. Massoli, “Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets,” Appl. Opt. 37, 3227-3235 (1998).
  26. D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702-1705 (1991).
  27. K. M. Lee, P. T. Leung, and K. M. Pang, “Iterative perturbation scheme for morphology-dependent resonances in dielectric spheres,” J. Opt. Soc. Am. A 15, 1383-1393 (1998).
  28. P. L. Marston and E. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
    [CrossRef]
  29. W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
    [CrossRef]
  30. H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
    [CrossRef]
  31. J. A. Adam and P. Laven, “Rainbows from inhomogeneous transparent spheres: a ray-theoretic approach,” Appl. Opt. 46, 922-929 (2007).
    [CrossRef]
  32. C. L. Brockman and N. G. Alexopoulos, “Geometrical optics of inhomogeneous particles; glory ray and the rainbow revisited,” Appl. Opt. 16, 166-174 (1977).
  33. A. Y. Perelman, “Scattering by particles with radially variable refractive indices,” Appl. Opt. 35, 5452-5460 (1996).

2008 (2)

2007 (1)

2005 (1)

1998 (2)

1997 (1)

1996 (3)

1994 (2)

1992 (1)

1991 (2)

D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702-1705 (1991).

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for a multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

1990 (2)

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef]

D. W. Mackowski, R. A. Altenkirch, and M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551-1559 (1990).

1989 (1)

W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
[CrossRef]

1985 (2)

1984 (1)

P. L. Marston and E. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
[CrossRef]

1981 (1)

1980 (1)

1978 (1)

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229-2233 (1978).
[CrossRef]

1977 (1)

1976 (1)

1969 (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82-124 (1969).
[CrossRef]

Ackerman, T. P.

Adam, J. A.

Alexopoulos, N. G.

Altenkirch, R. A.

Arnott, W. P.

W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
[CrossRef]

Barber, P. W.

D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702-1705 (1991).

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef]

Bhandari, R.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 127, 128, and 478.

Brockman, C. L.

Chowdhury, D. Q.

Chylek, P.

Gonzalez, R. C.

R. C. Gonzalez, Digital Image Processing, 3rd ed. (Pearson/Prentice Hall, 2008), pp. 299-302.

Gouesbet, G.

Grehan, G.

Guo, L. X.

Hill, S. C.

D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702-1705 (1991).

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef]

Hirleman, E. D.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 127, 128, and 478.

Johnson, B. R.

Kai, L.

Kiehl, J. T.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229-2233 (1978).
[CrossRef]

Ko, M. K. W.

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229-2233 (1978).
[CrossRef]

Lai, H. M.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef]

Lam, C. C.

Laven, P.

Lee, K. M.

Leung, P. T.

Lock, J. A.

Mackowski, D. W.

Marston, P. L.

W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
[CrossRef]

P. L. Marston and E. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
[CrossRef]

Massoli, P.

Menguc, M. P.

Nussenzveig, H. M.

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82-124 (1969).
[CrossRef]

Pang, K. M.

Pendleton, J. D.

Perelman, A. Y.

Pinnick, R. G.

Reithmuller, M. L.

Ren, K. F.

Schneider, M.

Toon, O. B.

Trinh, E.

P. L. Marston and E. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
[CrossRef]

van Beeck, J. P. A. J.

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 176-178.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 209-210.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 124-125.

Wang, Y. P.

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for a multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

Wiscombe, W. J.

Wu, Z. S.

Young, K.

C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585-1592 (1992).

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef]

Appl. Opt. (15)

C. L. Brockman and N. G. Alexopoulos, “Geometrical optics of inhomogeneous particles; glory ray and the rainbow revisited,” Appl. Opt. 16, 166-174 (1977).

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505-1509 (1980).

O. B. Toon and T. P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. 20, 3657-3660 (1981).

R. Bhandari, “Scattering coefficients for a multilayered sphere: analytic expressions and algorithms,” Appl. Opt. 24, 1960-1967 (1985).

D. W. Mackowski, R. A. Altenkirch, and M. P. Menguc, “Internal absorption cross sections in a stratified sphere,” Appl. Opt. 29, 1551-1559 (1990).

L. Kai and P. Massoli, “Scattering of electromagnetic-plane waves by radially inhomogeneous spheres: a finely stratified sphere model,” Appl. Opt. 33, 501-511 (1994).

M. Schneider and E. D. Hirleman, “Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry,” Appl. Opt. 33, 2379-2388 (1994).

P. Massoli, “Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets,” Appl. Opt. 37, 3227-3235 (1998).

B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286-3296 (1996).

A. Y. Perelman, “Scattering by particles with radially variable refractive indices,” Appl. Opt. 35, 5452-5460 (1996).

J. P. A. J. van Beeck and M. L. Reithmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259-2266 (1996).

Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Grehan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36, 5188-5198 (1997).

P. Chylek, J. D. Pendleton, and R. G. Pinnick, “Internal and near-surface scattered field of a spherical particle at resonant conditions,” Appl. Opt. 24, 3940-3942 (1985).

J. A. Lock, “Debye series analysis of scattering of a plane wave by a spherical Bragg grating,” Appl. Opt. 44, 5594-5603 (2005).
[CrossRef]

J. A. Adam and P. Laven, “Rainbows from inhomogeneous transparent spheres: a ray-theoretic approach,” Appl. Opt. 46, 922-929 (2007).
[CrossRef]

J. Acoust. Soc. Am. (1)

W. P. Arnott and P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427-1440 (1989).
[CrossRef]

J. Math. Phys. (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82-124 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

Nature (1)

P. L. Marston and E. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529-531 (1984).
[CrossRef]

Phys. Rev. A (2)

P. Chylek, J. T. Kiehl, and M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229-2233 (1978).
[CrossRef]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef]

Radio Sci. (1)

Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for a multilayered sphere: recursive algorithms,” Radio Sci. 26, 1393-1401 (1991).
[CrossRef]

Other (6)

R. C. Gonzalez, Digital Image Processing, 3rd ed. (Pearson/Prentice Hall, 2008), pp. 299-302.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 124-125.

M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 365, Eq. (9.3.1); p. 437, Eq. (10.1.1).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983), pp. 127, 128, and 478.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 176-178.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981), pp. 209-210.

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