Abstract

We examine scattering of a family of initially parallel diagonally incident rays by a dielectric circular cylinder and show that the interior and exterior caustics that occur are qualitatively identical to those produced at normal incidence. We find, however, that (1) varying the plane-wave tilt angle has the same effect on the caustics as varying the refractive index of the cylinder at normal incidence and (2) high-order interior caustics are visible because of larger internal-reflection Fresnel coefficients at diagonal incidence than at normal incidence. We also observe noncaustic ray trajectories produced by the sharp peaking of internal-reflection Fresnel coefficients at large ray impact parameters, as well as another class of internal caustics produced by scattering from inhomogeneities in our glass cylinder.

© 1997 Optical Society of America

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References

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  1. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), p. 222, Fig. 6.13.
  2. D. S. Benincasa, P. W. Barber, J.-Z. Zhang, W.-F. Hsieh, R. K. Chang, “Spatial distribution of the internal and near-field intensities of large cylindrical and spherical scatterers,” Appl. Opt. 26, 1348–1356 (1987).
    [CrossRef] [PubMed]
  3. K. Sassen, “Angular scattering and rainbow formation in pendant drops,” J. Opt. Soc. Am. 69, 1083–1089 (1979).
    [CrossRef]
  4. C. W. Chan, W. K. Lee, “Measurement of a liquid refractive index by using high-order rainbows,” J. Opt. Soc. Am. B 13, 532–535 (1996).
    [CrossRef]
  5. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  6. J. D. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(1), 138–144 (1977).
    [CrossRef]
  7. J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
    [CrossRef]
  8. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
    [CrossRef]
  9. J. A. Lock, “Theory of the observations made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
    [CrossRef] [PubMed]
  10. H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
    [CrossRef] [PubMed]
  11. C. F. Bohren, A. B. Fraser, “Newton’s zero-order rainbow: unobservable or nonexistent,” Am. J. Phys. 59, 325–326 (1991).
    [CrossRef]
  12. A. E. Shapiro, “Comment on Newton’s zero-order rainbow: unobservable or nonexistent,” Am. J. Phys. 60, 749–750 (1992).
    [CrossRef]
  13. J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
    [CrossRef]
  14. V. Srivastava, M. A. Jarzembski, “Laser-induced stimulated Raman scattering in the forward direction of a droplet: comparison of Mie theory with geometrical optics,” Opt. Lett. 16, 126–128 (1991).
    [CrossRef] [PubMed]
  15. J. A. Lock, E. A. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. A 8, 1541–1552 (1991).
    [CrossRef]
  16. H. M. Lai, P. T. Leung, K. L. Poon, K. Young, “Characterization of the internal energy density in Mie scattering,” J. Opt. Soc. Am. A 8, 1553–1558 (1991).
    [CrossRef]
  17. D. Q. Chowdhury, P. W. Barber, S. C. Hill, “Energy-density distribution inside large nonabsorbing spheres using Mie theory and geometrical optics,” Appl. Opt. 31, 3518–3523 (1992).
    [CrossRef] [PubMed]
  18. M. A. Jarzembski, V. Srivastava, “Electromagnetic field enhancement in small liquid droplets using geometrical optics,” Appl. Opt. 28, 4962–4965 (1989).
    [CrossRef] [PubMed]
  19. J.-G. Xie, T. E. Ruekgauer, J. Gu, R. L. Armstrong, R. G. Pinnick, “Observations of Descartes ring stimulated Raman scattering in micrometer-sized water droplets,” Opt. Lett. 16, 1310–1312 (1991).
    [CrossRef] [PubMed]
  20. P. Chylek, M. A. Jarzembski, N. Y. Chou, R. G. Pinnick, “Effect of size and material of liquid spherical particles on laser-induced breakdown,” Appl. Phys. Lett. 49, 1475–1477 (1986).
    [CrossRef]
  21. R. G. Pinnick, P. Chylek, M. Jarzembski, E. Creegan, V. Srivastava, G. Fernandez, J. D. Pendleton, A. Biswas, “Aerosol-induced laser breakdown thresholds: wavelength dependence,” Appl. Opt. 27, 987–996 (1988).
    [CrossRef] [PubMed]
  22. J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
    [CrossRef]
  23. J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London, Ser. A 438, 397–417 (1992).
    [CrossRef]
  24. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature (London) 312, 529–531 (1984).
    [CrossRef]
  25. P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
    [CrossRef] [PubMed]
  26. H. J. Simpson, P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468–3473, 3547 (1991).
    [CrossRef] [PubMed]
  27. G. Kaduchak, P. L. Marston, H. J. Simpson, “E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt. 33, 4691–4696 (1994).
    [CrossRef] [PubMed]
  28. G. Kaduchak, P. L. Marston, “Hyperbolic umbilic and E6 diffraction catastrophes associated with the secondary rainbow of oblate water drops: observations with laser illumination,” Appl. Opt. 33, 4697–4701 (1994).
    [CrossRef] [PubMed]
  29. P. L. Marston, G. Kaduchak, “Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander’s dark band,” Appl. Opt. 33, 4702–4713 (1994).
    [CrossRef] [PubMed]
  30. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. 35, 515–531 (1996).
    [CrossRef] [PubMed]
  31. J. F. Owen, R. K. Chang, P. W. Barber, “Internal electric field distributions of a dielectric cylinder at resonance wavelengths,” Opt. Lett. 6, 540–542 (1981).
    [CrossRef] [PubMed]
  32. A. Steinhardt, L. Fukshansky, “Geometrical optics approach to the intensity distribution infinite cylindrical media,” Appl. Opt. 26, 3778–3789 (1987).
    [CrossRef] [PubMed]
  33. J. A. Lock, C. L. Adler, “Debye-series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1316–1328 (1997).
    [CrossRef]
  34. J. B. Keller, H. B. Keller, “Determination of reflected and transmitted fields by geometrical optics,” J. Opt. Soc. Am. 40, 48–52 (1950).
    [CrossRef]
  35. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 264, Fig. 6.3.
  36. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 200, Fig. 8.4.
  37. Y. Takano, M. Tanaka, “Phase matrix and cross sections for single scattering by circular cylinders: a comparison of ray optics and wave theory,” Appl. Opt. 19, 2781–2793 (1980).
    [CrossRef] [PubMed]
  38. The glass cylinder was provided, polished, and frosted by Ferguson’s Cut Glass Originals, 4292 Pearl Road, Cleveland, Ohio 44109.
  39. G. P. Können, “Appearance of supernumeraries of the secondary rainbow in rain showers,” J. Opt. Soc. Am. A 4, 810–816 (1987).
    [CrossRef]

1997 (1)

1996 (2)

1994 (4)

1993 (1)

1992 (3)

D. Q. Chowdhury, P. W. Barber, S. C. Hill, “Energy-density distribution inside large nonabsorbing spheres using Mie theory and geometrical optics,” Appl. Opt. 31, 3518–3523 (1992).
[CrossRef] [PubMed]

J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London, Ser. A 438, 397–417 (1992).
[CrossRef]

A. E. Shapiro, “Comment on Newton’s zero-order rainbow: unobservable or nonexistent,” Am. J. Phys. 60, 749–750 (1992).
[CrossRef]

1991 (7)

1989 (1)

1988 (1)

1987 (4)

1986 (1)

P. Chylek, M. A. Jarzembski, N. Y. Chou, R. G. Pinnick, “Effect of size and material of liquid spherical particles on laser-induced breakdown,” Appl. Phys. Lett. 49, 1475–1477 (1986).
[CrossRef]

1985 (1)

1984 (2)

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

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

1981 (1)

1980 (2)

1979 (1)

1977 (1)

J. D. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(1), 138–144 (1977).
[CrossRef]

1976 (1)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

1950 (1)

Adler, C. L.

Armstrong, R. L.

Barber, P. W.

Benincasa, D. S.

Biswas, A.

Bohren, C. F.

C. F. Bohren, A. B. Fraser, “Newton’s zero-order rainbow: unobservable or nonexistent,” Am. J. Phys. 59, 325–326 (1991).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 200, Fig. 8.4.

Chan, C. W.

Chang, R. K.

Chou, N. Y.

P. Chylek, M. A. Jarzembski, N. Y. Chou, R. G. Pinnick, “Effect of size and material of liquid spherical particles on laser-induced breakdown,” Appl. Phys. Lett. 49, 1475–1477 (1986).
[CrossRef]

Chowdhury, D. Q.

Chylek, P.

R. G. Pinnick, P. Chylek, M. Jarzembski, E. Creegan, V. Srivastava, G. Fernandez, J. D. Pendleton, A. Biswas, “Aerosol-induced laser breakdown thresholds: wavelength dependence,” Appl. Opt. 27, 987–996 (1988).
[CrossRef] [PubMed]

P. Chylek, M. A. Jarzembski, N. Y. Chou, R. G. Pinnick, “Effect of size and material of liquid spherical particles on laser-induced breakdown,” Appl. Phys. Lett. 49, 1475–1477 (1986).
[CrossRef]

Creegan, E.

Fernandez, G.

Fraser, A. B.

C. F. Bohren, A. B. Fraser, “Newton’s zero-order rainbow: unobservable or nonexistent,” Am. J. Phys. 59, 325–326 (1991).
[CrossRef]

Fukshansky, L.

Gu, J.

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), p. 222, Fig. 6.13.

Hill, S. C.

Hovenac, E. A.

Hsieh, W.-F.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 200, Fig. 8.4.

Jarzembski, M.

Jarzembski, M. A.

Kaduchak, G.

Keller, H. B.

Keller, J. B.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 264, Fig. 6.3.

Können, G. P.

Lai, H. M.

Lee, W. K.

Leung, P. T.

Lock, J. A.

Marston, P. L.

McCollum, T. A.

J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

Nye, J. F.

J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London, Ser. A 438, 397–417 (1992).
[CrossRef]

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

Owen, J. F.

Pendleton, J. D.

Pinnick, R. G.

Poon, K. L.

Ruekgauer, T. E.

Sassen, K.

Shapiro, A. E.

A. E. Shapiro, “Comment on Newton’s zero-order rainbow: unobservable or nonexistent,” Am. J. Phys. 60, 749–750 (1992).
[CrossRef]

Simpson, H. J.

G. Kaduchak, P. L. Marston, H. J. Simpson, “E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt. 33, 4691–4696 (1994).
[CrossRef] [PubMed]

H. J. Simpson, P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468–3473, 3547 (1991).
[CrossRef] [PubMed]

Srivastava, V.

Steinhardt, A.

Takano, Y.

Tanaka, M.

Trinh, E. H.

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

van de Hulst, H. C.

Walker, J. D.

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
[CrossRef]

J. D. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(1), 138–144 (1977).
[CrossRef]

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Wang, R. T.

Xie, J.-G.

Young, K.

Zhang, J.-Z.

Am. J. Phys. (4)

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

C. F. Bohren, A. B. Fraser, “Newton’s zero-order rainbow: unobservable or nonexistent,” Am. J. Phys. 59, 325–326 (1991).
[CrossRef]

A. E. Shapiro, “Comment on Newton’s zero-order rainbow: unobservable or nonexistent,” Am. J. Phys. 60, 749–750 (1992).
[CrossRef]

J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
[CrossRef]

Appl. Opt. (13)

D. Q. Chowdhury, P. W. Barber, S. C. Hill, “Energy-density distribution inside large nonabsorbing spheres using Mie theory and geometrical optics,” Appl. Opt. 31, 3518–3523 (1992).
[CrossRef] [PubMed]

M. A. Jarzembski, V. Srivastava, “Electromagnetic field enhancement in small liquid droplets using geometrical optics,” Appl. Opt. 28, 4962–4965 (1989).
[CrossRef] [PubMed]

D. S. Benincasa, P. W. Barber, J.-Z. Zhang, W.-F. Hsieh, R. K. Chang, “Spatial distribution of the internal and near-field intensities of large cylindrical and spherical scatterers,” Appl. Opt. 26, 1348–1356 (1987).
[CrossRef] [PubMed]

J. A. Lock, “Theory of the observations made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
[CrossRef] [PubMed]

H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
[CrossRef] [PubMed]

R. G. Pinnick, P. Chylek, M. Jarzembski, E. Creegan, V. Srivastava, G. Fernandez, J. D. Pendleton, A. Biswas, “Aerosol-induced laser breakdown thresholds: wavelength dependence,” Appl. Opt. 27, 987–996 (1988).
[CrossRef] [PubMed]

H. J. Simpson, P. L. Marston, “Scattering of white light from levitated oblate water drops near rainbows and other diffraction catastrophes,” Appl. Opt. 30, 3468–3473, 3547 (1991).
[CrossRef] [PubMed]

G. Kaduchak, P. L. Marston, H. J. Simpson, “E6 diffraction catastrophe of the primary rainbow of oblate water drops: observations with white-light and laser illumination,” Appl. Opt. 33, 4691–4696 (1994).
[CrossRef] [PubMed]

G. Kaduchak, P. L. Marston, “Hyperbolic umbilic and E6 diffraction catastrophes associated with the secondary rainbow of oblate water drops: observations with laser illumination,” Appl. Opt. 33, 4697–4701 (1994).
[CrossRef] [PubMed]

P. L. Marston, G. Kaduchak, “Generalized rainbows and unfolded glories of oblate drops: organization for multiple internal reflections and extension of cusps into Alexander’s dark band,” Appl. Opt. 33, 4702–4713 (1994).
[CrossRef] [PubMed]

J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. II. Transmission and cross-polarization effects,” Appl. Opt. 35, 515–531 (1996).
[CrossRef] [PubMed]

A. Steinhardt, L. Fukshansky, “Geometrical optics approach to the intensity distribution infinite cylindrical media,” Appl. Opt. 26, 3778–3789 (1987).
[CrossRef] [PubMed]

Y. Takano, M. Tanaka, “Phase matrix and cross sections for single scattering by circular cylinders: a comparison of ray optics and wave theory,” Appl. Opt. 19, 2781–2793 (1980).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

P. Chylek, M. A. Jarzembski, N. Y. Chou, R. G. Pinnick, “Effect of size and material of liquid spherical particles on laser-induced breakdown,” Appl. Phys. Lett. 49, 1475–1477 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

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

Nature (London) (2)

J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature (London) 312, 531–532 (1984).
[CrossRef]

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

Opt. Lett. (4)

Proc. R. Soc. London, Ser. A (1)

J. F. Nye, “Rainbows from ellipsoidal water drops,” Proc. R. Soc. London, Ser. A 438, 397–417 (1992).
[CrossRef]

Sci. Am. (2)

J. D. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(1), 138–144 (1977).
[CrossRef]

J. D. Walker, “Mysteries of rainbows, notably their rare supernumerary arcs,” Sci. Am. 242(6), 174–184 (1980).
[CrossRef]

Other (4)

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), p. 222, Fig. 6.13.

The glass cylinder was provided, polished, and frosted by Ferguson’s Cut Glass Originals, 4292 Pearl Road, Cleveland, Ohio 44109.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), p. 264, Fig. 6.3.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 200, Fig. 8.4.

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Figures (10)

Fig. 1
Fig. 1

A plane wave with propagation direction k^i0 in the xz plane and making an angle ξ with the x axis is incident on a dielectric circular cylinder of radius a.

Fig. 2
Fig. 2

Trajectory of the incident ray k^i0 and the transmitted ray k^t0 as seen from (a) the top of the cylinder and (b) the side of the cylinder. k^i0xy and k^t0xy are the normalized projections of k^i0 and k^t0 in the xy plane.

Fig. 3
Fig. 3

p=1 focusing caustic as seen from the top of the cylinder for (a) n=1.807, (b) n=2.0, and (c) n=2.188. These refractive indices were chosen so as to correspond to the experimental observations of Figs. 7(a), (b), and (c), respectively, be low. For n<2.0 the cusp point focal line occurs outside the cylinder, and for n>2.0 it occurs inside the cylinder.

Fig. 4
Fig. 4

Interior cusp caustics for n=2.0 for (a) p=1, (b) p=2, (c) p=3, (d) p=4, and (e) p=5.

Fig. 5
Fig. 5

(a) Photograph of the rod's upper frosted end at ξe =45° at which the p=1 cusp point focal line touches the rod surface. (b) Composite of the p=1 and 3 caustics and the 2 p¯5 noncaustic ray trajectories over a 4° interval centered on the highest-intensity member of the interior p-ray family.

Fig. 6
Fig. 6

(a) Photograph of the rod's upper frosted end at ξe =45° and with two black cards blocking all but a relatively narrow band near the top of the rod. (b) Composite of the 1p 3 caustics.

Fig. 7
Fig. 7

Unfolding of the p=1 cusp caustic for (a) ξ=38°, (b) ξe=45°, and (c) ξ=50°. A composite of the p=1 and 2 caustics for n=2.0 and the interior line source 1+2 caustic is illustrated in (d) and is to be compared with the photograph in (b).    

Fig. 8
Fig. 8

Interior line source caustics 1+p for (a) p=2, (b) p=3, and (c) p=4.

Fig. 9
Fig. 9

(a) Photograph of the rod's upper frosted end at ξe =45° and with a black card blocking only the very top of the rod. (b) Composite of the p=1 caustic, and 3̅ noncaustic ray trajectory, and the 1+p interior line source caustics for 2 p4.

Fig. 10
Fig. 10

(a) Photograph of the rod's upper frosted end at ξe =45° and with a black card blocking a substantial fraction of the upper portion of the rod. (b) Composite of the 1+p interior line source caustics for 2p5.

Tables (1)

Tables Icon

Table 1 Intensity of the Paraxial Rays [I(ϕi0=0)] Contributing to the Cusp Points, Angle of Incidence of the Rays with Maximum Intensity (ϕi0max), and Maximum Ray Intensity (Imax) for Interior Ray Families 1p6 at Normal Incidence and with Tilt Angle ξe Corresponding to n=2.0 for n=1.484 and an Unpolarized Incident Beam

Equations (41)

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

k^i0=(cos ξ)u^x-(sin ξ)u^z,
n^0=-(cos ϕi0)u^x+(sin ϕi0)u^y.
cos θi0=-k^i0n^0=cos ξ cos ϕi0,
n sin θt0=sin θi0.
k^t0=1n k^i0+1n cos θi0-cos θt0n^0=1n cos ξ-cos ϕi01n cos θi0-cos θt0u^x+(sin ϕi0)1n cos θi0-cos θt0u^y-1n (sin ξ)u^z.
cos ϕt0=-k^t0xyn^0,
n sin γ=sin ξ,
k^t0=[cos γ cos(ϕi0-ϕt0)]u^x-[cos γ sin(ϕi0-ϕt0)]u^y-(sin γ)u^z.
n sin ϕt0=sin ϕi0,
n=n cos γ/cos ξ.
Θ=(p-1)π+2ϕi0-2pϕt0.
Ei0=E0u^y=E0u^i0,
Bi0=E0c [(sin ξ)u^x+(cos ξ)u^z],
Ei0μ=E0μ[(sin ξ)u^x+(cos ξ)u^y]=E0μu^i0μ,
Bi0μ=-E0μc u^y,
TE^i0=k^i0×n^0/sin θi0,TM^i0=TE^i0×k^i0,
TM^i0=(cos χ)u^i0-(sin χ)u^i0μ,
TE^i0=(sin χ)u^i0+(cos χ)u^i0μ,
TM^i0TE^i0=R(-χ)u^i0u^i0μ,
cos χ=sin ϕi0/sin θi0,
sin χ=sin ξ cos ϕi0/sin θi0
R(ψ)=cos ψ-sin ψsin ψcos ψ.
Et0TMEt0TE=t0TM00t0TE Ei0TMEi0TE
Er0TMEr0TE=r0TM00r0TE Ei0TMEi0TE
TE^t0=k^t0×n^0/sin θt0,TM^t0=TE^t0×k^t0.
k^i1=k^t0,
TE^i1=k^i1×n^1/sin θi1,TM^i1=TE^i1×k^i1,
TM^i1TE^i1=R(η)TM^t0TE^t0,
cos η=(sin2 ϕt0-sin2 γ cos2 ϕt0)/sin2 θt0,
sin η=sin γ sin(2ϕt0)/sin2 θt0.
ErpErpμ=R(-χ)r0TM00r0TER(-χ)E0E0μforp=0,
EtpEtpμ=R(-χ)tpTM00tpTE j=1p-1 R(η)rjTM00rjTER(η)×t0TM00t0TER(-χ)E0E0μforp1,
Einterior,pEinterior,pμ=R(σ)j=1p-1rjTM00rjTER(η)×t0TM00t0TER(-χ)E0E0μ,
cos σ=sin ϕt0/sin θt0,
sin σ=sin γ cos ϕt0/sin θt0,
x=(-1)p2p-1-n a,y=0
x=na2(n-1),y=0
cos ϕi0=n2-1p2-11/2.
sin ξe=4-n231/2.
n=1.484±0.002.
cos ξe=ba 1-a22b21-a24b2-1/2(n2-1)1/2.

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