Abstract

We use the semiclassical limit of electromagnetic wave scattering theory to determine the properties of the exterior caustics of a diagonally incident plane wave scattered by an infinitely long homogeneous dielectric circular cylinder in both the near zone and the far zone. The transmission caustic has an exterior/interior cusp transition as the tilt angle of the incident beam is increased, and each of the rainbow caustics has a far-zone rainbow/exterior cusp transition and an exterior/interior cusp transition as the incident beam tilt angle is increased. We experimentally observe and analyze both transitions of the first-order rainbow. We also compare the predictions of the semiclassical approximation with those of ray theory and exact electromagnetic wave scattering theory.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980); in particular, Sect. 3, pp. 277–297
    [CrossRef]
  2. P. L. Marston, “Geometrical and catastrophe methods in scattering,” Phys. Acoust. 21, 1–234 (1992); in particular, Sect. 4.8, Fig. 76, p. 187.
  3. C. L. Adler, J. A. Lock, B. R. Stone, C. J. Garcia, “High-order interior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1305–1315 (1997).
    [CrossRef]
  4. 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]
  5. C. M. Mount, D. B. Thiessen, P. L. Marston, “Scattering observations for tilted transparent fibers: evolution of the Airy caustics with cylinder tilt and the caustic merging transition,” Appl. Opt. 37, 1534–1539 (1998).
    [CrossRef]
  6. P. L. Marston, “Descartes glare points in scattering by icicles: color photographs and a tilted dielectric cylinder model of caustic and glare point evolution,” Appl. Opt. 37, 1551–1556 (1998).
    [CrossRef]
  7. F. J. Blonigen, P. L. Marston, “Backscattering enhancements for tilted solid plastic cylinders in water due to the caustic merging transition: observation and theory,” J. Acoust. Soc. Am. 107, 689–698 (2000).
    [CrossRef] [PubMed]
  8. J. A. Lock, T. A. McCollum, “Further thoughts on Newton’s zero-order rainbow,” Am. J. Phys. 62, 1082–1089 (1994).
    [CrossRef]
  9. K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
    [CrossRef]
  10. M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972); in particular, Sects. 6.1 and 6.2, pp. 335–364.
    [CrossRef]
  11. N. Roth, K. Anders, A. Frohn, “Refractive-index measurements for the correction of particle sizing methods,” Appl. Opt. 30, 4960–4965 (1991).
    [CrossRef] [PubMed]
  12. J. P. A. J. van Beeck, M. L. Reithmuller, “Nonintrusive measurements of temperature and size of single falling raindrops,” Appl. Opt. 34, 1633–1639 (1995).
    [CrossRef] [PubMed]
  13. X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, G. Grehan, “Characterization of initial disturbances in a liquid jet by rainbow sizing,” Appl. Opt. 37, 8498–8503 (1998).
    [CrossRef]
  14. Ref. 1, p. 263.
  15. W. J. Humphreys, Physics of the Air (Dover, New York, 1964), pp. 476–492.
  16. Ref. 1, Appendix 2, pp. 339–342.
  17. J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A 14, 640–652 (1997).
    [CrossRef]
  18. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. (9.2.3), p. 447.
  19. P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908), reprinted and translated into English in P. L. Marston, ed., Geometrical Aspects of Scattering (SPIE Press, Bellingham, Wash., 1994), pp. 198–204.
  20. Ref. 18, Eq. (10.4.32), p. 364.
  21. T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).
  22. V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
    [CrossRef]
  23. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979); plates on pp. 1193, 1194.
    [CrossRef]
  24. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. (N.Y.) 10, 82–124 (1969).
    [CrossRef]
  25. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. (N.Y.) 10, 125–176 (1969).
    [CrossRef]
  26. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, New York, 1992), pp. 101–116.
  27. Ref. 18, Eqs. (9.3.15)–(9.3.22), pp. 366–367.
  28. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 208–209.
  29. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  30. R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
    [CrossRef] [PubMed]
  31. This result is implicit in 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]

2000 (1)

F. J. Blonigen, P. L. Marston, “Backscattering enhancements for tilted solid plastic cylinders in water due to the caustic merging transition: observation and theory,” J. Acoust. Soc. Am. 107, 689–698 (2000).
[CrossRef] [PubMed]

1998 (3)

1997 (3)

1995 (1)

1994 (2)

1992 (1)

P. L. Marston, “Geometrical and catastrophe methods in scattering,” Phys. Acoust. 21, 1–234 (1992); in particular, Sect. 4.8, Fig. 76, p. 187.

1991 (2)

1980 (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980); in particular, Sect. 3, pp. 277–297
[CrossRef]

1979 (1)

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979); plates on pp. 1193, 1194.
[CrossRef]

1976 (1)

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

1974 (1)

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

1972 (1)

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972); in particular, Sects. 6.1 and 6.2, pp. 335–364.
[CrossRef]

1969 (2)

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

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. (N.Y.) 10, 125–176 (1969).
[CrossRef]

1959 (1)

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
[CrossRef]

1946 (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

1908 (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908), reprinted and translated into English in P. L. Marston, ed., Geometrical Aspects of Scattering (SPIE Press, Bellingham, Wash., 1994), pp. 198–204.

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. (9.2.3), p. 447.

Adler, C. L.

Anders, K.

Berry, M. V.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980); in particular, Sect. 3, pp. 277–297
[CrossRef]

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972); in particular, Sects. 6.1 and 6.2, pp. 335–364.
[CrossRef]

Blonigen, F. J.

F. J. Blonigen, P. L. Marston, “Backscattering enhancements for tilted solid plastic cylinders in water due to the caustic merging transition: observation and theory,” J. Acoust. Soc. Am. 107, 689–698 (2000).
[CrossRef] [PubMed]

Corbin, F.

Debye, P.

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908), reprinted and translated into English in P. L. Marston, ed., Geometrical Aspects of Scattering (SPIE Press, Bellingham, Wash., 1994), pp. 198–204.

Ford, K. W.

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
[CrossRef]

Frohn, A.

Garcia, C. J.

Gouesbet, G.

Grehan, G.

Han, X.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964), pp. 476–492.

Kaduchak, G.

Khare, V.

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

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]

Mount, C. M.

Mount, K. E.

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972); in particular, Sects. 6.1 and 6.2, pp. 335–364.
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979); plates on pp. 1193, 1194.
[CrossRef]

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

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

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. (N.Y.) 10, 125–176 (1969).
[CrossRef]

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, New York, 1992), pp. 101–116.

Pearcey, T.

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Reithmuller, M. L.

Ren, K. F.

Roth, N.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. (9.2.3), p. 447.

Stone, B. R.

Thiessen, D. B.

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980); in particular, Sect. 3, pp. 277–297
[CrossRef]

van Beeck, J. P. A. J.

van de Hulst, H. C.

R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
[CrossRef] [PubMed]

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 208–209.

Walker, J. D.

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

Wang, R. T.

Wheeler, J. A.

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
[CrossRef]

Wu, Z.

Am. J. Phys. (2)

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

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

Ann. Phys. (N.Y.) (1)

K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
[CrossRef]

Appl. Opt. (7)

J. Acoust. Soc. Am. (1)

F. J. Blonigen, P. L. Marston, “Backscattering enhancements for tilted solid plastic cylinders in water due to the caustic merging transition: observation and theory,” J. Acoust. Soc. Am. 107, 689–698 (2000).
[CrossRef] [PubMed]

J. Math. Phys. (N.Y.) (2)

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

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. (N.Y.) 10, 125–176 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979); plates on pp. 1193, 1194.
[CrossRef]

J. Opt. Soc. Am. A (3)

Philos. Mag. (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Phys. Acoust. (1)

P. L. Marston, “Geometrical and catastrophe methods in scattering,” Phys. Acoust. 21, 1–234 (1992); in particular, Sect. 4.8, Fig. 76, p. 187.

Phys. Rev. Lett. (1)

V. Khare, H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[CrossRef]

Phys. Z. (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908), reprinted and translated into English in P. L. Marston, ed., Geometrical Aspects of Scattering (SPIE Press, Bellingham, Wash., 1994), pp. 198–204.

Prog. Opt. (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980); in particular, Sect. 3, pp. 277–297
[CrossRef]

Rep. Prog. Phys. (1)

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972); in particular, Sects. 6.1 and 6.2, pp. 335–364.
[CrossRef]

Other (8)

Ref. 18, Eq. (10.4.32), p. 364.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Eq. (9.2.3), p. 447.

Ref. 1, p. 263.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964), pp. 476–492.

Ref. 1, Appendix 2, pp. 339–342.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, New York, 1992), pp. 101–116.

Ref. 18, Eqs. (9.3.15)–(9.3.22), pp. 366–367.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 208–209.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Plane wave of wavelength λ incident with the angle ξ on an infinitely long circular cylinder of radius a.

Fig. 2
Fig. 2

(a) p=1 portion of the exact near-zone scattered intensity for normal incidence, λ=0.6328 μm, a=10.07 μm, and n=1.33. The shades of black and white correspond to large and small intensities, respectively. The position of the p=1 caustic focal line is r/a=2. (b) The p=1 portion of the exact scattered intensity in the region between the cylinder surface and the caustic focal line. The white and dashed curves are the position of the p=1 caustic as predicted by the semiclassical approximation and ray theory, respectively.

Fig. 3
Fig. 3

p=2 rainbow caustic for a plane wave whose angle of incidence is less than that of the caustic merging transition.

Fig. 4
Fig. 4

p=2 rainbow angle θrainbow as a function of r/a, as predicted by the semiclassical approximation (solid curve), and the asymptotic r rainbow angle θrainbow() (dashed line) for n=1.5. The prediction of ray theory is the set of triangles, the prediction of exact electromagnetic wave theory fitted to the square of an Airy integral is the set of open circles, and a straight-line ray trajectory at the angle θrainbow() is the set of solid circles.

Fig. 5
Fig. 5

Supernumerary periodicity parameter h as predicted by the semiclassical approximation as a function of r/a for n=1.5. The prediction of exact electromagnetic wave theory fitted to the square of an Airy integral by using the first and second Airy peaks and the first and third Airy peaks is given by the set of solid and open circles, respectively. Dashed line as in Fig. 4.

Fig. 6
Fig. 6

p=2 rainbow caustic for a plane wave whose angle of incidence is equal to that of the caustic merging transition.

Fig. 7
Fig. 7

( p-1)-order rainbow angle as a function of n in the vicinity of the caustic merging transition. The sections AA, BB, and CC correspond to the plane-wave angle of incidence less than, equal to, and greater than that of the transition.

Fig. 8
Fig. 8

p=2 rainbow caustic for a plane wave whose angle of incidence is greater than that of the caustic merging transition and less than that of the exterior/interior cusp transition. The caustic focal line is AA.

Fig. 9
Fig. 9

p=2 caustic merging transition for a nominally 15.9mm-diameter glass rod with refractive index 1.474 and illuminated by a plane wave with tilt angles (a) ξ=ξc-3°, (b) ξ=ξc, and (c) ξ=ξc+8° with ξc=46.58°±0.08°. In (a) the beam is incident from the right, the rod is at the left, and the caustic illuminates a white piece of cardboard placed perpendicular to the axis of the rod at its base. The rainbow caustic has a far-zone scattering angle θrainbow()178.5°. In (b) and (c), the beam is incident from the right, the dark region to the left of the rod is its shadow, and the exterior and interior caustics illuminate a piece of tissue paper held firmly in a frame perpendicular to the axis of the rod at its base.

Fig. 10
Fig. 10

p=2 exterior/interior cusp caustic transition for a nominally 15.9-mm-diameter glass rod with refractive index 1.474 and illuminated by a plane wave with the tilt angles (a) ξ<ξc, (b) ξ=ξc, and (c) ξ>ξc with ξc=67.83°±0.08°. The beam is incident from the right, and the caustics are observed by using the method described in Figs. 9(b) and 9(c).

Equations (85)

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

x=ka cos ξ,
y=nx,
n=(n2-sin2 ξ)1/2/cos ξ.
Escatt(r, θ, z)
=iE0exp(-ikz sin ξ)×l=-{u^r[-(l/kr)sec ξ Pl(r, θ)al-sin ξ Ql(r, θ)ql]+u^θ[Ql(r, θ)al-(l/kr)tan ξ Pl(r, θ)ql]+u^z[-cos ξ Pl(r, θ)ql]},
cBscatt(r, θ, z)
=iE0exp(-ikz sin ξ)×l=-{u^r[sin ξ Ql(r, θ)al-(l/kr)sec ξ Pl(r, θ)ql]+u^θ[(l/kr)tan ξ Pl(r, θ)al+Ql(r, θ)ql]+u^z[cos ξ Pl(r, θ)al]},
Pl(r, θ)=il+1Hl(1)(kr cos ξ)exp(ilθ),
Ql(r, θ)ilHl(1)(kr cos ξ)exp(ilθ).
Escatt(r, θ, z)
=iE0exp(-ikz sin ξ)×l=-{u^r[sin ξ Ql(r, θ)bl-(l/kr)sec ξ Pl(r, θ)ql]+u^θ[(l/kr)tan ξ Pl(r, θ)bl+Ql(r, θ)ql]+u^z[cos ξ Pl(r, θ)bl]},
cBscatt(r, θ, z)
=iE0exp(-ikz sin ξ)×l=-{u^r[(l/kr)sec ξ Pl(r, θ)bl+sin ξ Ql(r, θ)ql]+u^θ[-Ql(r, θ)bl+(l/kr)tan ξ Pl(r, θ)ql]+u^z[cos ξ Pl(r, θ)ql]}.
Hl(1)(u)(2/πu)1/2exp[i(u-lπ/2-π/4)],
Pl(r, θ)Ql(r, θ)(2/πkr cos ξ)1/2exp[i(kr cos ξ+lθ-π/4)].
Hl(1, 2)(u)(2/πu)1/2(1-l2/u2)-1/4×exp{±i[u(1-l2/u2)1/2+l arcsin(l/u)-lπ/2-π/4]},
Hl(1,2)(u)±i(2/πu)1/2(1-l2/u2)1/4
×exp{±i[u(1-l2/u2)1/2
+l arcsin(l/u)-lπ/2-π/4]},
Pl(r, θ)Ql(r, θ)(2/πkr cos ξ)1/2×exp(ix{(r/a)[1-w2/(r/a)2]1/2+w arcsin[w/(r/a)]+wθ}+iπ/4),
w=l/x.
[al(sin ξ)]p[bl(sin ξ)]p[ql(sin ξ)]p=-(12)Tl21(Rl11) p-1Tl12,
Tl,21Tl,μμ21Tl,μ21Tl,μ21exp(iϕl21),
Rl,11Rl,μμ11Rl,μ11Rl,μ11exp(iϕl11),
Tl,12Tl,μμ12Tl,μ12Tl,μ12exp(iϕl21),
ϕl21=y(1-l2/y2)1/2+l arcsin(l/y)-x(1-l2/x2)1/2-l arcsin(l/x),
ϕl11=2y(1-l2/y2)1/2+2l arcsin(l/y)-lπ.
[al]p[bl]p[ql]pexp{i[2ϕl21+( p-1)ϕl11]}.
EscattcBscattl=-Alexp(iΦl)-dw A(w)exp[iΦ(w)],
Φ(w)=x{2pn(1-w2/n2)1/2+2pw arcsin(w/n)-2(1-w2)1/2-2w arcsin w+(r/a)×[1-w2/(r/a)2]1/2+w arcsin[w/(r/a)]+w[θ-( p-1)π]}.
EscattcBscattA(w0)-dw exp[iΦ(w)].
dΦ(w)/dw=0,
θ=( p-1)π+2 arcsin w-2p arcsin(w/n)-arcsin[w/(r/a)].
X=(24kr cos ξ)1/2[(n-1)/n][(f-r)/a]/D1/2,
Y=(24)1/4(kr cos ξ)3/4θ/D1/4,
D=[2(r/a)3(n3-1)/n3]-1.
f/a=n/[2(n-1)].
θcaustic=±(8/3)[(n-1)/n]3/2[(f-rcaustic)/a]3/2/D1/2.
rcaustic/a=f/a-δ
θcaustic±16(n-1)5/2(3n)-3/2
×(-n2+3n-1)-1/2δ3/2,
ycaustic/a=±(8/33/2)(n-1)3/2n-1/2
×(-n2+3n-1)-1[(f-xcaustic)/a]3/2.
xcaustic/a=(-1)p+1[cos(θi-2pθr)-R cos(2θi-2pθr)],
ycaustic/a=(-1)p[sin(θi-2pθr)-R sin(2θi-2pθr)],
sin θi=nsin θr,
R=(cos θi)(ncos θr-2p cos θi)/[2(ncos θr-p cos θi)].
cos θi=[(n2-1)/3]1/2.
dθ/dw=0=(1-wr2)-1/2-( p/n)(1-wr2/n2)-1/2-[1/(2r/a)][1-wr2/(r/a)2]-1/2.
wr=[( p2-n2)/( p2-1)]1/2,
θrainbow(r/a)=( p-1)π+2 arcsin wr-2p arcsin(wr/n)-arcsin[wr/(r/a)].
w=wr+ε,
θ=θrainbow+Δ.
EscattcBscattAi(-x2/3Δ/h1/3).
h(r/a)=wr(1-wr2)-3/2-( pwr/n3)(1-wr2/n2)-3/2-(wr/2)(r/a)-3[1-wr2/(r/a)2]-3/2.
h=( p2-1)2( p2-n2)1/2/[ p2(n2-1)3/2],
n>[sin(π/2p)]-1.
X=(12nx)1/2( p-n)/(n3-p)1/2,
Y=(12)1/4(nx)3/4[θ-( p-1)π]/(n3-p)1/4.
θrainbow=( p-1)π±(25/2/3)( p-n)3/2/(n3-p)1/2.
n=p-ε,
θrainbow=( p-1)π-25/3ε3/2/[3p1/2( p2-1)1/2]-21/2(7p2-3)ε5/2/[5p3/2( p2-1)3/2]+ ,
θPearcey=( p-1)π-25/3ε3/2/[3p1/2( p2-1)1/2]-23/2p1/2ε5/2/( p2-1)3/2+ ,
ErainbowAi{-x2/3p1/2×(θ-θrainbow)/[21/6( p2-1)1/6ε1/6]},
EPearceyAi{-x2/3p1/2×(θ-θPearcey)/[21/6( p2-1)1/6ε1/6]}.
θrainbow=π±K1/2(ξc-ξ)3/2
ξ=ξc-δ
 ε=δ( p2-1)( p2-n2)1/2/[p(n2-1)1/2]+O(δ2),
K=32( p2-1)2( p2-n2)3/2/[9p4(n2-1)3/2].
cos ξc=[(n2-1)/3]1/2.
 cos ξc=(n2-1)1/2[2(b/a)2-1]/[4(b/a)2-1]1/2.
X=(24kr cos ξ)1/2[(n-p)/n][(f-r)/a]/D1/2,
Y=(24)1/4(kr cos ξ)3/4[θ-( p-1)π]/D1/4,
D=[2(r/a)3(n3-p)/n3]-1,
f/a=n/[2(n-p)].
θcaustic(r)=( p-1)π±(8/3)[(n-p)/n]3/2×[(f-rcaustic)/a]3/2/D1/2.
rcaustic/a=f/a-δ
θcaustic( p-1)π±(16/3)(n-p)3n-3/2×[4(n-p)3-n3+p]-1/2δ3/2,
cos ξc=[(n2-1)/15]1/2.
cos ξc=(n2-1)1/2[8(b/a)4-8(b/a)2+1]/[64(b/a)6-64(b/a)4+16(b/a)2-1]1/2.
Ai(X)[31/3/(2π)]-du exp[i(u3+31/3Xu)].
P(X, Y)-du exp[i(u4+Xu2+Yu)].
Y=±(-2X/3)3/2.
Y=(-2X/3)3/2-Δ
P(X, Y)=21/2π(3|X|)-1/6exp[iX2/12-i(Δ/2)(2|X|/3)1/2]×Ai[-Δ(24|X|)-1/6]+[π/(3|X|)]1/2×exp(-2iX2/3-iπ/4),

Metrics