Abstract

The scattering far zone for light transmitted through a sphere following p-1 internal reflections by a family of near-grazing incident rays is subdivided into a lit region and a shadow region. The sharpness of the ray theory transition between the lit and the shadow regions is smoothed in wave theory by radiation shed by electromagnetic surface waves. It is shown that when higher-order terms in the physical optics approximation to the phase of the partial-wave scattering amplitudes are included, the transition between the lit and the shadow regions becomes a two-ray-to-zero-ray transition, called a superweak caustic in analogy to the more familiar scattering caustics and weak scattering caustics. One of the merged rays is a tunneling ray.

© 2003 Optical Society of America

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References

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  1. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
    [CrossRef]
  2. P. L. Marston, E. H. Trinh, “Hyperbolic umbilic diffraction catastrophe and rainbow scattering from spheroidal drops,” Nature 312, 529–531 (1984).
    [CrossRef]
  3. J. F. Nye, “Rainbow scattering from spheroidal drops—an explanation of the hyperbolic umbilic foci,” Nature 312, 531–532 (1984).
    [CrossRef]
  4. N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
    [CrossRef] [PubMed]
  5. P. L. Marston, “Critical angle scattering by a bubble: physical-optics approximation and observations,” J. Opt. Soc. Am. 69, 1205–1211 (1979).
    [CrossRef]
  6. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  7. H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (New York) 34, 23–95 (1965).
    [CrossRef]
  8. J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
    [CrossRef]
  9. The interpretation of the second stationary-phase point in terms of the tunneling ray was briefly discussed in J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
    [CrossRef]
  10. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 203–206.
  11. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading Mass., 1987), pp. 94–96.
  12. Ref. 10, pp. 206–207.
  13. Ref. 10, pp. 119–126.
  14. E. A. Hovenac, J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
    [CrossRef]
  15. Ref. 10, pp. 208–209.
  16. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington D.C., 1964), pp. 366–367, Eqs. (9.3.15, 9.3.16, 9.3.19, 9.3.20, 9.3.23, 9.3.24, 9.3.27, 9.3.28).
  17. Ref. 16, p. 446.
  18. Ref. 10, p. 213.
  19. K. W. Ford, J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. (N.Y.) 7, 259–286 (1959).
    [CrossRef]
  20. Ref. 10, p. 212.
  21. M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), eq. (6.12).
    [CrossRef]
  22. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969), Figs. 7, 10.
    [CrossRef]
  23. Ref. 16, p. 446, Eq. (10.4.9), and pp. 448–449, Eqs. (10.4.60, 10.4.64).
  24. Ref. 16, p. 478, Table 10.13.
  25. Ref. 16, p. 449, Eq. (10.4.63).
  26. A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975). It should be noted that Figs. 3b and 3c in Ref. 8 are in errorin that the incident ray should strike the centrifugal barrier tangentially and then tunnel radially to the sphere surface.
    [CrossRef]
  27. H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonance,” Comments At. Mol. Phys. 23, 175–187 (1989).
  28. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
    [CrossRef]
  29. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979), Fig. 2.
    [CrossRef]
  30. 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]
  31. H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
    [CrossRef] [PubMed]
  32. P. L. Marston, “Cusp diffraction catastrophe from spheroids: generalized rainbows and inverse scattering,” Opt. Lett. 10, 588–590 (1985).
    [CrossRef] [PubMed]
  33. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]

1993 (2)

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[CrossRef]

B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
[CrossRef]

1992 (1)

1991 (2)

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
[CrossRef] [PubMed]

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

1989 (1)

H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonance,” Comments At. Mol. Phys. 23, 175–187 (1989).

1988 (1)

1987 (1)

1985 (1)

1984 (2)

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

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

1980 (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

1979 (2)

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979), Fig. 2.
[CrossRef]

P. L. Marston, “Critical angle scattering by a bubble: physical-optics approximation and observations,” J. Opt. Soc. Am. 69, 1205–1211 (1979).
[CrossRef]

1976 (1)

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

1975 (1)

A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975). It should be noted that Figs. 3b and 3c in Ref. 8 are in errorin that the incident ray should strike the centrifugal barrier tangentially and then tunnel radially to the sphere surface.
[CrossRef]

1972 (1)

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), eq. (6.12).
[CrossRef]

1969 (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969), Figs. 7, 10.
[CrossRef]

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

1965 (1)

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (New York) 34, 23–95 (1965).
[CrossRef]

1959 (1)

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

Berry, M. V.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), eq. (6.12).
[CrossRef]

Fiedler-Ferrari, N.

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
[CrossRef] [PubMed]

Ford, K. W.

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

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading Mass., 1987), pp. 94–96.

Hovenac, E. A.

Johnson, B. R.

Lock, J. A.

Love, J. D.

A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975). It should be noted that Figs. 3b and 3c in Ref. 8 are in errorin that the incident ray should strike the centrifugal barrier tangentially and then tunnel radially to the sphere surface.
[CrossRef]

Marston, P. L.

Mount, K. E.

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), eq. (6.12).
[CrossRef]

Nussenzveig, H. M.

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
[CrossRef] [PubMed]

H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonance,” Comments At. Mol. Phys. 23, 175–187 (1989).

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979), Fig. 2.
[CrossRef]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969), Figs. 7, 10.
[CrossRef]

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

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (New York) 34, 23–95 (1965).
[CrossRef]

Nye, J. F.

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

Snyder, A. W.

A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975). It should be noted that Figs. 3b and 3c in Ref. 8 are in errorin that the incident ray should strike the centrifugal barrier tangentially and then tunnel radially to the sphere surface.
[CrossRef]

Trinh, E. H.

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

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

van de Hulst, H. C.

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

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 203–206.

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]

Wiscombe, W. J.

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
[CrossRef] [PubMed]

Am. J. Phys. (2)

J. A. Lock, E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993).
[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]

Ann. Phys. (New York) (1)

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (New York) 34, 23–95 (1965).
[CrossRef]

Appl. Opt. (2)

Comments At. Mol. Phys. (1)

H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonance,” Comments At. Mol. Phys. 23, 175–187 (1989).

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975). It should be noted that Figs. 3b and 3c in Ref. 8 are in errorin that the incident ray should strike the centrifugal barrier tangentially and then tunnel radially to the sphere surface.
[CrossRef]

J. Math. Phys. (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969), Figs. 7, 10.
[CrossRef]

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. (2)

P. L. Marston, “Critical angle scattering by a bubble: physical-optics approximation and observations,” J. Opt. Soc. Am. 69, 1205–1211 (1979).
[CrossRef]

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079, 1193–1194 (1979), Fig. 2.
[CrossRef]

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

Nature (2)

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

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

Opt. Lett. (1)

Phys. Rev. A (1)

N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
[CrossRef] [PubMed]

Prog. Opt. (1)

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980).
[CrossRef]

Rep. Prog. Phys. (1)

M. V. Berry, K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315–397 (1972), eq. (6.12).
[CrossRef]

Other (12)

Ref. 16, p. 446, Eq. (10.4.9), and pp. 448–449, Eqs. (10.4.60, 10.4.64).

Ref. 16, p. 478, Table 10.13.

Ref. 16, p. 449, Eq. (10.4.63).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 203–206.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading Mass., 1987), pp. 94–96.

Ref. 10, pp. 206–207.

Ref. 10, pp. 119–126.

Ref. 10, pp. 208–209.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington D.C., 1964), pp. 366–367, Eqs. (9.3.15, 9.3.16, 9.3.19, 9.3.20, 9.3.23, 9.3.24, 9.3.27, 9.3.28).

Ref. 16, p. 446.

Ref. 10, p. 213.

Ref. 10, p. 212.

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Figures (7)

Fig. 1
Fig. 1

Magnitude of the modified Fock function f(s) of Eq. (26) numerically computed by using the contour of Fig. 10 of Ref. 7 on which the integrand damps most rapidly to zero as a function of the integration variable z. The dashed curve G is the geometrical-ray stationary-phase limit of relation (29), and the dashed curve S is thè surface-wave limit of relation (32) obtained by using the first five zeros of the Airy function.

Fig. 2
Fig. 2

Debye intensity for transmission, p=1, by a sphere with size parameter x=1000.0 and refractive index n=1.333 as a function of scattering angle for the amplitudes of Eqs. (16) and (17): j=1, (solid curve) and j=2 (dashed curve). The scattering angle θc1=82.79° separates the lit region from the shadow region in ray theory. The Debye intensity smoothes the transition between these two regions.

Fig. 3
Fig. 3

Trajectory of the p=1 geometrical ray G and the tunneling ray T with the same scattering angle. The extent of the centrifugal barrier surrounding the sphere and probed by the tunneling ray is indicated by the dashed line. The ray tunnels from a to b and from c to d through the centrifugal barrier.

Fig. 4
Fig. 4

Partial-wave number of the p=1 stationary-phase points of Eq. (51) as a function of scattering angle for x=1000.0, n=1.333, and j=2 (circles). The solid curve is the prediction of ray theory for the impact parameter of the transmitted geometrical ray. The the uppermost two open circles in the upper-right corner of the figure are tunneling-ray stationary-phase points.

Fig. 5
Fig. 5

Partial-wave number of the p=1 geometrical-ray and tunneling-ray stationary-phase points as a function of scattering angle for x=1000.0 and n=1.333. The solid circles are the stationary-phase points of the j=1 amplitude, and the open circles are the stationary-phase points of the j=2 amplitude. The lower and upper solid lines are the predictions of Eqs. (31) and (36), respectively.

Fig. 6
Fig. 6

Phase of Eq. (51) as a function of partial-wave number for p=1, x=1000.0, n=1.333, and j=1 (solid curve) and j=2 (dashed curve) for (a) θ=79.5°, (b) θ=81.0°, and (c) θ=82.75° illustrating the merging of the geometrical- and tunneling-ray stationary-phase points.

Fig. 7
Fig. 7

Partial-wave number of the p=2 stationary-phase points of Eq. (51) as a function of scattering angle for x=1000.0, n=1.333, and j=2 (circles). The curved solid line is the prediction of ray theory, and the straight slanted solid line at the top right of the figure is the prediction of Eq. (36) for the tunneling ray. Region C contains radiation associated with the complex ray on the shadow side of the rainbow, and region S contains radiation shed by electromagnetic surface waves on the shadow side of the critical scattering angle.

Equations (57)

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Ө=(p-1)π+2θi-2pθt,
sin(θi)=n sin(θt).
Escatt(p)(r, t)=-(iE0/kr)exp(ikr-iωt)×[-STM(p)(θ)cos(ϕ)uθ+STE(p)(θ)sin(ϕ)uϕ],
Sα(p)(θ)=x[sin(θi)cos(θi)/2 sin(θ)]1/2×|1-[p cos(θi)/n cos(θt)]|-1/2tα21(θi)×[rα11(θi)]p-1tα12(θi)×exp{2ix[pn cos(θt)-cos(θi)]+iξp},
xka.
θ=Ө-2πNif2πNӨ2πN+π2π(N+1)-Өif2πN+π<Ө<2π(N+1).
sin(θi)=1-δ,
tα21(θi)tα12(θi)=25/2β2[δ/(n2-1)]1/2+O(δ),
β=1for TEnfor TM.
rα11(θi)=1-O(δ1/2).
Ө=Өcp-23/2δ1/2+O(δ),
Өcp=2p arccos(1/n)
Sα(p)(θ)=β2x(Өcp-Ө)3/2[(n2-1)sin(θ)]-1/2×exp[2ipx(n2-1)1/2-ix(Өcp-Ө)+iξp],
ξp=exp[-iπp/2-iπ(N+1)]if2πNӨ2πN+πexp[-iπ(p+1)/2-iπ(N+1)]if2πN+π<Ө<2π(N+1).
Escat(r, t)=-i(E0/kr)exp(ikr-iωt)[-S2(θ)×cos(ϕ)uθ+S1(θ)sin(ϕ)uϕ],
S1(θ)=l=1[(2l+1)/l(l+1)][alπl(θ)+blτl(θ)],
S2(θ)=l=1[(2l+1)/l(l+1)][alτl(θ)+blπl(θ)],
albl=(1/2)1-Rl22-p=1Tl21(Rl11)p-1Tl12,
l+(1/2)=x+z(x/2)1/3,
jl(x)π1/2Ai(z)/(21/6x5/6),
nl(x)-π1/2Bi(z)/(21/6x5/6),
jl(x)-π1/221/6Ai(z)/x7/6,
nl(x)π1/221/6Bi(z)/x7/6,
jl(y)n1/2 cos(Y)/[(n2-1)1/4y],
nl(y)n1/2 sin(Y)/[(n2-1)1/4y],
jl(y)-(n2-1)1/4 sin(Y)/(n1/2y),
nl(y)(n2-1)1/4 cos(Y)/(n1/2y),
y=nx,
Y=-π/4+nx{1-[1+(z/2)(2/x)2/3]2/n2}1/2-x[1+(z/2)(2/x)2/3]arcsin({1-[1+(z/2)×(2/x)2/3]2/n2}1/2).
Y-π/4+x(n2-1)1/2-x arccos(1/n)-z(x/2)1/3×arccos(1/n)+(z2/4)(x/2)-1/3(n2-1)-1/2.
Sj(p)(θ)β2(8x)1/2[π3(n2-1)sin(θ)]-1/2×-dz[Bi(z)+iAi(z)]-2 exp[2ipY+ixӨ+izӨ(x/2)1/3+iπp/2+3πi/4+iξp].
f(s)=(1/2π)exp(iπ/6)×Γdz{Ai[z exp(2πi/3)]}-2 exp(isz),
Sj(p)(θ)β2[2x/π(n2-1)sin(θ)]1/2×exp[2ipx(n2-1)1/2+ix(Ө-Өcp)+iπ/4+iξp]f[(Ө-Өcp)(x/2)1/3].
z=-s2/4
f(s)π1/2|s|3/2 exp(-iπ/4-is3/12).
z=-(1/4)(Өcp-Ө)2(x/2)2/3
l+(1/2)=x-x(Өcp-Ө)2/8.
f(s)s exp(-iπ/6)n[Ai(-wn)]-2×exp[(-31/2swn/2)+i(swn/2)],
Ai(-wn)=0.
Sj(p)(θ)β221/6x5/6[π(n2-1)sin(θ)]-1/2×exp[2ipx(n2-1)1/2+ix(Ө-Өcp)+iπ/12+iξp]×(Ө-Өcp)n[Ai(-wn)]-2×exp[i(wn/2)(Ө-Өcp)(x/2)1/3]×exp[-(31/2wn/2)(Ө-Өcp)(x/2)1/3].
z=[(n2-1)1/2/p](Өcp-Ө)(x/2)2/3,
l+(1/2)=x+x(n2-1)1/2(Өcp-Ө)/2p.
Sj(p)(θ)β2x[(n2-1)sin(θ)]-1/2×exp[2ipx(n2-1)1/2-ix(Өcp-Ө)+iξp]×{(Өcp-Ө)3/2-(Өcp-Ө)1/2×[23/2(n2-1)1/2/p]×exp[-ix(n2-1)1/2(Өcp-Ө)2/4p]×exp[-2x(n2-1)3/4(Өcp-Ө)3/2/3p3/2]}.
b=a sin(θi)=a-δa,
Φopl=2x[pn cos(θt)-cos(θi)]=2px(n2-1)1/2-x(Өcp-Ө),
b=a sin(θi)=a+Δa,
θiπ/2-i(2Δ)1/2,
θtarcsin(1/n)+Δ/(n2-1)1/2.
tα21(Δ)2iβ(2Δ)1/2(n2-1)-1/2+O(Δ).
rα11(Δ)1-O(Δ1/2).
tα12(Δ)2β+O(Δ1/2)
Ө=Өcp-2pΔ/(n2-1)1/2,
Φopl=2pnx cos(θt)=2px(n2-1)1/2-x(Өcp-Ө),
l+(1/2)=kb=x+x(n2-1)1/2(Өcp-Ө)/2p,
Sα(p)(θ)=x[sin(θ)|dθ/dΔ|]-1/2Aα(Δ)tα21(Δ)×[rα11(Δ)]p-1tα12(Δ)Aα(Δ)exp(iΦopl+iξp).
Aα(Δ)exp(iπ/4)exp[-x(n2-1)3/4(Өcp-Ө)3/2/3p3/2]=exp(iπ/4)exp[-Δx(x/2)1/3]=exp(iπ/4)exp(-2z3/2/3)
Dl(p)(θ)=Tl21(Rl11)p-1Tl12 exp[i(l+12)θ]

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