Abstract

Diffraction effects are incorporated into a ray-based method for wave propagation, referred to as stable aggregates of flexible elements (SAFE). SAFE is based on the assignment of a Gaussian field contribution to each ray, where these contributions are not independent beam solutions of the wave equation. The effects of diffraction by planar opaque obstacles (within the Kirchhoff approximation) are accounted for by introducing rays emanating from the obstacle’s edges. The two leading asymptotic terms to the complex amplitudes for these contributions are derived. It is shown that this scheme leads to field estimates that remain valid and accurate at caustics and shadow boundaries, as illustrated by two examples, corresponding to a focused wave in free space and a field propagating in a layered inhomogeneous medium. For simplicity, two-dimensional propagation is considered.

© 2013 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  7. Several significant papers in this area are collected in R. C. Hansen, ed., Geometric Theory of Diffraction (IEEE, 1981).
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    [CrossRef]
  10. D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (Artech House, 1990).
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    [CrossRef]
  13. G. W. Forbes and M. A. Alonso, “Asymptotic estimation of the optical wave propagator. II. Relative validity,” J. Opt. Soc. Am. A 15, 1341–1354 (1998).
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  14. Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2nd ed. (Springer-Verlag, 1999), pp. 128–134.
  15. V. P. Maslov and M. V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics (Reidel, 1981).
  16. Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2nd ed. (Springer-Verlag, 1999), pp. 116–127.
  17. R. W. Ziolkowski and G. A. Deschamps, “Asymptotic evaluation of high-frequency fields near a caustic: an introduction to Maslov’s method,” Radio Sci. 19, 1001–1025 (1984).
    [CrossRef]
  18. V. M. Babich and M. M. Popov, “The Gaussian summation method (review),” Radiophys. Quantum Electron. 32, 1063–1081 (1989).
    [CrossRef]
  19. M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).
  20. B. Z. Steinberg, E. Heyman, and L. B. Felsen, “Phase-space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A 8, 41–59 (1991).
    [CrossRef]
  21. M. Katsav and E. Heyman, “Gaussian beam summation representation of a two-dimensional Gaussian beam diffraction by a half plane,” IEEE Trans. Antennas Propag. 55, 2247–2257 (2007).
    [CrossRef]
  22. M. Katsav and E. Heyman, “Gaussian beam summation representation of half plane diffraction: a full 3D formulation,” IEEE Trans. Antennas Propag. 57, 1081–1094 (2009).
    [CrossRef]
  23. H.-T. Chou and P. Pathak, “Fast Gaussian beam based synthesis of shaped reflector antennas for contoured beam applications,” IEE Proc. Microw. Antennas Propag. 151, 13–20 (2004).
    [CrossRef]
  24. E. T. Copson, Asymptotic Expansions (Cambridge University, 1965), pp. 91–94.
  25. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), pp. 399–410. Regular asymptotic expansions for integrals whose saddle points are near poles of the integrand have been treated by many authors dating back to the 1930s. For a comprehensive treatment including the main relevant references, see also [21].
  26. The proof requires the use of Liouville’s theorem, which implies that X′(ξ0,z)P¯′(P0,z)−X¯′(P0,z)P′(ξ0,z) is a constant of propagation, which by setting z=z0 is seen to be equal to X′(ξ0,z0).

2009 (1)

M. Katsav and E. Heyman, “Gaussian beam summation representation of half plane diffraction: a full 3D formulation,” IEEE Trans. Antennas Propag. 57, 1081–1094 (2009).
[CrossRef]

2007 (1)

M. Katsav and E. Heyman, “Gaussian beam summation representation of a two-dimensional Gaussian beam diffraction by a half plane,” IEEE Trans. Antennas Propag. 55, 2247–2257 (2007).
[CrossRef]

2004 (1)

H.-T. Chou and P. Pathak, “Fast Gaussian beam based synthesis of shaped reflector antennas for contoured beam applications,” IEE Proc. Microw. Antennas Propag. 151, 13–20 (2004).
[CrossRef]

2002 (1)

2001 (4)

1998 (2)

1991 (1)

1989 (1)

V. M. Babich and M. M. Popov, “The Gaussian summation method (review),” Radiophys. Quantum Electron. 32, 1063–1081 (1989).
[CrossRef]

1984 (1)

R. W. Ziolkowski and G. A. Deschamps, “Asymptotic evaluation of high-frequency fields near a caustic: an introduction to Maslov’s method,” Radio Sci. 19, 1001–1025 (1984).
[CrossRef]

1980 (1)

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).

1974 (1)

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

1962 (1)

Alonso, M. A.

Babich, V. M.

V. M. Babich and M. M. Popov, “The Gaussian summation method (review),” Radiophys. Quantum Electron. 32, 1063–1081 (1989).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).

Born, M.

M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference, and Diffraction of Light7th ed. (Cambridge University, 1999), pp. 142–144.

Chou, H.-T.

H.-T. Chou and P. Pathak, “Fast Gaussian beam based synthesis of shaped reflector antennas for contoured beam applications,” IEE Proc. Microw. Antennas Propag. 151, 13–20 (2004).
[CrossRef]

Copson, E. T.

E. T. Copson, Asymptotic Expansions (Cambridge University, 1965), pp. 91–94.

Deschamps, G. A.

R. W. Ziolkowski and G. A. Deschamps, “Asymptotic evaluation of high-frequency fields near a caustic: an introduction to Maslov’s method,” Radio Sci. 19, 1001–1025 (1984).
[CrossRef]

Fedoriuk, M. V.

V. P. Maslov and M. V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics (Reidel, 1981).

Felsen, L. B.

B. Z. Steinberg, E. Heyman, and L. B. Felsen, “Phase-space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A 8, 41–59 (1991).
[CrossRef]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), pp. 399–410. Regular asymptotic expansions for integrals whose saddle points are near poles of the integrand have been treated by many authors dating back to the 1930s. For a comprehensive treatment including the main relevant references, see also [21].

Forbes, G. W.

Heyman, E.

M. Katsav and E. Heyman, “Gaussian beam summation representation of half plane diffraction: a full 3D formulation,” IEEE Trans. Antennas Propag. 57, 1081–1094 (2009).
[CrossRef]

M. Katsav and E. Heyman, “Gaussian beam summation representation of a two-dimensional Gaussian beam diffraction by a half plane,” IEEE Trans. Antennas Propag. 55, 2247–2257 (2007).
[CrossRef]

B. Z. Steinberg, E. Heyman, and L. B. Felsen, “Phase-space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A 8, 41–59 (1991).
[CrossRef]

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 3rd ed. (Peter Peregrinus, 1986).

Katsav, M.

M. Katsav and E. Heyman, “Gaussian beam summation representation of half plane diffraction: a full 3D formulation,” IEEE Trans. Antennas Propag. 57, 1081–1094 (2009).
[CrossRef]

M. Katsav and E. Heyman, “Gaussian beam summation representation of a two-dimensional Gaussian beam diffraction by a half plane,” IEEE Trans. Antennas Propag. 55, 2247–2257 (2007).
[CrossRef]

Keller, J. B.

Kouyoumjian, R. G.

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

Kravtsov, Y. A.

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2nd ed. (Springer-Verlag, 1999), pp. 116–127.

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2nd ed. (Springer-Verlag, 1999), pp. 128–134.

Malherbe, J. A. G.

D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (Artech House, 1990).

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), pp. 399–410. Regular asymptotic expansions for integrals whose saddle points are near poles of the integrand have been treated by many authors dating back to the 1930s. For a comprehensive treatment including the main relevant references, see also [21].

Maslov, V. P.

V. P. Maslov and M. V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics (Reidel, 1981).

McNamara, D. A.

D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (Artech House, 1990).

Orlov, Y. I.

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2nd ed. (Springer-Verlag, 1999), pp. 128–134.

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2nd ed. (Springer-Verlag, 1999), pp. 116–127.

Pathak, P.

H.-T. Chou and P. Pathak, “Fast Gaussian beam based synthesis of shaped reflector antennas for contoured beam applications,” IEE Proc. Microw. Antennas Propag. 151, 13–20 (2004).
[CrossRef]

Pathak, P. H.

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

Pistorius, C. W. I.

D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (Artech House, 1990).

Popov, M. M.

V. M. Babich and M. M. Popov, “The Gaussian summation method (review),” Radiophys. Quantum Electron. 32, 1063–1081 (1989).
[CrossRef]

Steinberg, B. Z.

Wolf, E.

M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference, and Diffraction of Light7th ed. (Cambridge University, 1999), pp. 142–144.

Ziolkowski, R. W.

R. W. Ziolkowski and G. A. Deschamps, “Asymptotic evaluation of high-frequency fields near a caustic: an introduction to Maslov’s method,” Radio Sci. 19, 1001–1025 (1984).
[CrossRef]

IEE Proc. Microw. Antennas Propag. (1)

H.-T. Chou and P. Pathak, “Fast Gaussian beam based synthesis of shaped reflector antennas for contoured beam applications,” IEE Proc. Microw. Antennas Propag. 151, 13–20 (2004).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

M. Katsav and E. Heyman, “Gaussian beam summation representation of a two-dimensional Gaussian beam diffraction by a half plane,” IEEE Trans. Antennas Propag. 55, 2247–2257 (2007).
[CrossRef]

M. Katsav and E. Heyman, “Gaussian beam summation representation of half plane diffraction: a full 3D formulation,” IEEE Trans. Antennas Propag. 57, 1081–1094 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Express (1)

Optik (1)

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).

Proc. IEEE (1)

R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface,” Proc. IEEE 62, 1448–1461 (1974).
[CrossRef]

Radio Sci. (1)

R. W. Ziolkowski and G. A. Deschamps, “Asymptotic evaluation of high-frequency fields near a caustic: an introduction to Maslov’s method,” Radio Sci. 19, 1001–1025 (1984).
[CrossRef]

Radiophys. Quantum Electron. (1)

V. M. Babich and M. M. Popov, “The Gaussian summation method (review),” Radiophys. Quantum Electron. 32, 1063–1081 (1989).
[CrossRef]

Other (10)

Several significant papers in this area are collected in R. C. Hansen, ed., Geometric Theory of Diffraction (IEEE, 1981).

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 3rd ed. (Peter Peregrinus, 1986).

D. A. McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to the Uniform Geometrical Theory of Diffraction (Artech House, 1990).

M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference, and Diffraction of Light7th ed. (Cambridge University, 1999), pp. 142–144.

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2nd ed. (Springer-Verlag, 1999), pp. 128–134.

V. P. Maslov and M. V. Fedoriuk, Semiclassical Approximation in Quantum Mechanics (Reidel, 1981).

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes, and Wave Fields, 2nd ed. (Springer-Verlag, 1999), pp. 116–127.

E. T. Copson, Asymptotic Expansions (Cambridge University, 1965), pp. 91–94.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994), pp. 399–410. Regular asymptotic expansions for integrals whose saddle points are near poles of the integrand have been treated by many authors dating back to the 1930s. For a comprehensive treatment including the main relevant references, see also [21].

The proof requires the use of Liouville’s theorem, which implies that X′(ξ0,z)P¯′(P0,z)−X¯′(P0,z)P′(ξ0,z) is a constant of propagation, which by setting z=z0 is seen to be equal to X′(ξ0,z0).

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Figures (5)

Fig. 1.
Fig. 1.

(a) Incident ray family (dark gray curves) partially blocked by a planar obstacle (thick vertical line). A family of diffracted rays (light gray curves) emanates from the edge of the obstacle. (b) For z=z0, the effect in phase space of the obstacle is to truncate the PSC for the incident ray family (so the dashed section is removed) and to introduce a family of diffracted rays whose PSC is a vertical line. (c) For z>z0, both the truncated PSC for the incident ray family and the PSC for the diffracted ray family evolve according to the laws of geometrical optics. In (b) and (c) the intersection ray is indicated by a black dot.

Fig. 2.
Fig. 2.

Magnitude and phase of R(ρ) for 0|ρ|7 and for different values of arg(ρ).

Fig. 3.
Fig. 3.

(a) Incident (solid gray lines) and diffracted (dashed gray lines) ray families for a focused wave diffracted by an aperture of size 2a. (b)–(f) For 2a=20λ and γ=0.2/a, amplitudes of (b) the estimate’s contribution due to the incident ray family, (c) the contribution (times 10) due to the rays diffracted from the top edge, (d) the total basic field estimate, (e) the rigorously computed field, (f) the error of the basic field estimate (times 100), and (g) the estimate of this error calculated from the leading corrections for both the incident and diffracted rays (times 100).

Fig. 4.
Fig. 4.

(a) Magnitudes of the on-axis field (black) and its estimates (two shades of gray) corresponding to two values of γ, as well as the magnitudes of the errors for these estimates (dashed) magnified by a factor of 10. (b) For each of these two estimates, separate contributions due to the incident (solid) and the diffracted (dashed) ray families.

Fig. 5.
Fig. 5.

(a) Incident (solid gray lines) and diffracted (dashed gray lines) ray families for a field diffracted into a layered medium. (b)–(f) For λν=1/2 and γ=0.2/ν, amplitudes of (b) the estimate’s contribution due to the incident ray family, (c) the contribution (times 3) due to the rays diffracted from the bottom edge, (d) the total field estimate, (e) the rigorously computed field, and (f) the error of the field estimate (times 20).

Equations (88)

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[k22+n2(x,z)]U(x,z)=0,
X˙(ξ;z)=P(ξ;z)H(ξ;z),P˙(ξ;z)=12H(ξ;z)n2x[X(ξ;z),z],
H(ξ;z)=n2[X(ξ;z),z]P2(ξ;z)>0.
L˙(ξ;z)=n2[X(ξ;z),z]H(ξ;z),L(ξ;z)=P(ξ;z)x(ξ;z),
Uγ(x,z)=w(ξ,γ;z)g(ξ,γ;x,z)dξ.
g(ξ,γ;x,z)=k2πexp(kγ2[xX(ξ;z)]2+ik{L(ξ;z)+[xX(ξ;z)]P(ξ;z)}).
w(ξ,γ;z)=Y(ξ;z)A(ξ,γ;z)=Y(ξ;z)H(ξ;z)[a0(ξ)+1ika1(ξ,γ;z)+O(k2)],
Uγ(0)(x,z)=Y(ξ;z)H(ξ;z)a0(ξ)g(ξ,γ;x,z)dξ.
εγ(0)(z)=[|U(x,z)Uγ(0)(x,z)|2dx|U(x,z)|2dx]1/2=1k[|a1(ξ,γ;z)|2/Hdξ|a0(ξ)|2/Hdξ]1/2+O(k2).
Uγ(0)γ=O(k1)Uγ(0).
Uγi(x,z)=k2πΘ[X(ξ,z0)X0]w(z0)g(ξ,γ;x,z0)dξ.
ΔUγ(x)=k2π[Θ(XX0)Θ(xX0)]w×exp{kγ2(xX)2+ik[L+(xX)P]}dξ.
U¯γ(x)=ki2πA¯(ζ)exp{kγ2(xX0)2+ik[L0+(xX0)ζ]}dζ=ki2πexp[kγ2(xX0)2+ikL0]A¯(ζ)exp[ik(xX0)ζ]dζ,
A¯(ζ)=k2πiU¯γ(x)exp{kγ2(xX0)2ik[L0+(xX0)ζ]}dx.
A¯(ζ)=k2πiwexp[kγ2(X2X02)+ik(LL0+X0ζXP)]×[Θ(xX0)Θ(XX0)]exp{kx[γ(XX0)+i(Pζ)]}dxdξ.
[Θ(xX0)Θ(XX0)]exp{kx[γ(XX0)+i(Pζ)]}dx=1kγ(XX0)+ik(Pζ)([1Θ(XX0)]exp{kx[γ(XX0)+i(Pζ)]}|x+Θ(XX0)exp{kx[γ(XX0)+i(Pζ)]}|xexp{kX0[γ(XX0)+i(Pζ)]})=exp{kX0[γ(XX0)+i(Pζ)]}k[γ(XX0)+i(Pζ)].
A¯(ζ)=12πiwYY¯exp(kΩ)dξ,
Ω(ξ,ζ)γ2(X2X02)i(LL0+X0ζXP)X0(YY¯).
Ωξ=(XX0)Y,
Rm,n(ρ)12πrm(ρ+ir)n+1exp(r22)dr.
A¯(ζ)=12πi(w0+w1ε+12w2ε2+)[i(P0ζ)+Y1ε+12Y2ε2+16Y3ε3+]exp[k(12Ω2ε2+16Ω3ε3+)]dε=iw02π1ζP0+iY1εexp(k2Ω2ε2)×(1+w1εw0+w2ε22w0+)[1i12Y2ε2+16Y3ε3+ζP0+iY1ε14Y22ε4+(ζP0+iY1ε)2+]×[1k(16Ω3ε3+124Ω4ε4+)+k2(172Ω32ε4+)+]dε,
A¯(ζ)=iw02πY11ρ+irexp(r22)[1+1q(w1w0rΩ3r36Ω2iY22Y1r2ρ+ir)+O(q2)]dr.
A¯(ζ)=iw0Y1[R+k1/2C1/2+k1C1+O(k3/2)],
C1/21Ω2(w1w0R1,0Ω36Ω2R3,0iY22Y1R2,1)=HaX(aHY)|ξ=ξ0R1+iXPXP6(XY)3/2|ξ=ξ0R3.
H¯1/2(ζ)A¯(ζ)a¯0(ζ)a0(ξ0)iH¯(ζ)H(ξ0)Y(ξ0)R[kX(ξ0)Y(ξ0)(ζP0)],
R(ρ)=12exp(ρ22){σ[Re(ρ)]erf(ρ2)}.
a¯a¯0iH¯(ζ)H(ξ0)Y(ξ0)(a1ikR+a0C1/2k)|ξ=ξ0,
a¯1/2=ikH¯(ζ)H(ξ0)Y(ξ0)a0(ξ0)C1/2=H¯(ζ)XYH[H(a0HX)Q1ia0XPXP6XY3/2Q3]|ξ=ξ0,
a¯(ζ)=H¯1/2(ζ)A¯(ζ)=a¯0(ζ)+a¯1/2(ζ)(ik)1/2+.
UγT(x)=ξ0wgdξ.
0=(k22+n2)UγT=2iξ0HY{a˙k+12ik2[(acHY2)Da]}gdξ1k[wYc(xX)]|ξ=ξ0+O(k2),
cγ2+Y˙2+122n2x2(X,z),D1Hz(Y˙Y).
wY|ξ=ξ0+w¯Y¯|ζ=P0+w¯Y¯|ζ=P0=0,
a¯˙+12ik{a¯[(c¯H¯Y¯2)D¯]+a¯c¯H¯Y¯2}=O(k1),|ζP0|k/γ,
c¯γ2+Y¯˙2+122n2x2(X¯,z),D¯1H¯z(Y¯˙Y¯).
a¯(ζ,γ;z)=a¯0(ζ,γ;z)+a¯1/2(ζ,γ;z)(ik)1/2+.
a¯0(ζ,γ;z)=a0(ξ0)H¯(ζ;z0)H(ξ0;z0)κ¯F¯R[kF¯(ζP0)],
κ¯(γ;z0)=iX(ξ0;z0),F¯(γ;z0)=X(ξ0;z0)Y(ξ0;z0).
a¯0(ζ,γ;z0)=a0(ξ0)ζP0iH¯(ζ;z0)2πkH(ξ0;z0)X(ξ0;z0)+O(k1),|ζP0|γ/k.
a¯0(ζ,γ;z0)=a0(ξ0)ζP0H¯(ζ;z0)2πkH(ξ0;z0)κ¯+O(k1),|ζP0|1/|F¯k|.
κ¯=iX(ξ0;z0).
F¯(γ;z)=X(ξ0;z0)Y¯(P0;z)iY(ξ0;z).
c¯iH¯Y¯2|ζ=P0=iX(ξ0;z0)z[Y(ξ0;z)Y¯(P0;z)].
a¯0(ζ,γ;z0)=a0(ξ0)H¯(ζ;z0)Y¯(P0;z)H(ξ0;z0)Y(ξ0;z)R[kX(ξ0;z0)Y¯(P0;z)iY(ξ0;z)(ζP0)].
U¯γ(0)(x,z)=k2πa¯0Y¯H¯exp{kγ2(xX¯)2+ik[L¯+(xX¯)P¯]}dζ.
a¯1/2(ζ,γ;z)=a0(ξ0)H¯(ζ;z0)H(ξ0;z0)X(ξ0;z0)[F¯2(γ;z)κ¯1Q1iF¯4(γ;z)κ¯3Q3],
κ¯˙1=12(1H¯z(Y¯˙Y¯)+H¯{1H¯2Y¯2[γ2+Y¯˙2+122n2x2(X¯,z)]})|ζ=P0,
κ¯˙3=Y(ξ0;z)2X(ξ0;z0)Y¯(P0;z){1H¯Y¯2[γ2+Y¯˙2+122n2x2(X¯,z)]}|ζ=P0.
κ¯1(z0)=1XHXa0(a0HX)|z=z0ξ=ξ0ζ=P0,
κ¯3(z0)=XPXP6X3|z=z0ξ=ξ0ζ=P0.
UγTγ=ξ0{(wγwX2Y)12k[1Y(wY)]}gdξ+w2Y(xX)g|ξ0+O(k1).
a¯γ+12k[(H¯Y¯2)H¯Y¯4a¯a¯Y¯2]=O(k1).
κ¯1γ=iY¯(P0;z)Y¯3(P0;z),
κ¯3γ=iY(ξ0;z)Y¯(P0;z)X(ξ0;z0)Y¯4(P0;z).
n2(x,z)=νx,
U(x,0)={0,x<X,U0exp(ikP0x)XxX+,0,x>X+.
X(ξ;z)=ξν+P0zHνz24H2,P(ξ;z)=P0νz2H,L(ξ;z)=P0ξνξzHP0νz22H2+ν2z312H3,
X¯±(ζ;z)=X±+ζzH¯νz24H¯2,P¯±(ζ;z)=ζνz2H¯,L¯±(ζ;z)=X±P0X±zνH¯ζνz22H¯2+ν2z312H¯3,
U(x,z)=(k2ν)2/3Ai[(k2ν)1/3(xx)]exp(ikzνx)×U(x,0)Ai[(k2ν)1/3(xx)]dxdx,
Rm,n(ρ)12πrm(ρ+ir)n+1exp(r22)dr,m,n0.
R(ρ)R0,0(ρ)=12π1ρ+irexp(r22)dr=σ(ρ)2π0exp{uσ[Re(ρ)](ρ+ir)}duexp(r22)dr=σ(ρ)2π0exp{u22σ[Re(ρ)]uρ}du=12exp(ρ22){σ[Re(ρ)]erf(ρ2)},
erf(τ)2π0τexp(t2)dt.
Rm,n(ρ)=i2πrm1(ρ+irρ)(ρ+ir)n+1exp(r22)dr=i[ρRm1,n(ρ)Rm1,n1(ρ)],m1,n0,
Rm,n(ρ)=i2πnrmexp(r22)(ρ+ir)nrdr=i2πn1(ρ+ir)nr[rmexp(r22)]dr=inRm+1,n1(ρ)imnRm1,n1(ρ),m0,n1,
Rm,1(ρ)=12πrmexp(r22)dr={(m1)!!2π,meven0,moddm0,
Rm(ρ)12πρrm[1irρ+]exp(r22)dr={(m1)!!2πρ+O(ρ3),meven,im!!2πρ2+O(ρ4),modd.
R(ρ)=j=0{σ[Re(ρ)]ρ2j2(2j)!!12πρ2j+1(2j+1)!!}.
Rm(0)={0,meven,(m2)!!2πi,modd.
ddρRm(ρ)=Rm,1(ρ)=iRm+1(ρ)+{δ(ρ),m=0,imRm1,m1.
ρRm(ρ)=iRm+1(ρ)+{(m1)!!2π,meven,0,modd.
F^ρϑR(ρ)=12πR(ρ)exp(iρϑ)dρ=iR(ϑ),F^ρϑR1(ρ)=R1(ϑ).
F^ρϑRm(ρ)=iQm(ϑ),
Q0=R0,Q1=iR1,Q2=R2+R,Q3=iR3+3iR1.
Qm+1(ρ)=ρQm(ρ)+{12π,m=0,mQm1(ρ),m1.
a¯0(ζ,γ;z)=a0(ξ0)H¯(ζ;z0)H(ξ0;z0)F¯(γ;z)κ¯R,
a¯1/2(ζ,γ;z)=a0(ξ0)κ¯iH¯(ζ;z0)H(ξ0;z0)[F¯2(γ;z)K¯1(γ;z)R1+iF¯4(γ;z)κ¯3(γ;z)R3],
ta¯+12k[A(ζ)a¯+B(ζ)a¯]=O(k1),|ζP0|k/γ,
ta¯0=a0(ξ0)H¯(ζ;z0)H(ξ0;z0)κ¯[F¯tR+k(ζP0)F¯F¯tdRdρ]=a0(ξ0)H¯(ζ;z0)H(ξ0;z0)κ¯F¯t(R+ρdRdρ)=a0(ξ0)H¯(ζ;z0)H(ξ0;z0)κ¯F¯t(RR2),
a¯0=a0(ξ0)H¯(ζ;z0)H(ξ0;z0)κ¯F¯[kF¯dRdρ+H¯(ζ;z0)2H¯(ζ;z0)R]=a0(ξ0)H¯(ζ;z0)H(ξ0;z0)κ¯F¯[ikF¯R1+H¯(P0;z0)2H¯(P0;z0)R+O(k1/2)],
a¯0=a0(ξ0)H¯(ζ;z0)H(ξ0;z0)κ¯F¯2[kF¯(RR2)ikH¯(P0;z0)H¯(P0;z0)R1+O(k0)],
A(ζ)=A(P0)+A(P0)(ζP0)+=A(P0)+A(P0)ρkF¯+O(k1),
B(ζ)=B(P0)+B(P0)ρkF¯+O(k1).
B(P0)=2F¯tF¯3=t(1F¯2)=iX(ξ0;z0)t[Y(ξ0;z)Y¯(P0;z)],
ta¯1/2=a0iH¯(ζ;z0)H(ξ0;z0)κ¯[(F¯2K¯1t+3F¯F¯tK¯1)R1+(iF¯4κ¯3t+7iF¯3F¯tκ¯3F¯F¯tK¯1)R3iF¯3F¯tκ¯3R5],
a¯1/2=O(k1/2),
a¯1/2=ka0iH¯(ζ;z0)H(ξ0;z0)κ¯F¯4[(3K¯16iF¯2κ¯3)R1+(7iF¯2κ¯3K¯1)R3iF¯2κ¯3R5]+O(k1/2).
κ¯1t=i{A2[BH¯(ζ;z0)]H¯(ζ;z0)}|ζ=P0,
κ¯3t=12F¯2B(P0).

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