Abstract

We study the suitability of a tapered plane-wave incident field, using both the Gaussian and the more advanced Thorsos tapers for low-grazing-angle rough-surface scattering problems as well as the problem of propagation in the presence of a rough surface. For surface scattering problems it is known that as the angle of incidence approaches grazing incidence the tapered beam waist should be made larger; several criteria relating these two parameters have been proposed for both the Gaussian and the Thorsos tapers. Our two-dimensional scattering simulations with the oceanlike Pierson–Moskowitz surfaces show that when the width of the Gaussian or the Thorsos taper is fixed, the backscatter cross section for TE polarization is characterized by a distinctive and consistent anomalous jump as grazing incidence is approached. This observation has led to a refined version of one of the above-mentioned beam waist–angle of incidence criteria and its robustness is demonstrated. The approximate (non-Maxwellian) nature of the Thorsos–Gaussian taper also becomes evident in over-surface-propagation simulations with use of the boundary integral equation method. A certain inconsistency was observed between the surface field that we obtained by first defining the Thorsos–Gaussian-tapered field on a vertical plane and then propagating it to the surface and that obtained by defining the same tapered field directly on the surface. This effect, not previously appreciated, may be of importance when the rough-surface effects are rigorously incorporated into the propagation problem. We conclude with a detailed derivation of the Thorsos taper that points out all the approximations involved in it and the resulting limitations.

© 1999 Optical Society of America

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References

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  1. E. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
    [CrossRef]
  2. R. T. Marchand, G. S. Brown, “On the use of finite surfaces in the numerical prediction of rough surface scattering,” submitted to IEEE Trans. Antennas Propag.
  3. D. A. Kapp, G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antennas Propag. 44, 711–721 (1996).
    [CrossRef]
  4. L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
    [CrossRef]
  5. J. T. Johnson, “A numerical study of low grazing angle backscatter from ocean-like impedance surfaces with the canonical grid method,” IEEE Trans. Antennas Propag. 46, 114–120 (1998).
    [CrossRef]
  6. R. S. Awadallah, G. S. Brown, “Parabolic equation formulation via a singular perturbation technique and its application to scattering from irregular surfaces,” Waves Random Media 8, 315–328 (1998).
    [CrossRef]
  7. D. J. Donohue, H.-C. Ku, D. Thompson, “Application of iterative moment-method solutions to ocean surface radar scattering,” IEEE Trans. Antennas Propag. 46, 121–132 (1998).
    [CrossRef]
  8. R. S. Awadallah, G. S. Brown, “Electromagnetic wave scattering from a rough surface in a surface-based duct created by a linear-square refractive index profile,” in Proceedings of the 1997 Battlespace Atmospherics Conference (Space and Naval Warfare Systems Center, San Diego, Calif., 1998), pp. 519–528.
  9. H. D. Ngo, C. L. Rino, “Application of beam simulation to scattering at low grazing angles. 1. Methodology and validation,” Radio Sci. 29, 1365–1379 (1994).
    [CrossRef]
  10. D. A. Kapp, “A new method to calculate wave scattering from rough surfaces at low grazing angles,” Ph.D. dissertation (Virginia Polytechnic Institute and State University, Blacksburg, Va., 1995).
  11. R. T. Marchand, “Numerical study on the validity of the quasi-specular and two-scale models for rough surface parameter estimation: one dimensional surfaces,” Ph.D. dissertation (Virginia Polytechnic Institute and State University, Blacksburg, Va., 1996), pp. 87–89.
  12. A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J.1991), pp. 354–356.
  13. J. V. Toporkov, R. T. Marchand, G. S. Brown, “On the discretization of the integral equation describing scattering by rough conducting surfaces,” IEEE Trans. Antennas Propag. 46, 150–161 (1998).
    [CrossRef]
  14. E. Thorsos, “Acoustic scattering from ‘Pierson–Moskowitz’ sea surface,” J. Acoust. Soc. Am. 88, 335–349 (1990).
    [CrossRef]
  15. F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, Norwood, Mass., 1986), Vol. 2, pp. 816–817.
  16. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1994), pp. 354–355.
  17. J. V. Toporkov, “Study of electromagnetic scattering from randomly rough ocean-like surfaces using integral-equation-based numerical technique,” Ph.D. dissertation (Virginia Polytechnic Institute and State University, Blacksburg, Va., 1998), pp. 123–178.
  18. R. J. Adams, B. A. Davis, G. S. Brown, “Scattering from a closed body above the rough surface using the method of ordered multiple interactions,” presented at the IEEE-APS International Symposium and USNC/URSI National Radio Science Meeting, Atlanta, Ga., June 21–26, 1998.

1998

J. T. Johnson, “A numerical study of low grazing angle backscatter from ocean-like impedance surfaces with the canonical grid method,” IEEE Trans. Antennas Propag. 46, 114–120 (1998).
[CrossRef]

R. S. Awadallah, G. S. Brown, “Parabolic equation formulation via a singular perturbation technique and its application to scattering from irregular surfaces,” Waves Random Media 8, 315–328 (1998).
[CrossRef]

D. J. Donohue, H.-C. Ku, D. Thompson, “Application of iterative moment-method solutions to ocean surface radar scattering,” IEEE Trans. Antennas Propag. 46, 121–132 (1998).
[CrossRef]

J. V. Toporkov, R. T. Marchand, G. S. Brown, “On the discretization of the integral equation describing scattering by rough conducting surfaces,” IEEE Trans. Antennas Propag. 46, 150–161 (1998).
[CrossRef]

1996

D. A. Kapp, G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antennas Propag. 44, 711–721 (1996).
[CrossRef]

1995

L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
[CrossRef]

1994

H. D. Ngo, C. L. Rino, “Application of beam simulation to scattering at low grazing angles. 1. Methodology and validation,” Radio Sci. 29, 1365–1379 (1994).
[CrossRef]

1990

E. Thorsos, “Acoustic scattering from ‘Pierson–Moskowitz’ sea surface,” J. Acoust. Soc. Am. 88, 335–349 (1990).
[CrossRef]

1988

E. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

Adams, R. J.

R. J. Adams, B. A. Davis, G. S. Brown, “Scattering from a closed body above the rough surface using the method of ordered multiple interactions,” presented at the IEEE-APS International Symposium and USNC/URSI National Radio Science Meeting, Atlanta, Ga., June 21–26, 1998.

Awadallah, R. S.

R. S. Awadallah, G. S. Brown, “Parabolic equation formulation via a singular perturbation technique and its application to scattering from irregular surfaces,” Waves Random Media 8, 315–328 (1998).
[CrossRef]

R. S. Awadallah, G. S. Brown, “Electromagnetic wave scattering from a rough surface in a surface-based duct created by a linear-square refractive index profile,” in Proceedings of the 1997 Battlespace Atmospherics Conference (Space and Naval Warfare Systems Center, San Diego, Calif., 1998), pp. 519–528.

Brown, G. S.

R. S. Awadallah, G. S. Brown, “Parabolic equation formulation via a singular perturbation technique and its application to scattering from irregular surfaces,” Waves Random Media 8, 315–328 (1998).
[CrossRef]

J. V. Toporkov, R. T. Marchand, G. S. Brown, “On the discretization of the integral equation describing scattering by rough conducting surfaces,” IEEE Trans. Antennas Propag. 46, 150–161 (1998).
[CrossRef]

D. A. Kapp, G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antennas Propag. 44, 711–721 (1996).
[CrossRef]

R. T. Marchand, G. S. Brown, “On the use of finite surfaces in the numerical prediction of rough surface scattering,” submitted to IEEE Trans. Antennas Propag.

R. S. Awadallah, G. S. Brown, “Electromagnetic wave scattering from a rough surface in a surface-based duct created by a linear-square refractive index profile,” in Proceedings of the 1997 Battlespace Atmospherics Conference (Space and Naval Warfare Systems Center, San Diego, Calif., 1998), pp. 519–528.

R. J. Adams, B. A. Davis, G. S. Brown, “Scattering from a closed body above the rough surface using the method of ordered multiple interactions,” presented at the IEEE-APS International Symposium and USNC/URSI National Radio Science Meeting, Atlanta, Ga., June 21–26, 1998.

Chan, C. H.

L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
[CrossRef]

Davis, B. A.

R. J. Adams, B. A. Davis, G. S. Brown, “Scattering from a closed body above the rough surface using the method of ordered multiple interactions,” presented at the IEEE-APS International Symposium and USNC/URSI National Radio Science Meeting, Atlanta, Ga., June 21–26, 1998.

Donohue, D. J.

D. J. Donohue, H.-C. Ku, D. Thompson, “Application of iterative moment-method solutions to ocean surface radar scattering,” IEEE Trans. Antennas Propag. 46, 121–132 (1998).
[CrossRef]

Fung, A. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, Norwood, Mass., 1986), Vol. 2, pp. 816–817.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1994), pp. 354–355.

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J.1991), pp. 354–356.

Johnson, J. T.

J. T. Johnson, “A numerical study of low grazing angle backscatter from ocean-like impedance surfaces with the canonical grid method,” IEEE Trans. Antennas Propag. 46, 114–120 (1998).
[CrossRef]

Kapp, D. A.

D. A. Kapp, G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antennas Propag. 44, 711–721 (1996).
[CrossRef]

D. A. Kapp, “A new method to calculate wave scattering from rough surfaces at low grazing angles,” Ph.D. dissertation (Virginia Polytechnic Institute and State University, Blacksburg, Va., 1995).

Ku, H.-C.

D. J. Donohue, H.-C. Ku, D. Thompson, “Application of iterative moment-method solutions to ocean surface radar scattering,” IEEE Trans. Antennas Propag. 46, 121–132 (1998).
[CrossRef]

Marchand, R. T.

J. V. Toporkov, R. T. Marchand, G. S. Brown, “On the discretization of the integral equation describing scattering by rough conducting surfaces,” IEEE Trans. Antennas Propag. 46, 150–161 (1998).
[CrossRef]

R. T. Marchand, “Numerical study on the validity of the quasi-specular and two-scale models for rough surface parameter estimation: one dimensional surfaces,” Ph.D. dissertation (Virginia Polytechnic Institute and State University, Blacksburg, Va., 1996), pp. 87–89.

R. T. Marchand, G. S. Brown, “On the use of finite surfaces in the numerical prediction of rough surface scattering,” submitted to IEEE Trans. Antennas Propag.

Moore, R. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, Norwood, Mass., 1986), Vol. 2, pp. 816–817.

Ngo, H. D.

H. D. Ngo, C. L. Rino, “Application of beam simulation to scattering at low grazing angles. 1. Methodology and validation,” Radio Sci. 29, 1365–1379 (1994).
[CrossRef]

Pak, K.

L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
[CrossRef]

Rino, C. L.

H. D. Ngo, C. L. Rino, “Application of beam simulation to scattering at low grazing angles. 1. Methodology and validation,” Radio Sci. 29, 1365–1379 (1994).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1994), pp. 354–355.

Sangani, H.

L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
[CrossRef]

Thompson, D.

D. J. Donohue, H.-C. Ku, D. Thompson, “Application of iterative moment-method solutions to ocean surface radar scattering,” IEEE Trans. Antennas Propag. 46, 121–132 (1998).
[CrossRef]

Thorsos, E.

E. Thorsos, “Acoustic scattering from ‘Pierson–Moskowitz’ sea surface,” J. Acoust. Soc. Am. 88, 335–349 (1990).
[CrossRef]

E. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

Toporkov, J. V.

J. V. Toporkov, R. T. Marchand, G. S. Brown, “On the discretization of the integral equation describing scattering by rough conducting surfaces,” IEEE Trans. Antennas Propag. 46, 150–161 (1998).
[CrossRef]

J. V. Toporkov, “Study of electromagnetic scattering from randomly rough ocean-like surfaces using integral-equation-based numerical technique,” Ph.D. dissertation (Virginia Polytechnic Institute and State University, Blacksburg, Va., 1998), pp. 123–178.

Tsang, L.

L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
[CrossRef]

Ulaby, F. T.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, Norwood, Mass., 1986), Vol. 2, pp. 816–817.

IEEE Trans. Antennas Propag.

D. A. Kapp, G. S. Brown, “A new numerical method for rough surface scattering calculations,” IEEE Trans. Antennas Propag. 44, 711–721 (1996).
[CrossRef]

L. Tsang, C. H. Chan, K. Pak, H. Sangani, “Monte-Carlo simulations of large-scale problems of random rough surface scattering and applications to grazing incidence with the BMIA/canonical grid method,” IEEE Trans. Antennas Propag. 43, 851–859 (1995).
[CrossRef]

J. T. Johnson, “A numerical study of low grazing angle backscatter from ocean-like impedance surfaces with the canonical grid method,” IEEE Trans. Antennas Propag. 46, 114–120 (1998).
[CrossRef]

D. J. Donohue, H.-C. Ku, D. Thompson, “Application of iterative moment-method solutions to ocean surface radar scattering,” IEEE Trans. Antennas Propag. 46, 121–132 (1998).
[CrossRef]

J. V. Toporkov, R. T. Marchand, G. S. Brown, “On the discretization of the integral equation describing scattering by rough conducting surfaces,” IEEE Trans. Antennas Propag. 46, 150–161 (1998).
[CrossRef]

J. Acoust. Soc. Am.

E. Thorsos, “Acoustic scattering from ‘Pierson–Moskowitz’ sea surface,” J. Acoust. Soc. Am. 88, 335–349 (1990).
[CrossRef]

E. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,” J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

Radio Sci.

H. D. Ngo, C. L. Rino, “Application of beam simulation to scattering at low grazing angles. 1. Methodology and validation,” Radio Sci. 29, 1365–1379 (1994).
[CrossRef]

Waves Random Media

R. S. Awadallah, G. S. Brown, “Parabolic equation formulation via a singular perturbation technique and its application to scattering from irregular surfaces,” Waves Random Media 8, 315–328 (1998).
[CrossRef]

Other

R. T. Marchand, G. S. Brown, “On the use of finite surfaces in the numerical prediction of rough surface scattering,” submitted to IEEE Trans. Antennas Propag.

R. S. Awadallah, G. S. Brown, “Electromagnetic wave scattering from a rough surface in a surface-based duct created by a linear-square refractive index profile,” in Proceedings of the 1997 Battlespace Atmospherics Conference (Space and Naval Warfare Systems Center, San Diego, Calif., 1998), pp. 519–528.

D. A. Kapp, “A new method to calculate wave scattering from rough surfaces at low grazing angles,” Ph.D. dissertation (Virginia Polytechnic Institute and State University, Blacksburg, Va., 1995).

R. T. Marchand, “Numerical study on the validity of the quasi-specular and two-scale models for rough surface parameter estimation: one dimensional surfaces,” Ph.D. dissertation (Virginia Polytechnic Institute and State University, Blacksburg, Va., 1996), pp. 87–89.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J.1991), pp. 354–356.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive (Artech House, Norwood, Mass., 1986), Vol. 2, pp. 816–817.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1994), pp. 354–355.

J. V. Toporkov, “Study of electromagnetic scattering from randomly rough ocean-like surfaces using integral-equation-based numerical technique,” Ph.D. dissertation (Virginia Polytechnic Institute and State University, Blacksburg, Va., 1998), pp. 123–178.

R. J. Adams, B. A. Davis, G. S. Brown, “Scattering from a closed body above the rough surface using the method of ordered multiple interactions,” presented at the IEEE-APS International Symposium and USNC/URSI National Radio Science Meeting, Atlanta, Ga., June 21–26, 1998.

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Figures (13)

Fig. 1
Fig. 1

Average normalized radar cross section in the backscattered direction, showing anomalous jumps at LGA’s. TE (horizontal) polarization; average taken over 100 realizations of P-M surfaces with a wind speed of 5 m/s.

Fig. 2
Fig. 2

Anomalous patterns in the average backscattered NRCS at LGA’s for different types of taper. TE (horizontal) polarization; EM wavelength, λ1=23 cm; beam waist, g=136.5λ1. Average is taken over 100 realizations of P-M surfaces with a wind speed of 5 m/s.

Fig. 3
Fig. 3

Examples of the amplitude of the incident field in the mean surface plane (z=0) with different types of taper. Beam waist g=136.5λ1.

Fig. 4
Fig. 4

Effects of poor edge suppression by the integral taper in the average bistatic NRCS. The longer surface results in stronger edge suppression. θi=86°; TE (horizontal) polarization; EM wavelength, λ1=23 cm; g=136.5λ1. Average is taken over 100 realizations of P-M surfaces with a wind speed of 5 m/s.

Fig. 5
Fig. 5

Effects of surface length on the average bistatic NRCS for the Thorsos taper (this figure is similar to Fig. 4, in which the integral taper was studied). θi=86°; TE (horizontal) polarization; EM wavelength, λ1=23 cm; g=136.5λ1. Average is taken over 100 realizations of P-M surfaces with a wind speed of 5 m/s.

Fig. 6
Fig. 6

Example of the surface current for a single realization of P-M surface with a wind speed of 5 m/s and Thorsos taper. TE (horizontal) polarization; EM wavelength, λ1=23 cm; g=136.5λ1.

Fig. 7
Fig. 7

Example of the surface current for a single realization of P-M surface with an integral taper. TE (horizontal) polarization; EM wavelength, λ1=23 cm; g=136.5λ1.

Fig. 8
Fig. 8

Distortions in the average bistatic NRCS due to approximate taper at the prebreakdown point θi=84°. The results with integral taper serve as a check. TE (horizontal) polarization; EM wavelength, λ1=23 cm; beam waist, g=136.5λ1. Average is taken over 100 realizations of P-M surfaces with a wind speed of 5 m/s.

Fig. 9
Fig. 9

Average backscattered NRCS for TM (vertical) polarization. Unlike its TE (horizontal) counterpart in Fig. 2, it shows no anomalous patterns. EM wavelength, λ1=23 cm; beam waist g=136.5λ1. Average is taken over 100 realizations of P-M surfaces with a wind speed of 5 m/s.

Fig. 10
Fig. 10

Geometry of the problem of propagation along a rough surface.

Fig. 11
Fig. 11

Magnitude comparison of Kirchhoff currents JIPi, JTPi, and JTSi on a perfectly conducting flat surface.

Fig. 12
Fig. 12

Magnitude comparison between Kirchhoff currents JIPi and JISi on a perfectly conducting flat surface.

Fig. 13
Fig. 13

Scattered power (in decibels) from a perfectly conducting flat surface that is due to the Kirchhoff currents shown in Fig. 11.

Equations (32)

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

ψi(x, z)=T(x, z)exp(-jki·r)T(x, z)exp[-jk(x sin θi-z cos θi)].
TG(x, z)=exp-(x+z tan θi)2g2.
TT(x, z)=exp-(x+z tan θi)2g2-j k(x sin θi-z cos θi)(kg cos θi)2×2(x+z tan θi)2g2-1.
ψi(x, z)=1πΔθ-π/2π/2 exp-(θ-θi)2(Δθ)2exp(-jk·r)dθ,
kg cos θi1.
g>A2k(π/2-θi) cos θi,
W(K)=α4|K|3exp-βg2K2U4,
J(x)=Ji(x)+P(x, x)J(x)dx.
Ji(x)=2Hi(x, z)|z=ζ(x),
P(x, x)=2 G(x, x)n1+ζx2(x).
Ji(x)=2Ei(x, z)nz=ζ(x),
P(x, x)=-2 G(x, x)n1+ζx2(x).
σ0(θi, θs)=limr 2πr|ψs|21cos θiRe  jkψi(x, 0) ψi*(x, 0)zdx,
k·r=k[x sin θ-z cos θ]=k[x sin(θi+δ)-z cos(θi+δ)],
ψi(r)=1πΔθexp(-jkβ)-π/2-θiπ/2-θi exp-jkγ-1(Δθ)2×δ2-jkαδdδ,
α=x cos θi+z sin θi,
β=x sin θi-z cos θi,
γ=(1/2)(z cos θi-x sin θi)=-β/2.
η=jkγ+1/(Δθ)2,
ς=jkα,
ψi(r)=1πΔθexp(-jkβ)-π/2-θiπ/2-θi exp(-ηδ2-ςδ)dδ.
Re(η)π2-θi2>Cor(π/2-θi)2(Δθ)2>C,
g>2Ck(π/2-θi) cos θi.
ψi(r)=1πΔθexp(-jkβ)- exp(-ηδ2-ςδ)dδ.
ψi(r)=1Δθηexp(-jkβ)expς24η.
ψi(r)=exp(-jkβ)[k2γ2(Δθ)4+1]1/4exp-k2α2(Δθ)24[1+jkγ(Δθ)2]×exp-jkγ(Δθ)22.
[k2γ2(Δθ)4+1]1/41,
[1+jkγ(Δθ)2]-11-jkγ(Δθ)2,
exp(-jk·r)=exp(-jkx sin θ)
θ0=±π/2.
ψi(x, 0)=1πΔθexp-(π/2+θi)2(Δθ)2exp(jkx)×-π/2 exp-j kx2θ+π22dθ+exp-(π/2-θi)2(Δθ)2exp(-jkx)×-π/2exp+j kx2θ-π22dθ.
ψi(x, 0)=1Δθexp-(π/2-θi)2(Δθ)2 exp[-j(kx±π/4)]2k|x|.

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