Abstract

A number of proposed applications of electromagnetic waves require that the radiation beam maintain a high intensity over an appreciable propagation distance. These applications include, among others, power beaming, advanced radar, laser acceleration of particles, and directed-energy sources. The quest to achieve these objectives has led to a resurgence of research on diffraction theory. We present a survey and critique of the analyses and experimental tests of solutions of the wave equation in connection with so-called diffractionless and other directed radiation beams. The examples discussed include electromagnetic missiles, Bessel beams, electromagnetic directed-energy pulse trains, and electromagnetic bullets.

© 1991 Optical Society of America

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  1. W. T. Welford, Optics (Oxford U. Press, New York, 1981), Chap. 3.
  2. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
    [Crossref]
  3. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.
  4. G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–435 (1952).
    [Crossref]
  5. J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
    [Crossref]
  6. P. A. Belanger, “Packetlike solutions of the homogeneous-wave equation,” J. Opt. Soc. Am. A 1, 723–724 (1984).
    [Crossref]
  7. T. T. Wu, R. W. P. King, “Comment on ‘Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,’” J. Appl. Phys. 56, 2587–2588 (1984).
    [Crossref]
  8. A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
    [Crossref]
  9. T. T. Wu, H. Lehman, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
    [Crossref]
  10. P. Hillion, “More on focus wave modes in Maxwell equations,” J. Appl. Phys. 60, 2981–2982 (1986).
    [Crossref]
  11. P. Hillion, “Spinor focus wave modes,”J. Math. Phys. 28, 1743–1748 (1987).
    [Crossref]
  12. E. Heyman, B. Z. Steinberg, L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4, 2081–2091 (1987).
    [Crossref]
  13. L. B. Felsen, E. Heyman, “Discretized beam methods for focused radiation from distributed apertures,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 320–328 (1988).
    [Crossref]
  14. T. T. Wu, “Electromagnetic missiles,” J. Appl. Phys. 57, 2370–2373 (1985).
    [Crossref]
  15. H. M. Lee, “Rise and fall of directed transient: use of Mellin transformation in time domain problems,” Radio Sci. 22, 1102–1108 (1987).
    [Crossref]
  16. T. T. Wu, R. W. P. King, H. M. Shen, “Spherical lens as a launcher of electromagnetic missiles,” J. Appl. Phys. 62, 4036–4040 (1987).
    [Crossref]
  17. T. T. Wu, H. M. Shen, “Radiation of an electromagnetic pulse from the open end of a circular waveguide,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 329–337 (1988).
    [Crossref]
  18. H. M. Shen, “Experimental study of electromagnetic missiles,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 338–346 (1988).
    [Crossref]
  19. J. M. Myers, “Pulsed radiation from a parabolic reflector,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 347–356 (1988).
    [Crossref]
  20. T. T. Wu, R. W. P. King, H. M. Shen, “Generalized analysis of the spherical lens as a launcher of electromagnetic missiles,” J. Appl. Phys. 63, 5647–5653 (1988).
    [Crossref]
  21. T. T. Wu, R. W. P. King, H. M. Shen, “Circular cylindrical lens as a line-source electromagnetic-missile launcher,”IEEE Trans. Antennas Propag. 37, 39–44 (1989).
    [Crossref]
  22. J. M. Myers, H. M. Shen, T. T. Wu, H. E. Brandt, “Curved electromagnetic missiles,” J. Appl. Phys. 65, 2604–2610 (1989).
    [Crossref]
  23. H. M. Shen, T. T. Wu, “The transverse energy pattern of an electromagnetic missile from a circular current disk,” in Microwave and Particle Beam Sources and Directed Energy Concepts, H. E. Brandt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1061, 352–360 (1989).
    [Crossref]
  24. T. T. Wu, H. M. Shen, J. M. Myers, “A review of electromagnetic missiles,” in Microwave and Particle Beam Sources and Directed Energy Concepts, H. E. Brandt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1061, 370–379 (1989).
    [Crossref]
  25. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  26. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  27. D. DeBeer, S. R. Hartman, R. Friedberg, “Comment on ‘Diffraction-free beams,’” Phys. Rev. Lett. 59, 2611 (1987).
    [Crossref] [PubMed]
  28. J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
    [Crossref] [PubMed]
  29. R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,”J. Math. Phys. 26, 861–863 (1985).
    [Crossref]
  30. R. W. Ziolkowski, “New electromagnetic directed energy pulses,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 312–319 (1988).
    [Crossref]
  31. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
    [Crossref] [PubMed]
  32. R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
    [Crossref] [PubMed]
  33. R. W. Ziolkowski, “Localized transmission of wave energy,” in Microwave and Particle Beam Sources and Directed Energy Concepts, H. E. Brandt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1061, 395–402 (1989).
    [Crossref]
  34. A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1989).
    [Crossref]
  35. H. E. Moses, “The time-dependent inverse source problem for the acoustic and electromagnetic equations in the one- and three-dimensional cases,”J. Math. Phys. 25, 1905–1923 (1985).
    [Crossref]
  36. H. E. Moses, R. T. Prosser, “Initial conditions, sources, and currents for prescribed time-dependent acoustic and electromagnetic fields in three dimensions, part I: the inverse initial value problem. Acoustic and electromagnetic bullets, expanding waves, and imploding waves,”IEEE Trans. Antennas Propag. AP-34, 188–196 (1986).
    [Crossref]
  37. H. E. Moses, R. T. Prosser, “A refinement of the radon transform and its inverse,” Proc. R. Soc. London Ser. A 422, 343–349 (1989); “Exact solutions of the three-dimensional scalar wave equation and Maxwell’s equations from the approximate solutions in the wave zone through the use of the radon transform,” Proc. A. Soc. London ser. A 422, 351–365 (1989).
    [Crossref]
  38. H. E. Moses, R. T. Prosser, “Acoustic and electromagnetic bullets. Derivation of new exact solutions of the acoustic and Maxwell’s equations,” SIAM J. Appl. Math. 50, 1325–1340 (1990).
    [Crossref]
  39. B. B. Godfrey, “Diffraction-free microwave propagation,” Sensor and Simulation Notes 320 (Weapons Laboratory, Kirtland Air Force Base, N.M., 1989).
  40. Yu. Yu. Ananev, “Nondiffracting light waves,” Opt. Spectrosc. (USSR) 64, 722–723 (1988).
  41. R. Barakat, “Solution of the Luneberg apodization problems,”J. Opt. Soc. Am. 52, 264–275 (1962).
    [Crossref]
  42. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9.
  43. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).
  44. I. W. Kay, H. E. Moses, “The determination of the scattering potential from the spectral measure function. V. The Gelfand–Levitan equation for the three-dimensional scattering problem,” Nuovo Cimento 22, 689–705 (1961).
    [Crossref]
  45. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).
  46. W. H. Southwell, “Validity of the Fresnel approximation in the near field,”J. Opt. Soc. Am. 71, 7–14 (1981).
    [Crossref]

1990 (1)

H. E. Moses, R. T. Prosser, “Acoustic and electromagnetic bullets. Derivation of new exact solutions of the acoustic and Maxwell’s equations,” SIAM J. Appl. Math. 50, 1325–1340 (1990).
[Crossref]

1989 (6)

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[Crossref] [PubMed]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[Crossref] [PubMed]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1989).
[Crossref]

H. E. Moses, R. T. Prosser, “A refinement of the radon transform and its inverse,” Proc. R. Soc. London Ser. A 422, 343–349 (1989); “Exact solutions of the three-dimensional scalar wave equation and Maxwell’s equations from the approximate solutions in the wave zone through the use of the radon transform,” Proc. A. Soc. London ser. A 422, 351–365 (1989).
[Crossref]

T. T. Wu, R. W. P. King, H. M. Shen, “Circular cylindrical lens as a line-source electromagnetic-missile launcher,”IEEE Trans. Antennas Propag. 37, 39–44 (1989).
[Crossref]

J. M. Myers, H. M. Shen, T. T. Wu, H. E. Brandt, “Curved electromagnetic missiles,” J. Appl. Phys. 65, 2604–2610 (1989).
[Crossref]

1988 (3)

T. T. Wu, R. W. P. King, H. M. Shen, “Generalized analysis of the spherical lens as a launcher of electromagnetic missiles,” J. Appl. Phys. 63, 5647–5653 (1988).
[Crossref]

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[Crossref] [PubMed]

Yu. Yu. Ananev, “Nondiffracting light waves,” Opt. Spectrosc. (USSR) 64, 722–723 (1988).

1987 (7)

P. Hillion, “Spinor focus wave modes,”J. Math. Phys. 28, 1743–1748 (1987).
[Crossref]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

E. Heyman, B. Z. Steinberg, L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4, 2081–2091 (1987).
[Crossref]

H. M. Lee, “Rise and fall of directed transient: use of Mellin transformation in time domain problems,” Radio Sci. 22, 1102–1108 (1987).
[Crossref]

T. T. Wu, R. W. P. King, H. M. Shen, “Spherical lens as a launcher of electromagnetic missiles,” J. Appl. Phys. 62, 4036–4040 (1987).
[Crossref]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

D. DeBeer, S. R. Hartman, R. Friedberg, “Comment on ‘Diffraction-free beams,’” Phys. Rev. Lett. 59, 2611 (1987).
[Crossref] [PubMed]

1986 (2)

P. Hillion, “More on focus wave modes in Maxwell equations,” J. Appl. Phys. 60, 2981–2982 (1986).
[Crossref]

H. E. Moses, R. T. Prosser, “Initial conditions, sources, and currents for prescribed time-dependent acoustic and electromagnetic fields in three dimensions, part I: the inverse initial value problem. Acoustic and electromagnetic bullets, expanding waves, and imploding waves,”IEEE Trans. Antennas Propag. AP-34, 188–196 (1986).
[Crossref]

1985 (5)

T. T. Wu, “Electromagnetic missiles,” J. Appl. Phys. 57, 2370–2373 (1985).
[Crossref]

H. E. Moses, “The time-dependent inverse source problem for the acoustic and electromagnetic equations in the one- and three-dimensional cases,”J. Math. Phys. 25, 1905–1923 (1985).
[Crossref]

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,”J. Math. Phys. 26, 861–863 (1985).
[Crossref]

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[Crossref]

T. T. Wu, H. Lehman, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
[Crossref]

1984 (2)

T. T. Wu, R. W. P. King, “Comment on ‘Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,’” J. Appl. Phys. 56, 2587–2588 (1984).
[Crossref]

P. A. Belanger, “Packetlike solutions of the homogeneous-wave equation,” J. Opt. Soc. Am. A 1, 723–724 (1984).
[Crossref]

1983 (1)

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[Crossref]

1981 (1)

1962 (1)

1961 (1)

I. W. Kay, H. E. Moses, “The determination of the scattering potential from the spectral measure function. V. The Gelfand–Levitan equation for the three-dimensional scattering problem,” Nuovo Cimento 22, 689–705 (1961).
[Crossref]

1954 (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[Crossref]

1952 (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–435 (1952).
[Crossref]

Ananev, Yu. Yu.

Yu. Yu. Ananev, “Nondiffracting light waves,” Opt. Spectrosc. (USSR) 64, 722–723 (1988).

Barakat, R.

Belanger, P. A.

Besieris, I. M.

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1989).
[Crossref]

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[Crossref]

Brandt, H. E.

J. M. Myers, H. M. Shen, T. T. Wu, H. E. Brandt, “Curved electromagnetic missiles,” J. Appl. Phys. 65, 2604–2610 (1989).
[Crossref]

Brittingham, J. N.

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[Crossref]

Cook, B. D.

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[Crossref] [PubMed]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

DeBeer, D.

D. DeBeer, S. R. Hartman, R. Friedberg, “Comment on ‘Diffraction-free beams,’” Phys. Rev. Lett. 59, 2611 (1987).
[Crossref] [PubMed]

Durnin, J.

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[Crossref] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Felsen, L. B.

E. Heyman, B. Z. Steinberg, L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4, 2081–2091 (1987).
[Crossref]

L. B. Felsen, E. Heyman, “Discretized beam methods for focused radiation from distributed apertures,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 320–328 (1988).
[Crossref]

Friedberg, R.

D. DeBeer, S. R. Hartman, R. Friedberg, “Comment on ‘Diffraction-free beams,’” Phys. Rev. Lett. 59, 2611 (1987).
[Crossref] [PubMed]

Godfrey, B. B.

B. B. Godfrey, “Diffraction-free microwave propagation,” Sensor and Simulation Notes 320 (Weapons Laboratory, Kirtland Air Force Base, N.M., 1989).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

Hartman, S. R.

D. DeBeer, S. R. Hartman, R. Friedberg, “Comment on ‘Diffraction-free beams,’” Phys. Rev. Lett. 59, 2611 (1987).
[Crossref] [PubMed]

Heyman, E.

E. Heyman, B. Z. Steinberg, L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4, 2081–2091 (1987).
[Crossref]

L. B. Felsen, E. Heyman, “Discretized beam methods for focused radiation from distributed apertures,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 320–328 (1988).
[Crossref]

Hillion, P.

P. Hillion, “Spinor focus wave modes,”J. Math. Phys. 28, 1743–1748 (1987).
[Crossref]

P. Hillion, “More on focus wave modes in Maxwell equations,” J. Appl. Phys. 60, 2981–2982 (1986).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9.

Kay, I. W.

I. W. Kay, H. E. Moses, “The determination of the scattering potential from the spectral measure function. V. The Gelfand–Levitan equation for the three-dimensional scattering problem,” Nuovo Cimento 22, 689–705 (1961).
[Crossref]

King, R. W. P.

T. T. Wu, R. W. P. King, H. M. Shen, “Circular cylindrical lens as a line-source electromagnetic-missile launcher,”IEEE Trans. Antennas Propag. 37, 39–44 (1989).
[Crossref]

T. T. Wu, R. W. P. King, H. M. Shen, “Generalized analysis of the spherical lens as a launcher of electromagnetic missiles,” J. Appl. Phys. 63, 5647–5653 (1988).
[Crossref]

T. T. Wu, R. W. P. King, H. M. Shen, “Spherical lens as a launcher of electromagnetic missiles,” J. Appl. Phys. 62, 4036–4040 (1987).
[Crossref]

T. T. Wu, R. W. P. King, “Comment on ‘Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,’” J. Appl. Phys. 56, 2587–2588 (1984).
[Crossref]

Lee, H. M.

H. M. Lee, “Rise and fall of directed transient: use of Mellin transformation in time domain problems,” Radio Sci. 22, 1102–1108 (1987).
[Crossref]

Lehman, H.

T. T. Wu, H. Lehman, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
[Crossref]

Lewis, D. K.

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[Crossref] [PubMed]

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[Crossref] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Moses, H. E.

H. E. Moses, R. T. Prosser, “Acoustic and electromagnetic bullets. Derivation of new exact solutions of the acoustic and Maxwell’s equations,” SIAM J. Appl. Math. 50, 1325–1340 (1990).
[Crossref]

H. E. Moses, R. T. Prosser, “A refinement of the radon transform and its inverse,” Proc. R. Soc. London Ser. A 422, 343–349 (1989); “Exact solutions of the three-dimensional scalar wave equation and Maxwell’s equations from the approximate solutions in the wave zone through the use of the radon transform,” Proc. A. Soc. London ser. A 422, 351–365 (1989).
[Crossref]

H. E. Moses, R. T. Prosser, “Initial conditions, sources, and currents for prescribed time-dependent acoustic and electromagnetic fields in three dimensions, part I: the inverse initial value problem. Acoustic and electromagnetic bullets, expanding waves, and imploding waves,”IEEE Trans. Antennas Propag. AP-34, 188–196 (1986).
[Crossref]

H. E. Moses, “The time-dependent inverse source problem for the acoustic and electromagnetic equations in the one- and three-dimensional cases,”J. Math. Phys. 25, 1905–1923 (1985).
[Crossref]

I. W. Kay, H. E. Moses, “The determination of the scattering potential from the spectral measure function. V. The Gelfand–Levitan equation for the three-dimensional scattering problem,” Nuovo Cimento 22, 689–705 (1961).
[Crossref]

Myers, J. M.

J. M. Myers, H. M. Shen, T. T. Wu, H. E. Brandt, “Curved electromagnetic missiles,” J. Appl. Phys. 65, 2604–2610 (1989).
[Crossref]

J. M. Myers, “Pulsed radiation from a parabolic reflector,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 347–356 (1988).
[Crossref]

T. T. Wu, H. M. Shen, J. M. Myers, “A review of electromagnetic missiles,” in Microwave and Particle Beam Sources and Directed Energy Concepts, H. E. Brandt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1061, 370–379 (1989).
[Crossref]

Prosser, R. T.

H. E. Moses, R. T. Prosser, “Acoustic and electromagnetic bullets. Derivation of new exact solutions of the acoustic and Maxwell’s equations,” SIAM J. Appl. Math. 50, 1325–1340 (1990).
[Crossref]

H. E. Moses, R. T. Prosser, “A refinement of the radon transform and its inverse,” Proc. R. Soc. London Ser. A 422, 343–349 (1989); “Exact solutions of the three-dimensional scalar wave equation and Maxwell’s equations from the approximate solutions in the wave zone through the use of the radon transform,” Proc. A. Soc. London ser. A 422, 351–365 (1989).
[Crossref]

H. E. Moses, R. T. Prosser, “Initial conditions, sources, and currents for prescribed time-dependent acoustic and electromagnetic fields in three dimensions, part I: the inverse initial value problem. Acoustic and electromagnetic bullets, expanding waves, and imploding waves,”IEEE Trans. Antennas Propag. AP-34, 188–196 (1986).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

Sezginer, A.

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[Crossref]

Shaarawi, A. M.

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1989).
[Crossref]

Shen, H. M.

J. M. Myers, H. M. Shen, T. T. Wu, H. E. Brandt, “Curved electromagnetic missiles,” J. Appl. Phys. 65, 2604–2610 (1989).
[Crossref]

T. T. Wu, R. W. P. King, H. M. Shen, “Circular cylindrical lens as a line-source electromagnetic-missile launcher,”IEEE Trans. Antennas Propag. 37, 39–44 (1989).
[Crossref]

T. T. Wu, R. W. P. King, H. M. Shen, “Generalized analysis of the spherical lens as a launcher of electromagnetic missiles,” J. Appl. Phys. 63, 5647–5653 (1988).
[Crossref]

T. T. Wu, R. W. P. King, H. M. Shen, “Spherical lens as a launcher of electromagnetic missiles,” J. Appl. Phys. 62, 4036–4040 (1987).
[Crossref]

T. T. Wu, H. M. Shen, “Radiation of an electromagnetic pulse from the open end of a circular waveguide,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 329–337 (1988).
[Crossref]

H. M. Shen, “Experimental study of electromagnetic missiles,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 338–346 (1988).
[Crossref]

H. M. Shen, T. T. Wu, “The transverse energy pattern of an electromagnetic missile from a circular current disk,” in Microwave and Particle Beam Sources and Directed Energy Concepts, H. E. Brandt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1061, 352–360 (1989).
[Crossref]

T. T. Wu, H. M. Shen, J. M. Myers, “A review of electromagnetic missiles,” in Microwave and Particle Beam Sources and Directed Energy Concepts, H. E. Brandt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1061, 370–379 (1989).
[Crossref]

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.

Southwell, W. H.

Steinberg, B. Z.

Toraldo di Francia, G.

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–435 (1952).
[Crossref]

Welford, W. T.

W. T. Welford, Optics (Oxford U. Press, New York, 1981), Chap. 3.

Wu, T. T.

T. T. Wu, R. W. P. King, H. M. Shen, “Circular cylindrical lens as a line-source electromagnetic-missile launcher,”IEEE Trans. Antennas Propag. 37, 39–44 (1989).
[Crossref]

J. M. Myers, H. M. Shen, T. T. Wu, H. E. Brandt, “Curved electromagnetic missiles,” J. Appl. Phys. 65, 2604–2610 (1989).
[Crossref]

T. T. Wu, R. W. P. King, H. M. Shen, “Generalized analysis of the spherical lens as a launcher of electromagnetic missiles,” J. Appl. Phys. 63, 5647–5653 (1988).
[Crossref]

T. T. Wu, R. W. P. King, H. M. Shen, “Spherical lens as a launcher of electromagnetic missiles,” J. Appl. Phys. 62, 4036–4040 (1987).
[Crossref]

T. T. Wu, “Electromagnetic missiles,” J. Appl. Phys. 57, 2370–2373 (1985).
[Crossref]

T. T. Wu, H. Lehman, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
[Crossref]

T. T. Wu, R. W. P. King, “Comment on ‘Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,’” J. Appl. Phys. 56, 2587–2588 (1984).
[Crossref]

H. M. Shen, T. T. Wu, “The transverse energy pattern of an electromagnetic missile from a circular current disk,” in Microwave and Particle Beam Sources and Directed Energy Concepts, H. E. Brandt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1061, 352–360 (1989).
[Crossref]

T. T. Wu, H. M. Shen, J. M. Myers, “A review of electromagnetic missiles,” in Microwave and Particle Beam Sources and Directed Energy Concepts, H. E. Brandt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1061, 370–379 (1989).
[Crossref]

T. T. Wu, H. M. Shen, “Radiation of an electromagnetic pulse from the open end of a circular waveguide,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 329–337 (1988).
[Crossref]

Ziolkowski, R. W.

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1989).
[Crossref]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[Crossref] [PubMed]

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[Crossref] [PubMed]

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,”J. Math. Phys. 26, 861–863 (1985).
[Crossref]

R. W. Ziolkowski, “New electromagnetic directed energy pulses,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 312–319 (1988).
[Crossref]

R. W. Ziolkowski, “Localized transmission of wave energy,” in Microwave and Particle Beam Sources and Directed Energy Concepts, H. E. Brandt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1061, 395–402 (1989).
[Crossref]

IEEE Trans. Antennas Propag. (2)

T. T. Wu, R. W. P. King, H. M. Shen, “Circular cylindrical lens as a line-source electromagnetic-missile launcher,”IEEE Trans. Antennas Propag. 37, 39–44 (1989).
[Crossref]

H. E. Moses, R. T. Prosser, “Initial conditions, sources, and currents for prescribed time-dependent acoustic and electromagnetic fields in three dimensions, part I: the inverse initial value problem. Acoustic and electromagnetic bullets, expanding waves, and imploding waves,”IEEE Trans. Antennas Propag. AP-34, 188–196 (1986).
[Crossref]

J. Appl. Phys. (10)

J. M. Myers, H. M. Shen, T. T. Wu, H. E. Brandt, “Curved electromagnetic missiles,” J. Appl. Phys. 65, 2604–2610 (1989).
[Crossref]

T. T. Wu, R. W. P. King, H. M. Shen, “Generalized analysis of the spherical lens as a launcher of electromagnetic missiles,” J. Appl. Phys. 63, 5647–5653 (1988).
[Crossref]

T. T. Wu, R. W. P. King, “Comment on ‘Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,’” J. Appl. Phys. 56, 2587–2588 (1984).
[Crossref]

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[Crossref]

T. T. Wu, H. Lehman, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
[Crossref]

P. Hillion, “More on focus wave modes in Maxwell equations,” J. Appl. Phys. 60, 2981–2982 (1986).
[Crossref]

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[Crossref]

T. T. Wu, “Electromagnetic missiles,” J. Appl. Phys. 57, 2370–2373 (1985).
[Crossref]

T. T. Wu, R. W. P. King, H. M. Shen, “Spherical lens as a launcher of electromagnetic missiles,” J. Appl. Phys. 62, 4036–4040 (1987).
[Crossref]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1989).
[Crossref]

J. Math. Phys. (3)

H. E. Moses, “The time-dependent inverse source problem for the acoustic and electromagnetic equations in the one- and three-dimensional cases,”J. Math. Phys. 25, 1905–1923 (1985).
[Crossref]

P. Hillion, “Spinor focus wave modes,”J. Math. Phys. 28, 1743–1748 (1987).
[Crossref]

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,”J. Math. Phys. 26, 861–863 (1985).
[Crossref]

J. Opt. Soc. Am. (2)

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

Nuovo Cimento (1)

I. W. Kay, H. E. Moses, “The determination of the scattering potential from the spectral measure function. V. The Gelfand–Levitan equation for the three-dimensional scattering problem,” Nuovo Cimento 22, 689–705 (1961).
[Crossref]

Nuovo Cimento Suppl. (1)

G. Toraldo di Francia, “Super-gain antennas and optical resolving power,” Nuovo Cimento Suppl. 9, 426–435 (1952).
[Crossref]

Opt. Lett. (1)

Opt. Spectrosc. (USSR) (1)

Yu. Yu. Ananev, “Nondiffracting light waves,” Opt. Spectrosc. (USSR) 64, 722–723 (1988).

Phys. Rev. A (1)

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Evidence of localized wave transmission,” Phys. Rev. Lett. 62, 147–150 (1989).
[Crossref] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

D. DeBeer, S. R. Hartman, R. Friedberg, “Comment on ‘Diffraction-free beams,’” Phys. Rev. Lett. 59, 2611 (1987).
[Crossref] [PubMed]

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

H. E. Moses, R. T. Prosser, “A refinement of the radon transform and its inverse,” Proc. R. Soc. London Ser. A 422, 343–349 (1989); “Exact solutions of the three-dimensional scalar wave equation and Maxwell’s equations from the approximate solutions in the wave zone through the use of the radon transform,” Proc. A. Soc. London ser. A 422, 351–365 (1989).
[Crossref]

Radio Sci. (1)

H. M. Lee, “Rise and fall of directed transient: use of Mellin transformation in time domain problems,” Radio Sci. 22, 1102–1108 (1987).
[Crossref]

Rep. Prog. Phys. (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[Crossref]

SIAM J. Appl. Math. (1)

H. E. Moses, R. T. Prosser, “Acoustic and electromagnetic bullets. Derivation of new exact solutions of the acoustic and Maxwell’s equations,” SIAM J. Appl. Math. 50, 1325–1340 (1990).
[Crossref]

Other (14)

B. B. Godfrey, “Diffraction-free microwave propagation,” Sensor and Simulation Notes 320 (Weapons Laboratory, Kirtland Air Force Base, N.M., 1989).

R. W. Ziolkowski, “Localized transmission of wave energy,” in Microwave and Particle Beam Sources and Directed Energy Concepts, H. E. Brandt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1061, 395–402 (1989).
[Crossref]

R. W. Ziolkowski, “New electromagnetic directed energy pulses,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 312–319 (1988).
[Crossref]

H. M. Shen, T. T. Wu, “The transverse energy pattern of an electromagnetic missile from a circular current disk,” in Microwave and Particle Beam Sources and Directed Energy Concepts, H. E. Brandt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1061, 352–360 (1989).
[Crossref]

T. T. Wu, H. M. Shen, J. M. Myers, “A review of electromagnetic missiles,” in Microwave and Particle Beam Sources and Directed Energy Concepts, H. E. Brandt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1061, 370–379 (1989).
[Crossref]

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.

W. T. Welford, Optics (Oxford U. Press, New York, 1981), Chap. 3.

L. B. Felsen, E. Heyman, “Discretized beam methods for focused radiation from distributed apertures,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 320–328 (1988).
[Crossref]

T. T. Wu, H. M. Shen, “Radiation of an electromagnetic pulse from the open end of a circular waveguide,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 329–337 (1988).
[Crossref]

H. M. Shen, “Experimental study of electromagnetic missiles,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 338–346 (1988).
[Crossref]

J. M. Myers, “Pulsed radiation from a parabolic reflector,” in Microwave and Particle Beam Sources and Propagation, N. Rostoker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.873, 347–356 (1988).
[Crossref]

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), Chap. 9.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).

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Figures (7)

Fig. 1
Fig. 1

Radiation diffracted by an aperture in an opaque screen. The distance between the element of area dS′ on the aperture and the point of observation P is denoted by R.

Fig. 2
Fig. 2

Diffraction through a circular opening of radius d in an opaque screen. The point of observation P lies on the symmetry axis.

Fig. 3
Fig. 3

Intensity profile on the axis of symmetry of a circular aperture of radius d. The point of demarcation between near-field and far-field regions is ~d2/λ.

Fig. 4
Fig. 4

Bounded region A inside which there is a distribution of charge ρ(r, t) and current J(r, t). The origin of coordinate system 0 lies inside A . The distance between 0 and a point on the screen is R. The unit vector normal to the screen is denoted by n ^.

Fig. 5
Fig. 5

Backward and forward light cones of an event t, indicating the causal properties of the propagator G(x; t).

Fig. 6
Fig. 6

Simple cone consisting of two associated solid angles and the two-sheeted cone that they subtend. One solid angle lies in the positive hemisphere and the other in the negative hemisphere relative to the indicated direction of the z axis. Positively directed propagation proceeds from the negative hemisphere to the positive hemisphere.

Fig. 7
Fig. 7

Plot of |Ix|2, in expression (A8), versus distance along the axis of symmetry z. Parameters are X = 3.831 mm, λ = 6328 Å, and k = 41 mm−1: (a) penultimate lobe (nx = 49); (b) lobe that is halfway between the z axis and the edge of the aperture (nx = 25); (c) central lobe (nx = 0). The Z coordinate in (a) extends to 2 cm only.

Equations (77)

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Z R π w 0 2 / λ ,
θ Δ k / k .
w d + θ z .
Z = d / θ .
Z P 2 π d 2 / λ .
Z G 2 π w 0 2 / λ ,
ψ P ( r , t ) = ( i λ ) - 1 exp ( - i ω t ) aperture d S u ( r ) R - 1 exp ( i ω R / c ) ,
ψ P ( i λ z ) - 1 exp [ i ω ( z / c - t ) ] × aperture d S u ( x , y ) exp [ i ω c ( x 2 + y 2 2 z - x x + y y z ) ] .
z d 2 / λ ,
z < d 2 / λ ,
I 1 - cos { ω [ ( d 2 + z 2 ) 1 / 2 - z ] / c } .
( S , R ) = - d t screen d S n ^ · c ( E × B ) / 4 π ,
A ˜ ( ω ) = 2 π c f ˜ ( ω ) 0 d d r r ( r 2 + z 2 ) - 1 / 2 exp [ i ω c ( r 2 + z 2 ) 1 / 2 ] e ^ x .
U = c - 1 [ 1 + z ( z 2 + d 2 ) - 1 / 2 ] 0 d ω f ˜ ( ω ) 2 × ( 1 - cos { ω c [ ( z 2 + d 2 ) 1 / 2 - z ] } ) .
U 2 c 2 c z / d 2 d ω f ˜ ( ω ) 2 [ 1 - cos ( ω d 2 / 2 c z ) ] .
f ˜ ( ω ) [ 1 + ( ω / ω 0 ) 2 ] - ( 1 + 2 ) / 4 ,
( 2 - c - 2 2 t 2 ) ψ ( r , t ) = 0
ψ = exp [ i ( k z - ω t ) ] × θ 0 2 π + θ 0 d θ A ( θ ) exp [ i k ( x cos θ + y sin θ ) ]
( 2 - c - 2 2 / t 2 ) Ψ = 0
( 2 + 2 c 2 ξ τ - c - 2 2 / τ 2 ) Ψ = 0.
Ψ = ψ ( ξ , r , τ ) exp ( i ω ξ / c )
( 2 + 2 i ω c - 2 τ + 2 c 2 τ ξ - c - 2 2 τ 2 ) ψ = 0.
( 2 + 2 i ω c - 2 τ ) ψ 0.
ψ = C w 0 w exp [ - i tan - 1 ( τ / τ R ) - ( 1 - i τ / τ R ) r 2 / w 2 ] ,
w = w 0 [ 1 + ( τ / τ R ) 2 ] 1 / 2
τ R = ω w 0 2 / 2 c 2
ξ = z - c t ,             η = z + c t
( 2 + 4 2 η ξ ) Ψ = 0.
Ψ = ψ ( η , r ) exp ( i ω ξ / c )
( 2 + 4 i ω c η ) ψ = 0.
ψ = C w 0 w exp [ - i tan - 1 ( η / η R ) - ( 1 - i η / η R ) r 2 / w 2 ] ,
w = w 0 [ 1 + ( η / η R ) 2 ] 1 / 2 ,
η R = ω w 0 2 / c
Ψ k ( r , z , t ) = exp ( i k η ) exp ( - k r 2 / V ) 4 π i V ,
1 V = 1 A - i R ,             A = z 0 + ξ 2 / z 0 ,             R = ξ + z 0 2 / ξ ,
f ( r , z , t ) = 0 d k Ψ k ( r , z , t ) F ( k ) = 1 4 π i ( z 0 + i ξ ) 0 d k F ( k ) exp ( - k s ) .
s = - i η + r 2 z 0 + i ξ .
F ( k ) = 4 π i β H ( k - k c ) exp [ - ( k - k c ) L ] ,
f ( r , z , t ) = { β z 0 + i ξ exp [ i k c η - k c r 2 / ( z 0 + i ξ ) ] L + r 2 / ( z 0 + i ξ ) - i η } .
f ( r , z ) = β z 0 exp ( 2 i k c z - k c r 2 z 0 ) L + r 2 / z 0 - 2 i z .
Z MPS = 2 π w 0 d / λ .
( 2 - c - 2 2 / t 2 ) / f ( r , t ) = - ρ ( r , t ) ,
f + ( x , t ) = d x [ G ( x - x ; t - T ) t f + ( x , t = T ) + f + ( x , t = T ) t G ( x - x ; t - T ) ] .
f - ( x , t ) = d x [ G ( x - x ; t + T ) t f - ( x , t = - T ) + f - ( x , t = - T ) t G ( x - x ; t + T ) ] .
G ( x ; t ) = ½ sgn ( t ) H ( c 2 t 2 - x 2 ) = ½ [ H ( x + c t ) - H ( x - c t ) ] .
ρ ˜ ( k , t ) = ( 2 π ) - 1 / 2 - d x ρ ( x , t ) exp ( - i k x ) ,
f ˜ ( k , σ , t ) = f ˜ ( k , σ , - T ) exp [ - i σ c k ( t + T ) ] + i σ 2 k × exp ( - i σ c k t ) - T t d t ρ ˜ ( k , t ) exp ( i σ c k t ) ,
f ( x , t ) = ( 2 π ) - 1 / 2 σ - d k f ˜ ( k , σ , t ) exp ( i k x )
f ± ( x , t ) = ( 2 π ) - 1 / 2 σ - d k f ˜ ( k , σ , ± T ) × exp [ i k x + i σ c k ( ± T - t ) ] .
f ˜ ( k , σ , T ) = ( 8 π ) - 1 / 2 - dx × [ f + ( x , t = T ) + i σ k t f + ( x , t = T ) ] exp ( - i k x ) ,
s ( k , ω ) = ( 2 π ) - 1 / 2 - T T d t ρ ˜ ( k , t ) exp ( i ω t ) .
ρ ( x , t ) = ( 2 π ) - 1 - d k - d ω s ( k , ω ) exp ( i k x - i ω t ) .
f + ( x , t ) = sin [ k ( x - c t ) ] ,             - a < x - c t < a ,             k a = n π ,
ρ ( x , t ) = - [ k δ ( t ) cos ( k x ) - c - 1 δ ( t ) sin ( k x ) ] ,             - a < x < a .
f ( r , t ) = ( 2 π ) - 3 / 2 σ d 3 k exp ( i k · r - i σ c k t ) f ˜ ( k , σ ) ,
lim r exp ( i k · r ) = 2 π i k r sin θ [ - exp ( i k r ) δ ( θ - θ r ) δ ( ϕ - ϕ r ) + exp ( - i k r ) δ ( θ - θ r ) δ ( ϕ - ϕ r ) ] ,
f ( r , t ) = ( 8 π ) - 1 / 2 r - 1 Im 0 d k { exp [ i k ( r - c t ) ] f ˜ ( k , θ r , ϕ r ) - exp [ i k ( r + c t ) ] f ˜ ( k , θ r , ϕ r ) } .
F ( k , n ^ ) = d 3 r f ( r ) δ ( r · n ^ - k ) ,
f ( r , t ) = H ( σ - θ ) [ H ( r - a - c t ) - H ( r - b - c t ) ] / r ,             r ,
( 2 - c - 2 2 t 2 ) f = ( 2 r 2 - c - 2 2 t 2 ) g + 1 r 2 sin θ θ ( sin θ θ ) g .
( 2 - c - 2 2 t 2 ) f = h r 2 sin θ d d θ ( sin θ d d θ Y ) ,
r s 2 = d r d θ d ϕ f r 2 sin θ ( 2 r sin θ ) 2 / d r d θ d ϕ f r 2 sin θ ,
r s [ ( 8 - 9 cos σ + cos 3 σ ) / 3 ( 1 - cos σ ) ] 1 / 2 c t ,             t ,
f ( r , t ) = f a ( r , t ) - f b ( r , t ) ,
f a ( r , t ) = H ( σ - θ ) { H [ a + c t - r cos ( θ - σ ) ] - H ( a + c t - r ) } / r + ν a { H [ a + c t - r cos ( θ + σ ) ] - H [ a + c t - r cos ( θ - σ ) ] } / π r ,
ν a = cos - 1 [ ( cos σ - cos β a cos θ ) / sin β a sin θ ] ,             0 < ν a < π
β a = cos - 1 [ ( a + c t ) / r ] ,             0 < β a < π / 2 ,
ψ z - 1 0 d d r r J 0 ( k r ) exp ( i k r 2 / 2 z ) .
Z P = 2 π d 2 / λ .
ξ ± = ( k / 2 z ) 1 / 2 ( X ± k x z / k ) ,
I x = [ C ( ξ + ) + i S ( ξ + ) + C ( ξ - ) + i S ( ξ - ) ] exp ( - i k x 2 z / 2 k )
S ( t ) = ( 2 / π ) 1 / 2 0 t d t sin t 2 ,             C ( t ) = ( 2 / π ) 1 / 2 0 t d t cos t 2
z k ( X 2 , Y 2 ) / 2
z k ( X / k x , Y / k y ) .
I 1 + exp [ - 2 ( d / w 0 ) 2 ] - 2 exp [ - ( d / w 0 ) 2 ] cos ( k d 2 / 2 z ) 1 + ( z / Z R ) 2 ,
ξ ± = ( k / 2 z ) 1 / 2 [ X ± ( k x z / k - n x π / k x ) ] , η ± = ( k / 2 z ) 1 / 2 [ X ± ( k x z / k + n x π / k x ) ] ,
I x [ C ( ξ + ) + i S ( ξ + ) + C ( ξ - ) + i S ( ξ - ) + C ( η + ) + i S ( η + ) + C ( η - ) + i S ( η - ) ] ,

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