Abstract

Several new classes of localized solutions to the homogeneous scalar wave and Maxwell’s equations have been reported recently. Theoretical and experimental results have now clearly demonstrated that remarkably good approximations to these acoustic and electromagnetic localized-wave solutions can be achieved over extended near-field regions with finite-sized, independently addressable, pulse-driven arrays. We demonstrate that only the forward-propagating (causal) components of any homogeneous solution of the scalar-wave equation are actually recovered from either an infinite- or a finite-sized aperture in an open region. The backward-propagating (acausal) components result in an evanescent-wave superposition that plays no significant role in the radiation process. The exact, complete solution can be achieved only from specifying its values and its derivatives on the boundary of any closed region. By using those localized-wave solutions whose forward-propagating components have been optimized over the associated backward-propagating terms, one can recover the desirable properties of the localized-wave solutions over the extended near-field regions of a finite-sized, independently addressable, pulse-driven array. These results are illustrated with an extreme exampl—one dealing with the original solution, which is superluminal, and its finite aperture approximation, a slingshot pulse.

© 1993 Optical Society of America

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  1. R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
    [CrossRef]
  2. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
    [CrossRef] [PubMed]
  3. I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
    [CrossRef]
  4. 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]
  5. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  6. P. D. Einziger, S. Raz, “Wave solutions under complex space–time shifts,” J. Opt. Soc. Am. A 4, 3–10 (1987).
    [CrossRef]
  7. E. Heyman, L. B. Felsen, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
    [CrossRef]
  8. P. Hillion, “Spinor focus wave modes,” J. Math. Phys. 28, 1743–1748 (1987).
    [CrossRef]
  9. A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and the Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
    [CrossRef]
  10. A. M. Vengsarkar, I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “Closed-form, localized wave solutions in optical fiber waveguides,” J. Opt. Soc. Am. A 9, 937–949 (1992).
    [CrossRef]
  11. M. K. Tippett, R. W. Ziolkowski, “A bidirectional wave transformation of the cold plasma equations,” J. Math. Phys. 32, 488–492 (1991).
    [CrossRef]
  12. R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
    [CrossRef]
  13. R. Donnelly, R. W. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc.R. Soc. London Ser. A 437, 673–692 (1992).
    [CrossRef]
  14. R. W. Ziolkowski, D. K. Lewis, B. D. Cook, “Experimental verification of the localized wave transmission effect,” Phys. Rev. Lett. 62, 147–150 (1989).
    [CrossRef] [PubMed]
  15. R. W. Ziolkowski, D. K. Lewis, “Verification of the localized wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
    [CrossRef]
  16. E. Heyman, B. Z. Steinberg, L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4, 2081–2091 (1987).
    [CrossRef]
  17. E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
    [CrossRef]
  18. G. C. Sherman, A. J. Devaney, L. Mandel, “Plane-wave expansions of the optical field,” Opt. Commun. 6, 115–118 (1972).
    [CrossRef]
  19. A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
    [CrossRef]
  20. O. Yu. Zharii, “Relationship between traveling and inhomogeneous waves in the theory of transient wave processes,” Sov. Phys. Acoust. 36, 372–374 (1990) [Akust. Zh. 36, 659–664 (1990)].
  21. R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
    [CrossRef] [PubMed]
  22. R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” Trans.IEEE Antennas Propag. 40, 888–905 (1992).
    [CrossRef]
  23. P. Hillion, “Focus wave modes: remarks,” J. Opt. Soc. Am. A 8, 695 (1991).
    [CrossRef]
  24. D. S. Jones, The Theory of Electromagnetism (Pergamon, New York, 1964), pp. 38–42.
  25. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New York, 1973).
  26. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  27. J. B. Marion, Classical Electromagnetic Radiation (Academic, New York, 1965), Chap. 7.
  28. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Chap. 14.
  29. J. A. Waak, J. H. Spencer, K. J. Johnston, R. S. Simon, “Superluminal resupply of a stationary hot spot in 3C 395?” Astron. J. 90, 1989–1991 (1985).
    [CrossRef]
  30. R. M. Bevensee, BOMA Enterprises, Alamo, California (personal communications, April1991).
  31. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).
  32. R. Donnelly, Department of Electrical Engineering, Memorial University, St. Johns, Newfoundland (personal communication, December1991).
  33. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

1992 (3)

R. Donnelly, R. W. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc.R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” Trans.IEEE Antennas Propag. 40, 888–905 (1992).
[CrossRef]

A. M. Vengsarkar, I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “Closed-form, localized wave solutions in optical fiber waveguides,” J. Opt. Soc. Am. A 9, 937–949 (1992).
[CrossRef]

1991 (4)

P. Hillion, “Focus wave modes: remarks,” J. Opt. Soc. Am. A 8, 695 (1991).
[CrossRef]

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
[CrossRef] [PubMed]

M. K. Tippett, R. W. Ziolkowski, “A bidirectional wave transformation of the cold plasma equations,” J. Math. Phys. 32, 488–492 (1991).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
[CrossRef]

1990 (3)

O. Yu. Zharii, “Relationship between traveling and inhomogeneous waves in the theory of transient wave processes,” Sov. Phys. Acoust. 36, 372–374 (1990) [Akust. Zh. 36, 659–664 (1990)].

R. W. Ziolkowski, D. K. Lewis, “Verification of the localized wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and the Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

1989 (6)

E. Heyman, L. B. Felsen, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
[CrossRef]

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
[CrossRef]

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

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

I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[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]

1987 (4)

1985 (2)

J. A. Waak, J. H. Spencer, K. J. Johnston, R. S. Simon, “Superluminal resupply of a stationary hot spot in 3C 395?” Astron. J. 90, 1989–1991 (1985).
[CrossRef]

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

1973 (1)

A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

1972 (1)

G. C. Sherman, A. J. Devaney, L. Mandel, “Plane-wave expansions of the optical field,” Opt. Commun. 6, 115–118 (1972).
[CrossRef]

Besieris, I. M.

A. M. Vengsarkar, I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “Closed-form, localized wave solutions in optical fiber waveguides,” J. Opt. Soc. Am. A 9, 937–949 (1992).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and the Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[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]

Bevensee, R. M.

R. M. Bevensee, BOMA Enterprises, Alamo, California (personal communications, April1991).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Cook, B. D.

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

Devaney, A. J.

A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

G. C. Sherman, A. J. Devaney, L. Mandel, “Plane-wave expansions of the optical field,” Opt. Commun. 6, 115–118 (1972).
[CrossRef]

Donnelly, R.

R. Donnelly, R. W. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc.R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

R. Donnelly, Department of Electrical Engineering, Memorial University, St. Johns, Newfoundland (personal communication, December1991).

Durnin, J.

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

Eberly, J. H.

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

Einziger, P. D.

Felsen, L. B.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Heyman, E.

Hillion, P.

P. Hillion, “Focus wave modes: remarks,” J. Opt. Soc. Am. A 8, 695 (1991).
[CrossRef]

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Chap. 14.

Johnston, K. J.

J. A. Waak, J. H. Spencer, K. J. Johnston, R. S. Simon, “Superluminal resupply of a stationary hot spot in 3C 395?” Astron. J. 90, 1989–1991 (1985).
[CrossRef]

Jones, D. S.

D. S. Jones, The Theory of Electromagnetism (Pergamon, New York, 1964), pp. 38–42.

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).

Lewis, D. K.

R. W. Ziolkowski, D. K. Lewis, “Verification of the localized wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
[CrossRef]

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

Mandel, L.

G. C. Sherman, A. J. Devaney, L. Mandel, “Plane-wave expansions of the optical field,” Opt. Commun. 6, 115–118 (1972).
[CrossRef]

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New York, 1973).

Marion, J. B.

J. B. Marion, Classical Electromagnetic Radiation (Academic, New York, 1965), Chap. 7.

Miceli, J. J.

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

Raz, S.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Shaarawi, A. M.

A. M. Vengsarkar, I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “Closed-form, localized wave solutions in optical fiber waveguides,” J. Opt. Soc. Am. A 9, 937–949 (1992).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and the Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[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]

Sherman, G. C.

A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

G. C. Sherman, A. J. Devaney, L. Mandel, “Plane-wave expansions of the optical field,” Opt. Commun. 6, 115–118 (1972).
[CrossRef]

Simon, R. S.

J. A. Waak, J. H. Spencer, K. J. Johnston, R. S. Simon, “Superluminal resupply of a stationary hot spot in 3C 395?” Astron. J. 90, 1989–1991 (1985).
[CrossRef]

Spencer, J. H.

J. A. Waak, J. H. Spencer, K. J. Johnston, R. S. Simon, “Superluminal resupply of a stationary hot spot in 3C 395?” Astron. J. 90, 1989–1991 (1985).
[CrossRef]

Steinberg, B. Z.

Tippett, M. K.

M. K. Tippett, R. W. Ziolkowski, “A bidirectional wave transformation of the cold plasma equations,” J. Math. Phys. 32, 488–492 (1991).
[CrossRef]

Vengsarkar, A. M.

Waak, J. A.

J. A. Waak, J. H. Spencer, K. J. Johnston, R. S. Simon, “Superluminal resupply of a stationary hot spot in 3C 395?” Astron. J. 90, 1989–1991 (1985).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Zharii, O. Yu.

O. Yu. Zharii, “Relationship between traveling and inhomogeneous waves in the theory of transient wave processes,” Sov. Phys. Acoust. 36, 372–374 (1990) [Akust. Zh. 36, 659–664 (1990)].

Ziolkowski, R. W.

R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” Trans.IEEE Antennas Propag. 40, 888–905 (1992).
[CrossRef]

A. M. Vengsarkar, I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “Closed-form, localized wave solutions in optical fiber waveguides,” J. Opt. Soc. Am. A 9, 937–949 (1992).
[CrossRef]

R. Donnelly, R. W. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc.R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

M. K. Tippett, R. W. Ziolkowski, “A bidirectional wave transformation of the cold plasma equations,” J. Math. Phys. 32, 488–492 (1991).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
[CrossRef]

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
[CrossRef] [PubMed]

R. W. Ziolkowski, D. K. Lewis, “Verification of the localized wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and the Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

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

I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[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]

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

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

Astron. J. (1)

J. A. Waak, J. H. Spencer, K. J. Johnston, R. S. Simon, “Superluminal resupply of a stationary hot spot in 3C 395?” Astron. J. 90, 1989–1991 (1985).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. 37, 1604–1608 (1989).
[CrossRef]

J. Appl. Phys. (2)

R. W. Ziolkowski, D. K. Lewis, “Verification of the localized wave transmission effect,” J. Appl. Phys. 68, 6083–6086 (1990).
[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. (5)

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

M. K. Tippett, R. W. Ziolkowski, “A bidirectional wave transformation of the cold plasma equations,” J. Math. Phys. 32, 488–492 (1991).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[CrossRef]

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

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gordon and the Dirac equations,” J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

G. C. Sherman, A. J. Devaney, L. Mandel, “Plane-wave expansions of the optical field,” Opt. Commun. 6, 115–118 (1972).
[CrossRef]

Phys. Rev. A (2)

R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
[CrossRef] [PubMed]

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

Phys. Rev. Lett. (2)

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

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

Proc. IEEE (1)

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
[CrossRef]

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

R. Donnelly, R. W. Ziolkowski, “A method for constructing solutions of homogeneous partial differential equations: localized waves,” Proc.R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

SIAM Rev. (1)

A. J. Devaney, G. C. Sherman, “Plane-wave representations for scalar wave fields,” SIAM Rev. 15, 765–786 (1973).
[CrossRef]

Sov. Phys. Acoust. (1)

O. Yu. Zharii, “Relationship between traveling and inhomogeneous waves in the theory of transient wave processes,” Sov. Phys. Acoust. 36, 372–374 (1990) [Akust. Zh. 36, 659–664 (1990)].

Trans.IEEE Antennas Propag. (1)

R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” Trans.IEEE Antennas Propag. 40, 888–905 (1992).
[CrossRef]

Other (9)

D. S. Jones, The Theory of Electromagnetism (Pergamon, New York, 1964), pp. 38–42.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New York, 1973).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

J. B. Marion, Classical Electromagnetic Radiation (Academic, New York, 1965), Chap. 7.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Chap. 14.

R. M. Bevensee, BOMA Enterprises, Alamo, California (personal communications, April1991).

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986).

R. Donnelly, Department of Electrical Engineering, Memorial University, St. Johns, Newfoundland (personal communication, December1991).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

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Equations (123)

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{ Δ c t 2 } f ( r , t ) = 0 ,
g ( r , t ) = Σ d S , Ψ ( x , y , z = 0 , t R / c ) 1 4 π R ,
Ψ ( x , y , z , t R / c ) = [ z f ] + [ c t f ] ( z z ) R + [ f ] ( z z ) R 2 .
f ( r , t ) = 0 χ d χ d k z d ω A 0 ( χ , k z , ω ) J 0 ( χ ρ ) × exp [ i ( k z z ω t ) ] δ ( ω 2 [ χ 2 + k z 2 ] c 2 ) .
A 0 ( χ , k z , ω ) = 2 c C 0 [ χ , 1 2 ( ω c + k z ) , 1 2 ( ω c k z ) ] .
f ( r , t ) = 0 d χ χ J 0 ( χ ρ ) × 0 d k z { A 0 ( χ , k z , ω + ) 1 2 ω + exp [ i ( k z z ω + t ) ] + A 0 ( χ , k z , ω + ) 1 2 ω + exp [ + i ( k z z + ω + t ) ] } + 0 d χ χ J 0 ( χ ρ ) × 0 d k z { A 0 ( χ , k z , ω + ) 1 2 ω + exp [ i ( k z z + ω + t ) ] + A 0 ( χ , k z , ω + ) 1 2 ω + exp [ + i ( k z z ω + t ) ] } ,
A 0 ( χ , k z , ω ) = A 0 * ( χ , k z , ω ) .
f ( r , t ) = Re 0 d χ χ J 0 ( χ ρ ) 0 d k z A 0 [ χ , k z , ( χ 2 + k z 2 ) 1 / 2 c ] ( χ 2 + k z 2 ) 1 / 2 c × exp { i [ k z z ( χ 2 + k z 2 ) 1 / 2 ( c t ) ] } + Re 0 d χ χ J 0 ( χ ρ ) 0 d k z A 0 [ χ , k z , ( χ 2 + k z 2 ) 1 / 2 c ] ( χ 2 + k z 2 ) 1 / 2 c × exp { + i [ k z z + ( χ 2 + k z 2 ) 1 / 2 ( c t ) ] } f + ( r , t ) + f ( r , t ) ,
g = Σ [ f ] = Σ [ f + ] + Σ [ f ] f + ,
Σ [ J 0 ( χ ρ ) exp ( i ( k z z ω t ) ) ] J 0 ( χ ρ ) exp [ i ( k z z ω t ) ] ,
Σ [ J 0 ( χ ρ ) exp ( i ( k z z ω t ) ) ] 0 .
k z = { [ ( ω / c ) 2 χ 2 ] 1 / 2 for ω > χ c i [ χ 2 ( ω / c ) 2 ] 1 / 2 for | ω | < χ c [ ( ω / c ) 2 χ 2 ] 1 / 2 for ω < χ c .
f ( r , t ) = 0 d χ χ J 0 ( χ ρ ) { χ c d ω [ A 0 ( χ , [ ( ω / c ) 2 χ 2 ] 1 / 2 , ω ) | ( ω / c ) 2 χ 2 | 1 / 2 c 2 exp ( i { [ ( ω / c ) 2 χ 2 ] 1 / 2 z ω t } ) + A 0 ( χ , [ ( ω / c ) 2 χ 2 ] 1 / 2 , ω ) | ( ω / c ) 2 χ 2 | 1 / 2 c 2 exp ( + i { [ ( ω / c ) 2 χ 2 ] 1 / 2 z ω t } ) ] + 0 χ c d ω [ A 0 ( χ , i [ χ 2 ( ω / c ) 2 ] 1 / 2 , ω ) | χ 2 ( ω / c ) 2 | 1 / 2 c 2 exp ( + i ω t ) exp { [ χ 2 ( ω / c ) 2 ] 1 / 2 z } + A 0 ( χ , i [ χ 2 ( ω / c ) 2 ] 1 / 2 , ω ) | χ 2 ( ω / c ) 2 | 1 / 2 c 2 exp ( i ω t ) exp { [ χ 2 ( ω / c ) 2 ] 1 / 2 z } ] } = Re 0 d χ χ J 0 ( χ ρ ) χ c d ω A 0 ( χ , [ ( ω / c ) 2 χ 2 ] 1 / 2 , ω ) | ( ω / c ) 2 χ 2 | 1 / 2 c 2 exp ( i { [ ( ω / c ) 2 χ 2 ] 1 / 2 z ω t } ) + Re 0 d χ χ J 0 ( χ ρ ) / 0 χ c d ω A 0 ( χ , i [ ( χ 2 ω / c ) 2 ] 1 / 2 , ω ) | χ 2 ( ω / c ) 2 | 1 / 2 c 2 exp ( + i ω t ) exp { [ χ 2 ( ω / c ) 2 ] 1 / 2 z } .
f FWM ( ρ , z c t , z + c t ) = z 0 z 0 + i ( z c t ) exp [ i k ( z + c t ) ] × exp { k ρ 2 / [ z 0 + i ( z c t ) ] } .
f ̂ FWM ( ρ , z , ω ) = d t exp ( i ω t ) f FWM ( ρ , z c t , z + c t ) = { 2 π z 0 c exp [ k ( z 0 + 2 i z ) ] exp [ ω ( z 0 + i z ) / c ] J 0 ( 2 ρ { k [ ( ω / c ) k ] } 1 / 2 ) for ω > k c 0 for ω < k c .
Λ = 1 k z 0 = ( w 0 z 0 ) 2 = ( 2 π w 0 λ min ) 2 = ω max ω min .
f FWM ( ρ , z c t , z + c t ) = f + FWM ( ρ , z , t ) + f FWM ( ρ , z , t ) = 0 d χ χ J 0 ( χ ρ ) [ f ̂ + FWM ( χ , z , t ) + f ̂ FWM ( χ , z , t ) ] ,
f ̂ + FWM ( χ , z , t ) = 0 d k z π ( χ 2 + k z 2 ) 1 / 2 exp { ( z 0 / 2 ) [ ( χ 2 + k z 2 ) 1 / 2 + k z ] } × δ [ ( χ 2 + k z 2 ) 1 / 2 k z 2 k ] × exp { i [ k z z ( χ 2 + k z 2 ) 1 / 2 c t ] } = π 2 k exp [ i k ( z + c t ) ] × exp { ( χ 2 / 4 k ) [ z 0 + i ( z c t ) ] } H ( χ 2 k ) ,
f ̂ FWM ( χ , z , t ) = 0 d k z π ( χ 2 + k z 2 ) 1 / 2 exp { ( z 0 / 2 ) [ ( χ 2 + k z 2 ) 1 / 2 + k z ] } × δ [ ( χ 2 + k z 2 ) 1 / 2 + k z 2 k ] × exp { + i [ k z z + ( χ 2 + k z 2 ) 1 / 2 c t ] } = π 2 k exp [ i k ( z + c t ) ] × exp { ( χ 2 / 4 k ) [ z 0 + i ( z c t ) ] } H ( 2 k χ ) ,
f ̂ + FWM ( χ , z , t ) = χ c d ω π c [ ( ω / c ) 2 χ 2 ] 1 / 2 × exp ( ( z 0 / 2 ) { ( ω / c ) + [ ( ω / c ) 2 χ 2 ] 1 / 2 } ) × δ { ω c + [ ( ω / c ) 2 χ 2 ] 1 / 2 2 k } × exp ( i { [ ( ω / c ) 2 χ 2 ] 1 / 2 z ω t } ) = π 2 k exp [ i k ( z + c t ) ] × exp { ( χ 2 / 4 k ) [ z 0 + i ( z c t ) ] } H ( χ 2 k ) ,
f ̂ FWM ( χ , z , t ) = 0 χ c d ω π i c [ χ 2 ( ω / c ) 2 ] 1 / 2 × exp ( ( z 0 / 2 ) { ( ω / c ) i [ χ 2 ( ω / c ) 2 ] 1 / 2 } ) × δ { ω c + i [ χ 2 ( ω / c ) 2 ] 1 / 2 2 k } × exp ( + i ω t ) exp { [ χ 2 ( ω / c ) 2 ] 1 / 2 z } = π 2 k exp [ i k ( z + c t ) ] × exp { ( χ 2 / 4 k ) [ z 0 + i ( z c t ) ] } H ( 2 k χ ) .
| f + FWM ( 0 , z c t , z + c t ) | = | 0 d χ χ f ̂ + FWM ( χ , z , t ) | 2 π k z 0 2 k z 0 0 d χ χ exp [ χ 2 / ( k z 0 ) ] = π z 0 exp ( k z 0 ) ,
| f FWM ( 0 , z c t , z + c t ) | = | 0 d χ χ f ̂ FWM ( χ , z , t ) | 2 π k z 0 2 0 k z 0 d χ χ exp [ χ 2 / ( k z 0 ) ] = π z 0 [ 1 exp ( k z 0 ) ] .
| f + FWM | 2 | f FWM | 2 | exp ( k z 0 ) 1 exp ( k z 0 ) | 2 .
| f + FWM | 2 | f FWM | 2 | 1 ( 1 / Λ ) 1 / Λ | 2 Λ 2 .
υ ρ = ω k z = ( χ 2 + 4 k 2 ) c / 4 k ( χ 2 4 k 2 ) / 4 k = ( χ 2 + 4 k 2 χ 2 4 k 2 ) c .
υ g = ω k z = ω χ χ k z = χ 2 k c × 2 k χ = + c .
g ( r , t ) A d S 2 [ c t f ( x , y , z = 0 , t ) ] ( t = t R / c ) 4 π R .
g ( r , t ) ( + 2 i ω c ) 0 2 π d ϕ 0 a d ρ ρ J 0 ( χ ρ ) 4 π R × exp [ + i ω ( t R / c ) ] .
R = [ ρ 2 + ρ 2 2 ρ ρ cos ( ϕ ϕ ) + z 2 ] 1 / 2 z + ρ 2 + ρ 2 2 z ρ ρ z cos ( ϕ ϕ ) ,
g ( r , t ) ( + 2 i ω c ) exp [ + i ω ( t z / c ) ] exp [ i ( ω / c ) ( ρ 2 / 2 z ) ] 4 π z × 0 a d ρ ρ J 0 ( χ ρ ) exp [ i ( ω / c ) ( ρ 2 / 2 z ) ] × 0 2 π d ϕ exp [ + i ( ω / c ) ( ρ ρ / z ) cos ( ϕ ϕ ) ] = ( + 2 i ω c ) exp [ + i ω ( t z / c ) ] exp [ i ( ω / c ) ( ρ 2 / 2 z ) ] 2 z × 0 a d ρ ρ J 0 ( χ ρ ) J 0 ( ω c ρ z ρ ) exp [ i ( ω / c ) ( ρ 2 / 2 z ) ] .
0 exp ( i q 2 x 2 ) J 0 ( α x ) J 0 ( β x ) x d x = i 2 q 2 exp ( + i α 2 + β 2 4 q 2 ) J 0 ( α β 2 q 2 ) .
g ( r , t ) exp [ + i ω ( t z / c ) ] × exp [ + ( i / 2 ) ( c / ω ) z χ 2 ] J 0 ( χ ρ ) G ( r , t ) ,
G ( r , t ) = ( + 2 i ω c ) exp [ + i ω ( t z / c ) ] exp [ i ( ω / c ) ( ρ 2 / 2 z ) ] 2 z × 0 d ρ ρ J 0 ( χ ρ ) J 0 ( ω c ρ z ρ ) exp [ i ( ω / c ) ( ρ 2 / 2 z ) ] .
ω c ω 0 a z 2
z < ω c a 2 2 = π a 2 λ .
z ω c ω 0 a 2 π ω 0 a λ
g ( r , t ) exp [ + i ω ( t z / c ) ] exp [ + ( i / 2 ) ( c / ω ) z χ 2 ] J 0 ( χ ρ ) ,
ω c = ( χ 2 + k z 2 ) 1 / 2
k z = [ ( ω / c ) 2 χ 2 ] 1 / 2 ,
k z ω c [ 1 1 2 χ 2 ( ω / c ) 2 ] = ω c 1 2 c ω χ 2 .
k k z ( 1 + 1 2 χ 2 k 2 ) k z + 1 2 χ 2 k z
g ( r , t ) J 0 ( χ ρ ) exp [ i ( k z z ω t ) ] f ( r , t ) ;
Δ f ( r , t ) + 4 ξ η f ( r , t ) = 0 ,
f k ( r , t ) = exp ( i k η ) F k ( r , ξ ) ,
4 i k ξ F k + Δ F k = 0 .
f k ( ρ , ξ , η ) = 2 i k ξ exp ( i k η ) exp ( i k ρ 2 / ξ ) × 0 d r r exp ( i k r 2 / ξ ) J 0 ( 2 k ξ ρ r ) F k ( r , 0 ) = 2 i k η exp ( i k ξ ) exp ( i k ρ 2 ) / ξ × 0 d r r exp ( i k r 2 / ξ ) J 0 ( 2 k ξ ρ r ) F k ( r , 0 , 2 c t ) .
F k ( r , 0 ) = exp ( r 2 / w 0 2 ) .
f k ( ρ , c t , + c t ) = 2 i k c t exp [ i k ( c t ) ] exp ( i k ρ 2 / c t ) × 0 d r r exp ( i k r 2 / c t ) J 0 ( 2 k c t ρ r ) F k ( r , 0 ) .
F k ( r , 0 ) = i k c t exp [ i k ( c t ) ] exp ( + i k r 2 / c t ) × 0 d ρ ρ exp ( + i k ρ 2 / c t ) J 0 ( 2 k c t r ρ ) f k ( ρ , c t , + c t ) .
f k ( ρ , ξ , η ) = i k z exp ( i k ρ 2 / z ) exp ( i k z ) 0 d r r exp ( i k r 2 / z ) × J 0 ( 2 k z ρ r ) f k ( r , c t , + c t ) .
f k FWM ( r , c t , + c t ) = z 0 z 0 i c t exp ( + ik c t ) exp [ k r 2 / ( z 0 ict ) ] .
ϕ ( r , t ) = q { [ 1 ( υ / c m ) 2 ] ( x 2 + y 2 ) + ( z υ t ) 2 } 1 / 2 ,
A ( r , t ) = v c m ϕ ( r , t ) ,
E ( r , t ) = q [ 1 ( υ / c m ) 2 ] [ x x ̂ + y y ̂ + ( z υ t ) z ̂ ] { [ 1 υ / c m ) 2 ] ( x 2 + y 2 ) + ( z υ t ) 2 } 3 / 2 ,
H ( r , t ) = v c m × E ( r , t ) .
ϕ ( r , t ) = q { [ 1 ( υ / c m ) 2 ] ( x 2 + y 2 ) + [ ( z υ t ) i z 0 ] 2 } 1 / 2 ,
A ( r , t ) = v c m ϕ ( r , t ) ,
E ( r , t ) = q [ 1 ( υ / c m ) 2 ] { x x ̂ + y y ̂ + [ ( z υ t ) i z 0 ] z ̂ } { [ 1 ( υ / c m ) 2 ] ( x 2 + y 2 ) + [ ( z υ t ) i z 0 ] 2 } 3 / 2 ,
H ( r , t ) = v c m × E ( r , t ) .
[ 2 ( 1 / c m ) 2 t 2 ] f ( ρ , θ , z υ t ) = ρ 2 f + 1 ρ ρ f + 1 ρ 2 θ 2 f + z 2 f ( 1 c m ) 2 t 2 f = ρ 2 f + 1 ρ ρ f + 1 ρ 2 θ 2 f + γ 2 ζ 2 f = 0 ,
ζ = z υ t
γ 2 = 1 ( υ / c m ) 2 .
Δ f = 0 ,
Δ f τ 2 f = 0 .
η 2 Φ + 1 η η Φ + 1 η 2 θ 2 Φ + ζ 2 Φ = 0 .
Φ n m ( η , θ , ζ ) = h n m ( η , ζ ) ( η 2 + ξ 2 ) m / 2 { cos n θ sin n θ ,
η 2 ( η 2 h n m + ζ 2 h n m ) = n 2 h n m η η h n m ,
η η h n m + ζ ζ h n m = ( m 1 2 ) h n m .
Φ 00 ( η , θ , ζ ) = h 00 ( η 2 + ξ 2 ) 1 / 2 .
Φ 00 ( η , θ , ζ ) = i z 0 { ( κ ρ ) 2 + [ ( z υ t ) + i z 0 ] 2 } 1 / 2 = z 0 { ( κ ρ ) 2 + [ z 0 i ( z υ t ) ] 2 } 1 / 2 .
d z 0 2 π d θ 0 θ d ρ ρ | Φ 00 | 2 0 d ρ ρ { [ γ 2 ρ 2 + ( z υ t ) 2 z 0 2 ] 2 + 4 z 0 2 ( z υ t ) 2 } 1 / 2 = γ 2 2 ln { ξ + [ ξ 2 + 4 z 0 2 ( z υ t ) 2 ] 1 / 2 } ξ = ( z υ t ) 2 z 0 2 ξ = = .
C ( χ , α , β ) = 2 i π 2 z 0 1 + υ / c m exp { [ 2 z 0 / ( 1 + υ / c m ) ] α } χ 2 4 α β × δ [ β + α ( 1 υ / c m 1 + υ / c m ) ] ,
f ( ρ , θ , z , t ) = 0 d χ χ d α d β C ( χ , α , β ) J 0 ( χ ρ ) × exp ( i α ζ ) exp ( i β η ) δ ( α β χ 2 / 4 ) ,
f ( ρ , θ , z , t ) = 2 i π 2 z 0 1 + υ / c m 0 d χ χ d α exp ( i α ζ ) exp { i α [ ( 1 υ / c m ) / ( 1 + υ / c m ) ] η } exp { α [ 2 z 0 / ( 1 + υ / c m ) ] } χ 2 + 4 α 2 [ ( 1 υ / c m ) / ( 1 + υ / c m ) ] J 0 ( χ ρ ) = 4 i π 2 z 0 1 + υ / c m 0 d α cos { α [ ζ + η ( 1 υ / c m 1 + υ / c m ) i 2 z 0 1 + υ / c m ] } 0 d χ χ J 0 ( χ ρ ) χ 2 + 4 α 2 [ ( 1 υ / c m ) / ( 1 + υ / c m ) ] = 4 i π 2 z 0 1 + υ / c m 0 d α K 0 [ 2 α ( 1 υ / c m 1 + υ / c m ) 1 / 2 ρ ] cos { α [ ζ + η ( 1 υ / c m 1 + υ / c m ) i 2 z 0 1 + υ / c m ] } = i 2 z 0 1 + υ / c m { 4 ( 1 υ / c m 1 + υ / c m ) ρ 2 + [ ζ + η ( 1 υ / c m 1 + υ / c m ) i 2 z 0 1 + υ / c m ] 2 } 1 / 2 = i z 0 { γ 2 ρ 2 + [ ζ ( 1 + υ / c m ) / 2 + η ( 1 υ / c m ) / 2 i z 0 ] 2 } 1 / 2 = i z 0 [ γ 2 ρ 2 + ( z υ t i z 0 ) 2 ] 1 / 2 Φ 00 ( η , θ , ζ ) ,
C ( χ , α , β ) = 4 π 1 1 + υ / c m q χ 2 4 α β δ [ β + α ( 1 υ / c m 1 + υ / c m ) ] .
g ( r , t ) = Σ d S Ψ ( x , y , z = 0 , t R / c ) 4 π R + R d S Ψ ¯ ( x , y , z , t R / c ) 4 π R ,
Ψ ¯ ( x , y , z , t R / c ) = n ̂ · { [ f ] [ c t f ] R ̂ R [ f ] R ̂ R 2 } ,
R d S Ψ ¯ ( x , y , z , t R / c ) 4 π R = lim R d Ω R 2 Ψ ¯ ( x , y , z , t R / c ) 4 π R .
lim R ( n ̂ · R ̂ ) [ f ] d Ω 0 .
lim R { R [ ( n ̂ · ) f ( n ̂ · R ̂ ) c t f ] } 0 .
f ( ρ , z , t ) = J 0 ( χ ρ ) exp [ ± i ( k z z ω t ) ] ,
f + ( ρ , z , t ) = J 0 ( χ ρ ) exp [ + i ( k z z ω t ) ] ,
Σ [ f + ] = 1 4 π 0 2 π d ϕ 0 d ρ ρ 1 R × { [ z f + ] + [ c t f + ] z R + [ f + ] z R 2 } = exp ( i ω t ) 4 π 0 2 π d ϕ 0 d ρ ρ { i k z i ω c z R + z R 2 } × exp [ i ( ω / c ) R ] R J 0 ( χ ρ ) .
z { exp [ i ( ω / c ) R ] R } = ( i ω c z R 2 z R 3 ) exp [ i ( ω / c ) R ] ,
Σ [ f + ] = exp ( i ω t ) 4 π ( i k z Z + z Z + ) ,
Z ± = 0 2 π d ϕ 0 d ρ ρ exp [ ± i ( ω / c ) R ] R J 0 ( χ ρ ) .
R 2 = ρ 2 + ρ 2 2 ρ ρ cos ϕ + z 2 ,
exp ( ± i k R ) R = ± i 0 λ d λ ( k 2 λ 2 ) 1 / 2 exp [ ± i ( k 2 λ 2 ) 1 / 2 | z | ] × J 0 [ ρ 2 + ρ 2 2 ρ ρ cos ϕ ) 1 / 2 ] ,
Z ± = ± i 0 λ d λ [ ( ω / c ) 2 λ 2 ] 1 / 2 exp { ± i [ ( ω / c ) 2 λ 2 ] 1 / 2 | z | } × 0 d ρ ρ J 0 ( χ ρ ) × 0 2 π d ϕ J 0 [ λ ( ρ 2 + ρ 2 2 ρ ρ cos ϕ ) 1 / 2 ] .
J 0 [ λ ( ρ 2 + ρ 2 2 ρ ρ cos ϕ ) 1 / 2 ] = l = J l ( λ ρ ) J l ( λ ρ ) exp ( i l ϕ ) ,
0 2 π d ϕ J 0 [ λ ( ρ 2 + ρ 2 2 ρ ρ cos ϕ ) 1 / 2 ] = l = J l ( λ ρ ) J l ( λ ρ ) 0 2 π d ϕ exp ( i l ϕ ) = 2 π J 0 ( λ ρ ) J 0 ( λ ρ ) ,
Z ± = ± 2 π i 0 λ d λ [ ( ω / c ) 2 λ 2 ] 1 / 2 exp { ± i [ ( ω / c ) 2 λ 2 ] 1 / 2 | z | } × J 0 ( χ ρ ) 0 d ρ ρ J 0 ( χ ρ ) J 0 ( λ ρ ) .
0 d ρ ρ J 0 ( λ ρ ) J 0 ( χ ρ ) = δ ( λ χ ) λ ,
Z ± = ± 2 π i exp ( ± i k z | z | ) k z J 0 ( χ ρ ) .
Z ± ( ρ , z ) = J 0 ( χ ρ ) * exp [ ± i ( ω / c ) ( ρ 2 + z 2 ) 1 / 2 ] ( ρ 2 + z 2 ) 1 / 2 .
x , y { Z ± } ( k x , k y ) = x , y { J 0 ( χ ρ ) } ( κ ) × x , y { exp [ ± i ( ω / c ) ( ρ 2 + z 2 ) 1 / 2 ] ( ρ 2 + z 2 ) 1 / 2 } ( κ ) ,
x , y { J 0 ( χ ρ ) } ( κ ) = 2 π δ ( κ χ ) κ ,
Z ± ( ρ , z ) = J 0 ( χ ρ ) × x , y { exp [ ± i ( ω / c ) ( ρ 2 + z 2 ) 1 / 2 ] ( ρ 2 + z 2 ) 1 / 2 } ( χ ) .
x , y { exp [ ± i ( ω / c ) ( ρ 2 + z 2 ) 1 / 2 ] ( ρ 2 + z 2 ) 1 / 2 } = ± 2 π i exp { ± i | z | [ ( ω / c ) 2 χ 2 ] 1 / 2 } [ ( ω / c ) 2 χ 2 ] 1 / 2 ± 2 π i exp ( ± i k z | z | ) k z ,
z Z ± = 2 π sgn ( z ) exp ( ± i k z | z | ) 1 / 2 J 0 ( χ ρ ) .
Σ [ f + ] = exp ( i ω t ) 4 π [ i k z 2 π i k z ( 2 π ) ] exp ( i k z z ) J 0 ( χ ρ ) = exp [ i ( k z z ω t ) J 0 ( χ ρ ) f + .
f ( ρ , z , t ) = J 0 ( χ ρ ) exp [ + i ( k z z + ω t ) ] ,
Σ [ f ] = exp ( i ω t ) 4 π 0 2 π d ϕ 0 d ρ ρ × { i k z + i ω c z R + z R 2 } exp [ i ( ω / c ) R ] R J 0 ( χ ρ ) , = exp ( + i ω t ) 4 π [ i k z Z z Z ] , = exp ( + i ω t ) 4 π [ i k z 2 π i k z ( 2 π ) ] exp ( i k z z ) J 0 ( χ ρ ) , = 1 4 π [ 2 π ( 2 π ) ] exp [ ( i ( k z z + ω t ) ] J 0 ( χ ρ ) 0.
f ± ( ρ , z , t ) = J 0 ( χ ρ ) exp [ i ( k z z ω t ) ] ,
Σ [ f + ] f +
Σ [ f ] 0 .
Σ { exp [ ± i ( k z z ω t ) ] } exp [ ± i ( k z z ω t ) ]
Σ { exp [ ± i ( k z z + ω t ) ] } 0 .
Σ { J 0 ( χ ρ ) exp [ i ( k z z + ω t ) ] } J 0 ( χ ρ ) exp [ i ( k z z + ω t ) ]
Σ { J 0 ( χ ρ ) exp [ i ( k z z ω t ) ] } 0 .
f ̂ FWM ( ρ , z , ω ) = d t exp ( i ω t ) f FWM ( ρ , z c t , z + c t ) .
f ̂ FWM ( ρ , z , ω ) = z 0 exp ( i k z ) n = 0 ( k ρ 2 ) n n ! d t × exp [ i ( ω k c ) t ] 1 [ ( z 0 + i z ) i c t ) ] n + 1 .
d x exp ( i p x ) ( ξ i x ) n + 1 = H ( p ) 2 π p n exp ( ξ p ) n ! ,
J 0 ( 2 x ) = n = 0 ( 1 ) n x 2 n n ! n ! ,
f ̂ FWM ( ρ , z , ω ) = 2 π z 0 c exp ( i k z ) H ( ω k c ) × exp { [ ( z 0 + i z ) / c ] ( ω k c ) } × n = 0 ( k ρ 2 ) n n ! ( ω k c ) n c n n ! = 2 π z 0 c H ( ω k c ) exp [ k ( z 0 + 2 i z ) ] × exp [ ω ( z 0 + i z ) / c ] × J 0 { 2 ρ [ k ( ω c k ) ] 1 / 2 } .
d t exp ( i ω t ) f FWM * ( ρ , z c t , z + c t ) = 2 π z 0 c H [ ( ω + k c ) ] exp [ k ( z 0 + 2 i z ) ] × exp [ + ω ( z 0 + i z ) / c ] J 0 { 2 ρ [ k ( ω c + k ) ] 1 / 2 } f ̂ FWM ( ρ , z , ω ) .
4 i k ξ F k + Δ F k = 0 ,
F k ( ρ , ξ ) = d k ξ exp ( i 2 π k ξ ξ ) × R 2 d κ exp ( i 2 π κ · r ) Λ ( | κ | , k ξ ) δ ( 2 k ξ + π | κ | 2 / k ) = 2 π 0 d κ κ J 0 ( 2 π κ ρ ) Λ ( κ , π κ 2 / k ) exp ( i π 2 ξ κ 2 / k ) .
G k ( ρ , ξ ) = i k π exp ( i k ρ 2 / ξ ) ξ
G k ( ρ , ξ ) = i k π exp ( i k ρ 2 / | ξ | ) | ξ | ,
G k ( ρ , ξ ) = i k π exp ( i k ρ 2 / ξ ) ξ .
G k ( ρ , ξ = 0 ) = δ ( ρ ) 2 π ρ .

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