Abstract

Subluminal, luminal and superluminal localized wave solutions to the paraxial pulsed beam equation in free space are determined. A clarification is also made to recent work on pulsed beams of arbitrary speed which are solutions of a narrowband temporal spectrum version of the forward pulsed beam equation.

© 2004 Optical Society of America

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References

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  1. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
    [Crossref] [PubMed]
  2. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Localized wave representations of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
    [Crossref]
  3. I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Progr. Electromagn. Res. (PIER) 19, 1–48 (1998).
    [Crossref]
  4. E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell’s equations,” Physica A 252586–610 (1998).
    [Crossref]
  5. J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
    [Crossref]
  6. P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 65, 036612 1–12 (2004).
  7. S. Longhi, “Spatial-temporal Gauss-Laguerre waves in dispersive media,” Phys. Rev. E 68, 066612 1–6 (2003).
    [Crossref]
  8. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 1–4 (2003).
    [Crossref]
  9. R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
    [Crossref]
  10. E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
    [Crossref]
  11. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
    [Crossref]
  12. M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. soc. Am. B 16, 1468–1474 (1999).
    [Crossref]
  13. A. Erdelyi, Tables of Integral Transforms (Academic Press, New York, 1980), Vol. I.
  14. S. M. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E 59, 4630–4649 (1999).
    [Crossref]
  15. P. Saari, “Evolution of subcycle pulses in nonparaxial Gaussian beams,” Opt. Express,  8, 590–598 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-11-590.
    [Crossref] [PubMed]
  16. J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to the free-space wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
    [Crossref] [PubMed]
  17. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
    [Crossref]
  18. A. Wunsche, “Embedding of focus wave modes into a wider class of approximate wave functions,” J. Opt. Soc. Am. A 6, 1661–1668 (1989).
    [Crossref]
  19. I. M. Besieris, M. Abdel-Rahman, and A. M. Shaarawi, “Symplectic (nonseparable) spectra and novel, slowly decaying beam solutions to the complex parabolic equation,” URSI Digest, p. 281 (abstract), IEEE AP-S Intern. Symp. and URSI Natl. Meeting, Baltimore, MD, July 21–26 (1996).
  20. S. Longhi, “Gaussian pulsed beams with arbitrary speeds,” Opt. Express,  12, 935–940 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-935.
    [Crossref] [PubMed]
  21. P. A. Be′langer, “Lorentz transformations of packet-like solutions of the homogeneous wave equation,”J. Opt. Soc. Am. A 3, 541–542 (1986).
    [Crossref]

2004 (2)

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 65, 036612 1–12 (2004).

S. Longhi, “Gaussian pulsed beams with arbitrary speeds,” Opt. Express,  12, 935–940 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-935.
[Crossref] [PubMed]

2003 (3)

S. Longhi, “Spatial-temporal Gauss-Laguerre waves in dispersive media,” Phys. Rev. E 68, 066612 1–6 (2003).
[Crossref]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 1–4 (2003).
[Crossref]

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
[Crossref]

2001 (1)

2000 (1)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[Crossref]

1999 (2)

M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. soc. Am. B 16, 1468–1474 (1999).
[Crossref]

S. M. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E 59, 4630–4649 (1999).
[Crossref]

1998 (3)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[Crossref]

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Progr. Electromagn. Res. (PIER) 19, 1–48 (1998).
[Crossref]

E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell’s equations,” Physica A 252586–610 (1998).
[Crossref]

1994 (1)

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[Crossref]

1993 (1)

1992 (1)

J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to the free-space wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

1991 (1)

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

1989 (2)

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

A. Wunsche, “Embedding of focus wave modes into a wider class of approximate wave functions,” J. Opt. Soc. Am. A 6, 1661–1668 (1989).
[Crossref]

1986 (1)

Abdel-Rahman, M.

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Progr. Electromagn. Res. (PIER) 19, 1–48 (1998).
[Crossref]

I. M. Besieris, M. Abdel-Rahman, and A. M. Shaarawi, “Symplectic (nonseparable) spectra and novel, slowly decaying beam solutions to the complex parabolic equation,” URSI Digest, p. 281 (abstract), IEEE AP-S Intern. Symp. and URSI Natl. Meeting, Baltimore, MD, July 21–26 (1996).

Be'langer, P. A.

Besieris, I. M.

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Progr. Electromagn. Res. (PIER) 19, 1–48 (1998).
[Crossref]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[Crossref]

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

I. M. Besieris, M. Abdel-Rahman, and A. M. Shaarawi, “Symplectic (nonseparable) spectra and novel, slowly decaying beam solutions to the complex parabolic equation,” URSI Digest, p. 281 (abstract), IEEE AP-S Intern. Symp. and URSI Natl. Meeting, Baltimore, MD, July 21–26 (1996).

Chatzipetros, A.

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Progr. Electromagn. Res. (PIER) 19, 1–48 (1998).
[Crossref]

Conti, C.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 1–4 (2003).
[Crossref]

Di Trapani, P.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 1–4 (2003).
[Crossref]

Erdelyi, A.

A. Erdelyi, Tables of Integral Transforms (Academic Press, New York, 1980), Vol. I.

Fagerholm, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[Crossref]

Feng, S. M.

S. M. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E 59, 4630–4649 (1999).
[Crossref]

Fortin, M.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
[Crossref]

Friberg, A. T.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[Crossref]

Greenleaf, J. F.

J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to the free-space wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

Griebner, U.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
[Crossref]

Grunwald, R.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
[Crossref]

Hellwarth, R. W.

S. M. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E 59, 4630–4649 (1999).
[Crossref]

Heyman, E.

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[Crossref]

Jedrkiewicz, O.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 1–4 (2003).
[Crossref]

Kebbel, V.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
[Crossref]

Kummrow, A.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
[Crossref]

Longhi, S.

Lu, J. Y.

J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to the free-space wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

Neumann, U.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
[Crossref]

Nibbering, E. T. J.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
[Crossref]

Piche, M.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
[Crossref]

Piskarskas, A.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 1–4 (2003).
[Crossref]

Porras, M. A.

M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. soc. Am. B 16, 1468–1474 (1999).
[Crossref]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[Crossref]

Recami, E.

E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell’s equations,” Physica A 252586–610 (1998).
[Crossref]

Reivelt, K.

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 65, 036612 1–12 (2004).

Rini, M.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
[Crossref]

Rousseau, G.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
[Crossref]

Saari, P.

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 65, 036612 1–12 (2004).

P. Saari, “Evolution of subcycle pulses in nonparaxial Gaussian beams,” Opt. Express,  8, 590–598 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-11-590.
[Crossref] [PubMed]

Salo, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[Crossref]

Salomaa, M. M.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[Crossref]

Shaarawi, A. M.

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Progr. Electromagn. Res. (PIER) 19, 1–48 (1998).
[Crossref]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[Crossref]

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

I. M. Besieris, M. Abdel-Rahman, and A. M. Shaarawi, “Symplectic (nonseparable) spectra and novel, slowly decaying beam solutions to the complex parabolic equation,” URSI Digest, p. 281 (abstract), IEEE AP-S Intern. Symp. and URSI Natl. Meeting, Baltimore, MD, July 21–26 (1996).

Trillo, S.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 1–4 (2003).
[Crossref]

Trull, J.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 1–4 (2003).
[Crossref]

Valiulis, G.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 1–4 (2003).
[Crossref]

Winful, H. G.

S. M. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E 59, 4630–4649 (1999).
[Crossref]

Wunsche, A.

Ziolkowski, R. W.

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[Crossref]

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

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

IEEE Trans. Antennas Propag. (1)

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to the free-space wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

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

J. Opt. soc. Am. B (1)

Opt. Express (2)

Phys. Rev. A (2)

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

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piche, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wave packets,” Phys. Rev. A 67, 063820 1–5 (2003).
[Crossref]

Phys. Rev. E (5)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Unified description of nondiffracting X and Y waves,” Phys. Rev. E 62, 4261–4275 (2000).
[Crossref]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 65, 036612 1–12 (2004).

S. Longhi, “Spatial-temporal Gauss-Laguerre waves in dispersive media,” Phys. Rev. E 68, 066612 1–6 (2003).
[Crossref]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[Crossref]

S. M. Feng, H. G. Winful, and R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E 59, 4630–4649 (1999).
[Crossref]

Phys. Rev. Lett. (1)

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90, 170406 1–4 (2003).
[Crossref]

Physica A (1)

E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell’s equations,” Physica A 252586–610 (1998).
[Crossref]

Proc. IEEE (1)

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

Progr. Electromagn. Res. (PIER) (1)

I. M. Besieris, M. Abdel-Rahman, A. M. Shaarawi, and A. Chatzipetros, “Two fundamental representations of localized pulse solutions to the scalar wave equation,” Progr. Electromagn. Res. (PIER) 19, 1–48 (1998).
[Crossref]

Other (2)

A. Erdelyi, Tables of Integral Transforms (Academic Press, New York, 1980), Vol. I.

I. M. Besieris, M. Abdel-Rahman, and A. M. Shaarawi, “Symplectic (nonseparable) spectra and novel, slowly decaying beam solutions to the complex parabolic equation,” URSI Digest, p. 281 (abstract), IEEE AP-S Intern. Symp. and URSI Natl. Meeting, Baltimore, MD, July 21–26 (1996).

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Equations (96)

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( Δ ρ 2 + 2 z 2 + ω 2 c 2 ) u ̂ ( r , ω ) = 0 ; ρ = ( x , y ) ,
i z ν ̂ ± ( r , ω ) = ± c 2 ω 2 ν ̂ ± ( r , ω ) ,
u ± ( ρ , τ ± , z ) = R 1 d ω R 2 d κ exp [ i ( ω τ ± κ · ρ ) exp [ ± i ( c κ 2 z ) / ( 2 ω ) ] u ˜ 0 ( κ , ω ) ,
( ρ 2 2 c 2 τ ± z ) u ± ( ρ , τ ± , z ) = 0 .
u + ( ρ , τ + , z ; α ) = exp ( i 2 α c τ + ) ν + ( ρ , z ; α ) .
i 4 α z v + ( ρ , z ; α ) = ρ 2 v + ( ρ , z ; α ) ,
ν n ( ρ , z ; α ) = A 0 a 1 ( a 1 + iz ) n + 1 exp ( α ρ 2 a 1 + iz ) L n ( 0 ) ( α ρ 2 a 1 + iz ) ; n = 0 , 1 , 2 ,
u + ( ρ , z , τ + ) = 0 d α exp ( 2 i α c τ + ) v n ( ρ , z ; α ) F ˜ ( α ) .
u + ( ρ , z , τ + ) = A 0 a 1 ( a 1 + iz ) n + 1 Γ ( n + 1 ) n ! [ 2 c ( a 2 + i τ + ) ] n [ 2 c ( a 2 + i τ + ) + ρ 2 / ( a 1 + iz ) ] n + 1 .
t ( z ± 1 c t ) u ± ( ρ , z , t ) = ± c 2 ρ 2 u ± ( ρ , z , t ) .
2 ( v c 1 ) 2 ζ + η + u + ( ρ , ζ + , η + ) 2 v c ( v c 1 ) 2 η + u + ( ρ , ζ + , η + ) = ρ 2 u + ( ρ , ζ + , η + ) .
u + ( e ) ( ρ , ζ + , η + ; α , β , κ ) = exp ( i κ · ρ ) exp ( i α ζ + ) exp ( i β η + ) ,
κ 2 = 2 v c ( v c 1 ) β 2 + 2 ( v c 1 ) α β .
u + ( ρ , z , t ) = 0 d α 0 d β R 2 d κ u + ( e ) ( ρ , ζ + , η + ; α , β , κ )
× δ [ κ 2 2 v c ( v c 1 ) β 2 2 ( v c 1 ) α β ] u ˜ 0 ( α , β , κ ) ,
u + ( ρ , z , t ) = 0 d α 0 d β exp [ i ( α ζ + + β η + ) ] J 0 [ ρ 2 v c ( v c 1 ) β 2 + c v α β ] u ˜ 1 ( α , β ) ,
u + ( ρ , z , t ) = 0 d α v + ( ρ , z , t ; α ) F ˜ ( α ) ,
v + ( ρ , z , t , α ) = exp [ i α ( 1 c 2 v ) ( z v ph t ) ] [ 2 ( v c ) ( v c 1 ) ρ 2 + ( a 1 + i ( z vt ) ) 2 ] 1 / 2
× exp [ c α 2 v 2 ( v c ) ( v c 1 ) ρ 2 + ( α 1 + i ( z vt ) ) 2 ] ,
u + ( ρ , z , t ) = { 2 ( v / c ) [ ( v / c ) 1 ] ρ 2 + ( a 1 + i ( z vt ) ) 2 } 1 / 2 .
F ˜ ( α ) = { 0 , b > α > 0 , 1 Γ ( q ) ( α b ) q 1 exp [ a 2 ( α b ) ] , α b ; b , q 0 ,
u + ( ρ , z , t ) = [ 2 ( v / c ) [ ( v / c ) 1 ] ρ 2 + ( a 1 + i ( z vt ) ) 2 ] 1 / 2 exp ( ib λ ) [ c / ( 2 v ) 2 ( v / c ) [ ( v / c ) 1 ] ρ 2 + ( a 1 + i ( z vt ) ) 2 + ( a 2 i λ ) ] q
× exp { ( bc ) / ( 2 v ) 2 ( v / c ) [ ( v / c ) 1 ] ρ 2 + ( a 1 + i ( z vt ) ) 2 } ,
κ 2 = 2 v c ( 1 v c ) [ ( α c 2 v ) 2 β ¯ 2 ] ; β ¯ β + α c 2 v .
u + ( ρ , ζ + , η + ) = 0 d α ( α c ) / ( 2 v ) d β ¯ J 0 { ρ [ 2 v c ( 1 v c ) ] 1 / 2 [ ( α c 2 v ) 2 β ¯ 2 ] 1 / 2 }
× exp ( i α ζ + ) exp ( i β ¯ η + ) exp [ i η + ( α c ) / ( 2 v ) ] u ˜ 2 ( α , β ¯ ) .
u + ( ρ , z , t ) = 0 d α sin [ ( α c ) / ( 2 v ) 2 ( v / c ) [ 1 ( v / c ) ] ρ 2 + ( z vt ) 2 ] 2 ( v / c ) [ 1 ( v / c ) ] ρ 2 + ( z vt ) 2
× exp { i α [ 1 c / ( 2 v ) ] ( z v ph t ) } F ˜ ( α ) ,
2 ( 1 + v c ) 2 ζ η + u ( ρ , ζ , η + ) 2 v c ( 1 + v c ) 2 η + 2 u ( ρ , ζ , η + ) = ρ 2 u ( ρ , ζ , η + ) .
u ( e ) ( ρ , ζ , η + ; α , β , κ ) = exp ( i κ · ρ ) exp ( i α ζ ) exp ( i β η + ) ,
κ 2 = 2 v c ( 1 + v c ) β 2 + 2 ( 1 + v c ) α β ,
u ( ρ , z , t ) = 0 d α 0 d β R 2 d κ u ( e ) ( ρ , ζ , η + ; , α , β , κ )
× δ [ κ 2 2 v c ( v c 1 ) β 2 2 ( v c 1 ) α β ] u ˜ 0 ( α , β , κ ) ,
u ( ρ , z , t ) = 0 d α 0 d β exp [ i ( α ζ β η + ) ] J 0 [ ρ 2 v c ( 1 + v c ) β 2 + c v α β ] u ˜ 1 ( α , β ) .
u ( ρ , z , t ) = 0 d α v ( ρ , z , t ; α ) F ˜ ( α ) ,
v ( ρ , z , t ; α ) = exp [ i α ( 1 + c 2 v ) ( z + v ph t ) ] [ 2 v c ( 1 + v c ) ρ 2 + ( a 1 + i ( z vt ) ) 2 ] 1 / 2
× exp { ( c α ) / ( 2 v ) 2 ( v / c ) [ ( 1 + v / c ) ] ρ 2 + ( a 1 + i ( z vt ) ) 2 } ,
u ( ρ , z , t ) = exp { i ( 3 / 2 ) α 0 [ z + ( c / 3 ) t ] } [ 4 ρ 2 + ( a 1 + i ( z ct ) ) 2 ] 1 / 2
× exp { α 0 / ( 2 c ) 4 ρ 2 + ( a 1 + i ( z ct ) ) 2 } ,
u ( ρ , z , t ) = [ 4 ρ 2 + ( a 1 + i ( z ct ) ) 2 ] 1 / 2
β ( κ , ω ) β ( κ , ω 0 ) = ( c κ 2 ) / ( 2 ω 0 ) .
ψ ± ( ρ , z , t ) exp ( i ω 0 τ ± ) R 1 d Ω R 2 d κ exp ( i Ω τ ± ) exp ( i κ · ρ )
× exp [ ± i ( c κ 2 z ) / ( 2 ω 0 ) ] u ˜ 1 ( κ , Ω ) ,
ψ ± ( ρ , z , t ) = exp ( i ω 0 τ ± ) ϕ ± ( ρ , z , t ) ,
i ( z ± 1 c t ) ϕ ± ( ρ , z , t ) = ± 1 2 k 0 ρ 2 ϕ ± ( ρ , z , t ) ; k 0 ω 0 / c .
ϕ + ( ρ , z , t ) = f ( τ + ) Φ ( ρ , η + ) ,
i ( 1 v c ) η + Φ ( ρ , η + ) = 1 2 k 0 ρ 2 Φ ( ρ , η + ) .
ψ + ( ρ , τ + , η + ) = exp ( i ω 0 τ + ) f ( τ + ) Φ ( ρ , η + ) ,
i 4 σ ± Φ ( ρ , σ ± ) = ρ 2 Φ ( ρ , σ ± ) ,
ψ + ( ρ , τ + , σ ± ) = exp ( i ω 0 τ + ) f ( τ + ) Φ ( ρ , σ ± ) ,
ψ + ( mn ) ( x , y , τ + , σ ± ) = exp ( i ω 0 τ + ) f ( τ + ) exp [ x 2 / ( γ 1 + i σ ± ) ] exp [ y 2 / ( γ 2 + i σ ± ) ] ( γ 1 + i σ ± ) ( m + 1 ) / 2 ( γ 2 + i σ ± ) ( n + 1 ) / 2
× H m ( x / γ 1 + i σ ± ) H n ( y / γ 2 + i σ ± ) .
ψ + ( n ) ( ρ , τ + , σ ± ) = exp ( i ω 0 τ + ) f ( τ + ) γ 0 ( γ 0 + i σ ± ) n + 1 exp ( ρ 2 γ 0 + i σ ± ) L n ( 0 ) [ ρ 2 ( γ 0 + i σ ± ) ] .
ψ + ( 0 ) ( ρ , τ + , σ ± ) = exp ( i ω 0 τ + ) f ( τ + ) γ 0 γ 0 + i σ ± exp [ ρ 2 ( γ 0 + i σ ± ) ] .
ψ + ( 0 ) ( ρ , τ + , η + ) = exp ( i ω 0 τ + ) f ( τ + ) a a ± i η + exp ( ω 0 2 c v c 1 ρ 2 a ± i η + ) .
f ( τ + ) = exp ( τ + 2 4 T 2 ) = exp { 1 4 T 2 [ ( t z v ) ( v c vc ) z ] 2 }
ψ + ( ρ , τ + , σ ± ) = exp ( i ω 0 τ + ) f ( τ + ) γ 0 γ 0 + i σ ± J 0 ( γ 0 k 0 ρ sin θ γ 0 + i σ ± ) exp ( ρ 2 γ 0 + i σ ± )
× exp [ i σ ± 4 ( k 0 2 γ 0 sin 2 θ ) / ( γ 0 + i σ ± ) ] .
ϕ + ( ρ , τ + , σ z ) = f ( τ + ) Φ ( ρ , σ z ) ,
i 4 σ z Φ ( ρ , σ z ) = ρ 2 Φ ( ρ , σ z ) .
ψ + ( ρ , τ + , σ z ) = exp ( i ω 0 τ + ) f ( τ + ) Φ ( ρ , σ z ) .
u + ( ρ , z ; α ) = exp ( 2 i α c τ + ) a 1 ( a 1 + i z ) exp ( α ρ 2 a 1 + i z ) ,
u + ( ρ , z , t ) = 1 π 0 d α F ˜ ( α ) exp ( 2 i α c τ + ) a 1 ( a 1 + i z ) exp ( α ρ 2 a 1 + i z ) ,
u + ( ρ , z , t ) = a 1 ( a 1 + i z ) f ̂ ( t z / c i 1 2 c ρ 2 a 1 + i z ) ,
ψ + ( ρ , z , t ) = exp [ i ω 0 ( t z / c ) ] f ( t z / c ) γ 0 γ 0 i 2 z / k 0 exp ( ρ 2 γ 0 i 2 z / k 0 ) .
ϕ ( ρ , z , t ) = f ( τ ) Φ ( ρ , η + ) ,
i ( 1 + v c ) η + Φ ( ρ , η + ) = 1 2 k 0 ρ 2 Φ ( ρ , η + ) .
i 4 σ ¯ + Φ ( ρ , σ ¯ + ) = ρ 2 Φ ( ρ , σ ¯ + ) .
ψ ( ρ , τ , σ ¯ + ) = exp ( i ω 0 τ ) f ( τ ) Φ ( ρ , σ ¯ + ) .
ψ ( ρ , τ , σ z + ) = exp ( i ω 0 τ ) f ( τ ) Φ ( ρ , σ z + ) ; σ z + σ z = 2 z / k 0 ,
i ( z ± 1 c t ) ϕ ± ( ρ , z , t ) = ± 1 2 k ± ρ 2 ϕ ± ( ρ , z , t ) ; k ± γ ¯ ( 1 v / c ) k 0 .
ψ ± ( ρ , τ ± , σ ¯ z ) = exp [ i ω 0 γ ¯ ( 1 v c ) τ ± ] f [ γ ¯ ( 1 v c ) τ ± ] Φ ( ρ , σ ¯ z ) ;
τ ± t z c ; σ ¯ z 2 z k ± ,
ψ + ( ρ , τ + , σ ) = exp ( i ω 0 τ + ) f ( τ + ) Φ ( ρ , σ )
ψ ( ρ , τ , σ ¯ + ) = exp ( i ω 0 τ ) f ( τ ) Φ ( ρ , σ ¯ + ) .
ϕ ± ( ρ , ς ± , t ) = g ( ς ± ) Ψ ( ρ , t ) ,
i 4 σ t Ψ ( ρ , σ t ) = ρ 2 Ψ ( ρ , σ t ) ,
ψ ± ( ρ , ς , σ t ) = exp ( i k 0 ς ± ) g ( ς ± ) Ψ ( ρ , σ t )
ψ ± ( ρ , z , t ) = exp [ i k 0 ( z c t ) ] g ( z c t ) γ 0 γ 0 i ( 2 c t ) / k 0 exp ( ρ 2 γ 0 i ( 2 c t ) / k 0 ) .
ψ ± ( ρ , ς ± , σ t ) = exp [ i k 0 γ ( v c 1 ) ς ± ] g [ γ ( v c 1 ) ς ± ] Ψ ( ρ , σ t ) ;
ς ± z c t ; σ t ( 2 c t ) / k ¯ ± ,
ψ + ( ρ , ς + , σ + ) = exp ( i k 0 ς + ) g ( ς + ) Ψ ( ρ , σ + )
ψ ( ρ , ς , σ ¯ + ) = exp ( i k 0 ς ) g ( ς ) Ψ ( ρ , σ ¯ + )
i 4 χ ϕ ± ( ρ , χ ) = ρ 2 ϕ ± ( ρ , χ ) ,
ψ + ( ρ , z , t ) = exp ( i k 0 ς + ) ϕ + ( ρ , χ ) = exp [ i k 0 ( z c t ) ] ϕ + [ ρ , ( z + c t ) / k 0 ] .
ψ ( ρ , z , t ) = exp ( i k 0 ς ) ϕ ( ρ , χ + ) = exp [ i k 0 ( z + c t ) ] ϕ [ ρ , ( z + c t ) / k 0 ] .
ψ + ( ρ , z , t ) = k 0 k 0 i ( z + c t ) exp [ i k 0 ( z c t ) ] exp [ k 0 ρ 2 k 0 i ( z + c t ) ] ,
ψ ( ρ , z , t ) = k 0 k 0 + i ( z c t ) exp [ i k 0 ( z + c t ) ] exp [ k 0 ρ 2 k 0 + i ( z c t ) ] .
ψ ( ρ , τ , σ ¯ + ) = exp ( i ω 0 τ ) f ( τ ) Φ ( ρ , σ ¯ + ) ; τ = t + z c z , σ ¯ + = 2 ( z v t ) k 0 ( 1 + v / c ) .
ψ ( ρ , z , t ) = exp [ i k 0 ( z + c t ) ] Φ ( ρ , z c t k 0 ) .
ψ ( ρ , z , t ) = k 0 a + i ( z c t ) exp [ i k 0 ( z + c t ) ] exp [ k 0 ρ 2 a + i ( z c t ) ] , a > 0 .
ψ ( ρ , τ , σ ¯ z + ) = 1 a / k 0 + i σ ¯ z + exp [ i ω 0 γ ¯ ( 1 v c ) τ ] exp [ ρ 2 a / k 0 + i σ ¯ z + ]
τ = t + z c ; σ ¯ z + = 2 z k , k = γ ¯ ( 1 + v / c ) k 0 .
B p ( ρ , z , t ) = exp [ i ( β α ) ( z v ph t ) ] J 0 [ ρ 2 v c ( 1 + v c ) β 2 + c v α β ] ,
B e ( ρ , z , t ) = exp [ i k z ( z ω k z t ) ] J 0 [ ρ ( ω / c ) 2 k z 2 ] .
B p ( ρ , z , t ) = exp [ i β ( z v t ) ] J 0 [ ρ β 2 v c ( 1 + v c ) ]

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