Abstract

We study two different switching patterns of dynamic apertures illuminated by a focus-wave-mode excitation field. The two sources are chosen to be identical except that one is illuminated with use of a Gaussian time window, whereas the other uses a periodic function defined over the same time span. In spite of the similarity between the two dynamic apertures, the decay patterns of the amplitudes of their radiated fields are significantly different. A detailed analysis of the depletion of their spectral contents shows that the coupling between the spatial and temporal spectral components plays a decisive role in extending the range of the localization of the radiated pulses.

© 1996 Optical Society of America

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References

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  1. A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focused wave modes,”J. Math. Phys. 36, 5565–5587 (1995).
    [CrossRef]
  2. A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, S. M. Sedky, “Generation of approximate focused-wave-mode pulses from wide-band dynamic Gaussian apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
    [CrossRef]
  3. R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
    [CrossRef]
  4. R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,”J. Math. Phys. 26, 861–863 (1985).
    [CrossRef]
  5. A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semiinfinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1989).
    [CrossRef]
  6. P. Hillion, “Spinor focus wave modes,”J. Math. Phys. 28, 1743–1748 (1987).
    [CrossRef]
  7. A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “A novel approach to the synthesis of nondispersive wave packet solutions to the Klein–Gorden and the Dirac equations,”J. Math. Phys. 31, 2511–2519 (1990).
    [CrossRef]
  8. 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]
  9. M. K. Tippet, R. W. Ziolkowski, “A bidirectional wave transformation of the cold plasma equations,”J. Math Phys. 32, 488–492 (1991).
    [CrossRef]
  10. R. Donnelly, R. W. Ziolkowski, “A method of constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
    [CrossRef]
  11. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
    [CrossRef] [PubMed]
  12. R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representation of acoustic and electromagnetic radiation,” Proc. IEEE 79, 1371–1378 (1991).
    [CrossRef]
  13. R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,”IEEE Trans. Antennas Propag. 40, 888–905 (1992).
    [CrossRef]
  14. R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
    [CrossRef] [PubMed]
  15. R. W. Ziolkowski, J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
    [CrossRef]
  16. J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1993).
    [CrossRef]
  17. I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,”J. Math. Phys. 30, 1254–1269 (1989).
    [CrossRef]
  18. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  19. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Sec. 11.3.

1995 (2)

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focused wave modes,”J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, S. M. Sedky, “Generation of approximate focused-wave-mode pulses from wide-band dynamic Gaussian apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
[CrossRef]

1993 (2)

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1993).
[CrossRef]

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

1992 (4)

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

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

R. W. Ziolkowski, J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
[CrossRef]

1991 (3)

M. K. Tippet, 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 representation 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]

1990 (1)

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

1989 (3)

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semiinfinite, 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]

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

1987 (2)

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

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

1985 (1)

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

Besieris, I. M.

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focused wave modes,”J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, S. M. Sedky, “Generation of approximate focused-wave-mode pulses from wide-band dynamic Gaussian apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[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. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representation 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–Gorden and the Dirac equations,”J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional traveling 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, semiinfinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (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 (1993).
[CrossRef]

Donnelly, R.

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

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]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Sec. 11.3.

Hillion, P.

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

Judkins, J. B.

Miceli, J. J.

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

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Sec. 11.3.

Sedky, S. M.

Shaarawi, A. M.

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, S. M. Sedky, “Generation of approximate focused-wave-mode pulses from wide-band dynamic Gaussian apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
[CrossRef]

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focused wave modes,”J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of the exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[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. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representation 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–Gorden and the Dirac equations,”J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, R. W. Ziolkowski, “A bidirectional traveling 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, semiinfinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1989).
[CrossRef]

Tippet, M. K.

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

Vengsarkar, A. M.

Ziolkowski, R. W.

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focused wave modes,”J. Math. Phys. 36, 5565–5587 (1995).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, R. W. Ziolkowski, S. M. Sedky, “Generation of approximate focused-wave-mode pulses from wide-band dynamic Gaussian apertures,” J. Opt. Soc. Am. A 12, 1954–1964 (1995).
[CrossRef]

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

R. W. Ziolkowski, J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (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 of constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

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

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

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

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

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 traveling 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, semiinfinite, circular waveguide,” J. Appl. Phys. 65, 805–813 (1989).
[CrossRef]

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

IEEE Trans. Antennas Propag. (1)

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

J. Appl. Phys. (2)

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

J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1993).
[CrossRef]

J. Math Phys. (1)

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

J. Math. Phys. (5)

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–Gorden and the Dirac equations,”J. Math. Phys. 31, 2511–2519 (1990).
[CrossRef]

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

A. M. Shaarawi, R. W. Ziolkowski, I. M. Besieris, “On the evanescent fields and the causality of the focused wave modes,”J. Math. Phys. 36, 5565–5587 (1995).
[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. A (4)

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. (1)

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

Proc. IEEE (1)

R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Localized wave representation 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 of constructing solutions of homogeneous partial differential equations: localized waves,” Proc. R. Soc. London Ser. A 437, 673–692 (1992).
[CrossRef]

Other (1)

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Sec. 11.3.

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Figures (9)

Fig. 1
Fig. 1

Time windows of the excitation of the periodic and the Gaussian apertures.

Fig. 2
Fig. 2

Fourier spectra of the illuminations of the periodic and the Gaussian apertures.

Fig. 3
Fig. 3

Spatial spectrum of the illumination of the Gaussian aperture.

Fig. 4
Fig. 4

Temporal spectrum of the illumination of the Gaussian aperture.

Fig. 5
Fig. 5

Decay of the centroid of the periodic and the Gaussian pulses.

Fig. 6
Fig. 6

Depletion of the spatial spectrum of the Gaussian FWM pulse with distance.

Fig. 7
Fig. 7

Depletion of the spatial spectrum of the periodic FWM pulse with distance.

Fig. 8
Fig. 8

Power amplitude of the FWM central pulse on the aperture at z = ct = 0.

Fig. 9
Fig. 9

Power amplitudes of the periodic and the Gaussian FWM central pulses at different distances from the aperture.

Equations (22)

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Ψ ( ρ , t ) = Re { Ψ ˆ ( ρ , t ) } ,
Ψ ˆ ( ρ , t ) = 1 4 π ( a 1 ict ) exp [ β ρ 2 / ( a 1 i c t ) ] × exp ( i β c t ) exp ( t / 4 T 2 ) ,
Ψ ˆ ( ρ , t ) = 1 4 π ( a 1 i c t ) exp [ β ρ / ( a 1 i c t ) ] × exp ( i β c t ) cos 2 ( 5 π t / 8 T ) ,
ϕ ( χ , ω ) = d t 0 d ρ J 0 ( χ ρ ) ρ exp ( i ω t ) Ψ ˆ ( ρ , t ) .
ϕ ( χ , ω ) = 1 4 β δ ˆ G [ ω 0 ω 0 ( χ ) ; c T ] exp ( χ 2 a 1 / 4 β ) ,
δ ˆ G [ ω ω 0 ( χ ) ; c T ] = T π exp { T 2 [ ω ω 0 ( χ ) ] 2 }
ϕ ( χ , ω ) = 1 4 β δ ˆ P [ ω ω 0 ( χ ) ; c T ] exp ( χ 2 a 1 / 4 β ) ,
δ ˆ P [ ω ω 0 ( χ ) ; c T ] = 1 2 π sin { [ ω ω 0 ( χ ) ] 4 T } ( 1 [ ω ω 0 ( χ ) ] [ ω ω 0 ( χ ) ] { [ ω ω 0 ( χ ) ] 2 ( 5 π / 4 T ) 2 } )
ϕ s ( χ , t ) = 1 2 π 0 d ω ϕ ( χ , ω ) exp ( i ω t ) .
ϕ s ( χ , t ) = exp ( χ 2 a 1 / 4 β ) 8 π β exp [ i ω 0 ( χ ) t ] × exp ( t 2 / 4 T 2 ) .
ϕ s ( χ , t ) = exp ( χ 2 a 1 / 4 β ) 8 π β cos 2 ( 5 π t / 8 T ) × exp [ i ω 0 ( χ ) t ] , 4 T t 4 T .
ϕ t ( ρ , ω ) = 0 d χ χ J 0 ( χ ρ ) ϕ ( χ , ω ) .
ϕ t ( ρ , ω ) = 1 4 β 0 d χ χ J 0 ( χ ρ ) δ ˆ G [ ω ω 0 ( χ ) ; c T ] × exp ( χ 2 a 1 / 4 β ) .
ϕ t ( ρ , ω ) = 1 4 β 0 d χ χ J 0 ( χ ρ ) δ ˆ P [ ω ω 0 ( χ ) ; c T ] × exp ( χ 2 a 1 / 4 β ) .
Ψ i ( ρ , t ) = Re ( 1 2 π 0 d χ χ J 0 ( χ ρ ) 0 d ω ϕ ( χ , ω ) × exp { i [ ( ω / c ) 2 χ 2 ] z } exp ( i ω t ) ) z = 0 .
Ψ ( ρ , z , t ) = 1 4 π 0 2 π d ϕ 0 d ρ × ρ R [ z Ψ ( ρ , z , = 0 , t ) + z R 2 Ψ ( ρ , z = 0 , t ) + z R c t Ψ ( ρ , z = 0 , t ) ] t = t R / c ,
Ψ ( ρ , z , t ) = Re [ Ψ ˆ ( ρ , z , t ) ] ,
Ψ ˆ ( ρ , z , t ) = 1 2 π 0 d χ χ J 0 ( χ ρ ) 0 d ω ϕ ( χ , ω ) exp ( i ω t ) × exp { i [ ( ω / c ) 2 χ 2 ] z } .
Ψ ( ρ , z , t ) = 1 8 π β 0 d χ χ J 0 ( χ ρ ) 0 d ω × exp ( χ 2 a 1 / 4 β ) δ ˆ G [ ω ω 0 ( χ ) ; c T ] × cos { ω t [ ( ω / c ) 2 χ 2 ] z } .
ϕ s ( χ , z , t ) = 1 2 π 0 d ω ϕ ( χ , ω ) × cos { [ ( ω / c ) 2 χ 2 ] z ω t } .
ϕ s ( χ , z , t ) = exp ( χ 2 a 1 / 4 β ) 8 π β 0 d ω δ ˆ G [ ω ω 0 ( χ ) ; c T ] × cos { [ ( ω / c ) 2 χ 2 ] z ω t } .
ϕ s ( χ , t ) = exp ( χ 2 a 1 / 4 β ) 8 π β 0 d ω δ ˆ P [ ω ω 0 ( χ ) ; c T ] × cos { [ ( ω / c ) 2 χ 2 ] z ω t } .

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