Abstract

We are concerned with an expansion scheme for radiation from well-collimated pulsed aperture distribution that uses a new set of pulsed-beam (PB) basis functions. Several beam-expansion schemes have been introduced recently. They are based on local matching, within a self-consistent phase space format, of beam propagators to the source distribution. For well-collimated source distributions, greater efficiency may be obtained by matching wide beams to the entire aperture distribution, so that the propagators exhibit the same radiation properties as the entire aperture. This is the underlying strategy of the well-known Hermite–Gaussian-beam expansion of time-harmonic fields. In the present study a type of PB propagator, complex multipole PB’s (CMPB’s), is introduced and utilized in a similar expansion scheme for time-dependent radiation. CMPB’s are exact, highly localized, space–time wave-packet solutions that can be modeled analytically in terms of radiation from time-dependent complex multipoles as a generalization of the recently introduced complex source PB. The CMPB’s have paraxial profiles that involve transverse oscillations expressed in terms of Hermite polynomials and temporal derivatives; hence they form a biorthogonal set of basis functions for the expansion of collimated time-dependent fields. It is shown that the representation is most efficient if both the wave-front curvature and the collimation of the CMPB’s are matched to those of the aperture so that they exhibit the same far-field properties (e.g., diffraction angle and decay rate). We therefore introduce criteria to determine the collimation (Fresnel) length of a given pulsed source distribution and thereby for the optimal expansion parameters. To establish the near- and the far-field convergence properties of the expansion under matched and unmatched conditions, we derive closed-form expressions for the expansion amplitudes for a general class of time-dependent distributions and examine extensive numerical results.

© 1992 Optical Society of America

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References

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  1. L. B. Felsen, “Novel ways for tracking rays,” J. Opt. Soc. Am. A 2, 954–963 (1985).
    [CrossRef]
  2. M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
    [CrossRef]
  3. V. Cerveny, M. M. Popov, I. Psencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
    [CrossRef]
  4. I. T. Lu, L. B. Felsen, Y. Z. Ruan, “Spectral aspects of the Gaussian beam method: reflection from a homogeneous half space,” Geophys. J. R. Astron. Soc. 89, 915–922 (1987).
    [CrossRef]
  5. B. S. White, A. Noris, A. Bayliss, R. Burridge, “Some remarks on the Gaussian beam summation method,” Geophys. J. R. Astron. Soc. 89, 579–636 (1987).
    [CrossRef]
  6. E. Heyman, “Complex source pulsed beam expansion of transient radiation,” Wave Motion 11, 337–349 (1989).
    [CrossRef]
  7. M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–102 (1980).
  8. P. D. Einziger, S. Raz, M. Shapira, “Gabor representation and aperture theory,” J. Opt. Soc. Am. A 3, 508–522 (1986).
    [CrossRef]
  9. J. J. Maciel, L. B. Felsen, “Systematic study of fields due to extended apertures by Gaussian beam discretization,”IEEE Trans. Antennas Propag. 37, 884–892 (1989).
    [CrossRef]
  10. B. Z. Steinberg, E. Heyman, L. B. Felsen, “Phase space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A 8, 41–59 (1991).
    [CrossRef]
  11. B. Z. Steinberg, E. Heyman, L. B. Felsen, “Phase space beam summation for time-dependent radiation from large apertures: continuous parameterization,” J. Opt. Soc. Am. A 8, 943–958 (1991).
    [CrossRef]
  12. B. Z. Steinberg, E. Heyman, “Phase space beam summation for time-dependent radiation from large apertures: discretized parameterization,” J. Opt. Soc. Am. A 8, 959–966 (1991).
    [CrossRef]
  13. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,”J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  14. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  15. T. Takenaka, M. Yokota, U. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  16. A. J. Bogush, R. E. Elkins, “Gaussian beam expansions for large aperture antennas,”IEEE Trans. Propag. AP-34, 228–243 (1986).
    [CrossRef]
  17. E. Heyman, B. Z. Steinberg, “A spectral analysis of complex source pulsed beams,” J. Opt. Soc. Am. A 4, 473–480 (1987).
    [CrossRef]
  18. E. Heyman, L. B. Felsen, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
    [CrossRef]
  19. E. Heyman, B. Z. Steinberg, R. Iancunescu, “Electromagnetic complex source pulsed beam fields,”IEEE Trans. Antennas Propag. 38, 957–963 (1990).
    [CrossRef]
  20. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  21. J. W. Ra, H. Bertoni, L. B. Felsen, “Reflection and transmission of beams at dielectric interfaces,” SIAM J. Appl. Math. 24, 396–412 (1973).
    [CrossRef]
  22. L. B. Felsen, “Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams,” Symp. Matemat. Inst. Nazion. Alta Matemat. 18, 40–56 (1976).
  23. S. Y. Shin, L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,”J. Opt. Soc. Am. 67, 699–700 (1977).
    [CrossRef]
  24. E. Heyman, R. Iancunescu, “Pulsed beam reflection and transmissions at a dielectric interface: part I. Two dimensional fields,”IEEE Trans. Antennas Propag. 38, 1791–1800 (1990).
    [CrossRef]
  25. A. Erdelyi, ed., Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2, Chap. XI.
  26. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  27. E. Heyman, L. B. Felsen, “Non-dispersive closed form approximation for transient propagation and scattering of ray fields,” Wave Motion 7, 335–358 (1985).
    [CrossRef]

1991 (3)

1990 (2)

E. Heyman, B. Z. Steinberg, R. Iancunescu, “Electromagnetic complex source pulsed beam fields,”IEEE Trans. Antennas Propag. 38, 957–963 (1990).
[CrossRef]

E. Heyman, R. Iancunescu, “Pulsed beam reflection and transmissions at a dielectric interface: part I. Two dimensional fields,”IEEE Trans. Antennas Propag. 38, 1791–1800 (1990).
[CrossRef]

1989 (3)

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

E. Heyman, “Complex source pulsed beam expansion of transient radiation,” Wave Motion 11, 337–349 (1989).
[CrossRef]

J. J. Maciel, L. B. Felsen, “Systematic study of fields due to extended apertures by Gaussian beam discretization,”IEEE Trans. Antennas Propag. 37, 884–892 (1989).
[CrossRef]

1987 (3)

I. T. Lu, L. B. Felsen, Y. Z. Ruan, “Spectral aspects of the Gaussian beam method: reflection from a homogeneous half space,” Geophys. J. R. Astron. Soc. 89, 915–922 (1987).
[CrossRef]

B. S. White, A. Noris, A. Bayliss, R. Burridge, “Some remarks on the Gaussian beam summation method,” Geophys. J. R. Astron. Soc. 89, 579–636 (1987).
[CrossRef]

E. Heyman, B. Z. Steinberg, “A spectral analysis of complex source pulsed beams,” J. Opt. Soc. Am. A 4, 473–480 (1987).
[CrossRef]

1986 (2)

A. J. Bogush, R. E. Elkins, “Gaussian beam expansions for large aperture antennas,”IEEE Trans. Propag. AP-34, 228–243 (1986).
[CrossRef]

P. D. Einziger, S. Raz, M. Shapira, “Gabor representation and aperture theory,” J. Opt. Soc. Am. A 3, 508–522 (1986).
[CrossRef]

1985 (3)

1982 (2)

M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
[CrossRef]

V. Cerveny, M. M. Popov, I. Psencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

1980 (1)

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–102 (1980).

1977 (1)

1976 (1)

L. B. Felsen, “Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams,” Symp. Matemat. Inst. Nazion. Alta Matemat. 18, 40–56 (1976).

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1973 (2)

A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,”J. Opt. Soc. Am. 63, 1093–1094 (1973).
[CrossRef]

J. W. Ra, H. Bertoni, L. B. Felsen, “Reflection and transmission of beams at dielectric interfaces,” SIAM J. Appl. Math. 24, 396–412 (1973).
[CrossRef]

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Bastiaans, M. J.

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–102 (1980).

Bayliss, A.

B. S. White, A. Noris, A. Bayliss, R. Burridge, “Some remarks on the Gaussian beam summation method,” Geophys. J. R. Astron. Soc. 89, 579–636 (1987).
[CrossRef]

Bertoni, H.

J. W. Ra, H. Bertoni, L. B. Felsen, “Reflection and transmission of beams at dielectric interfaces,” SIAM J. Appl. Math. 24, 396–412 (1973).
[CrossRef]

Bogush, A. J.

A. J. Bogush, R. E. Elkins, “Gaussian beam expansions for large aperture antennas,”IEEE Trans. Propag. AP-34, 228–243 (1986).
[CrossRef]

Burridge, R.

B. S. White, A. Noris, A. Bayliss, R. Burridge, “Some remarks on the Gaussian beam summation method,” Geophys. J. R. Astron. Soc. 89, 579–636 (1987).
[CrossRef]

Cerveny, V.

V. Cerveny, M. M. Popov, I. Psencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Einziger, P. D.

Elkins, R. E.

A. J. Bogush, R. E. Elkins, “Gaussian beam expansions for large aperture antennas,”IEEE Trans. Propag. AP-34, 228–243 (1986).
[CrossRef]

Felsen, L. B.

B. Z. Steinberg, E. Heyman, L. B. Felsen, “Phase space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A 8, 41–59 (1991).
[CrossRef]

B. Z. Steinberg, E. Heyman, L. B. Felsen, “Phase space beam summation for time-dependent radiation from large apertures: continuous parameterization,” J. Opt. Soc. Am. A 8, 943–958 (1991).
[CrossRef]

J. J. Maciel, L. B. Felsen, “Systematic study of fields due to extended apertures by Gaussian beam discretization,”IEEE Trans. Antennas Propag. 37, 884–892 (1989).
[CrossRef]

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

I. T. Lu, L. B. Felsen, Y. Z. Ruan, “Spectral aspects of the Gaussian beam method: reflection from a homogeneous half space,” Geophys. J. R. Astron. Soc. 89, 915–922 (1987).
[CrossRef]

L. B. Felsen, “Novel ways for tracking rays,” J. Opt. Soc. Am. A 2, 954–963 (1985).
[CrossRef]

E. Heyman, L. B. Felsen, “Non-dispersive closed form approximation for transient propagation and scattering of ray fields,” Wave Motion 7, 335–358 (1985).
[CrossRef]

S. Y. Shin, L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,”J. Opt. Soc. Am. 67, 699–700 (1977).
[CrossRef]

L. B. Felsen, “Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams,” Symp. Matemat. Inst. Nazion. Alta Matemat. 18, 40–56 (1976).

J. W. Ra, H. Bertoni, L. B. Felsen, “Reflection and transmission of beams at dielectric interfaces,” SIAM J. Appl. Math. 24, 396–412 (1973).
[CrossRef]

Fukumitsu, U.

Heyman, E.

B. Z. Steinberg, E. Heyman, L. B. Felsen, “Phase space beam summation for time-dependent radiation from large apertures: continuous parameterization,” J. Opt. Soc. Am. A 8, 943–958 (1991).
[CrossRef]

B. Z. Steinberg, E. Heyman, “Phase space beam summation for time-dependent radiation from large apertures: discretized parameterization,” J. Opt. Soc. Am. A 8, 959–966 (1991).
[CrossRef]

B. Z. Steinberg, E. Heyman, L. B. Felsen, “Phase space beam summation for time-harmonic radiation from large apertures,” J. Opt. Soc. Am. A 8, 41–59 (1991).
[CrossRef]

E. Heyman, B. Z. Steinberg, R. Iancunescu, “Electromagnetic complex source pulsed beam fields,”IEEE Trans. Antennas Propag. 38, 957–963 (1990).
[CrossRef]

E. Heyman, R. Iancunescu, “Pulsed beam reflection and transmissions at a dielectric interface: part I. Two dimensional fields,”IEEE Trans. Antennas Propag. 38, 1791–1800 (1990).
[CrossRef]

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

E. Heyman, “Complex source pulsed beam expansion of transient radiation,” Wave Motion 11, 337–349 (1989).
[CrossRef]

E. Heyman, B. Z. Steinberg, “A spectral analysis of complex source pulsed beams,” J. Opt. Soc. Am. A 4, 473–480 (1987).
[CrossRef]

E. Heyman, L. B. Felsen, “Non-dispersive closed form approximation for transient propagation and scattering of ray fields,” Wave Motion 7, 335–358 (1985).
[CrossRef]

Iancunescu, R.

E. Heyman, B. Z. Steinberg, R. Iancunescu, “Electromagnetic complex source pulsed beam fields,”IEEE Trans. Antennas Propag. 38, 957–963 (1990).
[CrossRef]

E. Heyman, R. Iancunescu, “Pulsed beam reflection and transmissions at a dielectric interface: part I. Two dimensional fields,”IEEE Trans. Antennas Propag. 38, 1791–1800 (1990).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lu, I. T.

I. T. Lu, L. B. Felsen, Y. Z. Ruan, “Spectral aspects of the Gaussian beam method: reflection from a homogeneous half space,” Geophys. J. R. Astron. Soc. 89, 915–922 (1987).
[CrossRef]

Maciel, J. J.

J. J. Maciel, L. B. Felsen, “Systematic study of fields due to extended apertures by Gaussian beam discretization,”IEEE Trans. Antennas Propag. 37, 884–892 (1989).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Noris, A.

B. S. White, A. Noris, A. Bayliss, R. Burridge, “Some remarks on the Gaussian beam summation method,” Geophys. J. R. Astron. Soc. 89, 579–636 (1987).
[CrossRef]

Popov, M. M.

M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
[CrossRef]

V. Cerveny, M. M. Popov, I. Psencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

Psencik, I.

V. Cerveny, M. M. Popov, I. Psencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

Ra, J. W.

J. W. Ra, H. Bertoni, L. B. Felsen, “Reflection and transmission of beams at dielectric interfaces,” SIAM J. Appl. Math. 24, 396–412 (1973).
[CrossRef]

Raz, S.

Ruan, Y. Z.

I. T. Lu, L. B. Felsen, Y. Z. Ruan, “Spectral aspects of the Gaussian beam method: reflection from a homogeneous half space,” Geophys. J. R. Astron. Soc. 89, 915–922 (1987).
[CrossRef]

Shapira, M.

Shin, S. Y.

Siegman, A. E.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Steinberg, B. Z.

Takenaka, T.

White, B. S.

B. S. White, A. Noris, A. Bayliss, R. Burridge, “Some remarks on the Gaussian beam summation method,” Geophys. J. R. Astron. Soc. 89, 579–636 (1987).
[CrossRef]

Yokota, M.

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Geophys. J. R. Astron. Soc. (3)

V. Cerveny, M. M. Popov, I. Psencik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astron. Soc. 70, 109–128 (1982).
[CrossRef]

I. T. Lu, L. B. Felsen, Y. Z. Ruan, “Spectral aspects of the Gaussian beam method: reflection from a homogeneous half space,” Geophys. J. R. Astron. Soc. 89, 915–922 (1987).
[CrossRef]

B. S. White, A. Noris, A. Bayliss, R. Burridge, “Some remarks on the Gaussian beam summation method,” Geophys. J. R. Astron. Soc. 89, 579–636 (1987).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

J. J. Maciel, L. B. Felsen, “Systematic study of fields due to extended apertures by Gaussian beam discretization,”IEEE Trans. Antennas Propag. 37, 884–892 (1989).
[CrossRef]

E. Heyman, B. Z. Steinberg, R. Iancunescu, “Electromagnetic complex source pulsed beam fields,”IEEE Trans. Antennas Propag. 38, 957–963 (1990).
[CrossRef]

E. Heyman, R. Iancunescu, “Pulsed beam reflection and transmissions at a dielectric interface: part I. Two dimensional fields,”IEEE Trans. Antennas Propag. 38, 1791–1800 (1990).
[CrossRef]

IEEE Trans. Propag. (1)

A. J. Bogush, R. E. Elkins, “Gaussian beam expansions for large aperture antennas,”IEEE Trans. Propag. AP-34, 228–243 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (8)

Optik (Stuttgart) (1)

M. J. Bastiaans, “The expansion of an optical signal into a discrete set of Gaussian beams,” Optik (Stuttgart) 57, 95–102 (1980).

Phys. Rev. A (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

SIAM J. Appl. Math. (1)

J. W. Ra, H. Bertoni, L. B. Felsen, “Reflection and transmission of beams at dielectric interfaces,” SIAM J. Appl. Math. 24, 396–412 (1973).
[CrossRef]

Symp. Matemat. Inst. Nazion. Alta Matemat. (1)

L. B. Felsen, “Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams,” Symp. Matemat. Inst. Nazion. Alta Matemat. 18, 40–56 (1976).

Wave Motion (3)

E. Heyman, L. B. Felsen, “Non-dispersive closed form approximation for transient propagation and scattering of ray fields,” Wave Motion 7, 335–358 (1985).
[CrossRef]

M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion 4, 85–97 (1982).
[CrossRef]

E. Heyman, “Complex source pulsed beam expansion of transient radiation,” Wave Motion 11, 337–349 (1989).
[CrossRef]

Other (2)

A. Erdelyi, ed., Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2, Chap. XI.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

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Figures (13)

Fig. 1
Fig. 1

Relations between various time-domain and time-harmonic solutions. Here , D, and represent, respectively, the Fourier-transform relation, the transverse-derivatives operation, and the paraxial approximations. The following abbreviations are used (the number in parentheses denotes the equation or the relation in which the corresponding wave field is defined): CMB, complex multipole beam (1); CSB, complex-source beam (2); GB, Gaussian beam (11); HGB, Hermite–Gaussian beam (12); CSPB, complex-source pulsed beam (22); CMPB, complex multipole pulsed beam (21); PPB, paraxial pulsed beam (28); HPB, Hermite pulsed beam (33). No special name is given here to the important special case of complex multipole Green’s-function PB’s in relations (37) and (38). These solutions are also termed CSPB, CMPB, etc. and have the same functional form, with f + replaced by δ +.

Fig. 2
Fig. 2

Beam oblate spheroidal system. The wiggly line describes the source disk, where s(r) and therefore the field solutions exhibit a branch discontinuity.

Fig. 3
Fig. 3

Complex multipole pulsed beam GRm,n for (a) (m, n) = (0,0), (b) (m, n) = (1,0), and (c) (m, n) = (2,0). The plots depict cross-sectional cuts through the (x, z) plane time vt = 2b. Beam parameters: vt1 = 1.0005b (i.e., β = 0.0005b). All the axes are normalized with respect to b and v as indicated. In (a) the pulse profile away from the central peak is smooth. The lobe structure on the plots is due to the finite resolution associated with the graphics. The insets show time sections through the PB maxima.

Fig. 4
Fig. 4

Beam expansion with (a) zb < 0 and (b) zb > 0.

Fig. 5
Fig. 5

Overcollimated Laguerre GB synthesis for the Gaussian initial distribution [Eq. (58)] for (a) z = 0 (the initial field distribution) and (b) z = 5000λ ≃ 2Di. The plots compare the magnitude of the exact solution in Eq. (62) (solid curves) with the synthesized distribution from relation (56) with Eq. (60), using nN terms, with N = 0, 1, 2, or 3, marked by dashed curves with zero, one, two, or three dots, respectively. Initial field parameter: qi/λ = −i800π (i.e., Ri = ∞ and Ŵi/λ = 20). The Laguerre GB’s are overcollimated with the expansion parameter q0 = 2qi (i.e., R0 = ∞ and W ^ 0 = 2 W ^ i).

Fig. 6
Fig. 6

As in Fig. 5 but for undercollimated Laguerre GB’s with the expansion parameter q0 = qi/2 (i.e., R0 = ∞ and W ^ 0 = W ^ i / 2).

Fig. 7
Fig. 7

Examples for the initial real pulsed field distributions uiR = Re u i + described by Eq. (81) with Eqs. (82), for which the closed-form expansions for the expansion coefficients are derived. (a) A i = A i ( 1 ), qi = −i4000vT. (b) A i = A i ( 1 ), qi = (−1 − i)103vT. (c) A i = A i ( 3 ) with d = 100vT and qi = ∞. (d) A i = A i ( 2 ) with d = 100vT and qi = −i4000vT. All the axes are normalized with respect to T and v as follows: tt/T, ρρ/vT, and uu/T.

Fig. 8
Fig. 8

Expansion of the initial field distribution uiR in Fig. 7(c), using optimal HPB’s with q0 = −ib = −i4000vT. (a), (b), (c) Contributions of the n = 0, the n = 1, and the n = 2 terms, respectively. (d) Sum of all the terms up to n = N = 6. All the axes are normalized as in Fig. 7.

Fig. 9
Fig. 9

As in Fig. 8 but for b = 2000vT.

Fig. 10
Fig. 10

As in Fig. 8 but for b = 8000vT.

Fig. 11
Fig. 11

Expansion of the radiated field uR owing to the initial field distribution in Fig. 7(c). Observation time: t = 4.104T (radiated field is in the far zone z = 4.104vTDi). (a), (b), (c) Contributions of the n = 0, the n = 1, and the n = 2 terms, respectively. (d) Sum of all the terms up to n = N = 6. The axes are normalized as follows: uu/T, ρρ/vT, and zz/vT.

Fig. 12
Fig. 12

As in Fig. 11 but for b = 2000vT.

Fig. 13
Fig. 13

As in Fig. 11 but for b = 8000vT.

Equations (116)

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G ^ m , n ( r , r ) = x m y n G ^ ( r , r ) ,
G ^ ( r , r ) = G ^ 0 , 0 = exp ( i k s ) / 4 π s ,             k = ω / v ,
s ( r ) = [ ( x - x ) 2 + ( y - y ) 2 + ( z - z ) 2 ] 1 / 2
r = r b + i b ,
b = ( 0 , 0 , b ) ,             b > 0 ,             r b = ( 0 , 0 , 0 )
s ( r ) = ( ρ 2 + q 2 ) 1 / 2 ,
ρ = ( x 2 + y 2 ) 1 / 2 ,             q ( z ) = z - i b .
Re s 0 ,
Im s 0 ,
( x , y , z ) = b ( cos ϕ sin η cosh ξ , sin ϕ sin η cosh ξ , cos η sinh ξ ) ,
s ( r ) = b ( sinh ξ - i cos η ) .
s ± ( q + ρ 2 / 2 q )             for             z 0 ,             ρ q .
G ^ U ^ exp ( i k q - Q ^ 2 ρ 2 ) / 4 π q ,
G ^ m , n U ^ m , n x m y n U ^ = ( - Q ^ ) m + n H m ( Q ^ x ) H n ( Q ^ y ) U ^ ,
Q ^ = ( k / 2 i q ) 1 / 2
H m ( α ) = ( - 1 ) m exp ( α 2 ) α m exp ( - α 2 ) .
1 / q ( z ) 1 / R ( z ) + i / k W ^ 2 ( z ) ,
W ^ ( z ) = W ^ 0 [ 1 + ( z / b ) 2 ] 1 / 2 ,             W ^ 0 ( ω ) = b k ,
R ( z ) = z + b 2 / z ,             D = b ,             Θ ^ D ( ω ) = 1 / b k .
u + ( t ) = 1 π 0 d ω exp ( - i ω t ) u ^ ( ω ) ,             Im t 0.
u + ( t ) = u ( t ) + i H u ( t ) ,             t real ,
t = i t 1 ,             t 1 > 0
v t 1 - b β 0 .
F + m , n ( r , t ; r , t ) = x m y n F + ( r , t ; r , t ) ,
F + ( r , t ; r , t ) = f + ( t - t - s / v ) / 4 π s
F R m , n ( r , t ) Re F + m , n ,
F I m , n ( r , t ) Im F + m , n = H F R m , n
F + 1 , 0 = ( - x / s ) [ v - 1 f + ( 1 ) + s - 1 f + ] / 4 π s ,
F + 1 , 1 = ( x y / s 2 ) [ v - 2 f + ( 2 ) + 3 v - 1 s - 1 f + ( 1 ) + 3 s - 2 f + ] / 4 π s ,
F 2 , 0 + = ( x / s ) 2 [ v - 2 f + ( 2 ) + 3 v - 1 s - 1 f + ( 1 ) + 3 s - 2 f + ] / 4 π s - [ v - 1 s - 1 f + ( 1 ) + s - 2 f + ] / 4 π s ,
F + ( r , t ; r , t ) = f + ( t - s R / v - i γ ) / 4 π ( s R + i s I ) , γ ( r ) t 1 - s I ( r ) / v ,
F R = [ 4 π ( s R 2 + s I 2 ) ] - 1 { s R + s I H } f γ ( t - s R / v ) ,
f + ( t - i γ ) = f γ ( t ) + i H f y ( t ) ,             t real .
F + P + f + ( t - z / v - i β / v - ρ 2 / 2 v q ) / 4 π q ,
1 / q ( z ) = 1 / R ( z ) + i / I ( z ) ,
I ( z ) = b ( 1 + z 2 / b 2 ) 0 ,
f + ( j + 1 ) / f + ( j ) > v / s ,             j = 0 , 1 , ,
q > v T .
F + m , n P + m , n ( - Q ) m + n H m ( Q x ) H n ( Q y ) P + ,
Q [ ( 2 q v ) - 1 t ] 1 / 2
( - Q ) m H m ( Q x ) = h m ( m ) x m Q 2 m + h m - 2 ( m ) x m - 2 Q 2 m - 2 + + { h 0 ( m ) Q m m even h 1 ( m ) x Q m + 1 m odd ,
δ + ( t ) = { 1 / π i t Im t < 0 δ ( t ) + P / π i t Im t = 0
G + m , n x m y n G + U + m , n ( - Q ) m + n H m ( Q x ) H n ( Q y ) U + ,
G + δ + ( t - t - s / v ) / 4 π s U + ( 4 π q ) - 1 ( π i ) - 1 ( t - z / v - i β - ρ 2 / 2 v q ) - 1 .
W ( z ) = W 0 [ 1 + ( z / b ) 2 ] 1 / 2 , W 0 = 2 [ 2 β b ( 2 - 1 ) ] 1 / 2 = b Θ D , T 0 = 2 β / v ,             D 0 = b .
D 0 = K W 0 2 / v T 0 ,
K = [ 4 ( 2 - 1 ) ] - 1 1.9 / π .
r b = ( 0 , 0 , z b ) .
t = z b / v + i ( b + β ) / v ,             β 0.
q = q 0 + z ,             q 0 = - z b - i b .
d x 0 d y 0 V ^ m , n ( r 0 ) U ^ μ , ν ( r 0 ) = δ m , μ δ n , ν ,
V ^ m , n ( r 0 ) = ( m ! n ! ) - 1 ( - 2 Q ^ 0 ) 2 - m - n H m ( Q ^ 0 x 0 ) H n ( Q ^ 0 y 0 ) q 0 × exp ( - i k q 0 ) ,
q 0 q z = 0 = - z b - i b , Q ^ 0 Q ^ z = 0 = ( k / 2 i q 0 ) 1 / 2 .
u ^ i ( r 0 ) = m , n a ^ m , n U ^ m , n ( r 0 ) ,
a ^ m , n = d x 0 d y 0 u ^ i ( r 0 ) V ^ m , n ( r 0 ) .
u ^ ( r ) m , n a ^ m , n U ^ m , n ( r ) .
1 / q 0 = 1 / R 0 + i / k W ^ 0 2 .
U ^ L n ( r ) Q ^ 2 n L n ( Q ^ 2 ρ 2 ) U ^ ( r ) ,
L n ( α ) = exp ( + α ) α n [ α n exp ( - α ) ]
L n ( x 2 + y 2 ) = ( - 1 ) n l = 0 n H 2 l ( x ) H 2 ( n - l ) ( y ) ,
U ^ L n ( r ) = ( - 1 ) n l = 1 n U ^ 2 l , 2 ( n - l ) ( r ) .
0 d ρ 0 U ^ L n ( ρ 0 , z = 0 ) V ^ L ν ( ρ 0 ) = δ n , ν ,
V ^ L n ( ρ 0 ) = 8 π ( n ! ) - 2 Q ^ 0 2 - 2 n L n ( Q ^ 0 2 ρ 0 2 ) ρ 0 q 0 exp ( - i k q 0 ) ,
u ^ ( r ) n a ^ L n U ^ L n ( r ) ,
a ^ L n = 0 d ρ 0 u ^ i ( ρ 0 ) V ^ L n ( ρ 0 ) .
u ^ i ( ρ 0 ) = exp ( i k ρ 0 2 / 2 q i ) ,             q i = q i R + i q i I ,             q i L < 0.
1 / q i = 1 / R i + i / k W ^ i 2 .
a ^ L n = ( n ! ) - 2 Q ^ 0 - 2 n 4 π q 0 exp ( - i k q 0 ) j = 0 n j ! l j ( n ) ( q i / q 0 ) j + 1 ,
j = 0 n j ! l j ( n ) = δ j , 0
u ^ ( r ) = ( q i / q ) exp [ i k ( q - q i + ρ 2 / 2 q ) ] ,             q ( z ) q i + z ,
u + ( r , t ) m , n ½ a + m , n ( t ) U + m , n ( r , t ) ,
a + m , n ( t ) = ½ d x 0 d y 0 u + i ( r 0 , t ) V + m , n ( r 0 , t ) ,
V + ( r 0 , t ) = ( m ! n ! ) - 1 q 0 ( - 2 Q 0 ) 2 - m - n H m ( Q 0 x 0 ) H n ( Q 0 y 0 ) δ + ( t ) ,
Q 0 = [ ( 2 q 0 v ) - 1 t ] 1 / 2 .
½ d x 0 d y 0 V + m , n ( r 0 , t ) U + μ , ν ( r 0 , t ) = ½ δ m , μ δ n , ν δ + ( t ) .
a + m , n ( t ) = d x 0 d y 0 ( m ! n ! ) - 1 q 0 ( - 2 Q 0 ) 2 - m - n H m ( Q 0 x 0 ) × H n ( Q 0 y 0 ) u + i ( r 0 , t ) .
Q 0 - m H m ( Q 0 x 0 ) = h m ( m ) x 0 m + h m - 2 ( m ) x 0 m - 2 Q 0 - 2 + + { h 0 ( m ) Q 0 - m m even h 1 ( m ) x 0 Q 0 1 - m m odd ,
u + ( r , t ) m , n u + m , n ( r , t ) ,
u + m , n ( r , t ) = ( π m ! n ! ) - 1 ( q 0 / 4 q ) ( m + n ) / 2 ( t / 2 q v ) H m ( Q x ) H n ( Q y ) × d x 0 d y 0 H m ( Q 0 x 0 ) H n ( Q 0 y 0 ) × u + i ( r 0 , t - z / v - ρ 2 / 2 q v ) .
D i = K W i 2 / v T i ,
1 / q 0 = 1 / R 0 + i / I 0 .
u + ( r , t ) n ½ a + L n ( t ) U + L n ( r , t ) ,
U + L n ( r , t ) = Q 2 n L n ( Q 2 ρ 2 ) δ + ( t - z / v - i β - ρ 2 / 2 q v ) / 4 π q ,
a + L n ( t ) = 0 d ρ 0 ½ u + i ( ρ 0 , t ) V + L n ( ρ 0 , t ) ,
V + L n ( ρ 0 , t ) = 8 π q 0 ( n ! ) - 2 Q 0 2 - 2 n ρ 0 L n ( Q 0 2 ρ 0 2 ) δ + ( t ) ,
a + L n ( t ) = 0 d ρ 0 ρ 0 8 π q 0 ( n ! ) - 2 Q 0 2 - 2 n L n ( Q 0 2 ρ 0 2 ) u + i ( ρ 0 , t ) .
u + ( r , t ) n u + L n ( r , t ) ,
u + L n ( r , t ) = 2 ( n ! ) - 2 ( q 0 / q ) n + 1 L n ( Q 2 ρ 2 ) 0 d ρ 0 ρ 0 Q 0 2 L n ( Q 0 2 ρ 0 2 ) × u + i ( ρ 0 , t - z / v - ρ 2 / 2 q v ) .
u + i ( ρ 0 , t ) = A i ( ρ 0 ) δ + ( t - ρ 0 2 / 2 q i v - i T ) ,             q i = q i R + i q i I ,             q i I 0 ,
A i ( 1 ) ( ρ 0 ) = 1 ,
A i ( 2 ) ( ρ 0 ) = H ( d - ρ 0 ) ,
A i ( 3 ) ( ρ 0 ) = [ 1 - ( ρ 0 / d ) 2 ] H ( d - ρ 0 ) ,
1 / q i = 1 / R i + i / I i ,             I i > 0
W i = 2 [ 2 v T I i ( 2 - 1 ) ] 1 / 2 .
u + ( r , t ) = ( q i / q ) δ + ( t - z / v - i T - ρ 2 / 2 q v ) , q ( z ) = q i + z .
I n , p = 0 d d ρ 0 ρ 0 ( ρ 0 / d ) p Q 0 2 L n ( Q 0 2 ρ 0 2 ) × δ + ( t - τ - i T - ρ 0 2 / 2 v q i ) ,
τ = z / v + ρ 2 / 2 v q ,             q = q 0 + z ,             Q = [ ( 2 v q ) - 1 t ] 1 / 2 .
Q 0 2 L n ( Q 0 2 ρ 0 2 ) = j = 0 n l j ( n ) ( 2 v q 0 ) - j - 1 ρ 0 2 j t j + 1 ,
I n , p = ( i 2 v q i / π ) j = 0 n ( j + 1 ) ! l j ( n ) ( q i / q 0 ) j + 1 J j , p ,
J j , p ( d ) ( - 2 v q i ) - j - 2 0 d d ρ 0 ( ρ 0 / d ) 2 ρ 0 2 j + 1 × ( t - τ - i T - ρ 0 2 / 2 v q i ) - j - 2 .
J j , 0 ( d ) = - ( 4 v q i ) - 1 ( t - τ - i T ) - 1 ( j + 1 ) - 1 μ = 1 j + 1 ( j + 1 μ ) × ( - 1 ) μ [ ( t - τ - i T t - τ - i T - d 2 / 2 v q i ) μ - 1 ] ,
J j , 2 ( d ) = 1 2 d 2 { ln [ t - τ - i T - d 2 / 2 v q i t - τ - i T ] - μ = 1 j + 1 ( j + 1 μ ) × ( - 1 ) μ μ [ ( t - τ - i T t - τ - i T - d 2 / 2 v q i ) μ - 1 ] } .
J j , 0 ( d ) = - ( 4 v q i ) - 1 ( t - τ - i T ) - 1 ( j + 1 ) - 1 ,
μ = 1 j + 1 ( j + 1 μ ) ( - 1 ) μ = - 1.
u L n + = ( 4 v i q i / π ) ( n ! ) - 2 ( q 0 / q ) n + 1 ν = 0 n j = 0 n l ν ( n ) l j ( n ) ( j + 1 ) ! ( q i / q 0 ) j + 1 × ( ρ 2 / 2 v q ) ν t ν J j ,
J j = { J j , 0 ( d ) for A i ( 1 ) J j , 0 ( d ) for A i ( 2 ) J j , 0 ( d ) - J j , 2 ( d ) for A i ( 3 ) .
I n , p = j = 0 n l j ( n ) ( 2 v q 0 ) - j - 1 δ + ( j + 1 ) ( t - τ - i T ) × d 2 j + 2 ( 2 j + 2 + p ) - 1 ;
u + L n = ( n ! ) - 2 ( q 0 / q ) n + 1 ν = 0 n j = 0 n l ν ( n ) l j ( n ) ( 2 v q ) - ν ( 2 v q 0 ) - j - 1 × d 2 j + 2 ρ 2 ν δ + ( j + ν + 1 ) ( t - τ - i T ) K j ,
K j = { ( j + 1 ) - 1 for A i ( 2 ) ( j + 1 ) - 1 ( j + 2 ) - 1 for A i ( 3 ) .
H u ( t ) = - - P u ( t ) π ( t - t ) d t ,
h ^ = f ^ g ^ h + = ½ f + g + = f + g = f g + ,
t ( f + g + ) = ( t f + ) g + = f + t g + ,
H ( f g ) = ( H f ) g = f ( H g ) ,             H 2 f = - f .
½ δ + ( m ) ( t ) f + ( t ) = t m f + ( t ) .
δ + ( m ) ( t ) t m δ + ( 0 ) ( t ) = ( - 1 ) m m ! ( π i ) - 1 t - m - 1 .
δ + ( - 1 ) ( t ) - t δ + ( τ ) d τ = 1 + ( π i ) - 1 ln t ,             - π < Im ln t 0 ;

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