Abstract

Assuming a moving disk source distribution, the Riemann method and Gegenbauer's addition theorem in a mixed coordinate-space–phase-space formulation have been used to derive a zeroth-order Bessel–Gauss pulse solution to the inhomogeneous wave equation. In the region where τ2ρeff2+z2, the total solution can be separated into a wave that attenuates quickly and a Bessel–Gauss pulse that exists only after a certain time and continues for long times. In the region where z2τ2ρeff2+z2, only a short-time solution exists.

© 1997 Optical Society of America

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  1. P. L. Overfelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
    [CrossRef] [PubMed]
  2. J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
    [CrossRef]
  3. P. A. Belanger, “Packetlike solutions of the homogeneous wave equation,” J. Opt. Soc. Am. A 1, 723–724 (1984).
    [CrossRef]
  4. R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
    [CrossRef]
  5. R. Donnelly, R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. London Ser. A 440, 541–565 (1993).
    [CrossRef]
  6. P. L. Overfelt, “Continua of localized wave solutions via a complex similarity transformation,” Phys. Rev. E 47, 4430–4438 (1993).
    [CrossRef]
  7. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  8. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  9. M. Palmer, R. Donnelly, “Focused waves and the scalar wave equation,” J. Math. Phys. 34, 4007–4013 (1993).
    [CrossRef]
  10. P. L. Overfelt, “Focus wave mode solutions of the inhomogeneous n-dimensional scalar wave equation,” Phys. Rev. E 52, 4387–4392 (1995) (Part B).
    [CrossRef]
  11. V. V. Borisov, A. B. Utkin, “On formation of focus wave modes,” J. Phys. A: Math. Gen. 27, 2587–2590 (1994).
  12. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  13. P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–745 (1991).
    [CrossRef]
  14. V. V. Borisov, A. B. Utkin, “Electromagnetic fields produced by the spike pulse of hard radiation,” J. Phys. A: Math. Gen. 26, 4081–4085 (1993).
  15. V. V. Borisov, I. I. Simonenko, “Transient waves generated by a source on a circle,” J. Phys. A: Math. Gen. 27, 6243–6252 (1994).
  16. P. R. Garabedian, Partial Differential Equations (Wiley, New York, 1964).
  17. A. Gervois, H. Navelet, “Integrals of three Bessel functions and Legendre functions. I and II,” J. Math. Phys. 26, 633–655 (1985).
    [CrossRef]
  18. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1944), Chaps. 11 and 16.
  19. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

1995 (1)

P. L. Overfelt, “Focus wave mode solutions of the inhomogeneous n-dimensional scalar wave equation,” Phys. Rev. E 52, 4387–4392 (1995) (Part B).
[CrossRef]

1994 (2)

V. V. Borisov, A. B. Utkin, “On formation of focus wave modes,” J. Phys. A: Math. Gen. 27, 2587–2590 (1994).

V. V. Borisov, I. I. Simonenko, “Transient waves generated by a source on a circle,” J. Phys. A: Math. Gen. 27, 6243–6252 (1994).

1993 (4)

V. V. Borisov, A. B. Utkin, “Electromagnetic fields produced by the spike pulse of hard radiation,” J. Phys. A: Math. Gen. 26, 4081–4085 (1993).

R. Donnelly, R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. London Ser. A 440, 541–565 (1993).
[CrossRef]

P. L. Overfelt, “Continua of localized wave solutions via a complex similarity transformation,” Phys. Rev. E 47, 4430–4438 (1993).
[CrossRef]

M. Palmer, R. Donnelly, “Focused waves and the scalar wave equation,” J. Math. Phys. 34, 4007–4013 (1993).
[CrossRef]

1991 (2)

1987 (3)

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

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

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

1985 (2)

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

A. Gervois, H. Navelet, “Integrals of three Bessel functions and Legendre functions. I and II,” J. Math. Phys. 26, 633–655 (1985).
[CrossRef]

1984 (1)

1983 (1)

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

Belanger, P. A.

Borisov, V. V.

V. V. Borisov, I. I. Simonenko, “Transient waves generated by a source on a circle,” J. Phys. A: Math. Gen. 27, 6243–6252 (1994).

V. V. Borisov, A. B. Utkin, “On formation of focus wave modes,” J. Phys. A: Math. Gen. 27, 2587–2590 (1994).

V. V. Borisov, A. B. Utkin, “Electromagnetic fields produced by the spike pulse of hard radiation,” J. Phys. A: Math. Gen. 26, 4081–4085 (1993).

Brittingham, J. N.

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

Donnelly, R.

M. Palmer, R. Donnelly, “Focused waves and the scalar wave equation,” J. Math. Phys. 34, 4007–4013 (1993).
[CrossRef]

R. Donnelly, R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. London Ser. A 440, 541–565 (1993).
[CrossRef]

Durnin, J.

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

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]

Garabedian, P. R.

P. R. Garabedian, Partial Differential Equations (Wiley, New York, 1964).

Gervois, A.

A. Gervois, H. Navelet, “Integrals of three Bessel functions and Legendre functions. I and II,” J. Math. Phys. 26, 633–655 (1985).
[CrossRef]

Gori, F.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Kenney, C. S.

Miceli, J. J.

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

Navelet, H.

A. Gervois, H. Navelet, “Integrals of three Bessel functions and Legendre functions. I and II,” J. Math. Phys. 26, 633–655 (1985).
[CrossRef]

Overfelt, P. L.

P. L. Overfelt, “Focus wave mode solutions of the inhomogeneous n-dimensional scalar wave equation,” Phys. Rev. E 52, 4387–4392 (1995) (Part B).
[CrossRef]

P. L. Overfelt, “Continua of localized wave solutions via a complex similarity transformation,” Phys. Rev. E 47, 4430–4438 (1993).
[CrossRef]

P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–745 (1991).
[CrossRef]

P. L. Overfelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
[CrossRef] [PubMed]

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Palmer, M.

M. Palmer, R. Donnelly, “Focused waves and the scalar wave equation,” J. Math. Phys. 34, 4007–4013 (1993).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Simonenko, I. I.

V. V. Borisov, I. I. Simonenko, “Transient waves generated by a source on a circle,” J. Phys. A: Math. Gen. 27, 6243–6252 (1994).

Utkin, A. B.

V. V. Borisov, A. B. Utkin, “On formation of focus wave modes,” J. Phys. A: Math. Gen. 27, 2587–2590 (1994).

V. V. Borisov, A. B. Utkin, “Electromagnetic fields produced by the spike pulse of hard radiation,” J. Phys. A: Math. Gen. 26, 4081–4085 (1993).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1944), Chaps. 11 and 16.

Ziolkowski, R. W.

R. Donnelly, R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. London Ser. A 440, 541–565 (1993).
[CrossRef]

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

J. Appl. Phys. (1)

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

J. Math. Phys. (3)

A. Gervois, H. Navelet, “Integrals of three Bessel functions and Legendre functions. I and II,” J. Math. Phys. 26, 633–655 (1985).
[CrossRef]

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

M. Palmer, R. Donnelly, “Focused waves and the scalar wave equation,” J. Math. Phys. 34, 4007–4013 (1993).
[CrossRef]

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

J. Phys. A: Math. Gen. (3)

V. V. Borisov, A. B. Utkin, “Electromagnetic fields produced by the spike pulse of hard radiation,” J. Phys. A: Math. Gen. 26, 4081–4085 (1993).

V. V. Borisov, I. I. Simonenko, “Transient waves generated by a source on a circle,” J. Phys. A: Math. Gen. 27, 6243–6252 (1994).

V. V. Borisov, A. B. Utkin, “On formation of focus wave modes,” J. Phys. A: Math. Gen. 27, 2587–2590 (1994).

Opt. Commun. (1)

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Phys. Rev. A (1)

P. L. Overfelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
[CrossRef] [PubMed]

Phys. Rev. E (2)

P. L. Overfelt, “Focus wave mode solutions of the inhomogeneous n-dimensional scalar wave equation,” Phys. Rev. E 52, 4387–4392 (1995) (Part B).
[CrossRef]

P. L. Overfelt, “Continua of localized wave solutions via a complex similarity transformation,” Phys. Rev. E 47, 4430–4438 (1993).
[CrossRef]

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

R. Donnelly, R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. London Ser. A 440, 541–565 (1993).
[CrossRef]

Other (3)

P. R. Garabedian, Partial Differential Equations (Wiley, New York, 1964).

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1944), Chaps. 11 and 16.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

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Equations (53)

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2τ2-2z2-1ρρρ ρψ(ρ, z, τ)=f(ρ, z, τ),
f(ρ, z, t)=δ(τ-z)g(τ, z)h(z)J0(κρ)exp-βρ2a1.
f(ρ, 0, 0)=g(0, 0)J0(κρ)exp-βρ2a1,
Θ(s, z, τ)=0dρρJ0(sρ)Θ(ρ, z, τ),
Θ(ρ, z, τ)=0ds sJ0(sρ)Θ(s, z, τ),
f(s, z, τ)=a12βq(z, τ)exp-a1(s2+κ2)4βI0κ a1s2β,
q(z, τ)=δ(τ-z)g(z, τ)h(z).
ψ(s, z, τ)=120τdτzLZUdzf(s, z, τ)J0(su),
zL=z+τ-τ=τ-(τ-z),
zU=z-τ+τ=-τ+(τ+z).
ψ(ρ, z, τ)
=a14βexp-κ2a14β0τdτzLzUdzq(z, τ)×0ds s J0(ρs)J0(us)I0κ a1s2βexp-a1s24β.
J0(ρs)J0iκa1s2β=1π0πdγJ0(sω¯),
ω¯=ρ2-κa12β2-i κa1ρβcos γ1/2,
ψ(ρ, z, τ)
=a14πβexp-κ2a14β0πdγ0τdτzLzUdz q(z, τ)×0dssJ0(ω¯s)J0(us)exp-a1s24β,
ψ(ρ, z, τ)
=12πexp-βρ2a10πdγ exp(iκρ cos γ)×0τdτzLzUdz q(z, τ)J0i2βωu¯a1exp-βu2a1,
ξ1=τ-z;ξ2=τ+z,
ξ1=τ-z;ξ2=τ+z,
ψ(ρ, ξ1, ξ2)
=-14πexp-βa1(ρ2+ξ1ξ2)0πdγ exp(iκρ cos γ)×0ξ2dξ2g(0, ξ2)J0i2βω¯a1ξ1(ξ2-ξ2)×expβξ1ξ2a1.
g(0, ξ2)=exp(iβξ2),
vy=i 2β ω¯a1ξ1(ξ2-ξ2)
v=i 2β ω¯a1ξ1ξ2,y=1-ξ2ξ2,
w=2βξ21-iξ1a1,
ψ(ρ, ξ1, ξ2)
=-ξ22πexp-βa1(ρ2+ξ1ξ2)0πdγ exp(iκρ cos γ)×01dy y expiw2(1-y2)J0(vy),
ψ(ρ, ξ1, ξ2)
=-ξ22πwexp-βa1(ρ2+ξ1ξ2)×0πdγ exp(iκρ cos γ)[U1(w, v)+iU2(w, v)].
U1(w, v)=sinw2+v22w+V1(w, v),
U2(w, v)=-cosw2+v22w+V0(w, v),
ψ(ρ, ξ1, ξ2)=ia14πβ(a1-iξ1)exp-βa1(ρ2+ξ1ξ2)×(M1+M2+M3),
M1=0πdγ exp(iκρ cos γ)expiw2+v22w,
M2=i0πdγ V1(w, v)exp(iκρ cos γ),
M3=-0πdγ V0(w, v)exp(iκρ cos γ).
M1=expiβξ2a1(a1-iξ1)×expiξ1(a1-iξ1)κ2a14β-βρ2a1×0πdγ expiκρa1(a1-iξ1)cos γ
M1=expiβξ2a1(a1-iξ1)expiξ1(a1-iξ1)×κ2a14β-βρ2a1J0κρa1a1-iξ1.
ψj(ρ, ξ1, ξ2)=ia14πβ(a1-iξ1)×exp-βa1(ρ2+ξ1ξ2)Mj;
j=1, 2, 3,
ψ1(ρ, ξ1, ξ2)=ia14πβ(a1-iξ1)J0κρa1a1-iξ1×expiβξ2-βρ2(a1-iξ1)+ia1κ2ξ14β(a1-iξ1)=ψBG(ρ, ξ1, ξ2).
ww*>vv*
ξ1ξ2+a12ξ2ξ1>ρ4+κa12β4+12κρa1β2 cos 2γ1/2,
ρ4+κa12β4+12κρa1β2 cos 2γ1/2=ρeff2
τ2>ρeff2+z2=reff2.
ψ2+ψ3=ia14πβ(a1-iξ1)exp-βa1(ρ2+ξ1ξ2)×m=0 (-1)mw2m[i(-1)1+2mS1+2m-S2m],
Sν=πk=0 ζ(ν-k)/2k!2kh12kJν-k(ζ)×j=0kkjik-2jJk-2j(κρ),
ζ=4β2ξ1ξ2κ2β2-ρa12,
h1=i4β2ξ1ξ2κρβa1.
ψ=-a14πβ(a1-iξ1)exp-βa1(ρ2+ξ1ξ2)×m=0(-1)mw2m(S1+2m+i S2+2m),
Sν=πk=0 ζ-(ν+k)/2k!2k-h12kJν+k(ζ)×j=0kkjik-2jJk-2j(κρ),
A=-βa1(ρ2+τ2-z2).
z2τ2<ρeff2+z2

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