Abstract

Assuming a disk source distribution moving in a straight line along the z axis at some velocity slower than the speed of light, an approximate Bessel–Gauss pulse solution to the inhomogeneous wave equation has been determined. This approximate pulse propagates in a specific region of space–time and is a long-time duration or steady-state solution of the inhomogeneous wave equation. The localization properties of this approximate waveform depend on the normalized speed of the source distribution. For source speeds close to the speed of light, the waveform is highly localized. As the source speed decreases, the scalar wave becomes less localized along the direction of propagation.

© 1999 Optical Society of America

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References

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  1. P. L. Overfelt, “Generation of a Bessel–Gauss pulse from a moving disk source distribution,” J. Opt. Soc. Am. A 14, 1087–1091 (1997).
    [CrossRef]
  2. P. L. Overfelt, “Bessel–Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
    [CrossRef] [PubMed]
  3. J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
    [CrossRef]
  4. V. V. Borisov, I. I. Simonenko, “Formation of Bessel–Gauss focus wave modes,” Can. J. Phys. 75, 573–579 (1997).
  5. P. L. Overfelt, C. S. Kenney, “Can a Bessel–Gauss pulse be generated by a disk source distribution moving much slower than the speed of light?” in Proceedings of the 1998 International Symposium on Electromagnetic Theory (International Union of Radio Science, Aristotle University, Thessaloniki, Greece, May1998), Vol. 2, pp. 802–804.
  6. V. V. Borisov, I. I. Simonenko, “Transient waves generated by a source on a circle,” J. Phys. A Math. Gen. 27, 6243–6252 (1994).
    [CrossRef]
  7. V. V. Borisov, “Formation of the beamlike electromagnetic waves,” in Proceedings of the 1998 International Symposium on Electromagnetic Theory (International Union of Radio Science, Aristotle University, Thessaloniki, Greece, May1998), Vol. 2, pp. 599–601.
  8. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1944).
  9. 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).
    [CrossRef]
  10. V. V. Borisov, E. I. Ivanov, I. I. Simonenko, V. B. Smirnov, “On electromagnetic waves generated by the spike pulse of hard radiation,” presented at the 15th International Conference on Coherent and Nonlinear Optics (St. Petersburg, Russia, June 27–July 1, 1995).

1997

V. V. Borisov, I. I. Simonenko, “Formation of Bessel–Gauss focus wave modes,” Can. J. Phys. 75, 573–579 (1997).

P. L. Overfelt, “Generation of a Bessel–Gauss pulse from a moving disk source distribution,” J. Opt. Soc. Am. A 14, 1087–1091 (1997).
[CrossRef]

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).
[CrossRef]

1993

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).
[CrossRef]

1991

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

1983

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

Borisov, V. V.

V. V. Borisov, I. I. Simonenko, “Formation of Bessel–Gauss focus wave modes,” Can. J. Phys. 75, 573–579 (1997).

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

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).
[CrossRef]

V. V. Borisov, “Formation of the beamlike electromagnetic waves,” in Proceedings of the 1998 International Symposium on Electromagnetic Theory (International Union of Radio Science, Aristotle University, Thessaloniki, Greece, May1998), Vol. 2, pp. 599–601.

V. V. Borisov, E. I. Ivanov, I. I. Simonenko, V. B. Smirnov, “On electromagnetic waves generated by the spike pulse of hard radiation,” presented at the 15th International Conference on Coherent and Nonlinear Optics (St. Petersburg, Russia, June 27–July 1, 1995).

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]

Ivanov, E. I.

V. V. Borisov, E. I. Ivanov, I. I. Simonenko, V. B. Smirnov, “On electromagnetic waves generated by the spike pulse of hard radiation,” presented at the 15th International Conference on Coherent and Nonlinear Optics (St. Petersburg, Russia, June 27–July 1, 1995).

Kenney, C. S.

P. L. Overfelt, C. S. Kenney, “Can a Bessel–Gauss pulse be generated by a disk source distribution moving much slower than the speed of light?” in Proceedings of the 1998 International Symposium on Electromagnetic Theory (International Union of Radio Science, Aristotle University, Thessaloniki, Greece, May1998), Vol. 2, pp. 802–804.

Overfelt, P. L.

P. L. Overfelt, “Generation of a Bessel–Gauss pulse from a moving disk source distribution,” J. Opt. Soc. Am. A 14, 1087–1091 (1997).
[CrossRef]

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

P. L. Overfelt, C. S. Kenney, “Can a Bessel–Gauss pulse be generated by a disk source distribution moving much slower than the speed of light?” in Proceedings of the 1998 International Symposium on Electromagnetic Theory (International Union of Radio Science, Aristotle University, Thessaloniki, Greece, May1998), Vol. 2, pp. 802–804.

Simonenko, I. I.

V. V. Borisov, I. I. Simonenko, “Formation of Bessel–Gauss focus wave modes,” Can. J. Phys. 75, 573–579 (1997).

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

V. V. Borisov, E. I. Ivanov, I. I. Simonenko, V. B. Smirnov, “On electromagnetic waves generated by the spike pulse of hard radiation,” presented at the 15th International Conference on Coherent and Nonlinear Optics (St. Petersburg, Russia, June 27–July 1, 1995).

Smirnov, V. B.

V. V. Borisov, E. I. Ivanov, I. I. Simonenko, V. B. Smirnov, “On electromagnetic waves generated by the spike pulse of hard radiation,” presented at the 15th International Conference on Coherent and Nonlinear Optics (St. Petersburg, Russia, June 27–July 1, 1995).

Utkin, A. B.

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).
[CrossRef]

Watson, G. N.

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

Can. J. Phys.

V. V. Borisov, I. I. Simonenko, “Formation of Bessel–Gauss focus wave modes,” Can. J. Phys. 75, 573–579 (1997).

J. Appl. Phys.

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

J. Opt. Soc. Am. A

J. Phys. A Math. Gen.

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

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).
[CrossRef]

Phys. Rev. A

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

Other

P. L. Overfelt, C. S. Kenney, “Can a Bessel–Gauss pulse be generated by a disk source distribution moving much slower than the speed of light?” in Proceedings of the 1998 International Symposium on Electromagnetic Theory (International Union of Radio Science, Aristotle University, Thessaloniki, Greece, May1998), Vol. 2, pp. 802–804.

V. V. Borisov, E. I. Ivanov, I. I. Simonenko, V. B. Smirnov, “On electromagnetic waves generated by the spike pulse of hard radiation,” presented at the 15th International Conference on Coherent and Nonlinear Optics (St. Petersburg, Russia, June 27–July 1, 1995).

V. V. Borisov, “Formation of the beamlike electromagnetic waves,” in Proceedings of the 1998 International Symposium on Electromagnetic Theory (International Union of Radio Science, Aristotle University, Thessaloniki, Greece, May1998), Vol. 2, pp. 599–601.

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

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Figures (1)

Fig. 1
Fig. 1

Bessel–Gauss approximation as source speed decreases (τ=0, κ=3.0, a1=0.5, β1=0.1666). (a) β=0.98, (b) β=0.9, (c) β=0.8, and (d) β=0.7.

Equations (51)

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2τ2-2z2-1ρ ρ ρψ(ρ, z, τ)=f(ρ, z, τ),
f(ρ, z, τ)=δ(z-βτ)h(τ)g(τ, z)J0(κρ)exp(-β1ρ2/a1),
zβ=z-βτ1-β2;τβ=τ-βz1-β2
f(ρ, zβ, τβ)=q(zβ, τβ)J0(κρ)exp(-β1ρ2/a1),
q(zβ, τβ)=δ(zβ1-β2)hτβ+βzβ1-β2×gτβ+βzβ1-β2, zβ+βτβ1-β2.
ψ(ρ, zβ, τβ)=exp[(-a1κ2)/(4β1)]2π1-β2 0πdγ0τβdτβ×zβ-τβ+τβzβ+τβ-τβdzβδ(zβ)hτβ+βzβ1-β2×gτβ+βzβ1-β2, zβ+βτβ1-β2×exp-β1(u2+ϖ2)a1J0i2β1uϖa1,
ϖ2=ρ2-κa12β12-iκa1ρ cos γβ1,
u2=(τβ-τβ)2-(zβ-zβ)2.
gτβ1-β2=expiβ1τβ1-β2,
ψ(ρ, zβ, τβ)=exp(-β1ρ2/a1)2π1-β2×0πdγ exp(iκρ cos γ)0τβ-zβdτβ expiβ1τβ1-β2-β1u2a1×J0i2β1ϖua1,
u2=(τβ-τβ)2-zβ2.
ξ1β=τβ-zβ,
ξ2β=τβ+zβ,
ξ1β=ξ2β=τβ,
u2=ξ1βξ2β-(ξ1β+ξ2β)ξ1β+(ξ1β)2.
uξ1βξ2β(1-μξ1β)1/2,
μ=ξ1β+ξ2βξ1βξ2β.
ν=i2β1ϖa1 ξ1βξ2β,
y=(1-μξ1β)1/2,
ψ(ρ, zβ, τβ)exp-β1a1 (ρ2+ξ1βξ2β)πμ1-β2×0πdγ exp(iκρ cos γ)×01dy y expiw2 (1-y2)J0(νy)
w=2β1μ 11-β2-ia1 (ξ1β+ξ2β).
ψ(ρ, zβ, τβ)exp[-(β1/a1)(ρ2+ξ1βξ2β)]πμ1-β2×0πdγ exp(iκρ cos γ)[U1(w, ν)+iU2(w, ν)];|w|<|ν|,
ψ(ρ, zβ, τβ)exp[-(β1/a1)(ρ2+ξ1βξ2β)]iπμw1-β2×[M1+M2+M3];|w|>|ν|,
ψ1(ρ, zβ, τβ)
exp[-(β1/a1)(ρ2+ξ1βξ2β)]iπμw1-β2×expiw20πdγ exp(iκρ cos γ+iν2/2w),
ψ1(ρ, z, τ)ia12β1V J0κρa1V×exp-β1ρ2V+iκ2a1(τ-βz)2β1V+iβ1(τ2-z2)2(τ-βz);0<β<1,
V=[a1-i2(τ-βz)];0<β<1.
ψ1(ρ, z, τ)ia14β1V J0κa1ρV×exp-β1ρ2V+iκ2a1(τ-βz)4β1V+iβ1(τ2-z2)(τ-βz);0<β<1,
V=[a1-i(τ-βz)].
ξ1β2=ξ2β2=τβ20,
ξ1β2=(τβ-zβ)20.
z=βτ.
τ20
β1.
(τ-z)21
0<(τβ-zβ)<1.
0<(τ-z)<1-β1+β
0<(τ-z)21-β1+β,
2ψ1(ρ, z, τ)=2-2τ2ψ1(ρ, z, τ)=0.
2ψ1(ρ, z, τ)=(L1+L2)ψ1(ρ, z, τ)=0,
L1=2ρ2+1ρ ρ,
L2=2z2-2τ2.
(L1+L2)ψ1(ρ, z, τ)0.
L1ψ1(ρ, z, τ)=ia14β1 exp(G)4β1ρκa1J1[(κa1ρ)/V](V)3-4β1J0[(κa1ρ)/V](V)2-[(κa1)2-4(β1ρ)2] J0[(κa1ρ)/V](V)3,
G=-β1ρ2V+iκ2a1(τ-βz)4β1V+iβ1(τ2-z2)(τ-βz).
L2ψ1(ρ, z, τ)=ia14β1 exp(G)4β1J0[(κa1ρ)/V](V)2-4β1ρκa1J1[(κa1ρ)/V](V)3+[(κa1)2-4(β1ρ)2] J0[(κa1ρ)/V](V)3+(1-β2)q1(ρ, z, τ).
2ψ1(ρ, z, τ)=(1-β2) ia14β1 exp(G)q1(ρ, z, τ).
q1(ρ, z, τ)=J0κa1V2if1(ρ, z, τ)V-[f1(ρ, z, τ)]2+2(V)2+(κρa1)2(V)4+f2(ρ, z, τ)+J1[(κa1ρ)/V](V)3×-2ia1κρf1(ρ, z, τ)+(1-2a1κρ)V
f1(ρ, z, τ)=iβ1(τ2-z2)(τ-βz)2+i{β1ρ2-[(a1)2κ2]/(4β1)}(V)2,
f2(ρ, z, τ)=2iβ1(τ2-z2)(τ-βz)3+2{β1ρ2-[(a1)2κ2]/(4β1)}(V)3.
limβ1 2ψ1(ρ, z, τ)=0,

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