Abstract

A general concept of the vectorial spectral analysis of the nonstationary optical fields representing exact solutions to the Maxwell equations is proposed. The method provides the possibility of examining the free-space spatiotemporal evolution of the electromagnetic field in relation to its dependence on the composition of its temporal- and angular-frequency spectra. Particular attention is given to the electromagnetic pulses whose monochromatic components are nondiffracting fields.

© 1998 Optical Society of America

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References

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  1. N. George, S. Radic, “Theory for the propagation of short electromagnetic pulses,” Opt. Commun. 139, 1–6 (1997).
    [CrossRef]
  2. Z. Wang, Z. Xu, Z. Zhang, “Diffraction integral formulas of the pulsed wave field in the temporal domain,” Opt. Lett. 22, 354–356 (1997).
    [CrossRef] [PubMed]
  3. B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am. A 8, 705–717 (1991).
    [CrossRef]
  4. R. W. Ziolkowski, “Localized wave physics and engineering,” Phys. Rev. A 44, 3960–3984 (1991).
    [CrossRef] [PubMed]
  5. P. Hillion, “Distortion-free progressing waves,” Europhys. Lett. 33, 7–10 (1996).
    [CrossRef]
  6. J. Fagerholm, A. T. Friberg, D. P. Morgan, M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
    [CrossRef]
  7. A. T. Friberg, J. Fagerholm, M. Salomaa, “Space-frequency analysis of nondiffracting pulses,” Opt. Commun. 136, 207–212 (1997).
    [CrossRef]
  8. P. Hillion, “Nondispersive wave representation of photons,” Phys. Rev. A 172, 1–2 (1992).
  9. M. M. Wefers, K. A. Nelson, A. M. Weiner, “Multidimensional shaping of ultrafast optical waveforms,” Opt. Lett. 21, 746–748 (1996).
    [CrossRef] [PubMed]
  10. J. Lu, J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
    [CrossRef] [PubMed]
  11. J. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
    [CrossRef] [PubMed]
  12. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1961).
  13. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).
  14. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  15. H. Sonajalg, P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators,” Opt. Lett. 21, 1162–1164 (1996).
    [CrossRef] [PubMed]
  16. Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
    [CrossRef]
  17. R. D. Romea, W. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D 42, 1807–1818 (1990).
    [CrossRef]

1997

N. George, S. Radic, “Theory for the propagation of short electromagnetic pulses,” Opt. Commun. 139, 1–6 (1997).
[CrossRef]

Z. Wang, Z. Xu, Z. Zhang, “Diffraction integral formulas of the pulsed wave field in the temporal domain,” Opt. Lett. 22, 354–356 (1997).
[CrossRef] [PubMed]

A. T. Friberg, J. Fagerholm, M. Salomaa, “Space-frequency analysis of nondiffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

1996

P. Hillion, “Distortion-free progressing waves,” Europhys. Lett. 33, 7–10 (1996).
[CrossRef]

J. Fagerholm, A. T. Friberg, D. P. Morgan, M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

M. M. Wefers, K. A. Nelson, A. M. Weiner, “Multidimensional shaping of ultrafast optical waveforms,” Opt. Lett. 21, 746–748 (1996).
[CrossRef] [PubMed]

H. Sonajalg, P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators,” Opt. Lett. 21, 1162–1164 (1996).
[CrossRef] [PubMed]

1995

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

1992

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

J. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
[CrossRef] [PubMed]

P. Hillion, “Nondispersive wave representation of photons,” Phys. Rev. A 172, 1–2 (1992).

1991

B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am. A 8, 705–717 (1991).
[CrossRef]

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

1990

R. D. Romea, W. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D 42, 1807–1818 (1990).
[CrossRef]

Bouchal, Z.

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Fagerholm, J.

A. T. Friberg, J. Fagerholm, M. Salomaa, “Space-frequency analysis of nondiffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

J. Fagerholm, A. T. Friberg, D. P. Morgan, M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Friberg, A. T.

A. T. Friberg, J. Fagerholm, M. Salomaa, “Space-frequency analysis of nondiffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

J. Fagerholm, A. T. Friberg, D. P. Morgan, M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

George, N.

N. George, S. Radic, “Theory for the propagation of short electromagnetic pulses,” Opt. Commun. 139, 1–6 (1997).
[CrossRef]

Greenleaf, J. F.

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

J. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
[CrossRef] [PubMed]

Hafizi, B.

Hillion, P.

P. Hillion, “Distortion-free progressing waves,” Europhys. Lett. 33, 7–10 (1996).
[CrossRef]

P. Hillion, “Nondispersive wave representation of photons,” Phys. Rev. A 172, 1–2 (1992).

Kimura, W.

R. D. Romea, W. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D 42, 1807–1818 (1990).
[CrossRef]

Lu, J.

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

J. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
[CrossRef] [PubMed]

Morgan, D. P.

J. Fagerholm, A. T. Friberg, D. P. Morgan, M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Nelson, K. A.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

Olivi´k, M.

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Radic, S.

N. George, S. Radic, “Theory for the propagation of short electromagnetic pulses,” Opt. Commun. 139, 1–6 (1997).
[CrossRef]

Romea, R. D.

R. D. Romea, W. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D 42, 1807–1818 (1990).
[CrossRef]

Saari, P.

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Salomaa, M.

A. T. Friberg, J. Fagerholm, M. Salomaa, “Space-frequency analysis of nondiffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

J. Fagerholm, A. T. Friberg, D. P. Morgan, M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Sonajalg, H.

Sprangle, P.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1961).

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Wang, Z.

Wefers, M. M.

Weiner, A. M.

Xu, Z.

Zhang, Z.

Ziolkowski, R. W.

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

Europhys. Lett.

P. Hillion, “Distortion-free progressing waves,” Europhys. Lett. 33, 7–10 (1996).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control

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

J. Lu, J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
[CrossRef] [PubMed]

J. Mod. Opt.

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

N. George, S. Radic, “Theory for the propagation of short electromagnetic pulses,” Opt. Commun. 139, 1–6 (1997).
[CrossRef]

A. T. Friberg, J. Fagerholm, M. Salomaa, “Space-frequency analysis of nondiffracting pulses,” Opt. Commun. 136, 207–212 (1997).
[CrossRef]

Opt. Lett.

Phys. Rev. A

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

P. Hillion, “Nondispersive wave representation of photons,” Phys. Rev. A 172, 1–2 (1992).

Phys. Rev. D

R. D. Romea, W. Kimura, “Modeling of inverse Čerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D 42, 1807–1818 (1990).
[CrossRef]

Phys. Rev. E

J. Fagerholm, A. T. Friberg, D. P. Morgan, M. Salomaa, “Angular-spectrum representation of nondiffracting X waves,” Phys. Rev. E 54, 4347–4352 (1996).
[CrossRef]

Other

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1961).

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (Wiley, New York, 1991).

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

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Figures (5)

Fig. 1
Fig. 1

Electromagnetic pulse propagating with the temporal spread (sin θ=ω0 sin θ0/ω, ω0=2πe14 rad/s, τ=25 fs, and sin θ0=0.9): (a) and (b) spatiofrequency distribution of the spectral energy density of the electric field, (c) and (d) spatiotemporal distribution of the electric-field energy density at the plane z=0, and (e) and (f) plane z=0.3.

Fig. 2
Fig. 2

Nonspreading electromagnetic pulse (θ=θ0, ω0=2πe14 rad/s, and τ=25 fs): (a) and (b) spatiotemporal distribution of the electric-field energy density for θ0=0.4, (c) and (d) θ0=0.9.

Fig. 3
Fig. 3

Monochromatic component of the electromagnetic field with linearly polarized electric vector: (a) needle plot of the transverse part of the vector E¯(r, ω), (b) needle plot of the transverse part of the vector H¯(r, ω), (c) snapshot of the longitudinal component of the Poynting vector, (d) longitudinal component of the electric field |E¯z(r, ω)|2.

Fig. 4
Fig. 4

Monochromatic component of the electromagnetic field with azimuthally polarized electric vector: (a) needle plot of the transverse part of E¯(r, ω), (b) needle plot of the transverse part of H¯(r, ω), (c) snapshot of the longitudinal component of the Poynting vector.

Fig. 5
Fig. 5

Monochromatic component of the general TE field (s=1): (a) needle plot of the transverse part of E¯(r, ω), (b) needle plot of the transverse part of H¯(r, ω), (c) snapshot of the longitudinal component of the Poynting vector.

Equations (73)

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E(r, t)=(·Π(e))-μ0×Π(m)t-1c22Π(e)t2,
H(r, t)=(·Π(m))+0×Π(e)t-1c22Π(m)t2,
Π(j)=xpαp(j)Up+yqβq(j)Uq+zsγs(j)Us,
j=e, m,
Q(r, t)=-Q¯(r, ω)exp(iωt)dω,
Q¯(r, ω)=-q(K, ω)exp(-iK·r)dK,
k2-Kx2-Ky2-Kz2=0,
e(K, ω)=k2Π(e)-ωμ0 K×Π(m)-K(K·Π(e)),
h(K, ω)=k2Π(m)+ω0K×Π(e)-K(K·Π(m)).
Π(j)(K, ω)=Π(j)(K, ω)+Π(j)(K, ω),
K·Π(j)=0,j=e, m,
K×Π(j)=0,j=e, m.
e(K, ω)=k2Π(e)-ωμ0K×Π(m),
h(K, ω)=k2Π(m)+ω0K×Π(e).
e·K=0,
h·K=0,
e·h=0.
E·z=0,
H·z=0.
h·(K×z)=0,
e·(K×z)=0.
(K×Π(e))·(K×z)=0,
Π(m)·(K×z)=0.
Π(e)=0,
Π(m)×z=0.
Π(m)=0,
Π(e)×z=0.
e(K, ω)=-ωμ0z(m)(xKy-yKx),
h(K, ω)=-z(m)[Kz(xKx+yKy)-z(Kx2+Ky2)],
z(m)(K, ω)=12π3-¯z(m)(r, ω)exp(iK·r)dr.
E(r, t)=-μ0-ωz(m)×(xKy-yKx)exp[i(ωt-K·r)]dKdω,
H(r, t)=--z(m)[Kz(xKy+yKx)-zKT2]×exp[i(ωt-K·r)]dKdω.
Kx=KT cos ψ,Ky=KT sin ψ.
x=R cos ϕ,y=R sin ϕ,
E(R, ϕ, z, t)=-2πμ0i0--ωz(m)×(x sin ϕ-y cos ϕ)×J1(KTR)exp[i(ωt-Kzz)]×KT2dKTdKzdω,
H(R, ϕ, z, t)=-2πi0--z(m)×[Kz(x cos ϕ+y sin ϕ)J1(KTR)+izKTJ0(KTR)]×exp[i(ωt-Kzz)]KT2dKTdKzdω.
e(K, ω)=-z(e)[Kz(xKx+yKy)-zKT2],
h(K, ω)=ω0z(e)(xKy-yKx),
E(R, ϕ, z, t)=-2πi0--z(e)×[Kz(x cos ϕ+y sin ϕ)J1(KTR)+izKTJ0(KTR)]×exp[i(ωt-Kzz)]KT2dKTdKzdω,
H(R, ϕ, z, t)=2π0i0--ωz(e)×(x sin ϕ-y cos ϕ)×J1(KTR)exp[i(ωt-Kzz)]×KT2dKTdKzdω,
Uj(r, t)=-U¯j(r, ω)exp(iωt)dω,
U¯j(r, ω)=12π2-uj(KT, z, ω)×exp(-iKT·rT)dKT,
uj(KT, z, ω)=uj0(KT, ω)exp(-iKzz),
uj0(KT, ω)=12π3-Uj0(rT, t)×exp[-i(ωt-KT·rT)]drTdt.
Π(j)(KT, z, ω)=Π0(j)(KT, ω)exp(-iKzz),
Π0(j)(KT, ω)=xpαp(j)up0(KT, ω)+yqβq(j)uq0(KT, ω)+zsγs(j)us0(KT, ω).
e(KT, z, ω)=k2Π(e)-ωμ0KT×Π(m)+(·Π(e))-KT(KT·Π(e))-i[KT(·Π(e))+(KT·Π(e))+ωμ0×Π(m)],
h(KT, z, ω)=k2Π(m)+ω0KT×Π(e)+(·Π(m))-KT(KT·Π(m))-i[KT(·Π(m))+(KT·Π(m))-ω0×Π(e)],
e(KT, z, ω)=e0(KT, ω)exp(-iKzz),
h(KT, z, ω)=h0(KT, ω)exp(-iKzz),
e0(KT, ω)=k2Π0(e)-KT(KT·Π0(e))-KTKz(z·Π0(e))-ωμ0(Kzz×Π0(m)+KT×Π0(m))-z[Kz(KT·Π0(e))+Kz2(z·Π0(e))],
h0(KT, ω)=k2Π0(m)-KT(KT·Π0(m))-KTKz(z·Π0(m))+ω0(Kzz×Π0(e)+KT×Π0(e))-z[Kz(KT·Π0(m))+Kz2(z·Π0(m))].
U¯j(R, ϕ, z, ω)=Aj(R, ϕ, ω)exp[-ikj(ω)z],
U¯j(R, ϕ, z, ω)=002πuj(KT, ψ, ω)×exp[iKTR cos(ϕ-ψ)]exp(-iKzz)KTdKTdψ,
uj(KT, ψ, ω)=aj(KT, ψ, ω)δ(KT-K0).
kj(ω)=(k2-K02)1/2,
Aj(R, ϕ, ω)=02πaj(K0, ψ, ω)×exp[iK0R cos(ϕ-ψ)]K0dψ.
K0(ω)=k sin θ(ω).
Π0(j)(KT, ψ, ω)=xαp(j)up0(KT, ψ, ω)+yβq(j)uq0(KT, ψ, ω)+zγs(j)us0(KT, ψ, ω),j=e, m,
aj(KT, ψ, ω)=exp(ijψ)S(ω),
S(ω)=exp-(ω0-ω)2τ22,
E¯(R, ϕ, z, ω)=2πk3S(ω)×e¯(R, ϕ, ω)sin θ exp(-ikz cos θ),
ex¯(R, ϕ, ω)
=-ipαp(e) exp(ipϕ)(Jp-12{Jp-12[Jp+2 exp(i2ϕ)+Jp-2 exp(-i2ϕ)]}sin2 θ)-iq exp(iqϕ)×cμ0βq(m)Jq cos θ-14βq(e)[Jq+2 exp(i2ϕ)-Jq-2 exp(-i2ϕ)]sin2 θ+12is sin θ exp(isϕ)×{cμ0γs(m)[Js+1 exp(iϕ)+Js-1 exp(-iϕ)]+iγs(e)[Js+1 exp(iϕ)-Js-1 exp(-iϕ)]cos θ},
ey¯(R, ϕ, ω)
=ip exp(ipϕ)cμ0αp(m)Jp cos θ+i4αp(e)[Jp+2 exp(i2ϕ)-Jp-2 exp(-i2ϕ)]sin2 θ-iqβq(e) exp(iqϕ)(Jq-12{Jq+12[Jq+2 exp(i2ϕ)+Jq-2 exp(-i2ϕ)]}sin2 θ)+12is sin θ exp(isϕ)×{icμ0γs(m)[Js+1 exp(iϕ)-Js-1 exp(-iϕ)]-γs(e)[Js+1 exp(iϕ)+Js-1 exp(-iϕ)]cos θ},
ez¯(R, ϕ, ω)
=-12ip+1 exp(ipϕ)sin θ{icμ0αp(m)×[Jp+1 exp(iϕ)+Jp-1 exp(-iϕ)]-αp(e)[Jp+1 exp(iϕ)-Jp-1 exp(-iϕ)]cos θ}+12iq+1 exp(iqϕ)sin θ{cμ0βq(m)×[Jq+1 exp(iϕ)-Jq-1 exp(-iϕ)]+iβq(e)[Jq+1 exp(iϕ)+Jq-1 exp(-iϕ)]cos θ}+γs(e)isJs exp(isϕ)cos2 θ.
sin θ(ω)=ω0ωsin θ0,
Dω=2π d2kdω2.
Dω=2πωvg cos2 θ(ω),
vg=dωdk=c cos θ(ω).
E(R, ϕ, z, t)E[R, ϕ, t-(z/c)cos θ0],

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