Abstract

We study a family of nonseparable solutions of the three-plus-one-dimensional paraxial wave equation in free space that represent ultrashort waveforms localized in time and space. We use them to develop a simple model for the spatiotemporal structure of ultrashort pulsed light beams, with only one or a few field oscillations of a nonsinusoidal nature. In the many-oscillation limit the model approaches the widely used separable models of modulated sinusoidal oscillations in time and of a Gaussian beam in space.

© 1999 Optical Society of America

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  1. A. Stingl, C. Spielmann, F. Krausz, and R. Szipöcs, “Generation of 11-fs pulses from a Ti:sapphire laser without the use of prisms,” Opt. Lett. 19, 204–206 (1994).
    [CrossRef] [PubMed]
  2. D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
    [CrossRef]
  3. M. S. Pshenichnikov, W. P. De Boeij, and D. A. Wiersma, “Generation of 13-fs, 5-MW pulses from a cavity-dumped Ti:sapphire laser,” Opt. Lett. 19, 572–574 (1994).
    [CrossRef] [PubMed]
  4. P. U. Jepsen and S. R. Keiding, “Radiation patterns from lens-coupled terahertz antennas,” Opt. Lett. 20, 807–809 (1995).
    [CrossRef] [PubMed]
  5. A. E. Kaplan and P. L. Shkolnikov, “Electromagnetic bubbles and shock waves: unipolar, nonoscillating EM solitons,” Phys. Rev. Lett. 75, 2316–2319 (1995).
    [CrossRef] [PubMed]
  6. A. E. Kaplan, “Diffraction-induced transformation of near-cycle and subcycle pulses,” J. Opt. Soc. Am. B 15, 951–956 (1998).
    [CrossRef]
  7. E. M. Belenov and A. V. Nazarkin, “Transient diffraction and precursorlike effects in vacuum,” J. Opt. Soc. Am. A 11, 168–172 (1994).
    [CrossRef]
  8. I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
    [CrossRef]
  9. R. W. Ziolkowski and J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
    [CrossRef]
  10. Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
    [CrossRef]
  11. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
    [CrossRef]
  12. S. Feng and H. G. Winful, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998).
    [CrossRef]
  13. T. Brabec and F. Krausz, “Nonlinear optical propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
    [CrossRef]
  14. E. M. Belenov and A. V. Nazarkin, “Time-varying diffraction effects in the propagation of an electromagnetic pulse in vacuum,” JETP Lett. 53, 200–203 (1991).
  15. P. D. Einziger and S. Raz, “Wave solutions under complex-time shifts,” J. Opt. Soc. Am. A 4, 3–10 (1987).
    [CrossRef]
  16. See, for example, A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  17. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
  18. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
    [CrossRef] [PubMed]
  19. See, for example, A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1987).
  20. M. Karlsson, D. Anderson, M. Desaix, and M. Lisak, “Dynamic effects of Kerr nonlinearity and spatial diffraction on self-phase modulation of optical pulses,” Opt. Lett. 16, 1373–1375 (1991).
    [CrossRef] [PubMed]
  21. J. E. Rothenberg, “Space-time focusing: breakdown of the slowly varying envelope approximation in the self-focusing of femtosecond pulses,” Opt. Lett. 17, 1340–1342 (1992).
    [CrossRef]
  22. J. T. Manassah and B. Gross, “The spatiotemporal dynamics of a pulse propagating in the quasi-Kerr regime of a two-level resonant medium,” Opt. Commun. 136, 193–205 (1997).
    [CrossRef]
  23. J. K. Ranka and A. L. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).
    [CrossRef]
  24. A. B. Shvartsburg, Time-Domain Optics of Ultrashort Waveforms (Oxford U. Press, Oxford, 1996).

1998

1997

J. T. Manassah and B. Gross, “The spatiotemporal dynamics of a pulse propagating in the quasi-Kerr regime of a two-level resonant medium,” Opt. Commun. 136, 193–205 (1997).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear optical propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

1995

P. U. Jepsen and S. R. Keiding, “Radiation patterns from lens-coupled terahertz antennas,” Opt. Lett. 20, 807–809 (1995).
[CrossRef] [PubMed]

A. E. Kaplan and P. L. Shkolnikov, “Electromagnetic bubbles and shock waves: unipolar, nonoscillating EM solitons,” Phys. Rev. Lett. 75, 2316–2319 (1995).
[CrossRef] [PubMed]

1994

1992

1991

M. Karlsson, D. Anderson, M. Desaix, and M. Lisak, “Dynamic effects of Kerr nonlinearity and spatial diffraction on self-phase modulation of optical pulses,” Opt. Lett. 16, 1373–1375 (1991).
[CrossRef] [PubMed]

E. M. Belenov and A. V. Nazarkin, “Time-varying diffraction effects in the propagation of an electromagnetic pulse in vacuum,” JETP Lett. 53, 200–203 (1991).

1989

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

1987

1985

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

1984

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Anderson, D.

Auston, D. H.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Belenov, E. M.

E. M. Belenov and A. V. Nazarkin, “Transient diffraction and precursorlike effects in vacuum,” J. Opt. Soc. Am. A 11, 168–172 (1994).
[CrossRef]

E. M. Belenov and A. V. Nazarkin, “Time-varying diffraction effects in the propagation of an electromagnetic pulse in vacuum,” JETP Lett. 53, 200–203 (1991).

Brabec, T.

T. Brabec and F. Krausz, “Nonlinear optical propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Cheung, K. P.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Christov, I. P.

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

De Boeij, W. P.

Desaix, M.

Einziger, P. D.

Feng, S.

Gaeta, A. L.

Gross, B.

J. T. Manassah and B. Gross, “The spatiotemporal dynamics of a pulse propagating in the quasi-Kerr regime of a two-level resonant medium,” Opt. Commun. 136, 193–205 (1997).
[CrossRef]

Jepsen, P. U.

Judkins, J. B.

Kaplan, A. E.

A. E. Kaplan, “Diffraction-induced transformation of near-cycle and subcycle pulses,” J. Opt. Soc. Am. B 15, 951–956 (1998).
[CrossRef]

A. E. Kaplan and P. L. Shkolnikov, “Electromagnetic bubbles and shock waves: unipolar, nonoscillating EM solitons,” Phys. Rev. Lett. 75, 2316–2319 (1995).
[CrossRef] [PubMed]

Karlsson, M.

Keiding, S. R.

Kleinman, D. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Krausz, F.

Lin, Q.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Lisak, M.

Manassah, J. T.

J. T. Manassah and B. Gross, “The spatiotemporal dynamics of a pulse propagating in the quasi-Kerr regime of a two-level resonant medium,” Opt. Commun. 136, 193–205 (1997).
[CrossRef]

Nazarkin, A. V.

E. M. Belenov and A. V. Nazarkin, “Transient diffraction and precursorlike effects in vacuum,” J. Opt. Soc. Am. A 11, 168–172 (1994).
[CrossRef]

E. M. Belenov and A. V. Nazarkin, “Time-varying diffraction effects in the propagation of an electromagnetic pulse in vacuum,” JETP Lett. 53, 200–203 (1991).

Porras, M. A.

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

Pshenichnikov, M. S.

Ranka, J. K.

Raz, S.

Rothenberg, J. E.

Shkolnikov, P. L.

A. E. Kaplan and P. L. Shkolnikov, “Electromagnetic bubbles and shock waves: unipolar, nonoscillating EM solitons,” Phys. Rev. Lett. 75, 2316–2319 (1995).
[CrossRef] [PubMed]

Spielmann, C.

Stingl, A.

Szipöcs, R.

Valdmanis, J. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

Wang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Wiersma, D. A.

Winful, H. G.

Xu, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Zhang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Ziolkowski, R. W.

IEEE J. Quantum Electron.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

JETP Lett.

E. M. Belenov and A. V. Nazarkin, “Time-varying diffraction effects in the propagation of an electromagnetic pulse in vacuum,” JETP Lett. 53, 200–203 (1991).

Opt. Commun.

J. T. Manassah and B. Gross, “The spatiotemporal dynamics of a pulse propagating in the quasi-Kerr regime of a two-level resonant medium,” Opt. Commun. 136, 193–205 (1997).
[CrossRef]

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

Opt. Lett.

Phys. Rev. A

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

Phys. Rev. E

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

Phys. Rev. Lett.

T. Brabec and F. Krausz, “Nonlinear optical propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov radiation from femtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53, 1555–1558 (1984).
[CrossRef]

A. E. Kaplan and P. L. Shkolnikov, “Electromagnetic bubbles and shock waves: unipolar, nonoscillating EM solitons,” Phys. Rev. Lett. 75, 2316–2319 (1995).
[CrossRef] [PubMed]

Other

See, for example, A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1987).

See, for example, A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

A. B. Shvartsburg, Time-Domain Optics of Ultrashort Waveforms (Oxford U. Press, Oxford, 1996).

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Figures (4)

Fig. 1
Fig. 1

Power-spectrum pulses Re[it0/(t+it0)]α for four values of α.

Fig. 2
Fig. 2

Pulse form of the PSPB α=8, t0=2.5 fs (ωm=3.2 fs-1), z0=103 mm, at z=z0 and several distances r from the axis. The vertical scale is enhanced for larger r to facilitate a clearer comparison of the pulse forms.

Fig. 3
Fig. 3

Change of the pulse form of the PSPB α=4, t0=2.5 fs, z0=103 mm, as it propagates along the beam axis r=0.

Fig. 4
Fig. 4

(a) Poisson-spectrum pulses with growing numbers of nonsinusoidal oscillations and the same frequency ωm. (b) Energy distributions at the waist z=0 of the PSPB with z0=103 mm associated with the pulses of (a).

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

ΔE-1c22Et2=0
ΔE+2Ez2-2c2Ezt=0,
Ez1cEt,
ΔE=2c2Ezt.
ΔzcΔt.
Δψ-2i ω0cψz=2c2ψzt.
Δψ-2ik0 ψz=0,
E(r, z, t)=iz0qFt-r22cq,
fˆ(ω)=-f(t)exp(-iωt)dt,
F(t)=1π0fˆ(ω)exp(iωt)dω
E=iz0qexp-iω0r22cqexp(iω0t).
F(t)=exp-t2T2exp(iω0t)-i Imexp(iω0t)erfcTω02+i tTψ(t)exp(iω0t),
E(r, z, t)=iz0πq0fˆ(ω)exp-iωr22cqexp(iωt)dω,
R(z)=z[1+(z0/z)2].
r=const.[1+(z/z0)2]1/2,
W(r, z)=14πz02|q|20|fˆ(ω)|2 exp-ωz0r2c|q|2dω.
F(t)=it0t+it0α,
fˆ(ω)=πt0αΓ(α)|ω|α-1 exp(-|ω|t0).
T=e2/α-1t0
ωm=α/t0
f(t)=exp(-t2/T2)cos(ω0t)
E(r, z, t)=iz0qit0t-r2/2cq+it0α.
E(r, z, t)
=iz0qt0t0+r2z0/2c|q|2α×i[t0+(r2z0/2c|q|2)][t-(r2/2cR)]+i[t0+(r2z0/2c|q|2)]α,
T=e2/α-1t0+r2z02c|q|2,
ωm=αt0+r2z0/2c|q|2,
W(r, z)z02|q|2t0t0+(r2z0/2c|q|2)2α-1.
a(z)=a01+zz021/2,
a02=2ct0z0{exp[2/(2α-1)]-1},
θ02=2ct0z0{exp[2/(2α-1)]-1},
W(r, z)z02|q|2exp-2r2a2(z)
E(r, z, t)
=f0 1[a1+i(z-ct)]×1{r2/[a1+i(z-ct)]+a2-i(z+ct)},
E1=A1it0t-r2/2cq1+it0α
E2=A2it0t-r2/2cq2+it0α
q2=Aq1+BCq1+D,
A2A1=q1Aq1+B.

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