Abstract

Localized, stationary, X-type solutions of the paraxial wave equation are described. Unlike conventional X waves, these are luminal, envelope X waves sustained by a material medium because of the interplay of dispersion and diffraction.

© 2003 Optical Society of America

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References

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  1. J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef] [PubMed]
  2. J. Lu J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectron. Freq. Control 37, 438 (1990).
    [CrossRef]
  3. H. Sonajalg, M. Ratsep, and P. Saari, Opt. Lett. 22, 310 (1997).
    [CrossRef]
  4. J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. E 62, 4261 (2000).
    [CrossRef]
  5. F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
    [CrossRef]
  6. T. Wulle and S. Herminghaus, Phys. Rev. Lett. 70, 1401 (1993). One can find, for instance, hundreds of papers with key words “Bessel” and “second-harmonic.”
    [CrossRef] [PubMed]
  7. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, O. Jedrkiewicz, and J. Trull, arXiv:physics/0204066 (2002), http://xxx.lanl.gov.
  8. See, for instance, A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Reading, Mass., 1992).
  9. H. Sonajalg and P. Saari, Opt. Lett. 21, 1162 (1996).
    [CrossRef]
  10. M. A. Porras, Opt. Lett. 26, 1364 (2001).
    [CrossRef]
  11. G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science Conference (QELS 2001), Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper QPD10–1.

Bramati, A.

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science Conference (QELS 2001), Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper QPD10–1.

Conti, C.

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science Conference (QELS 2001), Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper QPD10–1.

Di Trapani, P.

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science Conference (QELS 2001), Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper QPD10–1.

Jedrkiewicz, O.

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science Conference (QELS 2001), Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper QPD10–1.

Kilius, J.

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science Conference (QELS 2001), Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper QPD10–1.

Minardi, S.

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science Conference (QELS 2001), Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper QPD10–1.

Moloney, J. V.

See, for instance, A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Reading, Mass., 1992).

Newell, A. C.

See, for instance, A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Reading, Mass., 1992).

Piskarskas, A.

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science Conference (QELS 2001), Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper QPD10–1.

Trillo, S.

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science Conference (QELS 2001), Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper QPD10–1.

Valiulis, G.

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science Conference (QELS 2001), Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper QPD10–1.

OSA Trends in Optics and Photonics Series (1)

G. Valiulis, J. Kilius, O. Jedrkiewicz, A. Bramati, S. Minardi, C. Conti, S. Trillo, A. Piskarskas, and P. Di Trapani, in Quantum Electronics and Laser Science Conference (QELS 2001), Vol. 57 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), paper QPD10–1.

Other (10)

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

J. Lu J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectron. Freq. Control 37, 438 (1990).
[CrossRef]

H. Sonajalg, M. Ratsep, and P. Saari, Opt. Lett. 22, 310 (1997).
[CrossRef]

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. E 62, 4261 (2000).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

T. Wulle and S. Herminghaus, Phys. Rev. Lett. 70, 1401 (1993). One can find, for instance, hundreds of papers with key words “Bessel” and “second-harmonic.”
[CrossRef] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, O. Jedrkiewicz, and J. Trull, arXiv:physics/0204066 (2002), http://xxx.lanl.gov.

See, for instance, A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Reading, Mass., 1992).

H. Sonajalg and P. Saari, Opt. Lett. 21, 1162 (1996).
[CrossRef]

M. A. Porras, Opt. Lett. 26, 1364 (2001).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Gray-scale plot of the amplitude of the eX wave with fˆΩ=πτ0 exp-τ0Ω and Aρ,σ=Reρ2+1+iσ2-1/2. Normalized coordinates are σ=τ/τ0 and ρ=r/r0=rk0k0/τ0. (b) Spot size r0 of eX waves of different carrier frequencies in air and fused silica. Spot sizes are for duration τ0=T0, so, for an eX wave with N cycles, r0 is N times larger. The curve for fused silica ends at its zero dispersion point ω01.48 fs-1.

Fig. 2
Fig. 2

(a) Solid curve, dispersion curve KΩ about ω0=1.6 fs-1 in fused silica. Dashed curve, second-order approximation k0k0Ω. Dotted curve, spectrum fˆΩ=exp-τ02Ω2/4 in arbitrary units, with τ0=7.383 fs. (b) Amplitude A of fused-silica exact eX wave of frequency ω0=1.6 fs-1 and the Gaussian spectrum in (a). Frequency and duration are such that L2/L3=1.0. σ and ρ are defined in Fig. 1.

Fig. 3
Fig. 3

Gray-scale plot of amplitude A of fused-silica eX waves with Gaussian spectrum fˆΩ=exp-τ02Ω2/4. (a), (b), (c), (d) L2/L3=0.1, 0.5, 1.0, 5.0, respectively. Normalized coordinates are defined in Fig. 1. (a) ω0=2.3 fs-1, τ0=8.196 fs; (b) ω0=1.6 fs-1, τ0=14.805 fs; (c) ω0=1.5 fs-1, τ0=48.592 fs; (d) ω0=1.49 fs-1, τ0=19.693 fs;

Equations (7)

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zA=i2k0ΔA-ik02τ2A,
Ar,τ=12π-dΩfˆΩJ0k0k0Ωrexp-iΩτ,
Ar,z,τ=12π-dΩ0dKKAˆK,ΩJ0Kr×exp-iK2z2k0expik0Ω2z2×exp-iΩτ,
Ar,τ=Reτ0k0k0r2+τ0+iτ2-1/2,
KΩ=2k0kω0+Ω-k0-k0Ω1/2,
Ar,τ=12πKΩ realdΩfˆΩJ0KΩr×exp-iΩτ
E=-dωfˆωJ0kωsin θr×exp-iωt+ikωcos θz

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