Abstract

We find a new family of solutions of the nonparaxial wave equation that represents ultrashort pulsed light beam propagation in free space. The spatial and temporal parts of these pulsed beams are separable; the spatial transverse part is described by a Bessel function and remains unchanged during propagation, but the temporal profile can be arbitrary. Therefore the pulsed beam exhibits diffraction-free behavior with no transverse spreading, but the temporal part changes as if in a dispensive medium; the change is dominated by what we call spatially induced group-velocity dispersion. The analytical and numerical investigations show that the even- and odd-order spatially induced dispersions partially compensate for each other so as to give rise to pulse spreading, weakening, asymmetry, and center shift.

© 2002 Optical Society of America

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References

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  1. See, e.g., I. N. Duling, ed., Compact Sources of Ultrashort Pulses (Cambridge U. Press, New York, 1995).
  2. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
    [CrossRef]
  3. M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16, 1468–1474 (1999).
    [CrossRef]
  4. S. Feng, H. G. Winful, R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998).
    [CrossRef]
  5. S. Feng, H. G. Winful, R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E 59, 4630–4649 (1999).
    [CrossRef]
  6. S. Feng, H. G. Winful, “Spatiotemporal transformation of isodiffracting ultrashort pulses by nondispersive quadratic phase media,” J. Opt. Soc. Am. A 16, 2500–2509 (1999).
    [CrossRef]
  7. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
    [CrossRef] [PubMed]
  8. A. E. Kaplan, “Diffraction-induced transformation of near-cycles and sub-cycles pulses,” J. Opt. Soc. Am. B 15, 951–956 (1998).
    [CrossRef]
  9. G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52–56 (1998).
    [CrossRef]
  10. G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. 167, 15–22 (1999).
  11. M. A. Porras, “Propagation of single-cycle pulse light beams in dispersive media,” Phys. Rev. A 60, 5069–5073 (1999).
  12. T. Brabec, F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
    [CrossRef]
  13. A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of nondiffracting pulses,” Opt. Commun. 136, 207–212 (1997).
    [CrossRef]
  14. J. Lu, J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free space scalar wave equation and their finite aperture realization,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
    [CrossRef]
  15. J. Lu, J. F. Greenleaf, “Experimental verification of X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 441–446 (1992).
    [CrossRef]
  16. M. A. Porras, R. Borghi, M. Santarsiero, “Few-optical-cycle Bessel–Gauss pulsed beams in free space,” Phys. Rev. E 62, 5729–5737 (2000).
    [CrossRef]
  17. J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  18. J. Durnin, “Exact solution for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  19. J. Durnin, J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
    [CrossRef] [PubMed]
  20. Z. Y. Liu, D. Y. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt. 45, 17–21 (1998).
    [CrossRef]
  21. A. E. Kaplan, P. L. Shkolnikov, “Electromagnetic ‘bubbles’ and shock waves: unipolar, nonoscillating EM solitons,” Phys. Rev. Lett. 75, 2316–2319 (1995).
    [CrossRef] [PubMed]
  22. See, e.g., M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, UK, 1998).
  23. See, e.g., G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, 1995).
  24. E. M. Belenov, A. V. Nazakin, “Transient diffraction and precursorlike effects in vacuum,” J. Opt. Soc. Am. A 11, 168–172 (1994).
    [CrossRef]

2000 (1)

M. A. Porras, R. Borghi, M. Santarsiero, “Few-optical-cycle Bessel–Gauss pulsed beams in free space,” Phys. Rev. E 62, 5729–5737 (2000).
[CrossRef]

1999 (5)

M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16, 1468–1474 (1999).
[CrossRef]

S. Feng, H. G. Winful, R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E 59, 4630–4649 (1999).
[CrossRef]

S. Feng, H. G. Winful, “Spatiotemporal transformation of isodiffracting ultrashort pulses by nondispersive quadratic phase media,” J. Opt. Soc. Am. A 16, 2500–2509 (1999).
[CrossRef]

G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. 167, 15–22 (1999).

M. A. Porras, “Propagation of single-cycle pulse light beams in dispersive media,” Phys. Rev. A 60, 5069–5073 (1999).

1998 (5)

S. Feng, H. G. Winful, R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998).
[CrossRef]

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

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52–56 (1998).
[CrossRef]

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

Z. Y. Liu, D. Y. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt. 45, 17–21 (1998).
[CrossRef]

1997 (2)

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

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

1995 (1)

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

1994 (1)

1992 (2)

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

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

1989 (1)

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

1988 (1)

1987 (2)

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

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

Agrawal, G. P.

G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. 167, 15–22 (1999).

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52–56 (1998).
[CrossRef]

See, e.g., G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, 1995).

Belenov, E. M.

Borghi, R.

M. A. Porras, R. Borghi, M. Santarsiero, “Few-optical-cycle Bessel–Gauss pulsed beams in free space,” Phys. Rev. E 62, 5729–5737 (2000).
[CrossRef]

Born, M.

See, e.g., M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, UK, 1998).

Brabec, T.

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

Durnin, J.

Eberly, J. H.

J. Durnin, J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[CrossRef] [PubMed]

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

Fagerholm, J.

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

Fan, D. Y.

Z. Y. Liu, D. Y. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt. 45, 17–21 (1998).
[CrossRef]

Feng, S.

Friberg, A. T.

A. T. Friberg, J. Fagerholm, M. M. Salomaa, “Space-frequency analysis of nondiffracting pulses,” Opt. Commun. 136, 207–212 (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 realization,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[CrossRef]

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

Hellwarth, R. W.

S. Feng, H. G. Winful, R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E 59, 4630–4649 (1999).
[CrossRef]

S. Feng, H. G. Winful, R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998).
[CrossRef]

Kaplan, A. E.

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

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

Krausz, F.

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

Liu, Z. Y.

Z. Y. Liu, D. Y. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt. 45, 17–21 (1998).
[CrossRef]

Lu, J.

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

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

Miceli, J.

J. Durnin, J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[CrossRef] [PubMed]

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

Nazakin, A. V.

Porras, M. A.

M. A. Porras, R. Borghi, M. Santarsiero, “Few-optical-cycle Bessel–Gauss pulsed beams in free space,” Phys. Rev. E 62, 5729–5737 (2000).
[CrossRef]

M. A. Porras, “Propagation of single-cycle pulse light beams in dispersive media,” Phys. Rev. A 60, 5069–5073 (1999).

M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16, 1468–1474 (1999).
[CrossRef]

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

Salomaa, M. M.

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

Santarsiero, M.

M. A. Porras, R. Borghi, M. Santarsiero, “Few-optical-cycle Bessel–Gauss pulsed beams in free space,” Phys. Rev. E 62, 5729–5737 (2000).
[CrossRef]

Shkolnikov, P. L.

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

Winful, H. G.

Wolf, E.

See, e.g., M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, UK, 1998).

Ziolkowski, R. W.

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

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (2)

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

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

J. Mod. Opt. (1)

Z. Y. Liu, D. Y. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt. 45, 17–21 (1998).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

Opt. Commun. (3)

G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun. 157, 52–56 (1998).
[CrossRef]

G. P. Agrawal, “Far-field diffraction of pulsed optical beams in dispersive media,” Opt. Commun. 167, 15–22 (1999).

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

Opt. Lett. (2)

Phys. Rev. A (2)

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

M. A. Porras, “Propagation of single-cycle pulse light beams in dispersive media,” Phys. Rev. A 60, 5069–5073 (1999).

Phys. Rev. E (3)

S. Feng, H. G. Winful, R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E 59, 4630–4649 (1999).
[CrossRef]

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

M. A. Porras, R. Borghi, M. Santarsiero, “Few-optical-cycle Bessel–Gauss pulsed beams in free space,” Phys. Rev. E 62, 5729–5737 (2000).
[CrossRef]

Phys. Rev. Lett. (3)

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

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

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

Other (3)

See, e.g., M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, UK, 1998).

See, e.g., G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, 1995).

See, e.g., I. N. Duling, ed., Compact Sources of Ultrashort Pulses (Cambridge U. Press, New York, 1995).

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Figures (4)

Fig. 1
Fig. 1

Value of L2/λ0 and L3/λ0 as a function of γ. It is shown that for the single-cycle case (m=1) the spatially dispersion is stronger than in the two-cycle case (m=2) and that when γ<45°, the higher dispersion is weaker than SIGVD.

Fig. 2
Fig. 2

Contour graph of L3/L2. Three curves at L3/L2=10, 1, 0.1 divide the γm plane into four parts, labeled L3L2, L3>L2, L3<L2, and L3L2.

Fig. 3
Fig. 3

Broadening factor of a Gaussian pulse resulting from the spatially induced dispersion in propagating along z, in which γ=0.01(rad), (a) m=1, and (b) m=2. Dashed curve, only β2 is considered; solid curve, both β2 and β3 are considered; dotted curve, all higher-order dispersions are taken into account. It can be seen that when the higher-order dispersions are considered, the broadening becomes quicker and more significant for the single-cycle pulse, whereas for the two-cycle case this change is weaker.

Fig. 4
Fig. 4

Pulse shapes at distance z=5L2; all other parameters are the same as in Fig. 3. It can be seen that when the higher-order dispersions are taken into account the pulse becomes asymmetric and the leading edge steepens.

Equations (14)

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2-1c22t2E(x, y, z, t)=0,
[2+k2(ω)]E˜(x, y, z, ω)=0,
E˜(r, z, ω)=J0(αr)exp[iβ(ω)z]S(ω),
α2+β2(ω)=k2(ω),
S(ω)=(2π)-1/2-A(0, t)exp(iωt)dt
E(r, z, t)=J0(αr)A(z, t),
1rrr rJ0(αr)=-α2J0(αr),
2z2-1c22t2A(z, t)=α2A(z, t).
2z2-1c22t2E(z, t)=μ02t2 PL.
z A˜(z, ω)=iβA˜(z, ω),
ζψ+i sgn(β2)2 τ2ψ-in=3sgn(βn)n!L2Ln (iτ)nψ=0,
β0=k0cos γ,β1=1c cos γ,
β2=-k0ω02tan2 γcos γ,β3=3k0ω03tan2 γcos3 γ ,,
L2=2πλ0m2cos γtan2 γ,L3=43 π2λ0m3cos3 γtan2 γ,

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