Abstract

Propagation characteristics of truncated Localized Waves propagating in dispersive silica and free space are numerically analyzed. It is shown that those characteristics are affected by the changes in the relation between the transverse spatial spectral components and the wave vector. Numerical experiments demonstrate that as the non-linearity of this relation gets stronger, the pulses propagating in silica become more immune to decay and distortion whereas the pulses propagating in free-space suffer from early decay and distortion.

© 2010 Optical Society of America

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  1. H. A. Willebrand, and B. S. Ghuman, "Fiber optics without fiber," IEEE Spectr. 38, 40-45 (2001).
    [CrossRef]
  2. L. B. Felsen, "Phase space issues in ultrawideband/short pulse wave modeling," in Ultra-Wideband, Short-Pulse Electromagnetics, H. Bertoni, L. Carin, and L. B. Felsen, eds. (Plenum Press, New York, 1993).
  3. J.-Y. Lu, J. Cheng, and B. Cameron, "Low sidelobe limited diffraction optical coherence tomography," in "Coherence Domain Optical Methods in Biomedical Science and Clinical Applications VI, Proc. of SPIE," vol. 4619, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds. (SPIE, 2006), vol. 4619, pp. 300-311.
  4. T. Ito, and S. Okazaki, "Pushing the limits of lithography," Nature 406, 1027-1031 (2000).
    [CrossRef] [PubMed]
  5. J. N. Brittingham, "Focus waves modes in homogeneous Maxwell’s equations: transverse electric mode," J. Appl. Phys. 54, 1179-1189 (1983).
    [CrossRef]
  6. J.-Y. Lu, and J. F. Greenleaf, "Nondiffracting X waves: exact solutions to free-space scalar wave equation and their infinite realizations," IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19-31 (1992).
    [CrossRef] [PubMed]
  7. R. W. Ziolkowski, "Exact solutions of the wave equation with complex source locations," J. Math. Phys. 26, 861-863 (1985).
    [CrossRef]
  8. A. M. Shaarawi, "Comparison of two localized wave fields generated from dynamic apertures," J. Opt. Soc. Am. A 14, 1804-1816 (1997).
    [CrossRef]
  9. E. Heyman, B. Z. Steinberg, and L. B. Felsen, "Spectral analysis of focus wave modes," J. Opt. Soc. Am. A 4, 2081-2091 (1987).
    [CrossRef]
  10. E. Heyman, "The focus wave mode: a dilemma with causality," IEEE Trans. Antenn. Propag. 37, 1604-1608 (1989).
    [CrossRef]
  11. H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized waves (J. Wiley & Sons, New York, NY, 2008).
    [CrossRef]
  12. A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, "On the evanescent fields and the causality of the focus wave modes," J. Math. Phys. 36, 5565-5587 (1995).
    [CrossRef]
  13. M. Zamboni-Rached, "Subluminal wave bullets: Exact localized subluminal solutions to the wave equations," Phys. Rev. A 77, 033824 (2008).
    [CrossRef]
  14. M. Zamboni-Rached, "Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components," Phys. Rev. A 79, 013816 (2009).
    [CrossRef]
  15. A. Sezginer, "A general formulation of focus wave modes," J. Appl. Phys. 57, 678-683 (1985).
    [CrossRef]
  16. T. T. Wu, and H. Lehmann, "Spreading of electromagnetic pulses," J. Appl. Phys. 58, 2064-2065 (1985).
    [CrossRef]
  17. R. W. Ziolkowski, "Localized transmission of electromagnetic energy," Phys. Rev. A 39, 2005-2033 (1989).
    [CrossRef] [PubMed]
  18. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
    [CrossRef]
  19. H. Sõnajalg, and P. Saari, "Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators," Opt. Lett. 21, 1162-1164 (1996).
    [CrossRef] [PubMed]
  20. M. A. Porras, "Diffraction-free and dispersion-free pulsed beam propagation in dispersive media," Opt. Lett. 26, 1364-1366 (2001).
    [CrossRef]
  21. S. Orlov, A. Piskarskas, and A. Stabinis, "Localized optical subcycle pulses in dispersive media," Opt. Lett. 27, 2167-2169 (2002).
    [CrossRef]
  22. M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, "Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth," Opt. Commun. 226, 15-23 (2003).
    [CrossRef]
  23. R. Donnelly, and R. W. Ziolkowski, "Designing localized waves," Proc. R. Soc. Lond. A 440, 541-565 (1993).
    [CrossRef]
  24. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, New York, 1995), 2nd ed.
  25. A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, "The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses," J. Phys. Math. Gen. 29, 5157-5179 (1996).
    [CrossRef]

2009 (1)

M. Zamboni-Rached, "Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components," Phys. Rev. A 79, 013816 (2009).
[CrossRef]

2008 (1)

M. Zamboni-Rached, "Subluminal wave bullets: Exact localized subluminal solutions to the wave equations," Phys. Rev. A 77, 033824 (2008).
[CrossRef]

2003 (1)

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, "Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth," Opt. Commun. 226, 15-23 (2003).
[CrossRef]

2002 (1)

S. Orlov, A. Piskarskas, and A. Stabinis, "Localized optical subcycle pulses in dispersive media," Opt. Lett. 27, 2167-2169 (2002).
[CrossRef]

2001 (2)

M. A. Porras, "Diffraction-free and dispersion-free pulsed beam propagation in dispersive media," Opt. Lett. 26, 1364-1366 (2001).
[CrossRef]

H. A. Willebrand, and B. S. Ghuman, "Fiber optics without fiber," IEEE Spectr. 38, 40-45 (2001).
[CrossRef]

2000 (1)

T. Ito, and S. Okazaki, "Pushing the limits of lithography," Nature 406, 1027-1031 (2000).
[CrossRef] [PubMed]

1997 (1)

A. M. Shaarawi, "Comparison of two localized wave fields generated from dynamic apertures," J. Opt. Soc. Am. A 14, 1804-1816 (1997).
[CrossRef]

1996 (2)

H. Sõnajalg, and P. Saari, "Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators," Opt. Lett. 21, 1162-1164 (1996).
[CrossRef] [PubMed]

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, "The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses," J. Phys. Math. Gen. 29, 5157-5179 (1996).
[CrossRef]

1995 (1)

A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, "On the evanescent fields and the causality of the focus wave modes," J. Math. Phys. 36, 5565-5587 (1995).
[CrossRef]

1993 (1)

R. Donnelly, and R. W. Ziolkowski, "Designing localized waves," Proc. R. Soc. Lond. A 440, 541-565 (1993).
[CrossRef]

1992 (1)

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

1989 (3)

E. Heyman, "The focus wave mode: a dilemma with causality," IEEE Trans. Antenn. Propag. 37, 1604-1608 (1989).
[CrossRef]

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

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

1987 (1)

E. Heyman, B. Z. Steinberg, and L. B. Felsen, "Spectral analysis of focus wave modes," J. Opt. Soc. Am. A 4, 2081-2091 (1987).
[CrossRef]

1985 (3)

R. W. Ziolkowski, "Exact solutions of the wave equation with complex source locations," J. Math. Phys. 26, 861-863 (1985).
[CrossRef]

A. Sezginer, "A general formulation of focus wave modes," J. Appl. Phys. 57, 678-683 (1985).
[CrossRef]

T. T. Wu, and H. Lehmann, "Spreading of electromagnetic pulses," J. Appl. Phys. 58, 2064-2065 (1985).
[CrossRef]

1983 (1)

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

Besieris, I. M.

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, "The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses," J. Phys. Math. Gen. 29, 5157-5179 (1996).
[CrossRef]

A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, "On the evanescent fields and the causality of the focus wave modes," J. Math. Phys. 36, 5565-5587 (1995).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

Brittingham, J. N.

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

Donnelly, R.

R. Donnelly, and R. W. Ziolkowski, "Designing localized waves," Proc. R. Soc. Lond. A 440, 541-565 (1993).
[CrossRef]

Felsen, L. B.

E. Heyman, B. Z. Steinberg, and L. B. Felsen, "Spectral analysis of focus wave modes," J. Opt. Soc. Am. A 4, 2081-2091 (1987).
[CrossRef]

Ghuman, B. S.

H. A. Willebrand, and B. S. Ghuman, "Fiber optics without fiber," IEEE Spectr. 38, 40-45 (2001).
[CrossRef]

Greenleaf, J. F.

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

Hernández-Figueroa, H. E.

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, "Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth," Opt. Commun. 226, 15-23 (2003).
[CrossRef]

Heyman, E.

E. Heyman, "The focus wave mode: a dilemma with causality," IEEE Trans. Antenn. Propag. 37, 1604-1608 (1989).
[CrossRef]

E. Heyman, B. Z. Steinberg, and L. B. Felsen, "Spectral analysis of focus wave modes," J. Opt. Soc. Am. A 4, 2081-2091 (1987).
[CrossRef]

Ito, T.

T. Ito, and S. Okazaki, "Pushing the limits of lithography," Nature 406, 1027-1031 (2000).
[CrossRef] [PubMed]

Lehmann, H.

T. T. Wu, and H. Lehmann, "Spreading of electromagnetic pulses," J. Appl. Phys. 58, 2064-2065 (1985).
[CrossRef]

Lu, J.-Y.

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

Nóbrega, K. Z.

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, "Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth," Opt. Commun. 226, 15-23 (2003).
[CrossRef]

Okazaki, S.

T. Ito, and S. Okazaki, "Pushing the limits of lithography," Nature 406, 1027-1031 (2000).
[CrossRef] [PubMed]

Orlov, S.

S. Orlov, A. Piskarskas, and A. Stabinis, "Localized optical subcycle pulses in dispersive media," Opt. Lett. 27, 2167-2169 (2002).
[CrossRef]

Piskarskas, A.

S. Orlov, A. Piskarskas, and A. Stabinis, "Localized optical subcycle pulses in dispersive media," Opt. Lett. 27, 2167-2169 (2002).
[CrossRef]

Porras, M. A.

M. A. Porras, "Diffraction-free and dispersion-free pulsed beam propagation in dispersive media," Opt. Lett. 26, 1364-1366 (2001).
[CrossRef]

Recami, E.

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, "Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth," Opt. Commun. 226, 15-23 (2003).
[CrossRef]

Saari, P.

H. Sõnajalg, and P. Saari, "Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators," Opt. Lett. 21, 1162-1164 (1996).
[CrossRef] [PubMed]

Sedky, S. M.

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, "The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses," J. Phys. Math. Gen. 29, 5157-5179 (1996).
[CrossRef]

Sezginer, A.

A. Sezginer, "A general formulation of focus wave modes," J. Appl. Phys. 57, 678-683 (1985).
[CrossRef]

Shaarawi, A. M.

A. M. Shaarawi, "Comparison of two localized wave fields generated from dynamic apertures," J. Opt. Soc. Am. A 14, 1804-1816 (1997).
[CrossRef]

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, "The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses," J. Phys. Math. Gen. 29, 5157-5179 (1996).
[CrossRef]

A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, "On the evanescent fields and the causality of the focus wave modes," J. Math. Phys. 36, 5565-5587 (1995).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

Sõnajalg, H.

H. Sõnajalg, and P. Saari, "Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators," Opt. Lett. 21, 1162-1164 (1996).
[CrossRef] [PubMed]

Stabinis, A.

S. Orlov, A. Piskarskas, and A. Stabinis, "Localized optical subcycle pulses in dispersive media," Opt. Lett. 27, 2167-2169 (2002).
[CrossRef]

Steinberg, B. Z.

E. Heyman, B. Z. Steinberg, and L. B. Felsen, "Spectral analysis of focus wave modes," J. Opt. Soc. Am. A 4, 2081-2091 (1987).
[CrossRef]

Willebrand, H. A.

H. A. Willebrand, and B. S. Ghuman, "Fiber optics without fiber," IEEE Spectr. 38, 40-45 (2001).
[CrossRef]

Wu, T. T.

T. T. Wu, and H. Lehmann, "Spreading of electromagnetic pulses," J. Appl. Phys. 58, 2064-2065 (1985).
[CrossRef]

Zamboni-Rached, M.

M. Zamboni-Rached, "Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components," Phys. Rev. A 79, 013816 (2009).
[CrossRef]

M. Zamboni-Rached, "Subluminal wave bullets: Exact localized subluminal solutions to the wave equations," Phys. Rev. A 77, 033824 (2008).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, "Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth," Opt. Commun. 226, 15-23 (2003).
[CrossRef]

Ziolkowski, R. W.

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, "The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses," J. Phys. Math. Gen. 29, 5157-5179 (1996).
[CrossRef]

A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, "On the evanescent fields and the causality of the focus wave modes," J. Math. Phys. 36, 5565-5587 (1995).
[CrossRef]

R. Donnelly, and R. W. Ziolkowski, "Designing localized waves," Proc. R. Soc. Lond. A 440, 541-565 (1993).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

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

R. W. Ziolkowski, "Exact solutions of the wave equation with complex source locations," J. Math. Phys. 26, 861-863 (1985).
[CrossRef]

IEEE Spectr. (1)

H. A. Willebrand, and B. S. Ghuman, "Fiber optics without fiber," IEEE Spectr. 38, 40-45 (2001).
[CrossRef]

IEEE Trans. Antenn. Propag. (1)

E. Heyman, "The focus wave mode: a dilemma with causality," IEEE Trans. Antenn. Propag. 37, 1604-1608 (1989).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

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

J. Appl. Phys. (3)

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

A. Sezginer, "A general formulation of focus wave modes," J. Appl. Phys. 57, 678-683 (1985).
[CrossRef]

T. T. Wu, and H. Lehmann, "Spreading of electromagnetic pulses," J. Appl. Phys. 58, 2064-2065 (1985).
[CrossRef]

J. Math. Phys. (3)

R. W. Ziolkowski, "Exact solutions of the wave equation with complex source locations," J. Math. Phys. 26, 861-863 (1985).
[CrossRef]

A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, "On the evanescent fields and the causality of the focus wave modes," J. Math. Phys. 36, 5565-5587 (1995).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, "A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation," J. Math. Phys. 30, 1254-1269 (1989).
[CrossRef]

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

A. M. Shaarawi, "Comparison of two localized wave fields generated from dynamic apertures," J. Opt. Soc. Am. A 14, 1804-1816 (1997).
[CrossRef]

E. Heyman, B. Z. Steinberg, and L. B. Felsen, "Spectral analysis of focus wave modes," J. Opt. Soc. Am. A 4, 2081-2091 (1987).
[CrossRef]

J. Phys. Math. Gen. (1)

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, "The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses," J. Phys. Math. Gen. 29, 5157-5179 (1996).
[CrossRef]

Nature (1)

T. Ito, and S. Okazaki, "Pushing the limits of lithography," Nature 406, 1027-1031 (2000).
[CrossRef] [PubMed]

Opt. Commun. (1)

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, "Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth," Opt. Commun. 226, 15-23 (2003).
[CrossRef]

Opt. Lett. (3)

H. Sõnajalg, and P. Saari, "Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators," Opt. Lett. 21, 1162-1164 (1996).
[CrossRef] [PubMed]

M. A. Porras, "Diffraction-free and dispersion-free pulsed beam propagation in dispersive media," Opt. Lett. 26, 1364-1366 (2001).
[CrossRef]

S. Orlov, A. Piskarskas, and A. Stabinis, "Localized optical subcycle pulses in dispersive media," Opt. Lett. 27, 2167-2169 (2002).
[CrossRef]

Phys. Rev. A (3)

M. Zamboni-Rached, "Subluminal wave bullets: Exact localized subluminal solutions to the wave equations," Phys. Rev. A 77, 033824 (2008).
[CrossRef]

M. Zamboni-Rached, "Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components," Phys. Rev. A 79, 013816 (2009).
[CrossRef]

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

Proc. R. Soc. Lond. A (1)

R. Donnelly, and R. W. Ziolkowski, "Designing localized waves," Proc. R. Soc. Lond. A 440, 541-565 (1993).
[CrossRef]

Other (4)

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, New York, 1995), 2nd ed.

H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized waves (J. Wiley & Sons, New York, NY, 2008).
[CrossRef]

L. B. Felsen, "Phase space issues in ultrawideband/short pulse wave modeling," in Ultra-Wideband, Short-Pulse Electromagnetics, H. Bertoni, L. Carin, and L. B. Felsen, eds. (Plenum Press, New York, 1993).

J.-Y. Lu, J. Cheng, and B. Cameron, "Low sidelobe limited diffraction optical coherence tomography," in "Coherence Domain Optical Methods in Biomedical Science and Clinical Applications VI, Proc. of SPIE," vol. 4619, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds. (SPIE, 2006), vol. 4619, pp. 300-311.

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Figures (6)

Fig. 1
Fig. 1

The spectrum of the pulses | ψ ¯ ˜ ( k ρ , 0 , ω ) | showing the coupling between the spatial kρ and temporal ω spectral components of the (a) X-wave (α = 0), (b) FWM with ( = 0.02), (c) FWM with ( = 2.0), (d) FWM with ( = 5.0), (e) FWM with ( = 10.0) and (f) FWM with ( = 20.0). Exact coupling relation, that is obtained by substituting Eq. (3) into Eq. (2) is shown in red circles for reference.

Fig. 2
Fig. 2

Comparison of the spatial spectral depletion of the pulses propagating in free-space and silica at z′ = 0, z′ = zD and z′ = 2zD for (a) X-wave (α = 0), (b) FWM with ( = 0.02), (c) FWM with ( = 2.0), (d) FWM with ( = 5.0), (e) FWM with ( = 10.0) and (f) FWM with ( = 20.0).

Fig. 3
Fig. 3

Comparison of the temporal spectral depletion of the truncated pulses in free-space and silica at z′ = 0, z′ = zD and z′ = 2zD for (a) X-wave, (b) FWM with ( = 0.02),(c) FWM with ( = 2.0), (d) FWM with ( = 5.0), (e) FWM with ( = 10.0) and (f) FWM with ( = 20.0).

Fig. 4
Fig. 4

Normalized intensities at the centroid of the truncated pulses vesrus the propagation distance in (a) free-space and (b) silica.

Fig. 5
Fig. 5

Percentage change of the FWHM of the pulses along the propagation distance from z = 0 to z = zD in (a) free-space and (b) silica.

Fig. 6
Fig. 6

Percentage change of the spot-size of the pulses along the propagation distance from z = 0 to z = zD in (a) free-space and (b) silica.

Tables (2)

Tables Icon

Table 1 Values of Bj and ωj for typical bulk fused silica [see Eq. (9)]

Tables Icon

Table 2 Field depth of X-wave and FWM pulses propagating in free-space and silica

Equations (11)

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ψ ( ρ , z , t ) = 0 k ρ d k ρ d k z d ω ψ ˜ ( k ρ , k z , ω ) J 0 ( k ρ ρ ) e i ( k z z ω t ) .
k ρ 2 + k z 2 = ( n ( ω ) ω c ) 2 ,
ω = α + v 0 k z ,
ψ ˜ ( k ρ , k z , ω ) = ψ ˜ ( ω ) δ ( k ρ ( n ( ω ) ω c ) 2 k z 2 ) δ ( k z ω α v 0 ) ,
ψ ( ρ , z , t ) = ω min ω max d ω ψ ˜ ( ω ) ξ 1 J 0 ( ξ 1 ρ ) e i ( ξ 2 z ω t ) , ξ 1 = ( n ( ω ) ω c ) 2 ξ 2 2 , ξ 2 = ω α v 0 .
ω min = α c max ( c n ( ω ) v 0 , ω max = α c min ( c n ( ω ) v 0 ) ,
f t ( t ) = { e ( t / T ) | t | 2 T 0 elsewhere and f ρ ( ρ ) = { e ( ρ / R ) ρ 2 R 0 elsewhere .
ψ ¯ ˜ ( k ρ , k z , ω = ψ ˜ ( k z ) f ˜ ρ ( k ρ χ 1 ) f ˜ t ( ω χ 2 ) .
n 2 ( ω ) = 1 + j = 1 N B j ω j 2 ω j 2 ω 2 ,
Φ ρ ( k ρ ; z ) = 0 ρ d ρ ψ ¯ ( ρ , z , z c 0 ) J 0 ( k ρ ρ ) .
Φ t ( ω ; z ) = dt ψ ¯ ( 0 , z , t z v 0 ) e i ω t .

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