Abstract

We propose a versatile and efficient technique for the formation of X-pulses in materials with a periodical gain/loss modulation on the wavelength scale. We show that in such materials the strong wave-vector anisotropy of amplification/attenuation of the Bloch modes enables the shaping of ultra-short light pulses around the edges of the first Brillouin zone. X-pulses generation is numerically demonstrated and the optimum conditions are derived; specific characteristics of X-pulses can be tailored by appropriate selection of the geometry and modulation depth.

©2012 Optical Society of America

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  1. J.-Y. Lu and 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(1), 19–31 (1992).
    [Crossref] [PubMed]
  2. H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (John Wiley and Sons, 2008).
  3. J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39(3), 441–446 (1992).
    [Crossref] [PubMed]
  4. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
    [Crossref]
  5. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987).
    [Crossref]
  6. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
    [Crossref] [PubMed]
  7. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
    [Crossref]
  8. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
    [Crossref]
  9. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [Crossref] [PubMed]
  10. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
    [Crossref]
  11. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
    [Crossref] [PubMed]
  12. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44(8), 592–597 (1954).
    [Crossref]
  13. S. Fujiwara, “Optical properties of conic surfaces: I. Reflecting cone,” J. Opt. Soc. Am. 52(3), 287–292 (1962).
    [Crossref]
  14. H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. 22(5), 310–312 (1997).
    [Crossref] [PubMed]
  15. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
    [Crossref] [PubMed]
  16. O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Generation and control of phase-locked conical wave packets in type-I seeded optical parametric amplification,” Phys. Rev. A 80(3), 033813 (2009).
    [Crossref]
  17. A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016608 (2006).
    [Crossref] [PubMed]
  18. D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96(19), 193901 (2006).
    [Crossref] [PubMed]
  19. D. Faccio, A. Averchi, A. Dubietis, P. Polesana, A. Piskarskas, P. D. Trapani, and A. Couairon, “Stimulated Raman X waves in ultrashort optical pulse filamentation,” Opt. Lett. 32(2), 184–186 (2007).
    [Crossref] [PubMed]
  20. C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
    [Crossref] [PubMed]
  21. K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain/loss profile,” Phys. Rev. A 80(1), 013821 (2009).
    [Crossref]
  22. M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain/loss modulation on a wavelength scale,” Phys. Rev. A 82(1), 013828 (2010).
    [Crossref]
  23. R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34(12), 1589–1617 (1987).
    [Crossref]
  24. C. Conti and S. Trillo, “Nonspreading wave packets in three dimensions formed by an ultracold bose gas in an optical lattice,” Phys. Rev. Lett. 92(12), 120404 (2004).
    [Crossref] [PubMed]
  25. S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B 70(23), 235123 (2004).
    [Crossref]
  26. S. Longhi, “Localized and nonspreading spatiotemporal Wannier wave packets in photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 016603 (2005).
    [Crossref] [PubMed]
  27. A. Di Falco, C. Conti, and S. Trillo, “Tunneling mediated by 2D+1 conical waves in a 1D lattice,” Phys. Rev. Lett. 101(1), 013601 (2008).
    [Crossref] [PubMed]
  28. B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge University Press, 1996).

2010 (3)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref] [PubMed]

M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain/loss modulation on a wavelength scale,” Phys. Rev. A 82(1), 013828 (2010).
[Crossref]

2009 (2)

K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain/loss profile,” Phys. Rev. A 80(1), 013821 (2009).
[Crossref]

O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Generation and control of phase-locked conical wave packets in type-I seeded optical parametric amplification,” Phys. Rev. A 80(3), 033813 (2009).
[Crossref]

2008 (1)

A. Di Falco, C. Conti, and S. Trillo, “Tunneling mediated by 2D+1 conical waves in a 1D lattice,” Phys. Rev. Lett. 101(1), 013601 (2008).
[Crossref] [PubMed]

2007 (2)

2006 (2)

A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016608 (2006).
[Crossref] [PubMed]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96(19), 193901 (2006).
[Crossref] [PubMed]

2005 (2)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

S. Longhi, “Localized and nonspreading spatiotemporal Wannier wave packets in photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 016603 (2005).
[Crossref] [PubMed]

2004 (2)

C. Conti and S. Trillo, “Nonspreading wave packets in three dimensions formed by an ultracold bose gas in an optical lattice,” Phys. Rev. Lett. 92(12), 120404 (2004).
[Crossref] [PubMed]

S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B 70(23), 235123 (2004).
[Crossref]

2003 (2)

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

1997 (2)

1992 (2)

J.-Y. Lu and 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(1), 19–31 (1992).
[Crossref] [PubMed]

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39(3), 441–446 (1992).
[Crossref] [PubMed]

1987 (3)

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

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34(12), 1589–1617 (1987).
[Crossref]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

1962 (1)

1954 (1)

Abdollahpour, D.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref] [PubMed]

Averchi, A.

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Botey, M.

M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain/loss modulation on a wavelength scale,” Phys. Rev. A 82(1), 013828 (2010).
[Crossref]

Bragheri, F.

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96(19), 193901 (2006).
[Crossref] [PubMed]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Christodoulides, D. N.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Clerici, M.

O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Generation and control of phase-locked conical wave packets in type-I seeded optical parametric amplification,” Phys. Rev. A 80(3), 033813 (2009).
[Crossref]

Conti, C.

A. Di Falco, C. Conti, and S. Trillo, “Tunneling mediated by 2D+1 conical waves in a 1D lattice,” Phys. Rev. Lett. 101(1), 013601 (2008).
[Crossref] [PubMed]

C. Conti and S. Trillo, “Nonspreading wave packets in three dimensions formed by an ultracold bose gas in an optical lattice,” Phys. Rev. Lett. 92(12), 120404 (2004).
[Crossref] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

Couairon, A.

D. Faccio, A. Averchi, A. Dubietis, P. Polesana, A. Piskarskas, P. D. Trapani, and A. Couairon, “Stimulated Raman X waves in ultrashort optical pulse filamentation,” Opt. Lett. 32(2), 184–186 (2007).
[Crossref] [PubMed]

A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016608 (2006).
[Crossref] [PubMed]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96(19), 193901 (2006).
[Crossref] [PubMed]

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Di Falco, A.

A. Di Falco, C. Conti, and S. Trillo, “Tunneling mediated by 2D+1 conical waves in a 1D lattice,” Phys. Rev. Lett. 101(1), 013601 (2008).
[Crossref] [PubMed]

Di Trapani, P.

O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Generation and control of phase-locked conical wave packets in type-I seeded optical parametric amplification,” Phys. Rev. A 80(3), 033813 (2009).
[Crossref]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96(19), 193901 (2006).
[Crossref] [PubMed]

A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016608 (2006).
[Crossref] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Dubietis, A.

D. Faccio, A. Averchi, A. Dubietis, P. Polesana, A. Piskarskas, P. D. Trapani, and A. Couairon, “Stimulated Raman X waves in ultrashort optical pulse filamentation,” Opt. Lett. 32(2), 184–186 (2007).
[Crossref] [PubMed]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96(19), 193901 (2006).
[Crossref] [PubMed]

A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016608 (2006).
[Crossref] [PubMed]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

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

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Faccio, D.

D. Faccio, A. Averchi, A. Dubietis, P. Polesana, A. Piskarskas, P. D. Trapani, and A. Couairon, “Stimulated Raman X waves in ultrashort optical pulse filamentation,” Opt. Lett. 32(2), 184–186 (2007).
[Crossref] [PubMed]

A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016608 (2006).
[Crossref] [PubMed]

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96(19), 193901 (2006).
[Crossref] [PubMed]

Fujiwara, S.

Gaizauskas, E.

A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016608 (2006).
[Crossref] [PubMed]

Greenleaf, J. F.

J.-Y. Lu and 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(1), 19–31 (1992).
[Crossref] [PubMed]

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39(3), 441–446 (1992).
[Crossref] [PubMed]

Herrero, R.

M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain/loss modulation on a wavelength scale,” Phys. Rev. A 82(1), 013828 (2010).
[Crossref]

K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain/loss profile,” Phys. Rev. A 80(1), 013821 (2009).
[Crossref]

Janner, D.

S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B 70(23), 235123 (2004).
[Crossref]

Jedrkiewicz, O.

O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Generation and control of phase-locked conical wave packets in type-I seeded optical parametric amplification,” Phys. Rev. A 80(3), 033813 (2009).
[Crossref]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref] [PubMed]

Longhi, S.

S. Longhi, “Localized and nonspreading spatiotemporal Wannier wave packets in photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 016603 (2005).
[Crossref] [PubMed]

S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B 70(23), 235123 (2004).
[Crossref]

Lu, J.-Y.

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39(3), 441–446 (1992).
[Crossref] [PubMed]

J.-Y. Lu and 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(1), 19–31 (1992).
[Crossref] [PubMed]

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

McLeod, J. H.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Papazoglou, D. G.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref] [PubMed]

Piskarskas, A.

D. Faccio, A. Averchi, A. Dubietis, P. Polesana, A. Piskarskas, P. D. Trapani, and A. Couairon, “Stimulated Raman X waves in ultrashort optical pulse filamentation,” Opt. Lett. 32(2), 184–186 (2007).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref] [PubMed]

Polesana, P.

Porras, M. A.

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96(19), 193901 (2006).
[Crossref] [PubMed]

Rätsep, M.

Reivelt, K.

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[Crossref]

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Rubino, E.

O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Generation and control of phase-locked conical wave packets in type-I seeded optical parametric amplification,” Phys. Rev. A 80(3), 033813 (2009).
[Crossref]

Saari, P.

Siviloglou, G. A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Sõnajalg, H.

Staliunas, K.

M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain/loss modulation on a wavelength scale,” Phys. Rev. A 82(1), 013828 (2010).
[Crossref]

K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain/loss profile,” Phys. Rev. A 80(1), 013821 (2009).
[Crossref]

Suntsov, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref] [PubMed]

Trapani, P. D.

Trillo, S.

A. Di Falco, C. Conti, and S. Trillo, “Tunneling mediated by 2D+1 conical waves in a 1D lattice,” Phys. Rev. Lett. 101(1), 013601 (2008).
[Crossref] [PubMed]

C. Conti and S. Trillo, “Nonspreading wave packets in three dimensions formed by an ultracold bose gas in an optical lattice,” Phys. Rev. Lett. 92(12), 120404 (2004).
[Crossref] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

Trull, J.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref] [PubMed]

Tzortzakis, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref] [PubMed]

Valiulis, G.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref] [PubMed]

Vilaseca, R.

K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain/loss profile,” Phys. Rev. A 80(1), 013821 (2009).
[Crossref]

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Zengerle, R.

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34(12), 1589–1617 (1987).
[Crossref]

Am. J. Phys. (1)

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[Crossref]

Contemp. Phys. (1)

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[Crossref]

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

J.-Y. Lu and 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(1), 19–31 (1992).
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J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39(3), 441–446 (1992).
[Crossref] [PubMed]

J. Mod. Opt. (1)

R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Opt. 34(12), 1589–1617 (1987).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Nat. Photonics (1)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Opt. Lett. (2)

Phys. Rev. A (3)

K. Staliunas, R. Herrero, and R. Vilaseca, “Subdiffraction and spatial filtering due to periodic spatial modulation of the gain/loss profile,” Phys. Rev. A 80(1), 013821 (2009).
[Crossref]

M. Botey, R. Herrero, and K. Staliunas, “Light in materials with periodic gain/loss modulation on a wavelength scale,” Phys. Rev. A 82(1), 013828 (2010).
[Crossref]

O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Generation and control of phase-locked conical wave packets in type-I seeded optical parametric amplification,” Phys. Rev. A 80(3), 033813 (2009).
[Crossref]

Phys. Rev. B (1)

S. Longhi and D. Janner, “X-shaped waves in photonic crystals,” Phys. Rev. B 70(23), 235123 (2004).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

S. Longhi, “Localized and nonspreading spatiotemporal Wannier wave packets in photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 016603 (2005).
[Crossref] [PubMed]

A. Couairon, E. Gaizauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(1), 016608 (2006).
[Crossref] [PubMed]

Phys. Rev. Lett. (9)

D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical emission, pulse splitting, and X-wave parametric amplification in nonlinear dynamics of ultrashort light pulses,” Phys. Rev. Lett. 96(19), 193901 (2006).
[Crossref] [PubMed]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010).
[Crossref] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003).
[Crossref] [PubMed]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

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[Crossref] [PubMed]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[Crossref]

A. Di Falco, C. Conti, and S. Trillo, “Tunneling mediated by 2D+1 conical waves in a 1D lattice,” Phys. Rev. Lett. 101(1), 013601 (2008).
[Crossref] [PubMed]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, “Nonlinear electromagnetic X waves,” Phys. Rev. Lett. 90(17), 170406 (2003).
[Crossref] [PubMed]

C. Conti and S. Trillo, “Nonspreading wave packets in three dimensions formed by an ultracold bose gas in an optical lattice,” Phys. Rev. Lett. 92(12), 120404 (2004).
[Crossref] [PubMed]

Other (2)

B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge University Press, 1996).

H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (John Wiley and Sons, 2008).

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Figures (3)

Fig. 1
Fig. 1

(a) Gain/loss response of a GLMM with a square lattice in wave-vector space, as obtained by PWE. (b) Formation of an X-shaped wave: convolution of the GLMM response and a Gaussian input pulse (dashed circle) with carrier frequency centered at the corner of the BZ. (c) Gain/loss response of a GLMM structure with rhombic lattice. Brighter regions correspond to higher gain (the imaginary part of the frequency), in adimensional Lx/c units; where c is the speed of light in vacuum and L, Lx,y denote the lattice parameters.

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Fig. 2
Fig. 2

Square (a), rhombic (b,c) and hexagonal (d) GLMM lattices formed by two (a,b) and three (c,d) spatial harmonics. Rhombic lattices could be envisaged as squeezed square and hexagonal lattices, respectively. Open circles in the insets represent direct lattice, while q1,2,3 denote the reciprocal lattice vectors.

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Fig. 3
Fig. 3

Spatial profiles and spatial spectra (insets) of the electric field (Ez) of light pulses propagated in free space (a), in GLMM with a square (b-d) and rhombic lattice with fixed Ly /λ = 1.075 (e,f). (g) Optimum conditions for X-pulses generation, as a function of longitudinal, Lx, and transverse, Ly, lattice periods. Symbols represent numerical PSTD results, whereas the solid black curve is the analytic solution from Eq. (4). Dashed lines refer to square and hexagonal lattices.

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Equations (4)

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∂ t D z = ∂ x H y − ∂ y H x  ,  ∂ t B x =− ∂ y E z  ,  ∂ t B y = ∂ x E z  ,
∂ t D z = ε 0 ε(x,y) ∂ t E z +σ(x,y) E z  ,  B x,y = μ 0 H x,y  ,
σ(x,y)= 1 N σ 0 ∑ m=1 N cos( q → m ⋅ r → )  ,
( q y k ) 2 + ( q x k ) 2 =2 ( q x k ) ,     L x λ = 1 1± 1− ( λ/L y ) 2  .

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