Abstract

We show that, in low-dimensional photonic bandgaps, wave diffraction resulting from localization in the translational-invariant directions is strongly influenced by the photonic band structure of the periodic crystal, leading to new kinds of wave localization. In particular, for a periodic layered structure we show that, close to a bandgap edge, diffraction is enhanced, with a transition from a parabolic diffraction curve—typical of isotropic media and supporting Gaussian beams—to hyperbolic or elliptic diffraction curves. In the last two cases localization in the form of stationary X-shaped or sinc-shaped waves is possible.

© 2004 Optical Society of America

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  1. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
    [CrossRef]
  2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
    [CrossRef] [PubMed]
  3. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
    [CrossRef]
  4. M. Notomi, Opt. Quantum Electron. 34, 133 (2002).
    [CrossRef]
  5. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, Phys. Rev. Lett. 91, 207401 (2003).
    [CrossRef]
  6. See, e.g., I. M. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, Prog. Electron. Res. (PIER) 19, 1 (1998), and references therein.
    [CrossRef]
  7. P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).
    [CrossRef]
  8. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
    [CrossRef]
  9. C. Conti and S. Trillo, Phys. Rev. Lett. 92, 120404 (2004).
    [CrossRef]
  10. M. A. Porras, S. Trillo, C. Conti, and P. Di Trapani, Opt. Lett. 28, 1090 (2003).
    [CrossRef] [PubMed]
  11. S. Longhi, Opt. Lett. 29, 147 (2004).
    [CrossRef] [PubMed]
  12. S. Longhi, Phys. Rev. E 69, 016606 (2004).
    [CrossRef]
  13. Neglecting the second-order derivative in Eq. (5) corresponds, for an isotropic medium, to the well-known paraxial approximation, which is valid provided that the transverse beam size of, e.g., a Gaussian beam is much larger than its Rayleigh range. As θ diverges, the paraxial approximation breaks down, and higher-order derivative terms should be included, marking a continuous transition from the parabolic to the hyperbolic (or elliptic) diffractive regimes.
  14. This is due to the fact that in the monochromatic regime the wave equation in isotropic media is always of the elliptic type, and only the inclusion of temporal degree of freedom may lead to a hyperbolic equation supporting X waves.
  15. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  16. Most usual X waves, such as those considered in Refs. 9 and 10, actually correspond to a spectral amplitude Fk⊥∝exp-αk⊥; we consider here a different spectral shape for better comparing the transition from parabolic to hyperbolic localization for a transverse Gaussian field distribution.

2004 (3)

C. Conti and S. Trillo, Phys. Rev. Lett. 92, 120404 (2004).
[CrossRef]

S. Longhi, Phys. Rev. E 69, 016606 (2004).
[CrossRef]

S. Longhi, Opt. Lett. 29, 147 (2004).
[CrossRef] [PubMed]

2003 (3)

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

M. A. Porras, S. Trillo, C. Conti, and P. Di Trapani, Opt. Lett. 28, 1090 (2003).
[CrossRef] [PubMed]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef]

2002 (2)

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef]

M. Notomi, Opt. Quantum Electron. 34, 133 (2002).
[CrossRef]

2000 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

1998 (2)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

See, e.g., I. M. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, Prog. Electron. Res. (PIER) 19, 1 (1998), and references therein.
[CrossRef]

1997 (1)

P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).
[CrossRef]

Abdel-Rahman, M.

See, e.g., I. M. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, Prog. Electron. Res. (PIER) 19, 1 (1998), and references therein.
[CrossRef]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Aydin, K.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef]

Besieris, I. M.

See, e.g., I. M. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, Prog. Electron. Res. (PIER) 19, 1 (1998), and references therein.
[CrossRef]

Bräuer, A.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef]

Chatzipetros, A.

See, e.g., I. M. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, Prog. Electron. Res. (PIER) 19, 1 (1998), and references therein.
[CrossRef]

Conti, C.

C. Conti and S. Trillo, Phys. Rev. Lett. 92, 120404 (2004).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

M. A. Porras, S. Trillo, C. Conti, and P. Di Trapani, Opt. Lett. 28, 1090 (2003).
[CrossRef] [PubMed]

Cubukcu, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef]

Di Trapani, P.

M. A. Porras, S. Trillo, C. Conti, and P. Di Trapani, Opt. Lett. 28, 1090 (2003).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Foteinopolou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef]

Jedrkiewicz, O.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Lederer, F.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef]

Longhi, S.

S. Longhi, Opt. Lett. 29, 147 (2004).
[CrossRef] [PubMed]

S. Longhi, Phys. Rev. E 69, 016606 (2004).
[CrossRef]

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Notomi, M.

M. Notomi, Opt. Quantum Electron. 34, 133 (2002).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Ozbay, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef]

Pertsch, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef]

Peschel, U.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef]

Piskarskas, A.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Porras, M. A.

Reivelt, K.

P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).
[CrossRef]

Saari, P.

P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).
[CrossRef]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Shaarawi, A.

See, e.g., I. M. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, Prog. Electron. Res. (PIER) 19, 1 (1998), and references therein.
[CrossRef]

Silberberg, Y.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Trillo, S.

C. Conti and S. Trillo, Phys. Rev. Lett. 92, 120404 (2004).
[CrossRef]

M. A. Porras, S. Trillo, C. Conti, and P. Di Trapani, Opt. Lett. 28, 1090 (2003).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Trull, J.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Valiulis, G.

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Zentgraf, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

M. Notomi, Opt. Quantum Electron. 34, 133 (2002).
[CrossRef]

Phys. Rev. B (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Phys. Rev. E (1)

S. Longhi, Phys. Rev. E 69, 016606 (2004).
[CrossRef]

Phys. Rev. Lett. (6)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, Phys. Rev. Lett. 91, 207401 (2003).
[CrossRef]

P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
[CrossRef]

C. Conti and S. Trillo, Phys. Rev. Lett. 92, 120404 (2004).
[CrossRef]

Prog. Electron. Res. (PIER) (1)

See, e.g., I. M. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, Prog. Electron. Res. (PIER) 19, 1 (1998), and references therein.
[CrossRef]

Other (4)

Neglecting the second-order derivative in Eq. (5) corresponds, for an isotropic medium, to the well-known paraxial approximation, which is valid provided that the transverse beam size of, e.g., a Gaussian beam is much larger than its Rayleigh range. As θ diverges, the paraxial approximation breaks down, and higher-order derivative terms should be included, marking a continuous transition from the parabolic to the hyperbolic (or elliptic) diffractive regimes.

This is due to the fact that in the monochromatic regime the wave equation in isotropic media is always of the elliptic type, and only the inclusion of temporal degree of freedom may lead to a hyperbolic equation supporting X waves.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Most usual X waves, such as those considered in Refs. 9 and 10, actually correspond to a spectral amplitude Fk⊥∝exp-αk⊥; we consider here a different spectral shape for better comparing the transition from parabolic to hyperbolic localization for a transverse Gaussian field distribution.

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