Abstract

We show that new families of two- and three-dimensional nondiffracting Bessel X waves are possible in linear bidispersive optical systems. These X waves can be observed in both bulk and waveguide configurations as well as in photonic crystal lattices that simultaneously exhibit normal and anomalous dispersive–diffractive properties in different spatial or spatiotemporal coordinates.

© 2004 Optical Society of America

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  1. J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef] [PubMed]
  2. J. Y. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
    [CrossRef]
  3. M. A. Porras, Opt. Lett. 26, 1364 (2001).
    [CrossRef]
  4. J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. E 62, 4261 (2000).
    [CrossRef]
  5. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, J. Opt. Soc. Am. A 10, 75 (1993).
    [CrossRef]
  6. P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).
    [CrossRef]
  7. H. Sonajalg, M. Ratsep, and P. Saari, Opt. Lett. 22, 310 (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, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
    [CrossRef]
  10. J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, Phys. Rev. Lett. 77, 3783 (1996).
    [CrossRef] [PubMed]
  11. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
    [CrossRef] [PubMed]
  12. J. Hudock, N. K. Efremidis, and D. N. Christodoulides, Opt. Lett. 29, 268 (2004).
    [CrossRef] [PubMed]
  13. The X-wave solution of Eq. (4) can be further generalized by shifting the real independent variable x in the complex domain, i.e., x+iσ. Similar shifts can also be used in Eq. (6).
  14. L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Pramana J. Phys. 57, 1041 (2001).
    [CrossRef]

2004 (1)

2003 (2)

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, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

2001 (2)

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Pramana J. Phys. 57, 1041 (2001).
[CrossRef]

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

2000 (2)

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

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

1997 (2)

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

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

1996 (1)

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, Phys. Rev. Lett. 77, 3783 (1996).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

J. Y. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
[CrossRef]

1987 (1)

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

Aitchison, J. S.

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

Besieris, I. M.

Christodoulides, D. N.

Conti, C.

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, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

Crasovan, L.-C.

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Pramana J. Phys. 57, 1041 (2001).
[CrossRef]

Di Trapani, P.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

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

Durnin, J.

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

Eberly, J. H.

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

Efremidis, N. K.

Eisenberg, H. S.

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

Fagerholm, J.

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

Friberg, A. T.

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

Gaeta, A. L.

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, Phys. Rev. Lett. 77, 3783 (1996).
[CrossRef] [PubMed]

Greenleaf, J. F.

J. Y. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
[CrossRef]

Hudock, J.

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]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

Lu, J. Y.

J. Y. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
[CrossRef]

Malomed, B. A.

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Pramana J. Phys. 57, 1041 (2001).
[CrossRef]

Miceli, J. J.

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

Mihalache, D.

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Pramana J. Phys. 57, 1041 (2001).
[CrossRef]

Morandotti, R.

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

Piskarskas, A.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

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.

Ranka, J. K.

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, Phys. Rev. Lett. 77, 3783 (1996).
[CrossRef] [PubMed]

Ratsep, M.

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]

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

Salo, J.

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

Salomaa, M. M.

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

Schirmer, R. W.

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, Phys. Rev. Lett. 77, 3783 (1996).
[CrossRef] [PubMed]

Shaarawi, A. M.

Silberberg, Y.

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

Sonajalg, H.

Trillo, S.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

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]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

Valiulis, G.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

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

Ziolkowski, R. W.

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

J. Y. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19 (1992).
[CrossRef]

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

Opt. Lett. (3)

Phys. Rev. E (1)

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

Phys. Rev. Lett. (6)

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, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef]

J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, Phys. Rev. Lett. 77, 3783 (1996).
[CrossRef] [PubMed]

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

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

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

Pramana J. Phys. (1)

L.-C. Crasovan, B. A. Malomed, and D. Mihalache, Pramana J. Phys. 57, 1041 (2001).
[CrossRef]

Other (1)

The X-wave solution of Eq. (4) can be further generalized by shifting the real independent variable x in the complex domain, i.e., x+iσ. Similar shifts can also be used in Eq. (6).

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Figures (4)

Fig. 1
Fig. 1

(a) Intensity profile of an initial Gaussian beam on top of a constant background and (b) the resulting X-shaped diffraction pattern at z=2.

Fig. 2
Fig. 2

Intensity profiles of 2D X waves as given by Eq. (4) with a=c=1 and m values of (a) 0, (b) 1, (c) 2, (d) 3.

Fig. 3
Fig. 3

Phase of a 2D X-wave solution given by Eq. (4) with a=c=1 and m values of (a) 0 and (b) 1.

Fig. 4
Fig. 4

Isointensity profile of a 3D X-wave solution given by Eq. (6) with a=c=1 and (a) l=m=0 and (b) l=1 and m=0.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

iψz+12k2ψx2+2ψy2-k22ψτ2=0,
iψz+dx2ψx2+dy2ψy2=0,
iψZ+2ψX2-2ψY2=0,
iψZ+2ψX2+2ψY2-2ψT2=0.
ψ=AKmaX2+iY+c21/2×expim tan-1iY+cXexpia2Z,=AKmaX2+iY+c21/2×iX-Y-ciX+Y+cm/2 expia2Z,
UXX-UYY=0.
ψ=AKl+1/2arr1/2Ylmθ,ϕexpia2z,

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