Abstract

Linear defect modes in one-dimensional photonic lattices are studied theoretically. For negative (repulsive) defects, various localized defect modes are found. The strongest confinement of the defect modes appears when the lattice intensity at the defect site is nonzero rather than zero. When launched at small angles into such a defect site of the lattice, a Gaussian beam can be trapped and undergo snake oscillations under appropriate conditions.

© 2005 Optical Society of America

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2004 (5)

D. K. Campbell, S. Flach, and Y. S. Kivshar, Phys. Today 57(1), 43 (2004).
[CrossRef]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef]

J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, Opt. Lett. 29, 1662 (2004).
[CrossRef] [PubMed]

2003 (4)

2001 (1)

A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef]

1999 (1)

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
[CrossRef]

1998 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

1994 (1)

Aceves, A. B.

Aitchison, J. S.

R. Morandotti, H. S. Eisenberg, D. Dandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, Opt. Lett. 28, 834 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Bezryadina, A.

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Campbell, D. K.

D. K. Campbell, S. Flach, and Y. S. Kivshar, Phys. Today 57(1), 43 (2004).
[CrossRef]

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

Chen, Z.

J. Yang, I. Makasyuk, A. Bezryadina, and Z. Chen, Opt. Lett. 29, 1662 (2004).
[CrossRef] [PubMed]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef]

Christodoulides, D. N.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

Dandelik, D.

De Angelis, C.

Efremidis, N. K.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

Eisenberg, H. S.

R. Morandotti, H. S. Eisenberg, D. Dandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, Opt. Lett. 28, 834 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Eugenieva, E. D.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef]

Flach, S.

D. K. Campbell, S. Flach, and Y. S. Kivshar, Phys. Today 57(1), 43 (2004).
[CrossRef]

Fleischer, J. W.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

Iwanow, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Kartashov, Y. V.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef]

Kivshar, Y.

Kivshar, Y. S.

D. K. Campbell, S. Flach, and Y. S. Kivshar, Phys. Today 57(1), 43 (2004).
[CrossRef]

Kivshar, Yu. S.

A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef]

Krolikowski, W.

Lederer, F.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Makasyuk, I.

Martin, H.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Min, Y.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Modotto, D.

Morandotti, R.

R. Morandotti, H. S. Eisenberg, D. Dandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, Opt. Lett. 28, 834 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Neshev, D.

Ostrovskaya, E.

Pertsch, T.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Peschel, U.

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
[CrossRef]

Schiek, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Segev, M.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

R. Morandotti, H. S. Eisenberg, D. Dandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, and J. S. Aitchison, Opt. Lett. 28, 834 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

Sohler, W.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Sorel, M.

Stanley, C. R.

Stegeman, G. I.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Sukhorukov, A. A.

A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef]

Torner, L.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef]

Trillo, S.

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef]

Wabnitz, S.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Yang, J.

Appl. Phys. Lett. (1)

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Appl. Phys. Lett. 75, 1348 (1999).
[CrossRef]

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Opt. Lett. (4)

Phys. Rev. Lett. (6)

A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 (1998).
[CrossRef]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef]

Phys. Today (1)

D. K. Campbell, S. Flach, and Y. S. Kivshar, Phys. Today 57(1), 43 (2004).
[CrossRef]

Other (1)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

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Figures (4)

Fig. 1
Fig. 1

(a) Lattice intensity profile with I 0 = 3 and ϵ = 1 . (b) Applied dc field parameter E 0 versus the defect eigenvalues μ; the shaded regions are Bloch bands. (c)–(e) Three defect modes at ( E 0 , μ ) = ( 1.2 , 1.604 ) , ( 3.5 , 5.812 ) , ( 7.5 , 7.997 ) , marked by circles in (b). The shaded stripes indicate the locations of the lattice’s peak intensities.

Fig. 2
Fig. 2

(a) Lattice intensity pattern with I 0 = 3 and ϵ = 1 . (b) Evolution of an exact defect mode [shown in Fig. 1(e)] at E 0 = 7.5 . (c)–(e) Evolutions of a Gaussian beam at E 0 values of 5, 7.5, and 10, respectively.

Fig. 3
Fig. 3

(a) Defect strength ϵ versus the defect eigenvalues μ. (b) Intensity profile I L ( x ) of the photonic lattice with ϵ = 0.5 . (c)–(e) Three defect modes of the first bandgap with ( ϵ , μ ) , marked by circles in (a).

Fig. 4
Fig. 4

Evolution of a Gaussian beam launched at zero (b) and nonzero (c) angles into the defect site of a photonic lattice shown in (a). Here I 0 = 3 , E 0 = 6 , and ϵ = 0.5 in Eq. (1). The initial phase gradient in (c) is k = 1 .

Equations (1)

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i U z + U x x E 0 1 + I L ( x ) U = 0 .

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