Abstract

We study tunable refraction of light in one-dimensional periodic lattices induced optically in a photorefractive crystal. We observe experimentally both positive and negative refraction of beams that selectively excite the first or second spectral bands of the periodic lattice, and we demonstrate tunability of the output beam position by dynamically adjusting the lattice depth. At higher laser intensities, beam broadening due to diffraction can be suppressed through nonlinear self-focusing while preserving the general steering properties.

© 2005 Optical Society of America

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References

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  1. D. R. Smith, W. J. Padilla, D. C. Vier, S. C.Nemat Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
    [Crossref] [PubMed]
  2. C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Koltenbah, and M. Tanielian, Phys. Rev. Lett. 90, 107401 (2003).
    [Crossref]
  3. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, Nature 423, 604 (2003).
    [Crossref] [PubMed]
  4. P. St.J. Russell, T. A. Birks, and F. D.Lloyd Lucas, in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds. (Plenum, 1995), pp. 585–633.
    [Crossref]
  5. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003).
    [Crossref]
  6. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, Opt. Lett. 28, 710 (2003).
    [Crossref] [PubMed]
  7. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
    [Crossref]
  8. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, Opt. Lett. 30, 174 (2005).
    [Crossref] [PubMed]
  9. D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
    [Crossref]
  10. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
    [Crossref]

2005 (1)

2004 (1)

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[Crossref]

2003 (5)

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Koltenbah, and M. Tanielian, Phys. Rev. Lett. 90, 107401 (2003).
[Crossref]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, Nature 423, 604 (2003).
[Crossref] [PubMed]

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

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, Opt. Lett. 28, 710 (2003).
[Crossref] [PubMed]

2002 (1)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[Crossref]

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C.Nemat Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

Aitchison, J. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

Assanto, G.

Aydin, K.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, Nature 423, 604 (2003).
[Crossref] [PubMed]

Birks, T. A.

P. St.J. Russell, T. A. Birks, and F. D.Lloyd Lucas, in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds. (Plenum, 1995), pp. 585–633.
[Crossref]

Brzdakiewicz, K. A.

Carmon, T.

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

Christodoulides, D. N.

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

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[Crossref]

Cubukcu, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, Nature 423, 604 (2003).
[Crossref] [PubMed]

Efremidis, N. K.

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

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[Crossref]

Eisenberg, H. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[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]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[Crossref]

Foteinopoulou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, Nature 423, 604 (2003).
[Crossref] [PubMed]

Fratalocchi, A.

Greegor, R. B.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Koltenbah, and M. Tanielian, Phys. Rev. Lett. 90, 107401 (2003).
[Crossref]

Hanna, B.

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[Crossref]

Karpierz, M. A.

Kivshar, Y.

Kivshar, Yu. S.

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[Crossref]

Koltenbah, B. E.C.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Koltenbah, and M. Tanielian, Phys. Rev. Lett. 90, 107401 (2003).
[Crossref]

Krolikowski, W.

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[Crossref]

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, Opt. Lett. 28, 710 (2003).
[Crossref] [PubMed]

Li, K.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Koltenbah, and M. Tanielian, Phys. Rev. Lett. 90, 107401 (2003).
[Crossref]

Lucas, F. D.Lloyd

P. St.J. Russell, T. A. Birks, and F. D.Lloyd Lucas, in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds. (Plenum, 1995), pp. 585–633.
[Crossref]

Mandelik, D.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

Morandotti, R.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

Nasser, S. C.Nemat

D. R. Smith, W. J. Padilla, D. C. Vier, S. C.Nemat Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

Neshev, D.

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[Crossref]

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, Opt. Lett. 28, 710 (2003).
[Crossref] [PubMed]

Ostrovskaya, E.

Ozbay, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, Nature 423, 604 (2003).
[Crossref] [PubMed]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C.Nemat Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

Parazzoli, C. G.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Koltenbah, and M. Tanielian, Phys. Rev. Lett. 90, 107401 (2003).
[Crossref]

Russell, P. St.J.

P. St.J. Russell, T. A. Birks, and F. D.Lloyd Lucas, in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds. (Plenum, 1995), pp. 585–633.
[Crossref]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C.Nemat Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

Sears, S.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[Crossref]

Segev, M.

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

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[Crossref]

Silberberg, Y.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

Smith, D. R.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C.Nemat Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

Soukoulis, C. M.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, Nature 423, 604 (2003).
[Crossref] [PubMed]

Sukhorukov, A. A.

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[Crossref]

Tanielian, M.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Koltenbah, and M. Tanielian, Phys. Rev. Lett. 90, 107401 (2003).
[Crossref]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C.Nemat Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

Nature (1)

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, Nature 423, 604 (2003).
[Crossref] [PubMed]

Opt. Lett. (2)

Phys. Rev. E (1)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, Phys. Rev. E 66, 046602 (2002).
[Crossref]

Phys. Rev. Lett. (5)

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 083905 (2004).
[Crossref]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003).
[Crossref]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C.Nemat Nasser, and S. Schultz, Phys. Rev. Lett. 84, 4184 (2000).
[Crossref] [PubMed]

C. G. Parazzoli, R. B. Greegor, K. Li, B. E.C. Koltenbah, and M. Tanielian, Phys. Rev. Lett. 90, 107401 (2003).
[Crossref]

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

Other (1)

P. St.J. Russell, T. A. Birks, and F. D.Lloyd Lucas, in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds. (Plenum, 1995), pp. 585–633.
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic of the beam refraction associated with different Bloch modes: at the top and bottom of the first band (FB) and top of second band (SB). (b), (c) Bloch-wave dispersion curves in (b) shallow ( Δ n = 0.3 × 10 3 ) and (c) deeper ( Δ n = 0.45 × 10 3 ) lattices. (d) Effective diffraction coefficients at the band edges versus lattice index contrast Δ n compared with bulk diffraction D. (e), (f) Numerical results for refraction of beams associated with the top of the first and second bands in lattices corresponding to (b), (c) for α = 1 mrad .

Fig. 2
Fig. 2

(a) Shift of the output position of beams associated with three different Bloch waves versus the lattice tilt (measured in air). (b) Characteristic beam profiles for a tilt of 3 mrad. In (a) and (b) the dashed line corresponds to the shift of the lattice. The bias voltage is 1.5 kV.

Fig. 3
Fig. 3

(a) Measured output shift versus bias voltage at low power for beams corresponding to three different Bloch modes. The lattice tilt is 2 mrad (measured in air). (b) Output beam intensity profiles (at 1.8 kV) corresponding to the top of the second band for low (top) and high (bottom) power. In (a) and (b) the dashed line indicates the shift of the lattice.

Equations (1)

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i E z + D 2 E x 2 + F ( x α z , E 2 ) E = 0 ,

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