Abstract

We investigate light propagation and self-localization in a voltage-controlled array of channel waveguides realized in undoped nematic liquid crystals. We report on discrete diffraction and solitons, as well as all-optical angular steering and the formation of multiband vector breathers. The results and are in excellent agreement with both coupled mode theory and full numerical simulations.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]
  7. A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
    [CrossRef]
  8. H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
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  9. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
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  11. A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
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    [CrossRef] [PubMed]
  24. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90, 253902 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  26. J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. 91, 143907 (2003).
    [CrossRef] [PubMed]
  27. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
    [CrossRef] [PubMed]
  28. R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
  31. M. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E 66, 036603 (2002).
    [CrossRef]
  32. G. Assanto, M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003).
    [CrossRef]
  33. C. Conti, M. Peccianti, G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
    [CrossRef] [PubMed]
  34. M. Peccianti, C. Conti, G. Assanto, A. de Luca, C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002).
    [CrossRef]
  35. M. Peccianti, G. Assanto, “Signal readdressing by steering of spatial solitons in bulk nematic liquid crystals,” Opt. Lett. 26, 1690–1692 (2002).
    [CrossRef]
  36. A. Fratalocchi, G. Assanto, K. A. Brzda̢kiewicz, M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. 29, 1530–1532 (2004).
    [CrossRef] [PubMed]
  37. I. C. Khoo, Liquid crystals: physical properties and optical phenomena (Wiley & Sons, New York, 1995).
  38. D. A. Dumm, A. Fukuda, G. R. Luckhurst, Physical properties of liquid crystals: nematics (INSPEC, London, 2001).

2004

R. A. Vicencio, M. I. Molina, Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).
[CrossRef]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

A. Fratalocchi, G. Assanto, K. A. Brzda̢kiewicz, M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. 29, 1530–1532 (2004).
[CrossRef] [PubMed]

2003

G. Assanto, M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003).
[CrossRef]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, D. N. Christodoulides, “Multiband Vector Lattice Solitons,” Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90, 253902 (2003).
[CrossRef] [PubMed]

J. Hudock, P. G. Kevrekidis, B. A. Malomed, D. N. Christodoulides, “Discrete vector solitons in two-dimensional nonlinear waveguide arrays: solutions, stability, and dynamics,” Phys. Rev. E 67, 056618 (2003).
[CrossRef]

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[CrossRef]

D. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, “Generation and stability of discrete gap solitons,” Opt. Lett. 28, 2345–2347 (2003).
[CrossRef] [PubMed]

2002

M. Peccianti, G. Assanto, “Signal readdressing by steering of spatial solitons in bulk nematic liquid crystals,” Opt. Lett. 26, 1690–1692 (2002).
[CrossRef]

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

M. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E 66, 036603 (2002).
[CrossRef]

M. Peccianti, C. Conti, G. Assanto, A. de Luca, C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002).
[CrossRef]

2001

D. A. Dumm, A. Fukuda, G. R. Luckhurst, Physical properties of liquid crystals: nematics (INSPEC, London, 2001).

D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001).
[CrossRef] [PubMed]

2000

H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

M. Peccianti, G. Assanto, A. de Luca, C. Umeton, I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

1999

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

1997

T. Peschel, R. Muschall, F. Lederer, “Power-controlled beam steering in non equidistant arrays of nonlinear waveguides,” Opt. Comm. 136, 16–21 (1997).
[CrossRef]

1996

A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

O. Bang, P. D. Miller, “Exploiting discreteness for switching in waveguide arrays,” Opt. Lett. 21, 1105–1107 (1996).
[CrossRef] [PubMed]

W. Krolikowsky, Y. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–880 (1996).
[CrossRef]

1995

1991

1990

N. Finlayson, G. I. Stegeman, “Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler,” Appl. Phys. Lett. 56, 2276–2278 (1990).
[CrossRef]

1988

1985

1982

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

1973

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[CrossRef]

Aceves, A. B.

A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Aitchison, J. S.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90, 253902 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

Assanto, G.

C. Conti, M. Peccianti, G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

A. Fratalocchi, G. Assanto, K. A. Brzda̢kiewicz, M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. 29, 1530–1532 (2004).
[CrossRef] [PubMed]

G. Assanto, M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003).
[CrossRef]

M. Peccianti, C. Conti, G. Assanto, A. de Luca, C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002).
[CrossRef]

M. Peccianti, G. Assanto, “Signal readdressing by steering of spatial solitons in bulk nematic liquid crystals,” Opt. Lett. 26, 1690–1692 (2002).
[CrossRef]

M. Peccianti, G. Assanto, A. de Luca, C. Umeton, I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

Bang, O.

Brauer, A.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

Brzda?kiewicz, K. A.

Christodoulides, D.

D. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

Christodoulides, D. N.

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

J. Hudock, P. G. Kevrekidis, B. A. Malomed, D. N. Christodoulides, “Discrete vector solitons in two-dimensional nonlinear waveguide arrays: solutions, stability, and dynamics,” Phys. Rev. E 67, 056618 (2003).
[CrossRef]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, D. N. Christodoulides, “Multiband Vector Lattice Solitons,” Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001).
[CrossRef] [PubMed]

D. N. Christodoulides, R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988).
[CrossRef] [PubMed]

Cohen, O.

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, D. N. Christodoulides, “Multiband Vector Lattice Solitons,” Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

Conti, C.

C. Conti, M. Peccianti, G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

M. Peccianti, C. Conti, G. Assanto, A. de Luca, C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002).
[CrossRef]

de Angelis, C.

A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

de Luca, A.

M. Peccianti, C. Conti, G. Assanto, A. de Luca, C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002).
[CrossRef]

M. Peccianti, G. Assanto, A. de Luca, C. Umeton, I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

Dumm, D. A.

D. A. Dumm, A. Fukuda, G. R. Luckhurst, Physical properties of liquid crystals: nematics (INSPEC, London, 2001).

Efremidis, N. K.

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

Eisenberg, H. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90, 253902 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

Eugenieva, E. D.

D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001).
[CrossRef] [PubMed]

Finlayson, N.

N. Finlayson, G. I. Stegeman, “Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler,” Appl. Phys. Lett. 56, 2276–2278 (1990).
[CrossRef]

Fleischer, J. W.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, D. N. Christodoulides, “Multiband Vector Lattice Solitons,” Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

Fratalocchi, A.

Fukuda, A.

D. A. Dumm, A. Fukuda, G. R. Luckhurst, Physical properties of liquid crystals: nematics (INSPEC, London, 2001).

Garmire, E.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[CrossRef]

Garvin, H. L.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[CrossRef]

Hudock, J.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

J. Hudock, P. G. Kevrekidis, B. A. Malomed, D. N. Christodoulides, “Discrete vector solitons in two-dimensional nonlinear waveguide arrays: solutions, stability, and dynamics,” Phys. Rev. E 67, 056618 (2003).
[CrossRef]

Hunsperger, R. G.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[CrossRef]

Iwanow, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

Joseph, R. I.

Karpierz, M.

M. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E 66, 036603 (2002).
[CrossRef]

Karpierz, M. A.

Kevrekidis, P. G.

J. Hudock, P. G. Kevrekidis, B. A. Malomed, D. N. Christodoulides, “Discrete vector solitons in two-dimensional nonlinear waveguide arrays: solutions, stability, and dynamics,” Phys. Rev. E 67, 056618 (2003).
[CrossRef]

Khoo, I. C.

M. Peccianti, G. Assanto, A. de Luca, C. Umeton, I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

I. C. Khoo, Liquid crystals: physical properties and optical phenomena (Wiley & Sons, New York, 1995).

Kivshar, Y. S.

R. A. Vicencio, M. I. Molina, Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, “Generation and stability of discrete gap solitons,” Opt. Lett. 28, 2345–2347 (2003).
[CrossRef] [PubMed]

W. Krolikowsky, Y. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–880 (1996).
[CrossRef]

Krolikowsky, W.

Lederer, F.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

D. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

T. Peschel, R. Muschall, F. Lederer, “Power-controlled beam steering in non equidistant arrays of nonlinear waveguides,” Opt. Comm. 136, 16–21 (1997).
[CrossRef]

A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

C. Schmidt-Hattenberger, U. Trutschel, F. Lederer, “Nonlinear switching in multiple-core couplers,” Opt. Lett. 16, 294–296 (1991).
[CrossRef] [PubMed]

Luckhurst, G. R.

D. A. Dumm, A. Fukuda, G. R. Luckhurst, Physical properties of liquid crystals: nematics (INSPEC, London, 2001).

Malomed, B. A.

J. Hudock, P. G. Kevrekidis, B. A. Malomed, D. N. Christodoulides, “Discrete vector solitons in two-dimensional nonlinear waveguide arrays: solutions, stability, and dynamics,” Phys. Rev. E 67, 056618 (2003).
[CrossRef]

Mandelik, D.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90, 253902 (2003).
[CrossRef] [PubMed]

Matsumoto, M.

Meier, J.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

Miller, P. D.

Min, Y.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Molina, M. I.

R. A. Vicencio, M. I. Molina, Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).
[CrossRef]

Morandotti, R.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90, 253902 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

Muschall, R.

T. Peschel, R. Muschall, F. Lederer, “Power-controlled beam steering in non equidistant arrays of nonlinear waveguides,” Opt. Comm. 136, 16–21 (1997).
[CrossRef]

A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Peccianti, M.

C. Conti, M. Peccianti, G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

G. Assanto, M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003).
[CrossRef]

M. Peccianti, C. Conti, G. Assanto, A. de Luca, C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002).
[CrossRef]

M. Peccianti, G. Assanto, “Signal readdressing by steering of spatial solitons in bulk nematic liquid crystals,” Opt. Lett. 26, 1690–1692 (2002).
[CrossRef]

M. Peccianti, G. Assanto, A. de Luca, C. Umeton, I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

Pertsch, T.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

Peschel, T.

T. Peschel, R. Muschall, F. Lederer, “Power-controlled beam steering in non equidistant arrays of nonlinear waveguides,” Opt. Comm. 136, 16–21 (1997).
[CrossRef]

A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Peschel, U.

T. Pertsch, U. Peschel, F. Lederer, “All-optical switching in quadratically nonlinear waveguide arrays,” Opt. Lett. 28, 102–104 (2003).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals, (Springer-Verlag, Berlin, 2001).

Schiek, R.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Schmidt-Hattenberger, C.

Schwartz, T.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, D. N. Christodoulides, “Multiband Vector Lattice Solitons,” Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

Segev, M.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, D. N. Christodoulides, “Multiband Vector Lattice Solitons,” Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

Silberberg, Y.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90, 253902 (2003).
[CrossRef] [PubMed]

D. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

Sohler, W.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

Somekh, S.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[CrossRef]

Stegeman, G.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

Stegeman, G. I.

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

N. Finlayson, G. I. Stegeman, “Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler,” Appl. Phys. Lett. 56, 2276–2278 (1990).
[CrossRef]

Sukhorukov, A. A.

A. A. Sukhorukov, Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, “Generation and stability of discrete gap solitons,” Opt. Lett. 28, 2345–2347 (2003).
[CrossRef] [PubMed]

Trillo, S.

A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

S. Trillo, S. Wabnitz, “Coupling instability and power-induced switching with two-core dual-polarizations fiber nonlinear couplers,” J. Opt. Soc. Am. B 5, 483–491 (1985).
[CrossRef]

Trutschel, U.

Umeton, C.

M. Peccianti, C. Conti, G. Assanto, A. de Luca, C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002).
[CrossRef]

M. Peccianti, G. Assanto, A. de Luca, C. Umeton, I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

Vicencio, R. A.

R. A. Vicencio, M. I. Molina, Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).
[CrossRef]

Wabnitz, S.

A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

S. Trillo, S. Wabnitz, “Coupling instability and power-induced switching with two-core dual-polarizations fiber nonlinear couplers,” J. Opt. Soc. Am. B 5, 483–491 (1985).
[CrossRef]

Yariv, A.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[CrossRef]

Zentgraf, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

Appl. Phys. Lett.

M. Peccianti, G. Assanto, A. de Luca, C. Umeton, I. C. Khoo, “Electrically assisted self-confinement and waveguiding in planar nematic liquid crystal cells,” Appl. Phys. Lett. 77, 7–9 (2000).
[CrossRef]

N. Finlayson, G. I. Stegeman, “Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler,” Appl. Phys. Lett. 56, 2276–2278 (1990).
[CrossRef]

M. Peccianti, C. Conti, G. Assanto, A. de Luca, C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002).
[CrossRef]

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[CrossRef]

T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, F. Lederer, “Beam steering in waveguide arrays,” Appl. Phys. Lett. 80, 3247–3249 (2002).
[CrossRef]

IEEE J. Quantum Electron.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-18, 1580–1583 (1982).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[CrossRef]

G. Assanto, M. Peccianti, “Spatial solitons in nematic liquid crystals,” IEEE J. Quantum Electron. 39, 13–21 (2003).
[CrossRef]

J. Opt. Soc. Am. B

Nature

D. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef] [PubMed]

Opt. Comm.

T. Peschel, R. Muschall, F. Lederer, “Power-controlled beam steering in non equidistant arrays of nonlinear waveguides,” Opt. Comm. 136, 16–21 (1997).
[CrossRef]

Opt. Lett.

Phys. Rev. E

A. B. Aceves, C. de Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

M. Karpierz, “Solitary waves in liquid crystalline waveguides,” Phys. Rev. E 66, 036603 (2002).
[CrossRef]

R. A. Vicencio, M. I. Molina, Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E 70, 026602 (2004).
[CrossRef]

J. Hudock, P. G. Kevrekidis, B. A. Malomed, D. N. Christodoulides, “Discrete vector solitons in two-dimensional nonlinear waveguide arrays: solutions, stability, and dynamics,” Phys. Rev. E 67, 056618 (2003).
[CrossRef]

Phys. Rev. Lett.

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Discrete vector solitons in Kerr nonlinear waveguide arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

R. Iwanow, R. Schiek, G. I. Stegeman, T. Pertsch, F. Lederer, Y. Min, W. Sohler, “Observation of discrete quadratic solitons,” Phys. Rev. Lett. 93, 113902 (2004).
[CrossRef] [PubMed]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, D. N. Christodoulides, “Multiband Vector Lattice Solitons,” Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90, 253902 (2003).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001).
[CrossRef] [PubMed]

Other

I. C. Khoo, Liquid crystals: physical properties and optical phenomena (Wiley & Sons, New York, 1995).

D. A. Dumm, A. Fukuda, G. R. Luckhurst, Physical properties of liquid crystals: nematics (INSPEC, London, 2001).

K. Sakoda, Optical Properties of Photonic Crystals, (Springer-Verlag, Berlin, 2001).

Supplementary Material (2)

» Media 1: MOV (1123 KB)     
» Media 2: MOV (222 KB)     

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Figures (8)

Fig. 1 .
Fig. 1 .

eft) Sketch of the NLC waveguide array. Right) Typical transverse index distribution for a bias of 1V.

Fig. 2.
Fig. 2.

(1123 KB) Eigenvalue spectrum of the array (left), width of first and second gaps (center) and corresponding refractive index (cross sections at the center of the cell, x=0) for different biases. (Right) Gap 1 and gap 2 are between bands 0–1 and bands 1–2, respectively.

Fig. 3.
Fig. 3.

Experimental response of the NLC array with Λ=8µm: (a) discrete diffraction in (y,z) for P=1mW; (b) assembled photographs of light propagation in z 0<z<z 1 versus bias and (c) versus input power; (d) photograph of a discrete soliton excited by P=10mW and propagating along the input channel.

Fig. 4.
Fig. 4.

(222 KB) Animation showing light propagation in the plane (y,z), evolving from diffraction to discrete soliton generation as the power injected in a single channel ramps from 0.1.to 1.0mW. Here V=0.74V (z-and y-axes units are in µm).

Fig. 5.
Fig. 5.

BPM simulations of discrete beam steering: (a) linear (P L =0.1mW) and (b) nonlinear (P H =2.0mW) light propagation in (y,z); (c) corresponding cross-section of the output intensity distribution in z=2mm.

Fig. 6.
Fig. 6.

Experimental observations of angular beam steering: (a) discrete diffraction of a tilted beam for P L =1mW; (b) observation of nonlinear steering for P H =7mW; (c) profiles of the output intensity in z=2mm.

Fig. 7.
Fig. 7.

Numerical (BPM) experiments on vectors breather in NLC: (a) Floquet-Bloch profiles (density plots) with corresponding refractive index distributions for V=1V (contour lines) and ky=1 in band 0 (top) and band 1 (bottom); (b) light propagation of FB modes belonging to band 0 (top) and band 1 (bottom) for P=0.2mW; (c) a vector breather generated by superimposing band 0 and band 1 excitations (b) with a total P=0.4mW.

Fig. 8.
Fig. 8.

(a) Low power P=0.2mW and (b) high power P=7mW light propagation in the NLC array excited by a Gaussian beam with w y =5µm launched between waveguides. (c) Breather period (blue dots and error bars) and calculated width of gap 1 (red solid line) versus applied bias.

Equations (4)

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

K 2 θ + Δ ε RF E x 2 2 sin 2 θ = 0
x [ ( ε cos 2 θ + ε sin 2 θ ) x V ] + ε 2 y 2 V = 0
E FB = Π k ( x , y ) exp j ( k y y k z z )
[ 2 + 2 j k y y + k 0 2 n 2 ( x , y ) k y 2 ] Π k ( x , y ) = k z 2 Π k ( x , y )

Metrics