Abstract

Coupled mode theory for waveguide arrays is extended to next-nearest neighbor interactions using propagation equations. Both lateral diffraction and propagation of Floquet-Bloch waves are altered respectively by extra coupling and non-orthogonality between isolated waveguide modes. The analytical formula describing the distortions of the diffraction relation is validated by direct numerical simulation for weakly coupled InP and GaAs shallow ridge waveguides and for strongly coupled Si-SiO2 buried strip waveguides. The impact of extended coupled mode theory on propagation and diffraction design in waveguide arrays is discussed with reference to available experimental work.

© 2010 OSA

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  1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
    [CrossRef] [PubMed]
  2. J. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-06-1780 .
    [CrossRef] [PubMed]
  3. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
    [CrossRef] [PubMed]
  4. T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83(23), 4752–4755 (1999).
    [CrossRef]
  5. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optics Bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
    [CrossRef]
  6. F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling modulation in an optical waveguide system,” Phys. Rev. Lett. 101(14), 143602 (2008).
    [CrossRef] [PubMed]
  7. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
    [CrossRef] [PubMed]
  8. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
    [CrossRef] [PubMed]
  9. K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16(14), 10309–10314 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-14-10309 .
    [CrossRef] [PubMed]
  10. D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87(23), 233901 (2001).
    [CrossRef] [PubMed]
  11. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “All-optical switching and beam steering in tunable waveguide arrays,” Appl. Phys. Lett. 86(5), 051112 (2005).
    [CrossRef]
  12. R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(2), 026602 (2004).
    [CrossRef] [PubMed]
  13. A. L. Jones, “Coupling of optical fibers and scattering in fibers,” J. Opt. Soc. Am. 55(3), 261–269 (1965).
    [CrossRef]
  14. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008) (and references therein).
    [CrossRef]
  15. N. Belabas, S. Bouchoule, I. Sagnes, J. A. Levenson, C. Minot, and J. M. Moison, “Confining light flow in weakly coupled waveguide arrays by structuring the coupling constant: towards discrete diffractive optics,” Opt. Express 17(5), 3148–3156 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-5-3148 .
    [CrossRef] [PubMed]
  16. J. M. Moison, N. Belabas, C. Minot, and J. A. Levenson, “Discrete photonics in waveguide arrays,” Opt. Lett. 34(16), 2462–2464 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-16-2462 .
    [CrossRef] [PubMed]
  17. A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” N. J. Phys. 10(10), 103020 (2008).
    [CrossRef]
  18. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
    [CrossRef]
  19. W. P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11(3), 963–983 (1994).
    [CrossRef]
  20. L. Eyges and P. Wintersteiner, “Modes of an array of dielectric waveguides,” J. Opt. Soc. Am. 71, 1351–1360 (1981), http://www.opticsinfobase.org/abstract.cfm?URI=josa-71-11-1351 .
  21. A. Kaplan and S. Ruschin, “Characterization and performance evaluation of coupled multiwaveguide arrays,” J. Lightwave Technol. 17(10), 1884–1889 (1999), http://jlt.osa.org/abstract.cfm?URI=JLT-17-10-1884 .
    [CrossRef]
  22. M. L. Cooper and S. Mookherjea, “Numerically-assisted coupled-mode theory for silicon waveguide couplers and arrayed waveguides,” Opt. Express 17(3), 1583–1599 (2009), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-17-03-1583 .
    [CrossRef] [PubMed]
  23. G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, “Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 1046608 (2002).
    [CrossRef]
  24. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
    [CrossRef]
  25. T. Pertsch, T. Zentgraf, U. Perchel, A. Brauer, and F. Lederer, “Anomalous refraction diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
    [CrossRef] [PubMed]
  26. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 83, 4752–4754 (1999).
  27. B. E. A. Saleh, and M. C. Teich, “Fundamentals of Photonics,” Wiley-Interscience, 2007.

2009 (3)

2008 (5)

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” N. J. Phys. 10(10), 103020 (2008).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008) (and references therein).
[CrossRef]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling modulation in an optical waveguide system,” Phys. Rev. Lett. 101(14), 143602 (2008).
[CrossRef] [PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16(14), 10309–10314 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-14-10309 .
[CrossRef] [PubMed]

2007 (1)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

2005 (2)

2004 (1)

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(2), 026602 (2004).
[CrossRef] [PubMed]

2003 (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

2002 (2)

G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, “Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 1046608 (2002).
[CrossRef]

T. Pertsch, T. Zentgraf, U. Perchel, A. Brauer, and F. Lederer, “Anomalous refraction diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

2001 (1)

D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87(23), 233901 (2001).
[CrossRef] [PubMed]

2000 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[CrossRef] [PubMed]

1999 (4)

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83(23), 4752–4755 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optics Bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 83, 4752–4754 (1999).

A. Kaplan and S. Ruschin, “Characterization and performance evaluation of coupled multiwaveguide arrays,” J. Lightwave Technol. 17(10), 1884–1889 (1999), http://jlt.osa.org/abstract.cfm?URI=JLT-17-10-1884 .
[CrossRef]

1998 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

1994 (1)

1985 (1)

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

1981 (1)

1965 (1)

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optics Bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 83, 4752–4754 (1999).

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

Alfimov, G. L.

G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, “Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 1046608 (2002).
[CrossRef]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008) (and references therein).
[CrossRef]

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “All-optical switching and beam steering in tunable waveguide arrays,” Appl. Phys. Lett. 86(5), 051112 (2005).
[CrossRef]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Bartal, G.

Belabas, N.

Bouchoule, S.

Boyd, A.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

Brauer, A.

T. Pertsch, T. Zentgraf, U. Perchel, A. Brauer, and F. Lederer, “Anomalous refraction diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83(23), 4752–4755 (1999).
[CrossRef]

Brzdakiewicz, K. A.

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “All-optical switching and beam steering in tunable waveguide arrays,” Appl. Phys. Lett. 86(5), 051112 (2005).
[CrossRef]

Buljan, H.

Christodoulides, D. N.

K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16(14), 10309–10314 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-14-10309 .
[CrossRef] [PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008) (and references therein).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87(23), 233901 (2001).
[CrossRef] [PubMed]

Cohen, O.

Cooper, M. L.

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83(23), 4752–4755 (1999).
[CrossRef]

Dreisow, F.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling modulation in an optical waveguide system,” Phys. Rev. Lett. 101(14), 143602 (2008).
[CrossRef] [PubMed]

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” N. J. Phys. 10(10), 103020 (2008).
[CrossRef]

Efremidis, N.

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optics Bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 83, 4752–4754 (1999).

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83(23), 4752–4755 (1999).
[CrossRef]

Eugenieva, E. D.

D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87(23), 233901 (2001).
[CrossRef] [PubMed]

Eyges, L.

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Fleischer, J.

Fratalocchi, A.

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “All-optical switching and beam steering in tunable waveguide arrays,” Appl. Phys. Lett. 86(5), 051112 (2005).
[CrossRef]

Freedman, B.

Hardy, A.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

Heinrich, M.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” N. J. Phys. 10(10), 103020 (2008).
[CrossRef]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling modulation in an optical waveguide system,” Phys. Rev. Lett. 101(14), 143602 (2008).
[CrossRef] [PubMed]

Huang, W. P.

Jones, A. L.

Kaplan, A.

Karpierz, M. A.

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “All-optical switching and beam steering in tunable waveguide arrays,” Appl. Phys. Lett. 86(5), 051112 (2005).
[CrossRef]

Kevrekidis, P. G.

G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, “Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 1046608 (2002).
[CrossRef]

Kip, D.

Kivshar, Y. S.

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(2), 026602 (2004).
[CrossRef] [PubMed]

Konotop, V. V.

G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, “Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 1046608 (2002).
[CrossRef]

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008) (and references therein).
[CrossRef]

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” N. J. Phys. 10(10), 103020 (2008).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Perchel, A. Brauer, and F. Lederer, “Anomalous refraction diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83(23), 4752–4755 (1999).
[CrossRef]

Levenson, J. A.

Longhi, S.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling modulation in an optical waveguide system,” Phys. Rev. Lett. 101(14), 143602 (2008).
[CrossRef] [PubMed]

Makris, K. G.

Mandelik, D.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 83, 4752–4754 (1999).

Manela, O.

Minot, C.

Moison, J. M.

Molina, M. I.

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(2), 026602 (2004).
[CrossRef] [PubMed]

Mookherjea, S.

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optics Bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 83, 4752–4754 (1999).

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

Nolte, S.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” N. J. Phys. 10(10), 103020 (2008).
[CrossRef]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling modulation in an optical waveguide system,” Phys. Rev. Lett. 101(14), 143602 (2008).
[CrossRef] [PubMed]

Peleg, O.

Perchel, U.

T. Pertsch, T. Zentgraf, U. Perchel, A. Brauer, and F. Lederer, “Anomalous refraction diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

Pertsch, T.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” N. J. Phys. 10(10), 103020 (2008).
[CrossRef]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling modulation in an optical waveguide system,” Phys. Rev. Lett. 101(14), 143602 (2008).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Perchel, A. Brauer, and F. Lederer, “Anomalous refraction diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83(23), 4752–4755 (1999).
[CrossRef]

Peschel, U.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” N. J. Phys. 10(10), 103020 (2008).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optics Bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Ruschin, S.

Sagnes, I.

Salerno, M.

G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, “Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 1046608 (2002).
[CrossRef]

Schwartz, T.

Segev, M.

Silberberg, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008) (and references therein).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optics Bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 83, 4752–4754 (1999).

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008) (and references therein).
[CrossRef]

Streifer, W.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

Szameit, A.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” N. J. Phys. 10(10), 103020 (2008).
[CrossRef]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling modulation in an optical waveguide system,” Phys. Rev. Lett. 101(14), 143602 (2008).
[CrossRef] [PubMed]

Trompeter, H.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” N. J. Phys. 10(10), 103020 (2008).
[CrossRef]

Tünnermann, A.

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” N. J. Phys. 10(10), 103020 (2008).
[CrossRef]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling modulation in an optical waveguide system,” Phys. Rev. Lett. 101(14), 143602 (2008).
[CrossRef] [PubMed]

Vicencio, R. A.

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(2), 026602 (2004).
[CrossRef] [PubMed]

Wintersteiner, P.

Zentgraf, T.

T. Pertsch, T. Zentgraf, U. Perchel, A. Brauer, and F. Lederer, “Anomalous refraction diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “All-optical switching and beam steering in tunable waveguide arrays,” Appl. Phys. Lett. 86(5), 051112 (2005).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. (2)

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

N. J. Phys. (1)

A. Szameit, H. Trompeter, M. Heinrich, F. Dreisow, U. Peschel, T. Pertsch, S. Nolte, F. Lederer, and A. Tünnermann, “Fresnel’s laws in discrete optical media,” N. J. Phys. 10(10), 103020 (2008).
[CrossRef]

Nature (2)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446(7131), 52–55 (2007).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rep. (1)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008) (and references therein).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Switching of discrete optical solitons in engineered waveguide arrays,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(2), 026602 (2004).
[CrossRef] [PubMed]

G. L. Alfimov, P. G. Kevrekidis, V. V. Konotop, and M. Salerno, “Wannier functions analysis of the nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4), 1046608 (2002).
[CrossRef]

Phys. Rev. Lett. (9)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998).
[CrossRef]

T. Pertsch, T. Zentgraf, U. Perchel, A. Brauer, and F. Lederer, “Anomalous refraction diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 83, 4752–4754 (1999).

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83(23), 4752–4755 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optics Bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling modulation in an optical waveguide system,” Phys. Rev. Lett. 101(14), 143602 (2008).
[CrossRef] [PubMed]

D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87(23), 233901 (2001).
[CrossRef] [PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[CrossRef] [PubMed]

Other (1)

B. E. A. Saleh, and M. C. Teich, “Fundamentals of Photonics,” Wiley-Interscience, 2007.

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Figures (10)

Fig. 1
Fig. 1

Description of prototypal systems considered.

Fig. 2
Fig. 2

Variation of the effective indices of the isolated ridge waveguide nirw with ridge width Lr for the standard structure in InP (a) and GaAs (b) systems: fundamental mode (green symbols), mode 1 (orange), mode 2 (red), with horizontal polarization in full lines and vertical one in dashed lines. Planar waveguide levels nlpw and nupw are shown by dark green horizontal lines and their middle by light green ones. The supermode bands for C=1.3mm−1, Lr=3µm (InP) and C=0.3mm−1, Lr=4µm (GaAs) together with the levels of a 7-ridge array are illustrated by violet symbols. In (c) all curves are shown to coincide when drawn in reduced coordinates.

Fig. 3
Fig. 3

Horizontal section of the fundamental mode of the isolated waveguide around its center, in the InP system with Lr=3µm. Blue (resp. violet) symbols for horizontal (resp. vertical) polarization. O = left linear scale; Δ = right logarithmic scale. Lines are adjustments by exponential decay curves outside the ridge, with Ldecay=2.2µm (resp. 1.5µm) for horizontal (resp. vertical) polarization.

Fig. 4
Fig. 4

Model parameters as a function of ridge spacing S for horizontal polarization in InP-based 7-ridge structure with Lr=3µm. Calc, mod, fit in the captions indicate respectively the results of the calculation from the isolated mode, an exponential approximation of this mode, and the values giving the best fit of the FEM levels by the model diffraction relation.

Fig. 5
Fig. 5

X section of the fundamental mode of the 7-ridge array in the InP system (blue circles), with Lr=3µm and S=10µm (a) or S=4µm (b). The blue line is an adjustment by a combination of individual modes (violet lines) merely described by an exponential tail with weights given by the CMT model (red squares).

Fig. 6
Fig. 6

FEM data on InP-based structures with Lr=3µm. (a) Fan-out diagram, effective indices versus ridge spacing S for horizontal (top) and vertical (bottom) polarizations. Dashed lines show the indices of planar waveguides. (b) Effective index versus reduced supermode wavenumber kx for S=6µm and various numbers of ridges N (2, 3, 4 to 13), horizontal polarization; the blue curve is a fit by the extended CMT model. (c) Effective index versus supermode number p, for N=7, horizontal polarization, and S values of 15 (deep red dots), 10, 8, 6, 4µm (violet dots). Dashed lines are predictions of Eq. (5) using calculated values for η, ξ, and ζ, but C values increased by a factor 1.5; full lines use η values decreased by the factor 0.7.

Fig. 7
Fig. 7

Si-based arrays of 5 waveguides. (a) Fan-out diagrams for vertical (V, top) and horizontal (H, bottom) polarizations. (b) Fit of FEM data for vertical polarization with calculated parameters C, η, ξ, and ζ.

Fig. 8
Fig. 8

Variation of η, ξ, ζ, and C/Cmax with reduced ridge period s=S/Levan for all waveguide structures considered. Only values validated by FEM data are shown. Lines indicate the predictions of Eqs. (A10) and A13.

Fig. 9
Fig. 9

(a) Variation of diffraction relation for various values of the reduced inter-ridge spacing s. (b) same with kz normalized to take the value of 2 at kx=0.

Fig. A1
Fig. A1

Schematic decomposition of the dielectric constant profile of shallow ridge waveguide arrays in vacuum in (X,Y) plane. Light blue and dark blue dashed vertical lines on the left hand side indicate interfaces where boundary conditions must be written respectively in “etched” and “ridge” planar waveguide regions.

Equations (20)

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m | m ' = w h o l e p l a n e M m * M m ' d X d Y m | n | m ' = n t h r i d g e sec t i o n M m * M m ' d X d Y
C = β v a c u u m 2 2 β i r w ( n r i d g e 2 1 ) 001 00
a m z + η ( a m 1 z + a m + 1 z ) = i ( a m 1 + a m + 1 + ξ a m + ζ a m 2 + ζ a m + 2 )
η = 01 00 ξ = 2 101 001 ζ = 1 ¯ 01 + 002 001
k z = 2 cos ( k x ) + ξ + 2 ζ cos ( 2 k x ) 1 + 2 η cos ( k x ) i.e. in real units n ( k x ) = n i r w + C β v a c u u m k z
z m [ ( 1 2 η ) a m a m * + η a m + a m + 1 2 a m * + a m + 1 * 2 + η a m + a m 1 2 a m * + a m 1 * 2 ] = 0
n ( p ) = n i r w + 2 C β v a c u u m cos ( k x ( p ) )   with   k x ( p ) = p + 1 N + 1 π 0 p < N
2 E ( X , Y , Z ) + ω 2 c 2 ε ( X , Y ) E ( X , Y , Z ) = 0
| x | w 2 , E cos p x x > w 2 , E e q ( x w 2 ) x < w 2 , E e q ( x + w 2 )
r R / E q w 2 = p w 2 tan p w 2 and p 2 + q 2 = β 1 R 2 β 1 E 2
E ( X , Y , Z ) = m a m ( Z ) M 1 ( X m s , Y ) e i β Z
a m Z + a m 1 Z 01 00 + a m + 1 Z 01 00 + a m 2 Z 02 00 + a m + 2 Z 02 00 = C { a m 1 + a m + 1 + a m 2 101 001 + a m 2 1 ¯ 01 + 002 001 + a m 2 1 ¯ 01 + 002 001 }
a m z + η ( a m 1 z + a m + 1 z ) = i ( a m 1 + a m + 1 + ξ a m + ζ a m 2 + ζ a m + 2 )
k z = 2 cos ( k x ) + ξ + 2 ζ cos ( 2 k x ) 1 + 2 η cos ( k x )
η = e q S q × 1 + q ( S w ) e q w + 4 q p 2 + q 2 e q w 2 [ q   sh q w 2 + p tan p w 2   ch q w 2 ] w 2 + 1 2 q + sin p w 2 p + cos q w 2 q × cos 2 p w 2
C = e q S cos 2 p w 2 × 2 p 2 + q 2 e q w 2 [ q sh q w 2 + p tan p w 2   ch q w 2 ] w 2 + 1 2 q + sin p w 2 p + cos q w 2 q × β v a c u u m 2 2 β 1 ( ε 1 1 ) mesa edge d Y g 1 E 2
η q w 0 ( 1 + q S ) e q S , ξ q w 0 2 e q S , ζ q w 0 2 e q S
C C m a x q w 0 e q S × 8 r R / E ε 1 1 ε 1 R ε 1 E × mesa edge d Y g 1 E 2
β 1 R 2 = β 1 E 2 + mesa edge d Y g 1 E ( ε 1 1 ) g 1 E
C C m a x q w 0 8 e q S 8 e S / L e v a n

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