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Discretizing light behaviour in linear and nonlinear waveguide lattices

Abstract

Light propagating in linear and nonlinear waveguide lattices exhibits behaviour characteristic of that encountered in discrete systems. The diffraction properties of these systems can be engineered, which opens up new possibilities for controlling the flow of light that would have been otherwise impossible in the bulk: these effects can be exploited to achieve diffraction-free propagation and minimize the power requirements for nonlinear processes. In two-dimensional networks of waveguides, self-localized states—or discrete solitons—can travel along 'wire-like' paths and can be routed to any destination port. Such possibilities may be useful for photonic switching architectures.

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Figure 1: Waveguide arrays and their diffraction behaviour.
Figure 2: Phase difference between adjacent waveguides (kxD).
Figure 3: Experimental results depicting optical Bloch oscillations in a 4.5 cm long thermo-optically tuned polymer array at a wavelength of 0.633 μm (ref. 14).
Figure 4: Array band structure and the associated Floquet–Bloch modes.
Figure 5: Experimental observation of discrete solitons.
Figure 6: Experimental results in 2D photorefractive lattices.
Figure 7: Possible applications of discrete solitons.

References

  1. Jones, A. L. Coupling of optical fibers and scattering in fibers. J. Opt. Soc. Am. 55, 261–271 (1965).

    ADS  Article  Google Scholar 

  2. Somekh, S., Garmire, E., Yariv, A, Garvin, H. L. & Hunsperger, R. G. Channel optical waveguide directional couplers. Appl. Phys. Lett. 22, 46–48 (1973).

    ADS  CAS  Article  Google Scholar 

  3. Christodoulides, D. N. & Joseph, R. I. Discrete self-focusing in nonlinear arrays of coupled wave-guides. Opt. Lett. 13, 794–796 (1988).

    ADS  CAS  Article  Google Scholar 

  4. Davydov, A. S. Theory of contraction of proteins under their excitation. J. Theor. Biol. 38, 559–569 (1973); Solitons and energy-transfer along protein molecules. J. Theor. Biol. 66, 377–387 (1977).

    CAS  Article  Google Scholar 

  5. Scott, A. C. & Macneil, L. Binding energy versus nonlinearity for a 'small' stationary soliton. Phys. Lett. A 98, 87–88 (1983).

    ADS  Article  Google Scholar 

  6. Lederer, F., Darmanyan, S. & Kobyakov, A. in Spatial Optical Solitons (eds Trillo, S. and Torruellas, W. E.) 269–292 (Springer-Verlag, New York, 2001).

    Book  Google Scholar 

  7. Aceves, A. B. et al. Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays. Phys. Rev. E 53, 1172–1189 (1996).

    ADS  CAS  Article  Google Scholar 

  8. Krolikowski, W. & Kivshar Yu, S. Soliton-based optical switching in waveguide arrays. J. Opt. Soc. Amer. B 13, 876–887 (1996).

    ADS  CAS  Article  Google Scholar 

  9. Kivshar Yu, S. Self-localization in arrays of defocusing wave-guides. Opt. Lett. 18, 1147–1149 (1993).

    ADS  Article  Google Scholar 

  10. Eisenberg, H. S., Silberberg, Y., Morandotti, R., Boyd, A. R. & Aitchison, J. S. Discrete spatial optical solitons in waveguide arrays. Phys. Rev. Lett. 81, 3383–3386 (1998).

    ADS  CAS  Article  Google Scholar 

  11. Eisenberg, H. S., Silberberg, Y., Morandotti, R. & Aitchison, J. S. Diffraction management. Phys. Rev. Lett. 85, 1863–1866 (2000).

    ADS  CAS  Article  Google Scholar 

  12. Pertsch, T., Zentgraf, T., Peschel, U., Brauer, A. & Lederer, F. Anomalous refraction and diffraction in discrete optical systems. Phys. Rev. Lett. 88, 093901 (2002).

    ADS  CAS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  14. Pertsch, T., Dannberg, P., Elflein, W., Brauer, A. & Lederer, F. Optical Bloch oscillations in temperature tuned waveguide arrays. Phys. Rev. Lett. 83, 4752–4755 (1999).

    ADS  CAS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  16. Pertsch, T. et al. Discrete solitons in quadratic nonlinear waveguide arrays, in OSA Trends in Optics and Photonics 80 (Optical Society of America, Washington, 2002).

    Google Scholar 

  17. Fleischer, J. W., Carmon, T., Segev, M., Efremidis, N. K. & Christodoulides, D. N. Observation of discrete solitons in optically-induced real time waveguide arrays. Phys. Rev. Lett. 90, 023902 (2003).

    ADS  Article  Google Scholar 

  18. Fleischer, J. W., Segev, M., Efremidis, N. K. & Christodoulides, D. N. Observation of two-dimensional discrete solitons in optically-induced nonlinear photonic lattices. Nature 422, 147–150 (2003).

    ADS  CAS  Article  Google Scholar 

  19. Yablonovitch, E. Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58, 2059–2062 (1987).

    ADS  CAS  Article  Google Scholar 

  20. John, S. Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 58, 2486–2489 (1987).

    ADS  CAS  Article  Google Scholar 

  21. Joannopoulos, J. D., Villeneuve, P. R. & Fan, S. Photonic crystals: putting a new twist on light. Nature 386, 143–149 (1997).

    ADS  CAS  Article  Google Scholar 

  22. Russel, P. Photonic crystal fibers. Science 299, 358–362 (2003).

    ADS  Article  Google Scholar 

  23. Sievers, A. J. & Takeno, S. Intrinsic localized modes in anharmonic crystals. Phys. Rev. Lett. 61, 970–973 (1988).

    ADS  CAS  Article  Google Scholar 

  24. Dahan, M. B., Peik, E., Reichel, J., Castin, Y. & Salomon, C. Bloch oscillations of atoms in an optical potential. Phys. Rev. Lett. 76, 4508–4511 (1996).

    ADS  Article  Google Scholar 

  25. Anderson, B. P. & Kasevich, M. A. Macroscopic quantum interference from atomic tunnel arrays. Science 282, 1686–1689 (1998).

    ADS  CAS  Article  Google Scholar 

  26. Trombettoni, A. & Smerzi, A. Discrete solitons and breathers with dilute Bose–Einstein condensates. Phys. Rev. Lett. 86, 2353–2356 (2001).

    ADS  CAS  Article  Google Scholar 

  27. Bloch, F. Über die Quantenmechanik der Elektronen in Kristallgittern. Z. Phys. 52, 555–600 (1928).

    ADS  CAS  Article  Google Scholar 

  28. Zener, C. A theory of the electrical breakdown of solid dielectrics. Proc. R. Soc. London Ser. A 145, 523–529 (1932).

    ADS  Article  Google Scholar 

  29. Waschke, C. et al. Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice. Phys. Rev. Lett. 70, 3319–3322 (1993).

    ADS  CAS  Article  Google Scholar 

  30. Peschel, U., Pertsch, T. & Lederer, F. Optical Bloch oscillations in waveguide arrays. Opt. Lett. 23, 1701–1703 (1998).

    ADS  CAS  Article  Google Scholar 

  31. Stegeman, G. I. & Segev, M. Optical spatial solitons and their interactions: universality and diversity. Science 286, 1518–1522 (1999).

    CAS  Article  Google Scholar 

  32. Ashcroft, N. & Mermin, N. Solid State Physics (Saunders, Philadelphia, 1976).

    MATH  Google Scholar 

  33. Ablowitz, M. J. & Clarkson, P. A. Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge Univ, Press, Cambridge, 1992).

    MATH  Google Scholar 

  34. Kivshar Yu, S., Krolikowski, W. & Chubykalo, O. A. Dark solitons in discrete lattices. Phys. Rev. E 50, 5020–5032 (1994).

    ADS  Article  Google Scholar 

  35. Morandotti, R., Eisenberg, H. S., Silberberg, Y., Sorel, M. & Aitchison, J. S. Self-focusing and defocusing in waveguide arrays. Phys. Rev. Lett. 86, 3296–3299 (2001).

    ADS  CAS  Article  Google Scholar 

  36. Morandotti, R., Peschel, U., Aitchison, J. S., Eisenberg, H. S., & Silberberg, Y. Dynamics of discrete solitons in optical waveguide arrays. Phys. Rev. Lett. 83, 2726–2729 (1999).

    ADS  CAS  Article  Google Scholar 

  37. Darmanyan, S., Kobyakov, A., Schmidt, E. & Lederer, F. Strongly localized vectorial modes in nonlinear waveguide arrays. Phys. Rev. E 57, 3520–3530 (1998).

    ADS  CAS  Article  Google Scholar 

  38. Ablowitz, M. J. & Musslimani, Z. H. Discrete diffraction managed spatial solitons. Phys. Rev. Lett. 87, 254102–254104 (2001).

    ADS  CAS  Article  Google Scholar 

  39. Sukhorukov, A. A. & Kivshar Yu, S. Discrete gap solitons in modulated waveguide arrays. Opt. Lett. 27, 2112–2114 (2002).

    ADS  Article  Google Scholar 

  40. Bang, O. & Miller, P. D. Exploiting discreteness for switching in waveguide arrays. Opt. Lett. 21, 1105–1107 (1996).

    ADS  CAS  Article  Google Scholar 

  41. Meier, J. et al. Phase-control beam interactions in Kerr-nonlinear waveguide arrays, in OSA Trends in Optics and Photonics 80 (Optical Society of America, Washington, 2002)

    Google Scholar 

  42. Meier, J. et al. Discrete vector Kerr spatial solitons in AlGaAs array waveguides, in OSA Trends in Optics and Photonics 80 (Optical Society of America, Washington, 2002)

    Google Scholar 

  43. Peschel, T., Peschel, U. & Lederer, F. Discrete bright solitary waves in quadratically nonlinear media. Phys. Rev. E 57, 1127–1133 (1998).

    ADS  CAS  Article  Google Scholar 

  44. Malomed, B. & Kevrekidis, P. G. Discrete vortex solitons. Phys. Rev. E 64, 26601 (2001).

    ADS  CAS  Article  Google Scholar 

  45. Christodoulides, D. N. & Efremidis, N. K. Discrete temporal solitons along a chain of nonlinear coupled microcavities embedded in photonic crystals. Opt. Lett. 27, 568–570 (2002).

    ADS  Article  Google Scholar 

  46. Iliew, R., Peschel, U. & Lederer, F. Light propagation via coupled defects in photonic crystals. Proc. Quantum Electronics Laser Science Conference (2002).

    Google Scholar 

  47. Efremidis, N. K., Sears, S., Christodoulides, D. N., Fleischer, J. W. & Segev, M. Discrete solitons in photorefractive optically induced photonic lattices. Phys. Rev. E 66, 046602 (2002).

    ADS  Article  Google Scholar 

  48. Pertsch, T. et al. Light localization in disordered two-dimensional fiber arrays. Phys. Rev. Lett. (in the press).

  49. Cheo, P. K., Liu, A. & King, G. G. A high-brightness laser beam from a phase-locked multicore Yb-doped fiber laser array. IEEE Photon. Technol. Lett. 13, 439–441 (2001).

    ADS  Article  Google Scholar 

  50. Efremidis, N. K. & Christodoulides, D. N. Discrete Ginzburg-Landau solitons. Phys. Rev. E 67, 026606 (2003).

    ADS  Article  Google Scholar 

  51. Yariv, A, Xu, Y, Lee, R. K. & Scherer, A. Coupled-resonator optical waveguide: a proposal and analysis. Opt. Lett. 24, 711–713 (1999).

    ADS  CAS  Article  Google Scholar 

  52. Mingaleev, S. F., Kivshar Yu, S. & Sammut, R. A. Long-range interaction and nonlinear localized modes in photonic crystal waveguides. Phys. Rev. E 62, 5777–5782 (2000).

    ADS  CAS  Article  Google Scholar 

  53. Christodoulides, D. N. &. Eugenieva, E. D. Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays. Phys. Rev. Lett. 87, 233901 (2001).

    ADS  CAS  Article  Google Scholar 

  54. Christodoulides, D. N. &. Eugenieva, E. D. Minimizing bending losses in two-dimensional discrete soliton networks. Opt. Lett. 26, 1876–1878 (2001).

    ADS  CAS  Article  Google Scholar 

  55. Eugenieva, E. D., Efremidis, N. K. & Christodoulides, D. N. Design of switching junctions for two-dimensional discrete soliton networks. Opt. Lett. 26, 1978–1980 (2001).

    ADS  CAS  Article  Google Scholar 

  56. Pertsch, T., Zentgraf, T., Peschel, U., Bräuer, A. & Lederer, F. Beam steering in waveguide arrays. Appl. Phys. Lett. 80, 3247–3249 (2002).

    ADS  CAS  Article  Google Scholar 

  57. Pertsch, T., Peschel, U. & Lederer, F. All-optical switching in quadratically nonlinear waveguide arrays. Optics Lett. 28, 102–104 (2003).

    ADS  Article  Google Scholar 

  58. Russell, P. St. J. Optics of Floquet-Bloch waves in dielectric gratings. Appl. Phys. B 39, 231–246 (1986).

    ADS  Article  Google Scholar 

  59. Christodoulides, D. N. & Eugenieva, E. D. Animations of the processes described in Figs 7a, b as obtained after numerically solving the underlying evolution equations. AIP. EPAPS: E-PRLTAO-87-018147 at 〈http://www.aip.org/pubservs/epaps.html〉>.

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Acknowledgements

The authors acknowledge useful discussions with Stewart Aitchison, George Stegeman, Mordechai Segev, Ulf Peschel, Roberto Morandotti, Hagai Eisenberg, Nikos Efremidis, Thomas Pertsch, D. Mandelik and Jared Hudock. The work of D.N.C. was supported by ARO MURI and of F.L. by the European Community grant (IST-2000-26005).

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Christodoulides, D., Lederer, F. & Silberberg, Y. Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature 424, 817–823 (2003). https://doi.org/10.1038/nature01936

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