OSA's Digital Library

Journal of the Optical Society of America B

Journal of the Optical Society of America B


  • Vol. 19, Iss. 4 — Apr. 1, 2002
  • pp: 812–821

Plane waves in periodic, quadratically nonlinear slab waveguides: stability and exact Fourier structure

Joel F. Corney and Ole Bang  »View Author Affiliations

JOSA B, Vol. 19, Issue 4, pp. 812-821 (2002)

View Full Text Article

Enhanced HTML    Acrobat PDF (472 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We consider the propagation of broad optical beams through slab waveguides with a purely quadratic nonlinearity and containing linear and nonlinear long-period quasi-phase-matching gratings. An exact Floquet analysis of the periodic, plane-wave solution shows that the periodicity can drastically alter the growth rate of the modulational instability but that it never completely removes the instability. The results are confirmed by direct numerical simulation as well as through a simpler, approximate theory for the averaged fields that accurately predicts the low-frequency part of the gain spectrum.

© 2002 Optical Society of America

OCIS Codes
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.5940) Nonlinear optics : Self-action effects
(230.4320) Optical devices : Nonlinear optical devices

Joel F. Corney and Ole Bang, "Plane waves in periodic, quadratically nonlinear slab waveguides: stability and exact Fourier structure," J. Opt. Soc. Am. B 19, 812-821 (2002)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28, 1691–1740 (1996). [CrossRef]
  2. K. Hayata and M. Koshiba, “Multidimensional solitons in quadratic nonlinear media,” Phys. Rev. Lett. 71, 3275–3278 (1993). [CrossRef] [PubMed]
  3. L. Torner, C. R. Menyuk, W. E. Torruellas, and G. I. Stegeman, “Two-dimensional solitons with second-order nonlinearities,” Opt. Lett. 20, 13–15 (1995). [CrossRef] [PubMed]
  4. L. Bergé, V. K. Mezentsev, J. J. Rasmussen, and J. Wyller, “Formation of stable solitons in quadratic nonlinear media,” Phys. Rev. A 52, R28–R31 (1995). [CrossRef] [PubMed]
  5. D. E. Pelinovsky, A. V. Buryak, and Y. S. Kivshar, “Instability of solitons governed by quadratic nonlinearities,” Phys. Rev. Lett. 75, 591–595 (1995). [CrossRef] [PubMed]
  6. B. A. Malomed, P. D. Drummond, H. He, A. Berntson, D. Anderson, and M. Lisak, “Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity,” Phys. Rev. E 56, 4725–4735 (1997). [CrossRef]
  7. R. Schiek, Y. Baek, and G. I. Stegeman, “One-dimensional spatial solitary waves due to cascaded second-order nonlinearities in planar waveguides,” Phys. Rev. E 53, 1138–1141 (1996). [CrossRef]
  8. W. E. Torruellas, Z. Wang, D. J. Hagan, E. W. Van Stryland, G. I. Stegeman, L. Torner, and C. R. Menyuk, “Observation of two-dimensional spatial solitary waves in a quadratic medium,” Phys. Rev. Lett. 74, 5036–5039 (1995). [CrossRef] [PubMed]
  9. X. Liu, K. Beckwitt, and F. Wise, “Two-dimensional optical spatiotemporal solitons in quadratic media,” Phys. Rev. E 62, 1328–1340 (2000). [CrossRef]
  10. E. Lopez Lago, V. Couderc, C. de Angelis, F. Gringoli, and A. Barthélémy, “Experimental demonstration of the self-trapping of a weak probe induced by a quadratic spatial soliton,” in Nonlinear Guided Waves and Their Applications, Vol. 55 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 379–381.
  11. V. Couderc, E. Lopez Lago, C. Simos, and A. Barthélémy, “Experiments in quadratic spatial soliton generation and steering in a noncollinear geometry,” Opt. Lett. 26, 905–907 (2001). [CrossRef]
  12. W. E. Torruellas, G. Assanto, B. L. Lawrence, R. A. Fuerst, and G. I. Stegeman, “All-optical switching by spatial walkoff compensation and solitary-wave locking,” Appl. Phys. Lett. 68, 1449–1451 (1996). [CrossRef]
  13. P. Di Trapani, W. Chinaglia, S. Minardi, A. Piskarskas, and G. Valiulis, “Observation of quadratic optical vortex solitons,” Phys. Rev. Lett. 84, 3843–3846 (2000). [CrossRef] [PubMed]
  14. S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation,” Opt. Lett. 20, 438–440 (1995). [CrossRef] [PubMed]
  15. R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997). [CrossRef]
  16. H. Fang, R. Malendevich, R. Schiek, and G. I. Stegeman, “Spatial modulational instability in one-dimensional lithium niobate slab waveguides,” Opt. Lett. 25, 1786–1788 (2000). [CrossRef]
  17. R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001). [CrossRef] [PubMed]
  18. O. Bang, Y. S. Kivshar, and A. Buryak, “Bright spatial solitons in defocusing Kerr media supported by cascaded nonlinearities,” Opt. Lett. 22, 1680–1682 (1997). [CrossRef]
  19. T. J. Alexander, A. V. Buryak, and Y. S. Kivshar, “Stabilization of dark and vortex parametric spatial solitons,” Opt. Lett. 23, 670–672 (1998). [CrossRef]
  20. C. B. Clausen, O. Bang, and Y. S. Kivshar, “Spatial solitons and induced Kerr effects in quasi-phase-matched quadratic media,” Phys. Rev. Lett. 78, 4749–4752 (1997). [CrossRef]
  21. O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, “Engineering competing nonlinearities,” Opt. Lett. 24, 1413–1415 (1999). [CrossRef]
  22. J. F. Corney and O. Bang, “Solitons in quadratic nonlinear photonic crystals,” Phys. Rev. E 64, 047601 (2001). [CrossRef]
  23. P. Di Trapani, A. Bramati, S. Minardi, W. Chinaglia, C. Conti, S. Trillo, J. Kilius, and G. Valiulis, “Focusing versus defocusing nonlinearities due to parametric wave mixing,” Phys. Rev. Lett. 87, 183902 (2001). [CrossRef]
  24. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992). [CrossRef]
  25. T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990). [CrossRef]
  26. B. Jaskorzynska, G. Arvidsson, and F. Laurell, “Periodic structures for phase-matching in second harmonic generation in titanium lithium niobate waveguides,” in Integrated Optical Circuit Engineering III, R. Th. Kersten, ed., Proc. SPIE 651, 221–228 (1986). [CrossRef]
  27. C. L. Tang and P. P. Bey, “Phase matching in second-harmonic generation using artificial periodic structure,” IEEE J. Quantum Electron. 9, 9–17 (1973). [CrossRef]
  28. D. V. Petrov, “Nonlinear phase shift by cascaded quasi-phase-matched second harmonic generation,” Opt. Commun. 131, 102–106 (1996). [CrossRef]
  29. J. F. Corney and O. Bang, “Modulational instability in periodic quadratic nonlinear materials,” Phys. Rev. Lett. 87, 133901 (2001). [CrossRef] [PubMed]
  30. A. Kobyakov, F. Lederer, O. Bang, and Y. S. Kivshar, “Nonlinear phase shift and all-optical switching in quasi-phase-matched quadratic media,” Opt. Lett. 23, 506–508 (1998). [CrossRef]
  31. O. Bang, T. W. Graversen, and J. F. Corney, “Accurate switching intensities and length scales in quasi-phase-matched materials,” Opt. Lett. 26, 1007–1009 (2001). [CrossRef]
  32. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962). [CrossRef]
  33. C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2)(2) cascading,” J. Opt. Soc. Am. B 11, 2434–2443 (1994). [CrossRef]
  34. O. Bang, “Dynamical equations for wave packets in material with both quadratic and cubic response,” J. Opt. Soc. Am. B 14, 51–61 (1997). [CrossRef]
  35. H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity,” Opt. Commun. 123, 394–402 (1996). [CrossRef]
  36. H. He, A. Arraf, C. M. de Sterke, P. D. Drummond, and B. A. Malomed, “Theory of modulational instability in Bragg gratings with quadratic nonlinearity,” Phys. Rev. E 59, 6064–6078 (1996). [CrossRef]
  37. T. Cretegny, T. Dauxois, S. Ruffo, and A. Torcini, “Localization and equipartition of energy in the β-FPU chain: chaotic breathers,” Physica D 121, 109–126 (1998). [CrossRef]
  38. A. Saher Helmy, D. C. Hutchings, T. C. Kleckner, J. H. Marsh, A. C. Bryce, J. M. Arnold, C. R. Stanley, J. S. Aitchison, C. T. A. Brown, K. Moutzouris, and M. Ebrahimzadeh, “Quasi phase matching in GaAs-AlAs superlattice waveguides through bandgap tuning by use of quantum-well intermixing,” Opt. Lett. 25, 1370–1372 (2000). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited