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Journal of the Optical Society of America B

Journal of the Optical Society of America B


  • Vol. 17, Iss. 6 — Jun. 1, 2000
  • pp: 932–941

Geometric analysis of optical frequency conversion and its control in quadratic nonlinear media

G. G. Luther, M. S. Alber, J. E. Marsden, and J. M. Robbins  »View Author Affiliations

JOSA B, Vol. 17, Issue 6, pp. 932-941 (2000)

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We analyze frequency conversion and its control among three light waves using a geometric approach that enables the dynamics of the waves to be visualized on a closed surface in three dimensions. It extends the analysis based on the undepleted-pump linearization and provides a simple way to understand the fully nonlinear dynamics. The Poincaré sphere has been used in the same way to visualize polarization dynamics. A geometric understanding of control strategies that enhance energy transfer among interacting waves is introduced, and the quasi-phase-matching strategy that uses microstructured quadratic materials is illustrated in this setting for both type I and II second-harmonic generation and for parametric three-wave interactions.

© 2000 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(190.0190) Nonlinear optics : Nonlinear optics
(190.2640) Nonlinear optics : Stimulated scattering, modulation, etc.
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

G. G. Luther, M. S. Alber, J. E. Marsden, and J. M. Robbins, "Geometric analysis of optical frequency conversion and its control in quadratic nonlinear media," J. Opt. Soc. Am. B 17, 932-941 (2000)

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  1. A. Yariv, Quantum Electronics (Wiley, New York, 1980).
  2. R. Boyd, Nonlinear Optics (Wiley, New York, 1988).
  3. A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Palo Alto, 1992).
  4. 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]
  5. K. C. Rustagi, S. C. Mehendale, and S. Menakshi, “Optical frequency conversion in quasi-phase-matched stacks of nonlinear crystals,” IEEE J. Quantum Electron. QE-18, 1029 (1982). [CrossRef]
  6. C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Rev. A 127, 14 (1988).
  7. S. Trillo, S. Wabnitz, R. Chisari, and G. Cappellini, “Two-wave mixing in a quadratic nonlinear medium: bifurcations, spatial instabilities, and chaos,” Opt. Lett. 17, 637–639 (1992). [CrossRef] [PubMed]
  8. C. J. McKinstrie and X. D. Cao, “Nonlinear detuning of three-wave interactions,” J. Opt. Soc. Am. B 10, 898–912 (1993). [CrossRef]
  9. J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, Vol. 17 of Texts in Applied Mathematics, 2nd ed. (Springer-Verlag, New York, 1999). [CrossRef]
  10. M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins, “Geometric phases, reduction and Lie–Poisson structure for the resonant three-wave interaction,” Physica D 123, 271–290 (1998). [CrossRef]
  11. M. S. Alber, G. G. Luther, J. E. Marsden, and J. M. Robbins, in Proceedings of the Fields Institute Conference in Honour of the 60th Birthday of Vladimir I. Arnol’d, Fields Institute Communications Series, E. Bierstone, B. Khesin, A. Khovanskii, and J. Marsden, eds. (Field Institute, Toronto, Ontario, Canada, 1999).
  12. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  13. D. David, D. D. Holm, and M. V. Tratnik, “Integrable and chaotic polarization dynamics in nonlinear optical beams,” Phys. Lett. A 137, 355–369 (1989). [CrossRef]
  14. N. N. Akhmediev and A. Ankiewicz, Solitons (Chapman & Hall, London, 1997).
  15. 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]
  16. L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995). [CrossRef]
  17. A. Kobyakov, U. Peschel, and F. Lederer, “Vectorial type-II interaction in cascaded quadratic nonlinearities—an analytical approach,” Opt. Commun. 124, 184–194 (1996). [CrossRef]
  18. A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: a comprehensive analytical study,” Phys. Rev. A 54, 3455–3471 (1996). [CrossRef] [PubMed]
  19. C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Signal enhancement in collinear four-wave mixing,” J. Opt. Soc. Am. B 7, 340–344 (1990). [CrossRef]

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