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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 27, Iss. 11 — Nov. 1, 2010
  • pp: 2347–2353

A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects

Zhengfu Xu and Gang Bao  »View Author Affiliations

JOSA A, Vol. 27, Issue 11, pp. 2347-2353 (2010)

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A numerical scheme is presented to solve the nonlinear Helmholtz (NLH) equation modeling second-harmonic generation (SHG) in photonic bandgap material doped with a nonlinear χ ( 2 ) effect and the NLH equation modeling wave propagation in Kerr type gratings with a nonlinear χ ( 3 ) effect in the one-dimensional case. Both of these nonlinear phenomena arise as a result of the combination of high electromagnetic mode density and nonlinear reaction from the medium. When the mode intensity of the incident wave is significantly strong, which makes the nonlinear effect non-negligible, numerical methods based on the linearization of the essentially nonlinear problem will become inadequate. In this work, a robust, stable numerical scheme is designed to simulate the NLH equations with strong nonlinearity.

© 2010 Optical Society of America

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(190.0190) Nonlinear optics : Nonlinear optics
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.3270) Nonlinear optics : Kerr effect

ToC Category:
Nonlinear Optics

Original Manuscript: June 30, 2010
Revised Manuscript: August 30, 2010
Manuscript Accepted: September 2, 2010
Published: October 4, 2010

Zhengfu Xu and Gang Bao, "A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects," J. Opt. Soc. Am. A 27, 2347-2353 (2010)

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  1. K. Yamamoto, K. Mizuuchi, Y. Kitaoka, and M. Kato, “Highly efficient quasi-phase-matched second-harmonic generation by frequency doubling of a high-frequency superimposed laser diode,” Opt. Lett. 20, 273–275 (1995). [CrossRef] [PubMed]
  2. D. Blanc, A. M. Bouchoux, C. Plumereau, A. Cachard, and J. F. Roux, “Phase-matched frequency doubling in an aluminum nitride waveguide with a tunable laser source,” Appl. Phys. Lett. 66, 659–661 (1995). [CrossRef]
  3. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–120 (1961). [CrossRef]
  4. M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25–32 (1994). [CrossRef]
  5. N. Bloembergen and A. J. Sievers, “Nonlinear optical properties of periodic laminar structures,” Appl. Phys. Lett. 17, 483–486 (1970). [CrossRef]
  6. P. Yeh, Optical Waves in Layered Media (Wiley, 1998).
  7. J.-P. Fouque, J. Garnier, G. Papanicolaou, and K. Solna, Wave Propagation and Time Reversal in Randomly Layered Medium (Springer, 2007).
  8. G. Bao and D. Dobson, “Second harmonic generation in nonlinear optical films,” J. Math. Phys. 35, 1623–1633 (1994). [CrossRef]
  9. N. Bloembergen, Nonlinear Optics (Benjamin, 1965).
  10. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. Bloemer, and C. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structure with deep gratings,” Phys. Rev. E 67, 016606 (2003). [CrossRef]
  11. A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, 1993).
  12. A. Suryanto, E. V. Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quantum Electron. 35, 313–332 (2003). [CrossRef]
  13. P. Tran, “Optical limiting and switching of short pulses by use of a nonlinear photonic bandgap structure with a defect,” J. Opt. Soc. Am. B 14, 2589–2595 (1997). [CrossRef]
  14. A. Taflove and S. C. HagnessComputational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).
  15. G. Baruch, G. Fibich, and S. Tsynkov, “High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension,” J. Comput. Phys. 227, 820–850 (2007). [CrossRef]
  16. M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A 56, 3166–3174 (1997). [CrossRef]

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