Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T16:19:27.191Z Has data issue: false hasContentIssue false
  • English
  • Français

Diffractive optics in nonlinear media with periodic structure

Published online by Cambridge University Press:  26 September 2008

Gang Bao  and
David C. Dobson
Gang Bao
Affiliation:
Department of Mathematics, University of Florida, Gainesville, FL 32611, USA
David C. Dobson
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The diffraction of time-harmonic waves in a nonlinear medium with periodic structure is studied in this paper. In particular second harmonic generation – an important phenomenon in nonlinear optics-is modelled. The model, derived from a general nonlinear system of Maxwell's equations, is shown to have a unique solution for all but a discrete number of frequencies. The problem is solved numerically by combining a method of finite elements and a fixed-point iteration scheme. Numerical experiments for some simple grating structures are presented and discussed.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 6 , December 1995 , pp. 573 - 590
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Reinisch, R. & Nevière, M. 1983 Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects. Phys. Rev. B, 28 (4), 1870–1885.CrossRefGoogle Scholar
[2] Reinisch, R., Neviere, M., Akhouayri, H., Coutaz, J., Maystre, D. & Pic, E. 1988 Grating enhanced second harmonic generation through electromagnetic resonances. Opt. Eng. 27 (11), 961–971.CrossRefGoogle Scholar
[3] R., Petit (ed.) 1980 Electromagnetic Theory of Gratings: Topics in Current Physics, Vol. 22. Springer-Verlag.Google Scholar
[4] Shen, Y. R. 1984 The Principles of Nonlinear Optics. Wiley.Google Scholar
[5] Chen, X. & Friedman, A. 1991 Maxwell's equations in a periodic structure. Trans. Am. Math. Soc. 323 (2), 465–507.Google Scholar
[6] Dobson, D. & Friedman, A. 1992 The time-harmonic Maxwell equations in a doubly periodic structure. J. Math. Anal. Appl., 166 (2), 507–528.CrossRefGoogle Scholar
[7] Goldstein, C. 1982 A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains. Math. Comp. 39, 309–324.CrossRefGoogle Scholar
[8] Lenoir, M. & Tounsi, A. 1988 The localized finite element method and its application to the two-dimensional sea-keeping problem. SIAM J. Numer. Anal. 25 (4), 729–751.CrossRefGoogle Scholar
[9] Fix, G. J. & Marin, S. P. 1978 Variational methods for underwater acoustic problems. J. Comput. Phys. 28, 253–270.Google Scholar
[10] Kriegsmann, G. 1982 Radiation conditions for wave guide problems. SIAM J. Sci. Stat. Comput. 3, 318–326.Google Scholar
[11] Bayliss, A. & Turkel, E. 1979 Boundary conditions for exterior acoustic problems. Rep. 79–7, ICASE, NASA Langley Research Center, Hampton, VA.Google Scholar
[12] Bao, G. & Dobson, D. 1994 Second harmonic generation in nonlinear optical films. J. Math. Phys. 35(4), 1622–1633.CrossRefGoogle Scholar
[13] Achdou, Y. & Pironneau, O. 1991 Optimization of a photocell. Opt. Control Appl. Meth. 12, 221–246.Google Scholar
[14] Dobson, D. 1993 Optimal design of periodic antireflective structures for the Helmholtz equation. Euro. J. Appl. Math. 4, 321–340.Google Scholar
[15] Bao, G.Finite element approximation of time harmonic waves in periodic structures. SIAM J. Numeric. Anal. (to appear).Google Scholar
[16] Friedman, A. 1990 Mathematics in Industrial Problems, Part 3. Springer-Verlag.Google Scholar
[17] Bloembergen, N. 1965 Nonlinear Optics. Benjamin, NY.Google Scholar
[18] Bao, G. & Dobson, D. 1993 Nonlinear optics in periodic diffraction structures. In: 2nd Int. Conf. Mathematical and Numerical Aspects of Wave Propagation (Kleinman, R. et al. , eds.). SIAM, Philadelphia, pp. 30–38.Google Scholar
[19] Born, M. & Wolf, E. 1980 Principle of Optics (Sixth ed). Pergamon.Google Scholar
[20] Taylor, M. 1981 Pseudodifferential Operators. Princeton University Press.CrossRefGoogle Scholar
[21] Kato, T. 1980 Perturbation Theory for Linear Operators (corrected second ed). Springer-Verlag.Google Scholar
[22] Hutson, V. & Pym, J. S. 1980 Applications of Functional Analysis and Operator Theory. Academic Press.Google Scholar