Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T22:52:21.617Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuation Finite Element Simulation of Second Harmonic Generation in Photonic Crystals

Published online by Cambridge University Press:  20 August 2015

Gang Bao ,
Zhengfu Xu  and
Jianhua Yuan
Gang Bao*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, China Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Zhengfu Xu*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Jianhua Yuan*
Affiliation:
Department of Mathematics, Beijing University of Posts and Telecommunications, Beijing 100876, China
*
Email:bao@math.msu.edu
Email:zhengfu@math.msu.edu
Corresponding author.Email:yuan_bupt@yahoo.com.cn
Get access

Abstract

A computational study on the enhancement of the second harmonic generation (SHG) in one-dimensional (1D) photonic crystals is presented. The mathematical model is derived from a nonlinear system of Maxwell’s equations, which partly overcomes the shortcoming of some existing models based on the undepleted pump approximation. We designed an iterative scheme coupled with the finite element method which can be applied to simulate the SHG in one dimensional nonlinear photonic band gap structures in our previous work. For the case that the nonlinearity is strong which is desirable to enhance the conversion efficiency, a continuation method is introduced to ensure the convergence of the iterative procedure. The convergence of our method is fast. Numerical experiments also indicate the conversion efficiency of SHG can be significantly enhanced when the frequencies of the fundamental and the second harmonic wave are tuned at the photonic band edges. The maximum total conversion efficiency available reaches more than 50% in all the cases studied.

Keywords

35Q6078M1065Z0578A6065L6042.65.Ky42.70.Qs02.70.Dh02.60.CbPhotonic crystalssecond harmonic generationphotonic band gapconversion efficiencyfinite element methodsfixed-point iterationscontinuation method
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 1 , July 2011 , pp. 57 - 69
Copyright
Copyright © Global Science Press Limited 2011

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]Joannopoulos, J. D., Meade, R. D. and Winn, J. N., Photonic Crystals, Molding the Flow of Light, Princeton Univ. Press, Princeton, NJ, 1995.Google Scholar
[2]Ammari, H., Kang, H. and Lee, H., Layer Potential Techniques in Spectral Analysis, Mathematical Surveys and Monographs, Volume 153, American Mathematical Society, Providence, 2009.CrossRefGoogle Scholar
[3]Ammari, H. and Hamdache, K., Global existence and regularity of solutions to a system of nonlinear Maxwell equations, J. Math. Anal. Appl., 286 (2003), 51–63.CrossRefGoogle Scholar
[4]Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1993.Google Scholar
[5]Bertolott, M.et al., Nanoscale Linear and Nonlinear Optics, Amer Inst of Physics, New York, 2001.Google Scholar
[6]Enoch, S. and Akhouayri, H., Seond-harmonic generation in multilayered devices: theoretical tools, J. Opt. Soc. Am. B., 15 (1998), 1030–1041.Google Scholar
[7]D’Aguanno, G.et al., Generalized coupled-mode theory for χ (2) interactions in finite multi-layered structures, J. Opt. Soc. m. B., 19 (2002), 2111–2121.Google Scholar
[8]D’Aguanno, G. and Centini, M.et al., Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structures with deep gratings, Phys. Rev. E., 67 (2003), 016606.CrossRefGoogle ScholarPubMed
[9]Bertolotti, M., Wave interactions in photonic band structures: an overview, J. pt. A. Pure. Appl. Opt., 8 (2006), S9–S32.Google Scholar
[10]Bao, G. and Dobson, D. C., Second harmonic genetation in nonlinear optical films, J. Math. Phys., 35 (1994), 1622–1633.CrossRefGoogle Scholar
[11]Yuan, J., Computing for second harmonic generation in one-dimensional nonlinear photonic crystals, Opt. ommun., 282(13) (2009), 2628–2633.Google Scholar
[12]Scalora, M. and Bloemer, M. J.et al., Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures, Phys. Rev. A., 56 (1997), 3166–3174.CrossRefGoogle Scholar
[13]Centini, M. and Scalora, M.et al., Dispersive properties of one-dimensional photonic band gap structures for second harmonic generation, J. Opt. A. Pure. Appl. Opt., 2 (2000), 121–126.Google Scholar
[14]Cao, H., Hall, D. B., Torkelson, J. M. and Cao, C.-Q.. Large enhancement of second harmonic generation in polymer films by microcavities, Appl. Phys. Lett., 76 (2000), 538–540.Google Scholar
[15]Shi, B., Jiang, Z. M. and Wang, X., Defective photonic crystals with greatly enhanced second-harmonic generation, Opt. Lett., 26 (2001), 1194–1196.CrossRefGoogle ScholarPubMed
[16]Centini, M. and D’Aguanno, G.et al., Non-phase-matched enhancement of second-harmonic generation in multilayer nonlinear structures with internal reflections, Opt. ett., 29 (2004), 1924–1926.Google Scholar
[17]Ren, F. and Li, R.et al., Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes, Phys. Rev. B., 70 (2004), 245109.Google Scholar