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Asymptotic Stability of an Eikonal Transformation Based ADI Method for the Paraxial Helmholtz Equation at High Wave Numbers

Published online by Cambridge University Press:  20 August 2015

Qin Sheng  and
Hai-Wei Sun
Qin Sheng*
Affiliation:
Center for Astrophysics, Space Physics and Engineering Research, Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA
Hai-Wei Sun*
Affiliation:
Department of Mathematics, University of Macau, Macao
*
Corresponding author.Email:qin_sheng@baylor.edu
Email:HSun@umac.mo
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Abstract

This paper concerns the numerical stability of an eikonal transformation based splitting method which is highly effective and efficient for the numerical solution of paraxial Helmholtz equation with a large wave number. Rigorous matrix analysis is conducted in investigations and the oscillation-free computational procedure is proven to be stable in an asymptotic sense. Simulated examples are given to illustrate the conclusion.

Keywords

65M0665M1265F3515A12Paraxial equationhighly oscillatory problemseikonal splittingasymptotic stabilitymatrix eigenvaluesspectral radius
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 4 , October 2012 , pp. 1275 - 1292
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Band, Y. B., Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers, John Wiley & Sons, West Sussex, 2006.Google Scholar
[2]Beauregard, M. A. and Sheng, Q., An adaptive splitting approach for the quenching solution of reaction-diffusion equations over nonuniform grids, reprint, 2011.Google Scholar
[3]Bristeau, M. O., Erhel, J., Féat, P., Glowinski, R. and Periaux, J., Solving the Helmholtz equation at high-wave numbers on a parallel computer with a shared virtual memory, Int. High Perfor. Comput. Appl., 9 (1995), 18–28.Google Scholar
[4]Chinni, V. R., Menyuk, C. R. and Wai, P. K., Accurate solution of the paraxial wave equation using Richard extrapolation, IEEE Photonics Tech. Lett., 6 (1994), 409–411.CrossRefGoogle Scholar
[5]Condon, M., Deaño, A. and Iserles, A., On highly oscillatory problems arising in electronic engineering, Math. Model. Numer. Anal., 43 (2009), 785–804.Google Scholar
[6]Engquist, B., Fokas, A., Hairer, E. and Iserles, A., Highly Oscillatory Problems, London Math. Soc., London, 2009.CrossRefGoogle Scholar
[7]Gonzalez, L., Guha, S., Rogers, J. W. and Sheng, Q., An effective z-stretching method for paraxial light beam propagation simulations, J. Comput. Phys., 227 (2008), 7264–7278.CrossRefGoogle Scholar
[8]Goodman, J. W., Introduction to Fourier Optics, Third Edition, Roberts & Company Publishers, Denver, 2004.Google Scholar
[9]Guha, S., Validity of the paraxial approximation in the focal region of a small-/-number lens, Optical Lett., 26 (2001), 1598–1600.Google Scholar
[10]Horn, R. and Johnson, C., Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.Google Scholar
[11]Jin, S., Wu, H., Yang, X. and Huang, Z., Bloch decomposition-based Gaussian beam method for the Schrödinger equation with periodic potentials, J. Comput. Phys., 229 (2009), 4869–4883.CrossRefGoogle Scholar
[12]Levy, M. F., Perfectly matched layer truncation for parabolic wave equation models, Proc. Royal Soc. Lond., 457 (2001), 2609–2624.Google Scholar
[13]Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics, John Wiley & Sons, New York, 1991.CrossRefGoogle Scholar
[14]Saleh, M. A., Banerjee, P. P., Carns, J., Cook, G. and Evans, D. R., Stimulated photorefractive backscatter leading to six-wave mixing and phase conjugation in iron-doped lithium niobate, Appl. Optics, 46 (2007), 6151–6160.Google Scholar
[15]Sheng, Q., Adaptive decomposition finite difference methods for solving singular problems-a review, Front. Math. China, 4 (2009), 599–626.Google Scholar
[16]Sheng, Q., Guha, S. and Gonzalez, L., An exponential transformation based splitting method for fast computations of highly oscillatory solutions, J. Comput. Appl. Math., 235 (2011), 4452–4463.CrossRefGoogle Scholar
[17]van der Aa, N. P., The Rigorous Coupled-Wave Analysis, Ph.D. dissertation, Faculteit Wiskunde & Informatica, Technische Universiteit Eindhoven, the Netherland, 2007.Google Scholar
[18]Zang, W. P., Tian, J. G., Liu, Z. B., Zhou, W. Y., Song, F. and Zhang, C. P., Local one-dimensional approximation for fast simulation of Z-scan measurements with an arbitrary beam, Appl. Optics, 43 (2004), 4408–4414. CrossRefGoogle ScholarPubMed