OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 28, Iss. 3 — Mar. 1, 2011
  • pp: 373–380

Exact transparent boundary condition for the parabolic equation in a rectangular computational domain

R. M. Feshchenko and A. V. Popov  »View Author Affiliations

JOSA A, Vol. 28, Issue 3, pp. 373-380 (2011)

View Full Text Article

Enhanced HTML    Acrobat PDF (603 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



In this paper, an exact three-dimensional transparent boundary condition for the parabolic wave equation in a rectangular computational domain is reported. It is a generalization of the well-known two-dimensional Basakov–Popov–Papadakis transparent boundary condition. It relates the boundary transversal derivative of the wave field at any given longitudinal position to the field values at all preceding computational steps. Several examples demonstrate propagation of light along simple structured optical fibers as well as in x-ray guiding structures. The proposed condition is simple and robust and can help to reduce the size of the computational domain considerably.

© 2011 Optical Society of America

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(060.2310) Fiber optics and optical communications : Fiber optics
(260.1960) Physical optics : Diffraction theory
(340.0340) X-ray optics : X-ray optics

ToC Category:
Physical Optics

Original Manuscript: October 18, 2010
Revised Manuscript: December 17, 2010
Manuscript Accepted: December 22, 2010
Published: February 23, 2011

R. M. Feshchenko and A. V. Popov, "Exact transparent boundary condition for the parabolic equation in a rectangular computational domain," J. Opt. Soc. Am. A 28, 373-380 (2011)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, 1965).
  2. F. D. Tappert, “The parabolic approximation method,” Lect. Notes Phys. 70, 224–287 (1977). [CrossRef]
  3. J. S. Papadakis, “Exact nonreflecting boundary conditions for parabolic type approximations in underwater acoustics,” J. Comput. Acoust. 2, 83–98 (1994). [CrossRef]
  4. V. A. Baskakov and A. V. Popov, “Implementation of transparent boundaries for numerical solution of the Schrödinger equation,” Wave Motion 14, 123–128 (1991). [CrossRef]
  5. A. V. Popov, “Accurate modelling of transparent boundaries in quasi-optics,” Radio Sci. 31, 1781–1790 (1996). [CrossRef]
  6. D. Yevick, T. Friese, and F. Schmidt, “A comparison of transparent boundary conditions for the Fresnel equation,” J. Comput. Phys. 168, 433–444 (2001). [CrossRef]
  7. X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schädle, “A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations,” Commun. Comput. Phys. 4, 729–796 (2008).
  8. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200(1994). [CrossRef]
  9. R. M. Feshchenko and A. V. Popov, “Exact transparent boundary condition for beam propagation in rectangular domain,” in Proceedings of 2010 12th International Conference on Transparent Optical Networks (ICTON) (IEEE, 2010), pp. 1–4. [CrossRef]
  10. Yu. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the parabolic wave equation to x-ray diffraction optics,” Opt. Commun. 118, 619–636 (1995). [CrossRef]
  11. V. V. Aristov, M. V. Grigoriev, S. M. Kuznetsov, L. G. Shabelnikov, V. A. Yunkin, M. Hoffmann, and E. Voges, “X-ray focusing by planar parabolic refractive lenses made of silicon,” Opt. Commun. 177, 33–38 (2000). [CrossRef]
  12. S. Lagomarsino, I. Bukreeva, and A. Cedola, “Theoretical analysis of x-ray waveguides,” in Modern Developments in X-Ray and Neutron Optics, A. Erko, ed., Vol. 137 of Springer Series in Optical Sciences (Springer, 2008), pp. 91–111.
  13. I. Bukreeva, D. Pelliccia, A. Cedola, F. Scarinci, M. Ilie, C. Giannini, L. De Caro, and S. Lagomarsino, “Analysis of tapered front-coupling x-ray waveguides,” J. Synchrotron Radiat. 17, 61–68 (2009). [CrossRef] [PubMed]
  14. J. Vila-Comamala, K. Jefimovs, J. Raabe, B. Kaulich, and C. David, “Silicon Fresnel zone plates for high heat load x-ray microscopy,” Microelectron. Eng. 85, 1241–1244(2008). [CrossRef]
  15. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interaction: photoabsorption, scattering, transmission and reflection at E=50–30,000eV, Z=1–92,” At. Data Nucl. Data Tables 54, 181–342 (1993). [CrossRef]
  16. R. Jamier, P. Viale, S. Fevrier, J.-M. Blondy, S. L. Semjonov,M. E. Likhachev, M. M. Bubnov, E. M. Dianov, V. F. Khopin, M. Y. Salganskii, and A. N. Guryanov, “Depressed-index-core singlemode bandgap fiber with very large effective area,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2006), paper OFC6. [PubMed]
  17. D. Yevick, J. Yu, and F. Schmidt, “Analytic studies of absorbing and impedance-matched boundary layers,” IEEE Photon. Technol. Lett. 9, 73–75 (1997). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3
Fig. 4 Fig. 5

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited