OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 30, Iss. 10 — Oct. 1, 2013
  • pp: 2111–2123

Explicit error bounds for the α-quasi-periodic Helmholtz problem

Natacha H. Lord and Anthony J. Mulholland  »View Author Affiliations

JOSA A, Vol. 30, Issue 10, pp. 2111-2123 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (418 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



This paper considers a finite element approach to modeling electromagnetic waves in a periodic diffraction grating. In particular, an a priori error estimate associated with the α-quasi-periodic transformation is derived. This involves the solution of the associated Helmholtz problem being written as a product of eiαx and an unknown function called the α-quasi-periodic solution. To begin with, the well-posedness of the continuous problem is examined using a variational formulation. The problem is then discretized, and a rigorous a priori error estimate, which guarantees the uniqueness of this approximate solution, is derived. In previous studies, the continuity of the Dirichlet-to-Neumann map has simply been assumed and the dependency of the regularity constant on the system parameters, such as the wavenumber, has not been shown. To address this deficiency, in this paper an explicit dependence on the wavenumber and the degree of the polynomial basis in the a priori error estimate is obtained. Since the finite element method is well known for dealing with any geometries, comparison of numerical results obtained using the α-quasi-periodic transformation with a lattice sum technique is then presented.

© 2013 Optical Society of America

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(050.1960) Diffraction and gratings : Diffraction theory
(050.2770) Diffraction and gratings : Gratings
(080.2720) Geometric optics : Mathematical methods (general)
(290.0290) Scattering : Scattering
(080.1753) Geometric optics : Computation methods

ToC Category:
Diffraction and Gratings

Original Manuscript: May 16, 2013
Revised Manuscript: August 27, 2013
Manuscript Accepted: August 27, 2013
Published: September 25, 2013

Natacha H. Lord and Anthony J. Mulholland, "Explicit error bounds for the α-quasi-periodic Helmholtz problem," J. Opt. Soc. Am. A 30, 2111-2123 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. Peters, M. Rüdiger, B. Bläsi, and W. Platzer, “Electro-optical simulation of diffraction in solar cells,” Opt. Express 18, A584–A593 (2010). [CrossRef]
  2. R. A. Potyrailo, H. Ghiradella, A. Vertiatchikh, J. R. Cournoyer, K. Dovidenko, and E. Olson, “Morpho butterfly wing scales demonstrate highly selective vapour response,” Nat. Photonics 1, 123–128 (2007). [CrossRef]
  3. T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Phys. Med. Biol 55, 3317–3323 (2010). [CrossRef]
  4. P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based x-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. USA 107, 13576–13581 (2010). [CrossRef]
  5. G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Science, Vol. 22 of SIAM Frontiers in Applied Mathematics (Springer, 2001).
  6. G. Bao and A. Zhou, “Analysis of finite dimensional approximations to a class of partial differential equations,” Math. Methods Appl. Sci. 27, 2055–2066 (2004). [CrossRef]
  7. R. Petit, Electromagnetic Theory of Gratings (Springer, 1980).
  8. G. Bao, “Numerical analysis of diffraction by periodic structures: TM polarization,” Numer. Math. 75, 1–16 (1996). [CrossRef]
  9. G. Bao, Z. Chen, and H. Wu, “Adaptive finite element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005). [CrossRef]
  10. N. Lord, “Analysis of electromagnetic waves in a periodic diffraction grating using a priori error estimates and a dual weighted residual method,” Ph.D. thesis (University of Strathclyde, 2012) ( http://www.mathstat.strath.ac.uk/research/phd_mphil_theses ).
  11. N. Lord and A. J. Mulholland, “Analysis of the α,0-quasi periodic transformation for a periodic diffraction grating,” (Department of Mathematics and Statistics, University of Strathclyde, 2011) ( http://www.mathstat.strath.ac.uk/research/reports/2011 ).
  12. S. Chandler-Wilde, “Boundary value problems for the Helmholtz equation in a half-plane,” in Mathematical and Numerical Aspects of Wave Propagation (SIAM, 1995), pp. 188–197.
  13. G. Bao, Y. Cao, and H. Yang, “Numerical solution of diffraction problems by a least squares FEM,” Math. Methods Appl. Sci. 23, 1073–1092 (2000). [CrossRef]
  14. E. Wolf, ed., Rigorous Vector Theories of Diffraction Gratings, Vol. 21 of Progress in Optics (North-Holland, 1984), pp. 1–67.
  15. P. G. Ciarlet, The Finite Element Method for Elliptic Equations (North-Holland, 1978).
  16. J. M. Melenk and S. Sauter, “Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions,” Math. Comput. 79, 1871–1914 (2010). [CrossRef]
  17. A. Lechleiter and D. Nguyen, “Volume integral equations for scattering from anisotropic diffraction gratings,” Math. Methods Appl. Sci. 36, 262–274 (2012).
  18. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Vol. 15 of Texts in Applied Mathematics (Springer, 2002).
  19. E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, 1989).
  20. D. Maystre, Electromagnetic Theory of Gratings (Springer, 1980).
  21. L. F. Richardson, Measure and Integration. A Concise Introduction to Real Analysis (Wiley, 2009).
  22. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Vol. 93 of Applied Mathematical Sciences (Springer-Verlag, 1992).
  23. A. Buffa, “Trace theorems for functional spaces related to Maxwell equations: an overview,” in Computational Electromagnetics, Vol. 28 of Lecture Notes in Computational Science and Engineering (Springer, 2002), pp. 23–34.
  24. Z. Ding, “A proof of the trace theorem of Sobolev spaces on Lipschitz domains,” Proc. Am. Math. Soc. 124, 591–601 (1996). [CrossRef]
  25. D. Braess, Finite Elements (Cambridge University, 1997).
  26. F. Ihlenburgh, Finite Element Analysis of Acoustic Scattering (Springer, 1998), Vol. 132.
  27. J. T. Oden and M. Ainsworth, A Posteriori Error Estimation in Finite Element Analysis (Wiley, 2000).
  28. K. Ito, Encyclopedic Dictionary of Mathematics, 2nd ed. (Massachusetts Institute of Technology, 1987).
  29. K. Yasumoto, T. Kushta, and H. Toyama, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004). [CrossRef]
  30. R. E. Coath, “Investigating the use of replica Morpho butterfly scales for colour displays,” Society 5, 1–9 (2007) ( http://printfu.org/blue+morpho+didius+butterfly ).
  31. G. Berger, K. Müller, C. Denz, I. Földvári, and A. Péter, “Digital data storage in a phase-encoded holographic memory system: data quality and security,” Proc. SPIE 4988, 104–111 (2003). [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.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited