Regularity for Maxwell eigenproblems in photonic crystal fibre modelling

详细信息    查看全文

• 作者：Monique Dauge (1)
  Richard A. Norton (2)
  Robert Scheichl (3)
  
  1. IRMAR ; Universit茅 de Rennes 1 ; Campus de Beaulieu ; 35042聽 ; Rennes Cedex ; France
  2. Department of Physics ; University of Otago ; PO Box 56 ; Dunedin ; 9054 ; New Zealand
  3. Department of Mathematical Sciences ; University of Bath ; Bath ; BA2 7AY ; UK
  
• 关键词：Sobolev regularity ; Maxwell eigenproblem ; Kondratiev鈥檚 method ; Photonics ; Photonic crystal fibres ; 35B65 ; 35B15 ; 35Q61 ; 78A48
• 刊名：BIT Numerical Mathematics
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：55
• 期：1
• 页码：59-80
• 全文大小：307 KB
• 参考文献：1. Ashcroft, N., Mermin, N.: Solid State Physics. Saunders College, Philadelphia (1976)
  2. Bramble, J.H., King, J.T.: A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. Comput. Math. 6, 109鈥?38 (1996)
  3. Chen, Z., Du, Q., Zou, J.: Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal. 37(5), 1542鈥?570 (2000) CrossRef
  4. Chen, Z., Zou, J.: Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. 79(2), 175鈥?02 (1998) CrossRef
  5. Costabel, M., Dauge, M.: Maxwell and Lam茅 eigenvalues on polyhedra. Math. Meth. Appl. Sci. 22, 243鈥?58 (1999). doi:10.1002/(SICI)1099-1476(199902)22:3<243::AID-MMA37>3.0.CO;2-0
  6. Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Rational Mech. Anal. 151(3), 221鈥?76 (2000) CrossRef
  7. Costabel, M., Dauge, M., Nicaise, S.: Singularities of Maxwell interface problems. M2AN Math. Model. Numer. Anal. 33(3), 627鈥?49 (1999) doi:10.1051/m2an:1999155
  8. Costabel, M., Stephan, E.: A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106(2), 367鈥?13 (1985) CrossRef
  9. Dauge, M.: Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)
  10. Dauge, M., Dular, P., Kr盲henb眉hl, L., P茅ron, V., Perrussel, R., Poignard, C.: Corner asymptotics of the magnetic potential in the eddy-current model. Tech. Rep. RR-8204. Math. Meth. Appl. Sci. (2013). doi:10.1002/mma.2947
  11. Dobson, D.C.: An efficient method for band structure calculations in 2D photonic crystals. J. Comput. Phys. 149(2), 363鈥?76 (1999) CrossRef
  12. Elschner, J., Hinder, R., Schmidt, G., Penzel, F.: Existence, uniqueness and regularity for solutions of the conical diffraction problem. Math. Models Methods Appl. Sci. 10(3), 317鈥?41 (2000) CrossRef
  13. Ferrando, A., Silvester, E., Miret, J.J., Andr茅s, P., Andr茅s, M.V.: Full vector analysis of a realistic photonic crystal fiber. Opt. Lett. 24(5), 276鈥?78 (1999) CrossRef
  14. Figotin, A., Klein, A.: Localization of classical waves ii: electromagnetic waves. Commun. Math. Phys. 184(2), 411鈥?41 (1997) CrossRef
  15. Fliss, S.: Analyse math茅matique et num茅rique de probl猫mes de propagation des ondes dans des milieux p茅riodiques infinis localement perturb茅s. These, Ecole Polytechnique X, France (2009). http://pastel.archivesouvertes.fr/pastel-00005464
  16. Giani, S., Graham, I.G.: Adaptive finite element methods for computing band gaps in photonic crystals. Numerische Mathematik 121(1), 31鈥?4 (2012) CrossRef
  17. Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)
  18. Grisvard, P.: Singularities in Boundary Value Problems, Recherches en Math茅matiques Appliqu茅es [Research in Applied Mathematics], vol. 22. Masson, Paris (1992)
  19. Joannopoulos, J., Johnson, S., Winn, J., Meade, R.: Photonic Crystals Molding the Flow of Light, 2nd edn. Princeton University Press, Princeton (2008)
  20. Knight, J.: Photonic crystal fibres. Nature 424, 847鈥?51 (2003) CrossRef
  21. Kondrat鈥檈v, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Mosc. Math. Soc. 16, 227鈥?13 (1967)
  22. Kuchment, P.: Floquet Theory for Partial Differential Equations, Operator Theory: Advances and Applications, vol. 60. Birkh盲user, Basel (1993)
  23. Kuchment, P.: Mathematical Modelling in Optical Science, Chap. 7, Frontiers in Applied Mathematics. pp. 207鈥?72. SIAM, Philadelphia (2001)
  24. Nicaise, S.: Polygonal Interface Problems. Methoden und Verfahren der Mathematischen Physik, vol. 39. Verlag Peter D. Lang, Frankfurt-am-Main (1993)
  25. Nicaise, S., S盲ndig, A.M.: General interface problems I/II. Math. Meth. Appl. Sci. 17, 395鈥?50 (1994) CrossRef
  26. Nicolet, A., Guenneau, S., Geuzaine, C., Zolla, F.: Modelling of electromagnetic waves in periodic media with finite elements. J. Comput. Appl. Math. 168, 321鈥?29 (2004) CrossRef
  27. Norton, R.: Numerical computation of band gaps in photonic crystal fibres. PhD thesis, University of Bath, England (2008)
  28. Norton, R., Scheichl, R.: Convergence analysis of planewave expansion methods for 2D Schr枚dinger operators with discontinuous periodic potentials. SIAM J. Numer. Anal. 47(6), 4356鈥?380 (2010)
  29. Norton, R.A., Scheichl, R.: Planewave expansion methods for photonic crystal fibres. Appl. Numer. Math. 63, 88鈥?04 (2013) CrossRef
  30. Pearce, G., Hedley, T., Bird, D.: Adaptive curvilinear coordinates in a plane-wave solution of Maxwell鈥檚 equations in photonic crystals. Phys. Rev. B 71, 195108 (2005)
  31. Petzoldt, M.: Regularity and error estimators for elliptic problems with discontinuous coefficients. PhD thesis, FU Berlin, Germany (2001)
  32. Petzoldt, M.: Regularity results for Laplace interface problems in two dimensions. Z. Anal. Anwendungen 20(2), 431鈥?55 (2001) CrossRef
  33. Pottage, J., Bird, D., Hedley, T., Birks, T., Knight, J., Russell, P.: Robust photonic band gaps for hollow core guidance in PCF made from high index glass. Opt. Express 11(22), 2854鈥?861 (2003)
  34. Russell, P.: Photonic crystal fibers. Science 299(5605), 358鈥?62 (2003) CrossRef
  35. Saitoh, K., Koshiba, M.: Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: Application to photonic crystal fibers. IEEE J. Quant. Elec. 38(7), 927鈥?33 (2002)
  36. Soussi, S.: Convergence of the supercell method for defect modes calculations in photonic crystals. SIAM J. Numer. Anal. 43(3), 1175鈥?201 (2005) CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Computational Mathematics and Numerical Analysis
  Numeric Computing
  Mathematics
  
• 出版者：Springer Netherlands
• ISSN：1572-9125

文摘

The convergence behaviour and the design of numerical methods for modelling the flow of light in photonic crystal fibres depend critically on an understanding of the regularity of solutions to time-harmonic Maxwell equations in a three-dimensional, periodic, translationally invariant, heterogeneous medium. In this paper we determine the strength of the dominant singularities that occur at the interface between materials. By modifying earlier regularity theory for polygonal interfaces we find that on each subdomain, where the material in the fibre is constant, the regularity of in-plane components of the magnetic field are \(H^{2-\eta }\) for all \(\eta > 0\) . This estimate is sharp in the sense that these components do not belong to \(H^2\) , in general. However, global regularity is restricted by the presence of an interface between these subdomains and the interface conditions imply only \(H^{3/2-\eta }\) regularity across the interface. The results are useful to anyone applying a numerical method such as a finite element method or a planewave expansion method to model photonic crystal fibres or similar materials.