PDE eigenvalue iterations with applications in two-dimensional photonic crystals

Altmann, Robert; Froidevaux, Marine

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 5, pp. 1751-1776.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

We consider PDE eigenvalue problems as they occur in two-dimensional photonic crystal modeling. If the permittivity of the material is frequency-dependent, then the eigenvalue problem becomes nonlinear. In the lossless case, linearization techniques allow an equivalent reformulation as an extended but linear and Hermitian eigenvalue problem, which satisfies a Gårding inequality. For this, known iterative schemes for the matrix case such as the inverse power or the Arnoldi method are extended to the infinite-dimensional case. We prove convergence of the inverse power method on operator level and consider its combination with adaptive mesh refinement, leading to substantial computational speed-ups. For more general photonic crystals, which are described by the Drude–Lorentz model, we propose the direct application of a Newton-type iteration. Assuming some a priori knowledge on the eigenpair of interest, we prove local quadratic convergence of the method. Finally, numerical experiments confirm the theoretical findings of the paper.

[1] G. Allaire and S.M. Kaber, Numerical Linear Algebra. Springer, New York (2008). | MR | Zbl

[2] R. Altmann, P. Henning and D. Peterseim, Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials. Math. Models Methods Appl. Sci. (M3AS). 30 (2020) 917–955. | DOI | MR | Zbl

[3] R. Altmann, P. Henning and D. Peterseim, The J-method for the Gross-Pitaevskii eigenvalue problem. Preprint (2019). | arXiv | MR

[4] P.M. Anselone and L.B. Rall, The solution of characteristic value-vector problems by Newton’s method. Numer. Math. 11 (1968) 38–45. | DOI | MR | Zbl

[5] K.-J. Bathe and E.L. Wilson, Solution methods for eigenvalue problems in structural mechanics. Int. J. Numer. Methods Eng. 6 (1973) 213–226. | DOI | Zbl

[6] F. Bozorgnia, Convergence of inverse power method for first eigenvalue of $p$-Laplace operator. Numer. Funct. Anal. Optim. 37 (2016) 1378–1384. | DOI | MR | Zbl

[7] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010). | MR | Zbl

[8] S. Burger, R. Klose, A. Schädle, F. Schmidt and L. Zschiedrich, Adaptive FEM solver for the computation of electromagnetic eigenmodes in 3D photonic crystal structures. In: Scientific Computing in Electrical Engineering, edited by A.M. Anile, G. Ali and G. Mascali. Springer, Berlin, Heidelberg (2006) 169–173. | DOI | Zbl

[9] Concepts Development Team, Webpage of Numerical C++ Library Concepts. Available from www.concepts.math.ethz.ch (2019).

[10] R. Courant, Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik. Math. Z. 7 (1920) 1–57. | DOI | JFM | MR

[11] G. Demesy and S. John, Solar energy trapping with modulated silicon nanowire photonic crystals. J. Appl. Phys. 112 (2012) 074326. | DOI

[12] P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer-Verlag Berlin Heidelberg (2004). | MR | Zbl

[13] W. Dörfler, A. Lechleiter, M. Plum, G. Schneider and C. Wieners, Photonic Crystals. Mathematical Analysis and Numerical Approximation, Springer-Verlag, Berlin (2011). | DOI | Zbl

[14] S. Eastman and D. Estep, A power method for nonlinear operators. Appl. Anal. 86 (2007) 1303–1314. | DOI | MR | Zbl

[15] C. Effenberger, Robust solution methods for nonlinear eigenvalue problems. Ph.D. thesis, EPFL, Lausanne (2013).

[16] C. Effenberger, D. Kressner and C. Engström, Linearization techniques for band structure calculations in absorbing photonic crystals. Int. J. Numer. Methods Eng. 89 (2012) 180–191. | DOI | MR | Zbl

[17] C. Engström, On the spectrum of a holomorphic operator-valued function with applications to absorptive photonic crystals. Math. Models Methods Appl. Sci. (M3AS) 20 (2010) 1319–1341. | DOI | MR | Zbl

[18] M.A. Erickson, R.S. Smith and A.J. Laub, Power methods for calculating eigenvalues and eigenvectors of spectral operators on Hilbert spaces. Int. J. Control 62 (1995) 1117–1128. | DOI | MR | Zbl

[19] P. Frauenfelder and C. Lage, Concepts – an object-oriented software package for partial differential equations. ESAIM: M2AN 36 (2002) 937–951. | DOI | Numdam | MR | Zbl

[20] M. Garcia-Vergara, G. Demésy and F. Zolla, Extracting an accurate model for permittivity from experimental data: hunting complex poles from the real line. Opt. Lett. 42 (2017) 1145–1148. | DOI

[21] S. Giani and I.G. Graham, Adaptive finite element methods for computing band gaps in photonic crystals. Numer. Math. 121 (2012) 31–64. | DOI | MR | Zbl

[22] Q. Gong and X. Hu editors, Photonic Crystals: Principles and Applications. CRC Press, Boca Raton, FL (2014). | DOI

[23] A. Günnel, R. Herzog and E. Sachs, A note on preconditioners and scalar products in Krylov subspace methods for self-adjoint problems in Hilbert space. Electron. Trans. Numer. Anal. 41 (2014) 13–20. | MR | Zbl

[24] T.-M. Huang, W.-W. Lin and V. Mehrmann, A Newton-type method with nonequivalence deflation for nonlinear eigenvalue problems arising in photonic crystal modeling. SIAM J. Sci. Comput. 38 (2016) B191–B218. | DOI | MR | Zbl

[25] J.D. Jackson, Classical Electrodynamics, 3rd edition. John Wiley & Sons, New York (1999). | MR | Zbl

[26] E. Jarlebring, W. Michiels and K. Meerbergen, A linear eigenvalue algorithm for the nonlinear eigenvalue problem. Numer. Math. 122 (2012) 169–195. | DOI | MR | Zbl

[27] J.D. Joannopoulos, S.G. Johnson, J.N. Winn and R.D. Meade, Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton, NJ (2008). | Zbl

[28] S. John, Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 58 (1987) 2486–2489. | DOI

[29] S. John, Why trap light? Nat. Mater. 11 (2012) 997–999. | DOI

[30] D. Kressner, A block Newton method for nonlinear eigenvalue problems. Numer. Math. 114 (2009) 355–372. | DOI | MR | Zbl

[31] P. Kuchment, The mathematics of photonic crystals. In: Mathematical Modeling in Optical Science. SIAM, Philadelphia, PA (2001) 207–272. | DOI | MR | Zbl

[32] J. López-Gómez and C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of Linear Operators. Birkhäuser Verlag, Basel (2007). | MR | Zbl

[33] M. Luo and Q.H. Liu, Three-dimensional dispersive metallic photonic crystals with a bandgap and a high cutoff frequency. J. Opt. Soc. Am. A 27 (2010) 1878–1884. | DOI

[34] V. Mehrmann and A. Miedlar, Adaptive computation of smallest eigenvalues of self-adjoint elliptic partial differential equations. Numer. Linear Algebr. 18 (2011) 387–409. | DOI | MR | Zbl

[35] V. Mehrmann and H. Voss, Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods. GAMM-Mitt. 27 (2004) 121–152. | DOI | MR | Zbl

[36] A. Miedlar, Inexact adaptive finite element methods for elliptic PDE eigenvalue problems. Ph.D. thesis, Technische Universität Berlin (2011).

[37] M.R. Osborne, A new method for the solution of eigenvalue problems. Comput. J. 7 (1964) 228–232. | DOI | MR | Zbl

[38] L.P. Pitaevskii and S. Stringari, Bose-Einstein Condensation. The Clarendon Press, Oxford University Press, Oxford (2003). | MR | Zbl

[39] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994). | DOI | MR | Zbl

[40] A. Raman and S. Fan, Perturbation theory for plasmonic modulation and sensing. Phys. Rev. B 83 (2011) 205131. | DOI

[41] Y. Saad, Numerical Methods for Large Eigenvalue Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2011). | DOI | MR | Zbl

[42] K. Schmidt and R. Kappeler, Efficient computation of photonic crystal waveguide modes with dispersive material. Opt. Express 18 (2010) 7307–7322. | DOI

[43] K. Schmidt and P. Kauf, Computation of the band structure of two-dimensional photonic crystals with $hp$–finite elements. Comput. Methods Appl. Mech. Eng. 198 (2009) 1249–1259. | DOI | MR | Zbl

[44] K. Schreiber, Nonlinear eigenvalue problems: Newton-type methods and nonlinear Rayleigh functionals. Ph.D. thesis, Technische Universität Berlin (2008). | Zbl

[45] P. Schulze, B. Unger, C. Beattie and S. Gugercin, Data-driven structured realization. Linear Algebr. Appl. 537 (2018) 250–286. | DOI | MR | Zbl

[46] A. Spence and C. Poulton, Photonic band structure calculations using nonlinear eigenvalue techniques. J. Comput. Phys. 204 (2005) 65–81. | DOI | MR | Zbl

[47] Y. Su and Z. Bai, Solving rational eigenvalue problems via linearization. SIAM J. Matrix Anal. Appl. 32 (2011) 201–216. | DOI | MR | Zbl

[48] K. Tsakmadis, In the limelight. Nat. Mater. 11 (2012) 1000–1001. | DOI

[49] H. Voss, A rational spectral problem in fluid–solid vibration. Electron. Trans. Numer. Anal. 16 (2003) 93–105. | MR | Zbl

[50] A. Weinstein and W. Stenger, Methods of Intermediate Problems for Eigenvalues. Academic Press, New York, London (1972). | MR | Zbl

[51] J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge (1987). | MR | Zbl

[52] E. Zeidler, Nonlinear Functional Analysis and its Applications II/A: Linear Monotone Operators, Springer-Verlag, New York, 1990. | Zbl