The present work is devoted to the clarification of the conditions necessary for step-by-step justification of the possibility of reduction of the homogeneous system of linear algebraic equations for the spectral problems of 2-D photonic crystals by the plane waves method. The issues related to the algorithms and the numerical solutions of these spectral problems are analyzed. The possibility of analytical regularization is investigated, and the ways to improve the convergence of the obtained results are identified.
2. Lourtioz, J.-M., H. Bensty, V. Berger, J. M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices, Springer, New York, 2005.
3. Johnson, S. G. and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Optics Express, Vol. 8, No. 3, 173-190, 2001.
4. Velychko, L. G., Y. K. Sirenko, and O. S. Velychko, "Time-domain analysis of open resonators. Analytical grounds," Progress In Electromagnetics Research, Vol. 61, 1-26, 2006.
5. Sirenko, Y. K., S. Strom, and N. P. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures. New Methods and Techniques, Springer, New York, 2007.
6. Velychko, L. G. and Y. K. Sirenko, "Controlled changes in spectra of open quasi-optical resonators," Progress In Electromagnetics Research B, Vol. 16, 85-105, 2009.
7. Kravchenko, V. F., Y. K. Sirenko, and K. Y. Sirenko, Electromagnetic Wave Transformation and Radiation by the Open Resonant Structures. Modelling and Analysis of Transient and Steady-State Processes, Fizmathlit, Moscow, 2011.
8. Vaynikko, G. M. and O. O. Karma, "On rapidity of convergence of approximate methods in problem of eigenvalues with nonlinear occurrence of parameter," Zhurnal Vychislitelnoy Matematiki I Matematicheskoy Fiziki, Vol. 14, No. 6, 1393-1408, 1974.
9. Hokhberg, I. Z. and M. G. Krein, Introduction into the Theory of Linear not Self-Adjoint Operators, Nauka, Moscow, 1965.
10. Hutson, V. C. L. and J. S. Pym, Applications of Functional Analysis and Operator Theory, Academic Press, New York, 1980.
11. Titchmarsh, E. C., Eigenfunction Expansions Associated with Second-Order Differential Equations, Clarendon Press, Oxford, 1958.
12. Shestopalov, V. P. and Y. K. Sirenko, Dynamic Theory of Gratings, Naukova Dumka, Kiev, 1989.
13. Keldysh, M. V., "On the completeness of eigenfunctions of some classes of non-selfadjoint linear operators," Russian Mathematical Surveys, Vol. 26, No. 4, 15-44, 1971.
14. Hokhberg, I. Z. and Y. I. Seagul, "Operator generalization of the theorem about logarithmic residue and the Rouche theorem," Matematicheckiy Sbornik, Vol. 84, No. 4, 607-629, 1971.
15. Reed, M. and B. Simon, Methods of Mathematical Physics. IV: Analysis of Operators, Academic Press, New York, 1978.
16. Hardy, G. H., J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, Cambridge, 1934.
17. Shestopalov, V. P., A. A. Kirilenko, and S. A. Masalov, Matrix Convolution-Type Equations in the Diffraction Theory, Naukova Dumka, Kiev, 1984.