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STURM-LIOUVILLE MATRIX EQUATION FOR THE STUDY OF ELECTROMAGNETIC-WAVES PROPAGATION IN LAYERED ANISOTROPIC MEDIA

By R. Pernas-Salomon and R. Perez-Alvarez

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Abstract:
We obtain a Sturm-Lioville matrix equation of motion (SLME) for the study of electromagnetic wave propagation in layered anisotropic structures. Conducting media were taken into account so that ohmic loss is considered. This equation can be treated using a 4×4 associated transfer matrix (T) in layered anisotropic structures, where the tensors: permittivity, permeability and the electric conductivity have a piecewise dependence on the coordinate perpendicular to the layered structure. We use the SLME eigenfunctions and eigenvalues to analyze qualitatively the numerical instability (Ωd problem) which potentially affects practical applications of the transfer matrix method. By means of the SLME coefficients we show analytically that T determinant value can be used to keep a check on the numerical accuracy of calculations. We derive equations to analyze wave propagation in linear layered isotropic structures. The SLME approach is applied on two typical layered structures to verify theoretical predictions and experimental results.

Citation:
R. Pernas-Salomon and R. Perez-Alvarez, "Sturm-Liouville Matrix Equation for the Study of Electromagnetic-Waves Propagation in Layered Anisotropic Media," Progress In Electromagnetics Research M, Vol. 40, 79-90, 2014.
doi:10.2528/PIERM14110504

References:
1. Perez-Alvarez, R. and F. Garcıa-Moliner, Transfer Matrix, Green Function and Related Techniques: Tools for the Study of Multilayer Heterostructures, Universitat Jaume I, Castellon de la Plana, Spain, 2004.

2. Trallero-Giner, C., R. Perez-Alvarez, and F. Garcıa-Moliner, Long Wave Polar Modes in Semiconductor Heterostructures, Elsevier Science, Oxford GB, Pergamon, 1998.

3. Tisseur, F. and K. Meerbergen, "The quadratic eigenvalue problem," SIAM Rev., Vol. 43, No. 2, 235-286, 2001.
doi:10.1137/S0036144500381988

4. Bonnet, G., "Orthotropic elastic media having a closed form expression of the Green tensor," Int. J. Solids Struct., Vol. 46, No. 5, 1240-1250, 2009.
doi:10.1016/j.ijsolstr.2008.10.033

5. Li, X. and M. Wang, "Three-dimensional Green’s functions for infinite anisotropic piezoelectric media," Int. J. Solids Struct., Vol. 44, No. 5, 1680-1684, 2007.
doi:10.1016/j.ijsolstr.2006.06.021

6. Bastard, G. and J. A. Brum, "Electronic states in semiconductor heterostructures," IEEE J. Quantum Elect., Vol. 22, No. 9, 1625-1644, 1986.
doi:10.1109/JQE.1986.1073186

7. Bastard, G., Wave Mechanics Applied to Semiconductor Heterostructures, Editions de Physique, Paris, 1989.

8. Oldano, C., "Electromagnetic-wave propagation in anisotropic stratified media," Phys. Rev. A, Vol. 40, No. 10, 6014-6020, 1989.
doi:10.1103/PhysRevA.40.6014

9. Berreman, D. W., "Optics in stratified and anisotropic media: 4 × 4-matrix formulation," J. Opt. Soc. Am., Vol. 62, No. 4, 502-510, 1972.
doi:10.1364/JOSA.62.000502

10. Krijn, M. P. C. M., "Electromagnetic wave propagation in stratified anisotropic media in the presence of sources," Opt. Lett., Vol. 17, No. 3, 163-165, 1992.
doi:10.1364/OL.17.000163

11. Jiaming, H. and Z. Lei, "Electromagnetic wave scatterings by anisotropic metamaterials: Generalized 4 × 4 transfer-matrix method," Phys. Rev. B, Vol. 77, No. 9, 094201-1-094201-12, 2008.

12. Calas, H., R. Rodriguez-Ramos, J. A. Otero, L. Leija, A. Ramos, and G. Monsivais, "Dispersion curves of shear horizontal wave surface velocities in multilayer piezoelectric systems," J. Appl. Phys., Vol. 107, No. 4, 044511-1-044511-9, 2010.
doi:10.1063/1.3305793

13. Zhang, V. Y. and V. Laude, "Unified and stable scattering matrix formalism for acoustic waves in piezoelectric stacks," J. Appl. Phys., Vol. 104, No. 6, 064916-1-064916-7, 2008.

14. Tan, E. L., "Matrix algorithms for modeling acoustic waves in piezoelectric multilayers," IEEE Trans. Ultrason., Ferroelect., Freq. Contr., Vol. 54, No. 10, 2016-2023, 2007.
doi:10.1109/TUFFC.2007.496

15. Tan, E. L., "Hybrid compliance-stiffness matrix method for stable analysis of elastic wave propagation in multilayered anisotropic media," J. Acoust. Soc. Am., Vol. 119, No. 1, 45-53, 53.
doi:10.1121/1.2139617

16. Lowe, M. J. S., "Matrix techniques for modeling ultrasonic waves in multilayered media," IEEE Trans. Ultrason., Ferroelect., Freq. Contr., Vol. 42, No. 4, 525-542, 1995.
doi:10.1109/58.393096

17. Higham, N. J., Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2002.
doi:10.1137/1.9780898718027

18. Hurewicz, V., Lectures on Ordinary Differential Equations, The MIT Press, Cambridge, MA, 1958.

19. Bibikov, Y. N., General Course on Ordinary Differential Equations (in Russian), Leningrad University Press, Leningrad, 1981.

20. Kong, J. A., Electromagnetic Wave Theory, EMW Publishing, Cambridge, Massachusetts, USA, 2008.

21. Aktsipetrov, O. A., T. V. Dolgova, I. V. Soboleva, and A. A. Fedyanin, "Anisotropic photonic crystals and microcavities based on mesoporous silicon," Phys. Solid State, Vol. 47, No. 1, 156-158, 2005.
doi:10.1134/1.1853468

22. Kovalev, D., G. Polisski, J. Diener, H. Heckler, N. Kunzner, V. Yu. Timoshenko, and F. Koch, "Strong in-plane birefringence of spatially nanostructured silicon," Appl. Phys. Lett., Vol. 78, No. 7, 916-918, 2001.
doi:10.1063/1.1343476

23. De la Mora, M. B., O. A. Jaramillo, R. Nava, J. Taguena-Mart´nez, and J. A. del Rıo, "Viability study of porous silicon photonic mirrors as secondary reflectors for solar concentration systems," Sol. Energy Mater. Sol. Cells, Vol. 93, No. 8, 1218-1224, 2009.
doi:10.1016/j.solmat.2009.01.007

24. Diesinger, H., A. Bsiesy, and R. Herino, "In situ measurement of the optical absorption coefficient of porous silicon," J. Appl. Phys., Vol. 89, No. 1, 221-225, 2001.
doi:10.1063/1.1328785


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