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Improved Formulation of Scattering Matrices for Semi-Analytical Methods That Is Consistent with Convention

By Raymond C. Rumpf
Progress In Electromagnetics Research B, Vol. 35, 241-261, 2011


The literature describing scattering matrices for semi-analytical methods almost exclusively contains inefficient formulations and formulations that deviate from long-standing convention in terms of how the scattering parameters are defined. This paper presents a novel and highly improved formulation of scattering matrices that is consistent with convention, more efficient to implement, and more versatile than what has been otherwise presented in the literature. Semi-analytical methods represent a device as a stack of layers that are uniform in the longitudinal direction. Scattering matrices are calculated for each layer and are combined into a single overall scattering matrix that describes propagation through the entire device. Free space gaps with zero thickness are inserted between the layers and the scattering matrices are made to relate fields which exist outside of the layers, but directly on their boundaries. This framework produces symmetric scattering matrices so only two parameters need to be calculated and stored instead of four. It also enables the scattering matrices to be arbitrarily interchanged and reused to describe longitudinally periodic devices more efficiently. Numerical results are presented that show speed and efficiency can be increased by more than an order of magnitude using the improved formulation.


Raymond C. Rumpf, "Improved Formulation of Scattering Matrices for Semi-Analytical Methods That Is Consistent with Convention," Progress In Electromagnetics Research B, Vol. 35, 241-261, 2011.


    1. Helfert, S. F. and R. Pregla, "The method of lines: A versatile tool for the analysis of waveguide structures," Electromagnetics, Vol. 22, 615-637, Taylor & Francis, New York, 2002.

    2. Jamid, H. A. and M. N. Akram, "Analysis of deep waveguide gratings: An efficient cascading and doubling algorithm in the method of lines framework," J. Lightwave Technol., Vol. 20, No. 7, 1204-1209, 2002.

    3. Moharam, M. G., E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary grating," J. Opt. Soc. Am. A, Vol. 12, No. 5, 1068-1076, 1995.

    4. Moharam, M. G., D. A. Pommet, E. B. Grann, and T. K. Gaylord, "Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: Enhanced transmittance matrix approach," J. Opt. Soc. Am. A, Vol. 12, No. 5, 1077-1086, 1995.

    5. Berreman, D. W., "Optics in stratified and anisotropic media: 4 × 4-matrix formulation," J. Opt. Soc. Am. A, Vol. 62, No. 4, 502-510, 1972.

    6. Pendry, J. B., "Photonic band structures," J. Modern Optics, Vol. 41, No. 2, 209-229, 1994.

    7. Li, Z.-Y. and L.-L. Lin, "Photonic band structures solved by a plane-wave-based transfer-matrix method," Phys. Rev. E, Vol. 67, 046607, 2003.

    8. Matthews, Jr., E. W., "The use of scattering matrices in microwave circuits," IRE Trans. on Microwave Theory and Techniques, 21-26, 1955.

    9. Carlin, H. J., "The scattering matrix in network theory," IRE Trans. On Circuit Theory, Vol. 3, No. 2, 88-97, 1956.

    10. Kurokawa, K., "Power waves and the scattering matrix," IEEE Trans. on Microwave Theory and Techniques, 194-202, 1965.

    11. Collin, R. E., Foundations for Microwave Engineering, 1st Ed., 170-182, McGraw Hill, New York, 1966.

    12. Pozar, D. M., Microwave Engineering, 3rd Ed., 174-183, Wiley, New York, 2005.

    13. Rizzi, P. A., Microwave Engineering Passive Circuits, 1st Ed., 168-176, Prentice Hall, New Jersey, 1988.

    14. Tan, E. L., "Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings," J. Modern Optics, Vol. 53, No. 4, 417-428, 2006.

    15. Li, L., "Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings," J. Opt. Soc. Am. A, Vol. 11, No. 11, 2829-2836, 1994.

    16. Li, L., "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A, Vol. 13, No. 5, 1024-1035, 1996.

    17. http://cp.literature.agilent.com/litweb/pdf/5989-6353EN.pdf..

    18. Borsboom, P.-P. and H. J. Frankena, "Field analysis of two-dimensional integrated optical gratings," J. Opt. Soc. Am. A, Vol. 12, No. 5, 1134-1141, 1995.

    19. Lin, L.-L., Z.-Y. Li, and K.-M. Ho, "Lattice symmetry applied in transfer-matrix methods for photonic crystals," J. Appl. Phys., Vol. 94, No. 2, 811-821, 2003.

    20. Ko, D. Y. K. and J. R. Sambles, "Scattering matrix method for propagation of radiation in stratified media: Attenuated total reflection studies of liquid crystals," J. Opt. Soc. Am. A, Vol. 5, No. 11, 1863-1866, 1988.

    21. Li, L., "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A, Vol. 13, No. 5, 1024-1035, 1996.

    22. Silberstein, E., P. Lalanne, J.-P. Hugonin, and Q. Cao, "Use of grating theories in integrated optics," J. Opt. Soc. Am. A, Vol. 18, No. 11, 2865-2875, 2001.

    23. Gralak, B., S. Enoch, and G. Tayeb, "From scattering or impedance matrices to Bloch modes of photonic crystals," J. Opt. Soc. Am. A, Vol. 19, No. 8, 1547-1554, 2002.

    24. Li, L., "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A, Vol. 20, No. 4, 655-660, 2003.

    25. Kim, H., I.-M. Lee, and B. Lee, "Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis," J. Opt. Soc. Am. A, Vol. 24, No. 8, 2313-2327, 2007.

    26. Tervo, J., M. Kuittinen, P. Vahimaa, J. Turunen, T. Aalto, P. Heimala, and M. Leppihalme, "Efficient bragg waveguide-grating analysis by quasi-rigorous approach based on redheffer's star product," Optics Commun., Vol. 198, 265-272, 2001.

    27. Green, A. A., E. Istrate, and E. H. Sargent, "Efficient design and optimization of photonic crystal waveguides and couplers: The interface diffraction method," Optics Express, Vol. 13, No. 19, 7304-7318, 2005.

    28. Lalanne, P. and E. Silberstein, "Fourier-modal methods applied to waveguide computational problems," Opt. Lett., Vol. 25, No. 15, 1092-1094, 2000.

    29. Mingaleev, S. F. and K. Busch, "Scattering matrix approach to large-scale photonic crystal circuits," Opt. Lett., Vol. 28, No. 8, 619-621, 2003.

    30. Whittaker, D. M. and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B, Vol. 60, No. 4, 2610-2618, 1999.

    31. Li, Z.-Y. and K. M. Ho, "Light propagation in semi-infinite photonic crystals and related waveguide structures," Phys. Rev. B, Vol. 68, 155-101, 2003.

    32. Liscidini, M., D. Gerace, L. C. Andreani, and J. E. Sipe, "Scattering-matrix analysis of periodically patterned multilayers with asymmetric unit cells and birefringent media," Phys. Rev. B, Vol. 77, 035324, 2008.

    33. Moharam, M. G. and A. B. Greenwell, "Efficient rigorous calculations of power flow in grating coupled surface-emitting devices," Proc. SPIE, Vol. 5456, 57-67, 2004.

    34. Freundorfer, A. P., "Optical vector network analyzer as a reflectometer," Appl. Opt., Vol. 33, No. 16, 3559-3561, 1994.

    35. Yee, K. S., "Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media," IEEE Trans. on Antennas and Propagation, Vol. 14, No. 8, 302-307, 1966.

    36. Schneider, J. B. and R. J. Kruhlak, "Dispersion of homogeneous and inhomogeneous waves in the yee finite-difference time-domain grid," IEEE Trans. on Microwave Theory and Techniques, Vol. 49, No. 2, 280-287, 2001.

    37. Rumpf, R. C. Design and optimization of nano-optical elements by coupling fabrication to optical behavior, 60-84 Ph.D. Dissertation, University of Central Florida, 2006.

    38. Li, L., "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A, Vol. 13, No. 9, 1870-1876, 1996.

    39. Li, L., "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am A, Vol. 14, No. 10, 2758-2767, 1997.

    40. Lalanne, P., "Improved formulation of the coupled-wave method for two-dimensional gratings," J. Opt. Soc. Am. A, Vol. 14, No. 7, 1592-1598, 1997.

    41. Götz, P., T. Schuster, K. Frenner, S. Rafler, and W. Osten, "Normal vector method for the RCWA with automated vector field generation," Optics Express, Vol. 16, No. 22, 17295-17301, 2008.

    42. Redheffer, R., Difference equations and functional equations in transmission-line theory, Modern Mathematics for the Engineer, Vol. 12, 282-337, E. F. Beckenbach, ed., McGraw-Hill, New York, 1961.

    43. Smith, D. R. and J. B. Pendry, "Homogenization of metamaterials by field averaging (invited paper)," J. Opt. Soc. Am. B, Vol. 23, No. 3, 391-403, 2006.

    44. Smith, D. R., S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B, Vol. 65, 195104, 2002.

    45. Chen, X., T. M. Grzegorczyk, B.-I. Wu, J. Pachaco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E, Vol. 70, 016608, 2004.

    46. Smith, D. R., D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E, Vol. 71, 036617, 2005.