volume | PIER Journals

Abstract

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.

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.
doi:10.1080/02726340290084166 Google Scholar

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.
doi:10.1109/JLT.2002.800350 Google Scholar

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.
doi:10.1364/JOSAA.12.001068 Google Scholar

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.
doi:10.1364/JOSAA.12.001077 Google Scholar

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.
doi:10.1364/JOSA.62.000502 Google Scholar

6. Pendry, J. B., "Photonic band structures," J. Modern Optics, Vol. 41, No. 2, 209-229, 1994.
doi:10.1080/09500349414550281 Google Scholar

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.
doi:10.1103/PhysRevE.67.046607 Google Scholar

8. Matthews, Jr., E. W., "The use of scattering matrices in microwave circuits," IRE Trans. on Microwave Theory and Techniques, 21-26, 1955.
doi:10.1109/TMTT.1955.1124941 Google Scholar

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

10. Kurokawa, K., "Power waves and the scattering matrix," IEEE Trans. on Microwave Theory and Techniques, 194-202, 1965.
doi:10.1109/TMTT.1965.1125964 Google Scholar

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

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

13. Rizzi, P. A., Microwave Engineering Passive Circuits, 1st Ed., 168-176, Prentice Hall, 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.
doi:10.1080/09500340500407701 Google Scholar

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.
doi:10.1364/JOSAA.11.002829 Google Scholar

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.
doi:10.1364/JOSAA.13.001024 Google Scholar

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.
doi:10.1364/JOSAA.12.001134 Google Scholar

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.
doi:10.1063/1.1587011 Google Scholar

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.
doi:10.1364/JOSAA.5.001863 Google Scholar

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.
doi:10.1364/JOSAA.13.001024 Google Scholar

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.
doi:10.1364/JOSAA.18.002865 Google Scholar

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.
doi:10.1364/JOSAA.19.001547 Google Scholar

24. Li, L., "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A, Vol. 20, No. 4, 655-660, 2003.
doi:10.1364/JOSAA.20.000655 Google Scholar

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.
doi:10.1364/JOSAA.24.002313 Google Scholar

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.
doi:10.1016/S0030-4018(01)01530-9 Google Scholar

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.
doi:10.1364/OPEX.13.007304 Google Scholar

28. Lalanne, P. and E. Silberstein, "Fourier-modal methods applied to waveguide computational problems," Opt. Lett., Vol. 25, No. 15, 1092-1094, 2000.
doi:10.1364/OL.25.001092 Google Scholar

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.
doi:10.1364/OL.28.000619 Google Scholar

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.
doi:10.1103/PhysRevB.60.2610 Google Scholar

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. Google Scholar

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.
doi:10.1103/PhysRevB.77.035324 Google Scholar

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.
doi:10.1117/12.549477 Google Scholar

34. Freundorfer, A. P., "Optical vector network analyzer as a reflectometer," Appl. Opt., Vol. 33, No. 16, 3559-3561, 1994.
doi:10.1364/AO.33.003559 Google Scholar

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. Google Scholar

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.
doi:10.1109/22.903087 Google Scholar

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.
doi:10.1364/JOSAA.13.001870 Google Scholar

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.
doi:10.1364/JOSAA.14.002758 Google Scholar

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.
doi:10.1364/JOSAA.14.001592 Google Scholar

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.
doi:10.1364/OE.16.017295 Google Scholar

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.
doi:10.1364/JOSAB.23.000391 Google Scholar

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.
doi:10.1103/PhysRevB.65.195104 Google Scholar

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.
doi:10.1103/PhysRevE.70.016608 Google Scholar

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.
doi:10.1103/PhysRevE.71.036617 Google Scholar

Professor Raymond C. Rumpf