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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 21, Iss. 3 — Mar. 1, 2004
  • pp: 393–402

Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization

Svetlana V. Boriskina, Phillip Sewell, Trevor M. Benson, and Alexander I. Nosich  »View Author Affiliations

JOSA A, Vol. 21, Issue 3, pp. 393-402 (2004)

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A fast and accurate method is developed to compute the natural frequencies and scattering characteristics of arbitrary-shape two-dimensional dielectric resonators. The problem is formulated in terms of a uniquely solvable set of second-kind boundary integral equations and discretized by the Galerkin method with angular exponents as global test and trial functions. The log-singular term is extracted from one of the kernels, and closed-form expressions are derived for the main parts of all the integral operators. The resulting discrete scheme has a very high convergence rate. The method is used in the simulation of several optical microcavities for modern dense wavelength-division-multiplexed systems.

© 2004 Optical Society of America

OCIS Codes
(000.3860) General : Mathematical methods in physics
(140.4780) Lasers and laser optics : Optical resonators
(230.5750) Optical devices : Resonators
(260.2110) Physical optics : Electromagnetic optics
(290.0290) Scattering : Scattering

Svetlana V. Boriskina, Phillip Sewell, Trevor M. Benson, and Alexander I. Nosich, "Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization," J. Opt. Soc. Am. A 21, 393-402 (2004)

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  1. C. Gmachl, F. Cappasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonances,” Science 280, 1556–1564 (1998).
  2. S. V. Boriskina and A. I. Nosich, “Radiation and absorption losses of the whispering-gallery-mode dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 47, 224–231 (1999).
  3. P. P. Absil, J. V. Hryniewicz, B. E. Little, F. G. Johnson, K. J. Ritter, and P.-T. Ho, “Vertically coupled microring resonators using polymer wafer bonding,” IEEE Photon. Technol. Lett. 13, 49–51 (2001).
  4. M. K. Chin, D. Y. Chu, and S.-T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302–3307 (1994).
  5. M. Fujita, A. Sakai, and T. Baba, “Ultrasmall and ultralow threshold GaInAsP–InP microdisk injection lasers: design, fabrication, lasing characteristics, and spontaneous emission factor,” IEEE J. Sel. Top. Quantum Electron. 15, 673–681 (1999).
  6. S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Tuning of elliptic whispering-gallery-mode microdisk waveguide filters,” J. Lightwave Technol. 21, 1987–1995 (2003).
  7. B.-J. Li and P.-L. Liu, “Numerical analysis of microdisk lasers with rough boundaries,” IEEE J. Quantum Electron. 35, 791–795 (1997).
  8. M. Fujita and T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253–1258 (2001).
  9. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A Pure Appl. Opt. 5, 53–60 (2003).
  10. Y.-Z. Huang, W.-H. Guo, and Q.-M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron. 37, 100–107 (2001).
  11. A. W. Poon, F. Courvoisier, and R. K. Chang, “Multimode resonances in square-shaped optical microcavities,” Opt. Lett. 26, 632–634 (2001).
  12. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, and J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add–drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
  13. D. Colton and R. Kress, Integral Equations Methods in Scattering Theory (Wiley, New York, 1983).
  14. N. Morita, N. Kumagai, and J. R. Mautz, Integral Equations Methods for Electromagnetics (Artech House, London, 1990).
  15. A. J. Burton and G. F. Miller, “The application of integral equation methods to the numerical solution of some exterior boundary-value problem,” Proc. R. Soc. London Ser. A 323, 201–210 (1971).
  16. S. Amini and S. M. Kirkup, “Solution of Helmholtz equation in the exterior domain by elementary boundary integral methods,” J. Comput. Phys. 118, 208–221 (1995).
  17. H. A. Shenk, “Improved integral formulation for acoustic radiation problems,” J. Acoust. Soc. Am. 44, 41–68 (1968).
  18. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
  19. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, and A. F. Peterson, “Modeling considerations for rigorous boundary element method analysis of diffractive optical elements,” J. Opt. Soc. Am. A 18, 1495–1506 (2001).
  20. C. Muller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).
  21. V. V. Soloduhov and E. N. Vasiliev, “Diffraction of a plane electromagnetic wave by a dielectric cylinder of arbitrary cross section,” Sov. Phys. Tech. Phys. 15, 32–36 (1970).
  22. V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 86, 414–439 (1990).
  23. R. F. Harrington, Field Computation by Moment Methods (Krieger, Malabar, Fla., 1968).
  24. K. E. Atkinson, The Numerical Solution of Boundary Integral Equations (Cambridge U. Press, Cambridge, UK, 1997).
  25. A. Hoekstra, J. Rahola, and P. Sloot, “Accuracy of internal fields in volume integral equation simulations of light scattering,” Appl. Opt. 37, 8482–8497 (1998).
  26. G. L. Hower, R. G. Olsen, J. D. Earls, and J. B. Schneider, “Inaccuracies in numerical calculation of scattering near natural frequencies of penetrable objects,” IEEE Trans. Antennas Propag. 41, 982–986 (1993).
  27. J. Saranen and G. Vainikko, “Trigonometric collocation method with product integration for boundary integral equations on closed curves,” SIAM J. Numer. Anal. 33, 1577–1596 (1996).
  28. M. Paulus and O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A 18, 854–861 (2001).
  29. A. I. Nosich, “The method of analytical regularization in wave-scattering and eigenvalue problems: foundations and review of solutions,” IEEE Antennas Propag. Mag. 41, 34–49 (1999).
  30. U. Lamp, K.-T. Schleicher, and W. L. Wendland, “The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations,” Numer. Math. 15, 15–38 (1985).
  31. K. Atkinson and A. Bogomolny, “The discrete Galerkin method for integral equations,” Math. Comput. 48, 595–616 (1987).
  32. A. V. Boriskin and A. I. Nosich, “Whispering-gallery and Luneburg-lens effects in a beam-fed circularly layered dielectric cylinder,” IEEE Trans. Antennas Propag. 50, 1245–1249 (2002).
  33. H. Cao, J. Y. Xu, W. H. Xiang, Y. Ma, S.-H. Chang, S. T. Ho, and G. S. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. 76, 3519–3521 (2000).
  34. M. Fujita and T. Baba, “Microgear laser,” Appl. Phys. Lett. 80, 2051–2053 (2002).

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