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

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

http://dx.doi.org/10.1364/JOSAA.21.000393

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### Abstract

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

**Citation**

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)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-3-393

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### References

- 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).
- 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).
- 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).
- 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).
- 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).
- 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).
- B.-J. Li and P.-L. Liu, “Numerical analysis of microdisk lasers with rough boundaries,” IEEE J. Quantum Electron. 35, 791–795 (1997).
- 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).
- J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A Pure Appl. Opt. 5, 53–60 (2003).
- 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).
- A. W. Poon, F. Courvoisier, and R. K. Chang, “Multimode resonances in square-shaped optical microcavities,” Opt. Lett. 26, 632–634 (2001).
- 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).
- D. Colton and R. Kress, Integral Equations Methods in Scattering Theory (Wiley, New York, 1983).
- N. Morita, N. Kumagai, and J. R. Mautz, Integral Equations Methods for Electromagnetics (Artech House, London, 1990).
- 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).
- 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).
- H. A. Shenk, “Improved integral formulation for acoustic radiation problems,” J. Acoust. Soc. Am. 44, 41–68 (1968).
- 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).
- 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).
- C. Muller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).
- 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).
- V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 86, 414–439 (1990).
- R. F. Harrington, Field Computation by Moment Methods (Krieger, Malabar, Fla., 1968).
- K. E. Atkinson, The Numerical Solution of Boundary Integral Equations (Cambridge U. Press, Cambridge, UK, 1997).
- 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).
- 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).
- 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).
- 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).
- 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).
- 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).
- K. Atkinson and A. Bogomolny, “The discrete Galerkin method for integral equations,” Math. Comput. 48, 595–616 (1987).
- 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).
- 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).
- M. Fujita and T. Baba, “Microgear laser,” Appl. Phys. Lett. 80, 2051–2053 (2002).

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