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

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  1. C. Gmachl, F. Cappasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, A. Y. Cho, “High-power directional emission from microlasers with chaotic resonances,” Science 280, 1556–1564 (1998).
    [CrossRef] [PubMed]
  2. S. V. Boriskina, 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).
    [CrossRef]
  3. P. P. Absil, J. V. Hryniewicz, B. E. Little, F. G. Johnson, K. J. Ritter, P.-T. Ho, “Vertically coupled microring resonators using polymer wafer bonding,” IEEE Photon. Technol. Lett. 13, 49–51 (2001).
    [CrossRef]
  4. M. K. Chin, D. Y. Chu, 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).
    [CrossRef]
  5. M. Fujita, A. Sakai, 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).
    [CrossRef]
  6. S. V. Boriskina, T. M. Benson, P. Sewell, A. I. Nosich, “Tuning of elliptic whispering-gallery-mode microdisk waveguide filters,” J. Lightwave Technol. 21, 1987–1995 (2003).
    [CrossRef]
  7. B.-J. Li, P.-L. Liu, “Numerical analysis of microdisk lasers with rough boundaries,” IEEE J. Quantum Electron. 35, 791–795 (1997).
  8. M. Fujita, T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253–1258 (2001).
    [CrossRef]
  9. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A Pure Appl. Opt. 5, 53–60 (2003).
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    [CrossRef]
  11. A. W. Poon, F. Courvoisier, R. K. Chang, “Multimode resonances in square-shaped optical microcavities,” Opt. Lett. 26, 632–634 (2001).
    [CrossRef]
  12. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add–drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
    [CrossRef]
  13. D. Colton, R. Kress, Integral Equations Methods in Scattering Theory (Wiley, New York, 1983).
  14. N. Morita, N. Kumagai, J. R. Mautz, Integral Equations Methods for Electromagnetics (Artech House, London, 1990).
  15. A. J. Burton, 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).
    [CrossRef]
  16. S. Amini, S. M. Kirkup, “Solution of Helmholtz equation in the exterior domain by elementary boundary integral methods,” J. Comput. Phys. 118, 208–221 (1995).
    [CrossRef]
  17. H. A. Shenk, “Improved integral formulation for acoustic radiation problems,” J. Acoust. Soc. Am. 44, 41–68 (1968).
    [CrossRef]
  18. D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
    [CrossRef]
  19. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, A. F. Peterson, “Modeling considerations for rigorous boundary element method analysis of diffractive optical elements,” J. Opt. Soc. Am. A 18, 1495–1506 (2001).
    [CrossRef]
  20. C. Muller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).
  21. V. V. Soloduhov, 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).
    [CrossRef]
  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, P. Sloot, “Accuracy of internal fields in volume integral equation simulations of light scattering,” Appl. Opt. 37, 8482–8497 (1998).
    [CrossRef]
  26. G. L. Hower, R. G. Olsen, J. D. Earls, J. B. Schneider, “Inaccuracies in numerical calculation of scattering near natural frequencies of penetrable objects,” IEEE Trans. Antennas Propag. 41, 982–986 (1993).
    [CrossRef]
  27. J. Saranen, G. Vainikko, “Trigonometric collocation method with product integration for boundary integral equations on closed curves,” SIAM J. Numer. Anal. 33, 1577–1596 (1996).
    [CrossRef]
  28. M. Paulus, 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).
    [CrossRef]
  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).
    [CrossRef]
  30. U. Lamp, K.-T. Schleicher, W. L. Wendland, “The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations,” Numer. Math. 15, 15–38 (1985).
    [CrossRef]
  31. K. Atkinson, A. Bogomolny, “The discrete Galerkin method for integral equations,” Math. Comput. 48, 595–616 (1987).
    [CrossRef]
  32. A. V. Boriskin, 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).
    [CrossRef]
  33. H. Cao, J. Y. Xu, W. H. Xiang, Y. Ma, S.-H. Chang, S. T. Ho, G. S. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. 76, 3519–3521 (2000).
    [CrossRef]
  34. M. Fujita, T. Baba, “Microgear laser,” Appl. Phys. Lett. 80, 2051–2053 (2002).
    [CrossRef]

2003 (2)

2002 (2)

A. V. Boriskin, 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).
[CrossRef]

M. Fujita, T. Baba, “Microgear laser,” Appl. Phys. Lett. 80, 2051–2053 (2002).
[CrossRef]

2001 (6)

M. Paulus, 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).
[CrossRef]

Y.-Z. Huang, W.-H. Guo, 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).
[CrossRef]

A. W. Poon, F. Courvoisier, R. K. Chang, “Multimode resonances in square-shaped optical microcavities,” Opt. Lett. 26, 632–634 (2001).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, A. F. Peterson, “Modeling considerations for rigorous boundary element method analysis of diffractive optical elements,” J. Opt. Soc. Am. A 18, 1495–1506 (2001).
[CrossRef]

M. Fujita, T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253–1258 (2001).
[CrossRef]

P. P. Absil, J. V. Hryniewicz, B. E. Little, F. G. Johnson, K. J. Ritter, P.-T. Ho, “Vertically coupled microring resonators using polymer wafer bonding,” IEEE Photon. Technol. Lett. 13, 49–51 (2001).
[CrossRef]

2000 (1)

H. Cao, J. Y. Xu, W. H. Xiang, Y. Ma, S.-H. Chang, S. T. Ho, G. S. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. 76, 3519–3521 (2000).
[CrossRef]

1999 (4)

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).
[CrossRef]

S. V. Boriskina, 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).
[CrossRef]

M. Fujita, A. Sakai, 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).
[CrossRef]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add–drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

1998 (2)

C. Gmachl, F. Cappasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, A. Y. Cho, “High-power directional emission from microlasers with chaotic resonances,” Science 280, 1556–1564 (1998).
[CrossRef] [PubMed]

A. Hoekstra, J. Rahola, P. Sloot, “Accuracy of internal fields in volume integral equation simulations of light scattering,” Appl. Opt. 37, 8482–8497 (1998).
[CrossRef]

1997 (2)

D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

B.-J. Li, P.-L. Liu, “Numerical analysis of microdisk lasers with rough boundaries,” IEEE J. Quantum Electron. 35, 791–795 (1997).

1996 (1)

J. Saranen, G. Vainikko, “Trigonometric collocation method with product integration for boundary integral equations on closed curves,” SIAM J. Numer. Anal. 33, 1577–1596 (1996).
[CrossRef]

1995 (1)

S. Amini, S. M. Kirkup, “Solution of Helmholtz equation in the exterior domain by elementary boundary integral methods,” J. Comput. Phys. 118, 208–221 (1995).
[CrossRef]

1994 (1)

M. K. Chin, D. Y. Chu, 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).
[CrossRef]

1993 (1)

G. L. Hower, R. G. Olsen, J. D. Earls, J. B. Schneider, “Inaccuracies in numerical calculation of scattering near natural frequencies of penetrable objects,” IEEE Trans. Antennas Propag. 41, 982–986 (1993).
[CrossRef]

1990 (1)

V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 86, 414–439 (1990).
[CrossRef]

1987 (1)

K. Atkinson, A. Bogomolny, “The discrete Galerkin method for integral equations,” Math. Comput. 48, 595–616 (1987).
[CrossRef]

1985 (1)

U. Lamp, K.-T. Schleicher, W. L. Wendland, “The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations,” Numer. Math. 15, 15–38 (1985).
[CrossRef]

1971 (1)

A. J. Burton, 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).
[CrossRef]

1970 (1)

V. V. Soloduhov, 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).

1968 (1)

H. A. Shenk, “Improved integral formulation for acoustic radiation problems,” J. Acoust. Soc. Am. 44, 41–68 (1968).
[CrossRef]

Absil, P. P.

P. P. Absil, J. V. Hryniewicz, B. E. Little, F. G. Johnson, K. J. Ritter, P.-T. Ho, “Vertically coupled microring resonators using polymer wafer bonding,” IEEE Photon. Technol. Lett. 13, 49–51 (2001).
[CrossRef]

Amini, S.

S. Amini, S. M. Kirkup, “Solution of Helmholtz equation in the exterior domain by elementary boundary integral methods,” J. Comput. Phys. 118, 208–221 (1995).
[CrossRef]

Atkinson, K.

K. Atkinson, A. Bogomolny, “The discrete Galerkin method for integral equations,” Math. Comput. 48, 595–616 (1987).
[CrossRef]

Atkinson, K. E.

K. E. Atkinson, The Numerical Solution of Boundary Integral Equations (Cambridge U. Press, Cambridge, UK, 1997).

Baba, T.

M. Fujita, T. Baba, “Microgear laser,” Appl. Phys. Lett. 80, 2051–2053 (2002).
[CrossRef]

M. Fujita, T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253–1258 (2001).
[CrossRef]

M. Fujita, A. Sakai, 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).
[CrossRef]

Bendickson, J. M.

Benson, T. M.

Bogomolny, A.

K. Atkinson, A. Bogomolny, “The discrete Galerkin method for integral equations,” Math. Comput. 48, 595–616 (1987).
[CrossRef]

Boriskin, A. V.

A. V. Boriskin, 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).
[CrossRef]

Boriskina, S. V.

S. V. Boriskina, T. M. Benson, P. Sewell, A. I. Nosich, “Tuning of elliptic whispering-gallery-mode microdisk waveguide filters,” J. Lightwave Technol. 21, 1987–1995 (2003).
[CrossRef]

S. V. Boriskina, 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).
[CrossRef]

Burton, A. J.

A. J. Burton, 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).
[CrossRef]

Cao, H.

H. Cao, J. Y. Xu, W. H. Xiang, Y. Ma, S.-H. Chang, S. T. Ho, G. S. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. 76, 3519–3521 (2000).
[CrossRef]

Cappasso, F.

C. Gmachl, F. Cappasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, A. Y. Cho, “High-power directional emission from microlasers with chaotic resonances,” Science 280, 1556–1564 (1998).
[CrossRef] [PubMed]

Chang, R. K.

Chang, S.-H.

H. Cao, J. Y. Xu, W. H. Xiang, Y. Ma, S.-H. Chang, S. T. Ho, G. S. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. 76, 3519–3521 (2000).
[CrossRef]

Chin, M. K.

M. K. Chin, D. Y. Chu, 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).
[CrossRef]

Cho, A. Y.

C. Gmachl, F. Cappasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, A. Y. Cho, “High-power directional emission from microlasers with chaotic resonances,” Science 280, 1556–1564 (1998).
[CrossRef] [PubMed]

Chu, D. Y.

M. K. Chin, D. Y. Chu, 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).
[CrossRef]

Colton, D.

D. Colton, R. Kress, Integral Equations Methods in Scattering Theory (Wiley, New York, 1983).

Courvoisier, F.

Earls, J. D.

G. L. Hower, R. G. Olsen, J. D. Earls, J. B. Schneider, “Inaccuracies in numerical calculation of scattering near natural frequencies of penetrable objects,” IEEE Trans. Antennas Propag. 41, 982–986 (1993).
[CrossRef]

Faist, J.

C. Gmachl, F. Cappasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, A. Y. Cho, “High-power directional emission from microlasers with chaotic resonances,” Science 280, 1556–1564 (1998).
[CrossRef] [PubMed]

Fan, S.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add–drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

Fujita, M.

M. Fujita, T. Baba, “Microgear laser,” Appl. Phys. Lett. 80, 2051–2053 (2002).
[CrossRef]

M. Fujita, T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253–1258 (2001).
[CrossRef]

M. Fujita, A. Sakai, 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).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Gmachl, C.

C. Gmachl, F. Cappasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, A. Y. Cho, “High-power directional emission from microlasers with chaotic resonances,” Science 280, 1556–1564 (1998).
[CrossRef] [PubMed]

Guo, W.-H.

Y.-Z. Huang, W.-H. Guo, 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).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (Krieger, Malabar, Fla., 1968).

Haus, H. A.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add–drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

Ho, P.-T.

P. P. Absil, J. V. Hryniewicz, B. E. Little, F. G. Johnson, K. J. Ritter, P.-T. Ho, “Vertically coupled microring resonators using polymer wafer bonding,” IEEE Photon. Technol. Lett. 13, 49–51 (2001).
[CrossRef]

Ho, S. T.

H. Cao, J. Y. Xu, W. H. Xiang, Y. Ma, S.-H. Chang, S. T. Ho, G. S. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. 76, 3519–3521 (2000).
[CrossRef]

Ho, S.-T.

M. K. Chin, D. Y. Chu, 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).
[CrossRef]

Hoekstra, A.

Hower, G. L.

G. L. Hower, R. G. Olsen, J. D. Earls, J. B. Schneider, “Inaccuracies in numerical calculation of scattering near natural frequencies of penetrable objects,” IEEE Trans. Antennas Propag. 41, 982–986 (1993).
[CrossRef]

Hryniewicz, J. V.

P. P. Absil, J. V. Hryniewicz, B. E. Little, F. G. Johnson, K. J. Ritter, P.-T. Ho, “Vertically coupled microring resonators using polymer wafer bonding,” IEEE Photon. Technol. Lett. 13, 49–51 (2001).
[CrossRef]

Huang, Y.-Z.

Y.-Z. Huang, W.-H. Guo, 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).
[CrossRef]

Joannopoulos, J. D.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add–drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

Johnson, F. G.

P. P. Absil, J. V. Hryniewicz, B. E. Little, F. G. Johnson, K. J. Ritter, P.-T. Ho, “Vertically coupled microring resonators using polymer wafer bonding,” IEEE Photon. Technol. Lett. 13, 49–51 (2001).
[CrossRef]

Khan, M. J.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add–drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

Kirkup, S. M.

S. Amini, S. M. Kirkup, “Solution of Helmholtz equation in the exterior domain by elementary boundary integral methods,” J. Comput. Phys. 118, 208–221 (1995).
[CrossRef]

Kress, R.

D. Colton, R. Kress, Integral Equations Methods in Scattering Theory (Wiley, New York, 1983).

Kumagai, N.

N. Morita, N. Kumagai, J. R. Mautz, Integral Equations Methods for Electromagnetics (Artech House, London, 1990).

Lamp, U.

U. Lamp, K.-T. Schleicher, W. L. Wendland, “The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations,” Numer. Math. 15, 15–38 (1985).
[CrossRef]

Li, B.-J.

B.-J. Li, P.-L. Liu, “Numerical analysis of microdisk lasers with rough boundaries,” IEEE J. Quantum Electron. 35, 791–795 (1997).

Little, B. E.

P. P. Absil, J. V. Hryniewicz, B. E. Little, F. G. Johnson, K. J. Ritter, P.-T. Ho, “Vertically coupled microring resonators using polymer wafer bonding,” IEEE Photon. Technol. Lett. 13, 49–51 (2001).
[CrossRef]

Liu, P.-L.

B.-J. Li, P.-L. Liu, “Numerical analysis of microdisk lasers with rough boundaries,” IEEE J. Quantum Electron. 35, 791–795 (1997).

Ma, Y.

H. Cao, J. Y. Xu, W. H. Xiang, Y. Ma, S.-H. Chang, S. T. Ho, G. S. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. 76, 3519–3521 (2000).
[CrossRef]

Mait, J. N.

Manolatou, C.

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add–drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

Martin, O. J. F.

Mautz, J. R.

N. Morita, N. Kumagai, J. R. Mautz, Integral Equations Methods for Electromagnetics (Artech House, London, 1990).

Miller, G. F.

A. J. Burton, 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).
[CrossRef]

Mirotznik, M. S.

Morita, N.

N. Morita, N. Kumagai, J. R. Mautz, Integral Equations Methods for Electromagnetics (Artech House, London, 1990).

Muller, C.

C. Muller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, Berlin, 1969).

Narimanov, E. E.

C. Gmachl, F. Cappasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, A. Y. Cho, “High-power directional emission from microlasers with chaotic resonances,” Science 280, 1556–1564 (1998).
[CrossRef] [PubMed]

Nockel, J. U.

C. Gmachl, F. Cappasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, A. Y. Cho, “High-power directional emission from microlasers with chaotic resonances,” Science 280, 1556–1564 (1998).
[CrossRef] [PubMed]

Nosich, A. I.

S. V. Boriskina, T. M. Benson, P. Sewell, A. I. Nosich, “Tuning of elliptic whispering-gallery-mode microdisk waveguide filters,” J. Lightwave Technol. 21, 1987–1995 (2003).
[CrossRef]

A. V. Boriskin, 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).
[CrossRef]

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).
[CrossRef]

S. V. Boriskina, 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).
[CrossRef]

Olsen, R. G.

G. L. Hower, R. G. Olsen, J. D. Earls, J. B. Schneider, “Inaccuracies in numerical calculation of scattering near natural frequencies of penetrable objects,” IEEE Trans. Antennas Propag. 41, 982–986 (1993).
[CrossRef]

Paulus, M.

Peterson, A. F.

Poon, A. W.

Prather, D. W.

Rahola, J.

Ritter, K. J.

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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

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V. V. Soloduhov, 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).

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C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add–drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

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[CrossRef]

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U. Lamp, K.-T. Schleicher, W. L. Wendland, “The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations,” Numer. Math. 15, 15–38 (1985).
[CrossRef]

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J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A Pure Appl. Opt. 5, 53–60 (2003).
[CrossRef]

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H. Cao, J. Y. Xu, W. H. Xiang, Y. Ma, S.-H. Chang, S. T. Ho, G. S. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. 76, 3519–3521 (2000).
[CrossRef]

Xu, J. Y.

H. Cao, J. Y. Xu, W. H. Xiang, Y. Ma, S.-H. Chang, S. T. Ho, G. S. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. 76, 3519–3521 (2000).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

H. Cao, J. Y. Xu, W. H. Xiang, Y. Ma, S.-H. Chang, S. T. Ho, G. S. Solomon, “Optically pumped InAs quantum dot microdisk lasers,” Appl. Phys. Lett. 76, 3519–3521 (2000).
[CrossRef]

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[CrossRef]

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IEEE J. Quantum Electron. (4)

B.-J. Li, P.-L. Liu, “Numerical analysis of microdisk lasers with rough boundaries,” IEEE J. Quantum Electron. 35, 791–795 (1997).

M. Fujita, T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253–1258 (2001).
[CrossRef]

Y.-Z. Huang, W.-H. Guo, 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).
[CrossRef]

C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, J. D. Joannopoulos, “Coupling of modes analysis of resonant channel add–drop filters,” IEEE J. Quantum Electron. 35, 1322–1331 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Fujita, A. Sakai, 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).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

P. P. Absil, J. V. Hryniewicz, B. E. Little, F. G. Johnson, K. J. Ritter, P.-T. Ho, “Vertically coupled microring resonators using polymer wafer bonding,” IEEE Photon. Technol. Lett. 13, 49–51 (2001).
[CrossRef]

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A. V. Boriskin, 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).
[CrossRef]

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H. A. Shenk, “Improved integral formulation for acoustic radiation problems,” J. Acoust. Soc. Am. 44, 41–68 (1968).
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M. K. Chin, D. Y. Chu, 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).
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[CrossRef]

J. Lightwave Technol. (1)

J. Opt. A Pure Appl. Opt. (1)

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A Pure Appl. Opt. 5, 53–60 (2003).
[CrossRef]

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K. Atkinson, A. Bogomolny, “The discrete Galerkin method for integral equations,” Math. Comput. 48, 595–616 (1987).
[CrossRef]

Numer. Math. (1)

U. Lamp, K.-T. Schleicher, W. L. Wendland, “The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations,” Numer. Math. 15, 15–38 (1985).
[CrossRef]

Opt. Lett. (1)

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[CrossRef]

Science (1)

C. Gmachl, F. Cappasso, E. E. Narimanov, J. U. Nockel, A. D. Stone, J. Faist, D. L. Sivco, A. Y. Cho, “High-power directional emission from microlasers with chaotic resonances,” Science 280, 1556–1564 (1998).
[CrossRef] [PubMed]

SIAM J. Numer. Anal. (1)

J. Saranen, G. Vainikko, “Trigonometric collocation method with product integration for boundary integral equations on closed curves,” SIAM J. Numer. Anal. 33, 1577–1596 (1996).
[CrossRef]

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V. V. Soloduhov, 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).

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Figures (11)

Fig. 1
Fig. 1

Geometry of the problem. Shaded area S is the cross section of an optical microcavity with parameters εc and μc, enclosed by an arbitrary smooth simple contour LS. A canonical circular contour of radius a and global and local coordinate systems are also shown.

Fig. 2
Fig. 2

Two-dimensional model of the microdisk with the effective refractive-index approximation for TE and TM polarizations.

Fig. 3
Fig. 3

Computational errors versus truncated matrix size for a TM-polarized plane-wave scattering from a superelliptic microcavity with parameters μ=1.1 and εc=10.24+0.001i: (a) the error in and off two WG-mode resonances of an elliptic (ν=1) cavity, kac(WG4,1)=1.845 and kac(WG11,1)=4.279; (b) the error for three superelliptic cavities (kac=4.3) for various values of the corner-sharpness parameter (corners are shown in the inset).

Fig. 4
Fig. 4

Validation of the MIEs solution: comparison of results generated by the MIEs with the analytical solution and experimental data. The inset shows an emission spectrum of an optically pumped GaAs microdisk 3 µm in diameter and 90 nm in thickness [Ref. 33, Fig. 2(b)]. The effective refractive index used in the 2-D computations was taken as nc=2.462.

Fig. 5
Fig. 5

Normalized wavelengths and Q factors of WG5,1± modes of a microgear cavity (ac=0.8 µm, neff=2.63, and ν=10) as a function of relative perturbation amplitude δ. The results are compared with those obtained by the FDTD technique (Ref. 7, Figs. 5 and 6) (circles). Solid curves (filled circles) and dashed curves (open circles) are the results for the enhanced and suppressed modes, respectively.

Fig. 6
Fig. 6

Wavelength dependence of the total power scattered from an elliptical microcavity (ac=0.95 µm, εc=10.24+0.001i, and μ=1.1) excited by a TM-polarized CSP beam. The FSR of the first-radial-order WG-mode resonances in the vicinity of λ=1.55 µm is 118 nm. The inset shows a schematic of the CSP beam (kb=10, β=160°, kx0=kacμ+kb+1, and ky0=0) grazing the rim of the microdisk.

Fig. 7
Fig. 7

Near-field intensity patterns (20% contours) of WG modes in the elliptical microcavity. Corresponding resonances in the normalized scattered power in Fig. 6 are marked as a and b. (a) First-radial-order WG11,1 mode (λ=1.395 µm, and Q=1.01×104), (b) second-radial-order WG7,2 mode (λ=1.497 µm and Q=2.91×102).

Fig. 8
Fig. 8

Wavelength dependence of the total power scattered from a recetrack microcavity (ac=1.2 µm, εc=10.24+0.001i, e=0.15, and μ=1) excited by a TM-polarized CSP beam. The FSR of the first-radial-order WG-mode resonances in the vicinity of λ=1.55 µm is 106 nm. The inset shows a schematic of the incident CSP beam (kb=10, β=270°, kx0=kac/2, and ky0=kac+kb+0.1).

Fig. 9
Fig. 9

Near-field intensity patterns (20% contours) of (a) WG and (b) bow-tie modes in the racetrack microcavity. Corresponding resonances in the normalized scattered power in Fig. 8 are marked as a and b. (a) First-radial-order WG12,1 mode (λ=1.587 µm and Q=1.32×103), (b) bow-tie WG7,2 mode (λ=1.357 µm and Q=2.66×102).

Fig. 10
Fig. 10

Wavelength dependence of the total power scattered from a square microcavity (ac=1.45 µm, εc=10.24+0.001i, ν=10, and μ=1) excited by a TM-polarized CSP beam. The FSR in the vicinity of λ=1.55 µm is 178 nm. The inset shows a schematic of the incident CSP beam (kb=10, β=135°, kx0=kac+kb+0.1, and ky0=0).

Fig. 11
Fig. 11

Near-field intensity patterns (20% contours) of the modes in the square microcavity corresponding to the resonances in the normalized scattered power marked as a and b in Fig. 10. (a) (λ=1.33 µm and Q=3.48×103), (b) (λ=1.425 µm and Q=5.41×102).

Equations (36)

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U(r)=U0(r)+Ue(r),rSUc(r),rS,
(Ue+U0)|LS=Uc|LS,  1αcUcnLS=1αeUen+U0nLS;
Uc(r)=LSUc(r)Gc(r, r)n-Uc(r)nGc(r, r)dl,    rS,
Ue(r)=LSUe(r)nGe(r, r)-Ue(r)Ge(r, r)ndl+U0(r),    rS,
ϕ(r)-LSϕ(r)Gcn-Gen-ψ(r)Gc-αcαeGedl=U0(r),
αe+αc2αcψ(r)-LSϕ(r)2Gcnn-2Genn-ψ(r)×Gcn-αcαeGendl=U0(r)n.
Ψ(θ)=1+i4πkeLSikθn·[ϕ(r)-U0(r)]-ψ(r)-U0(r)nexp(-ikθr)dl,
σs=2π02π|Ψ(θ)|2dθ.
ϕ(s)=02π[ϕ(s)(s, s)-ψ(s)B(s, s)]L(s)ds+U0(s),
ψ(s)=02π[ϕ(s)C(s, s)-ψ(s)D(s, s)]L(s)ds+U0(s)n,
limss2F(s, s)nn=k24π[εe ln(keR)-εc ln(kcR)].
L(s)ϕ(s)=2iπk(m)ϕmexp(ims),
L(s)ψ(s)=2iπ(m)ψmexp(ims)
am11ϕm+am12ψm+(n)(ϕnAmn11+ψnAmn12)=em1,
am21ϕm+am22ψm+(n)(ϕnAmn21+ψnAmn22)=em2,
am11=εcJmcHmc-εeJmeHme,am12=JmcHmc-JmeHme,am21=εeJmeHme-εcJmcHmc,am22=εeJmeHme-εcJmcHmc,
Amn11=-Amn/k-2(1-δm,n)Lm-n/iπk,Amn12=Bmn,Amn21=-Cmn/k2,Amn22=Dmn/k-2(1-δm,n)Lm-n/iπk,
Amn=1iπ202π02π[A(s, s)-A0(s, s)]exp(-ims)exp(ins)dsds.
em1=12π02πU0(s)exp(-ims)ds,
em2=12π02πU0(s)nexp(-ims)ds,
Lm=12π02πexp(-ims)L(s)ds,
z1z2+M11M12M21M22×z1z2=e1e2.
e(N)=z(N)-z(N+1)z(N)-1,z(N)=|n|N[|zn1(N)|2+|zn2(N)|2]-1/2.
fm=12π02πf(s)exp(-ims)ds12Ml=1Qwl exp(-imsˆl)×n=0M-1fsˆl+2πnMexp-2iπmnM,
Gj(r, r)=i4H0(1)(kjR),
Gj(r, r)n=-ikj4H1(1)(kjR)Rn,
2Gj(r, r)nn=-i4kj2H0(1)(kjR)RnRn+kjH1(1)(kjR)×2Rnn-1RRnRn.
Rn=-1L(s)Rdyds(x-x)-dxds(y-y),
2Rnn=-dydsdyds+dxdsdxdsL(s)L(s)R-1RRnRn.
Gj0(s, s)=i4H0(1)(2kjaR0),
Gj0(s, s)n=-ikj4R0H1(1)(2kjaR0),
2Gj0(s, s)nns=ikj2412kjaR0-1H1(1)(2kjaR0)-R02H0(1)(2kjaR0),
R0=sins-s2.
02πGj0(s, s)exp(ims)ds=iπ2JmjHmjexp(ims),
02πGj0(s, s)nexp(ims)ds=-12+ikjπ2JmjHmjexp(ims)=12+ikjπ2JmjHmjexp(ims),
02π2Gj0(s, s)nnexp(ims)ds=ikj2π2JmjHmjexp(ims).

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