Abstract

We develop a scattering matrix approach for the numerical calculation of resonant states and Q values of a nonideal optical disk cavity with an arbitrary shape and with an arbitrary varying refraction index. The developed method is applied to study the effect of surface roughness and inhomogeneity of the refraction index on Q values of microdisk cavities for lasing applications. We demonstrate that even small surface roughness (Δr ≲ λ/50) can lead to a drastic degradation of high-Q cavity modes by many orders of magnitude. The results of the numerical simulation are analyzed and explained in terms of wave reflection at a curved dielectric interface, combined with an examination of Poincaré surfaces of section and of Husimi distributions.

© 2004 Optical Society of America

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    [CrossRef]
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  3. S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds., Vol. 1 of Advanced Series in Applied Physics (World Scientific, Singapore, 1989), pp. 3–61.
  4. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  23. F. T. Smith, “Lifetime matrix in collision theory,” Phys. Rev. 118, 349–356 (1960).
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  24. M. Bauer, P. A. Mello, K. W. McVoy, “Time delay in nuclear reactions,” Z. Phys. A 293, 151–163 (1979).
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  25. Z. S. Wu, Y. P. Y. Wang, “Electromagnetic scattering for multilayered spheres: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
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  26. B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286–3296 (1996).
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    [CrossRef]
  28. M. Hentschel, H. Schomerus, “Fresnel laws at curved dielectric interfaces of microresonators,” Phys. Rev. E 65, 045603 (2002).
    [CrossRef]
  29. A. D. Stone, “Wave-chaotic optical resonators and lasers,” in Proceedings of the Nobel Symposium Quantum Chaos 2000, Phys. Scr. T90, 248–262 (2001).
  30. B. Crespi, G. Perez, S. J. Chang, “Quantum Poincaré sections for two-dimensional billiards,” Phys. Rev. E 47, 986–991 (1993).
    [CrossRef]
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    [CrossRef]

2003 (3)

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

S. V. Boriskina, T. M. Benson, P. Sewell, A. I. Nosich, “Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators,” Opt. Quantum Electron. 35, 545–559 (2003).
[CrossRef]

J. Wiersig, “Hexagonal dielectric resonators and microcrystal lasers,” Phys. Rev. A 67, 023807 (2003).
[CrossRef]

2002 (3)

M. Hentschel, H. Schomerus, “Fresnel laws at curved dielectric interfaces of microresonators,” Phys. Rev. E 65, 045603 (2002).
[CrossRef]

M. Hentschel, K. Richter, “Quantum chaos in optical systems: the annular billiar,” Phys. Rev. E 66, 056207 1–13 (2002).
[CrossRef]

R. C. Polson, Z. Vardeny, D. A. Chinn, “Multiple resonances in microdisk lasers of π-conjugated polymers,” Appl. Phys. Lett. 81, 1561–1563 (2002).
[CrossRef]

2001 (2)

M. Theander, T. Granlund, D. M. Johanson, A. Ruseckas, V. Sundström, M. R. Andersson, O. Inganäs, “Lasing in a microcavity with an oriented liquid-crystalline polyfluorene copolymer as active layer,” Adv. Mater. 13, 323–327 (2001).
[CrossRef]

A. D. Stone, “Wave-chaotic optical resonators and lasers,” in Proceedings of the Nobel Symposium Quantum Chaos 2000, Phys. Scr. T90, 248–262 (2001).

2000 (1)

C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O. Piquet, P. Blondy, D. Cros, O. Marty, “InAs quantum wires in InP-based microdiscs: mode identification and continuous wave room temperature laser operation,” J. Appl. Phys. 88, 6170–6174 (2000).
[CrossRef]

1999 (1)

B. Gayral, J. M. Gérard, A. Lemaı̂tre, C. Dupuis, L. Manin, J. L. Pelouard, “High-Q wet etched GaAS microdisks containing InAs quantum boxes,” Appl. Phys. Lett. 75, 1908–1910 (1999).
[CrossRef]

1998 (2)

M. Fujita, K. Inoshita, T. Bata, “Room temperature continuous wave lasing characteristics of GaInAsP/InP microdisk injection laser,” Electron. Lett. 34, 278–279 (1998).
[CrossRef]

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, O. Nalamasu, “Resonators and materials for organic lasers based on energy transfer,” IEEE J. Sel. Top. Quantum Electron. 4, 67–74 (1998).
[CrossRef]

1997 (1)

B.-J. Li, P.-L. Liu, “Analysis of far-field patterns of microdisk resonators by the finite-difference time-domain method,” IEEE J. Quantum Electron. 33, 1489–1491 (1997).
[CrossRef]

1996 (3)

P. A. Knipp, T. L. Reinecke, “Boundary-element method for the calculation of the electronic states in semiconductor nanostructures,” Phys. Rev. B 54, 1880–1891 (1996).
[CrossRef]

K. S. Yee, “Numerical solution of initial boundary-value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1996).

B. R. Johnson, “Light scattering by a multilayer sphere,” Appl. Opt. 35, 3286–3296 (1996).
[CrossRef] [PubMed]

1993 (2)

B. Crespi, G. Perez, S. J. Chang, “Quantum Poincaré sections for two-dimensional billiards,” Phys. Rev. E 47, 986–991 (1993).
[CrossRef]

Y. Yamamoto, R. E. Slusher, “Optical processes in microcavities,” Phys. Today 46, 66–72 (1993).
[CrossRef]

1992 (2)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1992).
[CrossRef]

1991 (1)

Z. S. Wu, Y. P. Y. Wang, “Electromagnetic scattering for multilayered spheres: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

1979 (1)

M. Bauer, P. A. Mello, K. W. McVoy, “Time delay in nuclear reactions,” Z. Phys. A 293, 151–163 (1979).
[CrossRef]

1975 (1)

A. V. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).
[CrossRef]

1971 (1)

P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

1960 (1)

F. T. Smith, “Lifetime matrix in collision theory,” Phys. Rev. 118, 349–356 (1960).
[CrossRef]

Andersson, M. R.

M. Theander, T. Granlund, D. M. Johanson, A. Ruseckas, V. Sundström, M. R. Andersson, O. Inganäs, “Lasing in a microcavity with an oriented liquid-crystalline polyfluorene copolymer as active layer,” Adv. Mater. 13, 323–327 (2001).
[CrossRef]

Bao, Z.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, O. Nalamasu, “Resonators and materials for organic lasers based on energy transfer,” IEEE J. Sel. Top. Quantum Electron. 4, 67–74 (1998).
[CrossRef]

Bata, T.

M. Fujita, K. Inoshita, T. Bata, “Room temperature continuous wave lasing characteristics of GaInAsP/InP microdisk injection laser,” Electron. Lett. 34, 278–279 (1998).
[CrossRef]

Bauer, M.

M. Bauer, P. A. Mello, K. W. McVoy, “Time delay in nuclear reactions,” Z. Phys. A 293, 151–163 (1979).
[CrossRef]

Benner, R. E.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds., Vol. 1 of Advanced Series in Applied Physics (World Scientific, Singapore, 1989), pp. 3–61.

Benson, T. M.

S. V. Boriskina, T. M. Benson, P. Sewell, A. I. Nosich, “Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators,” Opt. Quantum Electron. 35, 545–559 (2003).
[CrossRef]

Berggren, M.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, O. Nalamasu, “Resonators and materials for organic lasers based on energy transfer,” IEEE J. Sel. Top. Quantum Electron. 4, 67–74 (1998).
[CrossRef]

Blondy, P.

C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O. Piquet, P. Blondy, D. Cros, O. Marty, “InAs quantum wires in InP-based microdiscs: mode identification and continuous wave room temperature laser operation,” J. Appl. Phys. 88, 6170–6174 (2000).
[CrossRef]

Boriskina, S. V.

S. V. Boriskina, T. M. Benson, P. Sewell, A. I. Nosich, “Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators,” Opt. Quantum Electron. 35, 545–559 (2003).
[CrossRef]

Brault, J.

C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O. Piquet, P. Blondy, D. Cros, O. Marty, “InAs quantum wires in InP-based microdiscs: mode identification and continuous wave room temperature laser operation,” J. Appl. Phys. 88, 6170–6174 (2000).
[CrossRef]

Chang, R. K.

J. U. Nöckel, R. K. Chang, “2-d microcavities: theory and experiments,” in Cavity-Enhanced Spectroscopies, R. D. van Zee, J. P. Looney, eds., Vol. 40 of Experimental Methods in the Physical Sciences (Academic, San Diego, 2002), pp. 185–226.

Chang, S. J.

B. Crespi, G. Perez, S. J. Chang, “Quantum Poincaré sections for two-dimensional billiards,” Phys. Rev. E 47, 986–991 (1993).
[CrossRef]

Chinn, D. A.

R. C. Polson, Z. Vardeny, D. A. Chinn, “Multiple resonances in microdisk lasers of π-conjugated polymers,” Appl. Phys. Lett. 81, 1561–1563 (2002).
[CrossRef]

Crespi, B.

B. Crespi, G. Perez, S. J. Chang, “Quantum Poincaré sections for two-dimensional billiards,” Phys. Rev. E 47, 986–991 (1993).
[CrossRef]

Cros, D.

C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O. Piquet, P. Blondy, D. Cros, O. Marty, “InAs quantum wires in InP-based microdiscs: mode identification and continuous wave room temperature laser operation,” J. Appl. Phys. 88, 6170–6174 (2000).
[CrossRef]

Datta, S.

S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

Dodabalapur, A.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, O. Nalamasu, “Resonators and materials for organic lasers based on energy transfer,” IEEE J. Sel. Top. Quantum Electron. 4, 67–74 (1998).
[CrossRef]

Dupuis, C.

B. Gayral, J. M. Gérard, A. Lemaı̂tre, C. Dupuis, L. Manin, J. L. Pelouard, “High-Q wet etched GaAS microdisks containing InAs quantum boxes,” Appl. Phys. Lett. 75, 1908–1910 (1999).
[CrossRef]

Fujita, M.

M. Fujita, K. Inoshita, T. Bata, “Room temperature continuous wave lasing characteristics of GaInAsP/InP microdisk injection laser,” Electron. Lett. 34, 278–279 (1998).
[CrossRef]

Gayral, B.

B. Gayral, J. M. Gérard, A. Lemaı̂tre, C. Dupuis, L. Manin, J. L. Pelouard, “High-Q wet etched GaAS microdisks containing InAs quantum boxes,” Appl. Phys. Lett. 75, 1908–1910 (1999).
[CrossRef]

Gendry, M.

C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O. Piquet, P. Blondy, D. Cros, O. Marty, “InAs quantum wires in InP-based microdiscs: mode identification and continuous wave room temperature laser operation,” J. Appl. Phys. 88, 6170–6174 (2000).
[CrossRef]

Gérard, J. M.

B. Gayral, J. M. Gérard, A. Lemaı̂tre, C. Dupuis, L. Manin, J. L. Pelouard, “High-Q wet etched GaAS microdisks containing InAs quantum boxes,” Appl. Phys. Lett. 75, 1908–1910 (1999).
[CrossRef]

Granlund, T.

M. Theander, T. Granlund, D. M. Johanson, A. Ruseckas, V. Sundström, M. R. Andersson, O. Inganäs, “Lasing in a microcavity with an oriented liquid-crystalline polyfluorene copolymer as active layer,” Adv. Mater. 13, 323–327 (2001).
[CrossRef]

Hentschel, M.

M. Hentschel, H. Schomerus, “Fresnel laws at curved dielectric interfaces of microresonators,” Phys. Rev. E 65, 045603 (2002).
[CrossRef]

M. Hentschel, K. Richter, “Quantum chaos in optical systems: the annular billiar,” Phys. Rev. E 66, 056207 1–13 (2002).
[CrossRef]

Hill, S. C.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds., Vol. 1 of Advanced Series in Applied Physics (World Scientific, Singapore, 1989), pp. 3–61.

Inganäs, O.

M. Theander, T. Granlund, D. M. Johanson, A. Ruseckas, V. Sundström, M. R. Andersson, O. Inganäs, “Lasing in a microcavity with an oriented liquid-crystalline polyfluorene copolymer as active layer,” Adv. Mater. 13, 323–327 (2001).
[CrossRef]

Inoshita, K.

M. Fujita, K. Inoshita, T. Bata, “Room temperature continuous wave lasing characteristics of GaInAsP/InP microdisk injection laser,” Electron. Lett. 34, 278–279 (1998).
[CrossRef]

Johanson, D. M.

M. Theander, T. Granlund, D. M. Johanson, A. Ruseckas, V. Sundström, M. R. Andersson, O. Inganäs, “Lasing in a microcavity with an oriented liquid-crystalline polyfluorene copolymer as active layer,” Adv. Mater. 13, 323–327 (2001).
[CrossRef]

Johnson, B. R.

Katz, H. E.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, O. Nalamasu, “Resonators and materials for organic lasers based on energy transfer,” IEEE J. Sel. Top. Quantum Electron. 4, 67–74 (1998).
[CrossRef]

Knipp, P. A.

P. A. Knipp, T. L. Reinecke, “Boundary-element method for the calculation of the electronic states in semiconductor nanostructures,” Phys. Rev. B 54, 1880–1891 (1996).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, (Cambridge U. Press, Cambridge, 2002).

Laskowski, E.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, O. Nalamasu, “Resonators and materials for organic lasers based on energy transfer,” IEEE J. Sel. Top. Quantum Electron. 4, 67–74 (1998).
[CrossRef]

Lemai^tre, A.

B. Gayral, J. M. Gérard, A. Lemaı̂tre, C. Dupuis, L. Manin, J. L. Pelouard, “High-Q wet etched GaAS microdisks containing InAs quantum boxes,” Appl. Phys. Lett. 75, 1908–1910 (1999).
[CrossRef]

Letartre, X.

C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O. Piquet, P. Blondy, D. Cros, O. Marty, “InAs quantum wires in InP-based microdiscs: mode identification and continuous wave room temperature laser operation,” J. Appl. Phys. 88, 6170–6174 (2000).
[CrossRef]

Levi, A. F. J.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1992).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Li, B.-J.

B.-J. Li, P.-L. Liu, “Analysis of far-field patterns of microdisk resonators by the finite-difference time-domain method,” IEEE J. Quantum Electron. 33, 1489–1491 (1997).
[CrossRef]

Liu, P.-L.

B.-J. Li, P.-L. Liu, “Analysis of far-field patterns of microdisk resonators by the finite-difference time-domain method,” IEEE J. Quantum Electron. 33, 1489–1491 (1997).
[CrossRef]

Logan, R. A.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1992).
[CrossRef]

Love, J. D.

A. V. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).
[CrossRef]

Manin, L.

B. Gayral, J. M. Gérard, A. Lemaı̂tre, C. Dupuis, L. Manin, J. L. Pelouard, “High-Q wet etched GaAS microdisks containing InAs quantum boxes,” Appl. Phys. Lett. 75, 1908–1910 (1999).
[CrossRef]

Marty, O.

C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O. Piquet, P. Blondy, D. Cros, O. Marty, “InAs quantum wires in InP-based microdiscs: mode identification and continuous wave room temperature laser operation,” J. Appl. Phys. 88, 6170–6174 (2000).
[CrossRef]

McCall, S. L.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1992).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

McVoy, K. W.

M. Bauer, P. A. Mello, K. W. McVoy, “Time delay in nuclear reactions,” Z. Phys. A 293, 151–163 (1979).
[CrossRef]

Mello, P. A.

M. Bauer, P. A. Mello, K. W. McVoy, “Time delay in nuclear reactions,” Z. Phys. A 293, 151–163 (1979).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, (Cambridge U. Press, Cambridge, 2002).

Mohideen, U.

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1992).
[CrossRef]

Nalamasu, O.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, O. Nalamasu, “Resonators and materials for organic lasers based on energy transfer,” IEEE J. Sel. Top. Quantum Electron. 4, 67–74 (1998).
[CrossRef]

Nikolskaya, T. I.

V. V. Nikolsky, T. I. Nikolskaya, Decomposition Approach to the Problems of Electrodynamics (Nauka, Moscow, 1983; in Russian).

Nikolsky, V. V.

V. V. Nikolsky, T. I. Nikolskaya, Decomposition Approach to the Problems of Electrodynamics (Nauka, Moscow, 1983; in Russian).

Nöckel, J. U.

J. U. Nöckel, R. K. Chang, “2-d microcavities: theory and experiments,” in Cavity-Enhanced Spectroscopies, R. D. van Zee, J. P. Looney, eds., Vol. 40 of Experimental Methods in the Physical Sciences (Academic, San Diego, 2002), pp. 185–226.

Nosich, A. I.

S. V. Boriskina, T. M. Benson, P. Sewell, A. I. Nosich, “Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators,” Opt. Quantum Electron. 35, 545–559 (2003).
[CrossRef]

Pearton, S. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1992).
[CrossRef]

Pelouard, J. L.

B. Gayral, J. M. Gérard, A. Lemaı̂tre, C. Dupuis, L. Manin, J. L. Pelouard, “High-Q wet etched GaAS microdisks containing InAs quantum boxes,” Appl. Phys. Lett. 75, 1908–1910 (1999).
[CrossRef]

Perez, G.

B. Crespi, G. Perez, S. J. Chang, “Quantum Poincaré sections for two-dimensional billiards,” Phys. Rev. E 47, 986–991 (1993).
[CrossRef]

Piquet, O.

C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O. Piquet, P. Blondy, D. Cros, O. Marty, “InAs quantum wires in InP-based microdiscs: mode identification and continuous wave room temperature laser operation,” J. Appl. Phys. 88, 6170–6174 (2000).
[CrossRef]

Polson, R. C.

R. C. Polson, Z. Vardeny, D. A. Chinn, “Multiple resonances in microdisk lasers of π-conjugated polymers,” Appl. Phys. Lett. 81, 1561–1563 (2002).
[CrossRef]

Pottier, P.

C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O. Piquet, P. Blondy, D. Cros, O. Marty, “InAs quantum wires in InP-based microdiscs: mode identification and continuous wave room temperature laser operation,” J. Appl. Phys. 88, 6170–6174 (2000).
[CrossRef]

Reinecke, T. L.

P. A. Knipp, T. L. Reinecke, “Boundary-element method for the calculation of the electronic states in semiconductor nanostructures,” Phys. Rev. B 54, 1880–1891 (1996).
[CrossRef]

Richter, K.

M. Hentschel, K. Richter, “Quantum chaos in optical systems: the annular billiar,” Phys. Rev. E 66, 056207 1–13 (2002).
[CrossRef]

Ruseckas, A.

M. Theander, T. Granlund, D. M. Johanson, A. Ruseckas, V. Sundström, M. R. Andersson, O. Inganäs, “Lasing in a microcavity with an oriented liquid-crystalline polyfluorene copolymer as active layer,” Adv. Mater. 13, 323–327 (2001).
[CrossRef]

Sadiku, M. N. O.

M. N. O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, Boca Raton, Fla., 2001).

Schiortino, P.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, O. Nalamasu, “Resonators and materials for organic lasers based on energy transfer,” IEEE J. Sel. Top. Quantum Electron. 4, 67–74 (1998).
[CrossRef]

Schomerus, H.

M. Hentschel, H. Schomerus, “Fresnel laws at curved dielectric interfaces of microresonators,” Phys. Rev. E 65, 045603 (2002).
[CrossRef]

Seassal, C.

C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O. Piquet, P. Blondy, D. Cros, O. Marty, “InAs quantum wires in InP-based microdiscs: mode identification and continuous wave room temperature laser operation,” J. Appl. Phys. 88, 6170–6174 (2000).
[CrossRef]

Sewell, P.

S. V. Boriskina, T. M. Benson, P. Sewell, A. I. Nosich, “Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators,” Opt. Quantum Electron. 35, 545–559 (2003).
[CrossRef]

Slusher, R. E.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, O. Nalamasu, “Resonators and materials for organic lasers based on energy transfer,” IEEE J. Sel. Top. Quantum Electron. 4, 67–74 (1998).
[CrossRef]

Y. Yamamoto, R. E. Slusher, “Optical processes in microcavities,” Phys. Today 46, 66–72 (1993).
[CrossRef]

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1992).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

Smith, F. T.

F. T. Smith, “Lifetime matrix in collision theory,” Phys. Rev. 118, 349–356 (1960).
[CrossRef]

Snyder, A. V.

A. V. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).
[CrossRef]

Stone, A. D.

A. D. Stone, “Wave-chaotic optical resonators and lasers,” in Proceedings of the Nobel Symposium Quantum Chaos 2000, Phys. Scr. T90, 248–262 (2001).

Sundström, V.

M. Theander, T. Granlund, D. M. Johanson, A. Ruseckas, V. Sundström, M. R. Andersson, O. Inganäs, “Lasing in a microcavity with an oriented liquid-crystalline polyfluorene copolymer as active layer,” Adv. Mater. 13, 323–327 (2001).
[CrossRef]

Theander, M.

M. Theander, T. Granlund, D. M. Johanson, A. Ruseckas, V. Sundström, M. R. Andersson, O. Inganäs, “Lasing in a microcavity with an oriented liquid-crystalline polyfluorene copolymer as active layer,” Adv. Mater. 13, 323–327 (2001).
[CrossRef]

Timko, A.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, O. Nalamasu, “Resonators and materials for organic lasers based on energy transfer,” IEEE J. Sel. Top. Quantum Electron. 4, 67–74 (1998).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, (Cambridge U. Press, Cambridge, 2002).

Vardeny, Z.

R. C. Polson, Z. Vardeny, D. A. Chinn, “Multiple resonances in microdisk lasers of π-conjugated polymers,” Appl. Phys. Lett. 81, 1561–1563 (2002).
[CrossRef]

Viktorovitch, P.

C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O. Piquet, P. Blondy, D. Cros, O. Marty, “InAs quantum wires in InP-based microdiscs: mode identification and continuous wave room temperature laser operation,” J. Appl. Phys. 88, 6170–6174 (2000).
[CrossRef]

Wang, Y. P. Y.

Z. S. Wu, Y. P. Y. Wang, “Electromagnetic scattering for multilayered spheres: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Wiersig, J.

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

J. Wiersig, “Hexagonal dielectric resonators and microcrystal lasers,” Phys. Rev. A 67, 023807 (2003).
[CrossRef]

Wu, Z. S.

Z. S. Wu, Y. P. Y. Wang, “Electromagnetic scattering for multilayered spheres: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto, R. E. Slusher, “Optical processes in microcavities,” Phys. Today 46, 66–72 (1993).
[CrossRef]

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary-value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1996).

Adv. Mater. (1)

M. Theander, T. Granlund, D. M. Johanson, A. Ruseckas, V. Sundström, M. R. Andersson, O. Inganäs, “Lasing in a microcavity with an oriented liquid-crystalline polyfluorene copolymer as active layer,” Adv. Mater. 13, 323–327 (2001).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

R. C. Polson, Z. Vardeny, D. A. Chinn, “Multiple resonances in microdisk lasers of π-conjugated polymers,” Appl. Phys. Lett. 81, 1561–1563 (2002).
[CrossRef]

B. Gayral, J. M. Gérard, A. Lemaı̂tre, C. Dupuis, L. Manin, J. L. Pelouard, “High-Q wet etched GaAS microdisks containing InAs quantum boxes,” Appl. Phys. Lett. 75, 1908–1910 (1999).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
[CrossRef]

R. E. Slusher, A. F. J. Levi, U. Mohideen, S. L. McCall, S. J. Pearton, R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310–1312 (1992).
[CrossRef]

Electron. Lett. (1)

M. Fujita, K. Inoshita, T. Bata, “Room temperature continuous wave lasing characteristics of GaInAsP/InP microdisk injection laser,” Electron. Lett. 34, 278–279 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

B.-J. Li, P.-L. Liu, “Analysis of far-field patterns of microdisk resonators by the finite-difference time-domain method,” IEEE J. Quantum Electron. 33, 1489–1491 (1997).
[CrossRef]

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

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, O. Nalamasu, “Resonators and materials for organic lasers based on energy transfer,” IEEE J. Sel. Top. Quantum Electron. 4, 67–74 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary-value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1996).

IEEE Trans. Microwave Theory Tech. (1)

A. V. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).
[CrossRef]

J. Appl. Phys. (1)

C. Seassal, X. Letartre, J. Brault, M. Gendry, P. Pottier, P. Viktorovitch, O. Piquet, P. Blondy, D. Cros, O. Marty, “InAs quantum wires in InP-based microdiscs: mode identification and continuous wave room temperature laser operation,” J. Appl. Phys. 88, 6170–6174 (2000).
[CrossRef]

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).

Opt. Quantum Electron. (1)

S. V. Boriskina, T. M. Benson, P. Sewell, A. I. Nosich, “Highly efficient design of spectrally engineered whispering-gallery-mode microlaser resonators,” Opt. Quantum Electron. 35, 545–559 (2003).
[CrossRef]

Phys. Rev. (1)

F. T. Smith, “Lifetime matrix in collision theory,” Phys. Rev. 118, 349–356 (1960).
[CrossRef]

Phys. Rev. A (1)

J. Wiersig, “Hexagonal dielectric resonators and microcrystal lasers,” Phys. Rev. A 67, 023807 (2003).
[CrossRef]

Phys. Rev. B (1)

P. A. Knipp, T. L. Reinecke, “Boundary-element method for the calculation of the electronic states in semiconductor nanostructures,” Phys. Rev. B 54, 1880–1891 (1996).
[CrossRef]

Phys. Rev. D (1)

P. C. Waterman, “Symmetry, unitarity and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825–839 (1971).
[CrossRef]

Phys. Rev. E (3)

M. Hentschel, H. Schomerus, “Fresnel laws at curved dielectric interfaces of microresonators,” Phys. Rev. E 65, 045603 (2002).
[CrossRef]

M. Hentschel, K. Richter, “Quantum chaos in optical systems: the annular billiar,” Phys. Rev. E 66, 056207 1–13 (2002).
[CrossRef]

B. Crespi, G. Perez, S. J. Chang, “Quantum Poincaré sections for two-dimensional billiards,” Phys. Rev. E 47, 986–991 (1993).
[CrossRef]

Phys. Today (1)

Y. Yamamoto, R. E. Slusher, “Optical processes in microcavities,” Phys. Today 46, 66–72 (1993).
[CrossRef]

Proceedings of the Nobel Symposium Quantum Chaos 2000 (1)

A. D. Stone, “Wave-chaotic optical resonators and lasers,” in Proceedings of the Nobel Symposium Quantum Chaos 2000, Phys. Scr. T90, 248–262 (2001).

Radio Sci. (1)

Z. S. Wu, Y. P. Y. Wang, “Electromagnetic scattering for multilayered spheres: recursive algorithms,” Radio Sci. 26, 1393–1401 (1991).
[CrossRef]

Z. Phys. A (1)

M. Bauer, P. A. Mello, K. W. McVoy, “Time delay in nuclear reactions,” Z. Phys. A 293, 151–163 (1979).
[CrossRef]

Other (6)

J. U. Nöckel, R. K. Chang, “2-d microcavities: theory and experiments,” in Cavity-Enhanced Spectroscopies, R. D. van Zee, J. P. Looney, eds., Vol. 40 of Experimental Methods in the Physical Sciences (Academic, San Diego, 2002), pp. 185–226.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds., Vol. 1 of Advanced Series in Applied Physics (World Scientific, Singapore, 1989), pp. 3–61.

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, (Cambridge U. Press, Cambridge, 2002).

M. N. O. Sadiku, Numerical Techniques in Electromagnetics (CRC Press, Boca Raton, Fla., 2001).

V. V. Nikolsky, T. I. Nikolskaya, Decomposition Approach to the Problems of Electrodynamics (Nauka, Moscow, 1983; in Russian).

S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic geometry of a cavity with the refraction index n surrounded by air. The space is divided in three regions. In the inner (r < d) and outer regions (r > R), the refraction indices are constant. In the intermediate region d < r < R, the refraction index n is a function of both r and φ. The intermediate region is divided by N narrow concentric rings. In each ring the refraction coefficient is regarded as a function only of the angle and is given as n i = n i (φ).

Fig. 2
Fig. 2

Intermediate region divided by N concentric rings of width 2Δ; ρ i is the distance to the middle of the ith ring. States a i , a i+1 propagate (or decay) toward the ith boundary, whereas states b i , b i+1 propagate (or decay) away from this boundary. The ith boundary is defined as the boundary between the ith and the i + 1th rings.

Fig. 3
Fig. 3

Transmission coefficient T of a locally plane wave incident on a curved surface with the radii of curvature ρ as a function of the incidence angle χ calculated from Eq. (26). The angle of total internal reflection sin χ c = 0.56 (corresponding to n = 1.8). The inset shows the dependence of the average radius of local curvature due to boundary imperfections, ρ, subject to Δr for the present model of surface roughness.

Fig. 4
Fig. 4

Examples of nonideal cavities studied in the present paper in terms of (a) surface roughness and (b) inhomogeneous refraction index. (a) Radius of the disk R = 5 μm, n = 1.8, surface roughness Δr = 100 nm. (b) R = 5 μm, 〈n〉 = 1.8, Δn = 5%.

Fig. 5
Fig. 5

Dependencies Q = Q(λ) for two representative modes TM82,1 and TM55,7 for the cases of (a) different surface roughness Δr (R = 5 μm, n = 1.8) and (b) different refraction index inhomogeneities [〈n〉 = 1.8]. Note that in case (b) the resonances shift when Δn varies. For the sake of clarity, we plot all the resonances centered around their maxima of the corresponding ideal disk (i.e., Δn = 0). The broadening of the high-Q resonance TM82,1 is not discernible on the scale of the figure for all the values of Δn. Insets in (a) and (b) show the dependencies Q = Qr) and Q = Qn), respectively.

Fig. 6
Fig. 6

Poincaré surfaces of section for geometrical rays corresponding to the states q = 55 (a)–(c) and q = 82 (g)–(i) for the cavity with surface roughness Δr = 0, 20, 100 nm. The Husimi distributions for the states TM55,7 (d)–(f) and TM82,1 (j)–(l) for the same values of Δr used in the corresponding Poincaré SoS. Δχ ch indicates the broadening of the phase space due to the transition to the chaotic dynamics. Dashed lines show the angle of total internal reflection χ c .

Fig. 7
Fig. 7

Illustrative examples of intensity distribution E z for the resonant state TM55,7 in cavities with Δr = 0 (a), 20 nm (b), 100 nm (c) and with R = 5 μm, n = 1.8. Dashed lines indicate boundaries of the cavity.

Fig. 8
Fig. 8

The Husimi distributions for the states TM55,7 (a) and TM82,1 (b) for the cavity with the refraction index inhomogeneity Δn = 5%.

Equations (33)

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2r2+1rr+1r22φ2Ψr, φ+kn2Ψr, φ=0,
Ψout=q=-+AqHq2kr+BqHq1krexpiqφ,
B=SA,
SS=I,
Sqq=Sqq.
bibi+1=Siaiai+1,  1iN-1,
bNB=SNaNA.
Ψin=q=-+ aq0Jqnkrexpiqφ,
a0b1=S0a0a1.
bibi+2=S˜i,i+1aiai+2,S˜11i,i+1=S11i+S12iS11i+1I-S22iS11i+1-1S21i,S˜12i,i+1=S12iI-S11i+1S22i-1S12i+1,S˜21i,i+1=S21i+1I-S22iS11i+1-1S21i,S˜22i,i+1=S22i+1+S21i+1I-S22iS11i+1-1S22iS12i+1.
aB=S˜0,NaA.
S=S˜210,NI-S˜110,N-1S˜120,N+S˜220,N.
Q=icdSdkS=-icSdSdk.
τDqk=Qqq=icqdSqqdkSqq.
τDk=1Mq=1M τDqk=1MicTrdSdkS=1cMμ=1Mdθμdk=1cMdθdk,
Q=ωτDkres.
r2Rr2Rrr2+rRrRrr=-1Φφ2Φφφ2-k2n2r, φr2.
2Φiφφ2+ζi+ki2φρi2Φiφ=0,
2Ririri2+Ririri-ζiRiri=0,
Ψiri, φ=m=1amiexp-1/2+iγmir˜i+bmiexp-1/2-iγmir˜iΦmiφ,
Ψi+1ri+1, φ=m=1bmi+1exp-1/2+iγmi+1r˜i+1+ami+1exp-1/2-iγmi+1r˜i+1Φmi+1φ.
Ψir, φ=Ψi+1r, φ,1χi2φΨir, φr=1χi+12φΨi+1r, φr,
Si=ΛKA-1BKΛ-1.
Sqq=Hq2kr-ξJqnkr/JqnkrHq2krHq1kr-ξJqnkr/JqnkrHq1kr δqq,
q=nkR sin χ.
T=|TF|exp-23nkρsin2χcos2 χc-cos2 χ3/2,
Q=2nkR cos χT,
Hφ, χ=02πdφΨrsurf, φΦφ; φ, χ,
Φφ; φ, χ=πσ-1/4lexp-1/2σφ-φ+2πl2-ik sin χφ+2πl,
Λ11=I, Λ22mj=exp-1/2Δ1δmj,Λ12=Λ21=0,K11=I, K22mj=expiγmΔ1δmj,K12=K21=0,A=0V0,1-JU0,1P1, B=J-V0,10-U0,1Q1,Jmj=Jmn0kdδmj, Jmj=Jmn0kdδmj,V0,1mj=02πexp-imφΦj1φdφ,U0,1mj=1n0kρ102πχ02φχ12φexp-imφΦj1φdφ.
Λ11mj=exp1/2Δiδmj,Λ22mj=exp-1/2Δi+1δmj,Λ12=Λ21=0, Kmj=expiγmΔiδmj,A=-IVi,i+1-QiUi,i+1Pi+1,B=I-Vi,i+1Pi-Ui,i+1Qi+1,Vi,i+1mj=02πΦmiφ*Φji+1φdφ,Ui,i+1mj=ρiρi+102πχi2φχi+12φΦmiφ*Φji+1φdφ.
Λ11mj=exp1/2ΔNδmj,Λ22=I, Λ12=Λ21=0,K11mj=expiγmΔNδmj,K22=I, K12=K21=0,A=-IVN,N+1H1-QNUN,N+1H1,B=I-VN,N+1H2PN-UN,N+1H2,H1,2mj=H1,2kRδmj,Hm1,2mj=Hm1,2kRδmj,VN,N+1mj=02πΦmNφ* expijφdφ,UN,N+1mj=kρN 02πχN2φχN+12φΦj1φ* expijφdφ.
Pimj=-1/2+iγmiδmj,Qimj=-1/2-iγmiδmj, 1iN.

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