Abstract

We investigate the lasing spectra, threshold gain values, and emission directionalities for a two-dimensional microcavity laser with a “kite” contour. The cavity modes are considered accurately using the linear electromagnetic formalism of the lasing eigenvalue problem with exact boundary and radiation conditions. We develop a numerical algorithm based on the Muller boundary integral equations discretized using the Nystrom technique, which has theoretically justified and fast convergence. The influence of the deviation from the circular shape on the modal characteristics is studied numerically for the modes polarized in the cavity plane, demonstrating opportunities of directionality improvement together with preservation of a low threshold. These advantageous features are shown for the perturbed whispering-gallery modes of high-enough azimuth orders. Other modes can display improved directivities while suffering from drastically higher threshold levels. Experiments based on planar organic microcavity lasers confirm the coexistence of Fabry–Perot-like and whispering-gallery-like modes in kite-shaped cavities and show good agreement with the predicted far-field angular diagrams.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  44. M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
    [CrossRef]
  45. I. Gozhyk, G. Clavier, R. Méallet-Renault, M. Dvorko, R. Pansu, J.-F. Audibert, A. Brosseau, C. Lafargue, V. Tsvirkun, S. Lozenko, S. Forget, S. Chénais, C. Ulysse, J. Zyss, and M. Lebental, “Polarization properties of solid-state organic lasers,” Phys. Rev. A 86, 043817 (2012).
    [CrossRef]
  46. M. V. Balaban, R. Sauleau, T. M. Benson, and A. I. Nosich, “Accurate quantification of the Purcell effect in the presence of a dielectric microdisk of nanoscale thickness,” IET Micro Nano Lett. 6, 393–396 (2011).
    [CrossRef]

2012 (1)

I. Gozhyk, G. Clavier, R. Méallet-Renault, M. Dvorko, R. Pansu, J.-F. Audibert, A. Brosseau, C. Lafargue, V. Tsvirkun, S. Lozenko, S. Forget, S. Chénais, C. Ulysse, J. Zyss, and M. Lebental, “Polarization properties of solid-state organic lasers,” Phys. Rev. A 86, 043817 (2012).
[CrossRef]

2011 (6)

M. V. Balaban, R. Sauleau, T. M. Benson, and A. I. Nosich, “Accurate quantification of the Purcell effect in the presence of a dielectric microdisk of nanoscale thickness,” IET Micro Nano Lett. 6, 393–396 (2011).
[CrossRef]

Q. H. Song, L. Ge, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, A. Eberspacher, W. Fang, G. S. Solomon, and H. Cao, “Wavelength-scale deformed microdisk lasers,” Phys. Rev. A 84, 063843 (2011).
[CrossRef]

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

E. Y. Schmidt, N. V. Zorina, M. Y. Dvorko, N. I. Protsuk, K. V. Belyaeva, G. Clavier, R. Méallet-Renault, T. T. Vu, A. B. I. Mikhaleva, and B. A. Trofimov, “A general synthetic strategy for the design of new BODIPY fluorophores based on pyrroles with polycondensed aromatic and metallocene substituents,” Chem. Eur. J. 17, 3069–3073 (2011).
[CrossRef]

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–281 (2011).
[CrossRef]

J.-W. Ryu and M. Hentschel, “Designing coupled microcavity lasers for high-Q modes with unidirectional light emission,” Opt. Lett. 36, 1116–1118 (2011).
[CrossRef]

2010 (2)

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. USA 107, 22407–22412 (2010).
[CrossRef]

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shin, J. Unterhinninghofen, W. Fang, and G. S. Solomon, “Directional laser emission from a wavelength-scale chaotic microcavity,” Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef]

2009 (3)

2008 (3)

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[CrossRef]

S. R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A 77, 013804 (2008).
[CrossRef]

E. I. Smotrova, J. Ctyroky, T. M. Benson, P. Sewell, and A. I. Nosich, “Lasing frequencies and thresholds of the dipole-type supermodes in an active microdisk concentrically coupled with a passive microring,” J. Opt. Soc. Am. A 25, 2884–2892 (2008).
[CrossRef]

2007 (3)

C.-M. Lai, H.-M. Wu, P.-C. Huang, S.-L. Wang, and L.-H. Peng, “Single mode stimulated emission from prismlike gallium nitride submicron cavity,” Appl. Phys. Lett. 90, 1106–1108 (2007).
[CrossRef]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

J. L. Tsalamengas, “Exponentially converging Nystrom methods applied to the integral-integrodifferential equations of oblique scattering/hybrid mode propagation in presence of composite dielectric cylinders of arbitrary cross-section,” IEEE Trans. Antennas Propag. 55, 3239–3250 (2007).
[CrossRef]

2006 (5)

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Q-factor and emission pattern control of the whispering gallery modes in notched microdisk resonators,” IEEE J. Sel. Top. Quantum Electron. 12, 66–70 (2006).
[CrossRef]

J. Wiersig and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 73, 031802 (2006).
[CrossRef]

2005 (3)

T. Ben-Massaoud and J. Zyss, “Unidirectional laser emission from polymer-based spiral microdisks,” Appl. Phys. Lett. 86, 241110 (2005).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

F. Courvoisier, V. Boutou, J. P. Wolf, R. K. Chang, and J. Zyss, “Deciphering output coupling mechanisms in spiral microcavities with femtosecond light bullets,” Opt. Lett. 30, 738–740 (2005).
[CrossRef]

2004 (3)

2003 (1)

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

1998 (1)

1993 (1)

A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, “Directional light coupling from microdisk lasers,” Appl. Phys. Lett. 62, 561–563 (1993).
[CrossRef]

1992 (1)

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

1990 (2)

A. F. Peterson, “The ‘interior resonance’ problem associated with surface integral equations of electromagnetics: numerical consequences and a survey of remedies,” Electromagnetics 10, 293–312 (1990).
[CrossRef]

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

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Audet, R.

Audibert, J.-F.

I. Gozhyk, G. Clavier, R. Méallet-Renault, M. Dvorko, R. Pansu, J.-F. Audibert, A. Brosseau, C. Lafargue, V. Tsvirkun, S. Lozenko, S. Forget, S. Chénais, C. Ulysse, J. Zyss, and M. Lebental, “Polarization properties of solid-state organic lasers,” Phys. Rev. A 86, 043817 (2012).
[CrossRef]

Balaban, M. V.

M. V. Balaban, R. Sauleau, T. M. Benson, and A. I. Nosich, “Accurate quantification of the Purcell effect in the presence of a dielectric microdisk of nanoscale thickness,” IET Micro Nano Lett. 6, 393–396 (2011).
[CrossRef]

Belkin, M. A.

Belyaeva, K. V.

E. Y. Schmidt, N. V. Zorina, M. Y. Dvorko, N. I. Protsuk, K. V. Belyaeva, G. Clavier, R. Méallet-Renault, T. T. Vu, A. B. I. Mikhaleva, and B. A. Trofimov, “A general synthetic strategy for the design of new BODIPY fluorophores based on pyrroles with polycondensed aromatic and metallocene substituents,” Chem. Eur. J. 17, 3069–3073 (2011).
[CrossRef]

Ben-Massaoud, T.

T. Ben-Massaoud and J. Zyss, “Unidirectional laser emission from polymer-based spiral microdisks,” Appl. Phys. Lett. 86, 241110 (2005).
[CrossRef]

Benson, T.

E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

Benson, T. M.

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

M. V. Balaban, R. Sauleau, T. M. Benson, and A. I. Nosich, “Accurate quantification of the Purcell effect in the presence of a dielectric microdisk of nanoscale thickness,” IET Micro Nano Lett. 6, 393–396 (2011).
[CrossRef]

E. I. Smotrova, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical fields of the lowest modes in a uniformly active thin sub-wavelength spiral microcavity,” Opt. Lett. 34, 3773–3775 (2009).
[CrossRef]

E. I. Smotrova, J. Ctyroky, T. M. Benson, P. Sewell, and A. I. Nosich, “Lasing frequencies and thresholds of the dipole-type supermodes in an active microdisk concentrically coupled with a passive microring,” J. Opt. Soc. Am. A 25, 2884–2892 (2008).
[CrossRef]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Q-factor and emission pattern control of the whispering gallery modes in notched microdisk resonators,” IEEE J. Sel. Top. Quantum Electron. 12, 66–70 (2006).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Accurate simulation of 2-D optical microcavities with uniquely solvable boundary integral equations and trigonometric-Galerkin discretization,” J. Opt. Soc. Am. A 21, 393–402 (2004).
[CrossRef]

Bogomolny, E.

S. R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A 77, 013804 (2008).
[CrossRef]

Boriskina, S. V.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Q-factor and emission pattern control of the whispering gallery modes in notched microdisk resonators,” IEEE J. Sel. Top. Quantum Electron. 12, 66–70 (2006).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Accurate simulation of 2-D optical microcavities with uniquely solvable boundary integral equations and trigonometric-Galerkin discretization,” J. Opt. Soc. Am. A 21, 393–402 (2004).
[CrossRef]

Boutou, V.

Brosseau, A.

I. Gozhyk, G. Clavier, R. Méallet-Renault, M. Dvorko, R. Pansu, J.-F. Audibert, A. Brosseau, C. Lafargue, V. Tsvirkun, S. Lozenko, S. Forget, S. Chénais, C. Ulysse, J. Zyss, and M. Lebental, “Polarization properties of solid-state organic lasers,” Phys. Rev. A 86, 043817 (2012).
[CrossRef]

Byelobrov, V. O.

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

Cao, H.

Q. H. Song, L. Ge, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, A. Eberspacher, W. Fang, G. S. Solomon, and H. Cao, “Wavelength-scale deformed microdisk lasers,” Phys. Rev. A 84, 063843 (2011).
[CrossRef]

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shin, J. Unterhinninghofen, W. Fang, and G. S. Solomon, “Directional laser emission from a wavelength-scale chaotic microcavity,” Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef]

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
[CrossRef]

Capasso, F.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. USA 107, 22407–22412 (2010).
[CrossRef]

M. Hentschel, T.-Y. Kwon, M. A. Belkin, R. Audet, and F. Capasso, “Angular emission characteristics of quantum cascade spiral microlasers,” Opt. Express 17, 10335–10343 (2009).
[CrossRef]

Chang, R. K.

Chénais, S.

I. Gozhyk, G. Clavier, R. Méallet-Renault, M. Dvorko, R. Pansu, J.-F. Audibert, A. Brosseau, C. Lafargue, V. Tsvirkun, S. Lozenko, S. Forget, S. Chénais, C. Ulysse, J. Zyss, and M. Lebental, “Polarization properties of solid-state organic lasers,” Phys. Rev. A 86, 043817 (2012).
[CrossRef]

Clavier, G.

I. Gozhyk, G. Clavier, R. Méallet-Renault, M. Dvorko, R. Pansu, J.-F. Audibert, A. Brosseau, C. Lafargue, V. Tsvirkun, S. Lozenko, S. Forget, S. Chénais, C. Ulysse, J. Zyss, and M. Lebental, “Polarization properties of solid-state organic lasers,” Phys. Rev. A 86, 043817 (2012).
[CrossRef]

E. Y. Schmidt, N. V. Zorina, M. Y. Dvorko, N. I. Protsuk, K. V. Belyaeva, G. Clavier, R. Méallet-Renault, T. T. Vu, A. B. I. Mikhaleva, and B. A. Trofimov, “A general synthetic strategy for the design of new BODIPY fluorophores based on pyrroles with polycondensed aromatic and metallocene substituents,” Chem. Eur. J. 17, 3069–3073 (2011).
[CrossRef]

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer, 1998).

Courvoisier, F.

Cox, J. A.

Ctyroky, J.

Diehl, L.

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[CrossRef]

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shin, J. Unterhinninghofen, W. Fang, and G. S. Solomon, “Directional laser emission from a wavelength-scale chaotic microcavity,” Phys. Rev. Lett. 105, 103902 (2010).
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S.-K. Kim, S.-H. Kim, G.-H. Kim, H.-G. Park, D.-J. Shin, and Y.-H. Lee, “Highly directional emission from few-micron-size elliptical microdisks,” Appl. Phys. Lett. 84, 861–863 (2004).
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[CrossRef]

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Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
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A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, “Directional light coupling from microdisk lasers,” Appl. Phys. Lett. 62, 561–563 (1993).
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A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, “Directional light coupling from microdisk lasers,” Appl. Phys. Lett. 62, 561–563 (1993).
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S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289–291 (1992).
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M. V. Balaban, R. Sauleau, T. M. Benson, and A. I. Nosich, “Accurate quantification of the Purcell effect in the presence of a dielectric microdisk of nanoscale thickness,” IET Micro Nano Lett. 6, 393–396 (2011).
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E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
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S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Accurate simulation of 2-D 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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I. Gozhyk, G. Clavier, R. Méallet-Renault, M. Dvorko, R. Pansu, J.-F. Audibert, A. Brosseau, C. Lafargue, V. Tsvirkun, S. Lozenko, S. Forget, S. Chénais, C. Ulysse, J. Zyss, and M. Lebental, “Polarization properties of solid-state organic lasers,” Phys. Rev. A 86, 043817 (2012).
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S.-K. Kim, S.-H. Kim, G.-H. Kim, H.-G. Park, D.-J. Shin, and Y.-H. Lee, “Highly directional emission from few-micron-size elliptical microdisks,” Appl. Phys. Lett. 84, 861–863 (2004).
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A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, “Directional light coupling from microdisk lasers,” Appl. Phys. Lett. 62, 561–563 (1993).
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C.-M. Lai, H.-M. Wu, P.-C. Huang, S.-L. Wang, and L.-H. Peng, “Single mode stimulated emission from prismlike gallium nitride submicron cavity,” Appl. Phys. Lett. 90, 1106–1108 (2007).
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[CrossRef]

M. V. Balaban, R. Sauleau, T. M. Benson, and A. I. Nosich, “Accurate quantification of the Purcell effect in the presence of a dielectric microdisk of nanoscale thickness,” IET Micro Nano Lett. 6, 393–396 (2011).
[CrossRef]

E. I. Smotrova, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical fields of the lowest modes in a uniformly active thin sub-wavelength spiral microcavity,” Opt. Lett. 34, 3773–3775 (2009).
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E. Y. Schmidt, N. V. Zorina, M. Y. Dvorko, N. I. Protsuk, K. V. Belyaeva, G. Clavier, R. Méallet-Renault, T. T. Vu, A. B. I. Mikhaleva, and B. A. Trofimov, “A general synthetic strategy for the design of new BODIPY fluorophores based on pyrroles with polycondensed aromatic and metallocene substituents,” Chem. Eur. J. 17, 3069–3073 (2011).
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S. R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A 77, 013804 (2008).
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E. I. Smotrova, J. Ctyroky, T. M. Benson, P. Sewell, and A. I. Nosich, “Lasing frequencies and thresholds of the dipole-type supermodes in an active microdisk concentrically coupled with a passive microring,” J. Opt. Soc. Am. A 25, 2884–2892 (2008).
[CrossRef]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Q-factor and emission pattern control of the whispering gallery modes in notched microdisk resonators,” IEEE J. Sel. Top. Quantum Electron. 12, 66–70 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Accurate simulation of 2-D optical microcavities with uniquely solvable boundary integral equations and trigonometric-Galerkin discretization,” J. Opt. Soc. Am. A 21, 393–402 (2004).
[CrossRef]

Shim, J.-B.

Q. H. Song, L. Ge, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, A. Eberspacher, W. Fang, G. S. Solomon, and H. Cao, “Wavelength-scale deformed microdisk lasers,” Phys. Rev. A 84, 063843 (2011).
[CrossRef]

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S.-K. Kim, S.-H. Kim, G.-H. Kim, H.-G. Park, D.-J. Shin, and Y.-H. Lee, “Highly directional emission from few-micron-size elliptical microdisks,” Appl. Phys. Lett. 84, 861–863 (2004).
[CrossRef]

Shin, J.-B.

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shin, J. Unterhinninghofen, W. Fang, and G. S. Solomon, “Directional laser emission from a wavelength-scale chaotic microcavity,” Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef]

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T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–281 (2011).
[CrossRef]

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A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, “Directional light coupling from microdisk lasers,” Appl. Phys. Lett. 62, 561–563 (1993).
[CrossRef]

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[CrossRef]

Smotrova, E. I.

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

E. I. Smotrova, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical fields of the lowest modes in a uniformly active thin sub-wavelength spiral microcavity,” Opt. Lett. 34, 3773–3775 (2009).
[CrossRef]

E. I. Smotrova, J. Ctyroky, T. M. Benson, P. Sewell, and A. I. Nosich, “Lasing frequencies and thresholds of the dipole-type supermodes in an active microdisk concentrically coupled with a passive microring,” J. Opt. Soc. Am. A 25, 2884–2892 (2008).
[CrossRef]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

E. I. Smotrova and A. I. Nosich, “Thresholds of lasing and modal patterns of a limacon cavity analysed with Muller’s integral equations,” in Proceedings of International Conference On Laser and Optical Networks Modeling (IEEE, 2011), paper 083.

E. I. Smotrova and A. I. Nosich, “Directional light emission from a kite-shaped microcavity laser,” in Proceedings of International Conference on Transparent Optical Networks (IEEE, 2011), paper We.A4.4.

E. I. Smotrova and A. I. Nosich, “Simulation of lasing modes in a kite-shaped microcavity laser,” in Proceedings of International Conference on Advanced Optoelectronics and Lasers (IEEE, 2010), pp. 144–146.

Solomon, G. S.

Q. H. Song, L. Ge, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, A. Eberspacher, W. Fang, G. S. Solomon, and H. Cao, “Wavelength-scale deformed microdisk lasers,” Phys. Rev. A 84, 063843 (2011).
[CrossRef]

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shin, J. Unterhinninghofen, W. Fang, and G. S. Solomon, “Directional laser emission from a wavelength-scale chaotic microcavity,” Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef]

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
[CrossRef]

Song, Q.

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
[CrossRef]

Song, Q. H.

Q. H. Song, L. Ge, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, A. Eberspacher, W. Fang, G. S. Solomon, and H. Cao, “Wavelength-scale deformed microdisk lasers,” Phys. Rev. A 84, 063843 (2011).
[CrossRef]

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shin, J. Unterhinninghofen, W. Fang, and G. S. Solomon, “Directional laser emission from a wavelength-scale chaotic microcavity,” Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Stone, A. D.

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shin, J. Unterhinninghofen, W. Fang, and G. S. Solomon, “Directional laser emission from a wavelength-scale chaotic microcavity,” Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef]

H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, and A. D. Stone, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B 21, 923–934 (2004).
[CrossRef]

H. G. L. Schwefel, H. E. Tureci, A. D. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2004), pp. 415–496.

Tikhonov, A. N.

A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Dover, 1990).

Trofimov, B. A.

E. Y. Schmidt, N. V. Zorina, M. Y. Dvorko, N. I. Protsuk, K. V. Belyaeva, G. Clavier, R. Méallet-Renault, T. T. Vu, A. B. I. Mikhaleva, and B. A. Trofimov, “A general synthetic strategy for the design of new BODIPY fluorophores based on pyrroles with polycondensed aromatic and metallocene substituents,” Chem. Eur. J. 17, 3069–3073 (2011).
[CrossRef]

Tsalamengas, J. L.

J. L. Tsalamengas, “Exponentially converging Nystrom methods applied to the integral-integrodifferential equations of oblique scattering/hybrid mode propagation in presence of composite dielectric cylinders of arbitrary cross-section,” IEEE Trans. Antennas Propag. 55, 3239–3250 (2007).
[CrossRef]

Tsvirkun, V.

I. Gozhyk, G. Clavier, R. Méallet-Renault, M. Dvorko, R. Pansu, J.-F. Audibert, A. Brosseau, C. Lafargue, V. Tsvirkun, S. Lozenko, S. Forget, S. Chénais, C. Ulysse, J. Zyss, and M. Lebental, “Polarization properties of solid-state organic lasers,” Phys. Rev. A 86, 043817 (2012).
[CrossRef]

Tureci, H. E.

H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, and A. D. Stone, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B 21, 923–934 (2004).
[CrossRef]

H. G. L. Schwefel, H. E. Tureci, A. D. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2004), pp. 415–496.

Ulysse, C.

I. Gozhyk, G. Clavier, R. Méallet-Renault, M. Dvorko, R. Pansu, J.-F. Audibert, A. Brosseau, C. Lafargue, V. Tsvirkun, S. Lozenko, S. Forget, S. Chénais, C. Ulysse, J. Zyss, and M. Lebental, “Polarization properties of solid-state organic lasers,” Phys. Rev. A 86, 043817 (2012).
[CrossRef]

Unterhinninghofen, J.

Q. H. Song, L. Ge, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, A. Eberspacher, W. Fang, G. S. Solomon, and H. Cao, “Wavelength-scale deformed microdisk lasers,” Phys. Rev. A 84, 063843 (2011).
[CrossRef]

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shin, J. Unterhinninghofen, W. Fang, and G. S. Solomon, “Directional laser emission from a wavelength-scale chaotic microcavity,” Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef]

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. USA 107, 22407–22412 (2010).
[CrossRef]

Vu, T. T.

E. Y. Schmidt, N. V. Zorina, M. Y. Dvorko, N. I. Protsuk, K. V. Belyaeva, G. Clavier, R. Méallet-Renault, T. T. Vu, A. B. I. Mikhaleva, and B. A. Trofimov, “A general synthetic strategy for the design of new BODIPY fluorophores based on pyrroles with polycondensed aromatic and metallocene substituents,” Chem. Eur. J. 17, 3069–3073 (2011).
[CrossRef]

Wang, L.

Wang, Q. J.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. USA 107, 22407–22412 (2010).
[CrossRef]

Wang, S.-L.

C.-M. Lai, H.-M. Wu, P.-C. Huang, S.-L. Wang, and L.-H. Peng, “Single mode stimulated emission from prismlike gallium nitride submicron cavity,” Appl. Phys. Lett. 90, 1106–1108 (2007).
[CrossRef]

Wiersig, J.

Q. H. Song, L. Ge, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, A. Eberspacher, W. Fang, G. S. Solomon, and H. Cao, “Wavelength-scale deformed microdisk lasers,” Phys. Rev. A 84, 063843 (2011).
[CrossRef]

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shin, J. Unterhinninghofen, W. Fang, and G. S. Solomon, “Directional laser emission from a wavelength-scale chaotic microcavity,” Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef]

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. USA 107, 22407–22412 (2010).
[CrossRef]

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[CrossRef]

J. Wiersig and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 73, 031802 (2006).
[CrossRef]

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

Wolf, J. P.

Wu, H.-M.

C.-M. Lai, H.-M. Wu, P.-C. Huang, S.-L. Wang, and L.-H. Peng, “Single mode stimulated emission from prismlike gallium nitride submicron cavity,” Appl. Phys. Lett. 90, 1106–1108 (2007).
[CrossRef]

Yamanishi, M.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. USA 107, 22407–22412 (2010).
[CrossRef]

Yan, C.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. USA 107, 22407–22412 (2010).
[CrossRef]

Yu, N.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. USA 107, 22407–22412 (2010).
[CrossRef]

Zorina, N. V.

E. Y. Schmidt, N. V. Zorina, M. Y. Dvorko, N. I. Protsuk, K. V. Belyaeva, G. Clavier, R. Méallet-Renault, T. T. Vu, A. B. I. Mikhaleva, and B. A. Trofimov, “A general synthetic strategy for the design of new BODIPY fluorophores based on pyrroles with polycondensed aromatic and metallocene substituents,” Chem. Eur. J. 17, 3069–3073 (2011).
[CrossRef]

Zyss, J.

I. Gozhyk, G. Clavier, R. Méallet-Renault, M. Dvorko, R. Pansu, J.-F. Audibert, A. Brosseau, C. Lafargue, V. Tsvirkun, S. Lozenko, S. Forget, S. Chénais, C. Ulysse, J. Zyss, and M. Lebental, “Polarization properties of solid-state organic lasers,” Phys. Rev. A 86, 043817 (2012).
[CrossRef]

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[CrossRef]

T. Ben-Massaoud and J. Zyss, “Unidirectional laser emission from polymer-based spiral microdisks,” Appl. Phys. Lett. 86, 241110 (2005).
[CrossRef]

F. Courvoisier, V. Boutou, J. P. Wolf, R. K. Chang, and J. Zyss, “Deciphering output coupling mechanisms in spiral microcavities with femtosecond light bullets,” Opt. Lett. 30, 738–740 (2005).
[CrossRef]

Appl. Phys. Lett. (6)

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

A. F. J. Levi, R. E. Slusher, S. L. McCall, J. L. Glass, S. J. Pearton, and R. A. Logan, “Directional light coupling from microdisk lasers,” Appl. Phys. Lett. 62, 561–563 (1993).
[CrossRef]

S.-K. Kim, S.-H. Kim, G.-H. Kim, H.-G. Park, D.-J. Shin, and Y.-H. Lee, “Highly directional emission from few-micron-size elliptical microdisks,” Appl. Phys. Lett. 84, 861–863 (2004).
[CrossRef]

C.-M. Lai, H.-M. Wu, P.-C. Huang, S.-L. Wang, and L.-H. Peng, “Single mode stimulated emission from prismlike gallium nitride submicron cavity,” Appl. Phys. Lett. 90, 1106–1108 (2007).
[CrossRef]

T. Ben-Massaoud and J. Zyss, “Unidirectional laser emission from polymer-based spiral microdisks,” Appl. Phys. Lett. 86, 241110 (2005).
[CrossRef]

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[CrossRef]

Chem. Eur. J. (1)

E. Y. Schmidt, N. V. Zorina, M. Y. Dvorko, N. I. Protsuk, K. V. Belyaeva, G. Clavier, R. Méallet-Renault, T. T. Vu, A. B. I. Mikhaleva, and B. A. Trofimov, “A general synthetic strategy for the design of new BODIPY fluorophores based on pyrroles with polycondensed aromatic and metallocene substituents,” Chem. Eur. J. 17, 3069–3073 (2011).
[CrossRef]

Electromagnetics (1)

A. F. Peterson, “The ‘interior resonance’ problem associated with surface integral equations of electromagnetics: numerical consequences and a survey of remedies,” Electromagnetics 10, 293–312 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, “Optical theorem helps understand thresholds of lasing in microcavities with active regions,” IEEE J. Quantum Electron. 47, 20–30 (2011).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (4)

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Q-factor and emission pattern control of the whispering gallery modes in notched microdisk resonators,” IEEE J. Sel. Top. Quantum Electron. 12, 66–70 (2006).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. Benson, and P. Sewell, “Cold-cavity thresholds of microdisks with uniform and non-uniform gain: quasi-3D modeling with accurate 2D analysis,” IEEE J. Sel. Top. Quantum Electron. 11, 1135–1142 (2005).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. L. Tsalamengas, “Exponentially converging Nystrom methods applied to the integral-integrodifferential equations of oblique scattering/hybrid mode propagation in presence of composite dielectric cylinders of arbitrary cross-section,” IEEE Trans. Antennas Propag. 55, 3239–3250 (2007).
[CrossRef]

IET Micro Nano Lett. (1)

M. V. Balaban, R. Sauleau, T. M. Benson, and A. I. Nosich, “Accurate quantification of the Purcell effect in the presence of a dielectric microdisk of nanoscale thickness,” IET Micro Nano Lett. 6, 393–396 (2011).
[CrossRef]

J. Comput. Phys. (1)

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

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Laser Photon. Rev. (1)

T. Harayama and S. Shinohara, “Two-dimensional microcavity lasers,” Laser Photon. Rev. 5, 247–281 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Opt. Quantum Electron. (1)

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modeling,” Opt. Quantum Electron. 39, 1253–1272 (2007).
[CrossRef]

Phys. Rev. A (6)

Q. H. Song, L. Ge, J. Wiersig, J.-B. Shim, J. Unterhinninghofen, A. Eberspacher, W. Fang, G. S. Solomon, and H. Cao, “Wavelength-scale deformed microdisk lasers,” Phys. Rev. A 84, 063843 (2011).
[CrossRef]

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

S. R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A 77, 013804 (2008).
[CrossRef]

J. Wiersig and M. Hentschel, “Asymmetric scattering and nonorthogonal mode patterns in optical microspirals,” Phys. Rev. A 73, 031802 (2006).
[CrossRef]

Q. Song, W. Fang, B. Liu, S.-T. Ho, G. S. Solomon, and H. Cao, “Chaotic microcavity laser with high quality factor and unidirectional output,” Phys. Rev. A 80, 041807 (2009).
[CrossRef]

I. Gozhyk, G. Clavier, R. Méallet-Renault, M. Dvorko, R. Pansu, J.-F. Audibert, A. Brosseau, C. Lafargue, V. Tsvirkun, S. Lozenko, S. Forget, S. Chénais, C. Ulysse, J. Zyss, and M. Lebental, “Polarization properties of solid-state organic lasers,” Phys. Rev. A 86, 043817 (2012).
[CrossRef]

Phys. Rev. Lett. (2)

Q. H. Song, L. Ge, A. D. Stone, H. Cao, J. Wiersig, J.-B. Shin, J. Unterhinninghofen, W. Fang, and G. S. Solomon, “Directional laser emission from a wavelength-scale chaotic microcavity,” Phys. Rev. Lett. 105, 103902 (2010).
[CrossRef]

J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, “Whispering-gallery mode resonators for highly unidirectional laser action,” Proc. Natl. Acad. Sci. USA 107, 22407–22412 (2010).
[CrossRef]

Other (11)

H. G. L. Schwefel, H. E. Tureci, A. D. Stone, and R. K. Chang, “Progress in asymmetric resonant cavities: using shape as a design parameter in dielectric microcavity lasers,” in Optical Microcavities, K. Vahala, ed. (World Scientific, 2004), pp. 415–496.

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

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer, 1998).

E. I. Smotrova and A. I. Nosich, “Simulation of lasing modes in a kite-shaped microcavity laser,” in Proceedings of International Conference on Advanced Optoelectronics and Lasers (IEEE, 2010), pp. 144–146.

E. I. Smotrova and A. I. Nosich, “Directional light emission from a kite-shaped microcavity laser,” in Proceedings of International Conference on Transparent Optical Networks (IEEE, 2011), paper We.A4.4.

E. I. Smotrova and A. I. Nosich, “Thresholds of lasing and modal patterns of a limacon cavity analysed with Muller’s integral equations,” in Proceedings of International Conference On Laser and Optical Networks Modeling (IEEE, 2011), paper 083.

A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Dover, 1990).

A. V. Pogorelov, Differential Geometry (Noordhoff, 1974).

Z. T. Nazarchuk, Numerical Analysis of Wave Diffraction by Cylindrical Structures (Naukova Dumka, 1989) (in Russian).

Y. V. Gandel, Introduction to the Methods of Computation of Singular and Hyper-Singular Integrals (Kharkiv National University, 2001) (in Russian).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

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Figures (9)

Fig. 1.
Fig. 1.

Geometry of a uniformly active 2D dielectric resonator of arbitrary shape.

Fig. 2.
Fig. 2.

Relief of determinant (a) for a fully convex kite δ=0.165 and (b) for a partially concave δ=0.5. Other parameters are α=1.5 and N=50. Marks correspond to the field patterns in Figs. 4 and 5.

Fig. 3.
Fig. 3.

(a) Lasing frequencies and (b) thresholds as functions of the kite deformation parameter δ, for the doublet of quasi-WGH10,1 (green and black lines) and quasi-H1,4 (blue and red) modes, α=1.5, N=50. Marks I–IV correspond to similar marks in Figs. 2, 4, and 5.

Fig. 4.
Fig. 4.

Near- and far-field patterns of |Hz| for two modes that form the quasi-WGH10,1 doublet (I), (II) and quasi-H1,4 doublet (III), (IV) in a kite with contour parameter δ=0.165. These field patterns correspond to the marks in Figs. 2 and 3. Mode (I) is an odd WG-like one with κ=8.8511, γ=7.352*102, and D=3.86. Mode (II) is an even WG-like one with κ=8.8534, γ=7.076*102, and D=3.33. Mode (III) is an odd FP-like one with κ=8.8105, γ=8.884*102, and D=3.54. Mode (IV) is an even FP-like one with κ=8.733, γ=9.207*102, and D=2.8. Other parameters are α=1.5 and N=50.

Fig. 5.
Fig. 5.

Same as in Fig. 4 in a kite with contour parameter δ=0.5. These field patterns correspond to marks in Figs. 2 and 3. Mode (I) is an odd FP-like one with κ=9.0367, γ=9.655*102, and D=3.49. Mode (II) is an even FP-like one with κ=8.9111, γ=0.1022, and D=3.23. Mode (III) is an odd volume one with κ=8.7076, γ=0.135, and D=4.08. Mode (IV) is an even volume one with κ=8.3764, γ=0.1117, and D=3.95. Mode (V) is an odd HS-like one with κ=9.0652, γ=8.1136*102, and D=5.65. Other parameters are α=1.5 and N=50.

Fig. 6.
Fig. 6.

Relief of determinant (a) for the fully convex kite δ=0.165 and (b) for a zoomed domain shown with dashed lines. Other parameters are α=1.5 and N=100. Marks correspond to the field patterns in Fig. 7.

Fig. 7.
Fig. 7.

Near- and far-field patterns of |Hz| for the WG-like, FP-like, and volume modes. These field patterns correspond to the marks in Fig. 6. Mode (I) is an even WG-like one with κ=23.6404, γ=9.7644*103, and D=6.78. Mode (II) is an even FP-like one with κ=23.3738, γ=3.5438*102, and D=5.48. Mode (III) is an odd FP-like one with κ=23.4871, γ=3.4565*102, and D=5.73. Mode (IV) is an odd volume one with κ=23.7649, γ=3.8783*102, and D=6.7. Other parameters are α=1.5, δ=0.165, and N=100.

Fig. 8.
Fig. 8.

Experimental emission characteristics of a microlaser with a=70μm and δ=0.165. (a) Spectrum recorded in the direction 60°. (b) Fourier transform of the spectrum in (a). The horizontal axis is labeled in units of an, which is the optical length of the corresponding circle radius. (c), (d) Emission patterns for FP-like mode and WG-mode (WGM), respectively.

Fig. 9.
Fig. 9.

Slightly oblique observation of light emission from a kite-shaped microlaser. (a) Scheme of the in-plane configuration. (b) Experimental image in real colors during lasing. The boundaries of the cavity and the layer-free part are emphasized with white lines.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

(Δ+kj2)Uj(r⃗)=0,r⃗=(x,y)Dj,
Ui(r⃗)=Ue(r⃗);ηiUi(r⃗)/n=ηeUe(r⃗)/n,
Uj(r⃗)=Γ(Uj(r⃗)Gj(r⃗,r⃗)nGj(r⃗,r⃗)Uj(r⃗)n)dl,r⃗Dj,
φ(r⃗)+Γφ(r⃗)A(r⃗,r⃗)dlΓψ(r⃗)B(r⃗,r⃗)dl=0.
ηi+ηe2ηeψ(r⃗)+Γφ(r⃗)C(r⃗,r⃗)dlΓψ(r⃗)D(r⃗,r⃗)dl=0.
A(r⃗,r⃗)=Gi(r⃗,r⃗)/nGe(r⃗,r⃗)/n,
B(r⃗,r⃗)=Gi(r⃗,r⃗)(ηi/ηe)Ge(r⃗,r⃗),
C(r⃗,r⃗)=2Gi(r⃗,r⃗)/nn2Ge(r⃗,r⃗)/nn,
D(r⃗,r⃗)=Gi(r⃗,r⃗)/n(ηi/ηe)Ge(r⃗,r⃗)/n.
Gj(r⃗,r⃗)/n=(i/4)kjH1(1)(kjR)(R⃗·n⃗)/R,
Gj(r⃗,r⃗)/n=(i/4)kjH1(1)(kjR)(R⃗·n⃗)/R,
2Gj(r⃗,r⃗)nn=ikj24H2(1)(kjR)(R⃗·n⃗)(R⃗·n⃗)R2+ikj4H1(1)(kjR)(n⃗·n⃗)R.
Gj(t,t)/n=χ0(t)/2,
B(t,τ)tτ(1/2π)[1ηi/ηe]lnR,
C(t,τ)tτ(1/2π)(ki2ke2)lnR.
F(t,τ)=F1(t,τ)ln[4sin2tτ2]+F2(t,τ),F=A,B,C,D,
A1(t,τ)=(1/4π)[kiJ1(kiR)keJ1(keR)](R⃗·n⃗)/R,B1(t,τ)=(1/4π)[J0(kiR)(ηi/ηe)J0(keR)],C1(t,τ)=(1/4π)[ki2J2(kiR)ke2J2(keR)](R⃗·n⃗)(R⃗·n⃗)/R2(1/4π)[kiJ1(kiR)keJ1(keR)](n⃗·n⃗)/R,D1(t,τ)=(1/4π)[kiJ1(kiR)(ηi/ηe)keJ1(keR)](R⃗·n⃗)/R.
02πln[4sin2tτ2]F1(t,τ)f(τ)L(τ)dτp=02N1Pp(N)(t)F1(t,tp)f(tp)L(tp),
Pp(N)(t)=(2π/N)m=1N1cos[m(ttp)]/m(π/N2)cos[N(ttp)].
02πF2(t,τ)f(τ)L(τ)dτ(π/N)p=02N1F2(t,tp)f(tp)L(tp).
[I+A](ΦΨ)=0,
A=[ABCD].
A={(Pp(N)(ts)A1(ts,tp)+(π/N)A2(ts,tp))L(tp)}p,s=02N1,B={2ηe/(ηe+ηi)(Pp(N)(ts)B1(ts,tp)+(π/N)B2(ts,tp))L(tp)}p,s=02N1,C={(Pp(N)(ts)C1(ts,tp)+(π/N)C2(ts,tp))L(tp)}p,s=02N1,D={2ηe/(ηe+ηi)(Pp(N)(ts)D1(ts,tp)+(π/N)D2(ts,tp))L(tp)}p,s=02N1.
det[I+A(κ,γ)]=0.
x(t)=a(cost+δcos2tδ),y(t)=asint,
Ue(r,θ)=r(i+1)eikerΦ(θ)/(4πker),
Φ(θ)=Γ{iφke[n⃗·(cosθ,sinθ)]+(ηi/ηe)ψ}eike[r⃗·(cosθ,sinθ)]dl.
D=(2π/P)|Φ(θmax)|2,P=02π|Φ(θ)|2dθ,

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