Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities

H. E. Tureci; H. G. L. Schwefel; A. Douglas Stone; E. E. Narimanov

doi:10.1364/OE.10.000752

Optics Express
Vol. 10,
Issue 16,
pp. 752-776
(2002)
•https://doi.org/10.1364/OE.10.000752

Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities

H. E. Tureci, H. G. L. Schwefel, A. Douglas Stone, and E. E. Narimanov

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H. E. Tureci, H. G. L. Schwefel, A. Douglas Stone, and E. E. Narimanov, "Gaussian-optical approach to stable periodic orbit resonances of partially chaotic dielectric micro-cavities," Opt. Express 10, 752-776 (2002)

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Abstract

The quasi-bound modes localized on stable periodic ray orbits of dielectric micro-cavities are constructed in the short-wavelength limit using the parabolic equation method. These modes are shown to coexist with irregularly spaced “chaotic” modes for the generic case. The wavevector quantization rule for the quasi-bound modes is derived and given a simple physical interpretation in terms of Fresnel reflection; quasi-bound modes are explictly constructed and compared to numerical results. The effect of discrete symmetries of the resonator is analyzed and shown to give rise to quasi-degenerate multiplets; the average splitting of these multiplets is calculated by methods from quantum chaos theory.

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References

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Year
Author
Publication

R. K. Chang and A. K. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996).
H. Yokoyama, “Physics and device applications of optical microcavities,” Science 256, 66–70 (1992).
[Crossref] [PubMed]
A. D. Stone, “Wave-chaotic optical resonators and lasers,” Phys. Scr. T90, 248–262 (2001).
[Crossref]
C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power directional emission from microlasers with chaotic resonators,” Science 280, 1556–1564 (1998).
[Crossref] [PubMed]
J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[Crossref]
A. J. Campillo, J. D. Eversole, and H. B. Lin, “Cavity quantum electrodynamic enhancement of stimulated-emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]
H. B. Lin, J. D. Eversole, and A. J. Campillo, “Spectral properties of lasing microdroplets,” J. Opt. Soc. Am. B 9, 43–50 (1992).
[Crossref]
S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]
B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[Crossref]
S. X. Qian, J. B. Snow, H. M. Tzeng, and R. K. Chang, “Lasing droplets - highlighting the liquid-air interface by laser-emission,” Science 231, 486–488 (1986).
[Crossref] [PubMed]
A. W. Poon, F. Courvoisier, and R. K. Chang, “Multimode resonances in square-shaped optical microcavities,” Opt. Lett. 26, 632–634 (2001).
[Crossref]
I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, “Hexagonal microlasers based on organic dyes in nanoporous crystals,” Appl. Phys. B-Lasers Opt. 70, 335–343 (2000).
[Crossref]
A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[Crossref] [PubMed]
J. U. Nöckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609–1611 (1996).
[Crossref]
S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, “Long-wavelength λ = 10μm quadrupolar-shaped GaAs-AlGaAs microlasers,” IEEE J. Quantum Electron. 36, 458–464 (2000).
[Crossref]
S. Chang, R. K. Chang, A. D. Stone, and J. U. Nöckel, “Observation of emission from chaotic lasing modes in deformed microspheres: displacement by the stable-orbit modes,” J. Opt. Soc. Am. B-Opt. Phys. 17, 1828–1834 (2000).
[Crossref]
N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” Phys. Rev. Lett. 88, art. no.094 102 (2002).
[Crossref]
N. B. Rex, Regular and chaotic orbit Gallium Nitride microcavity lasers, Ph.D. thesis, Yale University (2001).
C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, “Kolmogorov-Arnold-Moser transition and laser action on scar modes in semiconductor diode lasers with deformed resonators,” Opt. Lett. 27, 824–826 (2002).
[Crossref]
S. B. Lee, J. H. Lee, J. S. Chang, H. J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, art. no.033903 (2002).
[Crossref] [PubMed]
E. J. Heller, “Bound-state eigenfunctions of classically chaotic hamiltonian-systems - Scars of periodic orbits,” Phys. Rev. Lett. 53, 1515–1518 (1984).
[Crossref]
H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, and A. D. Stone, “Dramatic shape sensitivity of emission patterns for similarly deformed cylindrical polymer lasers,” in QELS 2002 Technical Digest, (Baltimore, MD, 2002), pp. 24–25.
M. V. Berry, “Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard,” Eur. J. Phys. 2, 91–102 (1981).
[Crossref]
B. Li and M. Robnik, “Geometry of high-lying eigenfunctions in a plane billiard system having mixed type classical dynamics,” J. Phys. A 28, 2799–2818 (1995).
[Crossref]
J. B. Keller and S. I. Rubinow, “Asymptotic Solution of Eigenvalue Problems,” Ann. Phys. 9, 24–75 (1960).
[Crossref]
M. C. Gutzwiller, Chaos in classical and quantum mechanics (Springer, New York, USA, 1990).
W. H. Miller, “Semiclassical quantization of nonseparable systems: A new look at periodic orbit theory,” J. Chem. Phys. 63, 996–999 (1975).
[Crossref]
S. D. Frischat and E. Doron, “Quantum phase-space structures in classically mixed systems: A scattering approach,” J. Phys. A-Math. Gen. 30, 3613–3634 (1997).
[Crossref]
O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, “Signature of dynamical localization in the resonance width distribution of wave-chaotic dielectric cavities,” Phys. Rev. E 62, 2078–2084 (2000).
[Crossref]
J. U. Nöckel, “Angular momentum localization in oval billiards,” Phys. Scr. T90, 263–267 (2001).
[Crossref]
V. M. BabiČ and V. S. Buldyrev, Asymptotic Methods in Shortwave Diffraction Problems (Springer, New York, USA, 1991).
F. Laeri and J. U. Nöckel, Nanoporous compound materials for optical applications - Microlasers and microresonators, in Handbook of Advanced Electronic and Photonic Materials, H. S. Nalwa, ed. (Academic Press, San Diego, 2001).
[Crossref]
A. E. Siegman, Lasers (University Science Books, Mill Valley, California, 1986).
V. P. Maslov and M. V. Fedoriuk, Semiclassical Approximations in Quantum Mechanics (Reidel, Boston, USA, 1981).
[Crossref]
N. A. Chernikov, “System whose hamiltonian is a time-dependent quadratic form in x and p,” Sov Phys-Jetp Engl Trans 26, 603–608 (1968).
E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, T. S. S., and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]
J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math 24, 396–413 (1973).
[Crossref]
J. M. Robbins, “Discrete symmetries in periodic-orbit theory,” Phys. Rev. A. 40, 2128–2136 (1989).
[Crossref] [PubMed]
M. J. Davis and E. J. Heller, “Multidimensional wave functions from classical trajectories,” J. Chem. Phys. 75, 246 (1981).
[Crossref]
O. Bohigas, S. Tomsovic, and D. Ullmo, “Manifestations of classical phase space structures in quantum mechanics,” Phys. Rep. 223, 45 (1993).
[Crossref]
S. D. Frischat and E. Doron, “Semiclassical description of tunneling in mixed systems: case of the annular billiard,” Phys. Rev. Lett. 75, 3661 (1995).
[Crossref] [PubMed]
F. Leyvraz and D. Ullmo, “The level splitting distribution in chaos-assisted tunneling,” J. Phys. A 29, 2529 (1996).
[Crossref]
E. E. Narimanov, unpublished.
M. V. Berry, “Regular and irregular semiclassical wavefunctions,” J. Phys. A 10, 2083 (1977).
[Crossref]
H. E. Tureci and A. D. Stone, “Deviation from Snell’s law for beams transmitted near the critical angle: application to microcavity lasers,” Opt. Lett. 27, 7–9 (2002).
[Crossref]
E. E. Narimanov, G. Hackenbroich, P. Jacquod, and A. D. Stone, “Semiclassical theory of the emission properties of wave-chaotic resonant cavities,” Phys. Rev. Lett. 83, 4991–4994 (1999).
[Crossref]

2002 (5)

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

N. B. Rex, H. E. Tureci, H. G. L. Schwefel, R. K. Chang, and A. D. Stone, “Fresnel filtering in lasing emission from scarred modes of wave-chaotic optical resonators,” Phys. Rev. Lett. 88, art. no.094 102 (2002).
[Crossref]

C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, “Kolmogorov-Arnold-Moser transition and laser action on scar modes in semiconductor diode lasers with deformed resonators,” Opt. Lett. 27, 824–826 (2002).
[Crossref]

S. B. Lee, J. H. Lee, J. S. Chang, H. J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, art. no.033903 (2002).
[Crossref] [PubMed]

H. E. Tureci and A. D. Stone, “Deviation from Snell’s law for beams transmitted near the critical angle: application to microcavity lasers,” Opt. Lett. 27, 7–9 (2002).
[Crossref]

2001 (3)

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

A. D. Stone, “Wave-chaotic optical resonators and lasers,” Phys. Scr. T90, 248–262 (2001).
[Crossref]

J. U. Nöckel, “Angular momentum localization in oval billiards,” Phys. Scr. T90, 263–267 (2001).
[Crossref]

2000 (4)

O. A. Starykh, P. R. J. Jacquod, E. E. Narimanov, and A. D. Stone, “Signature of dynamical localization in the resonance width distribution of wave-chaotic dielectric cavities,” Phys. Rev. E 62, 2078–2084 (2000).
[Crossref]

I. Braun, G. Ihlein, F. Laeri, J. U. Nöckel, G. Schulz-Ekloff, F. Schuth, U. Vietze, O. Weiss, and D. Wohrle, “Hexagonal microlasers based on organic dyes in nanoporous crystals,” Appl. Phys. B-Lasers Opt. 70, 335–343 (2000).
[Crossref]

S. Gianordoli, L. Hvozdara, G. Strasser, W. Schrenk, J. Faist, and E. Gornik, “Long-wavelength λ = 10μm quadrupolar-shaped GaAs-AlGaAs microlasers,” IEEE J. Quantum Electron. 36, 458–464 (2000).
[Crossref]

S. Chang, R. K. Chang, A. D. Stone, and J. U. Nöckel, “Observation of emission from chaotic lasing modes in deformed microspheres: displacement by the stable-orbit modes,” J. Opt. Soc. Am. B-Opt. Phys. 17, 1828–1834 (2000).
[Crossref]

1999 (1)

E. E. Narimanov, G. Hackenbroich, P. Jacquod, and A. D. Stone, “Semiclassical theory of the emission properties of wave-chaotic resonant cavities,” Phys. Rev. Lett. 83, 4991–4994 (1999).
[Crossref]

1998 (3)

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

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, “Ultra-compact Si-SiO2 microring resonator optical channel dropping filters,” IEEE Photonics Technol. Lett. 10, 549–551 (1998).
[Crossref]

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, T. S. S., and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70, 1545–1554 (1998).
[Crossref]

1997 (2)

S. D. Frischat and E. Doron, “Quantum phase-space structures in classically mixed systems: A scattering approach,” J. Phys. A-Math. Gen. 30, 3613–3634 (1997).
[Crossref]

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[Crossref]

1996 (2)

J. U. Nöckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609–1611 (1996).
[Crossref]

F. Leyvraz and D. Ullmo, “The level splitting distribution in chaos-assisted tunneling,” J. Phys. A 29, 2529 (1996).
[Crossref]

1995 (3)

S. D. Frischat and E. Doron, “Semiclassical description of tunneling in mixed systems: case of the annular billiard,” Phys. Rev. Lett. 75, 3661 (1995).
[Crossref] [PubMed]

B. Li and M. Robnik, “Geometry of high-lying eigenfunctions in a plane billiard system having mixed type classical dynamics,” J. Phys. A 28, 2799–2818 (1995).
[Crossref]

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[Crossref] [PubMed]

1993 (1)

O. Bohigas, S. Tomsovic, and D. Ullmo, “Manifestations of classical phase space structures in quantum mechanics,” Phys. Rep. 223, 45 (1993).
[Crossref]

1992 (2)

H. Yokoyama, “Physics and device applications of optical microcavities,” Science 256, 66–70 (1992).
[Crossref] [PubMed]

H. B. Lin, J. D. Eversole, and A. J. Campillo, “Spectral properties of lasing microdroplets,” J. Opt. Soc. Am. B 9, 43–50 (1992).
[Crossref]

1991 (1)

A. J. Campillo, J. D. Eversole, and H. B. Lin, “Cavity quantum electrodynamic enhancement of stimulated-emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

1989 (1)

J. M. Robbins, “Discrete symmetries in periodic-orbit theory,” Phys. Rev. A. 40, 2128–2136 (1989).
[Crossref] [PubMed]

1986 (1)

S. X. Qian, J. B. Snow, H. M. Tzeng, and R. K. Chang, “Lasing droplets - highlighting the liquid-air interface by laser-emission,” Science 231, 486–488 (1986).
[Crossref] [PubMed]

1984 (1)

E. J. Heller, “Bound-state eigenfunctions of classically chaotic hamiltonian-systems - Scars of periodic orbits,” Phys. Rev. Lett. 53, 1515–1518 (1984).
[Crossref]

1981 (2)

M. V. Berry, “Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard,” Eur. J. Phys. 2, 91–102 (1981).
[Crossref]

M. J. Davis and E. J. Heller, “Multidimensional wave functions from classical trajectories,” J. Chem. Phys. 75, 246 (1981).
[Crossref]

1977 (1)

M. V. Berry, “Regular and irregular semiclassical wavefunctions,” J. Phys. A 10, 2083 (1977).
[Crossref]

1975 (1)

W. H. Miller, “Semiclassical quantization of nonseparable systems: A new look at periodic orbit theory,” J. Chem. Phys. 63, 996–999 (1975).
[Crossref]

1973 (1)

J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math 24, 396–413 (1973).
[Crossref]

1968 (1)

N. A. Chernikov, “System whose hamiltonian is a time-dependent quadratic form in x and p,” Sov Phys-Jetp Engl Trans 26, 603–608 (1968).

1960 (1)

J. B. Keller and S. I. Rubinow, “Asymptotic Solution of Eigenvalue Problems,” Ann. Phys. 9, 24–75 (1960).
[Crossref]

An, K.

BabiC, V. M.

V. M. BabiČ and V. S. Buldyrev, Asymptotic Methods in Shortwave Diffraction Problems (Springer, New York, USA, 1991).

Baillargeon, J. N.

Berry, M. V.

M. V. Berry, “Regularity and chaos in classical mechanics, illustrated by three deformations of a circular billiard,” Eur. J. Phys. 2, 91–102 (1981).
[Crossref]

M. V. Berry, “Regular and irregular semiclassical wavefunctions,” J. Phys. A 10, 2083 (1977).
[Crossref]

Bertoni, H. L.

J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math 24, 396–413 (1973).
[Crossref]

Bohigas, O.

O. Bohigas, S. Tomsovic, and D. Ullmo, “Manifestations of classical phase space structures in quantum mechanics,” Phys. Rep. 223, 45 (1993).
[Crossref]

Braun, I.

Brink, A. Maassen van den

Buldyrev, V. S.

V. M. BabiČ and V. S. Buldyrev, Asymptotic Methods in Shortwave Diffraction Problems (Springer, New York, USA, 1991).

Campillo, A. J.

H. B. Lin, J. D. Eversole, and A. J. Campillo, “Spectral properties of lasing microdroplets,” J. Opt. Soc. Am. B 9, 43–50 (1992).
[Crossref]

A. J. Campillo, J. D. Eversole, and H. B. Lin, “Cavity quantum electrodynamic enhancement of stimulated-emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

Capasso, F.

Chang, J. S.

Chang, R. K.

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

J. U. Nöckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609–1611 (1996).
[Crossref]

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[Crossref] [PubMed]

S. X. Qian, J. B. Snow, H. M. Tzeng, and R. K. Chang, “Lasing droplets - highlighting the liquid-air interface by laser-emission,” Science 231, 486–488 (1986).
[Crossref] [PubMed]

H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, and A. D. Stone, “Dramatic shape sensitivity of emission patterns for similarly deformed cylindrical polymer lasers,” in QELS 2002 Technical Digest, (Baltimore, MD, 2002), pp. 24–25.

Chang, S.

Chen, G.

J. U. Nöckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609–1611 (1996).
[Crossref]

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[Crossref] [PubMed]

Chernikov, N. A.

N. A. Chernikov, “System whose hamiltonian is a time-dependent quadratic form in x and p,” Sov Phys-Jetp Engl Trans 26, 603–608 (1968).

Ching, E. S. C.

Cho, A. Y.

Chu, S. T.

Courvoisier, F.

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

Davis, M. J.

M. J. Davis and E. J. Heller, “Multidimensional wave functions from classical trajectories,” J. Chem. Phys. 75, 246 (1981).
[Crossref]

Doron, E.

S. D. Frischat and E. Doron, “Quantum phase-space structures in classically mixed systems: A scattering approach,” J. Phys. A-Math. Gen. 30, 3613–3634 (1997).
[Crossref]

S. D. Frischat and E. Doron, “Semiclassical description of tunneling in mixed systems: case of the annular billiard,” Phys. Rev. Lett. 75, 3661 (1995).
[Crossref] [PubMed]

Eversole, J. D.

H. B. Lin, J. D. Eversole, and A. J. Campillo, “Spectral properties of lasing microdroplets,” J. Opt. Soc. Am. B 9, 43–50 (1992).
[Crossref]

A. J. Campillo, J. D. Eversole, and H. B. Lin, “Cavity quantum electrodynamic enhancement of stimulated-emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991).
[Crossref] [PubMed]

Faist, J.

Fedoriuk, M. V.

V. P. Maslov and M. V. Fedoriuk, Semiclassical Approximations in Quantum Mechanics (Reidel, Boston, USA, 1981).
[Crossref]

Felsen, L. B.

J. W. Ra, H. L. Bertoni, and L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math 24, 396–413 (1973).
[Crossref]

Foresi, J. S.

Frischat, S. D.

S. D. Frischat and E. Doron, “Quantum phase-space structures in classically mixed systems: A scattering approach,” J. Phys. A-Math. Gen. 30, 3613–3634 (1997).
[Crossref]

S. D. Frischat and E. Doron, “Semiclassical description of tunneling in mixed systems: case of the annular billiard,” Phys. Rev. Lett. 75, 3661 (1995).
[Crossref] [PubMed]

Gianordoli, S.

Gmachl, C.

Gornik, E.

Greene, W.

Grossman, H. L.

J. U. Nöckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609–1611 (1996).
[Crossref]

Gutzwiller, M. C.

M. C. Gutzwiller, Chaos in classical and quantum mechanics (Springer, New York, USA, 1990).

Hackenbroich, G.

Haus, H. A.

Heller, E. J.

E. J. Heller, “Bound-state eigenfunctions of classically chaotic hamiltonian-systems - Scars of periodic orbits,” Phys. Rev. Lett. 53, 1515–1518 (1984).
[Crossref]

M. J. Davis and E. J. Heller, “Multidimensional wave functions from classical trajectories,” J. Chem. Phys. 75, 246 (1981).
[Crossref]

Hvozdara, L.

Ihlein, G.

Ippen, E. P.

Jacquod, P.

Jacquod, P. R. J.

Keller, J. B.

J. B. Keller and S. I. Rubinow, “Asymptotic Solution of Eigenvalue Problems,” Ann. Phys. 9, 24–75 (1960).
[Crossref]

Kim, S. W.

Kimerling, L. C.

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

Laeri, F.

F. Laeri and J. U. Nöckel, Nanoporous compound materials for optical applications - Microlasers and microresonators, in Handbook of Advanced Electronic and Photonic Materials, H. S. Nalwa, ed. (Academic Press, San Diego, 2001).
[Crossref]

Lee, J. H.

Lee, S. B.

Leung, P. T.

Leyvraz, F.

F. Leyvraz and D. Ullmo, “The level splitting distribution in chaos-assisted tunneling,” J. Phys. A 29, 2529 (1996).
[Crossref]

Li, B.

B. Li and M. Robnik, “Geometry of high-lying eigenfunctions in a plane billiard system having mixed type classical dynamics,” J. Phys. A 28, 2799–2818 (1995).
[Crossref]

Lin, H. B.

H. B. Lin, J. D. Eversole, and A. J. Campillo, “Spectral properties of lasing microdroplets,” J. Opt. Soc. Am. B 9, 43–50 (1992).
[Crossref]

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[Crossref] [PubMed]

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[Crossref]

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J. U. Nöckel, “Angular momentum localization in oval billiards,” Phys. Scr. T90, 263–267 (2001).
[Crossref]

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[Crossref]

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[Crossref]

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[Crossref] [PubMed]

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

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[Crossref]

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[Crossref]

J. U. Nöckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609–1611 (1996).
[Crossref]

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
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H. E. Tureci and A. D. Stone, “Deviation from Snell’s law for beams transmitted near the critical angle: application to microcavity lasers,” Opt. Lett. 27, 7–9 (2002).
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[Crossref] [PubMed]

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[Crossref]

Opt. Lett. (4)

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

J. U. Nöckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609–1611 (1996).
[Crossref]

H. E. Tureci and A. D. Stone, “Deviation from Snell’s law for beams transmitted near the critical angle: application to microcavity lasers,” Opt. Lett. 27, 7–9 (2002).
[Crossref]

Phys. Rep. (1)

O. Bohigas, S. Tomsovic, and D. Ullmo, “Manifestations of classical phase space structures in quantum mechanics,” Phys. Rep. 223, 45 (1993).
[Crossref]

Phys. Rev. A. (1)

J. M. Robbins, “Discrete symmetries in periodic-orbit theory,” Phys. Rev. A. 40, 2128–2136 (1989).
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Phys. Rev. E (1)

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[Crossref] [PubMed]

A. Mekis, J. U. Nöckel, G. Chen, A. D. Stone, and R. K. Chang, “Ray chaos and q spoiling in lasing droplets,” Phys. Rev. Lett. 75, 2682–2685 (1995).
[Crossref] [PubMed]

S. D. Frischat and E. Doron, “Semiclassical description of tunneling in mixed systems: case of the annular billiard,” Phys. Rev. Lett. 75, 3661 (1995).
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Phys. Scr. (2)

A. D. Stone, “Wave-chaotic optical resonators and lasers,” Phys. Scr. T90, 248–262 (2001).
[Crossref]

J. U. Nöckel, “Angular momentum localization in oval billiards,” Phys. Scr. T90, 263–267 (2001).
[Crossref]

Rev. Mod. Phys. (1)

Science (3)

H. Yokoyama, “Physics and device applications of optical microcavities,” Science 256, 66–70 (1992).
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SIAM J. Appl. Math (1)

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E. E. Narimanov, unpublished.

N. B. Rex, Regular and chaotic orbit Gallium Nitride microcavity lasers, Ph.D. thesis, Yale University (2001).

R. K. Chang and A. K. Campillo, eds., Optical Processes in Microcavities (World Scientific, Singapore, 1996).

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A. E. Siegman, Lasers (University Science Books, Mill Valley, California, 1986).

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[Crossref]

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Fig. 1. Surface of section illustrating the different regions of phase space for a closed quadrupole billiard with boundary given by r(ϕ) = R(1 + ∊ cos 2ϕ) for ∊ = 0.072. Real-space ray trajectories corresponding to each region are indicated at right: a) A quasi-periodic, marginally stable orbit. b) A stable four-bounce “diamond” periodic orbit (surrounded by stability “islands” in the SOS) c) A chaotic ray trajectory. Orbits of type (b) have associated with them regular gaussian solutions as we will show below.

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Fig. 2. Black background gives the surface of section for the quadrupole at ∊ = 0.17 for which the four small islands correspond to a stable bow-tie shaped orbit (inset). A numerical solution of the Helmholtz equation for this resonator can be projected onto this surface of section via the Husimi transform[28] and is found to have high intensity (in false color scale) precisely on these islands, indicating that this is a mode associated with the bow-tie orbit.

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Fig. 3. (a) Vertical lines indicate wavevectors of bound states of the closed quadrupole resonator for ∊ = 0.17; no regular spacings are visible. (b) Spectrum weighted by overlap of the Husimi function of the solution with the bow-tie island as in Fig. 2. Note the emergence of regularly spaced levels with two main spacings Δk_long and Δk_trans . These spacings, indicated by the arrows, are calculated from the length of the bow-tie orbit and the associated Floquet phase (see Section 2.4 below). The color coding corresponds to the four possible symmetry types of the solutions (see Section 4 below). In the inset is a magnified view showing the splitting of quasi-degenerate doublets as discussed in Section 4.3. Note the pairing of the (+, +) and (+, -) symmetry types as predicted in section 4.2. The different symmetry pairs alternate every free spectral range (Δk_long ).

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Fig. 4. Illustration of the reduction of the Maxwell equation for an infinite dielectric cylinder to the 2D Helmholtz equation for the TM case (E field parallel to axis) and k _∥ = 0.

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Fig. 5. Coordinate system and variables used in the text displayed for the case of a quadrupolar boundary ∂D and the diamond four-bounce PO. A fixed coordinate system (X, Z) is attached to the origin. The “mobile” coordinate systems (x_m ,z_m ) are fixed on segments of the periodic orbit so that their respective z-axes are parallel to the segment, while their origins are set back a distance L_m (or nL_m for transmitted beam axes), so as to account for zeroth order phase accumulation between successive bounce-points. ξ ₁, ξ ₂ are the common local coordinates at each bounce (index m suppressed). Scaled coordinates are denoted by tildes, e.g. x̃_m = √kx_m . The coordinate transformations at each bounce m are given by z_i = L_m + ξ ₁ sin χ_i + ξ ₂ cos χ_i , z_r = L_m + ξ ₁ sin χ_i - ξ ₂ cos χ_i , z_t = nL_m + ξ ₁ sin χ_t + ξ ₂ cos χ_t and x_i = ξ ₁ cos χ_i - ξ ₂ sin χ_i , x_r = ξ ₁ cos χ_i + ξ ₂ sin χ_i , x_t = ξ ₁ cos χ_t - ξ ₂ sin χ_t , where i, r, t refer to the incident, transmitted and reflected solutions.

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Fig. 6. Intensity of the TM solution for a bow-tie mode plotted in a false color scale, (a) calculated numerically and (b) from the gaussian optical theory with parameters m = 100, φ = 2.11391, N_μ = 1 and N = 4. Note the excellent agreement of the quantized values for kR (R is the average radius of the quadrupole).

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Fig. 7. Schematic indicating a direct tunneling process (black arrow) and a chaos-assisted tunneling process (yellow arrow) which would contribute to splitting of bow-tie doublets.

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Fig. 8. The numerically determined splittings of bow-tie doublets for a closed quadrupole resonator with ε = 0.14 (black dots) vs kR; the red line denotes the prediction of Eq. (60) for the average splitting, the blue line an estimate of the splitting based on the “direct” coupling. Note the large enhancement due to chaos-assisted tunneling and the large fluctuations around the mean splitting.

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Tables (2)

Table 1. Comparison of the gaussian optical predictions

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Table 2. Illustration of the symmetry rules for the quadrupole

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Equations (72)

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(\nabla^{2} + k^{2}) E = 0

E (X, Z) = \sum_{m = 1}^{N} E_{m} (x_{m} (X, Z), z_{m} (X, Z))

E_{m} (x_{m}, z_{m}) = u_{m} (x_{m}, z_{m}) e^{ik z_{m}}

(\nabla_{m}^{2} + k^{2}) E_{m} = 0

E_{m} + E_{m + 1} |_{\partial D} = 0

u_{xx} + u_{zz} + 2 ik u_{z} = 0

u_{\tilde{x} \tilde{x}} + \frac{1}{k} u_{zz} + 2 i u_{z} = 0 .

ℒ u (\tilde{x}, z) = 0

u (x, z) = c A (z) \exp [\frac{i}{2} Ω (z) {\tilde{x}}^{2}]

Ω^{2} + Ω' = 0

A Ω + 2 A' = 0

Ω = \frac{Q' (z)}{Q (z)}

Q ″ = 0

\frac{Q'}{Q} + 2 \frac{A'}{A} = 0

\frac{c_{m}}{\sqrt{Q_{m} (z_{m})}} \exp (ik z_{m} + \frac{i}{2} Ω (z_{m}) {\tilde{x}}_{m}^{2})

+ \frac{c_{m + 1}}{\sqrt{Q_{m + 1} (z_{m + 1})}} \exp (ik z_{m + 1} + \frac{i}{2} Ω (z_{m + 1}) {\tilde{x}}_{m + 1}^{2}) |_{\partial D} = 0

k (l_{m} + \frac{1}{\sqrt{k}} {\tilde{ξ}}_{1} \sin χ_{m} - \frac{1}{k} \frac{{\tilde{ξ}}_{1}^{2}}{2 ρ_{m}} \cos χ_{m}) + \frac{1}{2} \frac{Q_{m}^{'}}{Q_{m}} {({\tilde{ξ}}_{1} \cos χ_{m} + \frac{1}{k} \frac{{\tilde{ξ}}_{1}^{2}}{2 ρ_{m}} \sin χ_{m})}^{2} =

k (l_{m} + \frac{1}{\sqrt{k}} {\tilde{ξ}}_{1} \sin χ_{m} + \frac{1}{k} \frac{{\tilde{ξ}}_{1}^{2}}{2 ρ_{m}} \cos χ_{m}) + \frac{1}{2} \frac{Q_{m + 1}^{'}}{Q_{m + 1}} {({\tilde{ξ}}_{1} \cos χ_{m} + \frac{1}{k} \frac{{\tilde{ξ}}_{1}^{2}}{2 ρ_{m}} \sin χ_{m})}^{2}

\frac{c_{m}}{\sqrt{Q_{m} (L_{m})}} + \frac{c_{m + 1}}{\sqrt{Q_{m + 1} (L_{m})}} = 0

(\begin{matrix} Q_{m + 1} \\ Q_{m + 1}^{'} \end{matrix}) = (\begin{matrix} 1 & 0 \\ - \frac{2}{ρ_{m} \cos χ_{m}} & 1 \end{matrix}) (\begin{matrix} Q_{m} \\ Q_{m}^{'} \end{matrix}) \equiv ℜ_{m} (\begin{matrix} Q_{m} \\ Q_{m}^{'} \end{matrix})

C_{m + 1} = e^{- i π} C_{m}

\prod = (\begin{matrix} Q_{1} & Q_{2} \\ P_{1} & P_{2} \end{matrix})

\frac{d \prod}{dz} = ℑ H \prod

(\begin{matrix} Q (z) \\ P (z) \end{matrix}) = \prod (z) h_{m}

(\begin{matrix} Q_{m} (z + l) \\ P_{m} (z + l) \end{matrix}) = \prod (l) (\begin{matrix} Q_{m} (z) \\ P_{m} (z) \end{matrix}) .

(\begin{matrix} Q_{m + 1} (z') \\ P_{m + 1} (z') \end{matrix}) = \prod (z' - L_{m}) ℜ_{m} \prod (L_{m} - z) (\begin{matrix} Q_{m} (z) \\ P_{m} (z) \end{matrix})

𝚳 (z) = \prod (z - L_{m - 1}) ℜ_{m - 1} \prod (l_{m - 1}) \dots \prod (l_{m + 1}) ℜ_{m} \prod (L_{m} - z)

E (x, z + L) = E (x, z) .

u (x, z + L) e^{ikL} = u (x, z) .

p q^{*} - q p^{*} = i .

u (\tilde{x}, z + L) = e^{- i \frac{φ}{2} - iπ (N_{μ} + N)} u (\tilde{x}, z)

kL = \frac{1}{2} φ + 2 π m + \mod_{2 π} [(N + N_{μ}) π]

N_{μ} = [\frac{1}{2 π i} \int_{0}^{L} d (\ln q (z))]

Λ (z) = - iq (z) \partial_{\tilde{x}} + p (z) \tilde{x}

Λ^{†} (z) = i q^{*} (z) \partial_{\tilde{x}} + p^{*} (z) \tilde{x} .

u^{(l)} (\tilde{x}, z) = {(Λ^{†})}^{l} u^{(0)}

kL = (l + \frac{1}{2}) φ + 2 π m + \mod_{2 π} [(N + N_{μ}) π]

u^{(l)} (\tilde{x}, z) = {(i \sqrt{\frac{q^{*} (z)}{2 q (z)}})}^{l} H_{l} (\sqrt{I_{m} [\frac{p (z)}{q (z)}] \tilde{x}}) u^{(0)} (\tilde{x}, z)

(\nabla^{2} + n {(r)}^{2} k^{2}) Ψ = 0

E_{i} + E_{r} |_{\partial D^{-}} = E_{t} |_{\partial D^{+}}

\partial_{n} E_{i} + \partial_{n} E_{r} |_{\partial D^{-}} = \partial_{n} E_{t} |_{\partial D^{+}}

Φ_{i} = nk (L_{m} + \frac{1}{\sqrt{nk}} {\tilde{ξ}}_{1} \sin χ - \frac{1}{nk} \frac{{\tilde{ξ}}_{1}^{2}}{2 ρ} \cos χ) + \frac{1}{2} \frac{Q_{i}^{'}}{Q_{i}} {({\tilde{ξ}}_{1} \cos χ + \frac{1}{nk} \frac{{\tilde{ξ}}_{1}^{2}}{2 ρ} \sin χ)}^{2}

Φ_{t} = k (n L_{m} + \frac{1}{\sqrt{nk}} {\tilde{ξ}}_{1} \sin χ_{t} - \frac{1}{nk} \frac{{\tilde{ξ}}_{1}^{2}}{2 ρ} \cos χ_{t}) + \frac{1}{2} \frac{Q_{t}^{'}}{Q_{t}} {({\tilde{ξ}}_{1} \cos χ_{t} + \frac{1}{nk} \frac{{\tilde{ξ}}_{1}^{2}}{2 ρ} \sin χ_{t})}^{2}

(\begin{matrix} Q_{t} \\ Q_{t}^{'} \end{matrix}) = (\begin{matrix} \frac{1}{μ} & 0 \\ - \frac{2 (1 - μ)}{ρ \cos χ} & n μ \end{matrix}) (\begin{matrix} Q_{i} \\ Q_{i}^{'} \end{matrix})

\frac{c_{i}}{\sqrt{Q_{i}}} + \frac{c_{r}}{\sqrt{Q_{r}}} = \frac{c_{t}}{\sqrt{Q_{t}}}

c_{i} + c_{r} = \sqrt{μ} c_{t}

c_{i} \partial_{n} Φ_{i} + c_{r} \partial_{n} Φ_{r} = \sqrt{μ} c_{t} \partial_{n} Φ_{t} .

n \sqrt{μ} (c_{i} - c_{r}) = c_{t}

c_{r} = \frac{nμ - 1}{nμ + 1} c_{i} .

{∣ c_{r} ∣}^{2} = \frac{{∣ n \cos χ_{i} - \cos χ_{t} ∣}^{2}}{{∣ n \cos χ_{i} + \cos χ_{t} ∣}^{2}} {∣ c_{i} ∣}^{2}

E (x, z + L) = E (x, z)

u (x, z + L) e^{inkL} = u (x, z)

nkL = \frac{1}{2} φ + 2 πm + \mod_{2 π} [(N + N_{μ}) π] - i \sum_{b = 1}^{N} \log [\frac{n μ_{b} - 1}{n μ_{n} + 1}] .

Re [nkL] = 2 πm + \mod_{2 π} [(N + N_{μ}) π] + \frac{φ}{2} + φ_{f}

Im [nkL] = - γ_{f} .

P_{m} E (x) = \frac{d_{m}}{∣ G ∣} \sum_{g \in G} χ_{m} (g) E (g x)

E_{(+ +)} = \frac{1}{4} (e_{1} + e_{2} + e_{3} + e_{4}), E_{(+ -)} = \frac{1}{4} (e_{1} + e_{2} - e_{3} - e_{4})

E_{(- +)} = \frac{1}{4} (e_{1} - e_{2} + e_{3} - e_{4}), E_{(- -)} = \frac{1}{4} (e_{1} - e_{2} - e_{3} + e_{4})

e_{1} = E (X, Z), e_{2} = E (- X, Z), e_{3} = E (X, - Z), e_{4} = E (- X, - Z) .

e_{1} = E (g x) = \frac{1}{\sqrt{q (z)}} \exp [ikz + \frac{i}{2} Ω (z) x^{2}]

e_{2} = E (g x) = e^{\frac{1}{2} i π} \frac{1}{\sqrt{e^{\frac{i φ}{2}} q (z)}} \exp [ik (z + \frac{L}{2} + \frac{i}{2} Ω (z) x^{2})] \equiv e^{i ζ} e_{1}

e_{3} = E (g x) = \frac{1}{\sqrt{q (ℓ_{1} - z)}} \exp [ik (ℓ_{1} - z) + \frac{i}{2} Ω (ℓ_{1} - z) x^{2}]

e_{4} = E (g x) = e^{\frac{1}{2} i π} \frac{1}{\sqrt{e^{\frac{i φ}{2}} q (ℓ_{1} - z)}} \exp [ik (\frac{L}{2} + ℓ_{1} - z) + \frac{i}{2} Ω (ℓ_{1} - z) x^{2}] \equiv e^{i ζ} e_{3}

E_{(+ +)} = \frac{1}{2} (e_{1} + e_{3}) e^{i \frac{ζ}{2}} \cos \frac{ζ}{2}

E_{(+ -)} = \frac{1}{2} (e_{1} - e_{3}) e^{i \frac{ζ}{2}} \cos \frac{ζ}{2}

E_{(- +)} = \frac{1}{2 i} (e_{1} + e_{3}) e^{i \frac{ζ}{2}} \sin \frac{ζ}{2}

E_{(- -)} = \frac{1}{2 i} (e_{1} + e_{3}) e^{i \frac{ζ}{2}} \sin \frac{ζ}{2}

E_{(+ +)}, E_{(+ -)} \propto \cos \frac{ζ}{2} = 0 m = 1,3,5, . . .

E_{(- +)}, E_{(- -)} \propto \sin \frac{ζ}{2} = 0 m = 0,2,4, . . .

Δ E_{CAT} ≃ \frac{{∣ V_{RC} ∣}^{2}}{E_{R} - E_{C}}

{∣ V_{RC} ∣}^{2} \propto \int d ϕ \int d \sin χ W_{C} (ϕ, \sin χ) W_{R} (ϕ, \sin χ)

〈 Δ E_{CAT} 〉 \propto \exp (- Ak R_{0})

index of refraction	numerical calculation	gaussian quantization rule
n = 2.0	nkR = 100.53788 + 0.49758i	nkR = 100.53858 + 0.49635i
n = 2.9	nkR = 100.59376 + 0.16185i	nkR = 100.53858 + 0.14178i
n = 5.1	nkR = 100.84716 + 0.00245i	nkR = 100.85111 + 0.00000i

symmetry-related	time-reversal	quasi-degeneracy	symmetry-pairing
1	2	2	[(++),(+-)] [(--),(-+)]
1	2	2	[(++),(--)] [(+-),(-+)]
2	2	4	[(++),(+-) [(-+),(--)]
2	1	2	[(++),(-+)] [(+-),(--)]
1	1	1	[(++)] [(+-)]

index of refraction	numerical calculation	gaussian quantization rule
n = 2.0	nkR = 100.53788 + 0.49758i	nkR = 100.53858 + 0.49635i
n = 2.9	nkR = 100.59376 + 0.16185i	nkR = 100.53858 + 0.14178i
n = 5.1	nkR = 100.84716 + 0.00245i	nkR = 100.85111 + 0.00000i

symmetry-related	time-reversal	quasi-degeneracy	symmetry-pairing
1	2	2	[(++),(+-)] [(--),(-+)]
1	2	2	[(++),(--)] [(+-),(-+)]
2	2	4	[(++),(+-) [(-+),(--)]
2	1	2	[(++),(-+)] [(+-),(--)]
1	1	1	[(++)] [(+-)]