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.

© 2002 Optical Society of America

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    [CrossRef]
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  38. J. M. Robbins, “Discrete symmetries in periodic-orbit theory,” Phys. Rev. A. 40, 2128–2136 (1989).
    [CrossRef] [PubMed]
  39. M. J. Davis and E. J. Heller, “Multidimensional wave functions from classical trajectories,” J. Chem. Phys. 75, 246 (1981).
    [CrossRef]
  40. O. Bohigas, S. Tomsovic, and D. Ullmo, “Manifestations of classical phase space structures in quantum mechanics,” Phys. Rep. 223, 45 (1993).
    [CrossRef]
  41. 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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  46. 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)

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)

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.

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]

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.

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]

Brink, A. Maassen van den

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]

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.

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]

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]

Chang, J. S.

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]

Chang, R. K.

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]

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

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.

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]

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.

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]

Cho, A. Y.

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]

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]

Chu, S. T.

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]

Courvoisier, F.

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.

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]

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]

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.

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]

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

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

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.

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

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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]

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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).
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[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]

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]

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]

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]

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.

Strasser, G.

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).
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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).
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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).
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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).
[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]

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.

Tzeng, H. M.

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).
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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]

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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]

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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]

Wohrle, D.

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).
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[CrossRef]

Ann. Phys. (1)

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

Appl. Phys. B-Lasers Opt. (1)

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).
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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]

IEEE Photonics Technol. Lett. (1)

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]

J. Chem. Phys. (2)

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

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

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

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

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).
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Figures (8)

Fig. 1.
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.

Fig. 2.
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.

Fig. 3.
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 Δklong and Δktrans . 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 (Δklong ).

Fig. 4.
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.

Fig. 5.
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 (xm ,zm ) 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 Lm (or nLm 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. m = √kxm . The coordinate transformations at each bounce m are given by zi = Lm + ξ 1 sin χi + ξ 2 cos χi , zr = Lm + ξ 1 sin χi - ξ 2 cos χi , zt = nLm + ξ 1 sin χt + ξ 2 cos χt and xi = ξ 1 cos χi - ξ 2 sin χi , xr = ξ 1 cos χi + ξ 2 sin χi , xt = ξ 1 cos χt - ξ 2 sin χt , where i, r, t refer to the incident, transmitted and reflected solutions.

Fig. 6.
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, Nμ = 1 and N = 4. Note the excellent agreement of the quantized values for kR (R is the average radius of the quadrupole).

Fig. 7.
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.

Fig. 8.
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.

Tables (2)

Tables Icon

Table 1. Comparison of the gaussian optical predictions

Tables Icon

Table 2. Illustration of the symmetry rules for the quadrupole

Equations (72)

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( 2 + k 2 ) E = 0
E ( X , Z ) = 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
( m 2 + k 2 ) E m = 0
E m + E m + 1 | D = 0
u xx + u zz + 2 ik u z = 0
u x ˜ x ˜ + 1 k u zz + 2 i u z = 0 .
u ( x ˜ , z ) = 0
u ( x , z ) = c A ( z ) exp [ i 2 Ω ( z ) x ˜ 2 ]
Ω 2 + Ω = 0
A Ω + 2 A = 0
Ω = Q ( z ) Q ( z )
Q = 0
Q Q + 2 A A = 0
c m Q m ( z m ) exp ( ik z m + i 2 Ω ( z m ) x ˜ m 2 )
+ c m + 1 Q m + 1 ( z m + 1 ) exp ( ik z m + 1 + i 2 Ω ( z m + 1 ) x ˜ m + 1 2 ) | D = 0
k ( l m + 1 k ξ ˜ 1 sin χ m 1 k ξ ˜ 1 2 2 ρ m cos χ m ) + 1 2 Q m Q m ( ξ ˜ 1 cos χ m + 1 k ξ ˜ 1 2 2 ρ m sin χ m ) 2 =
k ( l m + 1 k ξ ˜ 1 sin χ m + 1 k ξ ˜ 1 2 2 ρ m cos χ m ) + 1 2 Q m + 1 Q m + 1 ( ξ ˜ 1 cos χ m + 1 k ξ ˜ 1 2 2 ρ m sin χ m ) 2
c m Q m ( L m ) + c m + 1 Q m + 1 ( L m ) = 0
( Q m + 1 Q m + 1 ) = ( 1 0 2 ρ m cos χ m 1 ) ( Q m Q m ) m ( Q m Q m )
C m + 1 = e i π C m
= ( Q 1 Q 2 P 1 P 2 )
d dz = H
( Q ( z ) P ( z ) ) = ( z ) h m
( Q m ( z + l ) P m ( z + l ) ) = ( l ) ( Q m ( z ) P m ( z ) ) .
( Q m + 1 ( z ) P m + 1 ( z ) ) = ( z L m ) m ( L m z ) ( Q m ( z ) P m ( z ) )
𝚳 ( z ) = ( z L m 1 ) m 1 ( l m 1 ) ( l m + 1 ) m ( 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 ( x ˜ , z + L ) = e i φ 2 ( N μ + N ) u ( x ˜ , z )
kL = 1 2 φ + 2 π m + mod 2 π [ ( N + N μ ) π ]
N μ = [ 1 2 π i 0 L d ( ln q ( z ) ) ]
Λ ( z ) = iq ( z ) x ˜ + p ( z ) x ˜
Λ ( z ) = i q * ( z ) x ˜ + p * ( z ) x ˜ .
u ( l ) ( x ˜ , z ) = ( Λ ) l u ( 0 )
kL = ( l + 1 2 ) φ + 2 π m + mod 2 π [ ( N + N μ ) π ]
u ( l ) ( x ˜ , z ) = ( i q * ( z ) 2 q ( z ) ) l H l ( I m [ p ( z ) q ( z ) ] x ˜ ) u ( 0 ) ( x ˜ , z )
( 2 + n ( r ) 2 k 2 ) Ψ = 0
E i + E r | D = E t | D +
n E i + n E r | D = n E t | D +
Φ i = nk ( L m + 1 nk ξ ˜ 1 sin χ 1 nk ξ ˜ 1 2 2 ρ cos χ ) + 1 2 Q i Q i ( ξ ˜ 1 cos χ + 1 nk ξ ˜ 1 2 2 ρ sin χ ) 2
Φ t = k ( n L m + 1 nk ξ ˜ 1 sin χ t 1 nk ξ ˜ 1 2 2 ρ cos χ t ) + 1 2 Q t Q t ( ξ ˜ 1 cos χ t + 1 nk ξ ˜ 1 2 2 ρ sin χ t ) 2
( Q t Q t ) = ( 1 μ 0 2 ( 1 μ ) ρ cos χ n μ ) ( Q i Q i )
c i Q i + c r Q r = c t Q t
c i + c r = μ c t
c i n Φ i + c r n Φ r = μ c t n Φ t .
n μ ( c i c r ) = c t
c r = 1 + 1 c i .
c r 2 = 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 = 1 2 φ + 2 πm + mod 2 π [ ( N + N μ ) π ] i b = 1 N log [ n μ b 1 n μ n + 1 ] .
Re [ nkL ] = 2 πm + mod 2 π [ ( N + N μ ) π ] + φ 2 + φ f
Im [ nkL ] = γ f .
P m E ( x ) = d m G g G χ m ( g ) E ( g x )
E ( + + ) = 1 4 ( e 1 + e 2 + e 3 + e 4 ) , E ( + ) = 1 4 ( e 1 + e 2 e 3 e 4 )
E ( + ) = 1 4 ( e 1 e 2 + e 3 e 4 ) , E ( ) = 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 ) = 1 q ( z ) exp [ ikz + i 2 Ω ( z ) x 2 ]
e 2 = E ( g x ) = e 1 2 i π 1 e i φ 2 q ( z ) exp [ ik ( z + L 2 + i 2 Ω ( z ) x 2 ) ] e i ζ e 1
e 3 = E ( g x ) = 1 q ( 1 z ) exp [ ik ( 1 z ) + i 2 Ω ( 1 z ) x 2 ]
e 4 = E ( g x ) = e 1 2 i π 1 e i φ 2 q ( 1 z ) exp [ ik ( L 2 + 1 z ) + i 2 Ω ( 1 z ) x 2 ] e i ζ e 3
E ( + + ) = 1 2 ( e 1 + e 3 ) e i ζ 2 cos ζ 2
E ( + ) = 1 2 ( e 1 e 3 ) e i ζ 2 cos ζ 2
E ( + ) = 1 2 i ( e 1 + e 3 ) e i ζ 2 sin ζ 2
E ( ) = 1 2 i ( e 1 + e 3 ) e i ζ 2 sin ζ 2
E ( + + ) , E ( + ) cos ζ 2 = 0 m = 1,3,5 , . . .
E ( + ) , E ( ) sin ζ 2 = 0 m = 0,2,4 , . . .
Δ E CAT V RC 2 E R E C
V RC 2 d ϕ d sin χ W C ( ϕ , sin χ ) W R ( ϕ , sin χ )
Δ E CAT exp ( Ak R 0 )

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