Abstract

Theoretical study of free space coupling to high-Q whispering gallery modes (WGMs) are presented in circular and deformed microcavities. Both analytical solutions and asymptotic formulas are derived for a circular cavity. The coupling efficiencies at different coupling regimes for cylindrical incoming wave are discussed, and the maximum efficiency is estimated for the practical Gaussian beam excitation. In the case of a deformed cavity, the coupling efficiency can be higher than the circular cavity if the excitation beam can match the intrinsic emission which can be tuned by adjusting the degree of deformation. Employing an abstract model of slightly deformed cavity, we find that the asymmetric and peak like line shapes instead of the Lorentz-shape dip are universal in transmission spectra due to multi-wave interference, and the coupling efficiency cannot be estimated from the absolute depth of the dip. Our results provide guidelines for free space coupling in experiments, suggesting that the high-Q asymmetric resonator cavities (ARCs) can be efficiently excited through free space which will stimulate further experiments and applications of WGMs based on free space coupling.

© 2013 Optical Society of America

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2013 (2)

L. He, S. K. Ozdemir, and L. Yang, “Whispering gallery microcavity lasers,” Laser Photon. Rev. 7, 60–82 (2013).
[Crossref]

C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, G.-C. Guo, and Z.-F. Han, “High Q and unidirectional emission whispering gallery modes: principles and design,” IEEE J. Sel. Top. Quantum Electron., In press (2013).

2012 (5)

X.-F. Jiang, Y.-F. Xiao, C.-L. Zou, L. He, C.-H. Dong, B.-B. Li, Y. Li, F.-W. Sun, L. Yang, and Q. Gong, “Highly Unidirectional emission and ultralow-threshold lasing from on-chip ultrahigh-Q microcavities,” Adv. Mater. 24, OP260–OP264 (2012).
[Crossref] [PubMed]

E. Verhagen, S. Deleglise, S. Weis, A. Schliesser, and T. J. Kippenberg, “Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode,” Nature 482, 63–67 (2012).
[Crossref] [PubMed]

H. Moyses Nussenzveig, “The Science of the Glory,” Sci. Am. 306(1), 68–73 (2012)
[Crossref] [PubMed]

F.-J. Shu, C.-L. Zou, and F.-W. Sun, “Perpendicular coupler for whispering-gallery resonators,” Opt. Lett. 37, 3123–3125 (2012).
[Crossref] [PubMed]

B.-B. Li, Y.-F. Xiao, C.-L. Zou, X.-F. Jiang, Y.-C. Liu, F.-W. Sun, Y. Li, and Q. Gong, “Experimental controlling of Fano resonance in indirectly coupled whispering-gallery microresonators,” Appl. Phys. Lett. 100, 021108 (2012).
[Crossref]

2011 (3)

B.-B. Li, Y.-F. Xiao, C.-L. Zou, Y.-C. Liu, X.-F. Jiang, Y.-L. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98, 021116 (2011).
[Crossref]

Y.-Z. Yan, C.-L. Zou, S.-B. Yan, F.-W. Sun, Z. Ji, J. Liu, Y.-G. Zhang, L. Wang, C.-Y. Xue, and W.-D. Zhang, “Packaged silica microsphere-taper coupling system for robust thermal sensing application,” Opt. Express 19, 5753–5759 (2011).
[Crossref] [PubMed]

V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
[Crossref] [PubMed]

2010 (3)

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflugl, 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] [PubMed]

J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, and K. An, “Observation of resonance effects in the pump transmission of a chaotic microcavity,” Opt. Express 18, 26141–26148 (2010).
[Crossref] [PubMed]

J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
[Crossref] [PubMed]

2009 (9)

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]

C. Yan, Q. J. Wang, L. Diehl, M. Hentschel, J. Wiersig, N. Yu, C. Pflugl, F. Capasso, M. A. Belkin, T. Edamura, M. Yamanishi, and H. Kan, “Directional emission and universal far-field behavior from semiconductor lasers with limacon-shaped microcavity,” Appl. Phys. Lett. 94, 251101 (2009).
[Crossref]

S. Shinohara, M. Hentschel, J. Wiersig, T. Sasaki, and T. Harayama, “Ray-wave correspondence in limacon-shaped semiconductor microcavities,” Phys. Rev. A 80, 031801 (2009).
[Crossref]

M. Tomes, K. J. Vahala, and T. Carmon, “Direct imaging of tunneling from a potential well,” Opt. Express 17, 19160–19165 (2009).
[Crossref]

S.-B. Lee, J. Yang, S.-Y. Lee, S. Moon, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Evolution of emission mechanism in deformed microcavities,” International Conference on Transparent Optical Networks, Paper no. Tu.P.16 (2009).

Y.-F. Xiao, C.-H. Dong, C.-L. Zou, Z.-F. Han, L. Yang, and G.-C. Guo, “Low-threshold microlaser in a high-Q asymmetrical microcavity,” Opt. Lett. 34, 509–511 (2009).
[Crossref] [PubMed]

Y. S. Park and H. Wang, “Resolved-sideband and cryogenic cooling of an optomechanical resonator,” Nat. Phys. 5, 489–493 (2009).
[Crossref]

C.-H. Dong, C.-L. Zou, J.-M. Cui, Y. Yang, Z.-F. Han, and G.-C. Guo, “Ringing phenomenon in silica microspheres,” Chin. Opt. Lett. 7, 299–301 (2009).
[Crossref]

C.-H. Dong, C.-L. Zou, Y.-F. Xiao, J.-M. Cui, Z.-F. Han, and G.-C. Guo, “Modified transmission spectrum induced by two-mode interference in a single silica microsphere,” J. Phys. B 42, 215401 (2009).
[Crossref]

2008 (5)

C.-L. Zou, Y. Yang, C.-H. Dong, Y.-F. Xiao, X.-W. Wu, Z.-F. Han, and G.-C. Guo, “Taper-microsphere coupling with numerical calculation of coupled-mode theory,” J. Opt. Soc. Am. B 25, 1895–1898 (2008)
[Crossref]

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nature Methods 5, 591–596 (2008).
[Crossref] [PubMed]

S. Shinohara, T. Fukushima, and T. Harayama, “Light emission patterns from stadium-shaped semiconductor microcavity lasers,” Phys. Rev. A 77, 033807 (2008).
[Crossref]

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

J. Yang, S.-B. Lee, J.-B. Shim, S. Moon, S.-Y. Lee, S. W. Kim, J.-H. Lee, and K. An, “Enhanced nonresonant optical pumping based on turnstile transport in a chaotic microcavity laser,” Appl. Phys. Lett. 93, 061101 (2008).
[Crossref]

2007 (6)

S. C. Creagh, “Directional emission from weakly eccentric resonators,” Phys. Rev. Lett. 98, 153901 (2007).
[Crossref] [PubMed]

S. B. Lee, J. Yang, S. Moon, J. H. Lee, K. An, J. B. Shim, H. W. Lee, and S. W. Kim, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802 (2007).
[Crossref]

Y.-S. Park and H. Wang, “Radiation pressure driven mechanical oscillation in deformed silica microspheres via free-space evanescent excitation,” Opt. Express 15, 16471–16477 (2007).
[Crossref] [PubMed]

Y.-F. Xiao, C.-H. Dong, Z.-F. Han, G.-C. Guo, and Y.-S. Park, “Directional escape from a high-Q deformed microsphere induced by short CO2 laser pulses,” Opt. Lett. 32, 644–646 (2007).
[Crossref] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[Crossref]

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. 98, 213904 (2007).
[Crossref] [PubMed]

2006 (4)

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

I. S. Grudinin, V. S. Ilchenko, and L. Maleki, “Ultrahigh optical Q factors of crystalline resonators in the linear regime,” Phys. Rev. A 74, 063806 (2006).
[Crossref]

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

J. Wiersig and M. Hentschel, “Unidirectional light emission from high-Q modes in optical microcavities,” Phys. Rev. A 73, 031802 (2006).
[Crossref]

2005 (3)

V. A. Podolskiy and E. E. Narimanov, “Chaos-assisted tunneling in dielectric microcavities,” Opt. Lett. 30, 474–476 (2005).
[Crossref] [PubMed]

W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
[Crossref]

A. Chiba, H. Fujiwara, J.-I. Hotta, S. Takeuchi, and K. Sasaki, “Fano resonance in a multimode tapered fiber coupled with a microspherical cavity,” Appl. Phys. Lett. 86, 261106 (2005).
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2004 (2)

2003 (3)

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D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
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2002 (1)

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, 033903 (2002).
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2001 (1)

D. A. Steck, W. H. Oskay, and M. G. Raizen, “Observation of chaos-assisted tunneling between islands of stability,” Science 293, 274–278 (2001).
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2000 (3)

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
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M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to silica-microsphere whispering gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
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1999 (1)

1998 (2)

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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1995 (1)

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1994 (2)

M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
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1989 (1)

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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)
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J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
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J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, and K. An, “Observation of resonance effects in the pump transmission of a chaotic microcavity,” Opt. Express 18, 26141–26148 (2010).
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J. Yang, S.-B. Lee, J.-B. Shim, S. Moon, S.-Y. Lee, S. W. Kim, J.-H. Lee, and K. An, “Enhanced nonresonant optical pumping based on turnstile transport in a chaotic microcavity laser,” Appl. Phys. Lett. 93, 061101 (2008).
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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, 033903 (2002).
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H. J. Moon, Y. T. Chough, and K. An, “Cylindrical microcavity laser based on the evanescent-wave-coupled gain,” Phys. Rev. Lett. 85, 3161–3164 (2000).
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T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
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P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
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D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
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F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nature Methods 5, 591–596 (2008).
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V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
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C. Yan, Q. J. Wang, L. Diehl, M. Hentschel, J. Wiersig, N. Yu, C. Pflugl, F. Capasso, M. A. Belkin, T. Edamura, M. Yamanishi, and H. Kan, “Directional emission and universal far-field behavior from semiconductor lasers with limacon-shaped microcavity,” Appl. Phys. Lett. 94, 251101 (2009).
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T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
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V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
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M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to silica-microsphere whispering gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
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Cao, H.

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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W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
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Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflugl, 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).
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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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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, 033903 (2002).
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H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B 21, 923–934 (2004).
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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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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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Chen, G.

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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B.-B. Li, Y.-F. Xiao, C.-L. Zou, Y.-C. Liu, X.-F. Jiang, Y.-L. Chen, Y. Li, and Q. Gong, “Experimental observation of Fano resonance in a single whispering-gallery microresonator,” Appl. Phys. Lett. 98, 021116 (2011).
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Chiba, A.

A. Chiba, H. Fujiwara, J.-I. Hotta, S. Takeuchi, and K. Sasaki, “Fano resonance in a multimode tapered fiber coupled with a microspherical cavity,” Appl. Phys. Lett. 86, 261106 (2005).
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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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H. J. Moon, Y. T. Chough, and K. An, “Cylindrical microcavity laser based on the evanescent-wave-coupled gain,” Phys. Rev. Lett. 85, 3161–3164 (2000).
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D. Q. Chowdhury, S. C. Hill, and Md. Mohiuddin Mazumder, “Quality factors and effective-average modal gain or loss in inhomogeneous spherical resonators: application to two-photon absorption.” IEEE J. Quant. Electron. 29, 2553–2561 (1993).
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C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, G.-C. Guo, and Z.-F. Han, “High Q and unidirectional emission whispering gallery modes: principles and design,” IEEE J. Sel. Top. Quantum Electron., In press (2013).

C.-H. Dong, C.-L. Zou, Y.-F. Xiao, J.-M. Cui, Z.-F. Han, and G.-C. Guo, “Modified transmission spectrum induced by two-mode interference in a single silica microsphere,” J. Phys. B 42, 215401 (2009).
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C.-H. Dong, C.-L. Zou, J.-M. Cui, Y. Yang, Z.-F. Han, and G.-C. Guo, “Ringing phenomenon in silica microspheres,” Chin. Opt. Lett. 7, 299–301 (2009).
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J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104, 243601 (2010).
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T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
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T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
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P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
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C. Yan, Q. J. Wang, L. Diehl, M. Hentschel, J. Wiersig, N. Yu, C. Pflugl, F. Capasso, M. A. Belkin, T. Edamura, M. Yamanishi, and H. Kan, “Directional emission and universal far-field behavior from semiconductor lasers with limacon-shaped microcavity,” Appl. Phys. Lett. 94, 251101 (2009).
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C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, G.-C. Guo, and Z.-F. Han, “High Q and unidirectional emission whispering gallery modes: principles and design,” IEEE J. Sel. Top. Quantum Electron., In press (2013).

X.-F. Jiang, Y.-F. Xiao, C.-L. Zou, L. He, C.-H. Dong, B.-B. Li, Y. Li, F.-W. Sun, L. Yang, and Q. Gong, “Highly Unidirectional emission and ultralow-threshold lasing from on-chip ultrahigh-Q microcavities,” Adv. Mater. 24, OP260–OP264 (2012).
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C.-H. Dong, C.-L. Zou, Y.-F. Xiao, J.-M. Cui, Z.-F. Han, and G.-C. Guo, “Modified transmission spectrum induced by two-mode interference in a single silica microsphere,” J. Phys. B 42, 215401 (2009).
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C.-H. Dong, C.-L. Zou, J.-M. Cui, Y. Yang, Z.-F. Han, and G.-C. Guo, “Ringing phenomenon in silica microspheres,” Chin. Opt. Lett. 7, 299–301 (2009).
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C.-L. Zou, Y. Yang, C.-H. Dong, Y.-F. Xiao, X.-W. Wu, Z.-F. Han, and G.-C. Guo, “Taper-microsphere coupling with numerical calculation of coupled-mode theory,” J. Opt. Soc. Am. B 25, 1895–1898 (2008)
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Faist, J.

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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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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W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
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V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing optical information as a mechanical excitation in a silica optomechanical resonator,” Phys. Rev. Lett. 107, 133601 (2011).
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S. Lacey, H. Wang, D. H. Foster, and J. U. Nöckel, “Directional tunneling escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902(2003).
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A. Chiba, H. Fujiwara, J.-I. Hotta, S. Takeuchi, and K. Sasaki, “Fano resonance in a multimode tapered fiber coupled with a microspherical cavity,” Appl. Phys. Lett. 86, 261106 (2005).
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S. Shinohara, T. Fukushima, and T. Harayama, “Light emission patterns from stadium-shaped semiconductor microcavity lasers,” Phys. Rev. A 77, 033807 (2008).
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T. Harayama, T. Fukushima, P. Davis, P. O. Vaccaro, T. Miyasaka, T. Nishimura, and T. Aida, “Lasing on scar modes in fully chaotic microcavities,” Phys. Rev. E 67, 015207 (2003).
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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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M. L. Gorodetsky and V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
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V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
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S. Shinohara, T. Fukushima, and T. Harayama, “Light emission patterns from stadium-shaped semiconductor microcavity lasers,” Phys. Rev. A 77, 033807 (2008).
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S. Shinohara, M. Hentschel, J. Wiersig, T. Sasaki, and T. Harayama, “Ray-wave correspondence in limacon-shaped semiconductor microcavities,” Phys. Rev. A 80, 031801 (2009).
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Figures (4)

Fig. 1
Fig. 1

(a) Electric field distribution for a cylinder cavity critical coupling with m = 30 cylinder wave, and detuning δ = 0. (b) The normalized reflectivity and the total Q factor against the angular number m, with the refractive index n = 1.45+0.000117i. (c) and (d) are the intensity and phase of outgoing field for different coupling conditions, with κ1/κ0 = 0.2 (Red Dotted lines), 1.0 (Blue Dashed lines) and 5.0 (Black Solid lines).

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Fig. 2
Fig. 2

(a) Schematic illustration of a Gaussian beam incident to a circular microcavity. The direction of beam is defined by θ0, and the location of center is (r0, ϕ0) in the cylindrical coordinate. (b) The electric field intensity distribution when the incident Gaussian beam is resonant with the WGM, with krc = 30.10, r0 = 1.1rc, ϕ0 = π/2 and θ0 = 0.

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Fig. 3
Fig. 3

(a) Field distribution of directional emission of anti-clockwise WGM in a near circular cavity (The field is shown in Logarithm scale, and the intracavity field is not shown since the exact boundary shape is not known in our abstract model). The emission is in the form of Gaussian beam with dt = 1.2rc, wt = 0.1rc and direction is shown by arrows. (b) Field distribution when an on-resonance Gaussian beam coupling to the cavity in the over coupling regime with κ1 = 0.1κ0, d = 1.2, w = 0.2rc and direction is shown by arrows. (c) Beam matching parameters against the d for different w. (d) The far field intensity of outgoing wave when a Gaussian beam incident to a cylinder, for excitation frequency off-resonance and on-resonance at difference coupling conditions. (e), (f) and (g) are spectra detected at different angle (ϕ) in far field.

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Fig. 4
Fig. 4

(a) Field distribution of directional emission of anti-clockwise WGM in a near circular cavity (The field is shown in normal scale, and the intracavity field is not shown since the exact boundary shape is not known in our abstract model). The emission is in the form of Gaussian beam with dr = 0.9rc, wr = 0.1rc and direction is shown by arrows. (b) The field distribution when an on-resonance Gaussian beam coupling to the cavity in the over coupling regime with κ1 = 0.1κ0, d = 0.9, w = 0.2rc and direction is shown by arrows. (c) The far field intensity of outgoing wave when a Gaussian beam is incident to the ARC with Gaussian beam emission, for off-resonance and on-resonance under different coupling conditions.

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Equations (34)

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a m = H m ( 1 ) ( z ) H m ( 2 ) ′ ( z ) − H m ( 1 ) ′ ( z ) H m ( 2 ) z n p J m ′ ( n z ) H m ( 1 ) ( z ) − J m ( n z ) H m ( 1 ) ′ ( z ) c m ,
b m = − n p J m ′ ( n z ) H m ( 2 ) ( z ) − J m ( n z ) H m ( 2 ) ′ ( z ) n p J m ′ ( n z ) H m ( 1 ) ( z ) − J m ( n z ) H m ( 1 ) ′ ( z ) c m ,
n p J m ′ ( n z 0 ) H m ( 1 ) ( z 0 ) − J m ( n z 0 ) H m ( 1 ) ′ ( z 0 ) = 0 .
n p J m ′ ( n z ) H m ( 1 ) ( k z ) − J m ( n z ) H m ( 1 ) ′ ( z ) ≈ F ( z 0 ) Δ ,
a m ( δ ) = 1 i δ + κ 0 + κ 1 − 4 π z 0 r F ( z 0 r ) c m ,
b m ( δ ) = − i δ − κ 0 + κ 1 i δ + κ 0 + κ 1 F * ( z 0 r ) F ( z 0 r ) c m .
d d t E m = ( − i δ − κ 0 − κ 1 ) E m + 2 κ 0 E m in ,
E m = 2 κ 0 i δ + κ 0 + κ 1 E m in ,
E m out = − i δ − κ 0 + κ 1 i δ + κ 0 + κ 1 E m in .
E m = 2 κ ext i δ + κ int + κ ext E m in ,
G ( x 0 , y ) = e − ( y − y 0 ) 2 w 2
G ˜ ( θ ) = 1 2 π ∫ − ∞ ∞ e − i y k sin θ e − y 2 w 2 d y = w 2 π e − k 2 w 2 4 sin 2 θ ,
E ( r , ϕ ) = ∫ − k k w 2 π e − k 2 w 2 4 sin 2 θ e i k ( sin θ y + cos θ x ) d ( k sin θ ) ,
x = r cos ( ϕ + θ 0 ) − r 0 cos ( ϕ 0 + θ 0 ) ,
y = r sin ( ϕ + θ 0 ) − r 0 sin ( ϕ 0 + θ 0 ) .
e i k r cos ( ϕ + θ 0 − θ ) = ∑ m = − ∞ ∞ J m ( k r ) e i m [ π / 2 − ( ϕ + θ 0 − θ ) ] ,
c m = e i m ( π / 2 − θ 0 ) − i k q − ( k r 0 d − m ) 2 k 2 w 2 + i 2 k q 2 1 + i 2 q k w 2 ,
P G = π 2 k w 2 ω μ ,
P m = 2 ω μ .
η m = | c m | 2 P m P G .
η max = 2 π 1 k w .
ℋ = ℋ modes + ℋ int + ℋ pump .
ℋ modes = ∑ j h ¯ δ j ( A j , c † A j , c + A j , a † A j , a )
ℋ int = ∑ j ∑ k > j ( g j , k , c A j , c † A k , c + g j , k , a A j , a † A k , a + h . c . ) + ∑ j , k ( β j , k A j , c † A k , a + h . c . ) ,
ℋ pump = ∑ j , k ( i h j , m , c A j , c † u m in + i h j , m , a A j , a † u − m in + h . c . ) .
d d t A j , c = − i ∑ k > j g j , k A k , c − i ∑ k < j g k , j * A k , c − χ j A j , c + ∑ m h j , m u m in ,
A j , c | j ≥ 2 = − i g 1 , j * χ j A 1 , c + ∑ m = 1 ∞ h j , m χ j u m in .
d d t A 1 , c = − χ ˜ 1 A 1 , c + ∑ m = 1 ∞ h ˜ 1 , m u m in .
< A 1 , c > = ξ | h → | | u → | / χ ˜ 1 ,
u m out = − u m in + f m < A 1 , c > + ∑ j ≥ 2 ∑ m ′ h j , m * h j , m ′ χ j u m ′ in ,
u → ∝ h → * / | h → | .
h 1 , m = 𝔥 0 e − i m π / 2 − ( k d t − m ) 2 k 2 w t 2 + i φ m ,
∑ j ≥ 2 g 1 , j χ j h j , m e − i φ m = 𝔥 e e − i m π / 2 − ( k d t − m ) 2 k 2 − w t 2 .
g 1 , m χ m h m , m = 𝔥 e e − i m π / 2 − ( k d t − m ) 2 k 2 w t 2 + i φ m .

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