Abstract

Here we demonstrate the impacts of emission mechanisms on the light confinements in open systems. Taking the oval-shaped cavities as examples, we show that the enhancements in quality (Q) factors are usually associated with the universal emissions. When the coupled resonances have similar far field patterns, the Q factor of the long-lived resonance has the possibility to be enhanced by the coherent destruction at the decay channels. Otherwise, the Q factors of long-lived resonances are usually reduced around the level crossings.

© 2014 Optical Society of America

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  17. S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101, 2009).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  28. S. M. Xiao, Z. Y. Gu, S. Liu, Q. H. Song, “Direct modulation of microcavity emission via local perturbation,” Phys. Rev. A 88, 053833, 2013).
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    [CrossRef]
  30. S.-B. Lee, J. Yang, S. Moon, J.-H. Lee, K. An, J.-B. Shim, H. W. Lee, S. W. Kim, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75, 011802, 2007).
    [CrossRef]
  31. S. Shinohara, T. Harayama, “Signature of ray chaos in quasibound wave functions for a stadium-shaped dielectric cavity,” Phys. Rev. E 75, 036216, 2007).
    [CrossRef]
  32. M. Hentschel, H. Schomerus, R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
    [CrossRef]
  33. H. E. Türeci, H. G. L. Schwefel, P. Jacquod, A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. 47, 75–137 (2005).
    [CrossRef]
  34. L. Ge, Q. H. Song, B. Redding, H. Cao, “Extreme output sensitivity to subwavelength boundary deformation in microcavities,” Phys. Rev. A 87, 023833, 2013).
    [CrossRef]

2013 (4)

Q. H. Song, C. Zeng, S. M. Xiao, “Coherent destruction of dynamical tunneling in asymmetrical resonant cavities,” Phys. Rev. A 87, 013831, 2013).
[CrossRef]

Q. H. Song, L. Ge, J. Wiersig, H. Cao, “Formation of long-lived resonances in hexagonal cavities by strong coupling of superscar modes,” Phys. Rev. A 88, 023834, 2013).
[CrossRef]

S. M. Xiao, Z. Y. Gu, S. Liu, Q. H. Song, “Direct modulation of microcavity emission via local perturbation,” Phys. Rev. A 88, 053833, 2013).
[CrossRef]

L. Ge, Q. H. Song, B. Redding, H. Cao, “Extreme output sensitivity to subwavelength boundary deformation in microcavities,” Phys. Rev. A 87, 023833, 2013).
[CrossRef]

2012 (2)

J-W Ryu, S-Y. Lee, S. W. Kim, “Coupled nonidentical microdisks: Avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858, 2012).
[CrossRef]

Q. H. Song, L. Ge, B. Redding, H. Cao, “Channeling chaotic rays into waveguides for efficient collection of microcavity emission,” Phys. Rev. Lett. 108, 243902, 2012).
[CrossRef] [PubMed]

2010 (3)

Q. H. Song, H. Cao, “Improving optical confinement in nanostructures via external mode coupling,” Phys. Rev. Lett. 105, 053902, 2010).
[CrossRef] [PubMed]

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902, 2010).
[CrossRef] [PubMed]

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

2009 (2)

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

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101, 2009).
[CrossRef] [PubMed]

2008 (4)

M. Müller, I. Rotter, “Exceptional points in open quantum systems,” J. Phys. A 41, 244018, 2008).
[CrossRef]

Y. Kayanuma, K. Saito, “Coherent destruction of tunneling, dynamic localization, and the Landau-Zener formula,” Phys. Rev. A 77, 010101(R) (2008).
[CrossRef]

A. Bäcker, R. Ketzmerick, S. Löck, L. Schilling, “Regular-to-chaotic tunneling rates using a fictitious integrable system,” Phys. Rev. Lett. 100, 104101, 2008).
[CrossRef] [PubMed]

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

2007 (3)

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

S. Shinohara, T. Harayama, “Signature of ray chaos in quasibound wave functions for a stadium-shaped dielectric cavity,” Phys. Rev. E 75, 036216, 2007).
[CrossRef]

K. Srinivasan, O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system,” Nature 450, 862–866 (2007).
[CrossRef] [PubMed]

2006 (3)

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901, 2006).
[CrossRef]

M. Bhattacharya, C. Raman, “Detecting level crossings without looking at the spectrum,” Phys. Rev. Lett. 97, 140405, 2006).
[CrossRef] [PubMed]

J. Wiersig, M. Hentschel, “Unidirectional light emission from high-Q modes in optical microcavities,” Phys. Rev. Lett. 73, 031802(R) (2006).

2005 (1)

H. E. Türeci, H. G. L. Schwefel, P. Jacquod, A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. 47, 75–137 (2005).
[CrossRef]

2004 (3)

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

V. A. Podolskiy, E. E. Narimanov, W. Fang, H. Cao, “Chaotic microlasers based on dynamical localization,” Proc Natl Acad Sci USA 101, 10498–10500 (2004).
[CrossRef] [PubMed]

D. K. Ferry, R. Akis, J. P. Bird, “Einselection in action: decoherence and pointer states in open quantum dots,” Phys. Rev. Lett. 93, 026803, 2004).
[CrossRef] [PubMed]

2003 (2)

V. A. Podolskiy, E. E. Narimanov, “Semiclassical description of chaos-assisted tunneling,” Phys. Rev. Lett. 91, 263601, 2003).
[CrossRef]

M. Hentschel, H. Schomerus, R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
[CrossRef]

2000 (2)

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[CrossRef]

E. Persson, I. Rotter, H. J. Stockmann, M. Barth, “Observation of resonance trapping in an open microwave cavity,” Phys. Rev. Lett. 85, 2478–2481 (2000).
[CrossRef] [PubMed]

1998 (1)

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

1997 (1)

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

1995 (1)

J. Burgdorfer, X. Z. Yang, J. Muller, “Parametric variation of resonances for regular and chaotic scattering,” Chaos Solutions & Fractals 5, 1235–1273 (1995).
[CrossRef]

1991 (1)

F. Grossmann, T. Dittrich, P. Jung, P. Hanggi, “Coherent destruction of the tunneling,” Phys. Rev. Lett. 67, 516–519 (1991).
[CrossRef] [PubMed]

Akis, R.

D. K. Ferry, R. Akis, J. P. Bird, “Einselection in action: decoherence and pointer states in open quantum dots,” Phys. Rev. Lett. 93, 026803, 2004).
[CrossRef] [PubMed]

An, K.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101, 2009).
[CrossRef] [PubMed]

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

Bäcker, A.

A. Bäcker, R. Ketzmerick, S. Löck, L. Schilling, “Regular-to-chaotic tunneling rates using a fictitious integrable system,” Phys. Rev. Lett. 100, 104101, 2008).
[CrossRef] [PubMed]

Barth, M.

E. Persson, I. Rotter, H. J. Stockmann, M. Barth, “Observation of resonance trapping in an open microwave cavity,” Phys. Rev. Lett. 85, 2478–2481 (2000).
[CrossRef] [PubMed]

Ben-Messaoud, T.

Bhattacharya, M.

M. Bhattacharya, C. Raman, “Detecting level crossings without looking at the spectrum,” Phys. Rev. Lett. 97, 140405, 2006).
[CrossRef] [PubMed]

Bird, J. P.

D. K. Ferry, R. Akis, J. P. Bird, “Einselection in action: decoherence and pointer states in open quantum dots,” Phys. Rev. Lett. 93, 026803, 2004).
[CrossRef] [PubMed]

Burgdorfer, J.

J. Burgdorfer, X. Z. Yang, J. Muller, “Parametric variation of resonances for regular and chaotic scattering,” Chaos Solutions & Fractals 5, 1235–1273 (1995).
[CrossRef]

Cao, H.

Q. H. Song, L. Ge, J. Wiersig, H. Cao, “Formation of long-lived resonances in hexagonal cavities by strong coupling of superscar modes,” Phys. Rev. A 88, 023834, 2013).
[CrossRef]

L. Ge, Q. H. Song, B. Redding, H. Cao, “Extreme output sensitivity to subwavelength boundary deformation in microcavities,” Phys. Rev. A 87, 023833, 2013).
[CrossRef]

Q. H. Song, L. Ge, B. Redding, H. Cao, “Channeling chaotic rays into waveguides for efficient collection of microcavity emission,” Phys. Rev. Lett. 108, 243902, 2012).
[CrossRef] [PubMed]

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

Q. H. Song, H. Cao, “Improving optical confinement in nanostructures via external mode coupling,” Phys. Rev. Lett. 105, 053902, 2010).
[CrossRef] [PubMed]

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

V. A. Podolskiy, E. E. Narimanov, W. Fang, H. Cao, “Chaotic microlasers based on dynamical localization,” Proc Natl Acad Sci USA 101, 10498–10500 (2004).
[CrossRef] [PubMed]

Capasso, F.

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

Chang, R. K.

Cho, A. Y.

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

Dittrich, T.

F. Grossmann, T. Dittrich, P. Jung, P. Hanggi, “Coherent destruction of the tunneling,” Phys. Rev. Lett. 67, 516–519 (1991).
[CrossRef] [PubMed]

Faist, J.

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

Fang, W.

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

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

V. A. Podolskiy, E. E. Narimanov, W. Fang, H. Cao, “Chaotic microlasers based on dynamical localization,” Proc Natl Acad Sci USA 101, 10498–10500 (2004).
[CrossRef] [PubMed]

Ferry, D. K.

D. K. Ferry, R. Akis, J. P. Bird, “Einselection in action: decoherence and pointer states in open quantum dots,” Phys. Rev. Lett. 93, 026803, 2004).
[CrossRef] [PubMed]

Fukushima, T.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902, 2010).
[CrossRef] [PubMed]

Gamachl, C.

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

Ge, L.

Q. H. Song, L. Ge, J. Wiersig, H. Cao, “Formation of long-lived resonances in hexagonal cavities by strong coupling of superscar modes,” Phys. Rev. A 88, 023834, 2013).
[CrossRef]

L. Ge, Q. H. Song, B. Redding, H. Cao, “Extreme output sensitivity to subwavelength boundary deformation in microcavities,” Phys. Rev. A 87, 023833, 2013).
[CrossRef]

Q. H. Song, L. Ge, B. Redding, H. Cao, “Channeling chaotic rays into waveguides for efficient collection of microcavity emission,” Phys. Rev. Lett. 108, 243902, 2012).
[CrossRef] [PubMed]

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

Grossmann, F.

F. Grossmann, T. Dittrich, P. Jung, P. Hanggi, “Coherent destruction of the tunneling,” Phys. Rev. Lett. 67, 516–519 (1991).
[CrossRef] [PubMed]

Gu, Z. Y.

S. M. Xiao, Z. Y. Gu, S. Liu, Q. H. Song, “Direct modulation of microcavity emission via local perturbation,” Phys. Rev. A 88, 053833, 2013).
[CrossRef]

Hanggi, P.

F. Grossmann, T. Dittrich, P. Jung, P. Hanggi, “Coherent destruction of the tunneling,” Phys. Rev. Lett. 67, 516–519 (1991).
[CrossRef] [PubMed]

Harayama, T.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902, 2010).
[CrossRef] [PubMed]

S. Shinohara, T. Harayama, “Signature of ray chaos in quasibound wave functions for a stadium-shaped dielectric cavity,” Phys. Rev. E 75, 036216, 2007).
[CrossRef]

Heiss, W. D.

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[CrossRef]

Hentschel, M.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902, 2010).
[CrossRef] [PubMed]

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

J. Wiersig, M. Hentschel, “Unidirectional light emission from high-Q modes in optical microcavities,” Phys. Rev. Lett. 73, 031802(R) (2006).

M. Hentschel, H. Schomerus, R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
[CrossRef]

Ho, S. T.

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

Jacquod, P.

H. E. Türeci, H. G. L. Schwefel, P. Jacquod, A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. 47, 75–137 (2005).
[CrossRef]

Jung, P.

F. Grossmann, T. Dittrich, P. Jung, P. Hanggi, “Coherent destruction of the tunneling,” Phys. Rev. Lett. 67, 516–519 (1991).
[CrossRef] [PubMed]

Kayanuma, Y.

Y. Kayanuma, K. Saito, “Coherent destruction of tunneling, dynamic localization, and the Landau-Zener formula,” Phys. Rev. A 77, 010101(R) (2008).
[CrossRef]

Ketzmerick, R.

A. Bäcker, R. Ketzmerick, S. Löck, L. Schilling, “Regular-to-chaotic tunneling rates using a fictitious integrable system,” Phys. Rev. Lett. 100, 104101, 2008).
[CrossRef] [PubMed]

Kim, S. W.

J-W Ryu, S-Y. Lee, S. W. Kim, “Coupled nonidentical microdisks: Avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858, 2012).
[CrossRef]

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101, 2009).
[CrossRef] [PubMed]

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

Lee, H. W.

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

Lee, J.-H.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101, 2009).
[CrossRef] [PubMed]

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

Lee, S.-B.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101, 2009).
[CrossRef] [PubMed]

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

Lee, S.-Y.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101, 2009).
[CrossRef] [PubMed]

Lee, S-Y.

J-W Ryu, S-Y. Lee, S. W. Kim, “Coupled nonidentical microdisks: Avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858, 2012).
[CrossRef]

Liu, B. Y.

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

Liu, S.

S. M. Xiao, Z. Y. Gu, S. Liu, Q. H. Song, “Direct modulation of microcavity emission via local perturbation,” Phys. Rev. A 88, 053833, 2013).
[CrossRef]

Löck, S.

A. Bäcker, R. Ketzmerick, S. Löck, L. Schilling, “Regular-to-chaotic tunneling rates using a fictitious integrable system,” Phys. Rev. Lett. 100, 104101, 2008).
[CrossRef] [PubMed]

Moiseyev, N.

N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University, 2011).

Moon, S.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101, 2009).
[CrossRef] [PubMed]

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

Muller, J.

J. Burgdorfer, X. Z. Yang, J. Muller, “Parametric variation of resonances for regular and chaotic scattering,” Chaos Solutions & Fractals 5, 1235–1273 (1995).
[CrossRef]

Müller, M.

M. Müller, I. Rotter, “Exceptional points in open quantum systems,” J. Phys. A 41, 244018, 2008).
[CrossRef]

Narimanov, E. E.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902, 2010).
[CrossRef] [PubMed]

V. A. Podolskiy, E. E. Narimanov, W. Fang, H. Cao, “Chaotic microlasers based on dynamical localization,” Proc Natl Acad Sci USA 101, 10498–10500 (2004).
[CrossRef] [PubMed]

V. A. Podolskiy, E. E. Narimanov, “Semiclassical description of chaos-assisted tunneling,” Phys. Rev. Lett. 91, 263601, 2003).
[CrossRef]

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

Nöckel, J. U.

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

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

Painter, O.

K. Srinivasan, O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system,” Nature 450, 862–866 (2007).
[CrossRef] [PubMed]

Persson, E.

E. Persson, I. Rotter, H. J. Stockmann, M. Barth, “Observation of resonance trapping in an open microwave cavity,” Phys. Rev. Lett. 85, 2478–2481 (2000).
[CrossRef] [PubMed]

Podolskiy, V. A.

V. A. Podolskiy, E. E. Narimanov, W. Fang, H. Cao, “Chaotic microlasers based on dynamical localization,” Proc Natl Acad Sci USA 101, 10498–10500 (2004).
[CrossRef] [PubMed]

V. A. Podolskiy, E. E. Narimanov, “Semiclassical description of chaos-assisted tunneling,” Phys. Rev. Lett. 91, 263601, 2003).
[CrossRef]

Raman, C.

M. Bhattacharya, C. Raman, “Detecting level crossings without looking at the spectrum,” Phys. Rev. Lett. 97, 140405, 2006).
[CrossRef] [PubMed]

Redding, B.

L. Ge, Q. H. Song, B. Redding, H. Cao, “Extreme output sensitivity to subwavelength boundary deformation in microcavities,” Phys. Rev. A 87, 023833, 2013).
[CrossRef]

Q. H. Song, L. Ge, B. Redding, H. Cao, “Channeling chaotic rays into waveguides for efficient collection of microcavity emission,” Phys. Rev. Lett. 108, 243902, 2012).
[CrossRef] [PubMed]

Rex, N. B.

Rotter, I.

M. Müller, I. Rotter, “Exceptional points in open quantum systems,” J. Phys. A 41, 244018, 2008).
[CrossRef]

E. Persson, I. Rotter, H. J. Stockmann, M. Barth, “Observation of resonance trapping in an open microwave cavity,” Phys. Rev. Lett. 85, 2478–2481 (2000).
[CrossRef] [PubMed]

Ryu, J-W

J-W Ryu, S-Y. Lee, S. W. Kim, “Coupled nonidentical microdisks: Avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858, 2012).
[CrossRef]

Saito, K.

Y. Kayanuma, K. Saito, “Coherent destruction of tunneling, dynamic localization, and the Landau-Zener formula,” Phys. Rev. A 77, 010101(R) (2008).
[CrossRef]

Sasaki, T.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902, 2010).
[CrossRef] [PubMed]

Schilling, L.

A. Bäcker, R. Ketzmerick, S. Löck, L. Schilling, “Regular-to-chaotic tunneling rates using a fictitious integrable system,” Phys. Rev. Lett. 100, 104101, 2008).
[CrossRef] [PubMed]

Schomerus, H.

M. Hentschel, H. Schomerus, R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
[CrossRef]

Schubert, R.

M. Hentschel, H. Schomerus, R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
[CrossRef]

Schwefel, H. G. L.

Shim, J.-B.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101, 2009).
[CrossRef] [PubMed]

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

Shim, J-B.

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

Shinohara, S.

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902, 2010).
[CrossRef] [PubMed]

S. Shinohara, T. Harayama, “Signature of ray chaos in quasibound wave functions for a stadium-shaped dielectric cavity,” Phys. Rev. E 75, 036216, 2007).
[CrossRef]

Sivco, D. L.

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

Solomon, G. S.

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

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

Song, Q. H.

S. M. Xiao, Z. Y. Gu, S. Liu, Q. H. Song, “Direct modulation of microcavity emission via local perturbation,” Phys. Rev. A 88, 053833, 2013).
[CrossRef]

Q. H. Song, L. Ge, J. Wiersig, H. Cao, “Formation of long-lived resonances in hexagonal cavities by strong coupling of superscar modes,” Phys. Rev. A 88, 023834, 2013).
[CrossRef]

Q. H. Song, C. Zeng, S. M. Xiao, “Coherent destruction of dynamical tunneling in asymmetrical resonant cavities,” Phys. Rev. A 87, 013831, 2013).
[CrossRef]

L. Ge, Q. H. Song, B. Redding, H. Cao, “Extreme output sensitivity to subwavelength boundary deformation in microcavities,” Phys. Rev. A 87, 023833, 2013).
[CrossRef]

Q. H. Song, L. Ge, B. Redding, H. Cao, “Channeling chaotic rays into waveguides for efficient collection of microcavity emission,” Phys. Rev. Lett. 108, 243902, 2012).
[CrossRef] [PubMed]

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

Q. H. Song, H. Cao, “Improving optical confinement in nanostructures via external mode coupling,” Phys. Rev. Lett. 105, 053902, 2010).
[CrossRef] [PubMed]

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

Srinivasan, K.

K. Srinivasan, O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system,” Nature 450, 862–866 (2007).
[CrossRef] [PubMed]

Stockmann, H. J.

E. Persson, I. Rotter, H. J. Stockmann, M. Barth, “Observation of resonance trapping in an open microwave cavity,” Phys. Rev. Lett. 85, 2478–2481 (2000).
[CrossRef] [PubMed]

Stone, A. D.

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

H. E. Türeci, H. G. L. Schwefel, P. Jacquod, A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. 47, 75–137 (2005).
[CrossRef]

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

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

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

Türeci, H. E.

Unterhinninghofen, J.

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

Wiersig, J.

Q. H. Song, L. Ge, J. Wiersig, H. Cao, “Formation of long-lived resonances in hexagonal cavities by strong coupling of superscar modes,” Phys. Rev. A 88, 023834, 2013).
[CrossRef]

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

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

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901, 2006).
[CrossRef]

J. Wiersig, M. Hentschel, “Unidirectional light emission from high-Q modes in optical microcavities,” Phys. Rev. Lett. 73, 031802(R) (2006).

Xiao, S. M.

Q. H. Song, C. Zeng, S. M. Xiao, “Coherent destruction of dynamical tunneling in asymmetrical resonant cavities,” Phys. Rev. A 87, 013831, 2013).
[CrossRef]

S. M. Xiao, Z. Y. Gu, S. Liu, Q. H. Song, “Direct modulation of microcavity emission via local perturbation,” Phys. Rev. A 88, 053833, 2013).
[CrossRef]

Yang, J.

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101, 2009).
[CrossRef] [PubMed]

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

Yang, X. Z.

J. Burgdorfer, X. Z. Yang, J. Muller, “Parametric variation of resonances for regular and chaotic scattering,” Chaos Solutions & Fractals 5, 1235–1273 (1995).
[CrossRef]

Zeng, C.

Q. H. Song, C. Zeng, S. M. Xiao, “Coherent destruction of dynamical tunneling in asymmetrical resonant cavities,” Phys. Rev. A 87, 013831, 2013).
[CrossRef]

Zyss, J.

Chaos Solutions & Fractals (1)

J. Burgdorfer, X. Z. Yang, J. Muller, “Parametric variation of resonances for regular and chaotic scattering,” Chaos Solutions & Fractals 5, 1235–1273 (1995).
[CrossRef]

Europhys. Lett. (1)

M. Hentschel, H. Schomerus, R. Schubert, “Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond,” Europhys. Lett. 62, 636–642 (2003).
[CrossRef]

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

J. Phys. A (1)

M. Müller, I. Rotter, “Exceptional points in open quantum systems,” J. Phys. A 41, 244018, 2008).
[CrossRef]

Nature (2)

K. Srinivasan, O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system,” Nature 450, 862–866 (2007).
[CrossRef] [PubMed]

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

Phys. Rev. A (8)

Q. H. Song, C. Zeng, S. M. Xiao, “Coherent destruction of dynamical tunneling in asymmetrical resonant cavities,” Phys. Rev. A 87, 013831, 2013).
[CrossRef]

Y. Kayanuma, K. Saito, “Coherent destruction of tunneling, dynamic localization, and the Landau-Zener formula,” Phys. Rev. A 77, 010101(R) (2008).
[CrossRef]

L. Ge, Q. H. Song, B. Redding, H. Cao, “Extreme output sensitivity to subwavelength boundary deformation in microcavities,” Phys. Rev. A 87, 023833, 2013).
[CrossRef]

J-W Ryu, S-Y. Lee, S. W. Kim, “Coupled nonidentical microdisks: Avoided crossing of energy levels and unidirectional far-field emission,” Phys. Rev. A 79, 053858, 2012).
[CrossRef]

Q. H. Song, L. Ge, J. Wiersig, H. Cao, “Formation of long-lived resonances in hexagonal cavities by strong coupling of superscar modes,” Phys. Rev. A 88, 023834, 2013).
[CrossRef]

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

S. M. Xiao, Z. Y. Gu, S. Liu, Q. H. Song, “Direct modulation of microcavity emission via local perturbation,” Phys. Rev. A 88, 053833, 2013).
[CrossRef]

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

Phys. Rev. E (2)

S. Shinohara, T. Harayama, “Signature of ray chaos in quasibound wave functions for a stadium-shaped dielectric cavity,” Phys. Rev. E 75, 036216, 2007).
[CrossRef]

W. D. Heiss, “Repulsion of resonance states and exceptional points,” Phys. Rev. E 61, 929–932 (2000).
[CrossRef]

Phys. Rev. Lett. (14)

J. Wiersig, “Formation of long-lived, scarlike modes near avoided resonance crossings in optical microcavities,” Phys. Rev. Lett. 97, 253901, 2006).
[CrossRef]

S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103, 134101, 2009).
[CrossRef] [PubMed]

M. Bhattacharya, C. Raman, “Detecting level crossings without looking at the spectrum,” Phys. Rev. Lett. 97, 140405, 2006).
[CrossRef] [PubMed]

E. Persson, I. Rotter, H. J. Stockmann, M. Barth, “Observation of resonance trapping in an open microwave cavity,” Phys. Rev. Lett. 85, 2478–2481 (2000).
[CrossRef] [PubMed]

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

F. Grossmann, T. Dittrich, P. Jung, P. Hanggi, “Coherent destruction of the tunneling,” Phys. Rev. Lett. 67, 516–519 (1991).
[CrossRef] [PubMed]

A. Bäcker, R. Ketzmerick, S. Löck, L. Schilling, “Regular-to-chaotic tunneling rates using a fictitious integrable system,” Phys. Rev. Lett. 100, 104101, 2008).
[CrossRef] [PubMed]

V. A. Podolskiy, E. E. Narimanov, “Semiclassical description of chaos-assisted tunneling,” Phys. Rev. Lett. 91, 263601, 2003).
[CrossRef]

Q. H. Song, L. Ge, B. Redding, H. Cao, “Channeling chaotic rays into waveguides for efficient collection of microcavity emission,” Phys. Rev. Lett. 108, 243902, 2012).
[CrossRef] [PubMed]

D. K. Ferry, R. Akis, J. P. Bird, “Einselection in action: decoherence and pointer states in open quantum dots,” Phys. Rev. Lett. 93, 026803, 2004).
[CrossRef] [PubMed]

Q. H. Song, H. Cao, “Improving optical confinement in nanostructures via external mode coupling,” Phys. Rev. Lett. 105, 053902, 2010).
[CrossRef] [PubMed]

J. Wiersig, M. Hentschel, “Unidirectional light emission from high-Q modes in optical microcavities,” Phys. Rev. Lett. 73, 031802(R) (2006).

S. Shinohara, T. Harayama, T. Fukushima, M. Hentschel, T. Sasaki, E. E. Narimanov, “Chaos-assisted directional light emission from microcavity lasers,” Phys. Rev. Lett. 104, 163902, 2010).
[CrossRef] [PubMed]

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

Proc Natl Acad Sci USA (1)

V. A. Podolskiy, E. E. Narimanov, W. Fang, H. Cao, “Chaotic microlasers based on dynamical localization,” Proc Natl Acad Sci USA 101, 10498–10500 (2004).
[CrossRef] [PubMed]

Prog. Opt. (1)

H. E. Türeci, H. G. L. Schwefel, P. Jacquod, A. D. Stone, “Modes of wave-chaotic dielectric resonators,” Prog. Opt. 47, 75–137 (2005).
[CrossRef]

Science (1)

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

Other (1)

N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University, 2011).

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Figures (8)

Fig. 1
Fig. 1

The real (a, d) and imaginary (b, e) parts of E± in Eq. (2) as a function of Δ. Here E1 = Δ − 0.01i, and E2 = −0.001i. (a) and (b) show the behaviors of weak coupling with VW = 0.00001 + 0.000026i; (d) and (e) show the phenomena of strong coupling with VW = 0.001 + 0.00006i. (c) and (f) give the dependencies of EF on Im(VW), where Ei and Re(VW) are kept as constants.

Fig. 2
Fig. 2

The schematic picture (a) of oval cavity and its corresponding Poincaré surface of section. The horizontal and vertical axes correspond to the bouncing positions and incident angles. Here ε1 = 0.1, ε2 = 0.01, and ε3 = 0.011.

Fig. 3
Fig. 3

(a) The Q factors of resonances in oval shaped microcavity. (b)–(d) are the field patterns of modes i – iii marked in Fig. 3(a). Here the deformation parameters are ε1 = 0.1, ε2 = 0.01, ε3 = 0.0107. The quasi-whispering gallery modes are ignored because they can be excluded experimentally using selective pumping. In Fig. 3(a), all the resonances are nearly-degenerate doublets, which have odd and even symmetries with respect to the x-axes.

Fig. 4
Fig. 4

The resonant frequencies (a) and Q factors (b) of rectangle resonances and diamond modes (modes i and iii in Fig. 3(a)) as a function of ε3. Here n = 3.5, ε1 = 0.1, and ε2 = 0.01. (c) shows the field patterns (|Hz|) of modes 1–6 in Fig. 4(a). All the modes in this figure and in Figs. 6 and 8(a) below are nearly-degenerate doublets. Both even and odd symmetries have similar behaviors.

Fig. 5
Fig. 5

(a), (c), (e) are the Husimi maps of the resonances marked as 1, 2, 3 in Fig. 4(a). (b), (d), (f) are their corresponding far field patterns. The horizontal solid lines are the critical lines with sinχc = 1/n. Here n is 3.5.

Fig. 6
Fig. 6

The dependence of resonant frequencies (a) and Q factors (b) of rectangle resonance and diamond modes on the shape deformation parameter ε3. Here the cavity shape is the parameters are the same as Fig. 4 except for n = 2.0. (c) and (d) show the field patterns of rectangle resonance and diamond resonance in oval cavity with ε1 = 0.1, ε2 = 0.1, ε3 = 0.0108.

Fig. 7
Fig. 7

(a) and (c) are the Husimi maps of the rectangle resonance and diamond resonance. The cavity shape is the same as mode-1 and mode-2 in Fig. 4(b). (b) and (d) are their corresponding far field patterns. The horizontal solid lines are the critical lines with n = 2.0.

Fig. 8
Fig. 8

(a) The Q factors of rectangle resonances as a function of shape deformation at different refractive index from n = 2.0 – 3.5. (b) The dependence of the corresponding far field patterns on the refractive index. (c) Far field pattern of rectangle resonance at n = 2.5. The intensity along ϕFF = ±90° has been significantly reduced and becomes to be comparable with the direct tunneling. The shape deformation in (b) and (c) is 0.0108.

Equations (3)

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

H = ( E 1 W V E 2 ) ,
E ± = E 1 + E 2 2 ± ( E 1 E 2 ) 2 4 + V W .
2 ψ = n 2 ( x , y ) ω 2 c 2 ψ ,

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