Abstract

It is well known that the strongly deformed microcavity with fully chaotic ray dynamics cannot support high-Q modes due to its fast chaotic diffusion to the critical line of refractive emission. Here, we investigate how the Q factor is modified when two chaotic cavities are coupled, and show that some modes, whose Q factor is about 10 times higher than that of the corresponding single cavity, can exist. These abnormal high-Q modes are the result of an optimal combination of coupling and cavity geometry. As an example, in the coupled stadium-shaped microcavities, the mode pattern extends over both cavities such that it follows a whispering-gallery-type mode at both ends, whereas a big coupling spot forms at the closest contact of the two microcavities. The pattern of such a “rounded bow tie” mode allows the mode to have a high-Q factor. This mode pattern minimizes the leakage of light at both ends of the microcavities as the pattern at both ends is similar to the whispering gallery mode.

© 2014 Optical Society of America

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    [CrossRef]
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2012 (2)

J.-W. Ryu, M. Hentschel, and S. W. Kim, Phys. Rev. E 85, 056213 (2012).
[CrossRef]

J. Wiersig, Phys. Rev. A 85, 063838 (2012).
[CrossRef]

2011 (2)

F.-J. Shu, C.-L. Zou, F.-W. Sun, and Y.-F. Xiao, Phys. Rev. A 83, 053835 (2011).
[CrossRef]

J.-W. Ryu and M. Hentschel, Opt. Lett. 36, 1116 (2011).
[CrossRef]

2010 (2)

J.-W. Ryu and M. Hentschel, Phys. Rev. A 82, 033824 (2010).
[CrossRef]

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
[CrossRef]

2009 (1)

S. B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, Phys. Rev. A 80, 011802R (2009).
[CrossRef]

2008 (1)

L. Shang, L. Liu, and L. Xu, Appl. Phys. Lett. 92, 071111 (2008).
[CrossRef]

2007 (1)

W. Fang and H. Cao, Appl. Phys. Lett. 91, 041108 (2007).
[CrossRef]

2006 (2)

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, Appl. Phys. Lett 88, 031108 (2006).
[CrossRef]

S. V. Boriskina, Opt. Lett. 31, 338 (2006).
[CrossRef]

2005 (1)

T. Harayama, S. Sunada, and K. S. Ikeda, Phys. Rev. A 72, 013803 (2005).
[CrossRef]

2003 (4)

T. Harayama, P. Davis, and K. S. Ikeda, Phys. Rev. Lett. 90, 063901 (2003).
[CrossRef]

J. Wiersig, J. Opt. A 5, 53 (2003).
[CrossRef]

M. Hentschel, H. Schomerus, and R. Schubert, Europhys. Lett. 62, 636 (2003).
[CrossRef]

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef]

1999 (1)

1998 (2)

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, Phys. Rev. Lett. 81, 2582 (1998).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef]

1997 (1)

J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

1993 (1)

Y. Yamamoto and R. E. Slusher, Phys. Today 46(6), 66 (1993).
[CrossRef]

1992 (1)

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

1979 (1)

L. A. Bunimovich, Commun. Math. Phys. 65, 295 (1979).
[CrossRef]

An, K.

S. B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, Phys. Rev. A 80, 011802R (2009).
[CrossRef]

Bayer, M.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, Phys. Rev. Lett. 81, 2582 (1998).
[CrossRef]

Boriskina, S. V.

Bunimovich, L. A.

L. A. Bunimovich, Commun. Math. Phys. 65, 295 (1979).
[CrossRef]

Cao, H.

W. Fang and H. Cao, Appl. Phys. Lett. 91, 041108 (2007).
[CrossRef]

Capasso, F.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
[CrossRef]

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef]

Cho, A. Y.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef]

Davis, P.

T. Harayama, P. Davis, and K. S. Ikeda, Phys. Rev. Lett. 90, 063901 (2003).
[CrossRef]

Diehl, L.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
[CrossRef]

Dremin, A. A.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, Phys. Rev. Lett. 81, 2582 (1998).
[CrossRef]

Edamura, T.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
[CrossRef]

Faist, J.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef]

Fang, W.

W. Fang and H. Cao, Appl. Phys. Lett. 91, 041108 (2007).
[CrossRef]

Forchel, A.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, Phys. Rev. Lett. 81, 2582 (1998).
[CrossRef]

Gmachl, C.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef]

Gutbrod, T.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, Phys. Rev. Lett. 81, 2582 (1998).
[CrossRef]

Harayama, T.

T. Harayama, S. Sunada, and K. S. Ikeda, Phys. Rev. A 72, 013803 (2005).
[CrossRef]

T. Harayama, P. Davis, and K. S. Ikeda, Phys. Rev. Lett. 90, 063901 (2003).
[CrossRef]

Hentschel, M.

J.-W. Ryu, M. Hentschel, and S. W. Kim, Phys. Rev. E 85, 056213 (2012).
[CrossRef]

J.-W. Ryu and M. Hentschel, Opt. Lett. 36, 1116 (2011).
[CrossRef]

J.-W. Ryu and M. Hentschel, Phys. Rev. A 82, 033824 (2010).
[CrossRef]

M. Hentschel, H. Schomerus, and R. Schubert, Europhys. Lett. 62, 636 (2003).
[CrossRef]

Hierle, R.

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, Appl. Phys. Lett 88, 031108 (2006).
[CrossRef]

Ikeda, K. S.

T. Harayama, S. Sunada, and K. S. Ikeda, Phys. Rev. A 72, 013803 (2005).
[CrossRef]

T. Harayama, P. Davis, and K. S. Ikeda, Phys. Rev. Lett. 90, 063901 (2003).
[CrossRef]

Kan, H.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
[CrossRef]

Kim, S. W.

J.-W. Ryu, M. Hentschel, and S. W. Kim, Phys. Rev. E 85, 056213 (2012).
[CrossRef]

S. B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, Phys. Rev. A 80, 011802R (2009).
[CrossRef]

J.-W. Ryu and S. W. Kim, “Localization induced by opening; why are most eigenmodes localized in chaotic microcavities?” arXiv:1305.6698 (2013).

Knipp, P. A.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, Phys. Rev. Lett. 81, 2582 (1998).
[CrossRef]

Kulakovskii, V. D.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, Phys. Rev. Lett. 81, 2582 (1998).
[CrossRef]

Lauret, J. S.

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, Appl. Phys. Lett 88, 031108 (2006).
[CrossRef]

Lebental, M.

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, Appl. Phys. Lett 88, 031108 (2006).
[CrossRef]

Lee, J.-H.

S. B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, Phys. Rev. A 80, 011802R (2009).
[CrossRef]

Lee, R. K.

Lee, S. B.

S. B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, Phys. Rev. A 80, 011802R (2009).
[CrossRef]

Lee, S.-Y.

S. B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, Phys. Rev. A 80, 011802R (2009).
[CrossRef]

S.-Y. Lee, Advances in Optical and Photonic Devices, (InTech, 2010).

Levi, A. F.

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Liu, L.

L. Shang, L. Liu, and L. Xu, Appl. Phys. Lett. 92, 071111 (2008).
[CrossRef]

Logan, R. A.

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

McCall, S. L.

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Moon, S.

S. B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, Phys. Rev. A 80, 011802R (2009).
[CrossRef]

Narimanov, E. E.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef]

Nöckel, J. U.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef]

J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

Pearton, S. J.

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Pflügl, C.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
[CrossRef]

Reinecke, T. L.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, Phys. Rev. Lett. 81, 2582 (1998).
[CrossRef]

Reithmaier, J. P.

M. Bayer, T. Gutbrod, J. P. Reithmaier, A. Forchel, T. L. Reinecke, P. A. Knipp, A. A. Dremin, and V. D. Kulakovskii, Phys. Rev. Lett. 81, 2582 (1998).
[CrossRef]

Ryu, J.-W.

J.-W. Ryu, M. Hentschel, and S. W. Kim, Phys. Rev. E 85, 056213 (2012).
[CrossRef]

J.-W. Ryu and M. Hentschel, Opt. Lett. 36, 1116 (2011).
[CrossRef]

J.-W. Ryu and M. Hentschel, Phys. Rev. A 82, 033824 (2010).
[CrossRef]

J.-W. Ryu and S. W. Kim, “Localization induced by opening; why are most eigenmodes localized in chaotic microcavities?” arXiv:1305.6698 (2013).

Scherer, A.

Schomerus, H.

M. Hentschel, H. Schomerus, and R. Schubert, Europhys. Lett. 62, 636 (2003).
[CrossRef]

Schubert, R.

M. Hentschel, H. Schomerus, and R. Schubert, Europhys. Lett. 62, 636 (2003).
[CrossRef]

Shang, L.

L. Shang, L. Liu, and L. Xu, Appl. Phys. Lett. 92, 071111 (2008).
[CrossRef]

Shim, J.-B.

S. B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, Phys. Rev. A 80, 011802R (2009).
[CrossRef]

Shu, F.-J.

F.-J. Shu, C.-L. Zou, F.-W. Sun, and Y.-F. Xiao, Phys. Rev. A 83, 053835 (2011).
[CrossRef]

Siegman, A.

A. Siegman, Lasers, (University Science, 1986).

Sivco, D. L.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef]

Slusher, R. E.

Y. Yamamoto and R. E. Slusher, Phys. Today 46(6), 66 (1993).
[CrossRef]

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Stone, A. D.

C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nöckel, A. D. Stone, J. Faist, D. L. Sivco, and A. Y. Cho, Science 280, 1556 (1998).
[CrossRef]

J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

Sun, F.-W.

F.-J. Shu, C.-L. Zou, F.-W. Sun, and Y.-F. Xiao, Phys. Rev. A 83, 053835 (2011).
[CrossRef]

Sunada, S.

T. Harayama, S. Sunada, and K. S. Ikeda, Phys. Rev. A 72, 013803 (2005).
[CrossRef]

Unterhinninghofen, J.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
[CrossRef]

Vahala, K. J.

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef]

Wang, Q. J.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
[CrossRef]

Wiersig, J.

J. Wiersig, Phys. Rev. A 85, 063838 (2012).
[CrossRef]

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
[CrossRef]

J. Wiersig, J. Opt. A 5, 53 (2003).
[CrossRef]

Xiao, Y.-F.

F.-J. Shu, C.-L. Zou, F.-W. Sun, and Y.-F. Xiao, Phys. Rev. A 83, 053835 (2011).
[CrossRef]

Xu, L.

L. Shang, L. Liu, and L. Xu, Appl. Phys. Lett. 92, 071111 (2008).
[CrossRef]

Xu, Y.

Yamamoto, Y.

Y. Yamamoto and R. E. Slusher, Phys. Today 46(6), 66 (1993).
[CrossRef]

Yamanishi, M.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
[CrossRef]

Yan, C.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
[CrossRef]

Yang, J.

S. B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, Phys. Rev. A 80, 011802R (2009).
[CrossRef]

Yariv, A.

Yu, N.

Q. J. Wang, C. Yan, N. Yu, J. Unterhinninghofen, J. Wiersig, C. Pflügl, L. Diehl, T. Edamura, M. Yamanishi, H. Kan, and F. Capasso, Proc. Natl. Acad. Sci. USA 107, 22407 (2010).
[CrossRef]

Zou, C.-L.

F.-J. Shu, C.-L. Zou, F.-W. Sun, and Y.-F. Xiao, Phys. Rev. A 83, 053835 (2011).
[CrossRef]

Zyss, J.

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, Appl. Phys. Lett 88, 031108 (2006).
[CrossRef]

Appl. Phys. Lett (1)

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, Appl. Phys. Lett 88, 031108 (2006).
[CrossRef]

Appl. Phys. Lett. (3)

W. Fang and H. Cao, Appl. Phys. Lett. 91, 041108 (2007).
[CrossRef]

L. Shang, L. Liu, and L. Xu, Appl. Phys. Lett. 92, 071111 (2008).
[CrossRef]

S. L. McCall, A. F. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Commun. Math. Phys. (1)

L. A. Bunimovich, Commun. Math. Phys. 65, 295 (1979).
[CrossRef]

Europhys. Lett. (1)

M. Hentschel, H. Schomerus, and R. Schubert, Europhys. Lett. 62, 636 (2003).
[CrossRef]

J. Opt. A (1)

J. Wiersig, J. Opt. A 5, 53 (2003).
[CrossRef]

Nature (2)

J. U. Nöckel and A. D. Stone, Nature 385, 45 (1997).
[CrossRef]

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (5)

F.-J. Shu, C.-L. Zou, F.-W. Sun, and Y.-F. Xiao, Phys. Rev. A 83, 053835 (2011).
[CrossRef]

J. Wiersig, Phys. Rev. A 85, 063838 (2012).
[CrossRef]

T. Harayama, S. Sunada, and K. S. Ikeda, Phys. Rev. A 72, 013803 (2005).
[CrossRef]

S. B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, Phys. Rev. A 80, 011802R (2009).
[CrossRef]

J.-W. Ryu and M. Hentschel, Phys. Rev. A 82, 033824 (2010).
[CrossRef]

Phys. Rev. E (1)

J.-W. Ryu, M. Hentschel, and S. W. Kim, Phys. Rev. E 85, 056213 (2012).
[CrossRef]

Phys. Rev. Lett. (2)

T. Harayama, P. Davis, and K. S. Ikeda, Phys. Rev. Lett. 90, 063901 (2003).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Q factors of resonant modes in a single stadium-shaped microcavity with n=1.45. Three selected intensity patterns of high-Q, mid-Q, and low-Q modes, as well as the resonator geometry, are shown as insets from top to bottom.

Fig. 2.
Fig. 2.

(a) Q-factors of resonant modes in coupled stadium-shaped microcavities with n=1.45 and d=0.1. The open red and black circles represent the modes with even and odd parities about the y axis, respectively. (b) The intensity patterns of highest Q modes (E1 mode). (c) Incident and (d) emerging generalized Husimi functions taken at the inner boundary of the left microcavity. The blue dashed line represents the critical angle for total internal reflection.

Fig. 3.
Fig. 3.

(a) Collection of ray trajectories corresponding to the high-Q modes. (b)–(d) Selected periodic orbits, which support the high-Q modes. (e) Poincaré surface of section for the left microcavity in the coupled stadium-shaped microcavities. The small red circles, black triangles, green diamonds, and blue circles represent red, black, green, and blue trajectories, respectively.

Fig. 4.
Fig. 4.

Q-factors of the E1 (red squares) and O1 (black circles) modes as a function of d. E1 corresponds to the mode pattern of Fig. 2(b). The EA(OA), EB(OB), and EC(OC) are the mode patterns at d, where Q-factors of the modes with even (odd) parity are locally highest. (Inset) The Q-factors of the modes around E1 and O1 as a function of d.

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