Abstract

Ultralong distance coupling between two deformed circular microcavities (DCMs) is studied. Different from traditional short distance tunneling coupling between microcavities, highly efficient free-space directional emission and excitation allow ultralong distance energy transfer between DCMs. In this paper, a novel unidirectional emission DCM, which shows directionality I40=0.54, is designed for materials of refractive index n=2.0. Compared with circular whispering gallery microcavities, the coupled unidirectional emission DCMs show modulations of resonance frequency even when the distance between cavities is much longer than wavelength. Also, the performance and properties of the ultralong distance interaction between DCMs are analyzed and studied by coupling mode theory in detail. The ultralong distance interaction between DCMs provides a new path to free-space-based optical interconnects between components in integrated photonic circuits.

© 2014 Optical Society of America

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    [CrossRef]
  2. J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, “Directional interacting whispering-gallery modes in coupled dielectric microdisks,” Phys. Rev. A 74, 013804 (2006).
    [CrossRef]
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    [CrossRef]
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  7. Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photonics Rev. 4, 179–191 (2010).
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  8. T. Grossmann, T. Wienhold, U. Bog, T. Beck, C. Friedmann, H. Kalt, and T. Mappes, “Polymeric photonic molecule super-mode lasers on silicon,” Light: Sci. Appl. 2, e82 (2013).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. C.-L. Zou, Y. Yang, C.-H. Dong, Y.-F. Xiao, Z.-F. Han, and G.-C. Guo, “Accurately calculating high quality factor of whispering-gallery modes with boundary element method,” J. Opt. Soc. Am. B 26, 2050–2053 (2009).
    [CrossRef]
  20. H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
    [CrossRef]
  21. E. I. Smotrova and A. I. Nosich, “Optical coupling of an active microdisk to a passive one: effect on the lasing thresholds of the whispering-gallery supermodes,” Opt. Lett. 38, 2059–2061 (2013).
    [CrossRef]
  22. M. D. Weed, C. G. Williams, P. J. Delfyett, and W. V. Schoenfeld, “Symmetry considerations for closed loop photonic crystal coupled resonators,” J. Lightwave Technol. 31, 1426–1432 (2013).
    [CrossRef]
  23. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with whispering-gallery modes,” Opt. Lett. 31, 921–923 (2006).
    [CrossRef]

2013

C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, Z.-F. Han, and G.-C. Guo, “High Q and unidirectional emission whispering gallery modes: principles and design,” IEEE J. Sel. Top. Quantum Electron. 19, 9000406 (2013).

T. Grossmann, T. Wienhold, U. Bog, T. Beck, C. Friedmann, H. Kalt, and T. Mappes, “Polymeric photonic molecule super-mode lasers on silicon,” Light: Sci. Appl. 2, e82 (2013).

E. Stock, F. Albert, C. Hopfmann, M. Lermer, C. Schneider, S. Hfling, A. Forchel, M. Kamp, and S. Reitzenstein, “On-chip quantum optics with quantum dot microcavities,” Adv. Mater. 25, 707–710 (2013).
[CrossRef]

M. D. Weed, C. G. Williams, P. J. Delfyett, and W. V. Schoenfeld, “Symmetry considerations for closed loop photonic crystal coupled resonators,” J. Lightwave Technol. 31, 1426–1432 (2013).
[CrossRef]

C.-L. Zou, F.-J. Shu, F.-W. Sun, Z.-J. Gong, Z.-F. Han, and G.-C. Guo, “Theory of free space coupling to high-Q whispering gallery modes,” Opt. Express 21, 9982–9995 (2013).
[CrossRef]

F.-J. Shu, C.-L. Zou, and A. F.-W. Sun, “Dynamic process of free space excitation of asymmetric resonant microcavity,” J. Lightwave Technol. 31, 1884–1889 (2013).
[CrossRef]

E. I. Smotrova and A. I. Nosich, “Optical coupling of an active microdisk to a passive one: effect on the lasing thresholds of the whispering-gallery supermodes,” Opt. Lett. 38, 2059–2061 (2013).
[CrossRef]

F.-J. Shu, C.-L. Zou, and F.-W. Sun, “An optimization method of asymmetric resonant cavities for unidirectional emission,” J. Lightwave Technol. 31, 2994–2998 (2013).
[CrossRef]

2012

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).

2011

2010

2009

2008

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

2007

S.-B. Lee, J. Yang, S. Moon, J.-H. Lee, and K. An, “Chaos-assisted nonresonant optical pumping of quadrupole-deformed microlasers,” Appl. Phys. Lett. 90, 041106 (2007).
[CrossRef]

J. J. Li, J. X. Wang, and Y. Z. Huang, “Mode coupling between first- and second-order whispering-gallery modes in coupled microdisks,” Opt. Lett. 32, 1563–1565 (2007).
[CrossRef]

2006

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with whispering-gallery modes,” Opt. Lett. 31, 921–923 (2006).
[CrossRef]

J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, “Directional interacting whispering-gallery modes in coupled dielectric microdisks,” Phys. Rev. A 74, 013804 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

2003

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[CrossRef]

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. Pure. Appl. Opt. 5, 53–60 (2003).

1991

H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[CrossRef]

Albert, F.

E. Stock, F. Albert, C. Hopfmann, M. Lermer, C. Schneider, S. Hfling, A. Forchel, M. Kamp, and S. Reitzenstein, “On-chip quantum optics with quantum dot microcavities,” Adv. Mater. 25, 707–710 (2013).
[CrossRef]

An, K.

S.-B. Lee, J. Yang, S. Moon, J.-H. Lee, and K. An, “Chaos-assisted nonresonant optical pumping of quadrupole-deformed microlasers,” Appl. Phys. Lett. 90, 041106 (2007).
[CrossRef]

Beck, T.

T. Grossmann, T. Wienhold, U. Bog, T. Beck, C. Friedmann, H. Kalt, and T. Mappes, “Polymeric photonic molecule super-mode lasers on silicon,” Light: Sci. Appl. 2, e82 (2013).

Benson, T. M.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with whispering-gallery modes,” Opt. Lett. 31, 921–923 (2006).
[CrossRef]

Bog, U.

T. Grossmann, T. Wienhold, U. Bog, T. Beck, C. Friedmann, H. Kalt, and T. Mappes, “Polymeric photonic molecule super-mode lasers on silicon,” Light: Sci. Appl. 2, e82 (2013).

Cui, J.-M.

C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, Z.-F. Han, and G.-C. Guo, “High Q and unidirectional emission whispering gallery modes: principles and design,” IEEE J. Sel. Top. Quantum Electron. 19, 9000406 (2013).

Delfyett, P. J.

Donegan, J. F.

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photonics Rev. 4, 179–191 (2010).
[CrossRef]

Dong, C. H.

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).

Dong, C.-H.

C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, Z.-F. Han, and G.-C. Guo, “High Q and unidirectional emission whispering gallery modes: principles and design,” IEEE J. Sel. Top. Quantum Electron. 19, 9000406 (2013).

C.-L. Zou, Y. Yang, C.-H. Dong, Y.-F. Xiao, Z.-F. Han, and G.-C. Guo, “Accurately calculating high quality factor of whispering-gallery modes with boundary element method,” J. Opt. Soc. Am. B 26, 2050–2053 (2009).
[CrossRef]

Forchel, A.

E. Stock, F. Albert, C. Hopfmann, M. Lermer, C. Schneider, S. Hfling, A. Forchel, M. Kamp, and S. Reitzenstein, “On-chip quantum optics with quantum dot microcavities,” Adv. Mater. 25, 707–710 (2013).
[CrossRef]

Friedmann, C.

T. Grossmann, T. Wienhold, U. Bog, T. Beck, C. Friedmann, H. Kalt, and T. Mappes, “Polymeric photonic molecule super-mode lasers on silicon,” Light: Sci. Appl. 2, e82 (2013).

Gong, Q.

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).

Gong, Z.-J.

Grossmann, T.

T. Grossmann, T. Wienhold, U. Bog, T. Beck, C. Friedmann, H. Kalt, and T. Mappes, “Polymeric photonic molecule super-mode lasers on silicon,” Light: Sci. Appl. 2, e82 (2013).

Guo, G. C.

Guo, G.-C.

Han, Z. F.

Han, Z.-F.

Haus, H. A.

H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[CrossRef]

He, L.

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).

Hentschel, M.

J. W. Ryu and M. Hentschel, “Designing coupled microcavity lasers for high-Q modes with unidirectional light emission,” Opt. Lett. 36, 1116–1118 (2011).
[CrossRef]

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

Hfling, S.

E. Stock, F. Albert, C. Hopfmann, M. Lermer, C. Schneider, S. Hfling, A. Forchel, M. Kamp, and S. Reitzenstein, “On-chip quantum optics with quantum dot microcavities,” Adv. Mater. 25, 707–710 (2013).
[CrossRef]

Hopfmann, C.

E. Stock, F. Albert, C. Hopfmann, M. Lermer, C. Schneider, S. Hfling, A. Forchel, M. Kamp, and S. Reitzenstein, “On-chip quantum optics with quantum dot microcavities,” Adv. Mater. 25, 707–710 (2013).
[CrossRef]

Huang, W.

H. A. Haus and W. Huang, “Coupled-mode theory,” Proc. IEEE 79, 1505–1518 (1991).
[CrossRef]

Huang, Y. Z.

Jiang, X. F.

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).

Kalt, H.

T. Grossmann, T. Wienhold, U. Bog, T. Beck, C. Friedmann, H. Kalt, and T. Mappes, “Polymeric photonic molecule super-mode lasers on silicon,” Light: Sci. Appl. 2, e82 (2013).

Kamp, M.

E. Stock, F. Albert, C. Hopfmann, M. Lermer, C. Schneider, S. Hfling, A. Forchel, M. Kamp, and S. Reitzenstein, “On-chip quantum optics with quantum dot microcavities,” Adv. Mater. 25, 707–710 (2013).
[CrossRef]

Kim, C.-M.

J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, “Directional interacting whispering-gallery modes in coupled dielectric microdisks,” Phys. Rev. A 74, 013804 (2006).
[CrossRef]

Lee, J.-H.

S.-B. Lee, J. Yang, S. Moon, J.-H. Lee, and K. An, “Chaos-assisted nonresonant optical pumping of quadrupole-deformed microlasers,” Appl. Phys. Lett. 90, 041106 (2007).
[CrossRef]

Lee, S.-B.

S.-B. Lee, J. Yang, S. Moon, J.-H. Lee, and K. An, “Chaos-assisted nonresonant optical pumping of quadrupole-deformed microlasers,” Appl. Phys. Lett. 90, 041106 (2007).
[CrossRef]

Lee, S.-Y.

J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, “Directional interacting whispering-gallery modes in coupled dielectric microdisks,” Phys. Rev. A 74, 013804 (2006).
[CrossRef]

Lermer, M.

E. Stock, F. Albert, C. Hopfmann, M. Lermer, C. Schneider, S. Hfling, A. Forchel, M. Kamp, and S. Reitzenstein, “On-chip quantum optics with quantum dot microcavities,” Adv. Mater. 25, 707–710 (2013).
[CrossRef]

Li, B. B.

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).

Li, J. J.

Li, Y.

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).

Liu, L.

Mappes, T.

T. Grossmann, T. Wienhold, U. Bog, T. Beck, C. Friedmann, H. Kalt, and T. Mappes, “Polymeric photonic molecule super-mode lasers on silicon,” Light: Sci. Appl. 2, e82 (2013).

Moon, S.

S.-B. Lee, J. Yang, S. Moon, J.-H. Lee, and K. An, “Chaos-assisted nonresonant optical pumping of quadrupole-deformed microlasers,” Appl. Phys. Lett. 90, 041106 (2007).
[CrossRef]

Nosich, A. I.

Park, Y.-J.

J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, “Directional interacting whispering-gallery modes in coupled dielectric microdisks,” Phys. Rev. A 74, 013804 (2006).
[CrossRef]

Rakovich, Y. P.

Y. P. Rakovich and J. F. Donegan, “Photonic atoms and molecules,” Laser Photonics Rev. 4, 179–191 (2010).
[CrossRef]

Reitzenstein, S.

E. Stock, F. Albert, C. Hopfmann, M. Lermer, C. Schneider, S. Hfling, A. Forchel, M. Kamp, and S. Reitzenstein, “On-chip quantum optics with quantum dot microcavities,” Adv. Mater. 25, 707–710 (2013).
[CrossRef]

Ryu, J. W.

Ryu, J.-W.

J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, “Directional interacting whispering-gallery modes in coupled dielectric microdisks,” Phys. Rev. A 74, 013804 (2006).
[CrossRef]

Schneider, C.

E. Stock, F. Albert, C. Hopfmann, M. Lermer, C. Schneider, S. Hfling, A. Forchel, M. Kamp, and S. Reitzenstein, “On-chip quantum optics with quantum dot microcavities,” Adv. Mater. 25, 707–710 (2013).
[CrossRef]

Schoenfeld, W. V.

Schwefel, H. G. L.

Sewell, P.

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Optical coupling of whispering-gallery modes of two identical microdisks and its effect on photonic molecule lasing,” IEEE J. Sel. Top. Quantum Electron. 12, 78–85 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with whispering-gallery modes,” Opt. Lett. 31, 921–923 (2006).
[CrossRef]

Shu, F.-J.

Smotrova, E. I.

Stock, E.

E. Stock, F. Albert, C. Hopfmann, M. Lermer, C. Schneider, S. Hfling, A. Forchel, M. Kamp, and S. Reitzenstein, “On-chip quantum optics with quantum dot microcavities,” Adv. Mater. 25, 707–710 (2013).
[CrossRef]

Sun, A. F.-W.

Sun, F. W.

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).

C. L. Zou, H. G. L. Schwefel, F. W. Sun, Z. F. Han, and G. C. Guo, “Quick root searching method for resonances of dielectric optical microcavities with the boundary element method,” Opt. Express 19, 15669–15678 (2011).
[CrossRef]

Sun, F.-W.

C.-L. Zou, F.-J. Shu, F.-W. Sun, Z.-J. Gong, Z.-F. Han, and G.-C. Guo, “Theory of free space coupling to high-Q whispering gallery modes,” Opt. Express 21, 9982–9995 (2013).
[CrossRef]

F.-J. Shu, C.-L. Zou, and F.-W. Sun, “An optimization method of asymmetric resonant cavities for unidirectional emission,” J. Lightwave Technol. 31, 2994–2998 (2013).
[CrossRef]

C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, Z.-F. Han, and G.-C. Guo, “High Q and unidirectional emission whispering gallery modes: principles and design,” IEEE J. Sel. Top. Quantum Electron. 19, 9000406 (2013).

Tu, X.

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[CrossRef]

Wang, J. X.

Weed, M. D.

Wienhold, T.

T. Grossmann, T. Wienhold, U. Bog, T. Beck, C. Friedmann, H. Kalt, and T. Mappes, “Polymeric photonic molecule super-mode lasers on silicon,” Light: Sci. Appl. 2, e82 (2013).

Wiersig, J.

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

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. Pure. Appl. Opt. 5, 53–60 (2003).

Williams, C. G.

Wu, X.

Wu, X.-W.

C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, Z.-F. Han, and G.-C. Guo, “High Q and unidirectional emission whispering gallery modes: principles and design,” IEEE J. Sel. Top. Quantum Electron. 19, 9000406 (2013).

Xiao, Y. F.

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).

Xiao, Y.-F.

Xu, L.

Yang, J.

S.-B. Lee, J. Yang, S. Moon, J.-H. Lee, and K. An, “Chaos-assisted nonresonant optical pumping of quadrupole-deformed microlasers,” Appl. Phys. Lett. 90, 041106 (2007).
[CrossRef]

Yang, L.

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).

Yang, Y.

C.-L. Zou, F.-W. Sun, C.-H. Dong, F.-J. Shu, X.-W. Wu, J.-M. Cui, Y. Yang, Z.-F. Han, and G.-C. Guo, “High Q and unidirectional emission whispering gallery modes: principles and design,” IEEE J. Sel. Top. Quantum Electron. 19, 9000406 (2013).

C.-L. Zou, Y. Yang, C.-H. Dong, Y.-F. Xiao, Z.-F. Han, and G.-C. Guo, “Accurately calculating high quality factor of whispering-gallery modes with boundary element method,” J. Opt. Soc. Am. B 26, 2050–2053 (2009).
[CrossRef]

Zou, C. L.

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).

C. L. Zou, H. G. L. Schwefel, F. W. Sun, Z. F. Han, and G. C. Guo, “Quick root searching method for resonances of dielectric optical microcavities with the boundary element method,” Opt. Express 19, 15669–15678 (2011).
[CrossRef]

Zou, C.-L.

Adv. Mater.

E. Stock, F. Albert, C. Hopfmann, M. Lermer, C. Schneider, S. Hfling, A. Forchel, M. Kamp, and S. Reitzenstein, “On-chip quantum optics with quantum dot microcavities,” Adv. Mater. 25, 707–710 (2013).
[CrossRef]

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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Figures (6)

Fig. 1.
Fig. 1.

(a) Near-field and (b) far-field distributions of TM37,1. Red–green–blue false colors indicate intensity from high to low in logarithmic scale.

Fig. 2.
Fig. 2.

Schematics of coupled DCMs.

Fig. 3.
Fig. 3.

Far-field coupling patterns of (a) TM37,1 of the DCM, (b) TM40,1, and (c) TM30,3 of the circular cavity. The (d) real and (e) imaginary parts of ΔkRs vary with gap L between two cavities.

Fig. 4.
Fig. 4.

Tracks of kR (a) theory and (b) simulation. Center cross indicates the location of resonant frequency without coupling.

Fig. 5.
Fig. 5.

(a) Values of g/κa from simulation results (solid blue line) and Eq. (7) with g(0)=1.4 (dashed red line). Time evolution of normalized energy of two coupled cavities. Energy in cavities (b) A and (d) B varies with time when g=0.68κa and θ=0,0.1π,0.2π,0.3π,0.4π,0.45π,0.5π. (c) Evolution of energy at different combination of parameters.

Fig. 6.
Fig. 6.

Time evolution of normalized energy of two mismatched cavities. Three columns (from left to right), weak, middle, and strong, represent detuning, respectively. Each row corresponds to a typical value of θ. Inset in (e): middle detuning with θ=0.75π.

Equations (10)

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B(ϕ)={R(1iaicosiϕ),π/2<ϕπ/2R(1ibicosiϕ),π/2<ϕ3π/2,
Im(kR)=Re(kR)2Q,
{ddta(t)=(iωaκa)a(t)+geiθb(t)ddtb(t)=(iωbκb)b(t)+geiθa(t),
M=(ω1γγω2)=(iωaκageiθgeiθiωbκb).
ω±=12(ω1+ω2±(ω1ω2)2+4γ2),
(cacb)=1|ω±ω1|2+|γ|2(γω±ω1).
g(L)=g(0)/L+2R.
Φ(r,t)=12[Φ+(r,0)eω+t+Φ(r,0)eωt],
Φ(r,t)=12[(eω+t+eωt)Ψa+(eω+teωt)Ψb].
Ea,b=|12(eω+t±eωt)|2=12e2κat[cosh(2gcosθ·t)±cos(2gsinθ·t)],

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