Abstract

When two microdisks are placed close to each other and the evanescent fields of their whispering gallery modes are overlapped, a strong coupling can be induced in the modes and lead to a doublet state. We attempt to evaluate the frequency splittings of the doublets by applying a semiclassical analysis in the regime of small wavelengths. Since a whispering gallery mode in a microdisk is a leaky mode, an established semiclassical method that deals with coupled closed systems is modified. As a result, we attain an analytic formula which can conveniently compute the frequency splittings of coupled whispering gallery modes. The derived formula is verified by demostrating a perfect agreement with numerical solutions of Maxwell’s equations.

© 2013 OSA

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References

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  1. K. Vahala, ed., Optical Microcavities(World Scientific, 2004).
  2. M. Benyoucef, J.-B. Shim, J. Wiersig, and O. G. Schmidt, “Quality-factor enhancement of supermodes in coupled microdisks,” Opt. Lett.36, 1317–1319 (2011).
    [CrossRef] [PubMed]
  3. M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
    [CrossRef]
  4. S. Preu, H. G. L. Schwefel, S. Malzer, G. H. Döhler, L. J. Wang, M. Hanson, J. D. Zimmerman, and A. C. Gossard, “Coupled whispering gallery mode resonators in the terahertz frequency range.” Opt. Express16, 7336–7343 (2008).
    [CrossRef] [PubMed]
  5. K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett.98, 213904 (2007).
    [CrossRef] [PubMed]
  6. U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B82, 094308 (2010).
    [CrossRef]
  7. M. Zhang, G. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett.109, 233906 (2012).
    [CrossRef]
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    [CrossRef]
  10. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun.181, 687–702 (2010).
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  14. M. Brack and R. K. Bhaduri, Semiclassical Physics (Westview, 2008).
  15. S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A34, 3791–3801 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. S. C. Creagh and M. White, “Differences between emission patterns and internal modes of optical resonators,” Phys. Rev. E85, 015201 (2012).
    [CrossRef]
  22. J.-B. Shim, J. Wiersig, and H. Cao, “Whispering gallery modes formed by partial barriers in ultrasmall deformed microdisks,” Phys. Rev. E84, 035202 (2011).
    [CrossRef]
  23. F. W. J. Oliver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions(Cambridge University, 2010).

2013 (1)

M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
[CrossRef]

2012 (2)

S. C. Creagh and M. White, “Differences between emission patterns and internal modes of optical resonators,” Phys. Rev. E85, 015201 (2012).
[CrossRef]

M. Zhang, G. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett.109, 233906 (2012).
[CrossRef]

2011 (3)

J.-B. Shim, J. Wiersig, and H. Cao, “Whispering gallery modes formed by partial barriers in ultrasmall deformed microdisks,” Phys. Rev. E84, 035202 (2011).
[CrossRef]

S.-Y. Lee and K. An, “Directional emission through dynamical tunneling in a deformed microcavity,” Phys. Rev. A83, 023827 (2011).
[CrossRef]

M. Benyoucef, J.-B. Shim, J. Wiersig, and O. G. Schmidt, “Quality-factor enhancement of supermodes in coupled microdisks,” Opt. Lett.36, 1317–1319 (2011).
[CrossRef] [PubMed]

2010 (2)

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B82, 094308 (2010).
[CrossRef]

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun.181, 687–702 (2010).
[CrossRef]

2008 (2)

2007 (1)

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

2006 (1)

2003 (1)

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A5, 53–60 (2003).
[CrossRef]

2001 (1)

S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A34, 3791–3801 (2001).
[CrossRef]

1993 (1)

1986 (1)

M. Wilkinson, “Tunnelling between tori in phase space,” Physica D21, 341–354 (1986).
[CrossRef]

1962 (1)

C. Herring, “Critique of the Heitler-London method of calculating spin couplings at large distances,” Rev. Mod. Phys.34, 631–645 (1962).
[CrossRef]

An, K.

S.-Y. Lee and K. An, “Directional emission through dynamical tunneling in a deformed microcavity,” Phys. Rev. A83, 023827 (2011).
[CrossRef]

Arfken, G. B.

G. B. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

Barkhofen, S.

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B82, 094308 (2010).
[CrossRef]

Barnard, A.

M. Zhang, G. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett.109, 233906 (2012).
[CrossRef]

Benyoucef, M.

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun.181, 687–702 (2010).
[CrossRef]

Bhaduri, R. K.

M. Brack and R. K. Bhaduri, Semiclassical Physics (Westview, 2008).

Bogomolny, E.

R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A77, 013804 (2008).
[CrossRef]

Boriskina, S. V.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Brack, M.

M. Brack and R. K. Bhaduri, Semiclassical Physics (Westview, 2008).

Cao, H.

J.-B. Shim, J. Wiersig, and H. Cao, “Whispering gallery modes formed by partial barriers in ultrasmall deformed microdisks,” Phys. Rev. E84, 035202 (2011).
[CrossRef]

Creagh, S. C.

S. C. Creagh and M. White, “Differences between emission patterns and internal modes of optical resonators,” Phys. Rev. E85, 015201 (2012).
[CrossRef]

S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A34, 3791–3801 (2001).
[CrossRef]

de Forges de Parny, L.

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B82, 094308 (2010).
[CrossRef]

Djellali, N.

R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A77, 013804 (2008).
[CrossRef]

Döhler, G. H.

Dubertrand, R.

R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A77, 013804 (2008).
[CrossRef]

Finn, M. D.

S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A34, 3791–3801 (2001).
[CrossRef]

Gossard, A. C.

Hanson, M.

Hargart, F.

M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
[CrossRef]

Herring, C.

C. Herring, “Critique of the Heitler-London method of calculating spin couplings at large distances,” Rev. Mod. Phys.34, 631–645 (1962).
[CrossRef]

Hossain, T.

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B82, 094308 (2010).
[CrossRef]

Hu, E.

M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
[CrossRef]

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun.181, 687–702 (2010).
[CrossRef]

Jetter, M.

M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
[CrossRef]

Jin, J.-M.

J.-M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley-IEEE, 2002).

Joannopoulos, J.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun.181, 687–702 (2010).
[CrossRef]

Johnson, B. R.

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun.181, 687–702 (2010).
[CrossRef]

Kobayashi, N.

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

Koroknay, E.

M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
[CrossRef]

Kuhl, U.

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B82, 094308 (2010).
[CrossRef]

Lebental, M.

R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A77, 013804 (2008).
[CrossRef]

Lee, S.-Y.

S.-Y. Lee and K. An, “Directional emission through dynamical tunneling in a deformed microcavity,” Phys. Rev. A83, 023827 (2011).
[CrossRef]

Lipson, M.

M. Zhang, G. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett.109, 233906 (2012).
[CrossRef]

Liu, T.-L.

M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
[CrossRef]

Malzer, S.

Manipatruni, S.

M. Zhang, G. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett.109, 233906 (2012).
[CrossRef]

McEuen, P.

M. Zhang, G. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett.109, 233906 (2012).
[CrossRef]

Michler, P.

M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
[CrossRef]

Mortessagne, F.

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B82, 094308 (2010).
[CrossRef]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun.181, 687–702 (2010).
[CrossRef]

Preu, S.

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun.181, 687–702 (2010).
[CrossRef]

Schmidt, O. G.

Schmit, C.

R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A77, 013804 (2008).
[CrossRef]

Schulz, W.-M.

M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
[CrossRef]

Schwefel, H. G. L.

Shim, J.-B.

M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
[CrossRef]

M. Benyoucef, J.-B. Shim, J. Wiersig, and O. G. Schmidt, “Quality-factor enhancement of supermodes in coupled microdisks,” Opt. Lett.36, 1317–1319 (2011).
[CrossRef] [PubMed]

J.-B. Shim, J. Wiersig, and H. Cao, “Whispering gallery modes formed by partial barriers in ultrasmall deformed microdisks,” Phys. Rev. E84, 035202 (2011).
[CrossRef]

Stöckmann, H.-J.

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B82, 094308 (2010).
[CrossRef]

Tomita, M.

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

Totsuka, K.

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

Tudorovskiy, T.

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B82, 094308 (2010).
[CrossRef]

Wang, L. J.

White, M.

S. C. Creagh and M. White, “Differences between emission patterns and internal modes of optical resonators,” Phys. Rev. E85, 015201 (2012).
[CrossRef]

Wiederhecker, G.

M. Zhang, G. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett.109, 233906 (2012).
[CrossRef]

Wiersig, J.

M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
[CrossRef]

M. Benyoucef, J.-B. Shim, J. Wiersig, and O. G. Schmidt, “Quality-factor enhancement of supermodes in coupled microdisks,” Opt. Lett.36, 1317–1319 (2011).
[CrossRef] [PubMed]

J.-B. Shim, J. Wiersig, and H. Cao, “Whispering gallery modes formed by partial barriers in ultrasmall deformed microdisks,” Phys. Rev. E84, 035202 (2011).
[CrossRef]

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A5, 53–60 (2003).
[CrossRef]

Wilkinson, M.

M. Wilkinson, “Tunnelling between tori in phase space,” Physica D21, 341–354 (1986).
[CrossRef]

Witzany, M.

M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Zhang, M.

M. Zhang, G. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett.109, 233906 (2012).
[CrossRef]

Zimmerman, J. D.

Comput. Phys. Commun. (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun.181, 687–702 (2010).
[CrossRef]

J. Opt. A (1)

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A5, 53–60 (2003).
[CrossRef]

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

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

J. Phys. A (1)

S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A34, 3791–3801 (2001).
[CrossRef]

New J. Phys. (1)

M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (2)

S.-Y. Lee and K. An, “Directional emission through dynamical tunneling in a deformed microcavity,” Phys. Rev. A83, 023827 (2011).
[CrossRef]

R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A77, 013804 (2008).
[CrossRef]

Phys. Rev. B (1)

U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B82, 094308 (2010).
[CrossRef]

Phys. Rev. E (2)

S. C. Creagh and M. White, “Differences between emission patterns and internal modes of optical resonators,” Phys. Rev. E85, 015201 (2012).
[CrossRef]

J.-B. Shim, J. Wiersig, and H. Cao, “Whispering gallery modes formed by partial barriers in ultrasmall deformed microdisks,” Phys. Rev. E84, 035202 (2011).
[CrossRef]

Phys. Rev. Lett. (2)

M. Zhang, G. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett.109, 233906 (2012).
[CrossRef]

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

Physica D (1)

M. Wilkinson, “Tunnelling between tori in phase space,” Physica D21, 341–354 (1986).
[CrossRef]

Rev. Mod. Phys. (1)

C. Herring, “Critique of the Heitler-London method of calculating spin couplings at large distances,” Rev. Mod. Phys.34, 631–645 (1962).
[CrossRef]

Other (6)

K. Vahala, ed., Optical Microcavities(World Scientific, 2004).

J.-M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley-IEEE, 2002).

M. Brack and R. K. Bhaduri, Semiclassical Physics (Westview, 2008).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

F. W. J. Oliver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions(Cambridge University, 2010).

G. B. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

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Figures (6)

Fig. 1
Fig. 1

Doublet states of the coupled whispering gallery modes, the mode numbers and the polarization of which are (m, N) = (21, 1) and TM respectively. The gap between the microdisks is 0.1 times radius R and the refractive index n is 1.5. (a) Bonding (kR = 16.632) and (b) Antibonding mode (kR = 16.571). The vacant line of intensity on the vertical symmetric axis is conspicuous.

Fig. 2
Fig. 2

(a) Effective confining potential of the radial motion in a microdisk. Here, the wave number k and the angular momentum l are assigned. The effective potential barrier in R < r < l/k separates the potential well and the free space. (b) Phase space portrait corresponding to the radial ray motion in (a). As the radial motions is oscillatory due to reflection at r = R, the manifold in phase space forms a closed area Sre.

Fig. 3
Fig. 3

WKB approximations of (a) resonances in a microdisk (nkR) and (b) their attenuation (Γ/(2nkR)) with changing refractive index (black points). (a) The rotational mode numbers are m = 20, 21, 22 and 23 from bottom to top. (b) m = 20 (bottom) and m = 23 (top). All the modes have TM polarization and 1 as the radial mode number in common. They show perfect agreement with solutions of Maxwell’s equations (dotted lines).

Fig. 4
Fig. 4

(a) TM (21, 1) WGM in a microdisk. The outer end of its evanescent region R < r < m/k is marked by green dotted line. (b) The ray dynamical trajectories corresponding to the mode in (a), the internal reflection angle of which is given by m/nkR. A single real trajectory (bright blue) runs around the cavity boundary. A trajectory in complex phase space (black dotted line) starts from the point where a real trajectory reflects at the boundary runs spirally around the disk. When it reaches the outer boundary of the evanescent region, it goes out along the straight line (red arrow).

Fig. 5
Fig. 5

Ray dynamics in two microdisks coupled via evanescent fields. The distance between the two centers of disks ( = O 1 O 2 ¯) is 2d. Two evanescent regions of TM (21,1)-WGMs are overlapped and their external boundaries intersect at the point A and B. The evanescent tunneling can be analyzed by means of two spiral trajectories (red line) which join smoothly on the line ∑.

Fig. 6
Fig. 6

(a) Splittings Δk of the TM (21,1)- and TM (23,1)-doublet states as a function of the half distance (d) between the two microdisks. The semiclassical approximation by Eq. (35) (squares and circles) and the numerical solution of Maxwell’s equations (dashed and dotted line) show a good agreement. (b) Minimum values of d, at which Eq. (35) is still valid for TM(m, 1)-WGMs.

Equations (48)

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( 2 + k 2 ) ψ = 0
Δ k ( d , k , m ) = 2 π n 2 k R 2 ( n 2 1 ) m 2 ( k R ) 2 m 2 ( k d ) 2 ( ( d R ) m m 2 ( k R ) 2 m m 2 ( k d ) 2 ) 2 m e 2 ( m 2 ( k d ) 2 m 2 ( k R ) 2 ) .
Δ E = 2 < ψ L | V L | ψ R > < ψ R | V R | ψ L > ,
Δ E = 2 Σ ( ψ L * ψ R ψ R ψ L * ) d s ,
ψ ( r ) = 1 2 π [ det ( 2 S r I ) ] 1 2 exp ( i S ( r , I ) )
Δ E ~ i ( 2 π ) 2 Σ d s D L D R ( S L + S R ) e i ( S R S L ) ,
D q = [ det ( 2 S q r I ) ] , q = L o r R .
S R x 1 S L x 1 = 0 ,
D total = D L D R / ( 2 ( S L S R ) x 1 2 ) .
Δ E = 2 ( 2 π ) 3 / 2 ω L ω R i { I R , I L } e i ( S L S R ) / .
H ( r , p ) = p 2 2 + V k ( r ) = E tot ,
V k ( r ) = { 0 ( | r | < R ) k 2 2 ( n 2 1 ) ( | r | > R ) .
H eff ( r , p r ) = p r 2 2 + l 2 2 r 2 + V k ( r ) ,
p r = ( n k ) 2 l 2 r 2 2 V k ( r ) .
δ m = 2 α m = 2 tan 1 ( ( m / n k R ) 2 1 / n 2 ν cos ( sin 1 ( m / n k R ) ) ) ,
ν = { 1 ( TM mode ) n 2 ( TE mode ) ,
S re = 2 N π + 2 α m + π 2 ,
n k R = N π + π / 4 + α m 1 ( m n k R ) 2 ( m n k R ) cos 1 ( m n k R ) .
Γ = 1 τ e S im ,
S im = 2 R m / k 2 ( m 2 2 r 2 + V ( r ) E tot ) d r = 2 m f ( k R m ) ,
f ( z ) = 1 z 1 t 2 t d t = 1 z 2 + ln ( 1 1 z 2 z ) ,
Γ = ( n k R ) 2 2 ( n k R ) 2 m 2 exp ( S im ) .
ψ ( r ) = A ψ sc ( r ) ,
ψ ( r ) = 1 2 π N m J m ( n k R ) H m ( 1 ) ( k R ) H m ( 1 ) ( k r ) e i m θ ,
N m = 0 R J m 2 ( n k r ) r d r .
N m = R 2 2 [ J m 2 ( n k R ) + J m 2 ( n k R ) ( 1 m 2 ( n k R ) 2 ) ] .
m 1 , k R 1 .
H m ( 1 ) ( k r ) ~ i 2 m π ( 1 1 ( k r / m ) 2 ) 1 4 e m f ( k r / m )
N m ~ J m 2 ( n k R ) ( n 2 1 ) R 2 2 n 2 .
ψ ( r , θ ) ~ n π ( n 2 1 ) R ( m 2 ( k R ) 2 m 2 ( k r ) 2 ) 1 4 e i m θ + m f ( k r m ) m f ( k R m ) .
| 2 S r I 2 S r l 2 S r θ I 2 S r θ l | = | ω r k r 0 0 1 r | = ω r k r ,
ψ s c ( r , θ ) = 1 2 π ω r r k r e i m θ + m f ( k r m ) m f ( k R m ) .
A = 2 n R π m 2 ( k R ) 2 ω r ( n 2 1 ) .
i { l L , l R } = 2 m 2 k 2 d 2
Δ k = 2 π n 2 k R 2 ( n 2 1 ) m 2 ( k R ) 2 m 2 ( k d ) 2 e 2 m ( f ( k d m ) f ( k R m ) ) .
Δ E = n 2 k Δ k
Δ θ = 1 2 ln [ ( m m 2 ( k d ) 2 m + m 2 ( k d ) 2 ) ( m + m 2 k 2 m m 2 k 2 ) ]
ν A m J m ( n k R ) = B m H m ( 1 ) ( k R ) n A m J m ( n k R ) = B m H m ( 1 ) ( k R ) ,
n J m ( n k R ) ν J m ( n k R ) = H m ( 1 ) ( k R ) H m ( 1 ) ( k R ) .
H m ( 1 ) ( k R ) = m k R H m ( 1 ) ( k R ) H m + 1 ( 1 ) ( k R ) ,
n J m ( n k R ) ν J m ( n k R ) = m k R H m + 1 ( 1 ) ( k R ) H m ( 1 ) ( k R ) .
H m ( 1 ) ( m z ) ~ i Y m ( m z ) .
Y m ( m z ) ~ ( 4 ζ ( z ) 1 z 2 ) 1 4 ( Bi ( m 2 3 ζ ( z ) ) m 1 3 ) ,
( d ζ d z ) 2 = 1 z 2 ζ z 2 .
2 3 ζ 3 2 ( z ) = z 1 1 t 2 t d t = ln ( 1 + 1 z 2 z ) 1 z 2 , z ( 0 , 1 ] .
d 2 w ( z ) d z 2 + z w ( z ) = 0 .
Bi ( z ) ~ exp ( 2 3 z 3 / 2 ) π z 1 / 4 .
Y m ( m z ) ~ 2 m π ( 1 1 z 2 ) 1 4 exp ( 2 m 3 ζ 3 2 ) .

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