Abstract

A new method based on the boundary element method (BEM) is developed to calculate high quality factors of whispering-gallery modes by studying the Poynting vector. The method has high calculation accuracy, which exceeds traditional methods with the same PC computation resource. To demonstrate the feasibility of the present method, circular and quadrupole microcavities are investigated with this method and compared with the results by analytical and finite element method (FEM) results, respectively.

© 2009 Optical Society of America

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    [CrossRef]
  2. R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).
    [CrossRef]
  3. J. U. Nockel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609-1611 (1996).
    [CrossRef]
  4. S.-B. Lee, J.-H. Lee, J.-S. Chang, S.-B. Lee, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
    [CrossRef] [PubMed]
  5. S. Lacy, H. Wang, D. H. Foster, and J. U. Nockel, “Directional tunneling escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
    [CrossRef]
  6. Y.-F. Xiao, C.-H. Dong, C.-L. Zou, Z.-F. Han, L. Yang, and G.-C. Guo, “Low-threshold microlaser in a high-Q asymmetrical microcavity,” Opt. Lett. 34, 509-511 (2009).
    [CrossRef] [PubMed]
  7. J. Wiersig and M. Hentschel, “Unidirectional light emission from high-Q modes in optical microcavities,” Phys. Rev. A 73, 031802(R) (2006).
    [CrossRef]
  8. J. Wiersig and M. Hentschel, “Combining directional light output and ultralow loss in deformed microdisks,” Phys. Rev. Lett. 100, 033901 (2008).
    [CrossRef] [PubMed]
  9. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A, Pure Appl. Opt. 5, 53-60 (2003).
    [CrossRef]
  10. A. V. Boriskin, S. V. Boriskina, A. Rolland, R. Sauleau, and A. I. Nosich, “Test of the FDTD accuracy in the analysis of the scattering resonances associated with high-Q whispering-gallery modes of a circular cylinder,” J. Opt. Soc. Am. A 25, 1169-1173 (2008).
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  12. A. I. Rahachou and I. V. Zozoulenko, “Scattering matrix approach to the resonant states and Q values of microdisk lasing cavities,” Appl. Opt. 43, 1761-1772 (2004).
    [CrossRef] [PubMed]
  13. S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
    [CrossRef]
  14. J. D. Jackson, Classical Electrodynamics, second ed. (Wiley, 1975).
  15. J.-W. Ryu, S. Rim, Y. J. Park, C.-M. Kim, and S.-Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531-3536 (2008).
    [CrossRef]
  16. J. Wiersig, S. W. Kim, and M. Hentschel, “Asymmetric scattering and non-orthogonal mode patterns in optical micro-spirals,” Phys. Rev. A 78, 053809 (2008).
    [CrossRef]

2009

2008

A. V. Boriskin, S. V. Boriskina, A. Rolland, R. Sauleau, and A. I. Nosich, “Test of the FDTD accuracy in the analysis of the scattering resonances associated with high-Q whispering-gallery modes of a circular cylinder,” J. Opt. Soc. Am. A 25, 1169-1173 (2008).
[CrossRef]

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

J.-W. Ryu, S. Rim, Y. J. Park, C.-M. Kim, and S.-Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531-3536 (2008).
[CrossRef]

J. Wiersig, S. W. Kim, and M. Hentschel, “Asymmetric scattering and non-orthogonal mode patterns in optical micro-spirals,” Phys. Rev. A 78, 053809 (2008).
[CrossRef]

2006

2004

A. I. Rahachou and I. V. Zozoulenko, “Scattering matrix approach to the resonant states and Q values of microdisk lasing cavities,” Appl. Opt. 43, 1761-1772 (2004).
[CrossRef] [PubMed]

S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

2003

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A, Pure Appl. Opt. 5, 53-60 (2003).
[CrossRef]

S. Lacy, H. Wang, D. H. Foster, and J. U. Nockel, “Directional tunneling escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
[CrossRef]

2002

S.-B. Lee, J.-H. Lee, J.-S. Chang, S.-B. Lee, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

1996

An, K.

S.-B. Lee, J.-H. Lee, J.-S. Chang, S.-B. Lee, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

Boriskin, A. V.

Boriskina, S. V.

Campillo, A. J.

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).
[CrossRef]

Chang, J.-S.

S.-B. Lee, J.-H. Lee, J.-S. Chang, S.-B. Lee, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

Chang, R. K.

Chen, G.

Chen, Q.

Dong, C.-H.

Foster, D. H.

S. Lacy, H. Wang, D. H. Foster, and J. U. Nockel, “Directional tunneling escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
[CrossRef]

Grossman, H. L.

Guo, G.-C.

Guo, W.-H.

Han, Z.-F.

Hentschel, M.

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

J. Wiersig, S. W. Kim, and M. Hentschel, “Asymmetric scattering and non-orthogonal mode patterns in optical micro-spirals,” Phys. Rev. A 78, 053809 (2008).
[CrossRef]

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

Huang, Y.-Z.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, second ed. (Wiley, 1975).

Kim, C.-M.

J.-W. Ryu, S. Rim, Y. J. Park, C.-M. Kim, and S.-Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531-3536 (2008).
[CrossRef]

S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

Kim, S. W.

J. Wiersig, S. W. Kim, and M. Hentschel, “Asymmetric scattering and non-orthogonal mode patterns in optical micro-spirals,” Phys. Rev. A 78, 053809 (2008).
[CrossRef]

S.-B. Lee, J.-H. Lee, J.-S. Chang, S.-B. Lee, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

Kurdoglyan, M. S.

S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

Lacy, S.

S. Lacy, H. Wang, D. H. Foster, and J. U. Nockel, “Directional tunneling escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
[CrossRef]

Lee, J.-H.

S.-B. Lee, J.-H. Lee, J.-S. Chang, S.-B. Lee, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

Lee, S.-B.

S.-B. Lee, J.-H. Lee, J.-S. Chang, S.-B. Lee, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

S.-B. Lee, J.-H. Lee, J.-S. Chang, S.-B. Lee, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

Lee, S.-Y.

J.-W. Ryu, S. Rim, Y. J. Park, C.-M. Kim, and S.-Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531-3536 (2008).
[CrossRef]

S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

Luo, X.-S.

Moon, H.-J.

S.-B. Lee, J.-H. Lee, J.-S. Chang, S.-B. Lee, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

Nockel, J. U.

S. Lacy, H. Wang, D. H. Foster, and J. U. Nockel, “Directional tunneling escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
[CrossRef]

J. U. Nockel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21, 1609-1611 (1996).
[CrossRef]

Nosich, A. I.

Park, Y. J.

J.-W. Ryu, S. Rim, Y. J. Park, C.-M. Kim, and S.-Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531-3536 (2008).
[CrossRef]

Rahachou, A. I.

Rim, S.

J.-W. Ryu, S. Rim, Y. J. Park, C.-M. Kim, and S.-Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531-3536 (2008).
[CrossRef]

S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

Rolland, A.

Ryu, J.-W.

J.-W. Ryu, S. Rim, Y. J. Park, C.-M. Kim, and S.-Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531-3536 (2008).
[CrossRef]

Sauleau, R.

Stone, A. D.

Vahala, K. J.

K. J. Vahala, Optical Microcavities (World Scientific, 2004).
[CrossRef]

Wang, H.

S. Lacy, H. Wang, D. H. Foster, and J. U. Nockel, “Directional tunneling escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
[CrossRef]

Wang, M.-Q.

Wiersig, J.

J. Wiersig, S. W. Kim, and M. Hentschel, “Asymmetric scattering and non-orthogonal mode patterns in optical micro-spirals,” Phys. Rev. A 78, 053809 (2008).
[CrossRef]

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

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

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A, Pure Appl. Opt. 5, 53-60 (2003).
[CrossRef]

Xiao, Y.-F.

Yang, L.

Yu, L.-J.

Zou, C.-L.

Zozoulenko, I. V.

Appl. Opt.

J. Opt. A, Pure Appl. Opt.

J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A, Pure Appl. Opt. 5, 53-60 (2003).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Lett. A

J.-W. Ryu, S. Rim, Y. J. Park, C.-M. Kim, and S.-Y. Lee, “Resonances in a circular dielectric cavity,” Phys. Lett. A 372, 3531-3536 (2008).
[CrossRef]

Phys. Rev. A

J. Wiersig, S. W. Kim, and M. Hentschel, “Asymmetric scattering and non-orthogonal mode patterns in optical micro-spirals,” Phys. Rev. A 78, 053809 (2008).
[CrossRef]

S.-Y. Lee, M. S. Kurdoglyan, S. Rim, and C.-M. Kim, “Resonance patterns in a stadium-shaped microcavity,” Phys. Rev. A 70, 023809 (2004).
[CrossRef]

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

Phys. Rev. Lett.

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

S.-B. Lee, J.-H. Lee, J.-S. Chang, S.-B. Lee, H.-J. Moon, S. W. Kim, and K. An, “Observation of scarred modes in asymmetrically deformed microcylinder lasers,” Phys. Rev. Lett. 88, 033903 (2002).
[CrossRef] [PubMed]

S. Lacy, H. Wang, D. H. Foster, and J. U. Nockel, “Directional tunneling escape from nearly spherical optical resonators,” Phys. Rev. Lett. 91, 033902 (2003).
[CrossRef]

Other

K. J. Vahala, Optical Microcavities (World Scientific, 2004).
[CrossRef]

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996).
[CrossRef]

J. D. Jackson, Classical Electrodynamics, second ed. (Wiley, 1975).

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Figures (4)

Fig. 1
Fig. 1

(a) Error of eigenvalues calculated by BEM compared to the analytical solutions versus the boundary element numbers (N). (b) Q factor calculated by the formula Q = Re ( k R ) 2 Im ( k R ) . The high-Q mode T M 56 , 1 (top) and low-Q mode T M 50 , 2 (bottom) are investigated.

Fig. 2
Fig. 2

Error of field distribution overlaps the analytical solution and BEM result with different boundary element numbers (N) for the T M 56 , 1 and T M 50 , 2 modes.

Fig. 3
Fig. 3

Q factor of WGMs calculated by the numerical Poynting vector method and BEM (dots) compares to the analytical solution (solid lines). From left to right, lines correspond to TM modes with radial quantum number q = 1 , 2, and 3, respectively.

Fig. 4
Fig. 4

Q factor of WGMs in a quadrupole cavity calculated by the numerical Poynting vector method (black squares) compares to the finite element method (FEM) (gray circles). Inset: the WGM field distribution in the quadrupole cavity with ε = 0 , 0.10, and 0.18.

Equations (12)

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

[ 2 + ( n k ) 2 ] ψ = 0 ,
ψ ( r ) = Γ d s [ v G ( s , r , n k ) ψ ( s ) G ( s , r , n k ) v ψ ( s ) ] ,
ψ ( i ) = j { [ Γ j d s v G ( s , r , n k ) ] ψ ( j ) + [ Γ j d s v G ( s , r , n k ) ] v ψ ( j ) } ,
M ( k ) ( ψ v ψ ) = 0 ,
n J m 1 ( n k R ) H m ( 1 ) ( k R ) J m ( n k R ) H m 1 ( 1 ) ( k R ) = 0 ,
F = | d Ω ψ A ψ B | d Ω ψ A ψ A d Ω ψ B ψ B ,
Q = ω I ( t ) d I ( t ) d t .
w ¯ = ε | E | 2 2 ,
d I ( t ) d t = S p ¯ v d l ,
p ¯ = ω 2 π 0 2 π ω 1 ω μ Re ( E ) × Im ( × E ) d t .
p ¯ n = 1 2 1 ω μ Im ( E * v E ) ,
Q = k 2 Ω n 2 | E | 2 d Ω S Im ( E * v E ) d l .

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