Abstract

The loss of high-Q whispering-gallery modes (WGMs) with lower azimuthal mode number [m(912)] in a circular cavity have been analyzed by using a two-dimensional finite-difference time domain method (2D FDTD) method employing Cartesian gridding and staircase approximation. The FDTD simulated Q-factors of these WGMs are generally lower than those of theoretical expectations. The variations of FDTD simulated Q-factors with spatial-calculation step size indicate that the FDTD results do not simply approximate to their theoretical expectation but jump unstably under the expectation. A loss estimation method similar to volume current method (VCM) is developed to explain the FDTD results and instability. This method calculates the “incoherent” scattering field of a scattering source under influence of cavity. Theoretical results coincident with the FDTD simulation are obtained, especially for transverse magnetic modes. As based on the developed method, the energy loss is affected by only a few harmonics of boundary fluctuation that cause the FDTD loss instability.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]

2008 (1)

2007 (2)

S. L. Qiu, J. X. Cai, Y. P. Li, and X. F. Han, “Mode frequency shifts and Q-factor changes in microflower cavity and its deformed cavity,” Opt. Commun. 277, 406-410 (2007).
[CrossRef]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

2006 (3)

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

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Directional emission, increased free spectral range and mode Q-factors in 2D wavelength-scale optical microcavity structures,” IEEE J. Sel. Top. Quantum Electron. 12, 1175-1182 (2006).
[CrossRef]

J. Yang and L. J. Guo, “Optical sensors based on active microcavities,” IEEE J. Sel. Top. Quantum Electron. 12, 143-147 (2006).
[CrossRef]

2005 (2)

2004 (4)

M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758-2760 (2004).
[CrossRef] [PubMed]

S. V. Boriskina, P. Sewell, T. M. Benson, and A. I. Nosich, “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Soc. Am. A 21, 393-402 (2004).
[CrossRef]

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693-3695 (2004).
[CrossRef]

S. Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C. M. Kim, “Quasi-scarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

2003 (4)

T. Ling, L. Y. Liu, Q. H. Song, L. Xu, and W. C. Wang, “Intense directional lasing from a deformed square-shaped organic-inorganic hybrid glass microring cavity,” Opt. Lett. 28, 1784-1786 (2003).
[CrossRef] [PubMed]

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

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

W. H. Guo, Y. Z. Huang, Q. Y. Li, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
[CrossRef]

2002 (1)

M. Hentschel1 and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
[CrossRef]

2001 (5)

W.-H. Guo, W.-J. Li, and Y.-Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[CrossRef]

M. Fujita and T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253-1258 (2001).
[CrossRef]

R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,” Appl. Opt. 40, 5742-5447 (2001).
[CrossRef]

A. W. Poon, F. Courvoisier, and R. K. Chang, “Multimode resonances in square-shaped optical microcavities,” Opt. Lett. 26, 632-634 (2001).
[CrossRef]

Y. Z. Huang, W. H. Guo, and Q. M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron. 37, 100-107 (2001).
[CrossRef]

2000 (2)

1999 (2)

S. A. Backes, J. R. A. Cleaver, A. P. Heberle, J. J. Baumberg, and K. Köhler, “Threshold reduction in pierced microdisk lasers,” Appl. Phys. Lett. 74, 176-178 (1999).
[CrossRef]

M. L. M. Balistreri, D. J. W. Klunder, F. C. Blom, A. Driessen, H. W. J. M. Hoekstra, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Visualizing the whispering gallery modes in a cylindrical optical microcavity,” Opt. Lett. 24, 1929-1831 (1999).
[CrossRef]

1997 (1)

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

1996 (3)

1994 (1)

M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302-3307 (1994).
[CrossRef]

1993 (2)

R. E. Slusher, A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

G. L. Hower, R. G. Olsen, J. D. Earls, and J. B. Schneider, “Inaccuracies in numerical calculation of scattering near natural frequencies of penetrable object,” IEEE Trans. Antennas Propag. 41, 982-986 (1993).
[CrossRef]

1992 (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289-291 (1992).
[CrossRef]

1990 (1)

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

1983 (1)

M. Kuznetsov and H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. 19, 1505-1514 (1983).
[CrossRef]

Agio, M.

Baba, T.

M. Fujita and T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253-1258 (2001).
[CrossRef]

Backes, S. A.

S. A. Backes, J. R. A. Cleaver, A. P. Heberle, J. J. Baumberg, and K. Köhler, “Threshold reduction in pierced microdisk lasers,” Appl. Phys. Lett. 74, 176-178 (1999).
[CrossRef]

Balistreri, M. L. M.

M. L. M. Balistreri, D. J. W. Klunder, F. C. Blom, A. Driessen, H. W. J. M. Hoekstra, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Visualizing the whispering gallery modes in a cylindrical optical microcavity,” Opt. Lett. 24, 1929-1831 (1999).
[CrossRef]

Barber, P. W.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Barclay, P. E.

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693-3695 (2004).
[CrossRef]

Baumberg, J. J.

S. A. Backes, J. R. A. Cleaver, A. P. Heberle, J. J. Baumberg, and K. Köhler, “Threshold reduction in pierced microdisk lasers,” Appl. Phys. Lett. 74, 176-178 (1999).
[CrossRef]

Benson, T. M.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Directional emission, increased free spectral range and mode Q-factors in 2D wavelength-scale optical microcavity structures,” IEEE J. Sel. Top. Quantum Electron. 12, 1175-1182 (2006).
[CrossRef]

S. V. Boriskina, P. Sewell, T. M. Benson, and A. I. Nosich, “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Soc. Am. A 21, 393-402 (2004).
[CrossRef]

Blom, F. C.

M. L. M. Balistreri, D. J. W. Klunder, F. C. Blom, A. Driessen, H. W. J. M. Hoekstra, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Visualizing the whispering gallery modes in a cylindrical optical microcavity,” Opt. Lett. 24, 1929-1831 (1999).
[CrossRef]

Boriskin, A. V.

Boriskina, S. V.

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]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Directional emission, increased free spectral range and mode Q-factors in 2D wavelength-scale optical microcavity structures,” IEEE J. Sel. Top. Quantum Electron. 12, 1175-1182 (2006).
[CrossRef]

S. V. Boriskina, P. Sewell, T. M. Benson, and A. I. Nosich, “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Soc. Am. A 21, 393-402 (2004).
[CrossRef]

Borselli, M.

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693-3695 (2004).
[CrossRef]

Boyd, R. W.

Cai, J. X.

S. L. Qiu, J. X. Cai, Y. P. Li, and X. F. Han, “Mode frequency shifts and Q-factor changes in microflower cavity and its deformed cavity,” Opt. Commun. 277, 406-410 (2007).
[CrossRef]

Cai, M.

Chang, R. K.

Chen, G.

Chin, M. K.

M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302-3307 (1994).
[CrossRef]

Choi, M.

S. Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C. M. Kim, “Quasi-scarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Chu, D. Y.

M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302-3307 (1994).
[CrossRef]

Chu, S. T.

Cleaver, J. R. A.

S. A. Backes, J. R. A. Cleaver, A. P. Heberle, J. J. Baumberg, and K. Köhler, “Threshold reduction in pierced microdisk lasers,” Appl. Phys. Lett. 74, 176-178 (1999).
[CrossRef]

Courvoisier, F.

Driessen, A.

M. L. M. Balistreri, D. J. W. Klunder, F. C. Blom, A. Driessen, H. W. J. M. Hoekstra, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Visualizing the whispering gallery modes in a cylindrical optical microcavity,” Opt. Lett. 24, 1929-1831 (1999).
[CrossRef]

Earls, J. D.

G. L. Hower, R. G. Olsen, J. D. Earls, and J. B. Schneider, “Inaccuracies in numerical calculation of scattering near natural frequencies of penetrable object,” IEEE Trans. Antennas Propag. 41, 982-986 (1993).
[CrossRef]

Fujita, M.

M. Fujita and T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253-1258 (2001).
[CrossRef]

Gorodetsky, M. L.

Grossman, H. L.

Grundmann, M.

T. Nobis and M. Grundmann, “Low-order optical whispering-gallery modes in hexagonal nanocavities,” Phys. Rev. A 72, 063806 (2005).
[CrossRef]

Guo, L. J.

J. Yang and L. J. Guo, “Optical sensors based on active microcavities,” IEEE J. Sel. Top. Quantum Electron. 12, 143-147 (2006).
[CrossRef]

Guo, W. H.

W. H. Guo, Y. Z. Huang, Q. Y. Li, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
[CrossRef]

Y. Z. Huang, W. H. Guo, and Q. M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron. 37, 100-107 (2001).
[CrossRef]

Guo, W.-H.

W.-H. Guo, W.-J. Li, and Y.-Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Han, X. F.

S. L. Qiu, J. X. Cai, Y. P. Li, and X. F. Han, “Mode frequency shifts and Q-factor changes in microflower cavity and its deformed cavity,” Opt. Commun. 277, 406-410 (2007).
[CrossRef]

Haus, H. A.

M. Kuznetsov and H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. 19, 1505-1514 (1983).
[CrossRef]

Heberle, A. P.

S. A. Backes, J. R. A. Cleaver, A. P. Heberle, J. J. Baumberg, and K. Köhler, “Threshold reduction in pierced microdisk lasers,” Appl. Phys. Lett. 74, 176-178 (1999).
[CrossRef]

Heebner, J. E.

Hentschel, M.

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

Hentschel1, M.

M. Hentschel1 and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
[CrossRef]

Hill, S. C.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Ho, S. T.

M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302-3307 (1994).
[CrossRef]

Hoekstra, H. W. J. M.

M. L. M. Balistreri, D. J. W. Klunder, F. C. Blom, A. Driessen, H. W. J. M. Hoekstra, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Visualizing the whispering gallery modes in a cylindrical optical microcavity,” Opt. Lett. 24, 1929-1831 (1999).
[CrossRef]

Hower, G. L.

G. L. Hower, R. G. Olsen, J. D. Earls, and J. B. Schneider, “Inaccuracies in numerical calculation of scattering near natural frequencies of penetrable object,” IEEE Trans. Antennas Propag. 41, 982-986 (1993).
[CrossRef]

Huang, Y. Z.

W. H. Guo, Y. Z. Huang, Q. Y. Li, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
[CrossRef]

Y. Z. Huang, W. H. Guo, and Q. M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron. 37, 100-107 (2001).
[CrossRef]

Huang, Y.-Z.

W.-H. Guo, W.-J. Li, and Y.-Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[CrossRef]

Ilchenko, V. S.

Kim, C. M.

S. Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C. M. Kim, “Quasi-scarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758-2760 (2004).
[CrossRef] [PubMed]

Klunder, D. J. W.

M. L. M. Balistreri, D. J. W. Klunder, F. C. Blom, A. Driessen, H. W. J. M. Hoekstra, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Visualizing the whispering gallery modes in a cylindrical optical microcavity,” Opt. Lett. 24, 1929-1831 (1999).
[CrossRef]

Köhler, K.

S. A. Backes, J. R. A. Cleaver, A. P. Heberle, J. J. Baumberg, and K. Köhler, “Threshold reduction in pierced microdisk lasers,” Appl. Phys. Lett. 74, 176-178 (1999).
[CrossRef]

Korterik, J. P.

M. L. M. Balistreri, D. J. W. Klunder, F. C. Blom, A. Driessen, H. W. J. M. Hoekstra, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Visualizing the whispering gallery modes in a cylindrical optical microcavity,” Opt. Lett. 24, 1929-1831 (1999).
[CrossRef]

Kuipers, L.

M. L. M. Balistreri, D. J. W. Klunder, F. C. Blom, A. Driessen, H. W. J. M. Hoekstra, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Visualizing the whispering gallery modes in a cylindrical optical microcavity,” Opt. Lett. 24, 1929-1831 (1999).
[CrossRef]

Kurdoglyan, M. S.

Kuznetsov, M.

M. Kuznetsov and H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. 19, 1505-1514 (1983).
[CrossRef]

Kwon, T. Y.

S. Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C. M. Kim, “Quasi-scarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Lai, H. M.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Lee, S. Y.

M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758-2760 (2004).
[CrossRef] [PubMed]

S. Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C. M. Kim, “Quasi-scarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Leung, P. T.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Levi, A. F. J.

R. E. Slusher, A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289-291 (1992).
[CrossRef]

Li, Q. Y.

W. H. Guo, Y. Z. Huang, Q. Y. Li, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
[CrossRef]

Li, W.-J.

W.-H. Guo, W.-J. Li, and Y.-Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[CrossRef]

Li, Y. P.

S. L. Qiu, J. X. Cai, Y. P. Li, and X. F. Han, “Mode frequency shifts and Q-factor changes in microflower cavity and its deformed cavity,” Opt. Commun. 277, 406-410 (2007).
[CrossRef]

Ling, T.

Little, B. E.

Liu, L. Y.

Logan, R. A.

R. E. Slusher, A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289-291 (1992).
[CrossRef]

McCall, S. L.

R. E. Slusher, A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289-291 (1992).
[CrossRef]

Mohammadi, A.

Nadgaran, H.

Nobis, T.

T. Nobis and M. Grundmann, “Low-order optical whispering-gallery modes in hexagonal nanocavities,” Phys. Rev. A 72, 063806 (2005).
[CrossRef]

Nockel, J. U.

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

Nöckel, J. U.

Nosich, A. I.

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]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Directional emission, increased free spectral range and mode Q-factors in 2D wavelength-scale optical microcavity structures,” IEEE J. Sel. Top. Quantum Electron. 12, 1175-1182 (2006).
[CrossRef]

S. V. Boriskina, P. Sewell, T. M. Benson, and A. I. Nosich, “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Soc. Am. A 21, 393-402 (2004).
[CrossRef]

Olsen, R. G.

G. L. Hower, R. G. Olsen, J. D. Earls, and J. B. Schneider, “Inaccuracies in numerical calculation of scattering near natural frequencies of penetrable object,” IEEE Trans. Antennas Propag. 41, 982-986 (1993).
[CrossRef]

Painter, O.

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693-3695 (2004).
[CrossRef]

M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett. 25, 1430-1432 (2000).
[CrossRef]

Pearton, S. J.

R. E. Slusher, A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289-291 (1992).
[CrossRef]

Poon, A. W.

Pryamikov, A. D.

Qiu, S. L.

S. L. Qiu, J. X. Cai, Y. P. Li, and X. F. Han, “Mode frequency shifts and Q-factor changes in microflower cavity and its deformed cavity,” Opt. Commun. 277, 406-410 (2007).
[CrossRef]

Richter, K.

M. Hentschel1 and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
[CrossRef]

Rim, S.

S. Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C. M. Kim, “Quasi-scarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

M. S. Kurdoglyan, S. Y. Lee, S. Rim, and C. M. Kim, “Unidirectional lasing from a microcavity with a rounded isosceles triangle shape,” Opt. Lett. 29, 2758-2760 (2004).
[CrossRef] [PubMed]

Rolland, A.

Ryu, J. W.

S. Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C. M. Kim, “Quasi-scarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Sauleau, R.

Savchenkov, A. A.

Schneider, J. B.

G. L. Hower, R. G. Olsen, J. D. Earls, and J. B. Schneider, “Inaccuracies in numerical calculation of scattering near natural frequencies of penetrable object,” IEEE Trans. Antennas Propag. 41, 982-986 (1993).
[CrossRef]

Sercel, P. C.

Sewell, P.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Directional emission, increased free spectral range and mode Q-factors in 2D wavelength-scale optical microcavity structures,” IEEE J. Sel. Top. Quantum Electron. 12, 1175-1182 (2006).
[CrossRef]

S. V. Boriskina, P. Sewell, T. M. Benson, and A. I. Nosich, “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Soc. Am. A 21, 393-402 (2004).
[CrossRef]

Slusher, R. E.

R. E. Slusher, A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289-291 (1992).
[CrossRef]

Smotrova, E. I.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

Song, Q. H.

Srinivasan, K.

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693-3695 (2004).
[CrossRef]

Stone, A. D.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Vahala, K. J.

van Hulst, N. F.

M. L. M. Balistreri, D. J. W. Klunder, F. C. Blom, A. Driessen, H. W. J. M. Hoekstra, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Visualizing the whispering gallery modes in a cylindrical optical microcavity,” Opt. Lett. 24, 1929-1831 (1999).
[CrossRef]

Wang, Q. M.

Y. Z. Huang, W. H. Guo, and Q. M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron. 37, 100-107 (2001).
[CrossRef]

Wang, W. C.

Wiersig, J.

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

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

Xu, L.

Yang, J.

J. Yang and L. J. Guo, “Optical sensors based on active microcavities,” IEEE J. Sel. Top. Quantum Electron. 12, 143-147 (2006).
[CrossRef]

Young, K.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

Yu, L. J.

W. H. Guo, Y. Z. Huang, Q. Y. Li, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (4)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode microdisk lasers,” Appl. Phys. Lett. 60, 289-291 (1992).
[CrossRef]

R. E. Slusher, A. F. J. Levi, S. L. McCall, S. J. Pearton, and R. A. Logan, “Threshold characteristics of semiconductor microdisk lasers,” Appl. Phys. Lett. 63, 1310-1312 (1993).
[CrossRef]

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693-3695 (2004).
[CrossRef]

S. A. Backes, J. R. A. Cleaver, A. P. Heberle, J. J. Baumberg, and K. Köhler, “Threshold reduction in pierced microdisk lasers,” Appl. Phys. Lett. 74, 176-178 (1999).
[CrossRef]

IEEE J. Quantum Electron. (4)

M. Fujita and T. Baba, “Proposal and finite-difference time-domain simulation of whispering gallery mode microgear cavity,” IEEE J. Quantum Electron. 37, 1253-1258 (2001).
[CrossRef]

M. Kuznetsov and H. A. Haus, “Radiation loss in dielectric waveguide structures by the volume current method,” IEEE J. Quantum Electron. 19, 1505-1514 (1983).
[CrossRef]

W. H. Guo, Y. Z. Huang, Q. Y. Li, and L. J. Yu, “Modes in square resonators,” IEEE J. Quantum Electron. 39, 1563-1566 (2003).
[CrossRef]

Y. Z. Huang, W. H. Guo, and Q. M. Wang, “Analysis and numerical simulation of eigenmode characteristics for semiconductor lasers with an equilateral triangle micro-resonator,” IEEE J. Quantum Electron. 37, 100-107 (2001).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Directional emission, increased free spectral range and mode Q-factors in 2D wavelength-scale optical microcavity structures,” IEEE J. Sel. Top. Quantum Electron. 12, 1175-1182 (2006).
[CrossRef]

J. Yang and L. J. Guo, “Optical sensors based on active microcavities,” IEEE J. Sel. Top. Quantum Electron. 12, 143-147 (2006).
[CrossRef]

IEEE Microw. Wirel. Compon. Lett. (1)

W.-H. Guo, W.-J. Li, and Y.-Z. Huang, “Computation of resonant frequencies and quality factors of cavities by FDTD technique and Padé approximation,” IEEE Microw. Wirel. Compon. Lett. 11, 223-225 (2001).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

G. L. Hower, R. G. Olsen, J. D. Earls, and J. B. Schneider, “Inaccuracies in numerical calculation of scattering near natural frequencies of penetrable object,” IEEE Trans. Antennas Propag. 41, 982-986 (1993).
[CrossRef]

J. Appl. Phys. (1)

M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of the spontaneous emission factor for microdisk lasers via the approximation of whispering gallery modes,” J. Appl. Phys. 75, 3302-3307 (1994).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

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 (2)

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

Nature (2)

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

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

Opt. Commun. (1)

S. L. Qiu, J. X. Cai, Y. P. Li, and X. F. Han, “Mode frequency shifts and Q-factor changes in microflower cavity and its deformed cavity,” Opt. Commun. 277, 406-410 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (8)

Opt. Quantum Electron. (1)

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, “Trends in microdisk laser research and linear optical modelling,” Opt. Quantum Electron. 39, 1253-1272 (2007).
[CrossRef]

Phys. Rev. A (3)

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187-5198 (1990).
[CrossRef] [PubMed]

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

T. Nobis and M. Grundmann, “Low-order optical whispering-gallery modes in hexagonal nanocavities,” Phys. Rev. A 72, 063806 (2005).
[CrossRef]

Phys. Rev. E (1)

M. Hentschel1 and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

S. Y. Lee, S. Rim, J. W. Ryu, T. Y. Kwon, M. Choi, and C. M. Kim, “Quasi-scarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Other (1)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Cited By

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Figures (7)

Fig. 1
Fig. 1

Cartesian Yee cells and 2D FDTD Cartesian discretization. (a) and (b) show two possible types of Cartesian Yee cells for 2D FDTD discretization. (c) Stairlike interface simulation of bended interface for 2D FDTD. (d) Our FDTD staircased CRC.

Fig. 2
Fig. 2

(a) Spectral response of CRC with radius 1 μ m and cavity refractive index of 3.2. (solid curve for TM modes and dashed-dotted curve for TE modes); Peaks are marked by WG ( m , n ) (m is the azimuthal mode number and n is the radial mode number, WGH, TM modes; WGE, TE mode); the inset shows split WGH ( 10 , 1 ) ; The split exists in other WGMs with an even azimuthal mode number. (b) Simulated Q-factors and resonant frequencies of WGE ( 9 , 1 ) versus spatial step size of square cell of our FDTD simulation.

Fig. 3
Fig. 3

Sketch for energy-loss estimation for a general 2D microcavity. Shadow region denotes existent region of refractive index perturbation (RIP). Also the existent region of “polarized” current source and the boundary perturbation (roughness) is turgidly shown.

Fig. 4
Fig. 4

Sketch for energy-loss estimation for a 2D CRC. D 1 denotes the region that lies between the inner boundary of the perturbed cavity and the outer boundary of the CRC. D 2 contrarily, Δ ρ 0 + ( φ ) , denotes the outer boundary of shadow region ( D ) and Δ ρ 0 ( φ ) inner boundary of D .

Fig. 5
Fig. 5

The z-component field distribution of WG ( 9 , 1 ) versus radial distance. For TE-polarized WGE ( 9 , 1 ) , the field comes very close to zero and varies rapidly with radial distance at the cavity boundary.

Fig. 6
Fig. 6

Boundary fluctuation of our FDTD discretization of a 2D CRC. The fluctuation is not nonrandom. Instead, it has cyclicity with period of π 2 radian.

Fig. 7
Fig. 7

Sketch for radiated power calculation, where Γ represents an infinite, far circular boundary. By integrating the radial energy flow density at the infinite, far circular boundary Γ radiated power can be obtained.

Tables (3)

Tables Icon

Table 1 Comparison between Theoretical Results and FDTD Simulation Results for WGH ( 9 , 1 ) a

Tables Icon

Table 2 Comparison between Theoretical Results and FDTD Simulation Results for WGH ( 12 , 1 ) a

Tables Icon

Table 3 Comparison between Theoretical Results and FDTD Simulation Results for WGE ( 10 , 1 ) a

Equations (50)

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

E x t = 1 ϵ H z y , E y t = 1 ϵ H z x ,
H z t = 1 μ 0 ( E y x E x y ) ,
H x t = 1 μ 0 E z y , H y t = 1 μ 0 E z x ,
E z t = 1 ϵ ( H y x H x y ) ,
F x t = 1 μ e F z y , F y t = 1 μ e F z x ,
F z t = 1 ϵ e ( F y x F x y ) ,
F x n + 1 2 ( i , j 1 2 ) = F x n 1 2 ( i , j 1 2 ) + Δ t μ e ( i , j 1 2 ) [ F z n ( i , j ) F z n ( i , j 1 ) δ y ] ,
F y n + 1 2 ( i 1 2 , j ) = F y n 1 2 ( i 1 2 , j ) + Δ t μ e ( i 1 2 , j ) [ F z n ( i , j ) F z n ( i 1 , j ) δ x ] ,
F z n + 1 ( i , j ) = F z n ( i , j ) + Δ t ϵ e ( i , j ) [ F y n + 1 2 ( i + 1 2 , j ) F y n + 1 2 ( i 1 2 , j ) δ x F x n + 1 2 ( i , j + 1 2 ) F x n + 1 2 ( i , j 1 2 ) δ y ] ,
{ 2 ψ + n 2 ( ρ ) k 2 ψ = 0 | ψ 1 | s = | ψ 2 | s | 1 α 1 ψ 1 n | s = | 1 α 2 ψ 2 n | s n ( ρ ) = { n 1 inner n 2 outer } , }
2 ψ + n 2 2 k 2 ψ = ( n 2 ( ρ ) n 2 2 ) k 2 ψ = Δ n 2 k 2 ψ .
2 ψ 1 + n 2 ( ρ ) k 2 ψ 1 = δ n 2 k 2 ψ 0 n 0 2 ( ρ ) ( k 2 k 0 2 ) ψ 0 ,
ψ 1 = k 0 2 D d D δ n 2 ψ 0 ( ρ ) G ( ρ , ρ ) ,
{ 2 G + n 2 ( ρ ) k 0 2 G = δ ( ρ ρ ) | G 1 | s = | G 2 | s | 1 α 1 G 1 n | s = | 1 α 2 G 2 n | s n ( ρ ) = { n 1 inner n 2 outer } , }
P rad = 1 2 η Γ d Γ | ψ | 2 = 1 2 η Γ d Γ | ψ 0 + ψ 1 | 2 ,
P rad = 1 2 η Γ d Γ [ | ψ 0 | 2 + | ψ 1 | 2 ] = P 0 + P sca .
Q 1 = P rad ω W = Q 0 1 + Q sca 1 ,
ψ ρ f ( φ ) exp ( j n 2 k ρ ) n 2 k ρ = f ( φ ) exp [ j n 2 Re ( k ) ρ ] n 2 k ρ × exp [ n 2 Im ( k ) ρ ] ,
ψ 1 = k 0 2 Δ n 2 { D 1 d D + D 2 d D } ψ 0 ( ρ ) G ( ρ , ρ ) .
ψ 1 = k 0 2 Δ n 2 0 2 π d φ { 0 R + Δ ρ 0 + ( φ ) + R R + Δ ρ 0 ( φ ) } d ρ ψ 0 ( ρ , φ ) G ( ρ ; ρ , φ ) .
{ ψ = f m ( ρ ) Φ m ( φ ) , f m ( ρ ) = c { J m ( n 1 k ρ ) , ρ < R c m H m ( 1 ) ( n 2 k ρ ) , ρ > R , } Φ m ( φ ) = { cos m φ sin m φ } or exp ( j m φ ) ( m = 0 , ± 1 , ) , }
| J m ( n 1 k R ) H m ( 1 ) ( n 2 k R ) n 1 α 1 J m ( n 1 k R ) n 2 α 2 H m ( 1 ) ( n 2 k R ) | = 0 ,
ψ 0 = f m ( ρ ) cos ( m φ + φ 0 ) ,
G 0 ( ρ , ρ ) = j 4 n A n ( ρ ) H n ( 1 ) ( n 2 k ρ ) exp [ j n ( φ φ ) ] ,
G in 0 ( | ρ , ρ | s ) = G out 0 ( | ρ , ρ | s ) α ,
G | ( ρ , ρ ) | ρ R , ρ D 2 G 0 in | ( ρ , ρ ) | ρ = R = G 0 out | ( ρ , ρ ) | ρ = R α .
G | ( ρ , ρ ) | ρ R , ρ D 1 G 0 out | ( ρ , ρ ) | ρ = R .
ψ 1 k 0 2 Δ n 2 0 2 π d φ Φ m ( φ ) { G 0 in ( ρ ; R , φ ) R R + Δ ρ 0 + ( φ ) + G 0 out ( ρ ; R , φ ) R R + Δ ρ 0 ( φ ) } d ρ f m ( ρ ) ,
f m β ( ρ ) f m ( R ) + f β m ( R ) ( ρ R ) ,
R R + Δ ρ 0 ( φ ) d ρ ρ f m ( ρ ) R [ f m ( R ) Δ ρ 0 ( φ ) + f m ( R ) Δ ρ 0 2 ( φ ) ] ,
ψ 1 = k 0 2 Δ n 2 π R n C n ( φ ) A n out f m ( R ) ξ n × exp ( j n 2 k 0 ρ ) n 2 k 0 ρ ,
ξ n = Δ ρ + ( n ) α + Δ ρ ( n ) + f out m ( R ) f m ( R ) Δ ρ + 2 ( n ) α + f in m ( R ) f m ( R ) Δ ρ 2 ( n ) ,
Δ ρ β ( n ) = 1 π 0 2 π d φ Δ ρ β ( φ ) Φ m ( φ ) exp ( j n φ ) = exp ( j φ 0 ) Δ ρ β m + n + exp ( j φ 0 ) Δ ρ β m + n ,
P sca = 1 2 η k 0 3 ( Δ n 2 ) 2 π 2 R 2 | f m ( R ) | 2 × n m l m | A n out ξ n | 2 0 2 π d φ C n ( φ ) C * l ( φ ) = η k 0 3 ( Δ n 2 ) 2 π 2 R 2 | f m ( R ) | 2 8 n 2 n m | A n out ξ n | 2 .
Q sca = 1 α 1 α 2 × 4 k 0 4 ( Δ n 2 ) 2 π R 2 × 1 n m | A n out ξ n | 2 × 1 f ¯ ,
f ¯ = | J m ( n 1 k 0 R ) | 2 ( n 1 k 0 ) 2 0 R d ρ ρ | J m ( n 1 k 0 ρ ) | 2 .
Δ ρ β n = { = Δ ρ β n n = 4 l , ( l = 0 , ± 1 , ) = 0 other } ,
F x = 1 j ω μ e F z y , F y = 1 j ω μ e F z x ,
S = 1 2 Re ( E * × H ) = 1 2 Re { 1 j ω μ e ( F z * F z y e y + F z * F z x e x ) } ,
F z f ( φ ) exp ( j n 2 k ρ ) n 2 k ρ , and F z ρ j n 2 k F z ,
P rad = Γ S d G = 1 2 Γ Re { 1 j ω μ e ( F z F z n ) } d Γ = n 2 2 C μ e Γ | F z | 2 d Γ = 1 2 η Γ | F z | 2 d Γ .
P rad = 1 2 η Γ | F z | 2 d Γ = 1 2 k 2 η 0 2 π d φ | f ( φ ) | 2 .
{ 2 G + n 2 ( ρ ) k 0 2 G = δ ( ρ ρ ) | G 1 | s = | G 2 | s | 1 α 1 G 1 n | s = | 1 α 2 G 2 n | s n ( ρ ) = { n 1 inner n 2 outer } , }
G ( ρ , ρ ) = j 4 H 0 ( 1 ) ( k 2 | ρ ρ | ) = 1 4 j { n H n ( 1 ) ( k 2 ρ ) J n ( k 2 ρ ) exp [ j n ( φ φ ) ] ( ρ < ρ ) n H n ( 1 ) ( k 2 ρ ) J n ( k 2 ρ ) exp [ j n ( φ φ ) ] ( ρ > ρ ) } ,
G 0 ( ρ , ρ ) = j 4 n A n ( ρ ) H n ( 1 ) ( n 2 k ρ ) exp ( j n ( φ φ ) ) ,
A n ( ρ ) = { J n ( n 2 k ρ ) + b n H n ( 1 ) ( n 2 k ρ ) A n out , ρ > R b n J n ( n 1 k ρ ) A n in , ρ < R } ,
b n = β 1 J n ( n 1 k R ) J n ( n 2 k R ) β 2 J n ( n 2 k R ) J n ( n 2 k R ) β 2 J n ( n 1 k R ) H n ( 1 ) ( n 2 k R ) β 1 J n ( n 2 k R ) H n ( 1 ) ( n 2 k R ) ,
b n = β 1 [ J n ( n 1 k R ) H n ( 1 ) ( n 2 k R ) J n ( n 2 k R ) H n ( 1 ) ( n 2 k R ) ] β 1 J n ( n 2 k R ) H n ( 1 ) ( n 2 k R ) β 2 J n ( n 1 k R ) H n ( 1 ) ( n 2 k R ) ,
G 0 ( ρ , ρ ) exp ( j n 2 k ρ ) n 2 k ρ n C n ( φ ) A n ( ρ ) exp ( j n φ ) ,
G in 0 ( ( | ρ , ρ | s 0 ) ) = G out 0 ( | ρ , ρ | s 0 ) α ,

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