Abstract

By taking advantage of axial symmetry of the planar whispering gallery microresonators, the three-dimensional (3D) problem of the resonator is reduced to a two-dimensional (2D) one; thus, only the cross section of the resonator needs to be analyzed. Then, the proposed formulation, which works based on a combination of the finite-elements method (FEM) and Fourier expansion of the fields, can be applied to the 2D problem. First, the axial field variation is expressed in terms of a Fourier series. Then, a FEM method is applied to the radial field variation. This formulation yields an eigenvalue problem with sparse matrices and can be solved using a well-known numerical technique. This method takes into account both the radiation loss and the dielectric loss; hence, it works efficiently either for high number or low number modes. Efficiency of the method was investigated by comparison of the results with those of commercial software.

© 2014 Optical Society of America

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References

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  1. E. Rivera-Perez, A. Diez, M. V. Andres, J. L. Cruz, and A. Rodriguez-Cobos, “Optical fiber whispering gallery modes resonances: applications,” in 15th International Conference on Transparent Optical Networks, Cartagena, Columbia (National Institute of Telecommunications, 2013), pp. 1–4.
  2. V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes—part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006).
    [CrossRef]
  3. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
    [CrossRef]
  4. S. C. Greedy, S. V. Boriskina, P. Sewell, and T. M. Benson, “Design and simulation pools for optical microresonators,” Proc. SPIE 4277, 21–32 (2001).
  5. T. M. Benson, A. Vukovic, J. G. Wykes, A. Al-Jarro, and P. Sewell, “Numerical simulations of 3D micro-resonators,” in 10th Anniversary International Conference on Transparent Optical Networks, Athens, Greece (National Institute of Telecommunications, 2008), pp. 54–57.
  6. A. Boriskin, S. Boriskina, A. Rolland, R. Sauleau, and A. 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]
  7. J. Krupka, D. Cros, M. Aubourg, and P. Guillon, “Study whispering gallery modes in anisotropic single-crystal dielectric resonators,” IEEE Trans. Microwave Theor. Tech. 42, 56–61 (1994).
    [CrossRef]
  8. A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525 (1991).
    [CrossRef]
  9. Y. Hao and C. J. Railton, “Analyzing electromagnetic structures with curved boundaries on Cartesian FDTD meshes,” IEEE Trans. Microwave Theor. Tech. 46, 82–88 (1998).
    [CrossRef]
  10. A. A. Kishk, M. R. Zunoubi, and D. Kajfez, “A numerical study of a dielectric disk antenna above grounded dielectric substrate,” IEEE Trans. Antennas Propag. 41, 813–821 (1993).
    [CrossRef]
  11. G. W. Hanson, “An efficient full-wave method for analysis of dielectric resonators possessing separable geometries immersed in inhomogeneous environments,” IEEE Trans. Microwave Theor. Tech. 48, 84–92 (2000).
    [CrossRef]
  12. A. J. Burton and G. F. Miller, “The application of integral equation methods to the numerical solution of some exterior boundary-value problems,” Proc. R. Soc. London Ser. A 323, 201–210 (1971).
    [CrossRef]
  13. S. V. Boriskina and A. I. Nosich, “Radiation, and absorption losses of the whispering-gallery-mode dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theor. Tech. 47, 224–231 (1999).
    [CrossRef]
  14. S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Tuning of elliptic whispering-gallery-mode microdisk waveguide filters,” J. Lightwave Technol. 21, 1987–1995 (2003).
    [CrossRef]
  15. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A Pure Appl. Opt. 5, 53–60 (2003).
    [CrossRef]
  16. D. Colton and R. Kress, Integral Equations Methods in Scattering Theory (Wiley, 1983).
  17. N. Morita, N. Kumagai, and J. R. Mautz, Integral Equations Methods for Electromagnetics (Artech House, 1990).
  18. K. Jiang and W. Huang, “Finite-difference-based mode-matching method for 3D waveguide structures under semivectorial approximation,” J. Lightwave Technol. 23, 4239–4248 (2005).
    [CrossRef]
  19. L. A. Francavilla, J. S. McLean, H. D. Foltz, and G. E. Crook, “Mode-matching analysis of top-hat monopole antennas loaded with radially layered dielectric,” IEEE Trans. Antennas Propag. 47, 179–185 (1999).
    [CrossRef]
  20. C. G. Wells and J. A. R. Ball, “Mode-matching analysis of a shielded rectangular dielectric-rod waveguide,” IEEE Trans. Microwave Theor. Tech. 53, 3169–3177 (2005).
    [CrossRef]
  21. S. W. Chen and K. A. Zaki, “Ring resonators loaded in waveguide and on substrate,” IEEE Trans. Microwave Theor. Tech. 39, 2069–2076 (1991).
    [CrossRef]
  22. L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217–223 (2005).
    [CrossRef]
  23. M. Mahmoud-Taheri, “Analysis of cylindrical cavity resonators loaded axisymmetrically with dielectric for multimode microwave filter design,” Ph.D. dissertation (University of Essex, 1989).
  24. D. M. Pozar, Microwave Engineering (Wiley, 1998).

2008

2006

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes—part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006).
[CrossRef]

2005

K. Jiang and W. Huang, “Finite-difference-based mode-matching method for 3D waveguide structures under semivectorial approximation,” J. Lightwave Technol. 23, 4239–4248 (2005).
[CrossRef]

C. G. Wells and J. A. R. Ball, “Mode-matching analysis of a shielded rectangular dielectric-rod waveguide,” IEEE Trans. Microwave Theor. Tech. 53, 3169–3177 (2005).
[CrossRef]

L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217–223 (2005).
[CrossRef]

2003

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Tuning of elliptic whispering-gallery-mode microdisk waveguide filters,” J. Lightwave Technol. 21, 1987–1995 (2003).
[CrossRef]

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

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

2001

S. C. Greedy, S. V. Boriskina, P. Sewell, and T. M. Benson, “Design and simulation pools for optical microresonators,” Proc. SPIE 4277, 21–32 (2001).

2000

G. W. Hanson, “An efficient full-wave method for analysis of dielectric resonators possessing separable geometries immersed in inhomogeneous environments,” IEEE Trans. Microwave Theor. Tech. 48, 84–92 (2000).
[CrossRef]

1999

L. A. Francavilla, J. S. McLean, H. D. Foltz, and G. E. Crook, “Mode-matching analysis of top-hat monopole antennas loaded with radially layered dielectric,” IEEE Trans. Antennas Propag. 47, 179–185 (1999).
[CrossRef]

S. V. Boriskina and A. I. Nosich, “Radiation, and absorption losses of the whispering-gallery-mode dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theor. Tech. 47, 224–231 (1999).
[CrossRef]

1998

Y. Hao and C. J. Railton, “Analyzing electromagnetic structures with curved boundaries on Cartesian FDTD meshes,” IEEE Trans. Microwave Theor. Tech. 46, 82–88 (1998).
[CrossRef]

1994

J. Krupka, D. Cros, M. Aubourg, and P. Guillon, “Study whispering gallery modes in anisotropic single-crystal dielectric resonators,” IEEE Trans. Microwave Theor. Tech. 42, 56–61 (1994).
[CrossRef]

1993

A. A. Kishk, M. R. Zunoubi, and D. Kajfez, “A numerical study of a dielectric disk antenna above grounded dielectric substrate,” IEEE Trans. Antennas Propag. 41, 813–821 (1993).
[CrossRef]

1991

A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525 (1991).
[CrossRef]

S. W. Chen and K. A. Zaki, “Ring resonators loaded in waveguide and on substrate,” IEEE Trans. Microwave Theor. Tech. 39, 2069–2076 (1991).
[CrossRef]

1971

A. J. Burton and G. F. Miller, “The application of integral equation methods to the numerical solution of some exterior boundary-value problems,” Proc. R. Soc. London Ser. A 323, 201–210 (1971).
[CrossRef]

Al-Jarro, A.

T. M. Benson, A. Vukovic, J. G. Wykes, A. Al-Jarro, and P. Sewell, “Numerical simulations of 3D micro-resonators,” in 10th Anniversary International Conference on Transparent Optical Networks, Athens, Greece (National Institute of Telecommunications, 2008), pp. 54–57.

Andres, M. V.

E. Rivera-Perez, A. Diez, M. V. Andres, J. L. Cruz, and A. Rodriguez-Cobos, “Optical fiber whispering gallery modes resonances: applications,” in 15th International Conference on Transparent Optical Networks, Cartagena, Columbia (National Institute of Telecommunications, 2013), pp. 1–4.

Aubourg, M.

J. Krupka, D. Cros, M. Aubourg, and P. Guillon, “Study whispering gallery modes in anisotropic single-crystal dielectric resonators,” IEEE Trans. Microwave Theor. Tech. 42, 56–61 (1994).
[CrossRef]

Ball, J. A. R.

C. G. Wells and J. A. R. Ball, “Mode-matching analysis of a shielded rectangular dielectric-rod waveguide,” IEEE Trans. Microwave Theor. Tech. 53, 3169–3177 (2005).
[CrossRef]

Benson, T. M.

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Tuning of elliptic whispering-gallery-mode microdisk waveguide filters,” J. Lightwave Technol. 21, 1987–1995 (2003).
[CrossRef]

S. C. Greedy, S. V. Boriskina, P. Sewell, and T. M. Benson, “Design and simulation pools for optical microresonators,” Proc. SPIE 4277, 21–32 (2001).

T. M. Benson, A. Vukovic, J. G. Wykes, A. Al-Jarro, and P. Sewell, “Numerical simulations of 3D micro-resonators,” in 10th Anniversary International Conference on Transparent Optical Networks, Athens, Greece (National Institute of Telecommunications, 2008), pp. 54–57.

Boriskin, A.

Boriskina, S.

Boriskina, S. V.

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Tuning of elliptic whispering-gallery-mode microdisk waveguide filters,” J. Lightwave Technol. 21, 1987–1995 (2003).
[CrossRef]

S. C. Greedy, S. V. Boriskina, P. Sewell, and T. M. Benson, “Design and simulation pools for optical microresonators,” Proc. SPIE 4277, 21–32 (2001).

S. V. Boriskina and A. I. Nosich, “Radiation, and absorption losses of the whispering-gallery-mode dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theor. Tech. 47, 224–231 (1999).
[CrossRef]

Burton, A. J.

A. J. Burton and G. F. Miller, “The application of integral equation methods to the numerical solution of some exterior boundary-value problems,” Proc. R. Soc. London Ser. A 323, 201–210 (1971).
[CrossRef]

Cangellaris, A. C.

A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525 (1991).
[CrossRef]

Chen, S. W.

S. W. Chen and K. A. Zaki, “Ring resonators loaded in waveguide and on substrate,” IEEE Trans. Microwave Theor. Tech. 39, 2069–2076 (1991).
[CrossRef]

Colton, D.

D. Colton and R. Kress, Integral Equations Methods in Scattering Theory (Wiley, 1983).

Crook, G. E.

L. A. Francavilla, J. S. McLean, H. D. Foltz, and G. E. Crook, “Mode-matching analysis of top-hat monopole antennas loaded with radially layered dielectric,” IEEE Trans. Antennas Propag. 47, 179–185 (1999).
[CrossRef]

Cros, D.

J. Krupka, D. Cros, M. Aubourg, and P. Guillon, “Study whispering gallery modes in anisotropic single-crystal dielectric resonators,” IEEE Trans. Microwave Theor. Tech. 42, 56–61 (1994).
[CrossRef]

Cruz, J. L.

E. Rivera-Perez, A. Diez, M. V. Andres, J. L. Cruz, and A. Rodriguez-Cobos, “Optical fiber whispering gallery modes resonances: applications,” in 15th International Conference on Transparent Optical Networks, Cartagena, Columbia (National Institute of Telecommunications, 2013), pp. 1–4.

Ctyroky, J.

L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217–223 (2005).
[CrossRef]

Diez, A.

E. Rivera-Perez, A. Diez, M. V. Andres, J. L. Cruz, and A. Rodriguez-Cobos, “Optical fiber whispering gallery modes resonances: applications,” in 15th International Conference on Transparent Optical Networks, Cartagena, Columbia (National Institute of Telecommunications, 2013), pp. 1–4.

Foltz, H. D.

L. A. Francavilla, J. S. McLean, H. D. Foltz, and G. E. Crook, “Mode-matching analysis of top-hat monopole antennas loaded with radially layered dielectric,” IEEE Trans. Antennas Propag. 47, 179–185 (1999).
[CrossRef]

Francavilla, L. A.

L. A. Francavilla, J. S. McLean, H. D. Foltz, and G. E. Crook, “Mode-matching analysis of top-hat monopole antennas loaded with radially layered dielectric,” IEEE Trans. Antennas Propag. 47, 179–185 (1999).
[CrossRef]

Greedy, S. C.

S. C. Greedy, S. V. Boriskina, P. Sewell, and T. M. Benson, “Design and simulation pools for optical microresonators,” Proc. SPIE 4277, 21–32 (2001).

Guillon, P.

J. Krupka, D. Cros, M. Aubourg, and P. Guillon, “Study whispering gallery modes in anisotropic single-crystal dielectric resonators,” IEEE Trans. Microwave Theor. Tech. 42, 56–61 (1994).
[CrossRef]

Hanson, G. W.

G. W. Hanson, “An efficient full-wave method for analysis of dielectric resonators possessing separable geometries immersed in inhomogeneous environments,” IEEE Trans. Microwave Theor. Tech. 48, 84–92 (2000).
[CrossRef]

Hao, Y.

Y. Hao and C. J. Railton, “Analyzing electromagnetic structures with curved boundaries on Cartesian FDTD meshes,” IEEE Trans. Microwave Theor. Tech. 46, 82–88 (1998).
[CrossRef]

Huang, W.

Hubalek, M.

L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217–223 (2005).
[CrossRef]

Ilchenko, V. S.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes—part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006).
[CrossRef]

Jiang, K.

Kajfez, D.

A. A. Kishk, M. R. Zunoubi, and D. Kajfez, “A numerical study of a dielectric disk antenna above grounded dielectric substrate,” IEEE Trans. Antennas Propag. 41, 813–821 (1993).
[CrossRef]

Kishk, A. A.

A. A. Kishk, M. R. Zunoubi, and D. Kajfez, “A numerical study of a dielectric disk antenna above grounded dielectric substrate,” IEEE Trans. Antennas Propag. 41, 813–821 (1993).
[CrossRef]

Kress, R.

D. Colton and R. Kress, Integral Equations Methods in Scattering Theory (Wiley, 1983).

Krupka, J.

J. Krupka, D. Cros, M. Aubourg, and P. Guillon, “Study whispering gallery modes in anisotropic single-crystal dielectric resonators,” IEEE Trans. Microwave Theor. Tech. 42, 56–61 (1994).
[CrossRef]

Kumagai, N.

N. Morita, N. Kumagai, and J. R. Mautz, Integral Equations Methods for Electromagnetics (Artech House, 1990).

Mahmoud-Taheri, M.

M. Mahmoud-Taheri, “Analysis of cylindrical cavity resonators loaded axisymmetrically with dielectric for multimode microwave filter design,” Ph.D. dissertation (University of Essex, 1989).

Matsko, A. B.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes—part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006).
[CrossRef]

Mautz, J. R.

N. Morita, N. Kumagai, and J. R. Mautz, Integral Equations Methods for Electromagnetics (Artech House, 1990).

McLean, J. S.

L. A. Francavilla, J. S. McLean, H. D. Foltz, and G. E. Crook, “Mode-matching analysis of top-hat monopole antennas loaded with radially layered dielectric,” IEEE Trans. Antennas Propag. 47, 179–185 (1999).
[CrossRef]

Miller, G. F.

A. J. Burton and G. F. Miller, “The application of integral equation methods to the numerical solution of some exterior boundary-value problems,” Proc. R. Soc. London Ser. A 323, 201–210 (1971).
[CrossRef]

Morita, N.

N. Morita, N. Kumagai, and J. R. Mautz, Integral Equations Methods for Electromagnetics (Artech House, 1990).

Nosich, A.

Nosich, A. I.

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Tuning of elliptic whispering-gallery-mode microdisk waveguide filters,” J. Lightwave Technol. 21, 1987–1995 (2003).
[CrossRef]

S. V. Boriskina and A. I. Nosich, “Radiation, and absorption losses of the whispering-gallery-mode dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theor. Tech. 47, 224–231 (1999).
[CrossRef]

Pozar, D. M.

D. M. Pozar, Microwave Engineering (Wiley, 1998).

Prkna, L.

L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217–223 (2005).
[CrossRef]

Railton, C. J.

Y. Hao and C. J. Railton, “Analyzing electromagnetic structures with curved boundaries on Cartesian FDTD meshes,” IEEE Trans. Microwave Theor. Tech. 46, 82–88 (1998).
[CrossRef]

Rivera-Perez, E.

E. Rivera-Perez, A. Diez, M. V. Andres, J. L. Cruz, and A. Rodriguez-Cobos, “Optical fiber whispering gallery modes resonances: applications,” in 15th International Conference on Transparent Optical Networks, Cartagena, Columbia (National Institute of Telecommunications, 2013), pp. 1–4.

Rodriguez-Cobos, A.

E. Rivera-Perez, A. Diez, M. V. Andres, J. L. Cruz, and A. Rodriguez-Cobos, “Optical fiber whispering gallery modes resonances: applications,” in 15th International Conference on Transparent Optical Networks, Cartagena, Columbia (National Institute of Telecommunications, 2013), pp. 1–4.

Rolland, A.

Sauleau, R.

Sewell, P.

S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich, “Tuning of elliptic whispering-gallery-mode microdisk waveguide filters,” J. Lightwave Technol. 21, 1987–1995 (2003).
[CrossRef]

S. C. Greedy, S. V. Boriskina, P. Sewell, and T. M. Benson, “Design and simulation pools for optical microresonators,” Proc. SPIE 4277, 21–32 (2001).

T. M. Benson, A. Vukovic, J. G. Wykes, A. Al-Jarro, and P. Sewell, “Numerical simulations of 3D micro-resonators,” in 10th Anniversary International Conference on Transparent Optical Networks, Athens, Greece (National Institute of Telecommunications, 2008), pp. 54–57.

Vahala, K. J.

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

Vukovic, A.

T. M. Benson, A. Vukovic, J. G. Wykes, A. Al-Jarro, and P. Sewell, “Numerical simulations of 3D micro-resonators,” in 10th Anniversary International Conference on Transparent Optical Networks, Athens, Greece (National Institute of Telecommunications, 2008), pp. 54–57.

Wells, C. G.

C. G. Wells and J. A. R. Ball, “Mode-matching analysis of a shielded rectangular dielectric-rod waveguide,” IEEE Trans. Microwave Theor. Tech. 53, 3169–3177 (2005).
[CrossRef]

Wiersig, J.

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

Wright, D. B.

A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525 (1991).
[CrossRef]

Wykes, J. G.

T. M. Benson, A. Vukovic, J. G. Wykes, A. Al-Jarro, and P. Sewell, “Numerical simulations of 3D micro-resonators,” in 10th Anniversary International Conference on Transparent Optical Networks, Athens, Greece (National Institute of Telecommunications, 2008), pp. 54–57.

Zaki, K. A.

S. W. Chen and K. A. Zaki, “Ring resonators loaded in waveguide and on substrate,” IEEE Trans. Microwave Theor. Tech. 39, 2069–2076 (1991).
[CrossRef]

Zunoubi, M. R.

A. A. Kishk, M. R. Zunoubi, and D. Kajfez, “A numerical study of a dielectric disk antenna above grounded dielectric substrate,” IEEE Trans. Antennas Propag. 41, 813–821 (1993).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes—part II: applications,” IEEE J. Sel. Top. Quantum Electron. 12, 15–32 (2006).
[CrossRef]

L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron. 11, 217–223 (2005).
[CrossRef]

IEEE Trans. Antennas Propag.

L. A. Francavilla, J. S. McLean, H. D. Foltz, and G. E. Crook, “Mode-matching analysis of top-hat monopole antennas loaded with radially layered dielectric,” IEEE Trans. Antennas Propag. 47, 179–185 (1999).
[CrossRef]

A. C. Cangellaris and D. B. Wright, “Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena,” IEEE Trans. Antennas Propag. 39, 1518–1525 (1991).
[CrossRef]

A. A. Kishk, M. R. Zunoubi, and D. Kajfez, “A numerical study of a dielectric disk antenna above grounded dielectric substrate,” IEEE Trans. Antennas Propag. 41, 813–821 (1993).
[CrossRef]

IEEE Trans. Microwave Theor. Tech.

G. W. Hanson, “An efficient full-wave method for analysis of dielectric resonators possessing separable geometries immersed in inhomogeneous environments,” IEEE Trans. Microwave Theor. Tech. 48, 84–92 (2000).
[CrossRef]

J. Krupka, D. Cros, M. Aubourg, and P. Guillon, “Study whispering gallery modes in anisotropic single-crystal dielectric resonators,” IEEE Trans. Microwave Theor. Tech. 42, 56–61 (1994).
[CrossRef]

S. V. Boriskina and A. I. Nosich, “Radiation, and absorption losses of the whispering-gallery-mode dielectric resonators excited by a dielectric waveguide,” IEEE Trans. Microwave Theor. Tech. 47, 224–231 (1999).
[CrossRef]

Y. Hao and C. J. Railton, “Analyzing electromagnetic structures with curved boundaries on Cartesian FDTD meshes,” IEEE Trans. Microwave Theor. Tech. 46, 82–88 (1998).
[CrossRef]

C. G. Wells and J. A. R. Ball, “Mode-matching analysis of a shielded rectangular dielectric-rod waveguide,” IEEE Trans. Microwave Theor. Tech. 53, 3169–3177 (2005).
[CrossRef]

S. W. Chen and K. A. Zaki, “Ring resonators loaded in waveguide and on substrate,” IEEE Trans. Microwave Theor. Tech. 39, 2069–2076 (1991).
[CrossRef]

J. Lightwave Technol.

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

Nature

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

Proc. R. Soc. London Ser. A

A. J. Burton and G. F. Miller, “The application of integral equation methods to the numerical solution of some exterior boundary-value problems,” Proc. R. Soc. London Ser. A 323, 201–210 (1971).
[CrossRef]

Proc. SPIE

S. C. Greedy, S. V. Boriskina, P. Sewell, and T. M. Benson, “Design and simulation pools for optical microresonators,” Proc. SPIE 4277, 21–32 (2001).

Other

T. M. Benson, A. Vukovic, J. G. Wykes, A. Al-Jarro, and P. Sewell, “Numerical simulations of 3D micro-resonators,” in 10th Anniversary International Conference on Transparent Optical Networks, Athens, Greece (National Institute of Telecommunications, 2008), pp. 54–57.

E. Rivera-Perez, A. Diez, M. V. Andres, J. L. Cruz, and A. Rodriguez-Cobos, “Optical fiber whispering gallery modes resonances: applications,” in 15th International Conference on Transparent Optical Networks, Cartagena, Columbia (National Institute of Telecommunications, 2013), pp. 1–4.

D. Colton and R. Kress, Integral Equations Methods in Scattering Theory (Wiley, 1983).

N. Morita, N. Kumagai, and J. R. Mautz, Integral Equations Methods for Electromagnetics (Artech House, 1990).

M. Mahmoud-Taheri, “Analysis of cylindrical cavity resonators loaded axisymmetrically with dielectric for multimode microwave filter design,” Ph.D. dissertation (University of Essex, 1989).

D. M. Pozar, Microwave Engineering (Wiley, 1998).

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Figures (11)

Fig. 1.
Fig. 1.

Cross section of a planar microdisk.

Fig. 2.
Fig. 2.

Triangular basis function used in the radial direction.

Fig. 3.
Fig. 3.

Typical modified cross section used for the simulation.

Fig. 4.
Fig. 4.

Field distribution of the first three modes of the lossless microdisk with an azimuth mode number of m=400. (a) Transverse magnetic field of EH1,400,1, (b) transverse magnetic field of EH2,400,1, and (c) transverse magnetic field of EH3,400,1.

Fig. 5.
Fig. 5.

(a) Resonance frequencies of the first three modes of the lossless microdisk with an azimuth mode number of m=400 versus the number of expansion basis functions. (b) Error percentage of first three modes of the lossless microdisk versus the penalty coefficient magnitude.

Fig. 6.
Fig. 6.

Cross section of the lossy microring.

Fig. 10.
Fig. 10.

Transverse magnetic distribution of the first three modes of the lossy microring with an azimuth mode number of m=400. (a) EH1,400,1, (b) EH2,400,1, and (c) EH3,400,1.

Fig. 7.
Fig. 7.

Wavelength error percentage caused by non-uniform meshing.

Fig. 8.
Fig. 8.

Simulation time for the microring with non-uniform meshing.

Fig. 9.
Fig. 9.

Cross section of the microring with ABC.

Fig. 11.
Fig. 11.

Transverse magnetic field distribution of the first three modes of the microdisk with an azimuth mode number of m=40 on a logarithmic scale. (a) HE1,40,1, (b) HE2,40,1, and (c) HE3,40,1.

Tables (5)

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Table 1. Resonance Wavelengths of the Microdisk (μm)

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Table 2. Resonance Wavelengths of the Lossy Microring (μm)

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Table 3. Calculated Quality Factors for the Microring

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Table 4. Resonance Wavelength of the Microdisk (μm) with m=40

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Table 5. Quality Factors of the Microdisk with m=40

Equations (17)

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Hz=cos(mφ)n=1Nq=1QAnqNn(r)sin(qπLz)Hr=cos(mφ)n=1Nq=1QBnqNn(r)cos(qπLz)Hφ=sin(mφ)n=1Nq=1QCnqNn(r)cos(qπLz).
k2=V(∇⃗×H⃗*)·εr1(∇⃗×H⃗)dv+αV|∇⃗·B⃗|2dvVH⃗*·(μrH⃗)dv.
k2=NUM(A,B,C,α)DEN(A,B,C)A=(A11A21A31AN1A12A22A32ANQ)TB=(B11B21B31BN1B12B22B32BNQ)TC=(C11C21C31CN1C12C22C32CNQ)TX=(ABC)
NUM=A*TWAAA+B*TWBBB+C*TWCCC+BTWBAA*+B*TWBAA+CTWCAA*+C*TWCAA+CTWCBB*+C*TWCBB*DEN=A*TW15A+B*TW15B+C*TW15C,
[L][X]=k02[R][X][L]=(WAAWBATWCATWBAWBBWCBTWCAWCBWCC)[R]=(W15000W15000W15).
Q=fr2fi,
εr=εr(kr(r)jσ(r)ω0ε0)kr(r)=1+rrminrmaxrmin(kmax21)σ(r)=rrminrmaxrminσ02,
σ0=1.5×105kmax=1.5rmax=6.5μmrmin=5.1μm.
k02=(NUM)Xi(DEN)Xi.
(x1x2xN)(XTWxyY)=WxyY.
X(XTWxyY)=WxyY.
Y(XTWxyY)=WxyTX.
(NUM)Ar=WAAA+WAATA*+WBATB+WBATB*+WCATC*+WCATC
1j(NUM)Ai=WAAAWAATA*+WBATB*WBATB+WCATC*WCATC
(DEN)Ar=W15A+W15TA*
1j(DEN)Ai=W15A+W15TA*.
(WAAWBATWCAT)(ABC)=k02W15A.

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