Abstract

This paper presents a new formulation for the fast and accurate three-dimensional vectorial analysis of dielectric whispering-gallery-mode (WGM) resonators using the finite element method (FEM). This method relies on two properties of the WGM resonators: (1) the axial symmetry to reduce the number of coordinate variables from three to two and (2) the use of the divergence equation for the magnetic field and its axial symmetry to express the azimuthal magnetic component of the field in terms of its transverse components. As a result, for our FEM analysis, only the transverse part of the magnetic Helmholtz equation is required, which is composed of two field variables and two coordinate variables. The method can use scalar elements without generating any spurious modes and without the need for a penalty function. This can substantially reduce the computational time and complexity. After verifying this FEM formulation through comparing the simulation results with the known analytical solutions for a few special resonators, a variety of WGM resonators in the silicon photonic platform are analyzed in terms of key parameters needed for designing integrated photonic structures.

© 2017 Optical Society of America

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References

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  1. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
    [Crossref]
  2. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
    [Crossref]
  3. Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-μm radius,” Opt. Express 16, 4309–4315 (2008).
    [Crossref]
  4. M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express 18, 19541–19557 (2010).
    [Crossref]
  5. S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photon. Rev. 6, 145–177 (2012).
    [Crossref]
  6. J.-F. Lee, G. M. Wilkins, and R. Mitra, “Finite-element analysis of axisymmetric cavity resonator using a hybrid edge element technique,” IEEE Trans. Microwave Theory Tech. 41, 1981–1987 (1993).
    [Crossref]
  7. R. Osegueda, J. Pierluissi, L. Gil, A. Revilla, G. Villalva, G. Dick, and D. S. Wang, “Azimuthally-dependent finite element solution to the cylindrical resonator,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (1994).
  8. S. M. Spillane, “Fiber-coupled ultra-high-q microresonators for nonlinear and quantum optics,” Ph.D. thesis (California Institute of Technology, 2004).
  9. M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microwave Theory Tech. 55, 1209–1218 (2007).
    [Crossref]
  10. M. Soltani, “Novel integrated silicon nanophotonic structures using ultra-high Q resonators,” Ph.D. thesis (Georgia Institute of Technology, 2009).
  11. G. B. Hocker and W. K. Burns, “Mode dispersion in diffused channel waveguides by the effective index method,” Appl. Opt. 16, 113–118 (1977).
    [Crossref]
  12. K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, “Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends,” Opt. Express 14, 11128–11141 (2006).
    [Crossref]
  13. M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000).
    [Crossref]
  14. B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. 32, 922–928 (1984).
    [Crossref]
  15. M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 33, 227–233 (1985).
    [Crossref]
  16. K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech. 34, 1120–1124 (1986).
    [Crossref]
  17. J.-M. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).
  18. S. Schiller, “Asymptotic expansion of morphological resonance frequencies in Mie scattering,” Appl. Opt. 32, 2181–2185 (1993).
    [Crossref]
  19. H. A. N. Hejase, “On the use of Davidenko’s method in complex root search,” IEEE Trans. Microwave Theory Tech. 41, 141–143 (1993).
    [Crossref]
  20. M. Soltani, S. Yegnanarayanan, Q. Li, and A. Adibi, “Systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quantum Electron. 46, 1158–1169 (2010).
    [Crossref]
  21. F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. 7, 371–373 (1997).
    [Crossref]
  22. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

2012 (1)

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photon. Rev. 6, 145–177 (2012).
[Crossref]

2010 (2)

M. Soltani, S. Yegnanarayanan, Q. Li, and A. Adibi, “Systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quantum Electron. 46, 1158–1169 (2010).
[Crossref]

M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express 18, 19541–19557 (2010).
[Crossref]

2008 (1)

2007 (2)

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
[Crossref]

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microwave Theory Tech. 55, 1209–1218 (2007).
[Crossref]

2006 (1)

2005 (1)

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[Crossref]

2000 (1)

1997 (1)

F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. 7, 371–373 (1997).
[Crossref]

1993 (3)

H. A. N. Hejase, “On the use of Davidenko’s method in complex root search,” IEEE Trans. Microwave Theory Tech. 41, 141–143 (1993).
[Crossref]

S. Schiller, “Asymptotic expansion of morphological resonance frequencies in Mie scattering,” Appl. Opt. 32, 2181–2185 (1993).
[Crossref]

J.-F. Lee, G. M. Wilkins, and R. Mitra, “Finite-element analysis of axisymmetric cavity resonator using a hybrid edge element technique,” IEEE Trans. Microwave Theory Tech. 41, 1981–1987 (1993).
[Crossref]

1986 (1)

K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech. 34, 1120–1124 (1986).
[Crossref]

1985 (1)

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 33, 227–233 (1985).
[Crossref]

1984 (1)

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. 32, 922–928 (1984).
[Crossref]

1977 (1)

Adibi, A.

M. Soltani, S. Yegnanarayanan, Q. Li, and A. Adibi, “Systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quantum Electron. 46, 1158–1169 (2010).
[Crossref]

M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express 18, 19541–19557 (2010).
[Crossref]

Beausoleil, R. G.

Burns, W. K.

Cai, H.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photon. Rev. 6, 145–177 (2012).
[Crossref]

Chen, H.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photon. Rev. 6, 145–177 (2012).
[Crossref]

Chew, W. C.

F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. 7, 371–373 (1997).
[Crossref]

Davies, J. B.

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. 32, 922–928 (1984).
[Crossref]

Dick, G.

R. Osegueda, J. Pierluissi, L. Gil, A. Revilla, G. Villalva, G. Dick, and D. S. Wang, “Azimuthally-dependent finite element solution to the cylindrical resonator,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (1994).

Eguchi, M.

K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech. 34, 1120–1124 (1986).
[Crossref]

Fattal, D.

Feng, S.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photon. Rev. 6, 145–177 (2012).
[Crossref]

Gil, L.

R. Osegueda, J. Pierluissi, L. Gil, A. Revilla, G. Villalva, G. Dick, and D. S. Wang, “Azimuthally-dependent finite element solution to the cylindrical resonator,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (1994).

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

Hayata, K.

K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech. 34, 1120–1124 (1986).
[Crossref]

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 33, 227–233 (1985).
[Crossref]

Hejase, H. A. N.

H. A. N. Hejase, “On the use of Davidenko’s method in complex root search,” IEEE Trans. Microwave Theory Tech. 41, 141–143 (1993).
[Crossref]

Hocker, G. B.

Jin, J.-M.

J.-M. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

Kakihara, K.

Kono, N.

Koshiba, M.

K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, “Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends,” Opt. Express 14, 11128–11141 (2006).
[Crossref]

M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737–743 (2000).
[Crossref]

K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech. 34, 1120–1124 (1986).
[Crossref]

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 33, 227–233 (1985).
[Crossref]

Lee, J.-F.

J.-F. Lee, G. M. Wilkins, and R. Mitra, “Finite-element analysis of axisymmetric cavity resonator using a hybrid edge element technique,” IEEE Trans. Microwave Theory Tech. 41, 1981–1987 (1993).
[Crossref]

Lei, T.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photon. Rev. 6, 145–177 (2012).
[Crossref]

Li, Q.

M. Soltani, S. Yegnanarayanan, Q. Li, and A. Adibi, “Systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quantum Electron. 46, 1158–1169 (2010).
[Crossref]

M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express 18, 19541–19557 (2010).
[Crossref]

Lipson, M.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[Crossref]

Luo, X.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photon. Rev. 6, 145–177 (2012).
[Crossref]

Mitra, R.

J.-F. Lee, G. M. Wilkins, and R. Mitra, “Finite-element analysis of axisymmetric cavity resonator using a hybrid edge element technique,” IEEE Trans. Microwave Theory Tech. 41, 1981–1987 (1993).
[Crossref]

Osegueda, R.

R. Osegueda, J. Pierluissi, L. Gil, A. Revilla, G. Villalva, G. Dick, and D. S. Wang, “Azimuthally-dependent finite element solution to the cylindrical resonator,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (1994).

Oxborrow, M.

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microwave Theory Tech. 55, 1209–1218 (2007).
[Crossref]

Pierluissi, J.

R. Osegueda, J. Pierluissi, L. Gil, A. Revilla, G. Villalva, G. Dick, and D. S. Wang, “Azimuthally-dependent finite element solution to the cylindrical resonator,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (1994).

Poon, A.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photon. Rev. 6, 145–177 (2012).
[Crossref]

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[Crossref]

Rahman, B. M. A.

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. 32, 922–928 (1984).
[Crossref]

Revilla, A.

R. Osegueda, J. Pierluissi, L. Gil, A. Revilla, G. Villalva, G. Dick, and D. S. Wang, “Azimuthally-dependent finite element solution to the cylindrical resonator,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (1994).

Saitoh, K.

Schiller, S.

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[Crossref]

Sekaric, L.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
[Crossref]

Soltani, M.

M. Soltani, S. Yegnanarayanan, Q. Li, and A. Adibi, “Systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quantum Electron. 46, 1158–1169 (2010).
[Crossref]

M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express 18, 19541–19557 (2010).
[Crossref]

M. Soltani, “Novel integrated silicon nanophotonic structures using ultra-high Q resonators,” Ph.D. thesis (Georgia Institute of Technology, 2009).

Spillane, S. M.

S. M. Spillane, “Fiber-coupled ultra-high-q microresonators for nonlinear and quantum optics,” Ph.D. thesis (California Institute of Technology, 2004).

Suzuki, M.

K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech. 34, 1120–1124 (1986).
[Crossref]

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 33, 227–233 (1985).
[Crossref]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

Teixeira, F. L.

F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. 7, 371–373 (1997).
[Crossref]

Tsuji, Y.

Villalva, G.

R. Osegueda, J. Pierluissi, L. Gil, A. Revilla, G. Villalva, G. Dick, and D. S. Wang, “Azimuthally-dependent finite element solution to the cylindrical resonator,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (1994).

Vlasov, Y.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
[Crossref]

Wang, D. S.

R. Osegueda, J. Pierluissi, L. Gil, A. Revilla, G. Villalva, G. Dick, and D. S. Wang, “Azimuthally-dependent finite element solution to the cylindrical resonator,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (1994).

Wilkins, G. M.

J.-F. Lee, G. M. Wilkins, and R. Mitra, “Finite-element analysis of axisymmetric cavity resonator using a hybrid edge element technique,” IEEE Trans. Microwave Theory Tech. 41, 1981–1987 (1993).
[Crossref]

Xia, F.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
[Crossref]

Xu, Q.

Q. Xu, D. Fattal, and R. G. Beausoleil, “Silicon microring resonators with 1.5-μm radius,” Opt. Express 16, 4309–4315 (2008).
[Crossref]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[Crossref]

Yegnanarayanan, S.

M. Soltani, S. Yegnanarayanan, Q. Li, and A. Adibi, “Systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quantum Electron. 46, 1158–1169 (2010).
[Crossref]

M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express 18, 19541–19557 (2010).
[Crossref]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

M. Soltani, S. Yegnanarayanan, Q. Li, and A. Adibi, “Systematic engineering of waveguide-resonator coupling for silicon microring/microdisk/racetrack resonators: theory and experiment,” IEEE J. Quantum Electron. 46, 1158–1169 (2010).
[Crossref]

IEEE Microwave Guided Wave Lett. (1)

F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. 7, 371–373 (1997).
[Crossref]

IEEE Trans. Microwave Theory Tech. (6)

H. A. N. Hejase, “On the use of Davidenko’s method in complex root search,” IEEE Trans. Microwave Theory Tech. 41, 141–143 (1993).
[Crossref]

J.-F. Lee, G. M. Wilkins, and R. Mitra, “Finite-element analysis of axisymmetric cavity resonator using a hybrid edge element technique,” IEEE Trans. Microwave Theory Tech. 41, 1981–1987 (1993).
[Crossref]

M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microwave Theory Tech. 55, 1209–1218 (2007).
[Crossref]

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguide solution by finite elements,” IEEE Trans. Microwave Theory Tech. 32, 922–928 (1984).
[Crossref]

M. Koshiba, K. Hayata, and M. Suzuki, “Improved finite-element formulation in terms of the magnetic field vector for dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 33, 227–233 (1985).
[Crossref]

K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech. 34, 1120–1124 (1986).
[Crossref]

J. Lightwave Technol. (1)

Laser Photon. Rev. (1)

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photon. Rev. 6, 145–177 (2012).
[Crossref]

Nat. Photonics (1)

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
[Crossref]

Nature (1)

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[Crossref]

Opt. Express (3)

Other (5)

J.-M. Jin, The Finite Element Method in Electromagnetics (Wiley, 1993).

M. Soltani, “Novel integrated silicon nanophotonic structures using ultra-high Q resonators,” Ph.D. thesis (Georgia Institute of Technology, 2009).

R. Osegueda, J. Pierluissi, L. Gil, A. Revilla, G. Villalva, G. Dick, and D. S. Wang, “Azimuthally-dependent finite element solution to the cylindrical resonator,” in 10th Annual Review of Progress in Applied Computational Electromagnetics (1994).

S. M. Spillane, “Fiber-coupled ultra-high-q microresonators for nonlinear and quantum optics,” Ph.D. thesis (California Institute of Technology, 2004).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Structure of a Si WGM ring resonator seated on a SiO 2 substrate on an SOI platform. The resonator can be covered by a cladding material. The axis of symmetry is at the center of the resonator and along the z direction. (b) Cross section of the resonator structure. When R in is zero, the resonator is a disk; otherwise, it is a ring or donut.

Fig. 2.
Fig. 2.

(a) Structure of a dielectric microsphere with a refractive index of n = 2 and a radius of 4 μm suspended in air. (b) Cross section of the electric field profile of E z corresponding to the TM polarization with azimuthal mode number m = 26 . (c) Comparison of the FEM and asymptotic methods for calculating resonance wavelength and radiative Q of the microsphere in (a) for different azimuthal mode numbers m . Good agreement suggests the accuracy of our proposed approach.

Fig. 3.
Fig. 3.

(a) Cross section of a Si microdisk resonator on a SiO 2 substrate, which is moderately meshed for FEM analysis. (b)–(g) Cross sections of the profiles of all six field components of one of the resonance modes; (h), (i) electric and magnetic energy densities of the same mode, respectively. This mode is the first-radial-order TE mode with m = 107 , because u E has one peak, and the electric field is predominantly in-plane ( E z is very small). For this simulation, m = 107 and the refractive indices of Si and SiO 2 are 3.475 and 1.444, respectively.

Fig. 4.
Fig. 4.

Effective indices ( n eff ) of first-, second-, and third-radial-order TE modes of the resonator with the structural parameters given in the captions of Fig. 3. The solid and dashed curves correspond to the simulation results using the FEM and the effective index method, respectively. The electric energy density of each mode is shown next to its corresponding dispersion of n eff versus wavelength.

Fig. 5.
Fig. 5.

Calculated FSR of the first- and second-radial-order TE modes of a Si microdisk resonator versus its diameter. All the simulations are for resonance wavelengths that exist in a range around 1550 nm. The microdisk has a thickness of 230 nm, and it is seated on a thick SiO 2 substrate and covered by air cladding.

Fig. 6.
Fig. 6.

Calculated effective index of the fundamental (or the first radial order) TE mode of a microring resonator versus its external diameter. For all the simulations, the microring has a thickness of 230 nm and a width of 500 nm, and it is seated on a thick SiO 2 substrate and covered by air cladding. The inset shows the cross section of the electric energy density profile for the first-radial-order mode of a microring with an external radius of 10 μm. The dashed line shows the effective index of a waveguide with the same cross section as that of the microring, calculated at the free-space wavelength of 1550 nm. For all the microring simulations, the effective index was calculated for one of the resonance wavelengths ( λ 0 ) that existed in a range of 1550 ± 20    nm .

Fig. 7.
Fig. 7.

Variation of the radiation Q of the first-radial-order TE mode of a Si microdisk resonator with a thickness of 230 nm with its radius for two different cases (i.e., suspended microdisk and microdisk buried in oxide). The markers correspond to the obtained simulation points. In all simulations, the radius of the microdisk is adjusted such that the resonance wavelength is in the range of 1550 ± 10    nm . The refractive indices of Si and SiO 2 are assumed to be 3.475 and 1.444, respectively.

Equations (29)

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( E ¯ H ¯ ) = ( E ρ ( ρ , z ) , E ϕ ( ρ , z ) , E z ( ρ , z ) H ρ ( ρ , z ) , H ϕ ( ρ , z ) , H z ( ρ , z ) ) exp ( i ω 0 t i m ϕ ) ,
× ( 1 n 2 × H ¯ ) = k 0 2 H ¯ ,
H ¯ = ( H ρ ρ ^ + H z z ^ ) + H ϕ ϕ ^ = H ¯ t + H ϕ ϕ ^ ,
ϕ F = i m F ,
H ϕ = t · ( ρ H ¯ t ) i m , ( m 0 )
1 ρ t × ( ρ 1 n 2 t × H ¯ t ) 1 n 2 ρ 2 t ( ρ t · ( ρ H ¯ t ) ) + m 2 n 2 ρ 2 H ¯ t = k 0 2 H ¯ t .
t × ( 1 n 2 t × H ¯ t ρ ) t × ( ρ ^ n 2 ρ × H ¯ t ρ ) 1 n 2 ρ t ( ρ t · H ¯ t ρ ) + m 2 n 2 ρ 2 H ¯ t ρ = k 0 2 H ¯ t ρ .
[ A ] [ H t ] = k 0 2 [ B ] [ H t ] ,
[ A ] = e = 1 N e Δ e w ¯ t · [ 1 ρ t × ( ρ 1 n 2 t × H ¯ t ) 1 n 2 ρ 2 t ( ρ t · ( ρ H ¯ t ) ) + m 2 n 2 ρ 2 H ¯ t ] d s ,
[ B ] = e = 1 N e Δ e w ¯ t · H ¯ t d s ,
w ¯ t = w ρ ( ρ , z ) ρ ^ + w z ( ρ , z ) z ^ ,
· ( ψ A ¯ ) = A ¯ · ψ + ψ · A ¯ , · ( A ¯ × B ¯ ) = B ¯ · × A ¯ A ¯ · × B ¯ , Δ · A ¯ d s = Δ A ¯ · n ^ d l ,
[ A ] = e = 1 N e Δ e ( t × w ¯ t ρ ) · [ ρ n 2 t × H ¯ t ] d s e = 1 N e Δ e [ w ¯ t n 2 × t × H ¯ t ] · n ^ d l + e = 1 N e Δ e ρ t · ( ρ H ¯ t ) t · w ¯ t n 2 ρ 2 d s e = 1 N e Δ e t · ( ρ H ¯ t ) w ¯ t n 2 ρ · n ^ d l + e = 1 N e Δ e m 2 n 2 ρ 2 w ¯ t · H ¯ t d s .
e = 1 N e Δ e [ w ¯ t n 2 × t × H ¯ t ] · n ^ d l = e = 1 N e Δ e [ w ¯ t × i ω 0 ε 0 E ϕ ϕ ^ ] · n ^ d l .
e = 1 N e Δ e t · ( ρ H ¯ t ) w ¯ t n 2 ρ · n ^ d l = e = 1 N e Δ e i m H ϕ w ¯ t n 2 ρ · n ^ d l .
( 1 ρ d d ρ ρ d d ρ m 2 ρ 2 + n eff 2 k 0 2 ) H z ( ρ ) = 0 ,
1 ρ ρ ( ρ H ρ ) i m ρ H ϕ + H z z = 0 ,
ρ ( ρ H ρ ) i m H ϕ + z ( ρ H z ) = 0 .
t · ( ρ H ¯ t ) = i m H ϕ ,
× H ¯ = 1 ρ ( i m H z ρ H ϕ z ) ρ ^ + 1 ρ ( ρ ( ρ H ϕ ) + i m H ρ ) z ^ + ( H ρ z H z ρ ) ϕ ^ .
× H ¯ = 1 ρ ( z ^ z + ρ ^ ρ ) × ( ρ H ϕ ϕ ^ ) + 1 ρ ( i m ϕ ^ ) × ( H ρ ρ ^ + H z z ^ ) + t × H ¯ t .
× H ¯ = 1 ρ t × ( ρ H ϕ ϕ ^ ) i m ϕ ^ ρ × H ¯ t + t × H ¯ t .
× A ¯ = 1 ρ t × ( ρ 1 ε t × H ¯ t ) i m ρ ϕ ^ × [ 1 ε ρ t × ( ρ H ϕ ϕ ^ ) i m ε ρ ϕ ^ × H ¯ t ] + t × [ 1 ε ρ t × ( ρ H ϕ ϕ ^ ) i m ε ρ ϕ ^ × H ¯ t ] .
1 ρ t × ( ρ 1 ε t × H ¯ t ) i m ρ ϕ ^ × ( 1 ε ρ t × ( ρ H ϕ ϕ ^ ) ) + m 2 ε ρ 2 H ¯ t = k 0 2 H ¯ t .
1 ρ t × ( ρ 1 ε t × H ¯ t ) i m ρ ϕ ^ × ( 1 ε ρ t × ( ρ t · ( ρ H ¯ t ) i m ϕ ^ ) ) + m 2 ε ρ 2 H ¯ t = k 0 2 H ¯ t .
1 ρ t × ( ρ 1 ε t × H ¯ t ) 1 ε ρ 2 t ( ρ t · ( ρ H ¯ t ) ) + m 2 ε ρ 2 H ¯ t = k 0 2 H ¯ t .
ε ¯ ¯ = ε Λ ¯ ¯ , μ ¯ ¯ = μ 0 Λ ¯ ¯ ,
Λ ¯ ¯ = [ ρ ˜ ρ s z s ρ 0 0 0 ρ ρ ˜ s z s ρ 0 0 0 ρ ˜ ρ s ρ s z ] .
1 ρ t × ( ρ 1 ε ϕ t × H ¯ t ) 1 [ ε ] t ρ 2 t ( ρ t · ( ρ [ μ ] t H ¯ t ) μ ϕ ) + m 2 [ ε ] t ρ 2 H ¯ t = k 0 2 [ μ ] t H ¯ t ,

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