Abstract

The problem of accurate calculation of eigenfrequencies in resonators of complex geometry is not only fundamental but also has many practical applications. In particular, a possibility for calculating the eigenfrequencies and geometry dependent dispersion of whispering gallery modes is important for optimization of dielectric microresonator-based Kerr frequency combs. In this case, the required anomalous second-order dispersion may be controlled by means of small shape variations of the resonator. Unfortunately, all uniform approximations for the eigenfrequencies do not reach the required precision for this purpose. We propose new approximations for spheroids, quartics, and toroids with better precision, which also allow for the estimation of the second-order dispersion. We also obtain analytical expressions for field distribution in microresonators and investigate the possibility of achieving better approximations by combining analytical and numerical methods.

© 2013 Optical Society of America

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References

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  1. V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering–gallery modes,” Phys. Lett. A 137, 393–397 (1989).
    [CrossRef]
  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. M. L. Gorodetsky, Optical Microresonators with Giant Quality-Factor (Fizmatlit, 2011).
  4. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  5. S. Schiller, “Asymptotic expansion of morphological resonance frequencies in Mie scattering,” Appl. Opt. 32, 2181–2185 (1993).
    [CrossRef]
  6. V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, “Microtorus: a high-finesse microcavity with whispering-gallery modes,” Opt. Lett. 26, 256–258 (2001).
    [CrossRef]
  7. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
    [CrossRef]
  8. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92, 043903 (2004).
    [CrossRef]
  9. M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultra-high-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103, 053901 (2009).
    [CrossRef]
  10. M. Oxborrow, “How to simulate the whispering-gallery-modes of dielectric microresonators in FEMLAB/COMSOL,” Proc. SPIE 6452, 64520J (2007).
    [CrossRef]
  11. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
    [CrossRef]
  12. P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
    [CrossRef]
  13. T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012).
    [CrossRef]
  14. T. Herr, V. Brash, M. L. Gorodetsky, and T. J. Kippenberg, “Soliton mode-locking in optical microresonators,” arXiv:1211.0733v1 (2012).
  15. A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics 5, 293–296 (2011).
    [CrossRef]
  16. I. S. Grudinin, L. Baumgartel, and N. Yu, “Frequency comb from a microresonator with engineered spectrum,” Opt. Express 20, 6604–6609 (2012).
    [CrossRef]
  17. M. Sumetsky, “Whispering-gallery bottle microcavities: the three-dimensional etalon,” Opt. Lett. 29, 8–10 (2004).
    [CrossRef]
  18. M. L. Gorodetsky and A. E. Fomin, “Eigenfrequencies and Q factor in the geometrical theory of whispering-gallery modes,” Quantum Electron. 37, 167–172 (2007).
    [CrossRef]
  19. M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39 (2006).
    [CrossRef]
  20. J. B. Keller and S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. 9, 24–75 (1960).
    [CrossRef]
  21. I. V. Komarov, L. I. Ponomarev, and S. J. Slavianov, Spheroidal and Coulomb Spheroidal Functions (Nauka, 1976; in Russian).
  22. M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).
  23. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

2012 (2)

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012).
[CrossRef]

I. S. Grudinin, L. Baumgartel, and N. Yu, “Frequency comb from a microresonator with engineered spectrum,” Opt. Express 20, 6604–6609 (2012).
[CrossRef]

2011 (3)

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics 5, 293–296 (2011).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

2009 (1)

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultra-high-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103, 053901 (2009).
[CrossRef]

2007 (2)

M. Oxborrow, “How to simulate the whispering-gallery-modes of dielectric microresonators in FEMLAB/COMSOL,” Proc. SPIE 6452, 64520J (2007).
[CrossRef]

M. L. Gorodetsky and A. E. Fomin, “Eigenfrequencies and Q factor in the geometrical theory of whispering-gallery modes,” Quantum Electron. 37, 167–172 (2007).
[CrossRef]

2006 (2)

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39 (2006).
[CrossRef]

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]

2004 (2)

M. Sumetsky, “Whispering-gallery bottle microcavities: the three-dimensional etalon,” Opt. Lett. 29, 8–10 (2004).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92, 043903 (2004).
[CrossRef]

2003 (1)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

2001 (1)

1993 (1)

1989 (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering–gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

1960 (1)

J. B. Keller and S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. 9, 24–75 (1960).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Baumgartel, L.

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering–gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

Brash, V.

T. Herr, V. Brash, M. L. Gorodetsky, and T. J. Kippenberg, “Soliton mode-locking in optical microresonators,” arXiv:1211.0733v1 (2012).

Del’Haye, P.

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef]

Fomin, A. E.

M. L. Gorodetsky and A. E. Fomin, “Eigenfrequencies and Q factor in the geometrical theory of whispering-gallery modes,” Quantum Electron. 37, 167–172 (2007).
[CrossRef]

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39 (2006).
[CrossRef]

Gavartin, E.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Gorodetsky, M. L.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

M. L. Gorodetsky and A. E. Fomin, “Eigenfrequencies and Q factor in the geometrical theory of whispering-gallery modes,” Quantum Electron. 37, 167–172 (2007).
[CrossRef]

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39 (2006).
[CrossRef]

V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, “Microtorus: a high-finesse microcavity with whispering-gallery modes,” Opt. Lett. 26, 256–258 (2001).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering–gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

M. L. Gorodetsky, Optical Microresonators with Giant Quality-Factor (Fizmatlit, 2011).

T. Herr, V. Brash, M. L. Gorodetsky, and T. J. Kippenberg, “Soliton mode-locking in optical microresonators,” arXiv:1211.0733v1 (2012).

Grudinin, I. S.

Hartinger, K.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012).
[CrossRef]

Herr, T.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

T. Herr, V. Brash, M. L. Gorodetsky, and T. J. Kippenberg, “Soliton mode-locking in optical microresonators,” arXiv:1211.0733v1 (2012).

Holwarth, R.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012).
[CrossRef]

Holzwarth, R.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Ilchenko, V. S.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics 5, 293–296 (2011).
[CrossRef]

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]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92, 043903 (2004).
[CrossRef]

V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, “Microtorus: a high-finesse microcavity with whispering-gallery modes,” Opt. Lett. 26, 256–258 (2001).
[CrossRef]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering–gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

Keller, J. B.

J. B. Keller and S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. 9, 24–75 (1960).
[CrossRef]

Kippenberg, T. J.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012).
[CrossRef]

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

T. Herr, V. Brash, M. L. Gorodetsky, and T. J. Kippenberg, “Soliton mode-locking in optical microresonators,” arXiv:1211.0733v1 (2012).

Komarov, I. V.

I. V. Komarov, L. I. Ponomarev, and S. J. Slavianov, Spheroidal and Coulomb Spheroidal Functions (Nauka, 1976; in Russian).

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Liang, W.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics 5, 293–296 (2011).
[CrossRef]

Maleki, L.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics 5, 293–296 (2011).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92, 043903 (2004).
[CrossRef]

V. S. Ilchenko, M. L. Gorodetsky, X. S. Yao, and L. Maleki, “Microtorus: a high-finesse microcavity with whispering-gallery modes,” Opt. Lett. 26, 256–258 (2001).
[CrossRef]

Matsko, A. B.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics 5, 293–296 (2011).
[CrossRef]

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]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92, 043903 (2004).
[CrossRef]

O’Shea, D.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultra-high-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103, 053901 (2009).
[CrossRef]

Oxborrow, M.

M. Oxborrow, “How to simulate the whispering-gallery-modes of dielectric microresonators in FEMLAB/COMSOL,” Proc. SPIE 6452, 64520J (2007).
[CrossRef]

Pöllinger, M.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultra-high-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103, 053901 (2009).
[CrossRef]

Ponomarev, L. I.

I. V. Komarov, L. I. Ponomarev, and S. J. Slavianov, Spheroidal and Coulomb Spheroidal Functions (Nauka, 1976; in Russian).

Rauschenbeutel, A.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultra-high-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103, 053901 (2009).
[CrossRef]

Riemensberger, J.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012).
[CrossRef]

Rubinow, S. I.

J. B. Keller and S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. 9, 24–75 (1960).
[CrossRef]

Savchenkov, A. A.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics 5, 293–296 (2011).
[CrossRef]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92, 043903 (2004).
[CrossRef]

Schiller, S.

Seidel, D.

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics 5, 293–296 (2011).
[CrossRef]

Slavianov, S. J.

I. V. Komarov, L. I. Ponomarev, and S. J. Slavianov, Spheroidal and Coulomb Spheroidal Functions (Nauka, 1976; in Russian).

Spillane, S. M.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Stegun, I. E.

M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Sumetsky, M.

Vahala, K. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Wang, C. Y.

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012).
[CrossRef]

Warken, F.

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultra-high-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103, 053901 (2009).
[CrossRef]

Yao, X. S.

Yu, N.

Ann. Phys. (1)

J. B. Keller and S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. 9, 24–75 (1960).
[CrossRef]

Appl. Opt. (1)

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

M. L. Gorodetsky and A. E. Fomin, “Geometrical theory of whispering gallery modes,” IEEE J. Sel. Top. Quantum Electron. 12, 33–39 (2006).
[CrossRef]

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]

Nat. Photonics (2)

T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Kerr combs with selectable central frequency,” Nat. Photonics 5, 293–296 (2011).
[CrossRef]

Nature (1)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Lett. A (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical whispering–gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

Phys. Rev. Lett. (3)

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett. 92, 043903 (2004).
[CrossRef]

M. Pöllinger, D. O’Shea, F. Warken, and A. Rauschenbeutel, “Ultra-high-Q tunable whispering-gallery-mode microresonator,” Phys. Rev. Lett. 103, 053901 (2009).
[CrossRef]

P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Octave spanning tunable frequency comb from a microresonator,” Phys. Rev. Lett. 107, 063901 (2011).
[CrossRef]

Proc. SPIE (1)

M. Oxborrow, “How to simulate the whispering-gallery-modes of dielectric microresonators in FEMLAB/COMSOL,” Proc. SPIE 6452, 64520J (2007).
[CrossRef]

Quantum Electron. (1)

M. L. Gorodetsky and A. E. Fomin, “Eigenfrequencies and Q factor in the geometrical theory of whispering-gallery modes,” Quantum Electron. 37, 167–172 (2007).
[CrossRef]

Science (1)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[CrossRef]

Other (6)

T. Herr, V. Brash, M. L. Gorodetsky, and T. J. Kippenberg, “Soliton mode-locking in optical microresonators,” arXiv:1211.0733v1 (2012).

M. L. Gorodetsky, Optical Microresonators with Giant Quality-Factor (Fizmatlit, 2011).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

I. V. Komarov, L. I. Ponomarev, and S. J. Slavianov, Spheroidal and Coulomb Spheroidal Functions (Nauka, 1976; in Russian).

M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1964).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

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Figures (2)

Fig. 1.
Fig. 1.

Precision of eigenfrequency calculation in spheroids for four different modes, depending on the oblateness. Points are relative errors of approximation (16), as compared to FEM simulations, lines—fitted polynomials (Appendix B).

Fig. 2.
Fig. 2.

Radial and axial distribution of the main component of electric field Er for TM500,500,1 mode in a toroid (R=30μm, r=2.5μm, n=1.446) and an equivalent spheroid (a=30μm, b=8.66μm) is compared to approximations obtained from Eqs. (25). Insets show simulated field distributions in the ρ,z plane, with marked cross sections along which the fields were taken.

Tables (2)

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Table 1. Comparison of Numerical Simulation (num) with Analytical Approximations

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Table 2. Numerical Fit for the Fundamental Mode

Equations (34)

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ρs(z)=a1z2b2μz4b4,
Ψe±iβ(z)dz±imϕR(ρ/ρs),
β(z)=k2y˜mq2ρs2(z),
β(z)dz=2z1z2k2y˜mq2ρs2(z)dz=2π(p+12),
k2ρs2=y˜2(1z2b2μz4b4)=y˜mq,zc2=b22μ(1+4μ(1y˜mq2/y˜2)1),2y˜abzczc(zc2z2)(1+(zc2+z2)μ/b4)1z2/b2μz4/b4dz=π(2p+1).
4y˜ηc2ba0π/21+μ(1+sin2ψ)ηc21ηc2sin2ψμηc4sin4ψcos2ψdψy˜mqηc2πba(1+5(1+μ)8ηc2+31+62μ13μ264ηc4+417+1251μ+231μ2+141μ31024ηc6+)=π(2p+1).
ηc2=τ(15(1+μ)8τ+19+38μ+63μ264τ2117+351μ+1531μ2+2041μ31024τ3+),
y˜y˜mq(1+12τ+1+3μ16τ21+6μ+17μ2128τ3+).
y˜pqαq(2)1/3+2p(ab)+a2b+320αq2(2)1/3+αap6(2)2/3+(αq3+101400+(1+3μ)(2p+1)2a232b2)(2)1,
x=d2[(ξ2s)(1η2)]1/2cos(ϕ),y=d2[(ξ2s)(1η2)]1/2sin(ϕ),z=d2ξη.
2k0Sη(η)|ηcηc=k0ndηcηc(ηc2η2)(ξc2sη2)1η2dη=2π(p+1/2),2k0Sξ(ξ)|ξcξs=k0ndξcξs(ξ2ξc2)(ξ2sηc2)ξ2sdξ=2π(q1/4),k0Sϕ(ϕ)|02π=2πy˜1ηc21ζc2=2πm.
ηc2=(2p+1)abm1[1βq(a2b2)2b2(m2)2/3a(5b2+a2)(2p+1)16b3(m2)1+],ζc2=βq(m2)2/3[1+3βq5(m2)2/3(2p+1)a36b3(m2)1+48βq2175(m2)4/3+],nk0a=mβq(m2)1/3+(2p+1)a2b+3βq220(m2)1/3βq12(2p+1)a3b3(m2)2/3+(βq31400+(2p+1)2a2(b2a2)32b4)(m2)1+.
1ρczczc1+gc2gc2ρc2gcdz=π(p+1/2)m,1ρczczs1+hc2hc2ρc2hcdz=π(q1/4)m,nk0ρc=m,
ηc(2p+1)acbcm[1ac(2p+1)(5(μc+1)b2+a2)8bc3m].
nk0amαq(m2)1/3+(2p+1)a2b+3αq220(m2)1/3αq12(2p+1)a3b3(m2)2/3+(αq31400+(2p+1)2a2(b2(1+3μ)a2)32b4)(m2)1.
y˜=nk0aαq(2)1/3+2p(ab)+a2b+3αq220(2)1/3αq122p(a3b3)+a3b3(2)2/3+(αq3+101400+(2p+1)2a2(b2(1+3μ)a2)32b4)(2)1.
δys=Pnn21αq122n3P(2P23)(n21)3/2(2)2/3
a=R,b=Rr,μ=Rr4r.
nk0Rmαq(m2)1/3+(2p+1)2(Rr)1/2+3αq220(m2)1/3αq(2p+1)12(Rr)3/2(m2)2/3+(αq31400(2p+1)2R(Rr)128r2)(m2)1+δys.
Rsph=hsph2+f2,hsph=1zc(rc2+(Rcrc)rc2zc2).
y˜pqnacωpq=y˜0q+d1p+12d2p2,d1abbαqa3b36b3(2)2/3+a2(b2(1+3μ)a2)4b4,d2=y˜p+1+y˜p12y˜pa2(b2(1+3μ)a2)2b41.
p0=(0,34μ22b),p1=(43a,12μ+762b),p2=(43a,12μ+762b),p3=(0,34μ22b).
R(ξ,c)Cξ[(λc2ξ2)(ξ2s)+sm2]1/4eik0S+c.c.,S(η,c)Cη[(1η2)(λsc2η2)m2]1/4eik0S+c.c.,λ=c2(ξc2s+sηc2).
Eχ(ρ<a)E0ez22b¯2m2b¯2/a¯21/4Jm(Tmqρa¯)eimϕ,Eχ(ρ>a)CE0ez22b¯2m2b¯2/a¯21/4×Hm(1)(Tmq2n2a2+m(n21)n2abρ)eimϕE0Pez22b¯2m2b¯2/a¯21/4Jm(Tmqaa¯)eimϕγ(ρa),
eθ˜2/2Hp(θ˜),θ˜zb¯(m2b¯2a¯214)1/4.
Eχ(ρ<a)E0ez2m2a¯b¯eimϕ×Ai((2m)1/3(Tmqρa¯m)),Eχ(ρ>a)E0Pez2m2a¯b¯eimϕeγ(ρa)×Ai((2m)1/3(Tmqaa¯m)).
E=ϵ0n222π0a¯|E|2ρdρdzϵ0n22E022ππa¯b¯ma¯22Jm2(Tm1),
Veff=n2|E|2dVmax(n2|E|2)π3/2a¯2a¯b¯mJm2(Tm1)Jm2(Tm1),
Veff15.12a¯a¯bm7/6,15.12a11/4r1/4m7/6.
Eexp[(ρam)22rr2z22rz2+imϕ],rz=a¯b¯m,rr=0.77a¯m2/3,am=Tm1λ2πn.
k0na=Tν,qPnn21+αq(32P2)Pn36(n21)3/2(ν2)2/3n4P(P1)(P2+P1)4(n21)2(ν2)1+O(ν4/3),
Tν,q=ναq(ν2)1/3+3αq220(ν2)1/3+αq3+101400(ν2)1αq(479αq340)504000(ν2)5/3αq2(20231αq3+55100)129360000(ν2)7/3+O(ν3).
k0na=Tm,qnP˜n21+αq(32P˜2)P˜n36(n21)3/2(m2)2/3n2P˜(P˜1)(P˜2n2+P˜n21)4(n21)2(m2)1+O(m3).
Δ(a/b,)=i=04j=46aij(ab)i(2)j/3.

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