Abstract

Optical whispering gallery modes (WGMs) of mm-sized axisymmetric resonators are well localized at the equator. Employing this distinctive feature, we obtain simple analytical relations for the frequencies and eigenfunctions of WGMs which include the major radius of the resonator and the curvature radius of the rim. Being compared with results of finite-element simulations, these relations show a high accuracy and practicability. High-precision free-spectral-range measurements with a millimeter-sized disc resonator made of MgF2 allow us to identify the WGMs and confirm the applicability of our analytical description.

© 2013 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  29. J. Li, H. Lee, K. Y. Yang, and K. J. Vahala, “Sideband spectroscopy and dispersion measurement in microcavities,” Opt. Express20, 26337–26344 (2012).
    [CrossRef] [PubMed]
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2012 (3)

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics6, 369–373 (2012).
[CrossRef]

M. L. Gorodetsky and Y. A. Demchenko, “Accurate analytical estimates of eigenfrequencies and dispersion in whispering-gallery spheroidal resonators,” Proc. SPIE8236, 823623 (2012).

J. Li, H. Lee, K. Y. Yang, and K. J. Vahala, “Sideband spectroscopy and dispersion measurement in microcavities,” Opt. Express20, 26337–26344 (2012).
[CrossRef] [PubMed]

2011 (2)

B. Sturman and I. Breunig, “Generic description of second-order nonlinear phenomena in whispering-gallery resonators,” J. Opt. Soc. Am. B28, 2465–2471 (2011).
[CrossRef]

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett.106, 143903 (2011).
[CrossRef] [PubMed]

2010 (2)

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105, 263904 (2010).
[CrossRef]

B. Sprenger, H. G. L. Schwefel, Z. H. Lu, S. Svitlov, and L. J. Wang, “CaF2whispering-gallery-mode-resonator stabilized-narrow-linewidth laser,” Opt. Lett.35, 2870–2871 (2010).
[CrossRef] [PubMed]

2009 (2)

2008 (1)

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett.100, 103905 (2008).
[CrossRef] [PubMed]

2007 (3)

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

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1218 (2007).
[CrossRef]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express15, 6768–6773 (2007).
[CrossRef] [PubMed]

2006 (3)

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun.265, 33–38 (2006).
[CrossRef]

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes Part II: Applications,” IEEE J. Quantum Electron.12, 15–32 (2006).

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

2005 (2)

Y. Louyer, D. Meschede, and A. Rauschenbeutel, “Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics,” Phys. Rev. A72, 31801 (2005).
[CrossRef]

H. G. L. Schwefel, A. D. Stone, and H. E. Tureci, “Polarization properties and dispersion relations for spiral resonances of a dielectric rod,” J. Opt. Soc. Am. B22, 2295–2307 (2005).
[CrossRef]

2004 (2)

T. J. Kippenberg, S. M. Spillane, D. K. Armani, and K. J. Vahala, “Ultralow-threshold microcavity Raman laser on a microelectronic chip,” Opt. Lett.29, 1224–1226 (2004).
[CrossRef] [PubMed]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett.92, 43903 (2004).
[CrossRef]

2003 (2)

K. J. Vahala, “Optical microcavities,” Nature424, 839–846 (2003).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
[CrossRef] [PubMed]

2002 (1)

A. N. Oraevsky, “Whispering-gallery waves,” Quantum Electron.32, 377–400 (2002).
[CrossRef]

1984 (1)

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. A. Stegun, Handbook of Mathematical Functions (Dover, 1972), p. 367.

Aiello, A.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105, 263904 (2010).
[CrossRef]

Andersen, U. L.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105, 263904 (2010).
[CrossRef]

Arcizet, O.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1218 (2007).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. G. Weber, Mathematical Methods for Physicists (Academic, 2001).

Armani, D. K.

T. J. Kippenberg, S. M. Spillane, D. K. Armani, and K. J. Vahala, “Ultralow-threshold microcavity Raman laser on a microelectronic chip,” Opt. Lett.29, 1224–1226 (2004).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
[CrossRef] [PubMed]

Babic, V. M.

V. M. Babic and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Springer, 1991).

Beckmann, T.

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett.106, 143903 (2011).
[CrossRef] [PubMed]

Breunig, I.

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett.106, 143903 (2011).
[CrossRef] [PubMed]

B. Sturman and I. Breunig, “Generic description of second-order nonlinear phenomena in whispering-gallery resonators,” J. Opt. Soc. Am. B28, 2465–2471 (2011).
[CrossRef]

Buldyrev, V. S.

V. M. Babic and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Springer, 1991).

Buse, K.

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett.106, 143903 (2011).
[CrossRef] [PubMed]

Carmon, T.

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett.100, 103905 (2008).
[CrossRef] [PubMed]

Chen, T.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics6, 369–373 (2012).
[CrossRef]

Del’Haye, P.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1218 (2007).
[CrossRef]

Demchenko, Y. A.

M. L. Gorodetsky and Y. A. Demchenko, “Accurate analytical estimates of eigenfrequencies and dispersion in whispering-gallery spheroidal resonators,” Proc. SPIE8236, 823623 (2012).

Dodge, M. J.

Elser, D.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105, 263904 (2010).
[CrossRef]

Fomin, A. E.

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

Fürst, J. U.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105, 263904 (2010).
[CrossRef]

Gorodetsky, M. L.

M. L. Gorodetsky and Y. A. Demchenko, “Accurate analytical estimates of eigenfrequencies and dispersion in whispering-gallery spheroidal resonators,” Proc. SPIE8236, 823623 (2012).

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

Grudinin, I. S.

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun.265, 33–38 (2006).
[CrossRef]

Haertle, D.

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett.106, 143903 (2011).
[CrossRef] [PubMed]

Holzwarth, R.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1218 (2007).
[CrossRef]

Huang, Y. P.

D. V. Strekalov, A. S. Kowligy, Y. P. Huang, and P. Kumar, “Optical sum-frequency generation in whispering gallery mode resonators,” arXiv:0695587 (2013).

Ilchenko, V. S.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express15, 6768–6773 (2007).
[CrossRef] [PubMed]

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun.265, 33–38 (2006).
[CrossRef]

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes Part II: Applications,” IEEE J. Quantum Electron.12, 15–32 (2006).

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett.92, 43903 (2004).
[CrossRef]

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, and A. B. Matsko, “Crystalline whispering gallery mode resonators in optics and photonics,” in Practical Applications of Microresonators in Optics and Photonics, A. B. Matsko, ed. (CRC Press, 2009).

Jeon, S.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics6, 369–373 (2012).
[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.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1218 (2007).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, D. K. Armani, and K. J. Vahala, “Ultralow-threshold microcavity Raman laser on a microelectronic chip,” Opt. Lett.29, 1224–1226 (2004).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
[CrossRef] [PubMed]

Kowligy, A. S.

D. V. Strekalov, A. S. Kowligy, Y. P. Huang, and P. Kumar, “Optical sum-frequency generation in whispering gallery mode resonators,” arXiv:0695587 (2013).

Kumar, P.

D. V. Strekalov, A. S. Kowligy, Y. P. Huang, and P. Kumar, “Optical sum-frequency generation in whispering gallery mode resonators,” arXiv:0695587 (2013).

Lee, H.

J. Li, H. Lee, K. Y. Yang, and K. J. Vahala, “Sideband spectroscopy and dispersion measurement in microcavities,” Opt. Express20, 26337–26344 (2012).
[CrossRef] [PubMed]

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics6, 369–373 (2012).
[CrossRef]

Leuchs, G.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105, 263904 (2010).
[CrossRef]

Li, J.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics6, 369–373 (2012).
[CrossRef]

J. Li, H. Lee, K. Y. Yang, and K. J. Vahala, “Sideband spectroscopy and dispersion measurement in microcavities,” Opt. Express20, 26337–26344 (2012).
[CrossRef] [PubMed]

Linnenbank, H.

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett.106, 143903 (2011).
[CrossRef] [PubMed]

Louyer, Y.

Y. Louyer, D. Meschede, and A. Rauschenbeutel, “Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics,” Phys. Rev. A72, 31801 (2005).
[CrossRef]

Lu, Z. H.

Maleki, L.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express15, 6768–6773 (2007).
[CrossRef] [PubMed]

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun.265, 33–38 (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, 43903 (2004).
[CrossRef]

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, and A. B. Matsko, “Crystalline whispering gallery mode resonators in optics and photonics,” in Practical Applications of Microresonators in Optics and Photonics, A. B. Matsko, ed. (CRC Press, 2009).

Marquardt, C.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105, 263904 (2010).
[CrossRef]

Matsko, A. B.

D. V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Efficient upconversion of subterahertz radiation in a high-Q whispering gallery resonator,” Opt. Lett.34, 713–715 (2009).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express15, 6768–6773 (2007).
[CrossRef] [PubMed]

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun.265, 33–38 (2006).
[CrossRef]

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes Part II: Applications,” IEEE J. Quantum Electron.12, 15–32 (2006).

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett.92, 43903 (2004).
[CrossRef]

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, and A. B. Matsko, “Crystalline whispering gallery mode resonators in optics and photonics,” in Practical Applications of Microresonators in Optics and Photonics, A. B. Matsko, ed. (CRC Press, 2009).

Meschede, D.

Y. Louyer, D. Meschede, and A. Rauschenbeutel, “Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics,” Phys. Rev. A72, 31801 (2005).
[CrossRef]

Oraevsky, A. N.

A. N. Oraevsky, “Whispering-gallery waves,” Quantum Electron.32, 377–400 (2002).
[CrossRef]

Oxborrow, M.

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett.100, 103905 (2008).
[CrossRef] [PubMed]

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

Painter, O.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics6, 369–373 (2012).
[CrossRef]

Rauschenbeutel, A.

Y. Louyer, D. Meschede, and A. Rauschenbeutel, “Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics,” Phys. Rev. A72, 31801 (2005).
[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.

D. V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Efficient upconversion of subterahertz radiation in a high-Q whispering gallery resonator,” Opt. Lett.34, 713–715 (2009).
[CrossRef] [PubMed]

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express15, 6768–6773 (2007).
[CrossRef] [PubMed]

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun.265, 33–38 (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, 43903 (2004).
[CrossRef]

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, and A. B. Matsko, “Crystalline whispering gallery mode resonators in optics and photonics,” in Practical Applications of Microresonators in Optics and Photonics, A. B. Matsko, ed. (CRC Press, 2009).

Schliesser, A.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1218 (2007).
[CrossRef]

Schwefel, H. G. L.

Spillane, S. M.

T. J. Kippenberg, S. M. Spillane, D. K. Armani, and K. J. Vahala, “Ultralow-threshold microcavity Raman laser on a microelectronic chip,” Opt. Lett.29, 1224–1226 (2004).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
[CrossRef] [PubMed]

Sprenger, B.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972), p. 367.

Steigerwald, H.

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett.106, 143903 (2011).
[CrossRef] [PubMed]

Stone, A. D.

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett.100, 103905 (2008).
[CrossRef] [PubMed]

H. G. L. Schwefel, A. D. Stone, and H. E. Tureci, “Polarization properties and dispersion relations for spiral resonances of a dielectric rod,” J. Opt. Soc. Am. B22, 2295–2307 (2005).
[CrossRef]

Strekalov, D.

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun.265, 33–38 (2006).
[CrossRef]

Strekalov, D. V.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105, 263904 (2010).
[CrossRef]

D. V. Strekalov, A. A. Savchenkov, A. B. Matsko, and N. Yu, “Efficient upconversion of subterahertz radiation in a high-Q whispering gallery resonator,” Opt. Lett.34, 713–715 (2009).
[CrossRef] [PubMed]

D. V. Strekalov, A. S. Kowligy, Y. P. Huang, and P. Kumar, “Optical sum-frequency generation in whispering gallery mode resonators,” arXiv:0695587 (2013).

Sturman, B.

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett.106, 143903 (2011).
[CrossRef] [PubMed]

B. Sturman and I. Breunig, “Generic description of second-order nonlinear phenomena in whispering-gallery resonators,” J. Opt. Soc. Am. B28, 2465–2471 (2011).
[CrossRef]

Svitlov, S.

Tureci, H. E.

Vahala, K. J.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics6, 369–373 (2012).
[CrossRef]

J. Li, H. Lee, K. Y. Yang, and K. J. Vahala, “Sideband spectroscopy and dispersion measurement in microcavities,” Opt. Express20, 26337–26344 (2012).
[CrossRef] [PubMed]

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett.100, 103905 (2008).
[CrossRef] [PubMed]

T. J. Kippenberg, S. M. Spillane, D. K. Armani, and K. J. Vahala, “Ultralow-threshold microcavity Raman laser on a microelectronic chip,” Opt. Lett.29, 1224–1226 (2004).
[CrossRef] [PubMed]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
[CrossRef] [PubMed]

K. J. Vahala, “Optical microcavities,” Nature424, 839–846 (2003).
[CrossRef] [PubMed]

Wang, L. J.

Weber, H. G.

G. B. Arfken and H. G. Weber, Mathematical Methods for Physicists (Academic, 2001).

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1218 (2007).
[CrossRef]

Yang, K. Y.

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics6, 369–373 (2012).
[CrossRef]

J. Li, H. Lee, K. Y. Yang, and K. J. Vahala, “Sideband spectroscopy and dispersion measurement in microcavities,” Opt. Express20, 26337–26344 (2012).
[CrossRef] [PubMed]

Yang, L.

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett.100, 103905 (2008).
[CrossRef] [PubMed]

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. Quantum Electron. (1)

V. S. Ilchenko and A. B. Matsko, “Optical resonators with whispering-gallery modes Part II: Applications,” IEEE J. Quantum Electron.12, 15–32 (2006).

IEEE J. Sel. Top. Quantum Electron. (1)

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

IEEE Trans. Microwave Theory Tech. (1)

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

J. Opt. Soc. Am. B (2)

Nat. Photonics (1)

H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics6, 369–373 (2012).
[CrossRef]

Nature (3)

K. J. Vahala, “Optical microcavities,” Nature424, 839–846 (2003).
[CrossRef] [PubMed]

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature450, 1214–1218 (2007).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Commun.265, 33–38 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. A (1)

Y. Louyer, D. Meschede, and A. Rauschenbeutel, “Tunable whispering-gallery-mode resonators for cavity quantum electrodynamics,” Phys. Rev. A72, 31801 (2005).
[CrossRef]

Phys. Rev. Lett. (4)

T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static envelope patterns in composite resonances generated by level crossing in optical toroidal microcavities,” Phys. Rev. Lett.100, 103905 (2008).
[CrossRef] [PubMed]

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett.105, 263904 (2010).
[CrossRef]

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett.106, 143903 (2011).
[CrossRef] [PubMed]

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Phys. Rev. Lett.92, 43903 (2004).
[CrossRef]

Proc. SPIE (1)

M. L. Gorodetsky and Y. A. Demchenko, “Accurate analytical estimates of eigenfrequencies and dispersion in whispering-gallery spheroidal resonators,” Proc. SPIE8236, 823623 (2012).

Quantum Electron. (1)

A. N. Oraevsky, “Whispering-gallery waves,” Quantum Electron.32, 377–400 (2002).
[CrossRef]

Other (5)

D. V. Strekalov, A. S. Kowligy, Y. P. Huang, and P. Kumar, “Optical sum-frequency generation in whispering gallery mode resonators,” arXiv:0695587 (2013).

L. Maleki, V. S. Ilchenko, A. A. Savchenkov, and A. B. Matsko, “Crystalline whispering gallery mode resonators in optics and photonics,” in Practical Applications of Microresonators in Optics and Photonics, A. B. Matsko, ed. (CRC Press, 2009).

V. M. Babic and V. S. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Springer, 1991).

G. B. Arfken and H. G. Weber, Mathematical Methods for Physicists (Academic, 2001).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972), p. 367.

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Figures (5)

Fig. 1
Fig. 1

Geometry of the problem; R and r are the major and minor radii of the resonator, while u, θ, and φ are the curvilinear coordinates, and ρ = ρ(u, θ) is the distance from the observation point (shown by the red dot) to the vertical rotational axis.

Fig. 2
Fig. 2

Resonant frequencies ωm,p,q/2π for a WGM resonator with R = 1 and r = 0.3 mm for m = 104. The circles correspond to Eq. (11), while the horizontal and vertical dashes represent the simulations for the h- and v-modes. Note different vertical scales in a) and b).

Fig. 3
Fig. 3

The normalized transverse field distributions for R = 1 mm, r = 0.3 mm, and m = 104. The upper and lower rows correspond to the modes with p = 0, q = 1 and p = 2, q = 2, respectively. The panel a) gives the analytical results in accordance with Eq. (10), the panel b) represents the numerical data for the largest field component of the v- and h-modes, and the panel c) shows the dependences E(x) along the white solid lines in a) and b). The dashed lines indicate the resonator rim.

Fig. 4
Fig. 4

a) Schematic of the experimental setup including the laser emitting at the angular frequency ω, the electro-optic modulator (EOM) driven at the angular frequency Ω, the coupling prism, the whispering gallery resonator (WGR), and the detector. b) Typical detector signal versus the frequency ω showing two side peaks with the distance |Ω − Δω| with respect to the main resonance. Here, Δω is the free spectral range.

Fig. 5
Fig. 5

a) Fraction of the transmission spectrum of our MgF2 resonator; b) Free spectral ranges determined for 31 considered modes (red filled circles). The blue lines indicate the model predictions for m = 16500, p = 0, and q = 1 to 5.

Equations (12)

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

ρ = R r + ( r u ) cos θ R ( 1 u R r 2 R θ 2 ) .
[ 1 h φ h θ ( u h θ h φ u + θ h φ h θ θ ) m 2 h φ 2 + k 2 ] E = 0 ,
( 2 u 2 + 2 r 2 θ 2 ) E + [ k 2 m 2 R 2 ( 1 + 2 u R + r θ 2 R ) ] E = 0.
d 2 Θ r 2 Θ d θ 2 r m 2 R 3 θ 2 = d 2 U U d u 2 k 2 + m 2 R 2 ( 1 + 2 u R ) = Λ ,
( d 2 d ξ 2 ξ 2 ) Θ = Λ ˜ Θ ,
Λ ˜ p = ( 2 p + 1 ) , Θ p = exp ( ξ 2 / 2 ) H p ( ξ ) ,
u m = R 2 1 / 3 m 2 / 3 , and δ = R 2 [ k 2 R 2 m 2 1 ( 2 p + 1 ) R m r ] ,
d 2 U d ζ 2 = ζ U , U = Ai ( ζ ) Ai ( u δ u m ) .
U m , q ( u ) = Ai ( u u m ζ q ) .
E m , p , q = C × e i m φ × e θ 2 / 2 θ m 2 H p ( θ / θ m ) × Ai ( u / u m ζ q ) ,
n ( ω ) ω = c m R [ 1 + ζ q 2 1 / 3 m 2 / 3 + ( 2 p + 1 ) R 2 m r ] .
n ^ n = 1 ζ q 2 1 / 3 m 2 / 3 ( 2 p + 1 ) R 2 m r .

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