Abstract

Axial symmetry is the cornerstone for theory and applications of high-Q optical whispering gallery resonators (WGRs). Nevertheless, research on birefringent crystalline material persistently pushes towards breaking this symmetry. We show theoretically and experimentally that the effect of broken axial symmetry, caused by optical anisotropy, is modest for the resonant frequencies and Q-factors of the WGR modes. Thus, the most important equatorial whispering gallery modes can be quantitatively described and experimentally identified. At the same time, the effect of broken axial symmetry on the light field distribution of the whispering gallery modes is typically very strong. This qualitatively modifies the phase-matching for the χ(2) nonlinear processes and enables broad-band second harmonic generation and optical parametric oscillation. The effect of weak geometric ellipticity in nominally symmetric WGRs is also considered. Altogether our findings pave the way for an extensive use of numerous birefringent (uniaxial and biaxial) crystals with broad transparency window and large χ(2) coefficients in nonlinear optics with WGRs.

© 2016 Optical Society of America

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References

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2015 (1)

J. Tang, J. Liu, C. Shang, C. Xie, H. Guo, K. Qian, C. Xue, and J. Liu, “Fabrication and spectral characterizations of high Q asymmetric resonant cavities,” Opt. Commun. 355, 269–273 (2015).
[Crossref]

2014 (3)

G. Schunk, J. U. Fürst, M. Förtsch, D. V. Strekalov, U. Vogl, F. Sedlmeir, H. G. L. Schwefel, G. Leuchs, and C. Marquardt, “Identifying modes of large whispering-gallery mode resonators from the spectrum and emission pattern,” Opt. Express 22, 30795–30806 (2014).
[Crossref]

P. S. Kuo, J. Bravo-Abad, and G. S. Solomon, “Second-harmonic generation using 4̄-quasi-phasematching in a GaAs whispering-gallery-mode microcavity,” Nat. Commun. 5, 3109 (2014).
[Crossref]

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

2013 (4)

2012 (2)

G. Lin, J. Fürst, D. V. Strekalov, I. S. Grudinin, and N. Yu, “High-Q UV whispering gallery mode resonators made of angle-cut BBO crystals,” Opt. Express 20, 21372–21378 (2012).
[Crossref] [PubMed]

M. L. Gorodetsky and Y. A. Demchenko, “Accurate analytical estimates of eigenfrequencies and dispersion in whipering-gallery spheroidal resonators,” Proc. SPIE 8236, 823623 (2012).
[Crossref]

2011 (4)

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

J. Fürst, D. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Quantum light from a whispering-gallery-mode disk resonator,” Phys. Rev. Lett. 106, 113901 (2011).
[Crossref] [PubMed]

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]

G. Kozyreff, J. L. Dominguez-Juarez, and J. Martorell, “Whispering gallery microresonators for second harmonic light generation from a low number of small molecules,” Nat. Commun. 2, 254 (2011).
[Crossref] [PubMed]

2008 (4)

G. Kozyreff, J. L. Dominguez-Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

F. Vollmer and A. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[Crossref] [PubMed]

Y. Dumeige, S. Trebanol, L. Ghişa, T. K. N. Nguyên, H. Tavernier, and P. Féron, “Determination of coupling regime of high-Q resonators and optical gain of highly selective amplifiers,” J. Opt. Soc. Am. B 25, 2073–2080 (2008).
[Crossref]

2007 (2)

Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, “Enhanced second-harmonic generation in AlGaAs microring resonators,” Opt. Lett. 32, 826–828 (2007).
[Crossref] [PubMed]

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 (3)

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

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

A. Matsko and V. Ilchenko, “Optical resonators with whispering-gallery modes, part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).
[Crossref]

2004 (1)

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] [PubMed]

2003 (1)

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

1997 (1)

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant cavities,” Nature 385, 45–47 (1997).
[Crossref]

1992 (1)

1989 (1)

V. Braginsky, M. Gorodetsky, and V. Il’chenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

1984 (1)

Abramowitz, M.

M. Abramowitz and A. Stergun, Handbook in Mathematical Functions (Dover Publications, 1972).

Aiello, A.

J. Fürst, D. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Quantum light from a whispering-gallery-mode disk resonator,” Phys. Rev. Lett. 106, 113901 (2011).
[Crossref] [PubMed]

Aitchison, J. S.

Andersen, U. L.

J. Fürst, D. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Quantum light from a whispering-gallery-mode disk resonator,” Phys. Rev. Lett. 106, 113901 (2011).
[Crossref] [PubMed]

Aoki, T.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

Arfken, G. B.

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

Arnold, A.

F. Vollmer and A. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[Crossref] [PubMed]

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]

Bowen, W. P.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

Braginsky, V.

V. Braginsky, M. Gorodetsky, and V. Il’chenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

Brasch, V.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Bravo-Abad, J.

P. S. Kuo, J. Bravo-Abad, and G. S. Solomon, “Second-harmonic generation using 4̄-quasi-phasematching in a GaAs whispering-gallery-mode microcavity,” Nat. Commun. 5, 3109 (2014).
[Crossref]

Breunig, I.

I. Breunig, B. Sturman, F. Sedlmeir, H. G. L. Schwefel, and K. Buse, “Whispering gallery modes at the rim of an axisymmetric optical resonator: analytical versus numerical description and comparison with experiment,” Opt. Express 21, 30683–30692 (2013).
[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]

Bristow, A. D.

Buse, K.

I. Breunig, B. Sturman, F. Sedlmeir, H. G. L. Schwefel, and K. Buse, “Whispering gallery modes at the rim of an axisymmetric optical resonator: analytical versus numerical description and comparison with experiment,” Opt. Express 21, 30683–30692 (2013).
[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]

Chak, P.

Dayan, B.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

Demchenko, Y. A.

M. L. Gorodetsky and Y. A. Demchenko, “Accurate analytical estimates of eigenfrequencies and dispersion in whipering-gallery spheroidal resonators,” Proc. SPIE 8236, 823623 (2012).
[Crossref]

Diddams, S. A.

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

Dodge, M. J.

Dominguez-Juarez, J. L.

G. Kozyreff, J. L. Dominguez-Juarez, and J. Martorell, “Whispering gallery microresonators for second harmonic light generation from a low number of small molecules,” Nat. Commun. 2, 254 (2011).
[Crossref] [PubMed]

G. Kozyreff, J. L. Dominguez-Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[Crossref]

Dumeige, Y.

Elser, D.

J. Fürst, D. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Quantum light from a whispering-gallery-mode disk resonator,” Phys. Rev. Lett. 106, 113901 (2011).
[Crossref] [PubMed]

Féron, P.

Fomin, A.

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

Förtsch, M.

Fürst, J.

Fürst, J. U.

Ghisa, L.

Gorodetsky, M.

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

V. Braginsky, M. Gorodetsky, and V. Il’chenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

Gorodetsky, M. L.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

M. L. Gorodetsky and Y. A. Demchenko, “Accurate analytical estimates of eigenfrequencies and dispersion in whipering-gallery spheroidal resonators,” Proc. SPIE 8236, 823623 (2012).
[Crossref]

Grudinin, I. S.

Guo, H.

J. Tang, J. Liu, C. Shang, C. Xie, H. Guo, K. Qian, C. Xue, and J. Liu, “Fabrication and spectral characterizations of high Q asymmetric resonant cavities,” Opt. Commun. 355, 269–273 (2015).
[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]

Hauer, M.

Herr, T.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Holzwarth, R.

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

Il’chenko, V.

V. Braginsky, M. Gorodetsky, and V. Il’chenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

Ilchenko, V.

A. Matsko and V. Ilchenko, “Optical resonators with whispering-gallery modes, part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).
[Crossref]

Ilchenko, V. S.

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] [PubMed]

Iyer, R.

Jost, J. D.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Kimble, H. J.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

Kippenberg, T. J.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

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

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

Kondratiev, N. M.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Kozyreff, G.

G. Kozyreff, J. L. Dominguez-Juarez, and J. Martorell, “Whispering gallery microresonators for second harmonic light generation from a low number of small molecules,” Nat. Commun. 2, 254 (2011).
[Crossref] [PubMed]

G. Kozyreff, J. L. Dominguez-Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[Crossref]

Kuo, P. S.

P. S. Kuo, J. Bravo-Abad, and G. S. Solomon, “Second-harmonic generation using 4̄-quasi-phasematching in a GaAs whispering-gallery-mode microcavity,” Nat. Commun. 5, 3109 (2014).
[Crossref]

Lam, C. C.

Leuchs, G.

Leung, P. T.

Lin, G.

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]

Liu, J.

J. Tang, J. Liu, C. Shang, C. Xie, H. Guo, K. Qian, C. Xue, and J. Liu, “Fabrication and spectral characterizations of high Q asymmetric resonant cavities,” Opt. Commun. 355, 269–273 (2015).
[Crossref]

J. Tang, J. Liu, C. Shang, C. Xie, H. Guo, K. Qian, C. Xue, and J. Liu, “Fabrication and spectral characterizations of high Q asymmetric resonant cavities,” Opt. Commun. 355, 269–273 (2015).
[Crossref]

Maleki, L.

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] [PubMed]

Marquardt, C.

Martorell, J.

G. Kozyreff, J. L. Dominguez-Juarez, and J. Martorell, “Whispering gallery microresonators for second harmonic light generation from a low number of small molecules,” Nat. Commun. 2, 254 (2011).
[Crossref] [PubMed]

G. Kozyreff, J. L. Dominguez-Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[Crossref]

Matsko, A.

A. Matsko and V. Ilchenko, “Optical resonators with whispering-gallery modes, part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).
[Crossref]

Matsko, A. B.

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] [PubMed]

Nguyên, T. K. N.

Nikogosyan, D. N.

D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005).

Nöckel, J. U.

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant cavities,” Nature 385, 45–47 (1997).
[Crossref]

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]

Parkins, A. S.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

Qian, K.

J. Tang, J. Liu, C. Shang, C. Xie, H. Guo, K. Qian, C. Xue, and J. Liu, “Fabrication and spectral characterizations of high Q asymmetric resonant cavities,” Opt. Commun. 355, 269–273 (2015).
[Crossref]

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D. Richards, Advanced Mathematical Methods with Maple (Cambridge University, 2002).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, “Fundamentals of photonics,” (Wiley, 1991).
[Crossref]

Savchenkov, A. A.

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] [PubMed]

Schunk, G.

Schwefel, H.

Schwefel, H. G. L.

Sedlmeir, F.

Shang, C.

J. Tang, J. Liu, C. Shang, C. Xie, H. Guo, K. Qian, C. Xue, and J. Liu, “Fabrication and spectral characterizations of high Q asymmetric resonant cavities,” Opt. Commun. 355, 269–273 (2015).
[Crossref]

Sipe, J. E.

Smirl, A. L.

Solomon, G. S.

P. S. Kuo, J. Bravo-Abad, and G. S. Solomon, “Second-harmonic generation using 4̄-quasi-phasematching in a GaAs whispering-gallery-mode microcavity,” Nat. Commun. 5, 3109 (2014).
[Crossref]

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]

Stergun, A.

M. Abramowitz and A. Stergun, Handbook in Mathematical Functions (Dover Publications, 1972).

Stone, A. D.

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant cavities,” Nature 385, 45–47 (1997).
[Crossref]

Strekalov, D.

J. Fürst, D. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Quantum light from a whispering-gallery-mode disk resonator,” Phys. Rev. Lett. 106, 113901 (2011).
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Strekalov, D. V.

Sturman, B.

I. Breunig, B. Sturman, F. Sedlmeir, H. G. L. Schwefel, and K. Buse, “Whispering gallery modes at the rim of an axisymmetric optical resonator: analytical versus numerical description and comparison with experiment,” Opt. Express 21, 30683–30692 (2013).
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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]

Tang, J.

J. Tang, J. Liu, C. Shang, C. Xie, H. Guo, K. Qian, C. Xue, and J. Liu, “Fabrication and spectral characterizations of high Q asymmetric resonant cavities,” Opt. Commun. 355, 269–273 (2015).
[Crossref]

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Teich, M. C.

B. E. A. Saleh and M. C. Teich, “Fundamentals of photonics,” (Wiley, 1991).
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T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

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

van Driel, H. M.

Vogl, U.

Vollmer, F.

F. Vollmer and A. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[Crossref] [PubMed]

Wang, C. Y.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Weber, H. G.

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

Wilcut, E.

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

Xie, C.

J. Tang, J. Liu, C. Shang, C. Xie, H. Guo, K. Qian, C. Xue, and J. Liu, “Fabrication and spectral characterizations of high Q asymmetric resonant cavities,” Opt. Commun. 355, 269–273 (2015).
[Crossref]

Xue, C.

J. Tang, J. Liu, C. Shang, C. Xie, H. Guo, K. Qian, C. Xue, and J. Liu, “Fabrication and spectral characterizations of high Q asymmetric resonant cavities,” Opt. Commun. 355, 269–273 (2015).
[Crossref]

Yang, Z.

Young, K.

Yu, N.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

G. Lin, J. U. Fürst, D. V. Strekalov, and N. Yu, “Wide-range cyclic phase matching and second harmonic generation in whispering gallery resonators,” Appl. Phys. Lett. 103, 181107 (2013).
[Crossref]

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

A. Matsko and V. Ilchenko, “Optical resonators with whispering-gallery modes, part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).
[Crossref]

M. Gorodetsky and A. 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 2-D 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. Commun. (2)

P. S. Kuo, J. Bravo-Abad, and G. S. Solomon, “Second-harmonic generation using 4̄-quasi-phasematching in a GaAs whispering-gallery-mode microcavity,” Nat. Commun. 5, 3109 (2014).
[Crossref]

G. Kozyreff, J. L. Dominguez-Juarez, and J. Martorell, “Whispering gallery microresonators for second harmonic light generation from a low number of small molecules,” Nat. Commun. 2, 254 (2011).
[Crossref] [PubMed]

Nat. Methods (1)

F. Vollmer and A. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[Crossref] [PubMed]

Nat. Photonics (1)

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014).
[Crossref]

Nature (3)

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

T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble, “Observation of strong coupling between one atom and a monolithic microresonator,” Nature 443, 671–674 (2006).
[Crossref] [PubMed]

J. U. Nöckel and A. D. Stone, “Ray and wave chaos in asymmetric resonant cavities,” Nature 385, 45–47 (1997).
[Crossref]

Opt. Commun. (1)

J. Tang, J. Liu, C. Shang, C. Xie, H. Guo, K. Qian, C. Xue, and J. Liu, “Fabrication and spectral characterizations of high Q asymmetric resonant cavities,” Opt. Commun. 355, 269–273 (2015).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Phys. Lett. A (1)

V. Braginsky, M. Gorodetsky, and V. Il’chenko, “Quality-factor and nonlinear properties of optical whispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[Crossref]

Phys. Rev. A (1)

G. Kozyreff, J. L. Dominguez-Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[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] [PubMed]

J. Fürst, D. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Quantum light from a whispering-gallery-mode disk resonator,” Phys. Rev. Lett. 106, 113901 (2011).
[Crossref] [PubMed]

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]

Proc. SPIE (1)

M. L. Gorodetsky and Y. A. Demchenko, “Accurate analytical estimates of eigenfrequencies and dispersion in whipering-gallery spheroidal resonators,” Proc. SPIE 8236, 823623 (2012).
[Crossref]

Science (2)

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: back-action at the mesoscale,” Science 321, 1172–1176 (2008).
[Crossref] [PubMed]

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

Other (5)

B. E. A. Saleh and M. C. Teich, “Fundamentals of photonics,” (Wiley, 1991).
[Crossref]

D. Richards, Advanced Mathematical Methods with Maple (Cambridge University, 2002).

D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005).

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

M. Abramowitz and A. Stergun, Handbook in Mathematical Functions (Dover Publications, 1972).

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Figures (7)

Fig. 1
Fig. 1 a) Geometric scheme of a WGR with broken axial symmetry. The x, y, z-axes are directed along the principal dielectric axes and ny > nz; the TE- and TH-modes are axially symmetric and asymmetric, respectively. b) Normalized dependence of Λm on β: The circles are obtained numerically for m = 103 without approximations; the open and filled circles correspond to Λ m + and Λ m . The circles calculated for m = 104 and the same β/m2 practically coincide with the shown ones. The solid line corresponds to Eq. (4).
Fig. 2
Fig. 2 a) Dependence f(λ) for several birefringent crystals and R = 1 mm within the transparency window on a semi-logarithmic scale; the letters in the parentheses indicate the crystallographic orientation of the WGR x-axis. b) Illustration of the discrete Fourier spectrum of the function Φm (φ) for f = 100.
Fig. 3
Fig. 3 Factor d eff / d = | Φ m p 2 Φ m s * | versus δm for different values of δf. Note the difference in the horizontal scales in the subfigures.
Fig. 4
Fig. 4 a) δm/δmmax versus λs within the transparency windows of different uniaxial crystals. Lines 1, 2, 3, and 4 correspond to negative crystals BBO, LB4, LN, and AGSe; line 5 refers to positive ZGP crystal. The symbols indicate the crystallographic orientation of the x-axis and polarization of the p-mode. b) The same for biaxial LBO crystal. Curves 1 to 4 correspond to different choices of the x-axis and polarization of the p-mode.
Fig. 5
Fig. 5 Local in φ fulfillment of the phase matching conditions n p = n s eff ( φ ) (subfigure a) and n s = n p eff ( φ ) (subfigure b) for the polarization schemes with TE and TH pump modes, respectively. Position of each horizontal solid line relative to its cos-like counterpart depends on λs. In the case of negative uniaxial crystals, nx,y = no and nz = ne < no, while for positive uniaxial crystals nx,z = no and ny = ne > no.
Fig. 6
Fig. 6 a) Geometry of a disk WGR with the major and minor radii R and r; the optic axis (red line) lies in the equatorial (horizontal) plane. b) Experimental setup comprising a tunable external-cavity diode laser (ECDL), a Fabry-Pérot interferometer (FPI), a coupling prism (P), the WGR, and two detectors (D). c) Transmission spectrum of the WGR ±15 GHz around the center frequency.
Fig. 7
Fig. 7 a) Maximum cross correlation (MCC) versus the center frequency νc and the major radius R. The absolute maximum occurs at νc = 287.92 THz and R = 1576.3 μm. b) The resonant frequencies for q = 1 . . . 10 (red) at R = 1576.3 μm around νc = 287.92 THz compared to the experimentally recorded transmission spectrum (blue).

Equations (13)

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

( 1 n y 2 2 y 2 + 1 n z 2 2 z 2 + k 2 ) H = 0 ,
y ^ = ( n y 2 n z 2 ) 1 / 2 R cosh u cos φ , z ^ = ( n y 2 n z 2 ) 1 / 2 R sinh u sin φ ,
Φ Φ + 2 β cos 2 φ = U U + 2 β cosh 2 u = Λ ,
Λ m ± ( β ) Λ m ( β ) = m 2 + β 2 2 m 2 .
d 2 U d ξ 2 = ( ξ ξ 0 ) U with ξ = ( 4 β sinh 2 u 0 ) 1 / 3 ( u 0 u ) and ξ 0 = 2 β cosh 2 u 0 Λ m ( 4 β sinh 2 u 0 ) 2 / 3 .
g 2 n 2 = m 2 + ξ q ( 2 n y n z ) 2 / 3 g 4 / 3 + g 4 ( n y 2 n z 2 ) 2 32 m 2 ,
v = v m , q = m c 2 π n ¯ R [ 1 + ξ q 2 1 / 3 m 2 / 3 + ( Δ n 4 n ¯ ) 2 ] .
Φ m exp [ i ( m φ f sin 2 φ ) ] .
f = β 2 m m Δ n 4 n ¯ π R Δ n 2 λ ,
exp ( i f sin 2 φ ) = j = J j ( f ) exp ( 2 i j φ )
| Φ m p 2 Φ m s * | = | J δ m / 2 ( δ f ) | .
2 m p m s = δ m = 0 , ± 2 , , ± 2 | δ f |
δ m δ m max 2 ( n p n s ) | Δ n p Δ n s | ,

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