Abstract

Extending Yariv’s generic approach to the description of optical microresonators, we describe the second- harmonic generation and the optical parametric oscillation in whispering-gallery resonators (WGRs). The output characteristics of these nonlinear processes are expressed in terms of conventional cavity/coupling parameters and nonlinear material coefficients. The found relations are relevant to the description and optimization of experiments with nonlinearly active WGRs at ultralow input light powers.

© 2011 Optical Society of America

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  1. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846(2003).
    [CrossRef] [PubMed]
  2. T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, “Fabrication, coupling and nonlinear optics of ultra-high-Q micro-sphere and chip-based toroid microcavities,” in Optical Microcavities, K.Vahala, ed. (World Scientific, 2004), Chap. 5.
    [CrossRef]
  3. A. B. Matsko, V. S. Ilchenko, R. L. Targat, A. A. Savchenkov, and L. Maleki, “Parametric optics with whispering-gallery modes,” Proc. SPIE 4969, 173–184 (2003).
    [CrossRef]
  4. A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes—Part I: Basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).
    [CrossRef]
  5. L. Maleki and A. B. Matsko, “Crystalline whispering gallery mode resonators in optics and photonics,” in Ferroelectric Crystals for Photonic Applications, P.Ferraro, S.Grilli, and P.De Natale, eds. (Springer, 2009), Chap. 13.
  6. 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]
  7. 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]
  8. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, Ch. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
    [CrossRef] [PubMed]
  9. J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, Ch. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett. 105, 263904 (2010).
    [CrossRef]
  10. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
    [CrossRef]
  11. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002).
    [CrossRef]
  12. D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. J 140, 177–188 (1993).
    [CrossRef]
  13. M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154 (1999).
    [CrossRef]
  14. M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
    [CrossRef] [PubMed]
  15. R. W. Boyd, Nonlinear Optics (Academic, 2008).
  16. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B 14, 2268–2294 (1997).
    [CrossRef]
  17. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).
  18. A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
    [CrossRef]
  19. V. S. Ilchenko, A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Low-threshold parametric nonlinear optics with quasi-phase-matched whispering-gallery modes,” J. Opt. Soc. Am. B 20, 1304–1308 (2003).
    [CrossRef]
  20. Y. Dumeige and P. Feron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006).
    [CrossRef]
  21. 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]
  22. A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109–124 (1966).
    [CrossRef]
  23. R. G. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Quantum Electron. 6, 215–223 (1970).
    [CrossRef]
  24. V. Berger, “Second-harmonic generation in monolithic cavities,” J. Opt. Soc. Am. B 14, 1351–1360 (1997).
    [CrossRef]

2011

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

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, Ch. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
[CrossRef] [PubMed]

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

2008

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]

2006

Y. Dumeige and P. Feron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006).
[CrossRef]

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes—Part I: Basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (2006).
[CrossRef]

2004

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

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

A. B. Matsko, V. S. Ilchenko, R. L. Targat, A. A. Savchenkov, and L. Maleki, “Parametric optics with whispering-gallery modes,” Proc. SPIE 4969, 173–184 (2003).
[CrossRef]

V. S. Ilchenko, A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Low-threshold parametric nonlinear optics with quasi-phase-matched whispering-gallery modes,” J. Opt. Soc. Am. B 20, 1304–1308 (2003).
[CrossRef]

2002

A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
[CrossRef]

2000

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
[CrossRef]

1999

1997

1993

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. J 140, 177–188 (1993).
[CrossRef]

1970

R. G. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Quantum Electron. 6, 215–223 (1970).
[CrossRef]

1966

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109–124 (1966).
[CrossRef]

Aiello, A.

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

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, Ch. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
[CrossRef] [PubMed]

Armani, D. K.

T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, “Fabrication, coupling and nonlinear optics of ultra-high-Q micro-sphere and chip-based toroid microcavities,” in Optical Microcavities, K.Vahala, ed. (World Scientific, 2004), Chap. 5.
[CrossRef]

Ashkin, A.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109–124 (1966).
[CrossRef]

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]

Berger, V.

Boyd, G. D.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109–124 (1966).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2008).

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]

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]

Cai, M.

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Dominguez-Juarez, J. L.

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.

Y. Dumeige and P. Feron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006).
[CrossRef]

Dziedzic, J. M.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” IEEE J. Quantum Electron. QE-2, 109–124 (1966).
[CrossRef]

Elser, D.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, Ch. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
[CrossRef] [PubMed]

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

Feron, P.

Y. Dumeige and P. Feron, “Whispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006).
[CrossRef]

Fürst, J. U.

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

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, Ch. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
[CrossRef] [PubMed]

Gorodetsky, M. L.

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]

Ilchenko, V. S.

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes—Part I: Basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (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] [PubMed]

A. B. Matsko, V. S. Ilchenko, R. L. Targat, A. A. Savchenkov, and L. Maleki, “Parametric optics with whispering-gallery modes,” Proc. SPIE 4969, 173–184 (2003).
[CrossRef]

V. S. Ilchenko, A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Low-threshold parametric nonlinear optics with quasi-phase-matched whispering-gallery modes,” J. Opt. Soc. Am. B 20, 1304–1308 (2003).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
[CrossRef]

M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154 (1999).
[CrossRef]

Ito, R.

Kippenberg, T. J.

T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, “Fabrication, coupling and nonlinear optics of ultra-high-Q micro-sphere and chip-based toroid microcavities,” in Optical Microcavities, K.Vahala, ed. (World Scientific, 2004), Chap. 5.
[CrossRef]

Kitamoto, A.

Kondo, T.

Kozyreff, G.

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]

Lassen, M.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, Ch. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
[CrossRef] [PubMed]

Leuchs, G.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, Ch. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
[CrossRef] [PubMed]

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

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]

Love, J. D.

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. J 140, 177–188 (1993).
[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]

A. B. Matsko, V. S. Ilchenko, R. L. Targat, A. A. Savchenkov, and L. Maleki, “Parametric optics with whispering-gallery modes,” Proc. SPIE 4969, 173–184 (2003).
[CrossRef]

V. S. Ilchenko, A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Low-threshold parametric nonlinear optics with quasi-phase-matched whispering-gallery modes,” J. Opt. Soc. Am. B 20, 1304–1308 (2003).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
[CrossRef]

L. Maleki and A. B. Matsko, “Crystalline whispering gallery mode resonators in optics and photonics,” in Ferroelectric Crystals for Photonic Applications, P.Ferraro, S.Grilli, and P.De Natale, eds. (Springer, 2009), Chap. 13.

Marquardt, Ch.

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

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, Ch. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
[CrossRef] [PubMed]

Martorell, J.

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. B.

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes—Part I: Basics,” IEEE J. Sel. Top. Quantum Electron. 12, 3–14 (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] [PubMed]

A. B. Matsko, V. S. Ilchenko, R. L. Targat, A. A. Savchenkov, and L. Maleki, “Parametric optics with whispering-gallery modes,” Proc. SPIE 4969, 173–184 (2003).
[CrossRef]

V. S. Ilchenko, A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Low-threshold parametric nonlinear optics with quasi-phase-matched whispering-gallery modes,” J. Opt. Soc. Am. B 20, 1304–1308 (2003).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
[CrossRef]

L. Maleki and A. B. Matsko, “Crystalline whispering gallery mode resonators in optics and photonics,” in Ferroelectric Crystals for Photonic Applications, P.Ferraro, S.Grilli, and P.De Natale, eds. (Springer, 2009), Chap. 13.

Min, B.

T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, “Fabrication, coupling and nonlinear optics of ultra-high-Q micro-sphere and chip-based toroid microcavities,” in Optical Microcavities, K.Vahala, ed. (World Scientific, 2004), Chap. 5.
[CrossRef]

Painter, O.

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Rowland, D. R.

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. J 140, 177–188 (1993).
[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]

A. B. Matsko, V. S. Ilchenko, R. L. Targat, A. A. Savchenkov, and L. Maleki, “Parametric optics with whispering-gallery modes,” Proc. SPIE 4969, 173–184 (2003).
[CrossRef]

V. S. Ilchenko, A. B. Matsko, A. A. Savchenkov, and L. Maleki, “Low-threshold parametric nonlinear optics with quasi-phase-matched whispering-gallery modes,” J. Opt. Soc. Am. B 20, 1304–1308 (2003).
[CrossRef]

A. B. Matsko, V. S. Ilchenko, A. A. Savchenkov, and L. Maleki, “Highly nondegenerate all-resonant optical parametric oscillator,” Phys. Rev. A 66, 043814 (2002).
[CrossRef]

Shirane, M.

Shoji, I.

Smith, R. G.

R. G. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Quantum Electron. 6, 215–223 (1970).
[CrossRef]

Spillane, S. M.

T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, “Fabrication, coupling and nonlinear optics of ultra-high-Q micro-sphere and chip-based toroid microcavities,” in Optical Microcavities, K.Vahala, ed. (World Scientific, 2004), Chap. 5.
[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]

Strekalov, D. V.

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

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, Ch. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
[CrossRef] [PubMed]

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]

Targat, R. L.

A. B. Matsko, V. S. Ilchenko, R. L. Targat, A. A. Savchenkov, and L. Maleki, “Parametric optics with whispering-gallery modes,” Proc. SPIE 4969, 173–184 (2003).
[CrossRef]

Vahala, K.

T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, “Fabrication, coupling and nonlinear optics of ultra-high-Q micro-sphere and chip-based toroid microcavities,” in Optical Microcavities, K.Vahala, ed. (World Scientific, 2004), Chap. 5.
[CrossRef]

Vahala, K. J.

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

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Yang, L.

T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, “Fabrication, coupling and nonlinear optics of ultra-high-Q micro-sphere and chip-based toroid microcavities,” in Optical Microcavities, K.Vahala, ed. (World Scientific, 2004), Chap. 5.
[CrossRef]

Yariv, A.

A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002).
[CrossRef]

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
[CrossRef]

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Electron. Lett.

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Figures (5)

Fig. 1
Fig. 1

(a) Schematic of coupling into and out of a WGR; a 1 and b 2 are the input amplitudes, while a 2 and b 1 are the output amplitudes; L is the circumference. (b) and (c) Illustration of coupling in and out for SHG and OPO; the indices p, s, and i correspond to the pump, signal, and idler waves.

Fig. 2
Fig. 2

Functions (a)  y ( x ) and (b)  x y 4 ( x ) ; the arrow in (b) indicates the maximum.

Fig. 3
Fig. 3

Contour plot of the function η s max ( r p , r s ) with r p = | κ p | 2 / α p L and r s = | κ s | 2 / α s L ; the numbers on the lines indicate the values of η s max .

Fig. 4
Fig. 4

Efficiency η p as a function of x for different values of r p = | κ p | 2 / α p L .

Fig. 5
Fig. 5

Dependences of η s + η i (curve 1), η p (curve 2), and the sum η p + η s + η i (curve 3) on ξ p for r p , s , i = 10 .

Equations (41)

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( a 2 b 1 ) = ( t κ κ * t * ) ( a 1 b 2 ) ,
b 2 = ( 1 0.5 α L ) e i k L b 1 ,
| b 1 | 2 | a 1 | 2 = | κ | 2 | e i ϕ 1 + ( α L + | κ | 2 ) / 2 | 2 , | a 2 | 2 | a 1 | 2 = | e i ϕ 1 + ( α L | κ | 2 ) / 2 | 2 | e i ϕ 1 + ( α L + | κ | 2 ) / 2 | 2 ,
| b 1 | 2 | a 1 | 2 = 4 | κ | 2 ( α L + | κ | 2 ) 2 , | a 2 | 2 | a 1 | 2 = ( α L | κ | 2 ) 2 ( α L + | κ | 2 ) 2 .
Q 1 = Q i 1 + Q c 1 ,
f 1 = f c 1 + f c 1 = ( α L + | κ | 2 ) / 2 π ,
a s 2 = κ s b s 2 , b s 1 = t s * b s 2 .
b ^ s + 0.5 α s b ^ s = i ν b ^ p 2 , b ^ p + 0.5 α p b ^ p = i ν b ^ s b ^ p * .
b ^ s 2 = ( 1 0.5 α s L ) b s 1 i ν L b p 1 2 , b ^ p 2 = ( 1 0.5 α p L ) b p 1 i ν L b s 1 b p 1 * .
b s 1 = 2 i ν L b p 1 2 | κ s | 2 + α s L i ν L f s π b p 1 2 .
0.5 ( | κ p | 2 + α p L ) b p 1 + i ν L b p 1 * b s 1 = | κ p | a p 1 .
b p 1 ( 1 f p + f s ν 2 L 2 | b p 1 | 2 π 2 ) = | κ p | π a p 1 .
y = π | b p 1 | | a p 1 | | κ p | f p , x = | κ p | 2 f p 3 f s ν 2 L 2 | a p 1 | 2 π 4 ,
( 1 + x y 2 ) y = 1.
η s = f s f p | κ s | 2 | κ p | 2 π 2 x y 4 ( x ) .
η s max = | κ s | 2 | κ p | 2 ( | κ s | 2 + α s L ) ( | κ p | 2 + α p L ) .
| a p 1 | max 2 = ( | κ p | 2 + α p L ) 3 ( | κ s | 2 + α s L ) 4 ν 2 L 2 | κ p | 2 .
a p 2 a p 1 = 1 2 | κ p | 2 y ( x ) | κ p | 2 + α p L .
b ^ p + 0.5 α p b ^ p = i ν ˜ b ^ s b ^ i , b ^ s + 0.5 α s b ^ s = i q s ν ˜ b ^ p b ^ i * , b ^ i + 0.5 α i b ^ i = i q i ν ˜ b ^ p b ^ s * ,
b ^ p 2 = ( 1 0.5 α p L ) b p 1 i ν ˜ L b s 1 b i 1 , b ^ s 2 = ( 1 0.5 α s L ) b s 1 i q s ν ˜ L b p 1 b i 1 * , b ^ i 2 = ( 1 0.5 α i L ) b i 1 i q i ν ˜ L b p 1 b s 1 * .
a p 2 = t p a p 1 + κ p b p 2 , b p 1 = κ p * a p 1 + t p * b p 2 .
a s 2 = κ s b s 2 , b s 1 = t s * b s 2 , a i 2 = κ i b i 2 , b i 1 = t i * b i 2 .
b s 1 = i q s f s ν ˜ L π b p 1 b i 1 * , b i 1 = i q i f i ν ˜ L π b p 1 b s 1 * .
b p 1 = f p π ( | κ p | a p 1 + i ν ˜ L b s 1 b i 1 ) .
b s 1 ( 1 + c s | b i 1 | 2 ) = i d s a p 1 b i 1 * , b i 1 ( 1 + c i | b s 1 | 2 ) = i d i a p 1 b s 1 * ,
| b s 1 , i 1 | 2 = q s , i f s , i f p | κ p | 2 π 2 | a p 1 | th 2 ( ξ p 1 ) ,
| a p 1 | th 2 = π 4 q s q i f s f i f p 2 | κ p | 2 ν ˜ 2 L 2
η s , i = q s , i ( r s , i 1 + 1 ) ( r p 1 + 1 ) × 4 ( ξ p 1 ) ξ p ,
a p 2 = 1 r p 1 + r p a p 1 2 i r p ν ˜ L 1 + r p b s 1 b i 1 .
η p = ( 1 2 ξ p 1 / 2 r p 1 + 1 ) 2 .
E = j E ^ j ( z ) Ψ j ( ρ ) e i ω t + i k z + c.c. ,
Ψ p E ^ p = i 2 ω p d eff c n p E ^ p * E ^ s Ψ p Ψ s , Ψ s E ^ s = i 2 ω p d eff c n s E ^ p 2 Ψ p 2 ,
σ j = Ψ j 2 ( ρ ) d ρ
E ^ j = b ^ j / 2 n j ϵ 0 c σ j .
ν = 2 ω p d eff σ s p p c ϵ 0 c n p σ p n s σ s ,
σ p p s = Ψ p 2 ( ρ ) Ψ s ( ρ ) d ρ .
ν ˜ = 2 ω p d eff σ p s i c ϵ 0 c n p n s n i σ p σ s σ i ,
| a p 1 | max 2 = P c × 1 f i , p 2 f i , s × ( r p + 1 ) 3 ( r s + 1 ) r p .
P c = ϵ 0 π 3 c 3 n p 2 n s ω p 2 L 2 d eff 2 σ p 2 σ s σ p p s 2 ,
| a p 1 | th 2 = P ˜ c × 1 f i , p f i , s f i , i × ( r p + 1 ) 2 ( r s + 1 ) ( r i + 1 ) r p ,
P ˜ c = ϵ 0 π 3 c 3 n p n s n i 4 q s q i ω p 2 L 2 d eff 2 σ p σ s σ i σ p s i 2 .

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