Abstract

Recently achieved radial poling of whispering-gallery resonators (WGRs) strongly extends the capabilities of tailoring the second-order nonlinear phenomena, such as second-harmonic generation and optical parametric oscillation, and transferring them to the range of low-power continuous-wave light sources. Owing to discreteness of the frequency spectrum, the resonance and phase-matching conditions for interacting waves cannot be fulfilled simultaneously in WGRs in the general case. Using Yariv’s generic approach to the description of WGR phenomena, we analyze two closely related issues: the possibilities to achieve the resonant and phase-matching conditions using the temperature tuning and the impact of detunings and phase mismatches on the nonlinear transformation efficiencies. It is shown that the radial poling provides important necessary conditions for the subsequent fine tuning to the nonlinear resonances. The requirements to the temperature tuning, as exemplified by the case of lithium niobate, are substantially dependent on the nonlinear process in question, the actual wavelength range, and the pump intensity.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
    [CrossRef]
  2. T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, Optical Microcavities, K. Vahala, ed. (World Scientific, 2004).
  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, Ferroelectric Crystals for Photonic Applications, P. Ferraro, S. Grilli, and P. De Natale, eds. (Springer, 2009).
  6. S. Weis, R. Riveire, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
    [CrossRef]
  7. 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]
  8. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
    [CrossRef]
  9. 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]
  10. 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]
  11. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [CrossRef]
  12. A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” J. Quantum Electron. QE-2, 109–124 (1966).
    [CrossRef]
  13. R. G. Smith, “Theory of intracavity optical second-harmonic generation,” IEEE J. Quantum Electron. 6, 215–223 (1970).
    [CrossRef]
  14. T. Debuisschert, A. Sizmann, E. Giacobino, and C. Fabre, “Type-II continuous-wave optical parametric oscillators: oscillation and frequency-tuning characteristics,” J. Opt. Soc. Am. B 10, 1668–1680 (1993).
    [CrossRef]
  15. V. Berger, “Second-harmonic generation in monolithic cavities,” J. Opt. Soc. Am. B 14, 1351–1360 (1997).
    [CrossRef]
  16. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000).
    [CrossRef]
  17. A. Yariv, “Critical coupling and its control in optical waveguide-ring resonator systems,” IEEE Photon. Technol. Lett. 14, 483–485 (2002).
    [CrossRef]
  18. B. Sturman and I. Breunig, “Generic description of second-order nonlinear phenomena in whispering-gallery resonators,” J. Opt. Soc. Am. B 28, 2465–2471 (2011).
    [CrossRef]
  19. D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proceedings J 140, 177–188 (1993).
    [CrossRef]
  20. 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]
  21. U. Schlarb and K. Betzler, “Influence of the defect structure on the refractive indices of undoped and Mg-doped lithium niobate,” Phys. Rev. B 50, 751–757 (1994).
    [CrossRef]
  22. O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
    [CrossRef]

2011 (2)

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]

B. Sturman and I. Breunig, “Generic description of second-order nonlinear phenomena in whispering-gallery resonators,” J. Opt. Soc. Am. B 28, 2465–2471 (2011).
[CrossRef]

2010 (3)

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

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]

S. Weis, R. Riveire, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

2008 (1)

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

2006 (1)

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 (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]

2003 (2)

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

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]

2002 (1)

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

2000 (1)

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

1999 (1)

1997 (1)

1994 (1)

U. Schlarb and K. Betzler, “Influence of the defect structure on the refractive indices of undoped and Mg-doped lithium niobate,” Phys. Rev. B 50, 751–757 (1994).
[CrossRef]

1993 (2)

T. Debuisschert, A. Sizmann, E. Giacobino, and C. Fabre, “Type-II continuous-wave optical parametric oscillators: oscillation and frequency-tuning characteristics,” J. Opt. Soc. Am. B 10, 1668–1680 (1993).
[CrossRef]

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

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

1970 (1)

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

1966 (1)

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

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]

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

Arcizet, O.

S. Weis, R. Riveire, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Arie, A.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

Armani, D. K.

T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, Optical Microcavities, K. Vahala, ed. (World Scientific, 2004).

Ashkin, A.

A. Ashkin, G. D. Boyd, and J. M. Dziedzic, “Resonant optical second harmonic generation and mixing,” 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]

Berger, V.

Betzler, K.

U. Schlarb and K. Betzler, “Influence of the defect structure on the refractive indices of undoped and Mg-doped lithium niobate,” Phys. Rev. B 50, 751–757 (1994).
[CrossRef]

Boyd, G. D.

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

Breunig, I.

B. Sturman and I. Breunig, “Generic description of second-order nonlinear phenomena in whispering-gallery resonators,” J. Opt. Soc. Am. B 28, 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]

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]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Debuisschert, T.

Deleglise, S.

S. Weis, R. Riveire, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Dziedzic, J. M.

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

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]

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

Fabre, C.

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[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]

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

Galun, E.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

Gavartin, E.

S. Weis, R. Riveire, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Gayer, O.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

Giacobino, E.

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]

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]

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]

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]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Kippenberg, T. J.

S. Weis, R. Riveire, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, Optical Microcavities, K. Vahala, ed. (World Scientific, 2004).

Lassen, M.

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

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

Love, J. D.

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

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[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]

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]

L. Maleki and A. B. Matsko, Ferroelectric Crystals for Photonic Applications, P. Ferraro, S. Grilli, and P. De Natale, eds. (Springer, 2009).

Marquardt, C.

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

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.

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]

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]

L. Maleki and A. B. Matsko, Ferroelectric Crystals for Photonic Applications, P. Ferraro, S. Grilli, and P. De Natale, eds. (Springer, 2009).

Min, B.

T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, Optical Microcavities, K. Vahala, ed. (World Scientific, 2004).

Riveire, R.

S. Weis, R. Riveire, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Rowland, D. R.

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

Sacks, Z.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[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]

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]

Schlarb, U.

U. Schlarb and K. Betzler, “Influence of the defect structure on the refractive indices of undoped and Mg-doped lithium niobate,” Phys. Rev. B 50, 751–757 (1994).
[CrossRef]

Schliesser, A.

S. Weis, R. Riveire, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Sizmann, A.

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, Optical Microcavities, K. Vahala, ed. (World Scientific, 2004).

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]

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]

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

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]

B. Sturman and I. Breunig, “Generic description of second-order nonlinear phenomena in whispering-gallery resonators,” J. Opt. Soc. Am. B 28, 2465–2471 (2011).
[CrossRef]

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, Optical Microcavities, K. Vahala, ed. (World Scientific, 2004).

Vahala, K. J.

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

Weis, S.

S. Weis, R. Riveire, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Yang, L.

T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, Optical Microcavities, K. Vahala, ed. (World Scientific, 2004).

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]

Appl. Phys. B (1)

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[CrossRef]

Electron. Lett. (1)

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

IEE Proceedings J (1)

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

IEEE J. Quantum Electron. (2)

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

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

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

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]

IEEE Photon. Technol. Lett. (1)

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

J. Opt. Soc. Am. B (4)

J. Quantum Electron. (1)

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

Nature (1)

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

Phys. Rev. B (1)

U. Schlarb and K. Betzler, “Influence of the defect structure on the refractive indices of undoped and Mg-doped lithium niobate,” Phys. Rev. B 50, 751–757 (1994).
[CrossRef]

Phys. Rev. Lett. (4)

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]

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

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]

Proc. SPIE (1)

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]

Science (1)

S. Weis, R. Riveire, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[CrossRef]

Other (2)

T. J. Kippenberg, S. M. Spillane, D. K. Armani, B. Min, L. Yang, and K. Vahala, Optical Microcavities, K. Vahala, ed. (World Scientific, 2004).

L. Maleki and A. B. Matsko, Ferroelectric Crystals for Photonic Applications, P. Ferraro, S. Grilli, and P. De Natale, eds. (Springer, 2009).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1.
Fig. 1.

(a) Geometric scheme of WGR excitation: The gray box is the coupler, a 1 and b 2 are the input amplitudes for this coupler, and a 2 and b 1 are the output amplitudes. Propagation of the internal wave along z is accompanied by linear and nonlinear losses. The WGR circumference is L = 2 π R , where R is the WGR major radius. The fields with the amplitudes a 1 , 2 and b 1 , 2 have the same frequency ω . If the internal wave is not pumped from outside, the input amplitude a 1 = 0 . (b) Corresponding wavevector spectrum: the mode number j is typically of the order of 10 4 , the width of the peak is α , and the detuning δ = k k j .

Fig. 2.
Fig. 2.

(a) Geometric scheme of a 75% radial poling and (b) corresponding spectrum of the reduction factor for N = 800 . (c) off-centered radial poling and (d) corresponding spectrum of the reduction factor for N = 800 and δ R / R = 0.01 . The gray bars in (b) and (d) refer to the perfect radial poling.

Fig. 3.
Fig. 3.

Dependence of the normalized internal pump power | y | 2 on the normalized external power x for δ ^ s = 0 , 5, and 20.

Fig. 4.
Fig. 4.

Dependence of 4 x | y ( x , δ ^ s ) | 4 / ( 1 + δ ^ s 2 ) on the normalized external pump power x for δ ^ s = 0 , 5, and 20.

Fig. 5.
Fig. 5.

Dependence of 4 x | y ( x , δ ^ s ) | 4 / ( 1 + δ ^ s 2 ) on the normalized detuning δ ^ s for x = 0.25 , 1, 10, and 50.

Fig. 6.
Fig. 6.

Dependence of the factor F ( ξ , δ ) on the normalized external pump power ξ = P / P 0 for δ ^ = 0 , 1 , 2, and 3.

Fig. 7.
Fig. 7.

(a) Dependence x s ( T ) for seven neighboring values of j p ; the circles (filled and open) indicate the WGR resonances. (b) Temperature dependence of the resonant pump wavelength λ j p ( T ) . The filled circles in (a) and (b) correspond to the WGR resonances that occur within the chosen tuning range of λ p .

Fig. 8.
Fig. 8.

Dependences x i ( x s ) (solid lines) for T = 30 ° C and 85° C and j p = 19410 . The filled circles indicate the solutions for δ x = δ x s = δ x i , while the arrows directed from the nearest open circle show the correspondence to the closest pair of integers j s , j i .

Fig. 9.
Fig. 9.

Nearly degenerate OPO: Detuning δ x versus T for four neighboring values of the pump number. The sequence of three numbers indicating the curves is j p , j s , j i .

Fig. 10.
Fig. 10.

Detuning δ x versus T for five sequential resonances. The cases (a) and (b) correspond to j p = 19411 and 19410, respectively. Lines 1–5 correspond to the following pairs ( j s , j p ) : (a) (13011, 5991), (13012, 5990), (13013, 5989), (13014, 5988), and (13015, 5987); (b) (13010, 5991), (13011, 5990), (13012, 5989), (13013, 5988), and (13014, 5987). For all combinations of j p , s , i , the difference j p j s j i is 409.

Equations (31)

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

S = j = S j exp ( i j z / R ) , with S j = S j * .
S j = i π j m = 0 N 1 ( 1 ) m e i j ( z m z 0 ) / R ,
a 2 = t a 1 + κ b 2 , b 1 = κ a 1 + t b 2 ,
b 1 a 1 = κ exp ( i k L ) exp ( i k L ) 1 + ( κ 2 + ε ) / 2 .
| b 1 | 2 | a 1 | 2 = 4 κ 2 ( κ 2 + ε ) 2 1 1 + ( 2 f ^ R δ j ) 2 ,
b ^ s + 0.5 α s b ^ s = i ν b ^ p 2 e i Δ k z , b ^ p + 0.5 α p b ^ p = i ν b ^ s b ^ p * e i Δ k z ,
b ^ s ( L ) = b ^ s ( 0 ) e 0.5 α s L i 0 L ν b ^ p 2 e 0.5 α s ( z L ) i Δ k z d z .
e i Δ k z = e i ( j s 2 j p ) z / R × e i δ s z .
b s 2 = b s 1 ( 1 α s L / 2 + i δ s L ) i ν eff L b p 1 2 ,
b p 2 = b p 1 ( 1 α p L / 2 ) i ν eff L b s 1 b p 1 * .
b s 1 = 2 i ν eff L b p 1 2 κ s 2 + α s L 2 i δ s L .
0.5 ( κ p 2 + α p L ) b p 1 + i ν eff L b s 1 b p 1 * = κ p a p 1 .
b p 1 [ 1 f ^ p + f ^ s ν eff 2 L 2 | b p 1 | 2 π 2 ( 1 i δ ^ s ) ] = κ p a p 1 π ,
x = κ p 2 f ^ p 3 f ^ s ν eff 2 L 2 | a p 1 | 2 π 4 , y = π b p 1 a p 1 κ p f ^ p .
y ( 1 + x | y | 2 1 i δ ^ s ) = 1 .
η s = 1 ( r p 1 + 1 ) ( r s 1 + 1 ) 4 x | y | 4 1 + δ ^ s 2 ,
x = 1 | y | 2 [ 1 + ( 1 + δ ^ s 2 ) ( 1 | y | 2 ) | y | 2 1 ] .
ε = α p L + 2 π x | y | 2 f ^ p ( 1 i δ ^ s ) .
b ^ p + 0.5 α p b ^ p = i ν ˜ b ^ s b ^ i e i Δ ˜ k z , b ^ s + 0.5 α s b ^ s = i q s ν ˜ b ^ p b ^ i * e i Δ ˜ k z , b ^ i + 0.5 α i b ^ i = i q i ν ˜ b ^ p b ^ s * e i Δ ˜ k z ,
b ^ p 2 = ( 1 0.5 α p L ) b ^ p 1 i ν ˜ e f f L b ^ s 1 b ^ i 1 , b ^ s 2 = ( 1 0.5 α s L ) b ^ s 1 i q s ν ˜ e f f L b ^ p 1 b ^ i 1 , * b ^ i 2 = ( 1 0.5 α i L ) b ^ i 1 i q i ν ˜ e f f L b ^ p 1 b ^ s 1 * .
b s 1 = i q s f ^ s ν ˜ eff L π ( 1 i δ ^ s ) b p 1 b i 1 * , b i 1 = i q i f ^ i ν ˜ eff L π ( 1 i δ ^ i ) b p 1 b s 1 * ,
e i Ψ 1 = i + δ ^ s = i + δ ^ i .
b p 1 = f ^ p π ( | κ p | a p 1 + i ν ˜ eff 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 * ,
c s , i = q s , i f ^ s , i f ^ p ν ˜ eff 2 L 2 π 2 ( 1 i δ ^ ) , d s , i = q s , i f ^ s , i f ^ p κ p ν ˜ eff L π 2 ( 1 i δ ^ ) ,
| b s 1 , i 1 | 2 = q s , i f ^ s , i f ^ p κ p 2 π 2 P 0 ( ξ δ ^ 2 1 ) ,
P 0 = π 4 q s q i f ^ s f ^ i f ^ p 2 κ p 2 ν ˜ eff 2 L 2 .
P th = P 0 ( 1 + δ ^ 2 ) .
η s , i = q s , i F ( ξ , δ ^ ) ( r s , i 1 + 1 ) ( r p 1 + 1 ) ,
x s n ( x s , T ) = 2 j p n ( j p , T ) .
j p n ( j p ) = x s n ( x s ) + x i n ( x i ) .

Metrics