Abstract

We show that plane-wave singly resonant optical parametric oscillators exhibit a temporal modulation instability when pumped a certain number of times above threshold. Previously, this instability threshold was predicted, with a model neglecting variations in pump power, to occur at around 4.61 times oscillation threshold. We consider here the full self-consistent interaction between pump, signal, and idler sidebands and find that in some spectral regions the instability threshold can be lower than previously predicted, in some cases even comparable to the oscillation threshold, preventing single mode operation at high conversion efficiency. We examine the behavior of the instability for typical regions of operation, and find that both group velocity mismatches and group velocity dispersion have a significant effect. The instability can be suppressed by a suitable choice of intracavity etalon, the design constraints of which are determined.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. E. Myers and W. R. Bosenberg, “Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators,” IEEE J. Quantum Electron. 33, 1663–1672 (1997).
    [CrossRef]
  2. W. R. Bosenberg, A. Drobshoff, J. I. Alexander, L. E. Myers, and R. L. Byer, “93% pump depletion, 3.5-W continuous-wave, singly resonant optical parametric oscillator,” Opt. Lett. 21, 1336–1338 (1996).
    [CrossRef] [PubMed]
  3. R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064–1068 (1968).
    [CrossRef]
  4. L. B. Kreuzer, “Single and multi-mode oscillation of the singly resonant optical parametric oscillator,” in Proceedings of the Joint Conference on Lasers and Opto-Electronics (Institution of Electrical and Radio Engineers, 1969), pp. 52–63.
  5. S. Zaske, D.-H. Lee, and C. Becher, “Green-pumped cw singly resonant optical parametric oscillator based on MgO: PPLN with frequency stabilization to an atomic resonance,” Appl. Phys. B 98, 729–735 (2010).
    [CrossRef]
  6. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
    [CrossRef]
  7. A. V. Smith, R. J. Gehr, and M. S. Bowers, “Numerical models of broad-bandwidth nanosecond optical parametric oscillators,” J. Opt. Soc. Am. B 16, 609–619 (1999).
    [CrossRef]
  8. A. V. Smith, “Bandwidth and group-velocity effects in nanosecond optical parametric amplifiers and oscillators,” J. Opt. Soc. Am. B 22, 1953–1965 (2005).
    [CrossRef]
  9. G. Arisholm, “Quantum noise initiation and macroscopic fluctuations in optical parametric oscillators,” J. Opt. Soc. Am. B 16, 117–127 (1999).
    [CrossRef]
  10. G. Arisholm, “General analysis of group velocity effects in collinear optical parametric amplifiers and generators,” Opt. Express 15, 6513–6527 (2007).
    [CrossRef] [PubMed]
  11. G. Arisholm, G. Rustad, and K. Stenersen, “Importance of pump-beam group velocity for backconversion in optical parametric oscillators,” J. Opt. Soc. Am. B 18, 1882–1890 (2001).
    [CrossRef]
  12. R. White, Y. He, B. Orr, M. Kono, and K. Baldwin, “Transition from single-mode to multimode operation of an injection-seeded pulsed optical parametric oscillator,” Opt. Express 12, 5655–5660 (2004).
    [CrossRef] [PubMed]
  13. R. T. White, Y. He, B. J. Orr, M. Kono, and K. G. H. Baldwin, “Control of frequency chirp in nanosecond-pulsed laser spectroscopy. 3. Spectrotemporal dynamics of an injection-seeded optical parametric oscillator,” J. Opt. Soc. Am. B 24, 2601–2609 (2007).
    [CrossRef]
  14. G. W. Baxter, J. G. Haub, and B. J. Orr, “Backconversion in a pulsed optical parametric oscillator: evidence from injection-seeded sidebands,” J. Opt. Soc. Am. B 14, 2723–2730 (1997).
    [CrossRef]
  15. D. J. Armstrong and A. V. Smith, “Tendency of nanosecond optical parametric oscillators to produce purely phase-modulated light,” Opt. Lett. 21, 1634–1636 (1996).
    [CrossRef] [PubMed]
  16. J. Bjorkholm, “Some effects of spatially nonuniform pumping in pulsed optical parametric oscillators,” IEEE J. Quantum Electron. 7, 109–118 (1971).
    [CrossRef]
  17. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  18. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989).
  19. G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  20. S. T. Yang, R. C. Eckardt, and R. L. Byer, “Power and spectral characteristics of continuous-wave parametric oscillators: the doubly to singly resonant transition,” J. Opt. Soc. Am. B 10, 1684–1695 (1993).
    [CrossRef]
  21. 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]
  22. R. Sowade, I. Breunig, I. Mayorga, J. Kiessling, C. Tulea, V. Dierolf, and K. Buse, “Continuous-wave optical parametric terahertz source,” Opt. Express 17, 22303–22310 (2009).
    [CrossRef]
  23. M. Lawrence, B. Willke, M. Husman, E. Gustafson, and R. Byer, “Dynamic response of a Fabry–Perot interferometer,” J. Opt. Soc. Am. B 16, 523–532 (1999).
    [CrossRef]
  24. R. Sowade, I. Breunig, J. Kiessling, and K. Buse, “Influence of the pump threshold on the single-frequency output power of singly resonant optical parametric oscillators,” Appl. Phys. B 96, 25–28 (2009).
    [CrossRef]
  25. R. O. Moore, G. Biondini, and W. L. Kath, “Self-induced thermal effects and modal competition in continuous-wave optical parametric oscillators,” J. Opt. Soc. Am. B 19, 802–811 (2002).
    [CrossRef]
  26. M. Vainio, M. Siltanen, J. Peltola, and L. Halonen, “Continuous-wave optical parametric oscillator tuned by a diffraction grating,” Opt. Express 17, 7702–7707 (2009).
    [CrossRef] [PubMed]
  27. L. Cheng, L. Cheng, J. Galperin, P. Hotsenpiller, and J. Bierlein, “Crystal growth and characterization of KTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
    [CrossRef]

2010 (1)

S. Zaske, D.-H. Lee, and C. Becher, “Green-pumped cw singly resonant optical parametric oscillator based on MgO: PPLN with frequency stabilization to an atomic resonance,” Appl. Phys. B 98, 729–735 (2010).
[CrossRef]

2009 (3)

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]

2007 (2)

2005 (1)

2004 (1)

2003 (1)

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[CrossRef]

2002 (1)

2001 (2)

1999 (3)

1997 (2)

G. W. Baxter, J. G. Haub, and B. J. Orr, “Backconversion in a pulsed optical parametric oscillator: evidence from injection-seeded sidebands,” J. Opt. Soc. Am. B 14, 2723–2730 (1997).
[CrossRef]

L. E. Myers and W. R. Bosenberg, “Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators,” IEEE J. Quantum Electron. 33, 1663–1672 (1997).
[CrossRef]

1996 (2)

1994 (1)

L. Cheng, L. Cheng, J. Galperin, P. Hotsenpiller, and J. Bierlein, “Crystal growth and characterization of KTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

1993 (1)

1989 (1)

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989).

1971 (1)

J. Bjorkholm, “Some effects of spatially nonuniform pumping in pulsed optical parametric oscillators,” IEEE J. Quantum Electron. 7, 109–118 (1971).
[CrossRef]

1969 (1)

L. B. Kreuzer, “Single and multi-mode oscillation of the singly resonant optical parametric oscillator,” in Proceedings of the Joint Conference on Lasers and Opto-Electronics (Institution of Electrical and Radio Engineers, 1969), pp. 52–63.

1968 (1)

R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064–1068 (1968).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Alexander, J. I.

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]

Arisholm, G.

Armstrong, D. J.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Baldwin, K.

Baldwin, K. G. H.

Baxter, G. W.

Becher, C.

S. Zaske, D.-H. Lee, and C. Becher, “Green-pumped cw singly resonant optical parametric oscillator based on MgO: PPLN with frequency stabilization to an atomic resonance,” Appl. Phys. B 98, 729–735 (2010).
[CrossRef]

Bierlein, J.

L. Cheng, L. Cheng, J. Galperin, P. Hotsenpiller, and J. Bierlein, “Crystal growth and characterization of KTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

Biondini, G.

Bjorkholm, J.

J. Bjorkholm, “Some effects of spatially nonuniform pumping in pulsed optical parametric oscillators,” IEEE J. Quantum Electron. 7, 109–118 (1971).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bosenberg, W. R.

L. E. Myers and W. R. Bosenberg, “Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators,” IEEE J. Quantum Electron. 33, 1663–1672 (1997).
[CrossRef]

W. R. Bosenberg, A. Drobshoff, J. I. Alexander, L. E. Myers, and R. L. Byer, “93% pump depletion, 3.5-W continuous-wave, singly resonant optical parametric oscillator,” Opt. Lett. 21, 1336–1338 (1996).
[CrossRef] [PubMed]

Bowers, M. S.

Breunig, I.

R. Sowade, I. Breunig, J. Kiessling, and K. Buse, “Influence of the pump threshold on the single-frequency output power of singly resonant optical parametric oscillators,” Appl. Phys. B 96, 25–28 (2009).
[CrossRef]

R. Sowade, I. Breunig, I. Mayorga, J. Kiessling, C. Tulea, V. Dierolf, and K. Buse, “Continuous-wave optical parametric terahertz source,” Opt. Express 17, 22303–22310 (2009).
[CrossRef]

Buse, K.

R. Sowade, I. Breunig, I. Mayorga, J. Kiessling, C. Tulea, V. Dierolf, and K. Buse, “Continuous-wave optical parametric terahertz source,” Opt. Express 17, 22303–22310 (2009).
[CrossRef]

R. Sowade, I. Breunig, J. Kiessling, and K. Buse, “Influence of the pump threshold on the single-frequency output power of singly resonant optical parametric oscillators,” Appl. Phys. B 96, 25–28 (2009).
[CrossRef]

Byer, R.

Byer, R. L.

Cerullo, G.

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[CrossRef]

Cheng, L.

L. Cheng, L. Cheng, J. Galperin, P. Hotsenpiller, and J. Bierlein, “Crystal growth and characterization of KTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

L. Cheng, L. Cheng, J. Galperin, P. Hotsenpiller, and J. Bierlein, “Crystal growth and characterization of KTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

De Silvestri, S.

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[CrossRef]

Dierolf, V.

Drobshoff, A.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Eckardt, R. C.

Galperin, J.

L. Cheng, L. Cheng, J. Galperin, P. Hotsenpiller, and J. Bierlein, “Crystal growth and characterization of KTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[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]

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]

Gehr, R. J.

Gustafson, E.

Halonen, L.

Harris, S. E.

R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064–1068 (1968).
[CrossRef]

Haub, J. G.

He, Y.

Hotsenpiller, P.

L. Cheng, L. Cheng, J. Galperin, P. Hotsenpiller, and J. Bierlein, “Crystal growth and characterization of KTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

Husman, M.

Kath, W. L.

Kiessling, J.

R. Sowade, I. Breunig, I. Mayorga, J. Kiessling, C. Tulea, V. Dierolf, and K. Buse, “Continuous-wave optical parametric terahertz source,” Opt. Express 17, 22303–22310 (2009).
[CrossRef]

R. Sowade, I. Breunig, J. Kiessling, and K. Buse, “Influence of the pump threshold on the single-frequency output power of singly resonant optical parametric oscillators,” Appl. Phys. B 96, 25–28 (2009).
[CrossRef]

Kono, M.

Kreuzer, L. B.

L. B. Kreuzer, “Single and multi-mode oscillation of the singly resonant optical parametric oscillator,” in Proceedings of the Joint Conference on Lasers and Opto-Electronics (Institution of Electrical and Radio Engineers, 1969), pp. 52–63.

Lawrence, M.

Lee, D.-H.

S. Zaske, D.-H. Lee, and C. Becher, “Green-pumped cw singly resonant optical parametric oscillator based on MgO: PPLN with frequency stabilization to an atomic resonance,” Appl. Phys. B 98, 729–735 (2010).
[CrossRef]

Mayorga, I.

Moore, R. O.

Myers, L. E.

L. E. Myers and W. R. Bosenberg, “Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators,” IEEE J. Quantum Electron. 33, 1663–1672 (1997).
[CrossRef]

W. R. Bosenberg, A. Drobshoff, J. I. Alexander, L. E. Myers, and R. L. Byer, “93% pump depletion, 3.5-W continuous-wave, singly resonant optical parametric oscillator,” Opt. Lett. 21, 1336–1338 (1996).
[CrossRef] [PubMed]

Orr, B.

Orr, B. J.

Peltola, J.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Rustad, G.

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]

Siltanen, M.

Smith, A. V.

Sowade, R.

R. Sowade, I. Breunig, J. Kiessling, and K. Buse, “Influence of the pump threshold on the single-frequency output power of singly resonant optical parametric oscillators,” Appl. Phys. B 96, 25–28 (2009).
[CrossRef]

R. Sowade, I. Breunig, I. Mayorga, J. Kiessling, C. Tulea, V. Dierolf, and K. Buse, “Continuous-wave optical parametric terahertz source,” Opt. Express 17, 22303–22310 (2009).
[CrossRef]

Stenersen, K.

Tulea, C.

Vainio, M.

White, R.

White, R. T.

Willke, B.

Yang, S. T.

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989).

Zaske, S.

S. Zaske, D.-H. Lee, and C. Becher, “Green-pumped cw singly resonant optical parametric oscillator based on MgO: PPLN with frequency stabilization to an atomic resonance,” Appl. Phys. B 98, 729–735 (2010).
[CrossRef]

Appl. Phys. B (3)

S. Zaske, D.-H. Lee, and C. Becher, “Green-pumped cw singly resonant optical parametric oscillator based on MgO: PPLN with frequency stabilization to an atomic resonance,” Appl. Phys. B 98, 729–735 (2010).
[CrossRef]

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]

R. Sowade, I. Breunig, J. Kiessling, and K. Buse, “Influence of the pump threshold on the single-frequency output power of singly resonant optical parametric oscillators,” Appl. Phys. B 96, 25–28 (2009).
[CrossRef]

IEEE J. Quantum Electron. (2)

L. E. Myers and W. R. Bosenberg, “Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators,” IEEE J. Quantum Electron. 33, 1663–1672 (1997).
[CrossRef]

J. Bjorkholm, “Some effects of spatially nonuniform pumping in pulsed optical parametric oscillators,” IEEE J. Quantum Electron. 7, 109–118 (1971).
[CrossRef]

J. Cryst. Growth (1)

L. Cheng, L. Cheng, J. Galperin, P. Hotsenpiller, and J. Bierlein, “Crystal growth and characterization of KTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

J. Opt. Soc. Am. B (9)

S. T. Yang, R. C. Eckardt, and R. L. Byer, “Power and spectral characteristics of continuous-wave parametric oscillators: the doubly to singly resonant transition,” J. Opt. Soc. Am. B 10, 1684–1695 (1993).
[CrossRef]

G. W. Baxter, J. G. Haub, and B. J. Orr, “Backconversion in a pulsed optical parametric oscillator: evidence from injection-seeded sidebands,” J. Opt. Soc. Am. B 14, 2723–2730 (1997).
[CrossRef]

G. Arisholm, “Quantum noise initiation and macroscopic fluctuations in optical parametric oscillators,” J. Opt. Soc. Am. B 16, 117–127 (1999).
[CrossRef]

M. Lawrence, B. Willke, M. Husman, E. Gustafson, and R. Byer, “Dynamic response of a Fabry–Perot interferometer,” J. Opt. Soc. Am. B 16, 523–532 (1999).
[CrossRef]

A. V. Smith, R. J. Gehr, and M. S. Bowers, “Numerical models of broad-bandwidth nanosecond optical parametric oscillators,” J. Opt. Soc. Am. B 16, 609–619 (1999).
[CrossRef]

G. Arisholm, G. Rustad, and K. Stenersen, “Importance of pump-beam group velocity for backconversion in optical parametric oscillators,” J. Opt. Soc. Am. B 18, 1882–1890 (2001).
[CrossRef]

R. O. Moore, G. Biondini, and W. L. Kath, “Self-induced thermal effects and modal competition in continuous-wave optical parametric oscillators,” J. Opt. Soc. Am. B 19, 802–811 (2002).
[CrossRef]

A. V. Smith, “Bandwidth and group-velocity effects in nanosecond optical parametric amplifiers and oscillators,” J. Opt. Soc. Am. B 22, 1953–1965 (2005).
[CrossRef]

R. T. White, Y. He, B. J. Orr, M. Kono, and K. G. H. Baldwin, “Control of frequency chirp in nanosecond-pulsed laser spectroscopy. 3. Spectrotemporal dynamics of an injection-seeded optical parametric oscillator,” J. Opt. Soc. Am. B 24, 2601–2609 (2007).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. (2)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064–1068 (1968).
[CrossRef]

Rev. Sci. Instrum. (1)

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[CrossRef]

Other (3)

L. B. Kreuzer, “Single and multi-mode oscillation of the singly resonant optical parametric oscillator,” in Proceedings of the Joint Conference on Lasers and Opto-Electronics (Institution of Electrical and Radio Engineers, 1969), pp. 52–63.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1989).

G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

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 (12)

Fig. 1
Fig. 1

Schematic of coupling between sidebands in the presence of the strong CW field solutions. The dotted red, dashed green, and solid blue arrows represent the idler, signal, and pump-carrier waves, respectively. The smaller arrows are the small sidebands, separated from their respective carrier frequencies by an arbitrary frequency Ω. Two of the six coupling processes are indicated with additional solid black arrows: the positive frequency sidebands of the idler and pump are coupled via the CW signal; and positive and negative sidebands of the idler and signal, respectively, are coupled via the CW pump.

Fig. 2
Fig. 2

Normalized net round-trip gain as a function of sideband frequency normalized to the idler-signal GVM for several pumping rates N with a 1064 nm pump and (a) 1550 nm signal for which the largest gain corresponds to AM and (b) 2450 nm signal for which the largest gain corresponds to PM.

Fig. 3
Fig. 3

Normalized net round-trip gain G s p as a function of pump-signal and idler-signal phase mismatches δ ν p s Ω L and δ ν i s Ω L , respectively, for N = ( π / 2 ) 2 . GVD is neglected, so the AM and PM eigenmodes are decoupled. (a) Gain for AM eigenmodes, (b) gain for PM eigenmodes.

Fig. 4
Fig. 4

MI threshold N t h as a function of idler-signal and pump-signal GVM ratio δ ν i s / δ ν p s for several values of the normalized signal GVD β s , calculated using Eqs. (4, 5, 9) assuming low cavity losses and negligible GVD or higher order dispersion terms at the idler or pump frequencies. Note that for negative GVM ratios the curves for these five GVD values are barely distinguishable on this scale.

Fig. 5
Fig. 5

Sideband gain for the case where AM and PM modes are coupled by GVD. The dashed lines correspond to the AM-like modes; solid lines to PM-like modes. The signal-GVD coefficient d 2 k / d ω 2 is varied for each frequency Ω such that there is a given signal-GVD phase ϕ GVD as defined in Eq. (21). (a) δ ν i s / δ ν p s = 0.25 and N = 2.5 , with unstable PM modes; (b) δ ν i s / δ ν p s = 0.25 and N = 5 , with unstable AM modes.

Fig. 6
Fig. 6

Sideband gain for the case where AM and PM modes are coupled by GVD, for different normalized signal GVD coefficients β s . The dashed lines correspond to the AM-like modes; solid lines to PM-like modes. The GVM ratio is δ ν i s / δ ν p s = 0.5 . (a) N = 5.5 , (b) N = 7.5 .

Fig. 7
Fig. 7

MI frequency Ω t h associated with the instability thresholds shown in Fig. 4, as a function of GVM ratio δ ν i s / δ ν p s for several values of the normalized signal GVD β s . GVM ratios for which N t h 10 are plotted. (a) δ ν i s / δ ν p s < 0 , (b) δ ν i s / δ ν p s > 0 . There is a discrete jump in Ω t h at δ ν i s / δ ν p s 0.63 for β s = 1 ; for other values of β s , Ω t h jumps between the β s = 0 (decoupled AM and PM) and β s = 2 (highly coupled AM and PM) curves at different GVM ratios, but is otherwise comparable.

Fig. 8
Fig. 8

AM content | η | 2 of the unstable mode at Ω = Ω t h and N = N t h as a function of GVM ratio δ ν i s / δ ν p s for several values of normalized signal GVD β s , for the modes whose properties are illustrated in Figs. 4, 7. | η | 2 is plotted for GVM ratios for which N t h 10 , as in Figs. 4, 7.

Fig. 9
Fig. 9

Effects of carrier mismatch on sideband amplification spectrum below the carrier-phase-matched instability threshold (i.e., N < N t h ), with β s = 0 . Solid lines denote G s p and dashed lines denote | a ̃ s + | 2 for the corresponding largest-gain eigenmode. (a) δ ν i s / δ ν p s = 0.5 , N = 1.1 ; (b) δ ν i s / δ ν p s = 0.5 , N = 2.5 ; (c) δ ν i s / δ ν p s = 0.5 , N = 2.5 .

Fig. 10
Fig. 10

Instability threshold for realistic SRO configurations of pump and resonant wavelength, using a 5 cm long PPLN QPM grating and R cav = 99 % . Included for comparison is an instability threshold spectrum when only GVM and GVD at the resonant wave are included. (a) 532 nm pumped, (b) 1064 nm pumped.

Fig. 11
Fig. 11

GVM ratio for some OPO configurations. Type 0 (eee) and type I (ooe) are chosen for (idler, signal, and pump) polarizations, and the GVM ratios are calculated for MgO : LiNbO 3 and KTP from Sellmeier relations [21, 27]. (a) 532 nm pumped, (b) 1064 nm pumped.

Fig. 12
Fig. 12

Instability suppression for the OPO described in the text with R cav = 99 % using etalons of increasing reflectance R e t , defined as the reflectance from a single facet of the assumed symmetric etalon and fixed free spectral range of 75 GHz.

Equations (36)

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

d E i d z = i κ i E s E p   exp ( i z = 0 z = z Δ k ( z ) d z ) ,
d E s d z = i κ s E i E p   exp ( i z = 0 z = z Δ k ( z ) d z ) ,
d E p d z = i κ p E i E s   exp ( i z = 0 z = z Δ k ( z ) d z ) ,
E j ( z ) = ω j n p n j ω p | E p 0 | A j ( z ) exp ( i z = 0 z = z Δ k ( z ) d z ) ,
d A i d z i Δ k ( z ) A i = i Γ A s A p ,
d A s d z i Δ k ( z ) A s = i Γ A i A p ,
d A p d z i Δ k ( z ) A p = i Γ A i A s ,
( sin ( γ L ) γ L ) 2 = 1 N ,
A i ( z ) i A s 0 | A s 0 | A p 0   sin ( γ z ) ,
A s ( z ) A s 0 ,
A p ( z ) A p 0   cos ( γ z ) ,
A i z i Δ k ( z ) A i + D ̂ i A i = i κ A s A p ,
A s z i Δ k ( z ) A s + D ̂ s A s = i κ A i A p ,
A p z i Δ k ( z ) A p + D ̂ p A p = i κ A i A s .
a i z + D ̂ i a i i Δ k a i = i κ [ A p ( 0 ) a s + A s ( 0 ) a p ] ,
a s z + D ̂ s a s i Δ k a s = i κ [ A p ( 0 ) a i + A i ( 0 ) a p ] ,
a p z + D ̂ p a p i Δ k a p = i κ [ A s ( 0 ) a i + A i ( 0 ) a s ] .
d v ̃ d z = M ( z ) v ̃ ,
i M ( z ) = [ K i , 0 0 κ A p ( 0 ) κ A s ( 0 ) 0 0 K i , + κ A p ( 0 ) 0 0 κ A s ( 0 ) 0 κ A p ( 0 ) K s , 0 κ A i ( 0 ) 0 κ A p ( 0 ) 0 0 K s , + 0 κ A i ( 0 ) κ A s ( 0 ) 0 κ A i ( 0 ) 0 K p , 0 0 κ A s ( 0 ) 0 κ A i ( 0 ) 0 K p , + ] .
v ̃ ( z ) = Φ ( z , z ) v ̃ ( z ) .
[ a s ( L ) , a s ( L ) + ] = Φ s [ a s ( 0 ) , a s ( 0 ) + ] + Φ p [ a p ( 0 ) , a p ( 0 ) + ] .
E ̃ ( L , ω ) = 1 2 A ̃ ( L , Ω ) exp [ i ( ϕ d ( Ω ) + ϕ env ) ] ,
ϕ cav = ϕ s + ϕ env + ϕ L = 2 m π ,
Φ r t ( Ω ) = e i ϕ d ( Ω ) R cav [ h ̃ ( Ω ) e i ϕ s 0 0 h ̃ ( Ω ) e i ϕ s ] Φ s ,
G s p = | λ Φ , r t | 2 1 1 R cav ,
v ̃ AM T = [ ( a ̃ i + a ̃ i , ) ( a ̃ s + + a ̃ s , ) ( a ̃ p + + a ̃ p , ) ] ,
v ̃ PM T = [ ( a ̃ i + + a ̃ i , ) ( a ̃ s + a ̃ s , ) ( a ̃ p + a ̃ p , ) ] .
M ( z ) = [ M AM 0 0 M PM ] ,
M AM = i [ δ ν i s Ω Γ   cos ( γ z ) γ Γ   cos ( γ z ) 0 i Γ   sin ( γ z ) γ i Γ   sin ( γ z ) δ ν p s Ω , ] ,
M PM = i [ δ ν i s Ω Γ   cos ( γ z ) γ Γ   cos ( γ z ) 0 i Γ   sin ( γ z ) γ i Γ   sin ( γ z ) δ ν p s Ω , ] ,
β s = | d 2 k d ω 2 | ω s π 2 δ ν i s 2 L 1 1 R cav ,
ϕ GVD = Ω 2 L 1 R cav | d 2 k d ω 2 | ω = ω s .
ϕ cav = k s ( ω s , T ) L ( T ) k 0 ( ω s ) L cav ( T ) + ϕ N L ( Δ k L ) ,
d ϕ cav d T = | ϕ cav T | ω s , Δ k L + | ϕ cav Δ k L | T , ω s d ( Δ k L ) d T + | ϕ cav ω s | T , Δ k L d ω s d T = 0 ,
d ( Δ k L ) d T = T ( Δ k L ) + ω s ( Δ k L ) d ω s d T = T ( Δ k L ) ω s ( Δ k L ) ω s ( ϕ cav ) 1 T ( ϕ cav ) 1 + ω s ( Δ k L ) ω s ( ϕ cav ) 1 Δ k L ( ϕ N L ) . .
| d ( Δ k L ) d T | T 0 = L 0 T ( Δ k ) + δ GVM [ T ( k s ) + ( k s k cav ) a T ] 1 δ GVM Δ k L ( ϕ N L ) ,

Metrics