Abstract

The signals in an optical parametric oscillator (OPO) build up from quantum noise. In pulsed OPO’s this can lead to fluctuations in such macroscopic signal properties as rise time, pulse energy, frequency spectrum, and transverse profile. The strength of these fluctuations is investigated by use of simulation models that include quantum noise, multiple longitudinal modes, dispersion, birefringence, arbitrary output coupling, and transverse pump profile. The results are compared with results obtained with classical deterministic models to find out how well such models can estimate expectation values of signal observables.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. J. Werner, M. G. Raymer, M. Beck, and P. D. Drummond, “Ultrashort pulsed squeezing by optical parametric amplification,” Phys. Rev. A 52, 4202–4213 (1995).
    [CrossRef] [PubMed]
  2. S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. Quantum Electron. QE-15, 415–431 (1979).
    [CrossRef]
  3. T. Debuisschert, J. Raffy, J. P. Pocholle, and M. Papuchon, “Intracavity optical parametric oscillator: study of the dynamics in pulsed regime,” J. Opt. Soc. Am. B 13, 1569–1587 (1996).
    [CrossRef]
  4. J. A. C. Terry, Y. Cui, Y. Yang, W. Sibbett, and M. H. Dunn, “Low-threshold operation of an all-solid-state KTP optical parametric oscillator,” J. Opt. Soc. Am. B 11, 758–769 (1994).
    [CrossRef]
  5. R. L. Byer and S. E. Harris, “Power and bandwidth of spontaneous parametric emission,” Phys. Rev. 168, 1064–1068 (1968).
    [CrossRef]
  6. B. R. Mollow, “Photon correlations in the parametric frequency splitting of light,” Phys. Rev. A 8, 2684–2694 (1973).
    [CrossRef]
  7. A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, “Comparison of a numerical model with measured performance of a seeded, nanosecond KTP optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2253–2260 (1995).
    [CrossRef]
  8. T. Schröder, K. J. Boller, and R. Wallenstein, “Spectral properties and numerical modelling of a critically phase-matched nanosecond LiB3O5 optical parametric oscillator,” Appl. Phys. B 58, 425–438 (1994).
    [CrossRef]
  9. A. Fix and R. Wallenstein, “Spectral properties of pulsed nanosecond optical parametric oscillators: experimental investigation and numerical analysis,” J. Opt. Soc. Am. B 13, 2484–2497 (1996).
    [CrossRef]
  10. 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]
  11. 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]
  12. R. Graham and H. Haken, “The quantum-fluctuations of the optical parametric oscillator. I.,” Z. Phys. 210, 276–302 (1968).
    [CrossRef]
  13. R. Graham, “Photon statistics of the optical parametric oscillator including the threshold region,” Z. Phys. 211, 469–482 (1968).
    [CrossRef]
  14. A. S. Lane, M. D. Reid, and D. F. Walls, “Quantum analysis of intensity fluctuations in the nondegenerate parametric oscillator,” Phys. Rev. A 38, 788–799 (1988).
    [CrossRef] [PubMed]
  15. S. Reynaud and A. Heidmann, “A semiclassical linear input output transformation for quantum fluctuations,” Opt. Commun. 71, 209–214 (1989).
    [CrossRef]
  16. L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
    [CrossRef] [PubMed]
  17. B. S. Abbott and S. Prasad, “Quantum noise and squeezing in an optical parametric oscillator with arbitrary output-mirror coupling,” Phys. Rev. A 45, 5039–5051 (1992).
    [CrossRef] [PubMed]
  18. L. A. Lugiato, A. Gatti, H. Ritsch, I. Marzoli, and G. L. Oppo, “Quantum images in nonlinear optics,” J. Mod. Opt. 44, 1899–1915 (1997).
    [CrossRef]
  19. I. A. Walmsley and M. G. Raymer, “Experimental study of the macroscopic quantum fluctuations of partially coherent Raman scattering,” Phys. Rev. A 33, 382–390 (1986).
    [CrossRef] [PubMed]
  20. S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam-pointing fluctuations in a gain-guided Raman amplifier,” Phys. Rev. A 45, 2031–2043 (1992).
    [CrossRef] [PubMed]
  21. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 9.
  22. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995), Chap. 2.
  23. D. T. Reid, M. Ebrahimzadeh, and W. Sibbett, “Design criteria and comparison of femtosecond optical parametric oscillators based on KTiOPO4 and RbTiOAsO4,” J. Opt. Soc. Am. B 12, 2168–2179 (1995).
    [CrossRef]
  24. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Heidelberg, 1991), Chap. 2.8.
  25. G. Arisholm, “General numerical methods for simulating second order nonlinear interactions in birefringent media,” J. Opt. Soc. Am. B 14, 2543–2549 (1997).
    [CrossRef]
  26. E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), Chap. 15.7.
  27. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 12.12.
  28. P. W. Milonni, The Quantum Vacuum (Academic, San Diego, Calif., 1994), Chap. 4.
  29. R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? A theory of single-quasimode laser operation,” Phys. Rev. A 7, 1788–1797 (1973).
    [CrossRef]
  30. Ref. 28, Chap. 8.12.
  31. R. G. Smith, “A study of factors affecting the performance of a continuously pumped doubly resonant optical parametric oscillator,” IEEE J. Quantum Electron. QE-9, 530–541 (1973).
    [CrossRef]

1997 (3)

1996 (3)

1995 (3)

1994 (2)

J. A. C. Terry, Y. Cui, Y. Yang, W. Sibbett, and M. H. Dunn, “Low-threshold operation of an all-solid-state KTP optical parametric oscillator,” J. Opt. Soc. Am. B 11, 758–769 (1994).
[CrossRef]

T. Schröder, K. J. Boller, and R. Wallenstein, “Spectral properties and numerical modelling of a critically phase-matched nanosecond LiB3O5 optical parametric oscillator,” Appl. Phys. B 58, 425–438 (1994).
[CrossRef]

1992 (2)

B. S. Abbott and S. Prasad, “Quantum noise and squeezing in an optical parametric oscillator with arbitrary output-mirror coupling,” Phys. Rev. A 45, 5039–5051 (1992).
[CrossRef] [PubMed]

S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam-pointing fluctuations in a gain-guided Raman amplifier,” Phys. Rev. A 45, 2031–2043 (1992).
[CrossRef] [PubMed]

1989 (1)

S. Reynaud and A. Heidmann, “A semiclassical linear input output transformation for quantum fluctuations,” Opt. Commun. 71, 209–214 (1989).
[CrossRef]

1988 (1)

A. S. Lane, M. D. Reid, and D. F. Walls, “Quantum analysis of intensity fluctuations in the nondegenerate parametric oscillator,” Phys. Rev. A 38, 788–799 (1988).
[CrossRef] [PubMed]

1986 (2)

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

I. A. Walmsley and M. G. Raymer, “Experimental study of the macroscopic quantum fluctuations of partially coherent Raman scattering,” Phys. Rev. A 33, 382–390 (1986).
[CrossRef] [PubMed]

1979 (1)

S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. Quantum Electron. QE-15, 415–431 (1979).
[CrossRef]

1973 (3)

B. R. Mollow, “Photon correlations in the parametric frequency splitting of light,” Phys. Rev. A 8, 2684–2694 (1973).
[CrossRef]

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? A theory of single-quasimode laser operation,” Phys. Rev. A 7, 1788–1797 (1973).
[CrossRef]

R. G. Smith, “A study of factors affecting the performance of a continuously pumped doubly resonant optical parametric oscillator,” IEEE J. Quantum Electron. QE-9, 530–541 (1973).
[CrossRef]

1968 (3)

R. Graham and H. Haken, “The quantum-fluctuations of the optical parametric oscillator. I.,” Z. Phys. 210, 276–302 (1968).
[CrossRef]

R. Graham, “Photon statistics of the optical parametric oscillator including the threshold region,” Z. Phys. 211, 469–482 (1968).
[CrossRef]

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

Abbott, B. S.

B. S. Abbott and S. Prasad, “Quantum noise and squeezing in an optical parametric oscillator with arbitrary output-mirror coupling,” Phys. Rev. A 45, 5039–5051 (1992).
[CrossRef] [PubMed]

Alford, W. J.

Arisholm, G.

Armstrong, D. J.

Baxter, G. W.

Beck, M.

M. J. Werner, M. G. Raymer, M. Beck, and P. D. Drummond, “Ultrashort pulsed squeezing by optical parametric amplification,” Phys. Rev. A 52, 4202–4213 (1995).
[CrossRef] [PubMed]

Boller, K. J.

T. Schröder, K. J. Boller, and R. Wallenstein, “Spectral properties and numerical modelling of a critically phase-matched nanosecond LiB3O5 optical parametric oscillator,” Appl. Phys. B 58, 425–438 (1994).
[CrossRef]

Bowers, M. S.

Brosnan, S. J.

S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. Quantum Electron. QE-15, 415–431 (1979).
[CrossRef]

Byer, R. L.

S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. Quantum Electron. QE-15, 415–431 (1979).
[CrossRef]

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

Cui, Y.

Debuisschert, T.

Drummond, P. D.

M. J. Werner, M. G. Raymer, M. Beck, and P. D. Drummond, “Ultrashort pulsed squeezing by optical parametric amplification,” Phys. Rev. A 52, 4202–4213 (1995).
[CrossRef] [PubMed]

Dunn, M. H.

Ebrahimzadeh, M.

Fix, A.

Gatti, A.

L. A. Lugiato, A. Gatti, H. Ritsch, I. Marzoli, and G. L. Oppo, “Quantum images in nonlinear optics,” J. Mod. Opt. 44, 1899–1915 (1997).
[CrossRef]

Graham, R.

R. Graham, “Photon statistics of the optical parametric oscillator including the threshold region,” Z. Phys. 211, 469–482 (1968).
[CrossRef]

R. Graham and H. Haken, “The quantum-fluctuations of the optical parametric oscillator. I.,” Z. Phys. 210, 276–302 (1968).
[CrossRef]

Haken, H.

R. Graham and H. Haken, “The quantum-fluctuations of the optical parametric oscillator. I.,” Z. Phys. 210, 276–302 (1968).
[CrossRef]

Hall, J. L.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

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.

Heidmann, A.

S. Reynaud and A. Heidmann, “A semiclassical linear input output transformation for quantum fluctuations,” Opt. Commun. 71, 209–214 (1989).
[CrossRef]

Kimble, H. J.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Kuo, S. J.

S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam-pointing fluctuations in a gain-guided Raman amplifier,” Phys. Rev. A 45, 2031–2043 (1992).
[CrossRef] [PubMed]

Lamb, W. E.

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? A theory of single-quasimode laser operation,” Phys. Rev. A 7, 1788–1797 (1973).
[CrossRef]

Lane, A. S.

A. S. Lane, M. D. Reid, and D. F. Walls, “Quantum analysis of intensity fluctuations in the nondegenerate parametric oscillator,” Phys. Rev. A 38, 788–799 (1988).
[CrossRef] [PubMed]

Lang, R.

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? A theory of single-quasimode laser operation,” Phys. Rev. A 7, 1788–1797 (1973).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, A. Gatti, H. Ritsch, I. Marzoli, and G. L. Oppo, “Quantum images in nonlinear optics,” J. Mod. Opt. 44, 1899–1915 (1997).
[CrossRef]

Marzoli, I.

L. A. Lugiato, A. Gatti, H. Ritsch, I. Marzoli, and G. L. Oppo, “Quantum images in nonlinear optics,” J. Mod. Opt. 44, 1899–1915 (1997).
[CrossRef]

Mollow, B. R.

B. R. Mollow, “Photon correlations in the parametric frequency splitting of light,” Phys. Rev. A 8, 2684–2694 (1973).
[CrossRef]

Oppo, G. L.

L. A. Lugiato, A. Gatti, H. Ritsch, I. Marzoli, and G. L. Oppo, “Quantum images in nonlinear optics,” J. Mod. Opt. 44, 1899–1915 (1997).
[CrossRef]

Orr, B. J.

Papuchon, M.

Pocholle, J. P.

Prasad, S.

B. S. Abbott and S. Prasad, “Quantum noise and squeezing in an optical parametric oscillator with arbitrary output-mirror coupling,” Phys. Rev. A 45, 5039–5051 (1992).
[CrossRef] [PubMed]

Raffy, J.

Raymer, M. G.

M. J. Werner, M. G. Raymer, M. Beck, and P. D. Drummond, “Ultrashort pulsed squeezing by optical parametric amplification,” Phys. Rev. A 52, 4202–4213 (1995).
[CrossRef] [PubMed]

S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam-pointing fluctuations in a gain-guided Raman amplifier,” Phys. Rev. A 45, 2031–2043 (1992).
[CrossRef] [PubMed]

I. A. Walmsley and M. G. Raymer, “Experimental study of the macroscopic quantum fluctuations of partially coherent Raman scattering,” Phys. Rev. A 33, 382–390 (1986).
[CrossRef] [PubMed]

Raymond, T. D.

Reid, D. T.

Reid, M. D.

A. S. Lane, M. D. Reid, and D. F. Walls, “Quantum analysis of intensity fluctuations in the nondegenerate parametric oscillator,” Phys. Rev. A 38, 788–799 (1988).
[CrossRef] [PubMed]

Reynaud, S.

S. Reynaud and A. Heidmann, “A semiclassical linear input output transformation for quantum fluctuations,” Opt. Commun. 71, 209–214 (1989).
[CrossRef]

Ritsch, H.

L. A. Lugiato, A. Gatti, H. Ritsch, I. Marzoli, and G. L. Oppo, “Quantum images in nonlinear optics,” J. Mod. Opt. 44, 1899–1915 (1997).
[CrossRef]

Schröder, T.

T. Schröder, K. J. Boller, and R. Wallenstein, “Spectral properties and numerical modelling of a critically phase-matched nanosecond LiB3O5 optical parametric oscillator,” Appl. Phys. B 58, 425–438 (1994).
[CrossRef]

Scully, M. O.

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? A theory of single-quasimode laser operation,” Phys. Rev. A 7, 1788–1797 (1973).
[CrossRef]

Sibbett, W.

Smith, A. V.

Smith, R. G.

R. G. Smith, “A study of factors affecting the performance of a continuously pumped doubly resonant optical parametric oscillator,” IEEE J. Quantum Electron. QE-9, 530–541 (1973).
[CrossRef]

Smithey, D. T.

S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam-pointing fluctuations in a gain-guided Raman amplifier,” Phys. Rev. A 45, 2031–2043 (1992).
[CrossRef] [PubMed]

Terry, J. A. C.

Wallenstein, R.

A. Fix and R. Wallenstein, “Spectral properties of pulsed nanosecond optical parametric oscillators: experimental investigation and numerical analysis,” J. Opt. Soc. Am. B 13, 2484–2497 (1996).
[CrossRef]

T. Schröder, K. J. Boller, and R. Wallenstein, “Spectral properties and numerical modelling of a critically phase-matched nanosecond LiB3O5 optical parametric oscillator,” Appl. Phys. B 58, 425–438 (1994).
[CrossRef]

Walls, D. F.

A. S. Lane, M. D. Reid, and D. F. Walls, “Quantum analysis of intensity fluctuations in the nondegenerate parametric oscillator,” Phys. Rev. A 38, 788–799 (1988).
[CrossRef] [PubMed]

Walmsley, I. A.

I. A. Walmsley and M. G. Raymer, “Experimental study of the macroscopic quantum fluctuations of partially coherent Raman scattering,” Phys. Rev. A 33, 382–390 (1986).
[CrossRef] [PubMed]

Werner, M. J.

M. J. Werner, M. G. Raymer, M. Beck, and P. D. Drummond, “Ultrashort pulsed squeezing by optical parametric amplification,” Phys. Rev. A 52, 4202–4213 (1995).
[CrossRef] [PubMed]

Wu, H.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Wu, L. A.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Yang, Y.

Appl. Phys. B (1)

T. Schröder, K. J. Boller, and R. Wallenstein, “Spectral properties and numerical modelling of a critically phase-matched nanosecond LiB3O5 optical parametric oscillator,” Appl. Phys. B 58, 425–438 (1994).
[CrossRef]

IEEE J. Quantum Electron. (2)

S. J. Brosnan and R. L. Byer, “Optical parametric oscillator threshold and linewidth studies,” IEEE J. Quantum Electron. QE-15, 415–431 (1979).
[CrossRef]

R. G. Smith, “A study of factors affecting the performance of a continuously pumped doubly resonant optical parametric oscillator,” IEEE J. Quantum Electron. QE-9, 530–541 (1973).
[CrossRef]

J. Mod. Opt. (1)

L. A. Lugiato, A. Gatti, H. Ritsch, I. Marzoli, and G. L. Oppo, “Quantum images in nonlinear optics,” J. Mod. Opt. 44, 1899–1915 (1997).
[CrossRef]

J. Opt. Soc. Am. B (7)

Opt. Commun. (1)

S. Reynaud and A. Heidmann, “A semiclassical linear input output transformation for quantum fluctuations,” Opt. Commun. 71, 209–214 (1989).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

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

Phys. Rev. A (7)

B. R. Mollow, “Photon correlations in the parametric frequency splitting of light,” Phys. Rev. A 8, 2684–2694 (1973).
[CrossRef]

M. J. Werner, M. G. Raymer, M. Beck, and P. D. Drummond, “Ultrashort pulsed squeezing by optical parametric amplification,” Phys. Rev. A 52, 4202–4213 (1995).
[CrossRef] [PubMed]

I. A. Walmsley and M. G. Raymer, “Experimental study of the macroscopic quantum fluctuations of partially coherent Raman scattering,” Phys. Rev. A 33, 382–390 (1986).
[CrossRef] [PubMed]

S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam-pointing fluctuations in a gain-guided Raman amplifier,” Phys. Rev. A 45, 2031–2043 (1992).
[CrossRef] [PubMed]

A. S. Lane, M. D. Reid, and D. F. Walls, “Quantum analysis of intensity fluctuations in the nondegenerate parametric oscillator,” Phys. Rev. A 38, 788–799 (1988).
[CrossRef] [PubMed]

B. S. Abbott and S. Prasad, “Quantum noise and squeezing in an optical parametric oscillator with arbitrary output-mirror coupling,” Phys. Rev. A 45, 5039–5051 (1992).
[CrossRef] [PubMed]

R. Lang, M. O. Scully, and W. E. Lamb, “Why is the laser line so narrow? A theory of single-quasimode laser operation,” Phys. Rev. A 7, 1788–1797 (1973).
[CrossRef]

Phys. Rev. Lett. (1)

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Z. Phys. (2)

R. Graham and H. Haken, “The quantum-fluctuations of the optical parametric oscillator. I.,” Z. Phys. 210, 276–302 (1968).
[CrossRef]

R. Graham, “Photon statistics of the optical parametric oscillator including the threshold region,” Z. Phys. 211, 469–482 (1968).
[CrossRef]

Other (7)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 9.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995), Chap. 2.

Ref. 28, Chap. 8.12.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Heidelberg, 1991), Chap. 2.8.

E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), Chap. 15.7.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 12.12.

P. W. Milonni, The Quantum Vacuum (Academic, San Diego, Calif., 1994), Chap. 4.

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 (11)

Fig. 1
Fig. 1

OPO model. The plane mirrors are placed directly at the ends of the nonlinear crystal. The reflectance (R) for each frequency is indicated below the mirrors. The different far-field signal profiles indicate fluctuations in beam shape and pointing direction. HR, highly reflecting; HT, highly transmitting.

Fig. 2
Fig. 2

Temporal walk-off in a long-pulsed OPO. (a) The real signal. The vertical lines indicate a round-trip time. (b) The periodic signal corresponding to a Fourier representation with the transform taken over the round-trip time shown in (a). (c) Signal (a) after temporal walk-off. (d) Periodic signal (b) after temporal walk-off. The result is equivalent to a cyclic shift within the round-trip interval.

Fig. 3
Fig. 3

(a) Conceptual two-stage model. The quantum stage yields an operator expression, which is used for generating random initial amplitudes for the classical stage. (b) Instead of the operator expression’s being computed explicitly, random amplitudes with the same joint distribution are generated in a semiclassical model with random noise inputs. The computer code for the two stages can be the same, but noise is not added in the second stage.

Fig. 4
Fig. 4

Pump pulse envelope. One round trip time is ≈0.24 ns. The envelope is asymmetric Gaussian with 10-ns rise time and 20-ns fall time, measured between the exp(-2) points.

Fig. 5
Fig. 5

Results from 3D simulations with a super-Gaussian pump beam with beam waist w0=2 mm and energy 106 mJ. (a) Near-field pump beam. (b), (c) Near-field signal fluence in two example pulses. (d) Mean (over ∼10 samples) of far-field signal fluence. (e), (f ) Far-field signal fluence in the same two example pulses as in (b) and (c), respectively.

Fig. 6
Fig. 6

Signal spectra from 3D simulations with noncritical phase matching. (a) Gaussian pump beam with w0=0.5 mm and peak fluence 1.2 J/cm2. Solid curve, mean spectrum; dashed and dotted curves, two individual examples. (b) Super-Gaussian pump beam with w0=2 mm and peak fluence 1.2 J/cm2. Conditions as in (a). (c) Spectra with peak pump fluence 1.2 J/cm2. Solid curve, Gaussian w0=0.5 mm pump beam; dashed curve, super-Gaussian w0=2 mm pump beam; dotted curve, plane-wave model. The three spectra are normalized to have equal areas. (d) Spectra with peak pump fluence 2.4 J/cm2. Conditions and normalization as in (c).

Fig. 7
Fig. 7

Spectra from 3D simulation with critical phase matching, 30-mrad walk-off for the idler, and a Gaussian pump beam with w0=0.5 mm and 9.4-mJ energy. For clarity, only the mean (solid curve) and a single example (dotted curve) are shown. This single example indicates large pulse-to-pulse fluctuations, and inspection of other examples confirms this.

Fig. 8
Fig. 8

Results from plane-wave (one-dimensional) simulations with a single-frequency pump. In parts (b)–(f ) the pump fluence is 1.5Φ0, where Φ0 is just above threshold fluence (see text). (a) Distribution of signal fluence for four different pump fluences, which are shown next to the traces. (b) Mean signal spectrum (solid curve) and two individual examples (dashed and dotted curves). (c) Mean signal intensity (solid curve) and two individual examples (dashed and dotted curves). The intensity was averaged over each round trip, so the rapid variation within the round trips cannot be seen. The two vertical lines indicate the positions of the round trips seen in detail in (e) and (f ). (d) Mean signals computed with exact temporal walk-off (solid curve) and cyclic temporal walk-off (dotted curve). (e), (f ) Mean signal (solid curves) and example signals (dotted and dashed curves) in round trips 65 and 100, computed with exact temporal walk-off. In round trip 65 the signals have not yet been smoothed by pump depletion.

Fig. 9
Fig. 9

Results from plane-wave simulations with a multimode pump (solid curves) compared with simulations with a single-mode pump (dotted curves). (a) Distribution of signal fluence for four different pump fluences, which are shown above the traces. (b) Mean signal spectra for pump fluence 1.5Φ0. (c) Mean signal spectra for pump fluence 3Φ0.

Fig. 10
Fig. 10

Results from plane-wave simulations with idler reflection, single-mode pump, and pump fluence 1.5Φ0. (a) Distribution of signal fluence for R1=0.01 and different relative phases (dashed, dotted, and dashed–dotted curves) and R1=0 (solid curve). (b) Mean signal spectra for R1=0.01 (dashed, dotted, and dashed–dotted curves) and R1=0 (solid curve). Conditions as in (a). (c) Distribution of signal fluences for R1=0.001 and R1=0. Conditions as in (a). (d) Mean signal spectra for R1=0.001 and R1=0. Conditions as in (a).

Fig. 11
Fig. 11

Plane-wave simulations with noise initiation, compared with a deterministic classical model. (a) Distribution of signal fluence [the same as in Fig. 8(a)]. The fluence from the classical model is indicated by dotted vertical lines. (b) Round-trip average signal intensity for the model with noise initiation (solid curves) and the classical model (dotted curves) for pulses with pump fluences of Φ0, 1.5Φ0, 2Φ0, and 3Φ0.

Tables (3)

Tables Icon

Table 1 Parameters Used in the Examples

Tables Icon

Table 2 Results from 3D Simulationsa

Tables Icon

Table 3 Relative Standard Deviation (σS) of Signal Fluence in the Plane-Wave Model with a Single-Mode (SLM) or Multimode (MLM) pump

Equations (14)

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

dΔkdω1=-dk1dω1+dk2dω2=ng,2-ng,1c=tw/L,
e(z, t) z+1vge(z, t) t+iβ22e(z, t) t2=0,
e¯(z, ω)=-e(z, t)exp(iω t)d t,
e¯(z, ω)z=iδ k(ω)e¯(z, ω),
δ k(ω)=ω/vg+βω2/2.
e3(z, t) z+1vg,3e3(z, t) t
=exp(-iΔkz)iμ0ω3c2n3p3(z, t),
p3(z, t)=20χeff e1(z, t)e2(z, t),
 e¯3(z, ω) z=iδ k3(ω)e¯3(z, ω)+iω3n3cχeff exp(-iΔkz)F (e1e2),
F(e1e2)=12π-e1(z, t)e2(z, t)exp(iω t)d t.
kz(j, kx, ky)=[k2(j, kx, ky)-kx2-ky2]1/2,
|k|=k(j, kx, ky)=2πn(j, kx, ky, kz)/λ,
 e¯3(z, ω, kx, ky) z=iδ k3,z(ω, kx, ky)e¯3+iω3n3cχeff exp(-iΔkz)F (e1e2),
eˆout=r1eˆin+t1eˆnoise,

Metrics