Abstract

We give numerical evidence that a singly resonant optical parametric oscillator is sensitive to noise in the single-pulse regime throughout the evolution of the fields. The noise induces pulse jitter that can be eliminated by use of either a small-amplitude injected field or an additional feedback cavity.

© 2000 Optical Society of America

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References

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  1. M. Cronin-Golomb, T. Honda, and K. Buse, eds., special feature on optical parametric devices, J. Opt. Soc. Am. B 12, 11 (1995).
  2. J. D. V. Khaydarov, J. H. Andrews, and K. D. Singer, “Pulse-compression mechanism in a synchronously pumped optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2199–2208 (1995).
    [CrossRef]
  3. L. Lefort, K. Puech, S. D. Butterworth, Y. P. Svirko, and D. C. Hanna, “Generation of femtosecond pulses from order-of-magnitude pulse compression in a synchronously pumped optical parametric oscillator based on periodically poled lithium niobate,” Opt. Lett. 24, 28–30 (1999).
    [CrossRef]
  4. G. Arisholm, “Quantum noise initiation and macroscopic fluctuations in optical parametric oscillators,” J. Opt. Soc. Am. B 16, 117–127 (1999).
    [CrossRef]
  5. P. M. W. French, “The generation of ultrashort laser-pulses,” Rep. Prog. Phys. 58, 169–262 (1995).
    [CrossRef]
  6. W. Forysiak and J. V. Moloney, “Dynamics of synchronously pumped mode-locked color-center lasers,” Phys. Rev. A 45, 3275–3288 (1992).
    [CrossRef] [PubMed]
  7. W. Forysiak and J. V. Moloney, “Mode-locked dynamics of synchronously pumped mode-locked color-center lasers—Fabry–Perot and ring geometries,” Phys. Rev. A 45, 8110–8120 (1992).
    [CrossRef] [PubMed]
  8. J. M. Catherall and G. H. C. New, “Role of spontaneous emission in the dynamics of mode locking by synchronous pumping,” IEEE J. Quantum Electron. QE-22, 1593–1599 (1986).
    [CrossRef]
  9. A. J. Scroggie, G. D’Alessandro, N. Langford, and G.-L. Oppo, “Pulse compression by slow saturable absorber action in an optical parametric oscillator,” Opt. Commun. 160, 119–124 (1999).
    [CrossRef]
  10. O. Svelto, Principles of Lasers, 3rd ed. (Plenum, New York, 1989).
  11. D. C. Hanna, L. Lefort, and K. Puech, Optoelectronics Research Centre, University of Southampton, Southampton, UK (personal communication, 1997).
  12. R. H. Hardin and F. D. Tappert, “Application of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423 (1973).
  13. C. Canuto, M. Yousuff Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1988).
  14. G. H. C. New, “Self-stabilization of synchronously mode-locked lasers,” Opt. Lett. 15, 1306–1308 (1990).
    [CrossRef] [PubMed]
  15. J. Q. Bi, W. Hodel, and H. P. Weber, “Numerical simulation of coherent photon seeding—A new technique to stabilize pumped mode-locked lasers,” Opt. Commun. 81, 408–418 (1991).
    [CrossRef]

1999 (3)

1995 (3)

M. Cronin-Golomb, T. Honda, and K. Buse, eds., special feature on optical parametric devices, J. Opt. Soc. Am. B 12, 11 (1995).

J. D. V. Khaydarov, J. H. Andrews, and K. D. Singer, “Pulse-compression mechanism in a synchronously pumped optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2199–2208 (1995).
[CrossRef]

P. M. W. French, “The generation of ultrashort laser-pulses,” Rep. Prog. Phys. 58, 169–262 (1995).
[CrossRef]

1992 (2)

W. Forysiak and J. V. Moloney, “Dynamics of synchronously pumped mode-locked color-center lasers,” Phys. Rev. A 45, 3275–3288 (1992).
[CrossRef] [PubMed]

W. Forysiak and J. V. Moloney, “Mode-locked dynamics of synchronously pumped mode-locked color-center lasers—Fabry–Perot and ring geometries,” Phys. Rev. A 45, 8110–8120 (1992).
[CrossRef] [PubMed]

1991 (1)

J. Q. Bi, W. Hodel, and H. P. Weber, “Numerical simulation of coherent photon seeding—A new technique to stabilize pumped mode-locked lasers,” Opt. Commun. 81, 408–418 (1991).
[CrossRef]

1990 (1)

1986 (1)

J. M. Catherall and G. H. C. New, “Role of spontaneous emission in the dynamics of mode locking by synchronous pumping,” IEEE J. Quantum Electron. QE-22, 1593–1599 (1986).
[CrossRef]

1973 (1)

R. H. Hardin and F. D. Tappert, “Application of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423 (1973).

Andrews, J. H.

Arisholm, G.

Bi, J. Q.

J. Q. Bi, W. Hodel, and H. P. Weber, “Numerical simulation of coherent photon seeding—A new technique to stabilize pumped mode-locked lasers,” Opt. Commun. 81, 408–418 (1991).
[CrossRef]

Buse, K.

M. Cronin-Golomb, T. Honda, and K. Buse, eds., special feature on optical parametric devices, J. Opt. Soc. Am. B 12, 11 (1995).

Butterworth, S. D.

Catherall, J. M.

J. M. Catherall and G. H. C. New, “Role of spontaneous emission in the dynamics of mode locking by synchronous pumping,” IEEE J. Quantum Electron. QE-22, 1593–1599 (1986).
[CrossRef]

Cronin-Golomb, M.

M. Cronin-Golomb, T. Honda, and K. Buse, eds., special feature on optical parametric devices, J. Opt. Soc. Am. B 12, 11 (1995).

D’Alessandro, G.

A. J. Scroggie, G. D’Alessandro, N. Langford, and G.-L. Oppo, “Pulse compression by slow saturable absorber action in an optical parametric oscillator,” Opt. Commun. 160, 119–124 (1999).
[CrossRef]

Forysiak, W.

W. Forysiak and J. V. Moloney, “Dynamics of synchronously pumped mode-locked color-center lasers,” Phys. Rev. A 45, 3275–3288 (1992).
[CrossRef] [PubMed]

W. Forysiak and J. V. Moloney, “Mode-locked dynamics of synchronously pumped mode-locked color-center lasers—Fabry–Perot and ring geometries,” Phys. Rev. A 45, 8110–8120 (1992).
[CrossRef] [PubMed]

French, P. M. W.

P. M. W. French, “The generation of ultrashort laser-pulses,” Rep. Prog. Phys. 58, 169–262 (1995).
[CrossRef]

Hanna, D. C.

Hardin, R. H.

R. H. Hardin and F. D. Tappert, “Application of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423 (1973).

Hodel, W.

J. Q. Bi, W. Hodel, and H. P. Weber, “Numerical simulation of coherent photon seeding—A new technique to stabilize pumped mode-locked lasers,” Opt. Commun. 81, 408–418 (1991).
[CrossRef]

Honda, T.

M. Cronin-Golomb, T. Honda, and K. Buse, eds., special feature on optical parametric devices, J. Opt. Soc. Am. B 12, 11 (1995).

Khaydarov, J. D. V.

Langford, N.

A. J. Scroggie, G. D’Alessandro, N. Langford, and G.-L. Oppo, “Pulse compression by slow saturable absorber action in an optical parametric oscillator,” Opt. Commun. 160, 119–124 (1999).
[CrossRef]

Lefort, L.

Moloney, J. V.

W. Forysiak and J. V. Moloney, “Mode-locked dynamics of synchronously pumped mode-locked color-center lasers—Fabry–Perot and ring geometries,” Phys. Rev. A 45, 8110–8120 (1992).
[CrossRef] [PubMed]

W. Forysiak and J. V. Moloney, “Dynamics of synchronously pumped mode-locked color-center lasers,” Phys. Rev. A 45, 3275–3288 (1992).
[CrossRef] [PubMed]

New, G. H. C.

G. H. C. New, “Self-stabilization of synchronously mode-locked lasers,” Opt. Lett. 15, 1306–1308 (1990).
[CrossRef] [PubMed]

J. M. Catherall and G. H. C. New, “Role of spontaneous emission in the dynamics of mode locking by synchronous pumping,” IEEE J. Quantum Electron. QE-22, 1593–1599 (1986).
[CrossRef]

Oppo, G.-L.

A. J. Scroggie, G. D’Alessandro, N. Langford, and G.-L. Oppo, “Pulse compression by slow saturable absorber action in an optical parametric oscillator,” Opt. Commun. 160, 119–124 (1999).
[CrossRef]

Puech, K.

Scroggie, A. J.

A. J. Scroggie, G. D’Alessandro, N. Langford, and G.-L. Oppo, “Pulse compression by slow saturable absorber action in an optical parametric oscillator,” Opt. Commun. 160, 119–124 (1999).
[CrossRef]

Singer, K. D.

Svirko, Y. P.

Tappert, F. D.

R. H. Hardin and F. D. Tappert, “Application of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423 (1973).

Weber, H. P.

J. Q. Bi, W. Hodel, and H. P. Weber, “Numerical simulation of coherent photon seeding—A new technique to stabilize pumped mode-locked lasers,” Opt. Commun. 81, 408–418 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. M. Catherall and G. H. C. New, “Role of spontaneous emission in the dynamics of mode locking by synchronous pumping,” IEEE J. Quantum Electron. QE-22, 1593–1599 (1986).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (2)

A. J. Scroggie, G. D’Alessandro, N. Langford, and G.-L. Oppo, “Pulse compression by slow saturable absorber action in an optical parametric oscillator,” Opt. Commun. 160, 119–124 (1999).
[CrossRef]

J. Q. Bi, W. Hodel, and H. P. Weber, “Numerical simulation of coherent photon seeding—A new technique to stabilize pumped mode-locked lasers,” Opt. Commun. 81, 408–418 (1991).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

W. Forysiak and J. V. Moloney, “Dynamics of synchronously pumped mode-locked color-center lasers,” Phys. Rev. A 45, 3275–3288 (1992).
[CrossRef] [PubMed]

W. Forysiak and J. V. Moloney, “Mode-locked dynamics of synchronously pumped mode-locked color-center lasers—Fabry–Perot and ring geometries,” Phys. Rev. A 45, 8110–8120 (1992).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

P. M. W. French, “The generation of ultrashort laser-pulses,” Rep. Prog. Phys. 58, 169–262 (1995).
[CrossRef]

SIAM Rev. (1)

R. H. Hardin and F. D. Tappert, “Application of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations,” SIAM Rev. 15, 423 (1973).

Other (3)

C. Canuto, M. Yousuff Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1988).

O. Svelto, Principles of Lasers, 3rd ed. (Plenum, New York, 1989).

D. C. Hanna, L. Lefort, and K. Puech, Optoelectronics Research Centre, University of Southampton, Southampton, UK (personal communication, 1997).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of a SPOPO. The shaded rectangle represents the χ(2) material. The pump field, at frequency ω1, is seeded by a periodic train of pulses with period TR. Only the field at frequency ω2 is resonated, while the field at frequency ω3 is regenerated from noise at each round trip.

Fig. 2
Fig. 2

(a) and (b) Intensity of the resonated field averaged over 200 round trips produced by use of two different FFT routines (solid and dashed curves) for (a) noise amplitude Ai=0 and (b) noise amplitude Ai=10-9 and time mismatch τ=-0.007 (equivalent to 1 ps). (c) Contour plot of the signal intensity as a function of time and round-trip number for the case illustrated in (b). (d) Intensity of the resonated field after 1000 round trips with Ai=10-9 (solid curve) and Ai=10-5 (dashed curve) for time mismatch τ=0.003 (equivalent to 0.4 ps). In Figs. 2(b) and 2(d) the two curves referred to are indistinguishable. Other parameter values: γ1=0.017, γ3=0.005, β1=-0.13×10-6, β2=-0.34×10-7, β3=0.15×10-6, ρi=θ=Δk=0, and R=0.15. In (a) the pump was 200% above threshold.

Fig. 3
Fig. 3

(a) Average width and (b) average energy of the resonated pulse as a function of the noise amplitudes, Ai=A, averaged over 200 round trips. (c) and (d) show the average signal intensity for Ai=10-6 and Ai=10-3, respectively. The contour plots in the insets show the signal intensity as a function of time and of the round-trip number. All parameters are as in Fig. 2.

Fig. 4
Fig. 4

(top) SPOPO with an additional feedback cavity. A fraction Rf of the signal intensity propagates across the feedback cavity and, at the next round trip, is superimposed on the leading edge of the signal pulse. (bottom) SPOPO with an additional seed field.

Fig. 5
Fig. 5

Profile of the signal intensity after 200 round trips in a SPOPO with a feedback cavity. In the inset the contour plot of the intensity shows that the pulse shape becomes stationary after a short transient. Parameter values: Rf=10-8, τf=0.021 (equivalent to 3 ps), Ai=10-7, and τ=-0.007 (equivalent to 1 ps). All other parameters are as in Fig. 2.

Fig. 6
Fig. 6

(a) Width and (b) energy of the resonated pulse as a function of the feedback reflectivity, Rf, averaged over 100 round trips. Ai=10-9, τf as in Fig. 5. All parameters are as in Fig. 2. Parts (c) to (f) show the average signal intensity for Log(Rf)={-12,-8,-6,-4}, respectively. Only in the case of (c) is the pulse time dependent.

Fig. 7
Fig. 7

(left) Average width and (right) average energy of the resonated pulse as a function of the noise amplitudes, Ai=A, averaged over 120 round trips in the cavity feedback scheme with the same values of Rf and τf as in Fig. 5. All parameters are as in Fig. 2.

Equations (24)

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F1(z, t)=μ12{E1(z, t)exp[i(k1z-ω1t)]+c.c.},
F2(z, t)=μ22{E2(z, t)exp[i(k2z-ω2t)]+c.c.},
F3(z, t)=μ32{E3(z, t)exp[i(k3z-ω3t)]+c.c.},
μi=cdeffLni+1ni+2ωi+1ωi+21/2,
(z+γ1t)E1=-(ρ1-iΔk)E1+iβ1ttE1-E2E3+A1S1(z, t),
zE2=-ρ2E2+iβ2ttE2+E1E3*+A2S2(z, t),
(z+γ3t)E3=-ρ3E3+iβ3ttE3+E1E2*+A3S3(z, t),
βi=-[vg(2)]22Ld2kdω2ω=ωi.
Si*(z, t)Sj(z, t)=δijδ(z-z)δ(t-t),
E1(0, t)=P(t),
E2(0, t)=exp(-iθ)RE2(1, t-Tc),
E3(0, t)=0.
E1(n)(0, t)=P(t),
E2(n)(0, t)=exp(-iθ)RE2(n-1)(1, t-Tc),
E3(n)(0, t)=0.
t(n)=t(n-1)-Tc,n>0,t(0)=t,n=0.
E1(n)[0, t(n)]=P[t(n)]=P(t-nTc)=P(t-nτ),
E2(n)[0, t(n)]=exp(-iθ)RE2(n-1)[1, t(n-1)-Tc]=exp(-iθ)RE2(n-1)[1, t(n)],
E3(n)[0, t(n)]=0,
E=-+|E2(1, t)|2dt,
tc=1E-+t|E2(1, t)|2dt,
W=-+(t2-tc2)|E2(1, t)|2dtE1/2.
E2(n)[0, t(n)]=exp(-iθ)RE2(n-1)[1, t(n)+τ]+RfE2(n-2)[1, t(n)+τf]
E2(n)[0, t(n)]=exp(-iθ)RE2(n)[1, t(n)+τ]+Es.

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