Abstract

A singly resonant synchronously pumped optical parametric oscillator (SPOPO) based on the experimental setup of J. Opt. Soc. Am. B 36, 131 (2019) [CrossRef]   is studied theoretically. Stable, oscillatory, and chaotic operation modes of the SPOPO are investigated. The need of the self- and cross-phase modulation terms in the theoretical model for the explanation of the instabilities is demonstrated. The theoretical values of the wavelengths of the signal spectrum maxima are found. The evidence of chaos by the calculation of the Lyapunov exponent is provided. The possibilities to avoid the instabilities are discussed.

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References

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  1. H. M. van Driel, “Synchronously pumped optical parametric oscillators,” Appl. Phys. B 60, 411–420 (1995).
    [Crossref]
  2. K. Ivanauskienė, I. Stasevičius, M. Vengris, and V. Sirutkaitis, “Pulse-to-pulse instabilities in synchronously pumped femtosecond optical parametric oscillator,” J. Opt. Soc. Am. B 36, 131–139 (2019).
    [Crossref]
  3. L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Il Nuovo Cimento 10, 959–977 (1988).
    [Crossref]
  4. D. T. Reid, “Ultra-broadband pulse evolution in optical parametric oscillators,” Opt. Express 19, 17979–17984 (2011).
    [Crossref]
  5. P.-S. Jian, W. E. Torruellas, M. Haelterman, S. Trillo, U. Peschel, and F. Lederer, “Solitons of singly resonant optical parametric oscillators,” Opt. Lett. 24, 400–402 (1999).
    [Crossref]
  6. B. Ruffing, A. Nebel, and R. Wallenstein, “All-solid-state cw mode-locked picosecond KTiOAsO4 (KTA) optical parametric oscillator,” Appl. Phys. B 67, 537–544 (1998).
    [Crossref]
  7. E. Gaižauskas, R. Grigonis, and V. Sirutkaitis, “Self- and cross-modulation effects in a synchronously pumped optical parametric oscillator,” J. Opt. Soc. Am. B 19, 2957–2966 (2002).
    [Crossref]
  8. N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).
  9. J. Seres and J. Hebling, “Nonstationary theory of synchronously pumped femtosecond optical parametric oscillators,” J. Opt. Soc. Am. B 17, 741–750 (2000).
    [Crossref]
  10. R. W. Boyd, Nonlinear Optics (Academic, 2003).
  11. D. N. Nikogosyan, Nonlinear Optical Crystals (Springer, 2005).
  12. R. Hamerly, A. Marandi, M. Jankowski, M. M. Fejer, Y. Yamamoto, and H. Mabuchi, “Reduced models and design principles for half-harmonic generation in synchronously pumped optical parametric oscillators,” Phys. Rev. A 94, 063809 (2016).
    [Crossref]
  13. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1989).
  14. M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65, 117–134 (1993).
    [Crossref]
  15. T. Steinle, M. Floess, A. Steinmann, V. Kumar, G. Cerullo, and H. Giessen, “Stimulated Raman scattering microscopy with an all-optical modulator,” Phys. Rev. Appl. 11, 044081 (2019).
    [Crossref]

2019 (2)

K. Ivanauskienė, I. Stasevičius, M. Vengris, and V. Sirutkaitis, “Pulse-to-pulse instabilities in synchronously pumped femtosecond optical parametric oscillator,” J. Opt. Soc. Am. B 36, 131–139 (2019).
[Crossref]

T. Steinle, M. Floess, A. Steinmann, V. Kumar, G. Cerullo, and H. Giessen, “Stimulated Raman scattering microscopy with an all-optical modulator,” Phys. Rev. Appl. 11, 044081 (2019).
[Crossref]

2016 (1)

R. Hamerly, A. Marandi, M. Jankowski, M. M. Fejer, Y. Yamamoto, and H. Mabuchi, “Reduced models and design principles for half-harmonic generation in synchronously pumped optical parametric oscillators,” Phys. Rev. A 94, 063809 (2016).
[Crossref]

2011 (1)

2002 (1)

2000 (1)

1999 (1)

1998 (1)

B. Ruffing, A. Nebel, and R. Wallenstein, “All-solid-state cw mode-locked picosecond KTiOAsO4 (KTA) optical parametric oscillator,” Appl. Phys. B 67, 537–544 (1998).
[Crossref]

1995 (1)

H. M. van Driel, “Synchronously pumped optical parametric oscillators,” Appl. Phys. B 60, 411–420 (1995).
[Crossref]

1993 (1)

M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65, 117–134 (1993).
[Crossref]

1988 (1)

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Il Nuovo Cimento 10, 959–977 (1988).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1989).

Akhmediev, N.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Cerullo, G.

T. Steinle, M. Floess, A. Steinmann, V. Kumar, G. Cerullo, and H. Giessen, “Stimulated Raman scattering microscopy with an all-optical modulator,” Phys. Rev. Appl. 11, 044081 (2019).
[Crossref]

Collins, J. J.

M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65, 117–134 (1993).
[Crossref]

De Luca, C. J.

M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65, 117–134 (1993).
[Crossref]

Fabre, C.

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Il Nuovo Cimento 10, 959–977 (1988).
[Crossref]

Fejer, M. M.

R. Hamerly, A. Marandi, M. Jankowski, M. M. Fejer, Y. Yamamoto, and H. Mabuchi, “Reduced models and design principles for half-harmonic generation in synchronously pumped optical parametric oscillators,” Phys. Rev. A 94, 063809 (2016).
[Crossref]

Floess, M.

T. Steinle, M. Floess, A. Steinmann, V. Kumar, G. Cerullo, and H. Giessen, “Stimulated Raman scattering microscopy with an all-optical modulator,” Phys. Rev. Appl. 11, 044081 (2019).
[Crossref]

Gaižauskas, E.

Giacobino, E.

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Il Nuovo Cimento 10, 959–977 (1988).
[Crossref]

Giessen, H.

T. Steinle, M. Floess, A. Steinmann, V. Kumar, G. Cerullo, and H. Giessen, “Stimulated Raman scattering microscopy with an all-optical modulator,” Phys. Rev. Appl. 11, 044081 (2019).
[Crossref]

Grigonis, R.

Haelterman, M.

Hamerly, R.

R. Hamerly, A. Marandi, M. Jankowski, M. M. Fejer, Y. Yamamoto, and H. Mabuchi, “Reduced models and design principles for half-harmonic generation in synchronously pumped optical parametric oscillators,” Phys. Rev. A 94, 063809 (2016).
[Crossref]

Hebling, J.

Horowicz, R. J.

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Il Nuovo Cimento 10, 959–977 (1988).
[Crossref]

Ivanauskiene, K.

Jankowski, M.

R. Hamerly, A. Marandi, M. Jankowski, M. M. Fejer, Y. Yamamoto, and H. Mabuchi, “Reduced models and design principles for half-harmonic generation in synchronously pumped optical parametric oscillators,” Phys. Rev. A 94, 063809 (2016).
[Crossref]

Jian, P.-S.

Kumar, V.

T. Steinle, M. Floess, A. Steinmann, V. Kumar, G. Cerullo, and H. Giessen, “Stimulated Raman scattering microscopy with an all-optical modulator,” Phys. Rev. Appl. 11, 044081 (2019).
[Crossref]

Lederer, F.

Lugiato, L. A.

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Il Nuovo Cimento 10, 959–977 (1988).
[Crossref]

Mabuchi, H.

R. Hamerly, A. Marandi, M. Jankowski, M. M. Fejer, Y. Yamamoto, and H. Mabuchi, “Reduced models and design principles for half-harmonic generation in synchronously pumped optical parametric oscillators,” Phys. Rev. A 94, 063809 (2016).
[Crossref]

Marandi, A.

R. Hamerly, A. Marandi, M. Jankowski, M. M. Fejer, Y. Yamamoto, and H. Mabuchi, “Reduced models and design principles for half-harmonic generation in synchronously pumped optical parametric oscillators,” Phys. Rev. A 94, 063809 (2016).
[Crossref]

Nebel, A.

B. Ruffing, A. Nebel, and R. Wallenstein, “All-solid-state cw mode-locked picosecond KTiOAsO4 (KTA) optical parametric oscillator,” Appl. Phys. B 67, 537–544 (1998).
[Crossref]

Nikogosyan, D. N.

D. N. Nikogosyan, Nonlinear Optical Crystals (Springer, 2005).

Oldano, C.

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Il Nuovo Cimento 10, 959–977 (1988).
[Crossref]

Peschel, U.

Reid, D. T.

Rosenstein, M. T.

M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65, 117–134 (1993).
[Crossref]

Ruffing, B.

B. Ruffing, A. Nebel, and R. Wallenstein, “All-solid-state cw mode-locked picosecond KTiOAsO4 (KTA) optical parametric oscillator,” Appl. Phys. B 67, 537–544 (1998).
[Crossref]

Seres, J.

Sirutkaitis, V.

Stasevicius, I.

Steinle, T.

T. Steinle, M. Floess, A. Steinmann, V. Kumar, G. Cerullo, and H. Giessen, “Stimulated Raman scattering microscopy with an all-optical modulator,” Phys. Rev. Appl. 11, 044081 (2019).
[Crossref]

Steinmann, A.

T. Steinle, M. Floess, A. Steinmann, V. Kumar, G. Cerullo, and H. Giessen, “Stimulated Raman scattering microscopy with an all-optical modulator,” Phys. Rev. Appl. 11, 044081 (2019).
[Crossref]

Torruellas, W. E.

Trillo, S.

van Driel, H. M.

H. M. van Driel, “Synchronously pumped optical parametric oscillators,” Appl. Phys. B 60, 411–420 (1995).
[Crossref]

Vengris, M.

Wallenstein, R.

B. Ruffing, A. Nebel, and R. Wallenstein, “All-solid-state cw mode-locked picosecond KTiOAsO4 (KTA) optical parametric oscillator,” Appl. Phys. B 67, 537–544 (1998).
[Crossref]

Yamamoto, Y.

R. Hamerly, A. Marandi, M. Jankowski, M. M. Fejer, Y. Yamamoto, and H. Mabuchi, “Reduced models and design principles for half-harmonic generation in synchronously pumped optical parametric oscillators,” Phys. Rev. A 94, 063809 (2016).
[Crossref]

Appl. Phys. B (2)

B. Ruffing, A. Nebel, and R. Wallenstein, “All-solid-state cw mode-locked picosecond KTiOAsO4 (KTA) optical parametric oscillator,” Appl. Phys. B 67, 537–544 (1998).
[Crossref]

H. M. van Driel, “Synchronously pumped optical parametric oscillators,” Appl. Phys. B 60, 411–420 (1995).
[Crossref]

Il Nuovo Cimento (1)

L. A. Lugiato, C. Oldano, C. Fabre, E. Giacobino, and R. J. Horowicz, “Bistability, self-pulsing and chaos in optical parametric oscillators,” Il Nuovo Cimento 10, 959–977 (1988).
[Crossref]

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

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

R. Hamerly, A. Marandi, M. Jankowski, M. M. Fejer, Y. Yamamoto, and H. Mabuchi, “Reduced models and design principles for half-harmonic generation in synchronously pumped optical parametric oscillators,” Phys. Rev. A 94, 063809 (2016).
[Crossref]

Phys. Rev. Appl. (1)

T. Steinle, M. Floess, A. Steinmann, V. Kumar, G. Cerullo, and H. Giessen, “Stimulated Raman scattering microscopy with an all-optical modulator,” Phys. Rev. Appl. 11, 044081 (2019).
[Crossref]

Physica D (1)

M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65, 117–134 (1993).
[Crossref]

Other (4)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1989).

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).

R. W. Boyd, Nonlinear Optics (Academic, 2003).

D. N. Nikogosyan, Nonlinear Optical Crystals (Springer, 2005).

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Figures (9)

Fig. 1.
Fig. 1. Schematic depiction of the singly resonant SPOPO.
Fig. 2.
Fig. 2. Dependence of the conversion efficiency to the signal wave on the roundtrip number. Stable operation. $ {E_{30}} = 4 \,\, {\rm nJ} $, $ {G_0} = - 400\,\,{{\rm fs}^2} $, $ l = 0 $. Three different noise realizations.
Fig. 3.
Fig. 3. Dependence of the output signal spectra profiles on the time delay parameter $ l $. Stable operation. $ {E_{30}} = 4 \,\, {\rm nJ} $, $ N = 1000 $. (a) $ {G_0} = - 400\,\,{{\rm fs}^2} $ ($ {G_s} = {G_0} + {G_c} \lt 0 $) and (b) $ {G_0} = - 200\,\,{{\rm fs}^2} $ ($ {G_s} \gt 0 $). Dashed lines: $ \Omega = 0 $ and $ \Omega = {\Omega _0} $, Eq. (7).
Fig. 4.
Fig. 4. Dependence of the (a, b) signal and (c, d) pump (a, c) spectrum and (b, d) temporal profiles on the roundtrip number. The intensity is normalized to the intensity of the input pump pulse. $ z = L $, inside the resonator. $ {E_{30}} = 6 \,\, {\rm nJ} $, $ l = 0 $, $ {G_0} = - 400\,\,{{\rm fs}^2} $.
Fig. 5.
Fig. 5. Dependence of the conversion efficiency to the signal wave on the roundtrip number. $ {G_0} = - 400\,\,{{\rm fs}^2} $. (a) $ {E_{30}} = 6 \,\, {\rm nJ} $, $ l = 0 $. (b) $ {E_{30}} = 8 \,\, {\rm nJ} $, $ l = 0 $. (c) $ {E_{30}} = 10.5 \,\, {\rm nJ} $, $ l = - 3 $.
Fig. 6.
Fig. 6. Poincaré map. Crosses, stable solutions; circles, maxima of oscillations. Dashed line: solutions of Eq. (1) without self- and cross-phase modulation terms (only stable solutions). $ {G_0} = - 400\,\,{{\rm fs}^2} $, $ l = 0 $.
Fig. 7.
Fig. 7. (a) Dependence of the conversion efficiency on the roundtrip number and (b) calculation of the Lyapunov exponent of the given data set by the use of the algorithm of Ref. [14]. Lag is equal to 1 and the embedded dimension $ m $ (from bottom to upper line): 1, 5, 50, 100, 200. $ d $ is the distance between the $ j $th pair of nearest neighbors after $ i $ discrete “time” steps and brackets $ \langle \ldots \rangle $ denote the averaging over all $ j $ [14]. $ {E_{30}} = 10.5 \,\, {\rm nJ} $, $ {G_0} = - 400\,\,{{\rm fs}^2} $, $ l = 0 $.
Fig. 8.
Fig. 8. Poincaré maps. Crosses, stable solutions; circles, maxima of oscillations. Dashed line in (a): $ {G_s} = {G_0} + {G_c} = 0 $. (a) $ l = 0 $. (b) $ {G_0} = - 400\,\,{{\rm fs}^2} $, $ {E_{30}} = 10.5 \,\,{\rm nJ} $.
Fig. 9.
Fig. 9. Two-parameter bifurcation diagram at $ l = 0 $. Crosses, stable solutions; open circles, one-period oscillations; filled circles, two-, three-, and four-period oscillations; red rectangles, more than four-period oscillations and chaos.

Equations (14)

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S 1 z = i k 1 S 1 + σ 1 P ^ 1 ( 2 ) + i γ 1 P ^ 1 ( 3 ) ,
S 2 z = i k 2 S 2 + σ 2 P ^ 2 ( 2 ) + i γ 2 P ^ 2 ( 3 ) ,
S 3 z = i k 3 S 3 σ 3 P ^ 3 ( 2 ) + i γ 3 P ^ 3 ( 3 ) ,
k 1 ( ω 10 ) + k 2 ( ω 20 ) = k 3 ( ω 30 ) ,
ω 10 + ω 20 = ω 30 .
E 1 ( t , 0 ) = a 0 ξ 1 ( t ) ,
E 2 ( t , 0 ) = 0 ,
E 3 ( t , 0 ) = a 3 exp ( 2 ln ( 2 ) t 2 τ 2 ) .
S 1 ( Ω , 0 ) | N + 1 = R 1 / 2 S 1 ( Ω , L ) | N exp ( i φ ( Ω ) ) ,
S 2 ( Ω , 0 ) | N + 1 = 0 ,
S 3 ( Ω , 0 ) | N + 1 = S 3 ( Ω , 0 ) | N = 1 ,
φ ( Ω ) = δ Ω + G 0 2 Ω 2 ,
Δ φ Ω = 0 ,
Ω 0 = ( G s + s i g n ( G s ) [ | G s | 2 2 h δ ] 1 / 2 ) / h .

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