## Growth dynamics of noise-sustained structures in nonlinear optical resonators

Optics Express, Vol. 3, Issue 2, pp. 63-70 (1998)

http://dx.doi.org/10.1364/OE.3.000063

Acrobat PDF (496 KB)

### Abstract

The existence of macroscopic noise-sustained structures in nonlinear optics is theoretically predicted and numerically observed, in the regime of convective instability. The advection-like term, necessary to turn the instability to convective for the parameter region where advection overwhelms the growth, can stem from pump beam tilting or birefringence induced walk-off. The growth dynamics of both noise-sustained and deterministic patterns is exemplified by means of movies. This allows to observe the process of formation of these structures and to confirm the analytical predictions. The amplification of quantum noise by several orders of magnitude is predicted. The qualitative analysis of the near- and far-field is given. It suffices to distinguish noise-sustained from deterministic structures; quantitative informations can be obtained in terms of the statistical properties of the spectra.

© Optical Society of America

## 1. Introduction

2. A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A **56**, 877 (1997). [CrossRef]

3. A. Gatti, L. A. Lugiato, G-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” opt. Express **1**, 21 (1997). [CrossRef] [PubMed]

4. A. Gatti and L. A. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A **52**, 1675 (1995). [CrossRef] [PubMed]

5. M. Hoyuelos, P. Colet, and M. San Miguel, “Fluctuations and correlations in polarization patterns of a Kerr medium,” to be published, Phys. Rev. E (1998). [CrossRef]

6. R. J. Deissler,“Noise-sustained structure, intermittency, and the Ginzburg-Landau equation,” J. Stat. Phys. **40**376 (1985). [CrossRef]

7. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. **79**, 3363 (1997). [CrossRef]

7. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. **79**, 3363 (1997). [CrossRef]

9. W. J. Firth, A. G. Scroggie, G. S. McDonald, and L. A. Lugiato, “Hexagonal patterns in optical bistability,” Phys. Rev. A **46**, R3609 (1992). [CrossRef] [PubMed]

10. L. A. Lugiato and F. Castelli, “Quantum noise reduction in a spatial dissipative structure,” Phys. Rev. Lett. **68**, 3284 (1992). [CrossRef] [PubMed]

11. K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. **91**, 82 (1992). [CrossRef]

12. G-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A **49**, 2028 (1994). [CrossRef] [PubMed]

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

## 2. Definitions and linear stability analysis

6. R. J. Deissler,“Noise-sustained structure, intermittency, and the Ginzburg-Landau equation,” J. Stat. Phys. **40**376 (1985). [CrossRef]

7. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. **79**, 3363 (1997). [CrossRef]

*defined by*

_{s}*λ*(

*,*

_{s}*F*)] in the complex vector space.

_{a}## 3. Kerr resonators

15. M. Haelterman and G. Vitrant, “Drift instability and spatiotemporal dissipative structures in a nonlinear Fabry-Perot resonator under oblique incidence,” J. Opt. Soc. Am. B **9**, 1563 (1992). [CrossRef]

16. G. Grynberg, “Drift instability and light-induced spin waves in an alkali vapor with feedback mirror,” Opt. Commun. **109**, 483 (1994). [CrossRef]

17. A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. **29**, 209 (1995). [CrossRef]

*A*(

*x,t*) is [7

**79**, 3363 (1997). [CrossRef]

15. M. Haelterman and G. Vitrant, “Drift instability and spatiotemporal dissipative structures in a nonlinear Fabry-Perot resonator under oblique incidence,” J. Opt. Soc. Am. B **9**, 1563 (1992). [CrossRef]

*α*

_{0}represents the tilt angle,

*η*the sign of the nonlinearity, Δ the cavity detuning and

*E*

_{0}the pump. Diffraction is represented by the first term on the r.h.s. and mirror losses by the first in the squared brackets (exact definitions can be found in [7

**79**, 3363 (1997). [CrossRef]

*ξ*(

*x,t*), Gaussian, with zero-mean and correlation ⟨

*ξ*(

*x,t*),

*ξ*

^{*}(

*x′*,

*t′*)⟩ = 2

*δ*(

*x*-

*x′*)

*δ*(

*t*-

*t′*), which is a standard semiclassical model of noise. For the linearized version of the Langevin equations of the optical parametric oscillator a similar term describes quantum noise in the Wigner representation, as considered in [2

2. A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A **56**, 877 (1997). [CrossRef]

*α*

_{0}= 0 and linearizing around a steady-state; in our case it can also account for thermal and input field fluctuations.

*A*

_{0}, solution of

*A*

_{0}[1 +

*iη*(Δ - |

*A*

_{0}|

^{2})] =

*E*

_{0}. For the same parameters we have integrated the equation for a pump amplitude slightly above this threshold and slightly below, in the regime of convective instability.

*A*) and far-field (the space Fourier transform of

*A*) of eq. (3) confirm that different, unstable regimes actually exist. In movie 1 the near field intensity time evolution (left side) can be observed, for the pump amplitude above the threshold of absolute instability.

*∈*= 0) because it is not necessary to generate the pattern. After a certain transitory a drifting structure is generated. In spite of the drift, the pattern tends to invade all the flat top region where the pump is above the threshold. The evolution of the far-field (right) shows that after the transitory, well defined harmonics are generated (due to the multiple wave mixing). Their line-width is scarcely influenced by the presence of noise as demonstrated in [7

**79**, 3363 (1997). [CrossRef]

^{-5}) and a pattern forms again. However, note that the structure, even for long times does not invade all the system but rather gathers its saturated value at a random spatial position. This is due to the fact that noise needs to drift for enough time in order to be amplified. When the noise source is turned off the pattern (after the delay due to the drifting) eventually disappears. In the far-field, the noticeable broadening of the spectral lines with respect to the previous case confirms the different, noisy, nature of the pattern observed.

## 4. Parametric oscillators

*A*

_{0}(

*x, y, t*) at frequency 2

*ω*

_{0}) and the signal (

*A*

_{1}(

*x, y, t*) at frequency

*ω*

_{0}) evolution is described by the following set of coupled equations [11

11. K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. **91**, 82 (1992). [CrossRef]

12. G-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A **49**, 2028 (1994). [CrossRef] [PubMed]

19. P. D. Drummond, K. J. Mc Neil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation: semiclassical theory,” Opt. Acta **27**, 321 (1980). [CrossRef]

*A*

_{0}=

*E*

_{0}/(1 +

*i*Δ

_{0}) ,

*A*

_{1}= 0. It turns out that it can become unstable along the signal component of the eigenvector,

*A*

_{1}, and thus it is necessary to consider only one linearized equation. We can calculate the predicted absolute instability thresholds through conditions (1,2) with

*λ*determined from eqs. (4). In summary, the stability diagram for the OPO is presented in figure 1 as a function of the signal detuning Δ

_{1}.

*q*= 0, i.e. is parallel to the x axis. This stems from the breaking of the rotational system due to the walk-off. The walk-off does not affect the growth rate but rather the spreading velocity of the perturbation. The first mode to become absolutely unstable is that which balances the advection with spreading [14].

_{x}*A*

_{1}is shown in movie 3, for the absolutely unstable, and in movie 4 for the convectively unstable regime. The pump

*A*

_{0}was a super-Gaussian beam and we show in the movie the central region of space where the pump was flat.

*q*= 0), in the convective regime two broadened arcs of the ring remain visible even for very large times (see figure 2b).

_{x}## 5. Conclusions

## References and links

1. | L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, and H. Wiedemann, “Quantum aspects of nonlinear optical patterns,” |

2. | A. Gatti, H. Wiedemann, L.A. Lugiato, I. Marzoli, G- L. Oppo, and S.M. Barnett, “A Langevin approach to quantum fluctuations and optical patterns formation,” Phys. Rev. A |

3. | A. Gatti, L. A. Lugiato, G-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” opt. Express |

4. | A. Gatti and L. A. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A |

5. | M. Hoyuelos, P. Colet, and M. San Miguel, “Fluctuations and correlations in polarization patterns of a Kerr medium,” to be published, Phys. Rev. E (1998). [CrossRef] |

6. | R. J. Deissler,“Noise-sustained structure, intermittency, and the Ginzburg-Landau equation,” J. Stat. Phys. |

7. | M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Convective noise-sustained structures in nonlinear optics,” Phys. Rev. Lett. |

8. | L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. A |

9. | W. J. Firth, A. G. Scroggie, G. S. McDonald, and L. A. Lugiato, “Hexagonal patterns in optical bistability,” Phys. Rev. A |

10. | L. A. Lugiato and F. Castelli, “Quantum noise reduction in a spatial dissipative structure,” Phys. Rev. Lett. |

11. | K. Staliunas, “Optical vortices during three-wave nonlinear coupling,” Opt. Commun. |

12. | G-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A |

13. | L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. |

14. | M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, “Two-dimensional noise-sustained structures in optical parametric oscillators,” to be published Phys. Rev. E (1998). |

15. | M. Haelterman and G. Vitrant, “Drift instability and spatiotemporal dissipative structures in a nonlinear Fabry-Perot resonator under oblique incidence,” J. Opt. Soc. Am. B |

16. | G. Grynberg, “Drift instability and light-induced spin waves in an alkali vapor with feedback mirror,” Opt. Commun. |

17. | A. Petrossian, L. Dambly, and G. Grynberg, “Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror,” Europhys. Lett. |

18. | N. Bloembergen, |

19. | P. D. Drummond, K. J. Mc Neil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation: semiclassical theory,” Opt. Acta |

**OCIS Codes**

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

**ToC Category:**

Focus Issue: Quantum structures in nonlinear optics and atomic physics

**History**

Original Manuscript: March 27, 1998

Published: July 20, 1998

**Citation**

Marco Santagiustina, Pere Colet, Maxi San Miguel, and Daniel Walgraef, "Growth dynamics of noise-sustained structures in
nonlinear optical resonators," Opt. Express **3**, 63-70 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-2-63

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### References

- L. A. Lugiato, S. Barnett, A. Gatti, I. Marzoli, G. L. Oppo, H. Wiedemann, "Quantum aspects of nonlinear optical patterns," Coherence and Quantum Optics VII (Plenum Press, New York, 1996), p 5.
- A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G- L. Oppo, S.M. Barnett, "A Langevin approach to quantum fluctuations and optical patterns formation," Phys. Rev. A 56, 877 (1997). [CrossRef]
- A. Gatti, L. A. Lugiato, G-L. Oppo, R. Martin, P. Di Trapani, A. Berzanskis, "From quantum to classical images," Opt. Express 1, 21 (1997). [CrossRef] [PubMed]
- A. Gatti, L. A. Lugiato, "Quantum images and critical uctuations in the optical parametric oscillator below threshold," Phys. Rev. A 52, 1675 (1995). [CrossRef] [PubMed]
- M. Hoyuelos, P. Colet, M. San Miguel, "Fluctuations and correlations in polarization patterns of a Kerr medium," to be published, Phys. Rev. E (1998). [CrossRef]
- R. J. Deissler,"Noise-sustained structure, intermittency, and the Ginzburg-Landau equation," J. Stat. Phys. 40, 376 (1985). [CrossRef]
- M. Santagiustina, P. Colet, M. San Miguel, D. Walgraef, "Convective noise-sustained structures in nonlinear optics," Phys. Rev. Lett. 79, 3363 (1997). [CrossRef]
- L. A. Lugiato, R. Lefever, "Spatial dissipative structures in passive optical systems," Phys. Rev. A 58, 2209 (1987).
- W. J. Firth, A. G. Scroggie, G. S. McDonald, L. A. Lugiato, "Hexagonal patterns in optical bistability," Phys. Rev. A 46, R3609 (1992). [CrossRef] [PubMed]
- L. A. Lugiato, F. Castelli, "Quantum noise reduction in a spatial dissipative structure," Phys. Rev. Lett. 68, 3284 (1992). [CrossRef] [PubMed]
- K. Staliunas, "Optical vortices during three-wave nonlinear coupling," Opt. Commun. 91, 82 (1992). [CrossRef]
- G-L. Oppo, M. Brambilla, L. A. Lugiato, "Formation and evolution of roll patterns in optical parametric oscillators," Phys. Rev. A 49, 2028 (1994). [CrossRef] [PubMed]
- L. A. Wu, H. J. Kimble, J. Hall, H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520 (1986). [CrossRef] [PubMed]
- M. Santagiustina, P. Colet, M. San Miguel, D. Walgraef, "Two-dimensional noise-sustained structures in optical parametric oscillators," to be published Phys. Rev. E (1998).
- M. Haelterman, G. Vitrant, "Drift instability and spatiotemporal dissipative structures in a nonlinear Fabry-Perot resonator under oblique incidence," J. Opt. Soc. Am. B 9, 1563 (1992). [CrossRef]
- G. Grynberg, "Drift instability and light-induced spin waves in an alkali vapor with feedback mirror," Opt. Commun. 109, 483 (1994). [CrossRef]
- A. Petrossian, L. Dambly, G. Grynberg, "Drift instability for a laser beam transmitted through a rubidium cell with feedback mirror," Europhys. Lett. 29, 209 (1995). [CrossRef]
- N. Bloembergen, Nonlinear Optics, (Benjamin Inc. Publ., Reading, 1965), Chapter 4.2. item Y. R. Shen, The principles of nonlinear optics, (Wiley, New York, 1984), Chapter 6.9.
- P. D. Drummond, K. J. Mc Neil, D. F. Walls, "Non-equilibrium transitions in sub/second harmonic generation: semiclassical theory," Opt. Acta 27, 321 (1980). [CrossRef]

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