Pattern formation and competition in nonlinear optics

We review the properties of optical spatial dissipative solitons (SDS). These are stable, self‐localized optical excitations sitting on a uniform, or quasi‐uniform, background in a dissipative environment like a nonlinear optical cavity. Indeed, in optics they are often termed “cavity solitons.” We discuss their dynamics and interactions in both ideal and imperfect systems, making comparison with experiments. SDS in lasers offer important advantages for applications. We review candidate schemes and the tremendous recent progress in semiconductor‐based cavity soliton lasers. We examine SDS in periodic structures, and we show how SDS can be quantitatively related to the locking of fronts. We conclude with an assessment of potential applications of SDS in photonics, arguing that best use of their particular features is made by exploiting their mobility, for example in all‐optical delay lines.

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Fundamentals and Applications of Spatial Dissipative Solitons in Photonic Devices

Beams of light interacting with nonlinear-optical materials naturally form complex transverse profiles. Self-focusing, filamentation, modulational instabilities, spontaneous pattern formation and symmetry breaking, nonlinear waveguiding, and transverse mode excitation are but some of the many phenomena that are observed in lasers, in optical bistability, and in the free interaction of beams. As an introduction to a special issue of the Journal of the Optical Society of America B on transverse effects in nonlinear-optical systems, we provide this overview of history and current work.

Overview of transverse effects in nonlinear-optical systems

Using a collective coordinate numerical optimization procedure, we construct ground-state configurations of interacting particle systems in various space dimensions so that the scattering of radiation exactly matches a prescribed pattern for a set of wave vectors. We show that the constructed ground states are, counterintuitively, disordered i.e., possess no long-range order in the infinite-volume limit. We focus on three classes of configurations with unique radiation scattering characteristics: i “stealth” materials, which are transparent to incident radiation at certain wavelengths; ii “super-ideal” gases, which scatter radiation identically to that of an ensemble of ideal gas configurations for a selected set of wave vectors; and iii “equi-luminous” materials, which scatter radiation equally intensely for a selected set of wave vectors. We find that ground-state configurations have an increased tendency to contain clusters of particles as one increases the prescribed luminosity. Limitations and consequences of this procedure are detailed. © 2008 American Institute of Physics. DOI: 10.1063/1.2961314

Classical disordered ground states: Super-ideal gases and stealth and equi-luminous materials

Extending general methods developed in synergetics to construct order-parameter equations we work out a systematic elimination procedure for the stable modes in nonlinear delay systems. It will be shown that in the vicinity of an instability the dynamical behavior of the infinite-dimensional delay system is approximately governed by a low-dimensional set of order-parameter equations which turn out to be of the form of ordinary differential equations, i.e., they no longer contain memory terms. The general formalism will then be compared with experimental data obtained from the nonlinear operation of a phase-locked loop where the finite propagation time of the signal in the feedback loop is taken into account.

Delay-induced instabilities in nonlinear feedback systems

Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium

The plane-wave dynamics of 3*omega => (2*omega, omega) subharmonic optical parametric oscillators containing a second harmonic generator of the idler wave omega is analyzed analytically by using the meanfield approximation and numerically by taking into account the field propagation inside the media. The resonant Chi(2):Chi(2) cascaded second-order nonlinearities induce a mutual injection-locking of the signal and idler waves that leads to coherent self phase-locking of the pump and subharmonic waves, freezing the phase diffusion noise. In case of signal-and-idler resonant devices, largely detuned sub-threshold states occur due to a subcritical bifurcation, broadening out the self-locking frequency range to a few cavity linewidths.

Theory of self-phase-locked optical parametric oscillators

Continuous-wave (cw) off-resonance rings and cw on-resonance enhancement (CORE) are observed as a result of an optical beam propagating through sodium vapor. They are interpreted in terms of diffraction-free encoding (DFE) followed by free-space propagation. They are discussed with respect to the three regimes of DFE, namely, pure phase encoding, on-resonance pure amplitude encoding leading to stripping, and the intermediate case involving both phase and amplitude encodings. First there are far-field interference rings displaying pure phase encoding. They are observed on both sides of resonance, as expected by theory. Second, there is CORE, which is a Fresnel diffraction manifestation resulting from the stripping of the beam profile by the nonlinear medium, and in the far field, the corresponding Fraunhofer patterns. CORE is observed in an inhomogeneously broadened system, and the role of Doppler broadening is theoretically studied. Finally, there are off-resonance rings involving both phase and amplitude encodings. They are numerically simulated and discussed with respect to DFE predictions.

Continuous-wave off-resonance rings and continuous-wave on-resonance enhancement

Multi-conical instability in the passive ring cavity: linear analysis

Labyrinthine structures often appear as the final steady state of pattern forming systems. Being disordered, they exhibit the same kind of short range positional order as the Newell-Pomeau turbulent crystal. Labyrinths can be seen as a limit case of the texture of disordered rolls with a coherence length of the same order as the wavelength. In the various two-dimensional model equations we looked at, labyrinths and parallel rolls are steady states for the same parameters, their occurrence depending on the initial conditions. Comparing the stability of these two structures, we find that in variational models their energy is very close, rolls always being more stable than labyrinths. For the nonvariational model we propose a numerical experiment which displays a well defined bifurcation from parallel rolls to labyrinths as the more stable state.

Elisabeth Ressayre

Papers

Self-focusing and spatial ringing of intense cw light propagating through a strong absorbing medium

Theory of self-phase-locked optical parametric oscillators

Continuous-wave off-resonance rings and continuous-wave on-resonance enhancement

Multi-conical instability in the passive ring cavity: linear analysis

Example of a chaotic crystal: The labyrinth