Файл: Staliunas K., Sanchez-Morcillo V.J. (eds.) Transverse Patterns in Nonlinear Optical Resonators(ST.pdf

12 1 Introduction

Fig. 1.6. Hexagonal patterns with di erent spatial scales observed in a photorefractive crystal with a single pump wave. From [203], c 1991 Optical Society of America

1.3.2 Single-Feedback-Mirror Configuration

The presence of a mirror introduces feedback into the system. Unlike the case in the previous schemes, here nonlinearity and di raction act at di erent spatial locations. The most typical configuration is formed by a thin slice of a Kerr medium and a mirror at some distance. Theoretical studies have predicted structures mainly with hexagonal symmetry [206, 207, 208, 209] (Fig. 1.7), although more complex solutions have been found [210, 214]. From the experimental side, various nonlinear media, such as atomic vapors [211, 212], and Kerr [213] and photorefractive [214] media have been used successfully. Also, this configuration led to the first realization of localized structures in nonlinear optics [215]. The dynamics and interaction of these localized structures have been extensively investigated [216, 217, 218, 219] (Fig. 1.8).

1.3.3 Optical Feedback Loops

Another configuration, somewhat between the single feedback mirror and the nonlinear resonator, is the feedback loop. In such a configuration, one has the possibility of acting on the field distribution on every round trip through the loop, continuously transforming the pattern distribution. Some typical two-dimensional transformations are the rotation, translation, scaling and filtering of the pattern. The first work obtained pattern formation by controlling the spatial scale and the topology of the transverse interaction of

14 1 Introduction

Fig. 1.10. Quasicrystal patterns with dodecagonal symmetry, with di erent spatial scales, together with the corresponding spatial spectra. From [224], c 1998 American Physical Society

the light field in a medium with cubic nonlinearity [220, 221, 222], by controlling the phase of the field with a spatial Fourier filter [223, 224] (Figs. 1.9 and 1.10), and by introducing a medium with a binary-type refractive nonlinear response [225].

A very versatile system is a feedback loop with a liquid-crystal light valve acting as a phase modulator with a Kerr-type nonlinearity. The conversion from a phase to an intensity distribution, required to close the feedback loop, can be performed by two means: by free propagation (di ractional feedback) [226, 227] or by interference with reflected waves (interferential feedback), as shown in Fig. 1.11 [228, 229, 230]. In both cases, a great variety of kaleidoscope-like patterns have been obtained theoretically and experimentally. The patterns can also be controlled by means of nonlocal interactions, via rotation [231, 232, 233] (Fig. 1.12) or translation [234, 235] of the signal in the feedback loop, giving rise to more exotic solutions such as quasicrys-

Fig. 1.11. Patterns in a liquid-crystal light valve in the interferential feedback configuration, for increasing translational nonlocality ∆x. The near field (top row) is shown together with the corresponding spectrum (bottom row). From [230], c 1998 American Physical Society

16 1 Introduction

subsequent chapters of the book. The derivation of the OPEs for class A and class C lasers is given in Chap. 2. For completeness, two techniques of derivation are given: one based on the adiabatic elimination of the fast variables, and one based on multiscale expansion techniques. Both procedures lead to the complex Swift–Hohenberg equation as the OPE for lasers. The CSH equation describes the spatio-temporal dynamics of the complex-valued order parameter, which is proportional to the envelope of the optical field. In Chap. 3, the OPEs for optical parametric oscillators and photorefractive oscillators (PROs) are derived. In the degenerate case, the resulting equation is shown to be the real Swift–Hohenberg equation, first obtained in a hydrodynamic context. For large pump detuning values, a generalized model including nonlinear resonance e ects is obtained. In the case of PROs, the adiabatic elimination technique is used to derive the CSH equation. The order parameter equations derived in Chaps. 2 and 3 divide nonlinear optical resonators into distinct classes, and thus allow one to study pattern formation phenomena without necessarily considering every nonlinear optical system separately; instead, one can consider classes of the systems.

Chapters 2 and 3 are devoted to the patterns of the first class of systems, that described by the CSH equation, i.e. lasers, photorefractive oscillators and nondegenerate OPOs. The localized patterns in this class of systems are optical vortices: these are zeros of the amplitude of the optical field, and are simultaneously singularities of the field phase. Optical vortices dominate the dynamics of the system in near-resonant cases (when the detuning is close to zero). The CGL equation in this near-resonant limit can be rewritten in a hydrodynamic form. Owing to this analogy between laser and hydrodynamics, the dynamics of the transverse distribution of the laser radiation are very similar to the dynamics of a superfluid. It is shown that optical vortices of the same topological charge rotate around one another; a pair of vortices of the same charge translate in parallel through the aperture of the laser or annihilate, depending on the parameters.

In Chap. 5, the limit of large or moderate detuning is considered. The CSH equation cannot be rewritten in a hydrodynamic form, but the dynamics of the fields can still be well interpreted by hydrodynamic means. For large detuning, tilted waves are excited. In hydrodynamic terms, flows with a velocity of fixed magnitude but arbitrary direction are favored. This results, in particular, in counterpropagating flows separated by vortex sheets. This also leads to optical vortices advected by the mean flow, and similar phenomena. Such phenomena are analyzed theoretically and demonstrated numerically. A pattern of square symmetry, called a square vortex lattice, consisting of four counterpropagating flows in the form of a cross, is also described and discussed.

In Chap. 6, the e ects of the curvature of the mirrors of the resonator are analyzed. The majority of theoretical investigations of pattern formation in nonlinear optics, including those in the largest part of this book, have been

performed by assuming a plane-mirror cavity model. However, in experiments resonators with curved mirrors are often used. Therefore a model of a laser with curved mirrors is introduced. The presence of curved mirrors results in an additional term in the order parameter equation, proportional to the total curvature of the mirrors in the resonator. This term produces a coordinatedependent (parabolic) phase shift of the order parameter during propagation in the resonator. The presence of the parabolic potential allows one to expand the field of the resonator in terms of the eigenfunctions (transverse modes) of the potential. Although this mode expansion is strictly valid for linear resonators only, the nonlinearity in the resonator results in a weakly nonlinear coupling of the complex amplitudes of the modes. As a result, an infinite set of coupled ordinary di erential equations for complex-valued mode amplitudes is derived. This gives an alternative way of investigating the transverse dynamics of a laser, by solving the equations for the mode amplitudes instead of solving the partial di erential equations. The technique of mode expansion is shown to be extremely useful when one is dealing with a small number of transverse modes. In particular, the transverse dynamics of class A lasers and photorefractive oscillators are considered; the phenomena of transverse mode pulling and locking are observed. Chapter 6 also deals with degenerate resonators, such as self-imaging and confocal resonators. In such resonators, the longitudinal mode separation is an integer multiple of the transverse mode separation. It is shown, by analysis of the corresponding ABCD matrices, that self-imaging resonators are equivalent to planar resonators of zero length. This insight opened up new possibilities for experimenting with transverse patterns in nonlinear optical systems, and allowed the first experimental realization of a number of phenomena predicted theoretically for nonlinear resonators.

Chapter 7 deals with patterns in class B lasers. Class B lasers are not describable by the CSH equation. Owing to the slowness of the population inversion, the order parameter equation in this case is not a single equation belonging to one of the classes defined above, but a system of two coupled equations, resembling those derived for excitatory or oscillatory chemical systems, where the (slow) population inversion plays the role of the recovery variable, and the fast optical field plays the role of the excitable variable. An analysis of such self-sustained spatio-temporal dynamics in a class B laser is performed. The vortices, which are stationary in a class A laser, perform selfsustained meandering in a class B laser, a phenomenon known as the “restless vortex”. Also, vortex lattices experience self-sustained oscillatory dynamics. Either the vortices in the lattice oscillate in such a way that neighboring vortices rotate in antiphase, thus resulting in an “optical” mode of vortex lattice oscillation, or the vortex lattice drifts spontaneously with a well-defined velocity, thus resulting in an “acoustic” oscillation mode.

The following chapters, Chaps. 8 to 11, are devoted to amplitude and phase domains, as well as amplitude and phase solitons in bistable nonlinear

18 1 Introduction

optical systems. The general theory of subcritical spatially extended systems is developed in Chap. 8, where two mechanisms of creation of subcriticality in optical resonators are described: one due to the presence of a saturable absorber, and one due to the presence of a nonlinear resonance. A discussion in terms of order parameter equations is given.

In Chap. 9, a theoretical description and experimental evidence of domain dynamics and spatial solitons in lasers containing a saturable absorber are presented. Two di erent resonator configurations are used: a self-imaging resonator where both nonlinearities (due to the gain and to saturable absorption) are placed at the same location on the optical axis of the resonator, and a self-imaging resonator where the two nonlinearities are placed at Fourierconjugated locations. For spatially coincident nonlinearities, the evolution of domains is demonstrated numerically and experimentally, with the eventual appearance of spatial solitons. For nonlinearities placed in conjugate locations in the resonator, the competition, mutual interaction and drift of solitons are investigated, also both theoretically and experimentally.

In Chap. 10, a subcriticality mechanism di erent from saturable absorption is studied, in this case related to the existence of a nonlinear resonance due to nonresonant pumping. As an example, the order parameter equation obtained in Chap. 3 for a degenerate OPO with a detuned pump is considered. The nonlinear resonance implies that the pattern wavenumber depends on the intensity of the radiation. With approriate values of the detuning, the nonlinear resonance can lead to bistability, and thus allow the excitation of amplitude domains and spatial solitons. Numerical results from the DOPO mean-field model are given for comparison.

In Chap. 11, the dynamics of phase domains in supercritical real-valued order parameter systems, such as the degenerate OPO, are analized. These systems should properly be described by the real Swift–Hohenberg equation. It is demonstrated that the domain boundaries, the lines of zero intensity separating domains of opposite phase, may contract or expand depending on the value of the resonator detuning. In this way, the domain boundaries behave as elastic ribbons, with the elasticity coe cient depending on the detuning. Contracting domains, observed for small values of the detuning, eventually disappear. Expanding domains are found for large values of the detuning, and their evolution results in labyrinthine structures. For intermediate values of the detuning, the contracting domain boundaries stop contracting at a particular radius. The latter scenario results in stable rings of domain boundaries, which are phase solitons. The experimental confirmation of the predicted phenomena is described.

In Chap. 12, the Turing pattern formation mechanism, typical of chemical reaction–di usion systems, is shown to exist also in nonlinear optics. The pattern formation mechanism described in most of the chapters of the book is based on an o -resonance excitation. The Turing mechanism, however, is based on the interplay between the di usion and/or di raction of interacting