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

41 Introduction

The success in understanding laser patterns initiated a search for spontaneous pattern formation in other nonlinear resonators. One of the most extensively studied systems has been photorefractive oscillators, where the theoretical backgrounds were laid [26], complicated structures experimentally observed [27, 28] and order parameter equations derived [29]. Intensive studies of pattern formation in passive, driven, nonlinear Kerr resonators were also performed [30, 31, 32, 33]. Also, the patterns in optical parametric oscillators received a lot of attention. The basic patterns were predicted [34, 35], and order parameter equations were derived in the degenerate [36, 37] and nondegenerate [38] regimes. The connection between the patterns formed in planarand curved-mirror resonators was treated in [39], where an order parameter equation description of weakly curved (quasi-plane) nonlinear optical resonators was given.

These are just a few examples. In the next section, the general characteristics of nonlinear resonators, and the state of the art are reviewed.

1.2 Patterns in Nonlinear Optical Resonators

The patterns discussed in the main body of the book are those appearing in nonlinear optical resonators only. This particular configuration is characterized by (1) strong feedback and (2) a mode structure, both due to the cavity. The latter also implies temporal coherence of the radiation. Thanks to the feedback, the system does not just perform a nonlinear transformation of the field distribution, where the fields at the output can be expressed as some nonlinear function of the fields at the input and of the boundary conditions. Owing to the feedback, the system can be considered as a nonlinear dynamical system with an ability to evolve, to self-organize, to break spontaneously the spatial translational symmetry, and in general, to show its “own” distributions not present in the initial or boundary conditions.

Nonlinear optical resonators can be classified in di erent ways: by the resonator geometry (planar or curved), by the damping rates of the fields (class A, B or C lasers), by the field–matter interaction process (active and passive systems) and in other ways. After order parameter equations were derived for various systems, a new type of classification became possible. One can distinguish several large groups of nonlinear resonators, each of which can be described by a common order parameter equation:

1.Laser-like nonlinear resonators, such as lasers of classes A and C, photorefractive oscillators, and nondegenerate optical parametric oscillators. They are described by the complex Swift–Hohenberg equation,

∂A

∂t

and show optical vortices as the basic localized structures, and tilted waves and square vortex lattices as the basic extended patterns.

2.Resonators with squeezed phase, such as degenerate optical parametric oscillators and degenerate four-wave mixers. They are described, in the most simplified way, by the real Swift–Hohenberg equation,

and show phase domains and phase solitons as the basic localized structures, and stripes and hexagons as the basic extended patterns.

3.Lasers with a slow population inversion D (class B lasers). They cannot be described by a single order parameter equation, but can be described by two coupled equations,

and their basic feature is self-sustained dynamics, in particular the “restless vortex”.

4.Subcritical nonlinear resonators, such as lasers with intracavity saturable absorbers or optical parametric oscillators with a detuned pump. The e ects responsible for the subcriticality give rise to additional terms in the order parameter equation, which in general has the form of a modified Swift–Hohenberg equation,

where F represents a nonlinear, nonlocal function of the fields. Its solutions can show bistability and, as consequence, such systems can support bistable bright spatial solitons.

This classification is used throughout this book as the starting point for studies of pattern formation in nonlinear optical resonators. The main advantage of this choice is that one can investigate dynamical phenomena not necessarily for a particular nonlinear resonator, but for a given class of systems characterized by a common order parameter equation, and consequently by a common manifold of phenomena.

In this sense, the patterns in nonlinear optics can be considered as related to other patterns observed in nature and technology, such as in Rayleigh–B´enard convection [40], Taylor–Couette flows [41], and in chemical [42] and biological [43] systems. The study of patterns in nonlinear resonators has been strongly influenced and profited from the general ideas of Haken’s synergetics [44] and Prigogine’s dissipative structures [45, 46]. On the other hand, the knowledge achieved about patterns in nonlinear resonators provides feedback to the general understanding of pattern formation and evolution in nature.

61 Introduction

Next we review the basic transverse patterns observable in a large variety of optical resonators. It is convenient to distinguish between two kinds of patterns: localized structures, and extended patterns in the form of spatially periodic structures.

1.2.1 Localized Structures: Vortices and Solitons

A transverse structure which enjoys great popularity and on which numerous studies have been performed, is the optical vortex, a localized structure with topological character, which is a zero of the field amplitude and a singularity of the field phase.

Although optical vortices have been mainly studied in systems where free propagation occurs in a nonlinear material (see Sect. 1.3), some works have treated the problem of vortex formation in resonators. As mentioned above, the early studies of these fascinating objects [15, 16, 17, 18, 19] strongly stimulated interest in studies of pattern formation in general. The existence of vortices indicates indirectly the analogy between optics and hydrodynamics [22, 47, 48, 49]. It has been shown that the presence of vortices may initiate or stimulate the onset of (defect-mediated) turbulence [27, 50, 51, 52, 53]. Vortices may exist as stationary isolated structures [54, 55] or be arranged in regular vortex lattices [17, 23, 28]. Also, nonstationary dynamics of vortices have been reported, both of single vortices [56, 57] and of vortex lattice structures [58]. Recently, optical vortex lattices have been experimentally observed in microchip lasers [59].

Another type of localized structure is spatial solitons, which are nontopological structures. Although such structures do not appear exclusively in optical systems [60, 61, 62], they are now receiving tremendous interest in the field of optics owing to possible technological applications. A spatial soliton in a dissipative system, being bistable, can carry a bit of information, and thus such solitons are very promising for applications in parallel storage and parallel information processing.

Spatial solitons excited in optical resonators are usually known as cavity solitons. Cavity solitons can be classified into two main categories: amplitude (bright and dark) solitons, and phase (dark-ring) solitons. Investigations of the formation of bright localized structures began with early work on bistable lasers containing a saturable absorber [63, 64] and on passive nonlinear resonators [65].

Amplitude solitons can be excited in subcritical systems under bistability conditions, and can be considered as homoclinic connections between the lower (unexcited) and upper (excited) states. They have been reported for a great variety of passive nonlinear optical resonators, such as degenerate [66, 67, 68] and nondegenerate [69, 70] optical parametric oscillators, and for second-harmonic generation [71, 72, 73] (Fig. 1.2), where the bistability was related to the existence of a nonlinear resonance [37]. In some systems, the interaction of solitons and their dynamical behavior have been studied [73,

Fig. 1.2. Interaction of two moving amplitude solitons in vectorial intracavity second-harmonic generation: (a) central collision, (b) noncentral collision. From [73], c 1998 American Physical Society

74, 75]. Resonators containing Kerr media also support amplitude solitons, as a result of either Kerr [76] or polarization (vectorial) [77] instabilities.

In active systems, bright solitons have been demonstrated in photorefractive oscillators [78, 79, 80] and in lasers containing saturable absorbers [81, 82] or an intracavity Kerr lens [83]. A promising system for practical applications is the vertical cavity surface emission laser (VCSEL), which forms a microresonator with a semiconductor as a nonlinear material. The theoretically predicted patterns for this system [84, 85, 86, 87, 88, 89] were recently experimentally confirmed in [90].

The required subcriticality condition is usually achieved by introducing an intracavity absorbing element. However, recently, stable solitons in the absence of an additional medium have been reported in cascade lasers [91].

Besides the amplitude solitons in subcritical nonlinear resonators, a different type of bistable soliton exists in supercritical resonators. Such systems are characterized by a broken phase symmetry of the order parameter, and solutions with only two possible phase values are allowed. In this case the solitons connect two homogeneous solutions of the same amplitude but of opposite phase. Such phase solitons, which are round, stable phase domains of minimum size, appear as a dark ring on a bright background. This novel type of optical soliton is now receiving a lot of interest, since the solitons are seemingly much easier to realize experimentally than their bright counterparts in subcritical systems.

One of the systems most investigated has been the degenerate optical parametric oscillator (DOPO), either in the one-dimensional case [92, 93] or in the more realistic case of two transverse dimensions [94, 95, 96, 97]. Also, the soliton formation process [98, 99, 100] and its dynamical behavior [101, 102] have been analyzed. Optical bistability in a passive cavity driven by a coherent external field is another example of a system supporting such phase

8 1 Introduction

Fig. 1.3. Phase domains and phase (darkring) solitons in a cavity four-wave-mixing experiment. From [115], c 1999 Optical Society of America

Fig. 1.4. Modulational instability of a straight domain boundary and formation of a finger pattern, in a type II degenerate optical parametric oscillator. The upper row shows the intensity, and the lower row the phase pattern. From [102], c 2001 American Physical Society

solitons [103, 104, 105, 106, 107]. Both the DOPO and systems showing optical bistability are systems described by a common order parameter equation, the real Swift–Hohenberg equation [108]. Systems with a higher order of nonlinearity, such as vectorial Kerr resonators, have also been shown to support phase solitons [109, 110, 111].

Phase solitons can form bound states, resulting in soliton aggregates or clusters [94, 112].

Phase solitons in a cavity are seemingly much easier to excite than their counterparts in subcritical systems. In fact, such phase solitons have already been experimentally demonstrated in degenerate four-wave mixers [113, 114, 115] (Figs. 1.3 and 1.4).

1.2.2 Extended Patterns

Besides the localized patterns, vortices and solitons, to which the book is mainly devoted, extended patterns in optical resonators have been also extensively studied. In optical resonators, two main categories of patterns can

be distinguished. One class of patterns appears in low-aperture systems, characterized by a small Fresnel number, such as a laser with curved mirrors. Since this is the most typical configuration of an optical cavity, this phenomenon was observed in the very first experimental realizations, although a systematic study was postponed to a later time [16]. The patterns of this kind are induced by the boundary conditions, and can be interpreted as a weakly nonlinear superposition of a small number of cavity modes of Gauss–Hermite or Gauss–Laguerre type.

Theoretical predictions based on modal expansions of the field [14, 116, 117] have been confirmed by a large number of experiments, some of them reported in [118, 119, 120, 121, 122]. Owing to the particular geometry of the cavity, this kind of pattern is almost exclusively optical. If the aperture is increased, the number of cavity modes excited can grow, and so the spatial complexity of the pattern grows[123].

The other class of extended optical patterns is typical of large-aperture resonators, formed by plane mirrors in a ring or a Fabry–P´erot configuration. The transverse boundary conditions have a weak influence on the system dynamics, in contrast to what happens in small-aperture systems. Consequently, the patterns found in these systems are essentially nonlinear, and the system dynamics can be reduced to the evolution of a single field, called the order parameter.

The simplest patterns in these systems consist of a single tilted or traveling wave (TW), which is the basic transverse solution in a laser [21], although more complex solutions formed by several TWs have been found [125, 124]. The predicted laser TW patterns have been observed in experiments with large-Fresnel-number cavities [126, 127, 128]. The TW solutions are also found in passive resonators described by the same order parameter equation, such as nondegenerate optical parametric oscillators (OPOs) [35, 129]. The e ect of an externally injected signal in a laser has been also studied [130, 131], showing the formation of more complex patterns, such as rolls or hexagons.

Roll, or stripe, patterns are commonplace for a large variety of nonlinear passive cavities, such as degenerate OPOs [34], four-wave mixers [37], systems showing optical bistability [31, 132] and cavities containing Kerr media [133]. Patterns with hexagonal symmetry are also frequently found in such resonators [134, 135]. Both types of pattern are familiar in hydrodynamic systems, such as systems showing Rayleigh–B´enard convection.

Another kind of traveling solution existing in optical resonators corresponds to spiral patterns, such as those found in lasers [136, 137] and in OPOs [138, 139], which are typical structures in chemical reaction–di usion systems.

When more complex models, including additional e ects, are considered, a larger variety of patterns, sometimes of exotic appearance, is found. Some such models generalize the above cited models by considering the existence

10 1 Introduction

of competition between di erent parametric processes [140, 141] or between scalar and vectorial instabilities [142], the walk-o e ect due to birefringence in the medium [143, 144, 145], or external temporal variation of the cavity parameters [146].

Some systems allow the simultaneous excitation of patterns with di erent wavenumbers. These systems form patterns with di erent periodicities that have been called quasicrystals [147, 148] and daisy patterns [149] (Fig. 1.5).

The experimental conditions for large-aspect-ratio resonators are not easy to achieve. Most of the experiments performed have studied multimode regimes involving high-order transverse modes. The formation of the patterns described above was reported in lasers [126, 127, 128] and OPOs [151, 152]. The observed patterns correspond well to the numerical solutions of large- aspect-ratio models. Conditions for boundary-free, essentially nonlinear patterns were obtained in [78, 153] with the use of self-imaging resonators, which allowed the experimenters to obtain Fresnel numbers of arbitrarily high value.

All the patterns reviewed above are two-dimensional, the light being distributed in the transverse space perpendicular to the resonator axis, and evolving in time. Recently, the possibility of three-dimensional patterns was demonstrated for OPOs [154], nonlinear resonators with Kerr media [155, 156], optical bistability [157] and second-harmonic generation [158].

Finally, the problem of the e ect of noise on the pattern formation properties of a nonlinear resonator has also been treated. One can expect that

Fig. 1.5. Experimentally observed hexagonal patterns with sixfold and twelvefold symmetry (quasipatterns), in a nonlinear optical system with continuous rotational symmetry. From [150], c 1999 American Physical Society

noise, which is present in every system, will bring about new features in the spatio-temporal dynamics of the system. First, noise can modify (shift) the threshold of pattern formation [159]. Second, owing to noise, the precursors of patterns can be seen below the pattern formation threshold [160, 161, 162]. While a noiseless pattern-forming system below the pattern formation threshold shows no pattern at all, since all perturbations decay, one observes in the presence of noise a particular form of spatially filtered noise, which in the field of nonlinear optics has been called “quantum patterns” when the noise is of quantum origin [163, 164, 165]. Above the pattern formation threshold, noise can also result in defects (dislocations or disclinations) of the patterns [166, 167].

1.3 Optical Patterns in Other Configurations

In parallel with the studies on nonlinear resonators, pattern formation problems have been considered in other optical configurations. These configurations can be divided into the following categories, according to their geometry and complexity.

1.3.1 Mirrorless Configuration

When an intense light beam propagates in a nonlinear medium, it can experience filamentation e ects, leading to periodic spatial distributions [168], or develop into self-trapped states of light, or solitons. The self-focusing action of the nonlinearity compensated by di raction results in self-sustained bright spatial solitons [12], which can exist as isolated states or form complex ensembles, sometimes interacting in a particle-like fashion [169, 170, 171, 172, 173, 174, 175]. Also, dark solitons [176, 177, 178, 179, 180, 181, 182, 183, 184] and optical vortices [185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196] have been described and experimentally observed. In such a mirrorless configuration feedback is absent, and one obtains not a spontaneous pattern formation, but just a nonlinear transformation of the input distribution. This nonlinear transformation can be very complicated, and can be described by complicated integro-di erential equations. However, every transformation remains a transformation, and without feedback it does not lead to spontaneous pattern formation. Some other mirrorless schemes, where optical pattern formation has been predicted, are based on the interaction of two counterpropagating pumping waves in a nonlinear medium. It has been shown that the waves that appear through nonlinear mixing processes have their lowest threshold at certain angles with respect to the pumping waves, and may result in a wide variety of patterns, either extended, such as rolls or hexagons [197, 198, 199, 200, 201, 202, 203, 204](Fig. 1.6), or localized [205]. Experimental confirmation has been obtained using various nonlinear media, such as atomic vapors and photorefractive crystals.