A review on nonlinear dynamics of single-mode lasers is presented. A special attention is paid to the properties of a strange attractor corresponding to a chaotic regime of a single-mode laser. The main topics of nonlinear laser dynamics are listed and discussed.

Fundamentals of geometric phase theory in general-type dynamical systems and some particular results on geometric phases in semiclassical and quantum models of laser physics and nonlinear optics are reviewed. The emphasis is made on the physical meaning rather than on formal mathematical background to make the presentation clear for a wide circle of researchers in the field of nonlinear dynamics.

Short review of the papers on transverse and polarization transverse pattern dynamics in lasers is presented. The theoretical and numerical models are discussed including cavities with image rotation and allowing for the Zeeman sublevels structure of the lasing transition. The results of numerical modelling of the field dynamics, oscillation regimes, and effect of the spontaneous emission on the pattern resulted are presented and discussed for the class A lasers with the polarization isotropic cavity and unlimited transverse mode number.

We have studied the chaotic bifurcation mechanisms in the laser model with a saturable absorber. It has been shown that the nonzero steady state eigenvalues influence the specific dynamics in the multistability conditions. The asymptotic properties of the periodic regimes have been investigated. We have found that the chaotic regimes arise as sequence of the period-doubling cascade, intermittence and heteroclinic bifurcation to the saddle-focus steady state.

CO₂ lasers are very sensitive to optical feedback and can be driven into a large variety of dynamical states by an external modulation of the feedback parameters. We report on the experimental results concerning instabilities of CO₂ lasers induced by either an additional passive resonator of periodically varying geometrical length or an refractive index modulation in the external cavity by an electro- optical modulator. Experimental results show the presence of stable periodic orbits of different periods, bistability of periodic orbits and chaotic attractors with complex transition scenarios.

The dynamics of passive mode locking is investigated numerically for unidirectional ring YAG-Nd lasers and dye lasers. The difference of the lasers' spectroscopic parameters determines the difference in their generation regimes: the periodic pulse trains for YAG-Nd lasers and the stationary pulse generation for the dye lasers. For dye lasers data are obtained for inner pulse structure dependence on the gain factor and for stochasticity of the generation regime if gain and absorption line widths are not equal.

For some human diseases both the blood and lymph microcirculation parameters change. These parameters are of great importance in diagnostics. As distinct from blood flow from the lymph motion is more complicated and diverse. Lymph flow in microvessels is non-stationary and randomly varies in time. At present, the existing measuring methods of these flows are not fully developed and are mostly of qualitative character. In this paper the blood and the lymph motion have been considered from the viewpoint of nonlinear dynamics. The speckle-interferometric method has been proposed for the investigation of the dynamic characteristics of biofluids. The method has been applied in the investigation of blood and lymph flows in microvessels. Using the method of the focused Gaussian beam diffraction, the following parameters were defined: V that is the value characterizing the time- averaged velocity of biofluid and (Sigma) _V that is the parameter indicating the deviation degree of the spectral envelope from the Gaussian curve. The value of (Sigma) _V contains an information about the velocity range in the flow and, besides, indicates the spatial-temporal velocity changes in region being investigated. Investigations of the lymph flow dynamics also have been carried out under the influence of the lymphotropic agent. It has been shown that lymph vessels may act at different stages when both the structure of the lymph flow and its temporal dynamics change qualitatively.

For the complex Lorenz model, which is one of basic laser models, it is shown that the phase space has the geometric structure associated with a fiber bundle. Using the equation of motion in the base space of the fiber bundle, we find the surfaces bounding the attractors in this space. The homoclinic `butterfly' responsible for the Lorenz-like attractor appearance is shown to correspond to a codimension-two bifurcation.

The relaxation frequencies of the system of two and more solid-state lasers with crossed channels of the generation are calculated. It is shown that, as in the multimode laser with a single beam, the coupling among modes of the system through the spatial hole burning in the inverted population of the active medium does not vary the value of the highest relaxation frequency of isolated cavities. Close to the relaxation frequencies the minima of the excitation threshold in the system of dynamic chaos are observed. The transition of the system to dynamic chaos obeys the Ruel- Takens-Newhouse scenario.

The results of computer modelling of the transverse field pattern dynamics in a unidirectional ring laser with the fast relaxed active medium of arbitrary optical thickness are presented. In the theoretical model the configuration of the cavity having the spherical mirrors and apertures is considered. Numerical results are obtained for the symmetrical cavity. The modes with radial and azimuthal indices of up to 10 and 9, respectively, were taken into consideration. The regimes obtained include the steady-state regimes with distorted modes, quasi-periodic oscillations regimes, lock-in regimes when variations of the laser gain and optical power of the mirrors do not lead to variations of the transverse mode beat frequency. The optical vortices (defects) and their dynamics were demonstrated.

In the work the spatial dynamics of a cylindrically symmetrical unidirectional ring laser with spherical mirrors is investigated. A prism inserted in the cavity is used to rotate the electric field by an angle (theta) . The value of the angle depends on prism orientation. The angle of rotation affects the mode frequencies in the empty cavity. The frequency-degenerate sets of modes in this cavity differ from the commonly known ones. For example, at a certain angle of rotation the frequency-degenerate set consists of the infinite number of modes. The spatial structures in the laser differ from the patterns in a laser without field rotation. We investigate laser dynamics for two models of the active medium: the two-level model and the model allowing for the rotational sublevels of the CO₂ molecule. The latter model leads to appearance of an additional dissipative term in the equations for the mode amplitudes. In some cases this term can change the system dynamics. The number of rotational levels influences on the rate with which the system relaxes to the steady state. Our results show that one can control the spatial structure formation by changing the angle of field rotation. The number of the rotational sublevels of the CO₂ molecule influences on the system dynamics.

Within the framework of the Akhmanov-Vorontsov model the transverse spatial-temporal dynamics of the phase shift u(x,y,t) of a laser beam (LB) is investigated numerically for the system with Kerr nonlinearity to which 2D feedback is applied. Principles of the analysis of the LB transverse dynamics are discussed that allow us to explain and predict the form of the phase shift u(x,y,t) depending on the parameters of the large-scale transformation of the field of LB in the feedback loop, the nonlinearity parameter, the diffusion coefficient and the relaxation time of molecules of the nonlinear medium (liquid crystal). The regimes of the periodic variation of the phase shift u(x,y,t) are described. The ways of the quantitative estimation of structurization processes are suggested.

The results of theoretical research of the propagation effect of an ultrashort optical pulse under the conditions of coherent interaction with two resonant transitions are presented. One of the transitions is tuned to exact resonance with the pulse carrier frequency; the other is distanced by much more than the pulse spectral width, thus the adiabatic following approximation may be applied to it. The interaction between the pulse and the far-from-resonance transition is described by the wave equation with the nonlinear index of refraction in the form of the Lorentz function of the field amplitude. This wave equation is solved together with the Bloch equations for the exact resonant transition. The solution in the form of the stationary phase-modulated pulse is obtained, and (pi) -pulse of self-induced transparency is the special case of this solution. The obtained solution has the area under the pulse envelope less than (pi) , and with increasing the absorption coefficient of the far-from-resonance transition the pulse area decreases. The effect can be used for reducing the transparency threshold of the resonant medium below (pi) .

The effect of the optical fiber inhomogeneities, giving rise to stochastization of the transmitted signal, is investigated in the canonical formulation. The nonlinear stage of the modulation instability of the pump wave in the three-wave approximation is considered. The analytical analysis of the equations for the wave amplitudes in the Hamiltonian form shows that a `stochastic layer', i.e. the region in which the phase trajectories behave chaotically, is formed in the phase space of a dynamic system even at a rather weak inhomogeneity of the fiber. The estimates indicate that the phenomenon under study is very likely to be observed in experiments.

Various methods of coherent optical quantification of the structure functions of the boundary field phase distributions in the case of fractal-like scattering objects are considered: (1) by using the analysis of the angular dependence of scattered light mean intensity; (2) by estimating the parameters of the structure function of the far-field intensity fluctuations of scattered light. It has been shown in the second case that the equality of the correlation exponents of the boundary field phase and of the far-field intensity takes place if scattering objects with the Gaussian phase statistics are illuminated by a collimated beam and a great number of statistically independent elementary scatterers within illuminated area is provided. Experiments with rough glass plates illuminated by the Gaussian beams have shown that the values of the correlation exponents of the boundary field phase fluctuations (which have been obtained by means of microinterferometric measurements) are in satisfactory agreement with the corresponding values of the far-field speckle intensity.

Professor Mark L.Katz was a well-known scientist and a recognized expert in the fields of luminescence, spectroscopy and laser physics. He began his career as a modest teacher at a village school, then he studied in Odessa University. His first research work was a study of luminescence phenomena. M.L.Katz proposed a new method of thermally induced luminescence for the study of electron processes in ionic crystals that became a classical technique in studies of local energy levels of solids.