This lecture gives a general overview of new possibilities offered by photonic crystals in nonlinear and ultrafast optics. The abilities of photonic band-gap (PBG) structures to guide and localize light, phase-match nonlinear interactions, and chirp, compress, and switch laser pulses are discussed. Several types of PBG structures allowing photonic band gaps to be produced in one and two dimensions are considered. In particular, the properties of a hollow corrugated planar waveguide formed by a diffraction grating and a mirror are examined and possible applications of such optical components combining the dispersion of a gas, a waveguide, and a PBG structure are discussed. Holey fibers are shown to substantially enhance nonlinear-optical interactions, increasing the efficiency of spectral broadening of short laser pulses and opening new horizons in optical frequency metrology.

The dynamics of actively mode-locked laser is investigated and the results of numerical simulations are presented. Different regimes corresponding to the relaxation oscillations and spiking are observed depending on the noise level and frequency detuning. The numerical results demonstrate in detail the processes of destroying synchronization and pulse-train instabilities.

Relaxation of the electronic energy in the active medium of the chemical oxygen-iodine laser (COIL) causes non- equilibrium population of vibrational levels of oxygen. It is found that the fraction of vibrationally excited oxygen can reach several of percent in the oxygen stream. The rate relaxation of vibrational energy of oxygen is limited by the exchange between oxygen at the first vibrational level and the bending mode of water. A simple formulas for the population of singlet oxygen vibrational levels at the output of a singlet-oxygen generator are found. It is noted that the EE energy transfer between singlet oxygen and molecular iodine stimulated by the vibrational excitation may be of considerable importance in the kinetics of iodine dissociation.

Experimental results are presented which have been obtained in the course of studying the effect of impurity levels on the kinetics of photoinduced electric field tuning in the bulk of homogeneous MDS structures with a steady leakage current.

The multicore fiber laser (MCFL), containing several single mode microcores on the circle inside the pump core, is a promising compact high brightness laser source. The problem is to synchronize the radiation of microcores having different propagation constant values at a given radiation frequency. We present a theory of phase-locking of MCFL with a mirror spaced at fraction of Talbot distance. The collective mode selection appears to occur due to spatial filtering at fractional Talbon distance in combination with radiation frequency selection from the spectral gain range. Good agreement is found between the theory and experiment.

The major directions of manipulations with micro objects by laser beams are considered. Using spiral beams can essentially extend the potential of manipulation. In ray optics approximation the calculation of the torque transmitted from a spiral beam to an absorbing particle is made in accordance with the wave optics. The experimental observation of torque transfer from non zero orbital moment beams to particles trapped into the laser beam focus is described.

The computer model of interaction of a tightly focused laser beam with transparent dielectric particles in the shape of ellipsoid of revolution of the sizes > 10(lambda) was created. For a zero Gaussian mode the interaction with latex particles weighed in water was simulated. For these particular particles the forces exerted by Gaussian beam in a longitudinal direction were calculated. It was revealed that the particles flattened out along the axis of the beam propagation were better trapped, than the longitudinal ones. It was found, that the behavior of the trapping force acting on the longitudinal particles is influenced by the total reflection arising at the light propagation inside a particle.

Vectorial Karhunen-Loeve modes were used for the description of the polarization transverse pattern dynamics of the Zeeman multitransverse mode layer with large Fresnel number and anisotropic cavity. Stokes parameter distribution of the laser field in the transverse plane define new 2D vector field. Catastrophe theory and Newton diagram method were used for the classification of the laser field based on the characterization of the 2D vector field behavior in the vicinity of its singular points in the transverse plane. The results of numerical experiments show that the regimes with rotating patterns are closely connected with A_n catastrophe functions of argument r equals (root)(x²+PLUy²), where x,y are the transverse coordinates. For the oscillating patterns the regimes can be classified using Newton diagrams which are also connected with corresponding polynomials.

Nonlinear optics is hardly considered on beam propagation in EDFAs. Hereby, we will study the effects of SRS on the distributed EDFAs pumped by 1480 nm wavelength. The results of numerical simulation show that the SRS in DEDFAs cannot be neglected due to its serious affects on noise and the spectral gain of amplifier. The numerical results will be demonstrated and the possibility of using SRS for gain broadening of the EDFAs will be described.

Optical properties of one-dimensional heterostructures having optical forbidden gap are described and photonic structures with n-i-p-i superlattices are designed for laser applications.

The gain/attenuation factor for lightwave propagating in 1D photonic band-gap (PBG) structure having layers with gain/losses was calculated using Kronig-Penny model. Special attention was given to the dispersion characteristics near the boundary of band gap. To distinguish between the directions of propagation, gain and attenuation the complex frequency of eigenwaves was used in the spirit of the theory of instabilities. It was shown the presence of bad gaps does not lead to considerable gain/loss enhancement and the gain is definitely absent within the band gaps. Possible mechanisms of gain enhancement were discussed.

On-resonance self-action of a beam with periodically modulated amplitude in a two-level saturable absorber is studied numerically. When the period of modulation is comparable with relaxation times, one cannot describe the medium by means of susceptibility, and the response of the medium to the amplitude variation is not immediate. We solve the Bloch equations in each point of the medium to find this response correctly. Transient behavior followed by stable oscillations of population difference and polarization vector is shown to depend strongly on the frequency and amplitude of modulation. Beam propagation is considered using a simplified model of a thin absorber followed by free space, the propagation distance being small compared with the modulation wavelength. Manifestations of non-stationary response of the medium in the output intensity behavior are analyzed.

Using method developed by us early for transformation of so called generalized Schrodinger equation with temporal derivation from nonlinear response, we consider self-action of femtosecond laser pulse in optical fiber with the help of computer simulation. For this problem, conservative difference scheme is proposed. An influence of time dispersion of nonlinear response on self-focusing of Gaussian pulse is investigated. A restriction of light intensity in focus due to influence of time dispersion of nonlinear response is shown.

Problem of high-efficiency second harmonic generation (SHG) of femtosecond laser powerful pulses under the condition of laser radiation self-action (cubic non-linearity) is considered using both analytical and numerical methods. The process is described by non-linear Schrodinger equation regarding second order dispersion, phase mismatching and cubic non-linearity. Analytical investigation of the problem in case of long pulses, as well as the numerical simulation, shows a possibility of essential growth (up to 60%) of SHG generation efficiency. This result is due to introduced phase shifts between the interacting waves in a definite section of media. High-intensity and efficient SHG is described.

In this paper, we normalize the nonlinear equation of pulse propagation and use it for dimensionless simulation of propagation of ultrashort soliton-like pulses in anomalous dispersion regime of optical fibers using beam propagation method. It is to observe and analyze the interaction of nonlinear effect of self phase modulation and group velocity dispersion on pulse compression and spectral broadening of such optical pulses in time and frequency domains, respectively.

On the basis of the master equation for the density matrix the dynamics of the collective spontaneous radiation of two (Lambda) - type three-level atoms interacting with two mode of quantum electromagnetic field in the damped cavity has been considered. The behavior of the photon numbers in modes and the atomic levels populations has been investigated in the case when only a single atom is excited at the initial time.

Small deviations from the stationary solutions of the Gross_Pitaevskii equation describing non-ground states of Bose-Einstein condensate of atoms in a harmonic trap are analyzed basing on the Bogoliubov equations and using both perturbation theory and direct numerical calculations. Non- vortex excited stationary that were not studied before are shown to possess new and unexpected features.

A technique is presented for treating strongly nonstationary and transient processes in optics, permitting one to take into account both types of competing with each other effects, quantum as well as coherent. The main equations for describing the interplay between these two kinds of effects are derived. The possibility of influencing coherent optical phenomena by preparing special quantum states of matter is discussed.

We compare exact and SU(2)-cluster approximate calculation schemes to determine dynamics of the second-harmonic generation model using its reformulation in terms of the polynomial Lie algebra su_pd(2). These schemes, based on diagonal representations of the model evolution operator (due to diagonalizing the model Hamiltonian) and realized in the algorithmic forms, enabled us to implement computer experiments exhibiting a satisfactory accuracy of the cluster approximations in a large range of characteristic model parameters.

The effect of recoil on the fine shift of the energy levels of the hydrogen-like atom is considered up to terms of the order (alpha) ⁶(mu) ³/(m₁m₂)(Beta) ln(Beta) ^-1. Contributions of this order from one-photon exchange between particles with unequal masses (m₁ not equal to m₂) are calculated analytically for the first time.

A three-body scattering problem is formulated in an adiabatic representation as a multi-channel spectral problem for a system of one-dimensional integral equations using the Schwinger variational functional. The stable Newtonian iteration schemes for calculating the eigenfunctions and eigenvalues which are phase shift and energy for continuous and discrete spectrum, correspondingly, are elaborated. Convergence and efficiency of the proposed schemes are demonstrated using the exact solvable model of three identical particles (bosons) on a line with pair attractive (delta) -potentials.

Electrical parameters of trimethylentinilsylan are calculated by solving the SCF equations in CNDO approximation. The results are in satisfactory agreement with appropriate parameters found from the solution of the inverse electrooptical problem.

The effect of coherent population trapping related to quantum interference of states now is well investigated for (Lambda) -systems in which the lower levels are generated by the hyperfine structure sublevels of the ground state of the atom. At present time the majority of experimental researches of resonances of CPT are made with alkaline atoms, for which hyperfine components of the ground state have typical interval of several gigahertz. We consider CPT in (Lambda) -systems in which the fine structure sublevels of the atom generate the lower levels. Our computer simulations have been made for the thallium atom for which the distance between sublevels of fine structure components is approximately 7792 cm^-1. Theoretical and computer simulations of dynamics and dispersions of population of the considered multilevel system allow to detect new peculiarities of effect and demonstrate the influence of orientation and strength of magnetic field.

Non-dispersing wave packet states of the electron in a hydrogen atom in constant magnetic and rotating electric fields are considered. These states, referred as the states of Satrurnian atom, may be presented as eigenstates of an effective Hamiltonian in the coordinate frame rotating with the electric field. We start from solving the eigenvalue problem in the plane zequals0 of the electron wave packet orbit. General approach valid for arbitrary z is formulated in terms of the solutions of a parametric eigenvalue problem. The results of 2D calculations are used to justify the rough approximation for the wave functions of the ground oscillatory and the continuum states. Using these functions the probabilities of induced and spontaneous transitions from the lower continuum state to the ground state of a Saturnian atom are roughly estimated in a very simple analytical way.

A new method of calculation of the values the fine shifts of energy levels of bound states is considered basing on the upgraded relativistic quasipotential equation. The appropriate perturbation theory permitting to calculate the shifts of the energy levels in hydrogen-like atoms with high accuracy is developed. The new logarithmic corrections of the order (alpha) ⁴ln(beta) ^-1, (alpha) ⁵(beta) ²Ln(beta) ^-1 are obtained and earlier results are confirmed.

Semiclassical model of double ionization of Helium atom by fast electron based on the classical trajectories for the ejected electrons is presented. The quantities needed to fit known experimental data are calculated. Typical examples of trajectories of the ejected electrons are presented also.