V. A. Trofimov

Conservative finite-difference scheme for the problem of propagation of a femtosecond pulse in a nonlinear photonic crystal with nonreflecting boundary conditions

Conservative finite-difference schemes are constructed for the problems of self-action of a femtosecond laser pulse and of second-harmonic generation in a one-dimensional nonlinear photonic crystal with nonreflecting boundary conditions. The invariants of the governing equations are found taking into account these conditions. Nonreflecting conditions substantially improve the efficiency of conservative finite-difference schemes used in the modeling of complex nonlinear effects in photonic crystals, which require much smaller steps in space and time than those used in the case of linear propagation. The numerical experiments performed show that the boundary reflects no more than 0.01% of the transmitted energy, which corresponds to the truncation error in the boundary conditions. The amplitude of the reflected pulse is less than that of the pulse transmitted through the boundary by two (and more) orders of magnitude. The simulation is based on the approach proposed by the authors for the given class of problems.

Soliton-like propagation of light pulses in a nonlinear photonic crystal

On the basis of computer simulation, we analyze the dynamics of interaction of subpulses localized inside certain layers of a one-dimensional nonlinear photonic crystal. Localized subpulses are shown to move and interact like solitons. Possible scenarios of their interaction are described: a spatially periodic regime, the merging of several subpulses with similar characteristics into a single higher intensity and more slowly propagating subpulse, and the formation of an immovable pulse in a layer of the photonic crystal. To verify the results of the computer simulation and to elucidate the nature of spatially localized structures forming, a comparison with the analytical solution for a medium with cubic nonlinearity is performed.

Second harmonic generation by femtosecond pulses under the condition of nonzero amplitude of the double-frequency wave

Second-harmonic generation by femtosecond pulses in a medium with simultaneous action of the second-and third-order nonlinearities under the condition of group matching of interacting waves is analyzed. The problem is solved analytically within the long-pulse approximation, and possible generation modes are identified. In particular, it is shown that the presence of even a small second-harmonic signal (with the intensity comprising no more than 1% of that of the main wave) enables the maximal efficiency of energy conversion to be attained well before the intensities of interacting waves are increased by more than one order of magnitude due to the compression of pulses. This result is practically significant since it allows one to avoid the optical breakdown of the medium.

Localization of light energy of a femtosecond laser pulse upon generation of the second harmonic in a one-dimensional nonlinear photonic crystal with alternating nonlinear response

A possibility of localization of the light energy upon generation of the second harmonic of a femtosecond laser pulse in a one-dimensional photonic crystal, some layers of which are either quadratic or cubic nonlinear, is demonstrated in computer simulation. It is shown that there exist light energy localization modes at one of the interacting frequencies and at both interacting frequencies simultaneously. The formation of solitons in separate layers of the photonic crystal is the basis of light energy localization. Logical operations of Boolean algebra can be implemented based on the discovered effect.

Restoration of a signal and its instantaneous spectral characteristics by the sliding window method

A new algorithm is suggested that makes it possible to gain information on the dynamics of the spectral lines of the response from a medium by singly measuring the integral characteristics of the signal. A technique of separating out algorithm-induced spurious frequencies that are present in the restored signal from true frequencies is described. The efficiency of the algorithm is demonstrated with processing the response of an L-cystine molecule subjected to a short-period terahertz pulse. The results are compared with those obtained by the Fourier-Gabor method, which was tried out in an earlier physical experiment.

Iterative Method for Finding the Eigenfunctions of a System of Two Schrödinger Equations with Combined Nonlinearity

An iterative method is proposed to determine the eigenfunctions of a system of two nonlinear Schrödinger equations governing the interaction of two femtosecond pulses in a medium with quadratic and cubic nonlinearity. The method produces soliton solutions of a new form for a wide range of nonlinearity coefficients corresponding to the first and second eigenvalues. A specially chosen initial approximation is required to determine the third and higher eigenfunctions.

On the possibility of light energy localization in a nonlinear photonic crystal

A possibility of the localization of light energy in a one-dimensional nonlinear photonic crystal is demonstrated by means of computer simulation. The energy localized in the crystal may reach up to 50% of the energy penetrating into it, depending on the femtosecond pulse length and intensity. The effect may be used for recording and storing information in three-dimensional optical memory units. The simulation is performed on the basis of an approach proposed by the authors for the class of problems under consideration.

An all-optical switch based on a one-dimensional layered structure with defocusing nonlinearity

The feasibility of an optical switch of femtosecond range controlled by a femtosecond light pulse acting on a nonlinear one-dimensional photonic crystal (a layered one-dimensional periodic structure) is demonstrated by computer simulation. The crystal comprises linear and weakly defocusing layers. It is shown that switching occurs when the peak intensity of the initial pulse exceeds a certain threshold value due to the dependence of the transmission-reflection bands of the photonic crystal on the nonlinearity of the medium.

Formation of a high-frequency subpulse upon propagation of a femtosecond pulse in a medium with a saturable potential

On the basis of computer simulation of propagation of femtosecond light pulses in terms of the one-dimensional nonstationary nonlinear Maxwell equations with a saturable restoring force, the possibility of formation of a sequence of subpulses with different carrier frequencies is shown. The formation of frequency-modulated pulses with an asymmetric distribution of spectral components with respect to the center of a local line is demonstrated. Appearance of such pulses indicates that a hysteretic dependence of the frequency of the corresponding local line on the amplitude of an external signal can be realized.

Efficient second harmonic generation by a femtosecond pulse far from phase matching

Highly efficient second harmonic generation by a femtosecond pulse under the condition of group matching in the absence of phase matching was obtained. Second-order dispersion and cubic nonlinearity of the propagation medium were considered. The cubic nonlinearity is necessarily required for the generation to be efficient. The results obtained open up a fundamental opportunity for the efficient cascade generation of optical harmonics to be realized.