Hysteretic behavior of spectral components of a femtosecond pulse passing through a nonlinear layer

The variation of the frequency at which the spectral intensity is maximal within a given time interval with the amplitude of an applied femtosecond pulse is simulated. The computer simulation is performed in the model of an optically thin layer for a medium with cubic nonlinearity and saturable restoring force. It is shown that the hysteretic dependence of the spectral line frequency on the applied pulse amplitude may take place for both one and several simultaneously generated harmonics.     

Nonlinear pulse propagation in a single- and a few-cycle regimes with Raman response

The propagation equation for a single- and a few-cycle pulses was derived in a cubic nonlinear medium including the Raman response. Using this equation, the propagation characteristics of a single- and a 4-cycle pulse, at 0.8 μm wavelength, were studied numerically in one spatial dimension. It was shown that Raman term does influence the propagation characteristics of a single-as well as a few-cycle pulses by counteracting the self-steepening effect.     

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.     

On the modulation instability of femtosecond light pulses propagating in an optical fiber

The well-known problem of the modulation instability of femtosecond light pulses propagating in a cubic nonlinear medium is considered on the basis of computer simulation with allowance made for the time dispersion of the nonlinear response. The dependence of the maximum perturbation frequency at which the instability takes place on the parameter characterizing the time dispersion of the nonlinearity is studied. The numerical experiments carried out verified the previous conclusion of the authors, based on an original approach to the analysis of propagation of a femtosecond pulse, that the frequency interval of the modulation instability can increase with increasing influence of the dispersion of the nonlinear response. This conclusion is at variance with the conclusion drawn by other researchers on the basis of linear analysis.     

Velocity of the maximum of the envelope of a frequency-modulated Gaussian pulse in an amplifying nonlinear medium

The variation in the velocity of propagation of a narrowband frequency-modulated pulse in a nonlinear medium with dispersion of gain or absorption is studied. It is shown that the self-phase modulation caused by a cubic nonlinearity of the Kerr type can “accelerate” optical pulses of a Gaussian shape up to superluminal velocities even in the case where an initial frequency modulation (chirp) is absent.     

On certain regularities of the response of a medium upon generation of the triple frequency of a few-period laser pulse in the case of its passage through an optically thin nonlinear layer

The response of a medium at the triple frequency under the action of few-period laser pulses is considered within the framework of the thin-optical-layer approximation for the case where the triple frequency is close to the natural frequency of the linear oscillator. It is shown that a bistable dependence of the polarization amplitudes on the external action amplitude simultaneously appears at both the natural frequency and the frequency of the external action. This allows one, in particular, to reveal the presence of bistability of the regime of third harmonic generation in a physical experiment by using the transmitted radiation of the acting pulse. A decrease in the duration of the pulse incident on the medium leads to an increase in the hysteresis loops. The effect of the absolute phase of a short pulse on the spectral composition of the response of the medium is studied. For pulses of medium duration, in addition to the resonant response of the medium, the presence of the dynamic response of the medium at the triple frequency was revealed, despite a detuning of the latter from the resonance. The presence of this frequency in the spectrum of the response of the medium results in the dependence of the resonant response of the medium on the absolute phase even for a sufficiently long pulse containing tens oscillations of the electric field strength under the pulse envelope. To obtain the dynamics of the spectral lines from the results of computer simulation, the Fourier-Gabor method is used, the applicability of which is demonstrated by the comparison of the results obtained on its basis with the corresponding analytical dependences.     

粒子数密度对飞秒Gauss型脉冲传播及光谱特性的影响

利用预估校正-时域有限差分(PC-FDTD)法求解全波Maxwell-Bloch方程,研究介质粒子数密度(N)对飞秒Gauss型激光脉冲在Λ型三能级原子介质中传播及光谱特性的影响.结果表明:小面积2π脉冲在不同N介质中都不发生分裂,脉冲频谱基本没有新的高频成分产生,随N增大中心频率附近光谱强度明显减小.面积4π脉冲,在N较大的稀疏介质及稠密介质中都产生分裂,在稀疏介质中随N增大频谱展宽幅度及高频成分强度增大,但在稠密介质中频谱展宽变小且远小于N较大时的稀疏介质情况.大面积8π脉冲,脉冲分裂情况与4π脉冲情况相似,但随N增大频谱展宽幅度及高频成分强度单调增大,且在稠密介质中的频谱展宽幅度及高频成分强度远大于N较小的稀疏介质情况.     

Diffraction-free propagation of a focused delta-pulsed beam

The propagation of short broadband pulses in a homogeneous medium is analyzed on the basis of the wave equation and in the parabolic approximation of that equation. It is shown that the diffraction spreading of a focused optical pulsed beam does not occur in the limiting case of a δ pulse.     

Mutual influence of the spectral components of perturbations on the modulation instability frequency interval in a cubic nonlinear medium

The influence of the shape of a Gaussian pulse propagating in a cubic nonlinear medium on the development of its modulation instability is analyzed, taking into account the time dispersion of the nonlinear response as well. Analytical dependences of the frequency interval of the perturbation development on the dispersion of the nonlinear response, the pulse duration, and the number of simultaneously acting perturbations, which affect each other due to the inhomogeneous pulse shape, are obtained. It is shown that, with increasing dispersion of nonlinear response, the frequency interval of the instability development initially narrows and then expands. Qualitative agreement between the analytical dependences and the results of computer simulation is demonstrated. The frequency interval of the modulation instability development in a medium without the nonlinear response dispersion that was previously determined by different authors under the assumption of “long” pulses is found to coincide with that obtained in this study for the Gaussian pulse under the assumption of the simultaneous action of a large number of perturbations.     

Breathing and beating dynamics of Gaussian beam in planar graded-index absorbing nonlinear waveguides

The spatial dynamics of laser beams in absorbing planar waveguides with a parabolic index profile in a saturable or cubic-quintic medium are calculated using the “collective variable approach” technique. In the absence of losses, we construct diagrams which define regions of self-focusing and self-diffractive beam propagation for both types of media. It is found that propagating pulses exhibit an oscillatory pattern, similar to breathing behavior in homogeneous media. If the incident pulse spatial profile and the center of the index profile are not aligned, the pulse oscillates around the index origin with a “beat” frequency that depends on the graded index. Both the breathing and the beat frequencies are also calculated for other graded-index profiles, such as those with additional higher-power terms, and are found to be extremely sensitive to the index profile. In media with linear and nonlinear absorption, we demonstrate the difference between the breathing behavior in graded-index and homogeneous waveguides.     

Amplitude and phase measurements of femtosecond pulse splitting in nonlinear dispersive media

Frequency-resolved optical gating is used to characterize the propagation of intense femtosecond pulses in a nonlinear, dispersive medium. The combined effects of diffraction, normal dispersion, and cubic nonlinearity lead to pulse splitting. The role of the phase of the input pulse is studied. The results are compared with the predictions of a three-dimensional nonlinear Schr?dinger equation.     

Propagation of elliptically polarized laser pulses in isotropic gyrotropic medium with relaxation cubic nonlinearity and anomalous frequency dispersion

The self-action of elliptically polarized Gaussian laser pulses in an isotropic gyrotropic medium with an anomalous frequency dispersion and cubic Kerr nonlinearity with a finite relaxation time on the order of the pulse duration is numerically studied. It is shown that, at the output of the medium, the pulse polarization nonmonotonically varies with time. The main peak of the pulse is additionally delayed compared to the time of passing the linear medium; the value of this delay significantly depends on the polarization of the incident pulse and achieves a maximum for incident pulses whose degree of ellipticity is equal to the ratio of the material constants characterizing the local and nonlocal nonlinear optical response of the medium. It seems promising to search for possible differences in relaxation times depending on the intensity of additions to the refractive indices of the right and left circularly polarized waves by investigating the time dependence of polarization characteristics at the output of the medium.     

Comparison of effects of relative carry-envelope phase on two-color few cycle pulses propagating in dense and dilute lambda-type mediums

In this paper we compared effects of relative carry-envelope phase (RCEP) of two-color few cycle pulses on propagation behavior and spectral property under the single-photon resonant condition in the dense and dilute lambda-type three-level atomic mediums. It is found that, in the dense medium, with propagation distance increasing, the compound pulse of the two-color few cycle pulses will split into two or three sub-pulses with different amplitudes and shapes, or does not split, and this is completely determined by value size of RCEP; with value of RCEP decreasing, range and strength of the spectrum (particularly the higher spectral components) increase obviously, the highest frequency is about eight times of the resonance frequency. In the corresponding dilute medium, effect of RCEP on propagating behavior and spectral property is much different from that in the dense medium; specially with value of RCEP decreasing, spectral strength increasing is evident but spectral range not, the highest frequency of the spectrum is only about two times of the resonance frequency.     

Analysis of ultrafast nonlinear phenomena��s influences on output optical pulses and four-wave mixing characteristics in semiconductor optical amplifiers

We have analysed the output pulse characteristics of semiconductor optical amplifier (SOA). It is shown that they can be modified due to the variation of input parameters, such as, gain, input pulsewidth, input pulse energy and effects imposed by the medium. Therefore, the influence of these parameters are analysed on the output pulse shape, spectrum, chirp and pulsewidth. We have used the nonlinear propagation equation taking into account the gain spectrum dynamics, gain saturation which depends on carrier depletion, carrier heating, spectral hole-burning, group velocity dispersion, self-phase modulation and two photon absorption. We have used the finite-difference beam propagation method to simulate the wave evolution both in time and spectral domain in the SOA. We have also simulated the four-wave mixing characteristics between pulses for various input pulses. An accurate output pulse shape can be achieved by controlling the mentioned parameters. To the authors knowledge, pulse shaping in co-propagation regime due to medium effect and input pulse shapes in presence of all nonlinear effects relevant to picosecond regime have been studied comprehensively, for the first time in this work.     

Soliton regimes of ultrashort pulse propagation through an array of asymmetric quantum objects

Nonlinear regimes of two-component ultrashort pulse propagation through an array of anisotropic quantum systems are analyzed theoretically under conditions when the slowly varying envelope approximation is inapplicable. It is shown that if the spectral width of the pulse is much larger than the quantum transition frequency, then the dynamics of its ordinary component can be described by a modified sine-Gordon equation that belongs to the class of integrable models but has never been used in physical applications. It is shown that the extraordinary pulse component responsible for the dynamic Stark shift of the transition frequency is coupled to its ordinary component by an algebraic relation. The soliton solutions to the derived equation are classified. Depending on the pulse duration, the solutions have the form of 2π pulses (kinks and antikinks) or steady traveling 0π pulses having no counterparts in isotropic models (neutral kinks). Interactions between the medium and solitons of different types and soliton collisions are analyzed.     

Dynamics of ultimately short pulses in partially absorbing media

Propagation of broad-band ultimately short light pulses in a partially absorbing medium is analyzed in the framework of the three-level model. Nonlinear wave equations are obtained describing propagation of light pulses in media with quadratic (all three transitions are allowed) or cubic (one of the transitions is forbidden) nonlinearity in the range of optical transparency or with the sine-Gordon-type nonlinearity in the region of absorption. Using the averaged variational principle, the approximate solutions of equations in the form of unipolar soliton-like signals are found and conditions of their transverse stability are determined. A stable propagation of a broad-band pulse is shown to be possible under conditions when monochromatic signals exhibit self-focusing.     

Influence of Polarization and Pulse Shape of Femtosecond Initial Laser Pulses on Spectral Broadening in Microstructure Fibers

We theoretically studied the influence of initial parameters of laser pulses, such as polarization, pulse shape and frequency chirp, on the broadening of spectrum during pulse propagation through microstructure fibers (MSFs). We utilized two coupled-mode equations based on the nonlinear Schrödinger equation using an intermediate-broadening model for a Raman response function, and the dispersion coefficients from 2nd to 7th orders for the slow and fast axes, respectively, of highly birefringent MSFs.     

Propagation of the ultrashort laser pulses through the quantum nonlinear medium with resonant properties

The propagation of ultrashort few-cycle laser pulses through a quantum nonlinear medium with resonant properties is studied using the method of slowly varying amplitudes with allowance for dispersion effects. A self-consistent system of nonlinear wave equation for the evolution of laser field and the nonstationary Schrödinger equation that determines the material polarization response is numerically solved. Specific features of the propagation of laser pulse caused by a relatively large spectral width and the nonadiabatic character of the laser pulse are established. The rise of the slowly dumping polarization at the eigenfrequency of the medium and the consequent stretching of the laser pulse are observed. The role of the resonant absorption of the laser energy in the medium is analyzed. The nonlinear spectral broadening is demonstrated for the laser pulse propagating through the medium.     

Modelling of nonlinear effects and the response of ultrasound contrast micro bubbles: simulation and experiment

The propagation of diagnostic ultrasonic imaging pulses in tissue and their interaction with contrast micro bubbles is a very complex physical process, which we assumed to be separable into three stages: pulse propagation in tissue, the interaction of the pulse with the contrast bubble, and the propagation of the scattered echo. The model driven approach is used to gain better knowledge of the complex processes involved. A simplified way of field simulation is chosen due to the complexity of the task and the necessity to estimate comparative contributions of each component of the process. Simulations are targeted at myocardial perfusion estimation. A modified method for spatial superposition of attenuated waves enables simulations of low intensity pulse pressure fields from weakly focused transducers in a nonlinear, attenuating, and liquid-like biological medium. These assumptions enable the use of quasi-linear calculations of the acoustic field. The simulations of acoustic bubble response are carried out with the Rayleigh-Plesset equation with the addition of radiation damping. Theoretical simulations with synthesised and experimentally sampled pulses show that the interaction of the excitation pulses with the contrast bubbles is the main cause of nonlinear scattering, and a 2-3 dB increase of second harmonic amplitude depends on nonlinear distortions of the incident pulse.     

Short chirp pulses in graded-index light guides

We consider the nonlinear process of propagation of a short chirp optical pulse in a light guide in which the refractive index of a fiber material strongly depends on the radial coordinate and weakly depends on the longitudinal coordinates. The chirped and strongly chirped pulses are distinguished by the relationship between the modulation depth and the spectrum width. The chirp-pulse propagation is simulated by a nonlinear wave equation solved asymptotically with respect to the small parameter prescribing the order of magnitude of the pulse amplitude. It is shown that the pulse propagation is three-scale. The phase of high-frequency filling and the field distribution in the fiber cross section are obtained as the eigenvalue and eigenfunction of a singular Sturm-Liouville problem, respectively. On the additional assumption of longitudinal inhomogeneity of the fiber, the general equation for the pulse envelope is reduced to the second Painlevé equation. It is shown that the linear modulation of the carrier frequency is not an independent characteristic of the propagation process, but varies in certain accordance with parameters of the transverse and longitudinal inhomogeneities of the fiber. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 1, pp. 64–71, January 2006.