We study the frequency up- and down-conversion based on the cascaded process which accompanies large phase mismatching between fundamental and intermediate waves. So, for example, we propose using the cascaded Second-Harmonic Generation (SHG) to implement the frequency conversion process, which is similar to that occurring in a medium with cubic susceptibility. At phase matching between the fundamental wave and the third-harmonic wave, Third Harmonic Generation (THG) occurs with high efficiency (94.5%). We demonstrate that the cascaded process may also influence negatively on the frequency conversion processes. SHG in a medium with combined quadratic and cubic nonlinear response accounting for weak THG at large phase mismatching demonstrates that in some cases, the second harmonic intensity evolution is different, whether we take into account of weak THG process or not. So, then the intensity at doubled frequency may be much lower in certain sections in comparison with those without accounting for THG. We study such an influence of weak THG both for pulses with long duration by developing analytical approach and for short pulses based on computer simulation.
One of very interesting phenomena under the frequency conversion appears, if incident intensity of basic wave is enough high. In this case due to self- and cross-modulation of basic wave and wave with doubled frequency a synchronous mode of laser pulses changing appears under certain condition as well as color soliton formation. For both processes the main role plays frequency modulation of the interacting waves. Therefore, investigation of the pulse chirp appearance is very important for color soliton formation. First, we investigate the phenomenon using the analytical approach without applying the basic wave energy non-depletion approximation. Based on original approach we derive the solution of Schr¨odinger equations, describing the SHG for femtosecond pulses, that describes correctly the energy conversion with propagation distance, which is much greater than the dispersion length and nonlinear length, for incident Gaussian shape (or other shape) of pulse with basic frequency. Using this solution we compare the frequency modulation of both interacting pulses for various incident temporal distribution of wave with basic frequency.
SHG is used in many practical applications such as a substance diagnostics, and imaging of various processes as well as for a frequency conversion of an optical pulse. As a rule, the frequency doubling is used also for a generation of the optical wave with tripled frequency by using the mixing of optical radiation with the basic and doubled frequencies. Nevertheless, THG can occur in a medium with cubic nonlinear response and in a medium with quadratic nonlinear response if the incident laser pulse intensity of the basic wave is enough high or if the intensities of the interacting waves increase until such values which are sufficient for occurring the cubic nonlinearity of a crystal. As a consequence, THG may occur. We investigate an efficiency of comb frequency generation if an intensity of the incident femtosecond pulse is sufficiently high .
Obviously, one of these generation processes occurs with a big phase mismatching. We investigate comb generation efficiency in a dependence of the phase mismatching between interacting waves and other parameters. As a result, we observe various regimes of comb generation. It should be emphasized that we find out the generation regime with synchronic changing of the maximal intensities of all waves. At the same time, shape of the appearing sub-pulses at these frequencies are similar to Gaussian one.
Our consideration is based on computer simulation with taking into account the SOD, phase mismatching, selfand cross-modulation for interacting waves.
SHG and SFG (SWG) and THG are used widely in many practical applications such as a substance diagnostics, and imaging of various physical, chemical and biological processes as well as for laser radiation frequency conversion. One of very interesting phenomena under the frequency conversion takes place if a basic wave incident intensity is enough high: a synchronic mode of the laser pulse intensities changing along a propagation coordinate appears under certain conditions. First of all, we investigate this phenomenon using the frame-work of long pulse duration approximation and plane wave approximation without applying the basic wave energy non-depletion approximation. Applying an original approach we derive the solution of Schr¨odinger equations describing the THG via a SHG process and summary frequency wave generation (SFG) process for femtosecond pulses. Among many modes of the frequency conversion process under consideration we found out analytically the mode corresponding to synchronous intensities changing for the interacting waves. We derive conditions of such mode realization in dependence of the problem parameters. After that we verify our analytical consideration using a computer simulation of the problem on the base of the corresponding Schr¨odinger equations. Computer simulation shown also a new phenomenon at three-wave interaction: interacting wave intensities changing with two (or more) oscillation periods.
SHG is used in many practical applications such as a substance diagnostics, and imaging of various processes as well as for frequency conversion. Well known that the frequency doubling is used also for a generation of tripled frequency wave due to mixing of optical radiation at basic frequency with optical radiation at doubled frequency. In this case an important role plays a relation between phases of interacting waves with basic frequency and doubled frequency. Therefore, a derivation of the corresponding law for phase evolutions is an urgent problem. Below we provide such derivation for a SHG of high intensive femtosecond pulse with taking into account an influence of a cubic nonlinear response on the frequency doubling. Using the frame-work of long pulse duration approximation and plane wave approximation as well as an original approach we derive the solution of Schrodinger equations describing the SHG for femtosecond pulse without using the basic wave energy non-depletion approximation. It should be stressed, that the frequency conversion in conditions under consideration possesses multi-stability: there are many modes of SHG efficiency. We write a wave phase evolution for each of the modes. The derived formulas are verified by computer simulation based on using the corresponding Schrodinger equations.
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