Phase effects at second harmonic generation in the layer media

An analysis is made of quasi-phase-matched interaction at second harmonic generation in a regular domain structure with accounts for losses and change of phases of all the interacting waves. The constant-intensity approximation of basic radiation is applied for this purpose, not to regular domain structure as a whole, but to each separately taken domain. It allows to carry out more strict analysis of quasi-synchronous interaction during the frequency conversion to the polydomain consisting of n layers, forming “grating” periods of modulation of the nonlinear susceptibility. With this, the values of complex amplitudes of basic radiation and second harmonic wave at the outlet of each domain are entrance values of the corresponding complex amplitudes to the following domain. The analytical expression is given for the case of n domains and the factors limiting the efficiency of the process of frequency conversion are analyzed. In the constant-intensity approximation, in contrast to the constant field approximation, the coherent length of domain depends on pump intensity. With increasing pump intensity the optimum length decreases. In a regular domain structure at frequency conversion from a layer to a layer, intensity of the basic radiation changes. Also the optimum length of domains at which conversion efficiency is maximal therefore changes. Thus, it is possible to obtain the high values of conversion frequency at the outlet of a regular domain structure by choice of optimum parameters of a task (length of domains, phase mismatch, pump intensity), as well as using the layers-domains of high quality.     

Third-harmonic generation in regular domain structures

The process of frequency conversion in regular domain structures is studied using the constant-intensity approximation. The investigations are carried out at values of complex amplitudes of the fundamental radiation and third harmonic at the output of each domain equal to the values of the corresponding complex amplitudes at the input of the subsequent domain. We show that the optimum length of each domain depends on the input pump intensity in the given domain. Thus, it is possible by choosing the optimum lengths of domains, phase mismatch, and pump intensity even at a low number of periods of nonlinear susceptibility modulation of the lattice to reach considerable values of conversion efficiency at the structure output in comparison with the traditional case of homogeneous nonlinear media.     

Cascade parametric light amplification with low-frequency pumping

Cascade parametric amplification in a regular domain structure is examined theoretically taking the variations in phase of all the interacting waves in a dissipative medium into account. Analytic expressions are obtained for the conversion efficiency of the laser energy with low-frequency pumping. The dependence of the parametric amplification efficiency on the number of layers is given. Ways of increasing the frequency conversion efficiency are discussed. The presence of a nonzero input intensity at the sum frequency is found to cause a nonlinear increase in the high-frequency signal at the output of the structure. As the losses of the interacting waves increase, both the frequency conversion efficiency and the optimum domain length decrease.     

Quasi-phase-matched sum-frequency generation in layered structures

A theory has been developed for quasi-phase-matched generation of a wave at a sum-frequency within a constant-intensity approximation. In contrast to a constant-field approximation, the constant-intensity approach has been found to give an optimum pump intensity value at which the conversion efficiency reaches its maximum. Analytical expressions have been derived for optimal values of the problem’s parameters. Ways to enhance the power of the coherent optical radiation are shown and analyzed. This will make it possible, in particular, to increase the power in an RGB-source color for which the sum-frequency generation is responsible.     

Efficiency of sum frequency generation by regular domain structures

We theoretically examine generation of sum-frequency radiation in a regular domain structure, taking into account phase changes for all the interacting waves in a dissipative medium. We show that in contrast to the constant-field approximation, in the constant-intensity approximation there is an optimal value for the longitudinal dimension (length) of the domains at which maximum conversion of the signal wave energy to the sum-frequency wave energy occurs. As the pump intensity increases, the optimal domain length decreases. As the losses of the interacting waves increase, both the conversion efficiency and the optimal domain dimensions decrease.     

Phase Effects of the Parametric Interaction in Metamaterials

We study the phase effects of the three-wave parametric interaction in metamaterials considering the negative refraction at the frequency of a signal wave. We analyze the efficiency of energy conversions between two direct waves with respect to the energy of the backward signal wave along with the dynamics of the signal-wave amplification in metamaterials. We show that there exist optimum values of the fundamental-wave intensity and the phase mismatch at which the efficiency of conversion is maximum. We obtain an analytic expression for the optimum value of the relative length of the metamaterial and present a numerical evaluation of the expected value for efficiency of the frequency parametric conversion in dielectric waveguides. A sufficient enhancement in the signal-wave amplification, which is possible at optimum values of the pump intensity and the metamaterial total length, leads to the parametric generation of the signal wave. Changing the frequency and power of the pump wave, one can realize a regular tuning of the frequencies of parametric converters.     

Theoretical studies on frequency doubling in glass optical fibers in constant-intensity approximation

Theoretical consideration of the second harmonic generation in the constant-intensity approximation with account of the inverse effect of the excited wave on a stimulation wave through account of the phase change of interaction waves is presented. The behavior of harmonic wave intensity in a fiber for different parameters of the problem has been studied, the self-action of the light both in glass optical fiber and inside the homogeneous medium has been analyzed. In this approximation a decrease of conversion efficiency and a change of parameters of the curves of synchronism versus the pump intensity, in particular, the change of positions of intensity minima that does not take place in the constant-field approximation, are observed. The propagation of a plane wave packet in the homogeneous medium with quadratic nonlinearity and in an optical fiber accompanied by the self-phase modulation and leading to the change of the Gaussian pulse spectrum is analyzed with account of the phase changes of all the interacting waves.     

Experimental observation of motion of edge dislocations in Ge/Ge<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>Si<Subscript>1–<Emphasis Type="Italic">x</Emphasis></Subscript>/Si(001) (<Emphasis Type="Italic">x</Emphasis> = 0.2–0.6) heterostructures

The rotating magnetohydrodynamic flows of a thin layer of astrophysical and space plasmas with a free surface in a vertical external magnetic field are considered in the shallow water approximation. The presence of a vertical external magnetic field changes significantly the dynamics of wave processes in an astrophysical plasma, in contrast to a neutral fluid and a plasma layer in an external toroidal magnetic field. There are three-wave nonlinear interactions in the case under consideration. Using the asymptotic method of multiscale expansions, we have derived nonlinear equations for the interaction of wave packets: three magneto- Poincare waves, three magnetostrophic waves, two magneto-Poincare and one magnetostrophic waves, and two magnetostrophic and one magneto-Poincare waves. The existence of decay instabilities and parametric amplification is predicted. We show that a magneto-Poincare wave decays into two magneto-Poincare waves, a magnetostrophic wave decays into two magnetostrophic waves, a magneto-Poincare wave decays into one magneto-Poincare and one magnetostrophic waves, and a magnetostrophic wave decays into one magnetostrophic and one magneto-Poincare waves. There are the following parametric amplification mechanisms: the parametric amplification of magneto-Poincare waves, the parametric amplification of magnetostrophic waves, the amplification of a magneto-Poincare wave in the field of a magnetostrophic wave, and the amplification of a magnetostrophic wave in the field of a magneto-Poincare wave. The instability growth rates and parametric amplification factors have been found for the corresponding processes.     

Rossby waves in the magnetic fluid dynamics of a rotating plasma in the shallow-water approximation

We have studied rotating magnetohydrodynamic flows of a thin layer of astrophysical plasma with a free boundary in the β-plane. Nonlinear interactions of the Rossby waves have been analyzed in the shallow-water approximation based on the averaging of the initial equations of the magnetic fluid dynamics of the plasma over the depth. The shallow-water magnetohydrodynamic equations have been generalized to the case of a plasma layer in an external vertical magnetic field. We have considered two types of the flow, viz., the flow in an external vertical magnetic field and the flow in the presence of a horizontal magnetic field. Qualitative analysis of the dispersion curves shows the presence of three-wave nonlinear interactions of the magnetic Rossby waves in both cases. In the particular case of zero external magnetic field, the wave dynamics in the layer of a plasma is analogous to the wave dynamics in a neutral fluid. The asymptotic method of multiscale expansions has been used for deriving the nonlinear equations of interaction in an external vertical magnetic field for slowly varying amplitudes, which describe three-wave interactions in a vertical external magnetic field as well as three-wave interactions of waves in a horizontal magnetic field. It is shown that decay instabilities and parametric wave amplification mechanisms exist in each case under investigation. The instability increments and the parametric gain coefficients have been determined for the relevant processes.     

Phase effects at second harmonic generation of powerful laser radiation in noncentrosymmetrical media

There has been developed the theory of second harmonic generation of the intensive laser fields in the existence of both quadratic and cubic polarization in medium in the constant-intensity approximation accounting for the reverse effect of the excited wave on the exciting one and simultaneously allowing us to take into account phase mismatch and the damping of all the interacting waves. It is shown that the changes of pump intensity through self-phase and cross-phase modulation processes effect optimum phase relationship between interacting waves, the change of the spatial beating period. The conditions of compensating undesirable phase shifts between interacting waves have been determined, the analytical expressions for calculation of optimum values of phase mismatch, length of noncentrosymmetrical medium and the spatial beating period are offered. It is shown that in the absence of linear phase mismatch with an increase of basic radiation intensity the spatial beating period is being reduced. The numerous analysis has been made of frequency doubling process efficiency for KDP and LiNbO₃ crystals.     

Optical parametric amplification beyond the slowly varying amplitude approximation

The coupled-wave equations describing optical parametric amplification (OPA) are usually solved in the slowly varying amplitude (SVA) approximation regime, in which the second-order derivatives of the signal and idler amplitudes are ignored and in fact the electromagnetic effects due to exit face of the medium is not involved. Here, an analytical plane-wave solution of these coupled-wave equations in a non-absorbing medium is presented. The solutions are derived beyond the SVA approximation up to order of κ/k (coupling constant over the wave number). The intensity distributions of the signal and the idler waves show a periodic behavior about their corresponding distributions of SVA-adapted solution. This behavior can be explained by the interference of the forward propagating signal (idler) wave and the corresponding backward one resulted from the reflection by the end face of the medium. Furthermore, this interference pattern in the medium can in turn serve as a periodic source for the next generations of the signal and idler waves. Therefore, the superposition of the waves, generated from different points of this periodic source, at the exit face of the medium shows an oscillatory behavior of the transmitted signal (idler) wave in terms of normalized coupling constant, κL. This study also shows that this effect is more considerable for high intensity pump beam, high relative refractive index and short length of the nonlinear medium.      

Causality and the velocity of acoustic signals in bubbly liquids

Several versions of the dispersion formula governing the acoustic propagation in bubbly liquids are shown to exhibit acausal behavior. The cause of this behavior is due to the inappropriate application of a low frequency approximation in the determination of the extinction of the signal from radiative scattering. Using a corrected causal formula, several principles of wave propagation in bubbly media consistent with the general theory of wave propagation in dispersive media are demonstrated: There exist two precursors to any finite signal. Both propagate without regard to the source characteristics at velocities, frequencies, and amplitudes dependent wholly upon the characteristics of the medium supporting the wave motion. The first travels at the infinite frequency phase velocity that is coincident with the infinite frequency limit of the group velocity. That part of a propagating wave oscillating at the source frequency arrives at a time determined by the signal velocity. Analogous to the well known signal velocity of electromagnetic wave propagation in conducting media, the value of the signal velocity depends on the detailed structure of the dispersion formula in the complex frequency plane.     

Exact treatment of the saturation behaviour of second-harmonic generation

We investigate the interaction between a fundamental plane wave and the second-harmonic wave propagating in a nonlinear medium. Of particular interest is the dependence of the second-harmonic amplitude and of the susceptibility on the amplitude of the fundamental wave in the stationary case. Therefore, spatial structures are assumed as simple as possible (amplitudes constant in space, phase matching direction). The susceptibilities are not calculated by iteration, as it was done so far, but exactly, using a classical model of matter (anharmonic oscillators). The exact solution is expanded into series for small field amplitudes and compared term by term with the results achieved by the usual methods, i.e. calculation of the susceptibilities by perturbation theory, parametric approximation, coupled amplitude approximation. The lowest term is identical with the parametric approximation. The additional term of the coupled amplitude approximation, however, is not consistent with the exact solution.     

Theorie der parametrischen Vier-Photonen-Wechselwirkung. II., Verstärkung und Oszillation in der Näherung des vorgegebenen Pumpfeldes)

The processes of parametric amplification and oscillation in a medium with cubic polarization is investigated in parametric approximation. For the case of amplification the threshold power of the pump wave is calculated. The influence of a finite phase mismatching and of the initial phases on the amplification process is discussed. For larger values of mismatching a periodical change between amplification and depletion may appear. For the cases of a single resonant oscillator (for the signal wave only), a double resonant oscillator (for both signal and idler wave) and a double resonant oscillator with the pump wave reflected at the second mirror the threshold power of the pump wave and the rise time are calculated. Characteristic differences between four-photon and three-photon parametric oscillators are discussed. A numerical estimation of the threshold power for some media with nonlinear susceptibilities known from literature leads to the conclusion that the realization of a four-photon parametric oscillator should be possible. In the appendix we discuss the influence of diffraction, focussing, birefrigence, nonstationarity and depletion of the pump wave on the studied process as well as the conditions under which these effects can be neglected.     

Visualization of phase conjugate ultrasound waves passed through inhomogeneous layer

Compensation of phase distortions of ultrasound beams by means of parametric phase conjugation is visualized. Quasi-plane and focused primary beams were distorted by a polymer aberration layer introduced between the primary wave source and the wave phase conjugator. It is demonstrated acousto-optically that, while the acoustic field is strongly irregular in the area between aberration layer and conjugator, the phase conjugate wave visibly reproduces the primary beams in the area between the layer and the primary wave source. The phenomenon is observed in supercritical mode of parametric amplification when intensity of phase conjugate wave is high enough for manifestations of acoustic nonlinearities in water.     

Parametric instability and Hamiltonian chaos in cavity semiclassical electrodynamics

We study the nonlinear dynamics of the interaction of two-level atoms and a selected mode of a high-Q cavity with frequency modulation analytically and numerically. In the absence of modulation, the corresponding semiclassical Heisenberg equations for the expectation values of the collective atomic observables and the field-mode amplitudes allow, in the rotating wave approximation and in the strong-coupling limit, an exact solution with arbitrary detuning. Using this solution, we detect the coherent effect of trapping of the population of atomic levels and of trapping of the number of photons in the cavity. The explanation for this effect lies in the destructive interference of the atomic dipoles and the field mode. The integrable version of the system of equations exhibits a separatrix near which a stochastic layer is formed when modulation is introduced. The width of the layer is found to gradually increase with degree of modulation, and finally it fills the entire energy-permissible volume of the phase space. We show that the rotating wave approximation does not hinder the formation of Hamiltonian chaos in cavity semiclassical electrodynamics. The calculation of the maximum Lyapunov indices of nonlinear (in this approximation) equations of motion as functions of the modulation frequency δ and the frequency of natural Rabi oscillations of the atom-field system, Ω, suggests that Hamiltonian chaos appears first in the area of the fundamental parametric resonance, δ/2Ω≃1. Parametric instability increases with increasing modulation and decreasing detuning from the atom-field resonance, generating at exact resonance new areas of chaos corresponding to multiple parametric resonances. The results of numerical experiments and estimates of the characteristic parameters show that Rydberg atoms placed in a high-Q microwave cavity are possible objects for observing parametric instability and dynamical chaos. Zh. éksp. Teor. Fiz. 115, 740–753 (February 1999)     

Quantum properties of optical images in coupled nondegenerate parametric processes

Quantum properties of parametric amplification and frequency conversion of an optical image in coupled parametric interactions are analyzed for the case of a remote object. The coupled-wave interaction consists of a traditional process of parametric amplification in the high-frequency pump field accompanied by the frequency mixing of generated and pump waves. The photon number density and signal-to-noise ratio at the interacting frequencies are studied in the image plane at the output of the nonlinear frequency converter in the fixed pump field approximation. The degree of entanglement of optical images at frequencies above and below the pump frequency is investigated.     

高阶色散对高双折射光纤增益谱的影响

根据参量放大和拉曼效应共同作用下的耦合波方程,在考虑高阶色散情况下,当输入泵浦波偏振方向同双折射轴成45时,通过引入拉曼增益的洛伦兹模型,研究了高双折射光纤中,参量放大和拉曼效应共同作用下的增益谱随相关参量的变化关系。结果表明:高双折射光纤中,在不同色散区,不同的输入参量（输入功率、群速度失配等）条件下,参量放大和拉曼效应共同作用下的增益谱受到高阶色散的影响,增益谱的结构、强度和谱宽产生了变化, 高阶色散对增益谱的影响不可忽略;可以利用增益谱在大群速度失配区域远离中心频率偏移的性质,提取T频率脉冲。     

Phase effects at second-harmonic generation in metamaterials

We develop the theory of second-harmonic generation in metamaterials where fundamental radiation is considered in the constant-intensity approximation taking into account the phase changes of interacting waves. We investigate the second-harmonic wave intensity in metamaterials at different parameters of the problem under consideration. We consider a self-action effect of the light wave in metamaterials in view of the phase changes of all the interacting waves and compare this effect with an analogous effect in usual homogeneous quadratic media. We show that it is possible to change the phase velocity of the pump wave by varying such parameters as the pump intensity, nonlinear medium length, and phase mismatch between the interacting waves. The metamaterials under consideration provide a possibility to change the phase of the fundamental radiation of order of magnitude higher than those in the usual homogeneous quadratic media.     

Dynamics of non-linear optical processes

We study the statistical properties of parametric processes (parametric amplification, frequency conversion, splitting of frequency, etc.) described by the trilinear Hamiltonian and of the second harmonic generation including the lossy mechanism and rotating terms. In a short-time approximation some conservation laws are derived and the existence of the Glauber-Sudarshan weighting function is discussed. It is shown in the second harmonic generation case that good coherence of the incident radiation is lost proportionally to the intensity in interaction while the second harmonic radiation has tendency to be coherent. The effects of the spontaneous emission of the medium and of the physical vacuum are also discussed. In this approximation the statistics is generally described by the superposition of coherent and chaotic fields with correlated components.