Nonreciprocity of losses and shift of the zero point in a laser gyroscope

Results of calculations of intensities, losses, and frequencies of counterpropagating waves in a ring resonator containing a weakly nonlinear active medium and an aperture are given. It is shown that inequalities of frequencies and intensities of generation of counterpropagating waves occur in such a resonator. The behavior of these inequalities is determined by the nonreciprocity of frequency-dependent losses of the counterpropagating waves.     

Superluminal parametric Doppler effect in insulators and in an electron-positron vacuum

It has been shown that counterpropagating electromagnetic waves with different frequencies generate two lattices in a cubically nonlinear medium, one of which moves at a superluminal velocity. When weak radiation reflects from the superluminal lattice, the wavefront is quasi-conjugated with distortions owing to the Doppler frequency shift. These effects occur both in insulators with fast nonlinearity and in an electron-positron vacuum.     

Resonant properties of a nonlinear dissipative layer excited by a vibrating boundary: Q-factor and frequency response

Simplified nonlinear evolution equations describing non-steady-state forced vibrations in an acoustic resonator having one closed end and the other end periodically oscillating are derived. An approach based on a nonlinear functional equation is used. The nonlinear Q-factor and the nonlinear frequency response of the resonator are calculated for steady-state oscillations of both inviscid and dissipative media. The general expression for the mean intensity of the acoustic wave in terms of the characteristic value of a Mathieu function is derived. The process of development of a standing wave is described analytically on the base of exact nonlinear solutions for different laws of periodic motion of the wall. For harmonic excitation the wave profiles are described by Mathieu functions, and their mean energy characteristics by the corresponding eigenvalues. The sawtooth-shaped motion of the boundary leads to a similar process of evolution of the profile, but the solution has a very simple form. Some possibilities to enhance the Q-factor of a nonlinear system by suppression of nonlinear energy losses are discussed.     

Wave interaction in acoustic resonators with and without hysteresis

Nonlinear interaction of counterpropagating waves in solids is considered by using a general approach taking into account only the cumulative (resonant) nonlinear perturbations giving a nonzero contribution over the period and, consecutively, potentially able to significantly modify the linear solution. Different stress-strain relations are addressed, including those with hysteresis which serve as basic models for the recent acoustic experiments with rock and metals. An important case of the interaction of counterpropagating waves with close amplitudes in a high-Q resonator (bar) with hysteresis is specially addressed and compared with the case of a ring resonator.     

Methods for eliminating the influence of nonlinear Kerr effect on the zero drift of fiber ring interferometers

Nonlinear Kerr effect leads to the appearance of a periodic structure in the saturated refractive index of an optical fiber, which corresponds to a standing structure formed by counterpropagating waves in the circuit of a fiber ring interferometer (FRI). If the intensities of counterpropagating waves are slightly different, their reflection from this periodic structure leads to the appearance of a phase shift of interference of counterpropagating waves unrelated to rotation at the FRI output. If a nonmonochromatic radiation source is used in the FRI system, only radiation rereflected from the middle of the circuit makes a contribution to the phase shift. A method for eliminating the influence of the nonlinear Kerr effect on the zero shift of fiber ring interferometers is proposed. This consists in making the middle of the circuit discontinuous. Numerical estimates are made.     

Application of the local mode theory to nonlinear waveguides

Local mode theory for two counterpropagating waves in nonlinear waveguides is formulated. This model allows us to describe the transverse effects as well as the counterpropagating waves interaction in optical waveguides. The results are applied to the analysis of the nonlinear Fabry-Perot waveguide resonator and are compared with the conventional coupled mode theory solution.     

Transmission of oscillations through a layer of a nonlinear elastic medium

The transmission of shear one-dimensional periodic perturbations through a layer of a nonlinearly elastic medium under the conditions close to resonance is considered. The layer separates two half-spaces consisting of a medium that is much more rigid, as compared to the medium in the layer. A system of differential equations is obtained for describing the slow variations in the amplitude and waveform of nonlinear strain and stress oscillations at the fixed boundary that occur because of the nonlinear properties of the medium while the other boundary performs arbitrary periodic motions in its plane. The period of these oscillations is close to the period of natural oscillations of the layer. It is shown that, in addition to continuous strain variations at the fixed boundary, strain variations containing strong discontinuities are possible. Relations at the discontinuities are obtained. The analogy between the equations derived for the case under study and the equations describing the propagation of strain waves in a homogeneous anisotropic elastic medium is pointed out.     

Splitting of oscillation frequencies of a ring gas laser caused by the diffraction lens effect: II. Nonlinear theory

An analysis is made of the problem of generating the fundamental mode in a laser cavity containing a weakly nonlinear active medium and an aperture. Frequency-dependent nonlinear loss of counterpropagating waves is calculated. The loss was found to have a jumpwise decrease at the boundary of the lasing region. An explanation is given regarding the mechanism responsible for the asymmetry of loss about the central frequency of the transition. It is shown that counterpropagating waves differ in loss, as well as in the phase velocity, which is one of the reasons of that the counterpropagating waves of a ring laser, without special nonreciprocal devices, have different frequencies and intensities.     

Lateral field instability and six-wave mixing in a diode laser with broad active area

The electromagnetic field inside a nonlinear active medium of a laser is considered as a system of counterpropagating waves. Such an approach changes radically an earlier studied behavior of the lateral field instability due to self-deformaion (or self-focusing). In our calculations we used an expression for a laser field in the form of two “strong” counterpropagating waves whose complex amplitudes have weak perturbations. Amplitude perturbations of each of the “strong” waves can be presented by two spatial harmonics corresponding to two weak perturbation waves with wave vectors making some tilted angle ±φ with the cavity axis. Thus six waves would participate in the interaction: two counterpropagating strong waves and two pairs of weak waves. Using this approach, we have developed a theory for the propagation of four “weak” perturbation waves in a nonlinear amplifying medium in the presence of two counterpropagating “strong” waves. It is shown that perturbation waves with tilted angle φ⋍0.5–1.2° inside the active region, and respecively, with the side lobes of the far-field pattern at ∼1.7–4°, have the greatest growth increment. These perturbation waves produce lateral intensity modulation with period 10–30 μm for the 0.85 μm lasing wavelength. The appearance of such waves corresponds to the instability threshold of a homogeneous lateral distribution of optical power in a diode laser. The present theory makes it possible to investigate the stability of the homogeneous lateral optical intensity distribution in a diode laser of any design. This allows one to choose a suitable design of a laser with a homogeneous lateral distribution at high radiation power. Translated from Preprint No. 43 (1992) of the Lebedev Physics Institute, Russian Academy of Sciences.     

Spatial distribution of the interacting Waves’ amplitudes under third harmonic generation in a negative-positive refractive medium

The process of third harmonic generation in a cubically nonlinear medium that is negatively refractive at the fundamental frequency and positively refractive at the third harmonic frequency is considered. For the stationary case, the amplitude distribution was obtained for waves interacting inside a sample for different values of phase mismatch. In the periodic regime of generation, it is shown that the amplitude of the fundamental wave inside the medium can exceed its value at the input, which is impossible in the standard case of harmonic generation in a medium positively refractive at both frequencies. The influence of energy losses in the sample on the spatial distribution of waves’ amplitudes of both frequencies has been studied.     

Nonlinear standing waves in a resonator with feedback control

An experimental study is presented to demonstrate that nonlinear effect on standing waves in a resonator can be reduced by a feedback loop responding to the second harmonic. The resonator was a cylindrical tube sealed at one end and driven by a horn driver unit at another end. The feedback control loop consisted of a pressure sensor, a frequency filter, a phase shifter, and an actuator. The results show that the waveform distortions can be eliminated and large amplitude sinusoidal pressure oscillations are obtained. A simple model is proposed for a qualitative discussion on the control mechanism, which shows that the feedback loop alters the imaginary part of the complex mode frequency so as to suppress (or enhance) the second harmonic.     

Wave convection regimes in a binary mixture in a modulated gravitational field

Nonlinear wave convection regimes are studied in a horizontal layer of an incompressible binary mixture with anomalous thermal diffusion in the gravitational field modulated with an arbitrary amplitude and finite frequency. Oscillation regimes are numerically simulated by the finite difference method for the case of a layer with impenetrable rigid boundaries, which better corresponds to experimental laboratory conditions. A qualitative difference is found in the dynamics of nonlinear quasi-periodic and subharmonic oscillations appearing in the initially stratified mixture and behaving as modulated and regular standing waves. The dependences of the intensity of convective flows on the modulation amplitude are obtained. The results of nonlinear calculations are compared with data on the boundaries of the equilibrium stability found from the linear theory. It is shown that a region of parameters exists where alternating action suppresses the convective motion.     

Electromagnetic radiation at twice the plasma frequency emitted from the region of interaction of two short laser pulses in a rarefied plasma

We study the electromagnetic radiation at twice the plasma frequency, which emerges because of the interaction of two identical counterpropagating short laser pulses in a rarefied plasma and caused by excitation of small-scale standing plasma waves in the pulse overlap region. The energy, spectral, and angular characteristics of radiation are investigated, and the dependence of these characteristics on the parameters of the laser pulses is analyzed. The possibility of applying this effect for diagnostics of localized plasma oscillations is discussed.     

Directionally asymmetrical bistability in a symmetrically pumped nonlinear ring interferometer

We consider a nonlinear ring resonator pumped symmetrically by two beams of equal intensities and opposite directions. We show that this system is characterized by a new directionally asymmetrical regime of multistability. This is due to the non-reciprocity of propagation of the counterpropagating waves in the resonator produced by a nonlinear index grating.     

Nonlinear phenomena accompanying the development of oscillations excited in a layer of a linear dissipative medium by finite displacements of its boundary

A method of describing oscillations in resonators on the basis of evolution equations is proposed. The latter are obtained by simplifying the functional equations under the assumption that the distortions of travelling waves within the resonator length are small, that the Mach number for the moving boundary oscillations is small, and that the frequency is close to one of the natural frequencies of the resonator. The problems of nonstationary oscillations of a layer with a moving boundary are solved. The law that should govern the wall oscillations to provide the development of steady-state linear resonance oscillations is determined. The shape of the resonance curve formed in the presence of a boundary nonlinearity is calculated. The method of matching of asymptotics is applied to the singularly perturbed problem with small dissipation. It is shown that a boundary nonlinearity leads to a distortion of the temporal profile of the standing wave and to the generation of higher harmonics in the process of the development of steady-state oscillations. In contrast to the classical linear problems where the resonance occurs at the coincidence of the external force frequency with one of the natural frequencies, in the case under study the resonance behavior is observed in frequency bands, which are wider the higher the amplitude of the boundary oscillations is.     

Harnessing caustics for wave-front sensing

Scintillation in measured wave fronts adds spurious dislocations and deformations to their reconstruction. The source of the problem is caustics formed by aberrations in intermediate planes. I propose to use intentional caustics to measure wave fronts under severe conditions such as low light level, fast scale variations, large aberrations, and discontinuities in the wave front. A simple realization is based on the Hartmann-Shack sensor, which samples the wave front with a lenslet array. Movement of the lenslets' foci is linear with slope changes. Here the lenslets are effectively formed in an acousto-optic device: Two standing waves are launched perpendicularly to the light beam and to each other. At some distance down the beam, each wave creates a comb of caustics, and the two orthogonal combs add up to an array of caustic spots. The spatial frequency of the array is linear with the temporal frequency of the standing sound waves. A simple Fourier demodulation scheme supplies the two wave-front gradients.     

Nonlinear Spherical Standing Waves in an Acoustically Excited Liquid Drop

Nonlinear evolution of a standing acoustic wave in a spherical resonator with a perfectly soft surface is analyzed. Quadratic approximation of nonlinear acoustics is used to analyze oscillations in the resonator by the slowly varying amplitude method for the standing wave harmonics and slowly varying profile method for the standing wave profile. It is demonstrated that nonlinear effects may lead to considerable increase in peak pressure at the center of the resonator. The proposed theoretical model is used to analyze the acoustic field in liquid drops of an acoustic fountain. It is shown that, as a result of nonlinear evolution, the peak negative pressure may exceed the mechanical strength of the liquid, which may account for the explosive instability of drops observed in experiments.     

Transverse instability threshold in counterpropagating light beams for a nonlinear medium with local photorefractive response

We derive the threshold conditions for the instability of counterpropagating waves in a nonlinear medium with local photorefractive response against the excitation of transverse small-angle structures. These conditions allow for all the important types of diffraction from refractive-index reflection gratings and are not limited to the case of strict frequency degeneracy of the waves. We study the dependence of the crystal-thickness threshold and the secondary wave emission angle on the crystal parameters and the pump conditions. We show that when the pump wave intensities differ considerably, excitation of standing light structures is replaced by excitation of traveling structures. Finally, we discuss the applications of the theory to experiments with the photorefractive crystals LiNbO₃ and LiTaO₃. Zh. éksp. Teor. Fiz. 111, 1611–1623 (May 1977)     

Dynamics of counterpropagating waves in parametrically driven systems: dispersion vs. advection

The dynamics of parametrically driven counterpropagating waves in a one-dimensional extended nearly conservative annular system are described by two coupled, damped, parametrically driven nonlinear Schrödinger (NLS) equations with opposite transport terms due to the group velocity, and small dispersion. The system is characterized by two length scales defined by a balance between (a) forcing and dispersion (the dispersive scale), and (b) forcing and advection at the group velocity (the transport scale). Both are large compared to the basic wavelength of the pattern. The dispersive scale plays an important role in the structure of solutions arising from secondary instabilities of frequency-locked spatially uniform standing waves (SW), and manifests itself both in traveling pulses or fronts and in extended spatio-temporal chaos, depending on the signs of the dispersion coefficient and nonlinearity.