A nonlinear theory of laser noise and coherence. I

We consider the interaction of a set of atoms at random lattice sites with a decaying resonator mode. The optical transition is supposed to possess a homogeneously broadened Lorentzian line. The pumping is taken into account explicitly as a stochastic process. After elimination of the atomic coordinates a second order nonlinear differential equation for the light amplitude is found. In between excitation collisions this equation can be solved exactly if the resonator width is large as compared to all other frequency differences. In contrast to linear theories there exists a marked threshold. Below it the amplitude decreases after each excitation exponentially and the linewidth turns out to be identical with those of previous authors (for instanceWagner andBirnbaum), if specialized to large cavity width. Above the threshold the light amplitude converges towards a stable value, whereas the phase undergoes some kind of undamped diffusion process. We then consider the general case with arbitrary cavity width. If the general equation of motion of the light amplitude is interpreted as that of a particle moving in two dimensions, it becomes clear that also in this case the amplitude oscillates above threshold around a stable value which is identical with that determined in previous papers byHaken andSauermann neglecting laser noise. This stable value may, however, undergo shifts, if there are slow systematic changes of the cavity width, inversion etc. On the other hand the phase still fluctuates in an undamped way. After splitting off the phase factor the equations can be linearized and solved explicitly. With these solutions simple examples of correlation functions are calculated in a semiclassical way, thus yielding expressions for the line width above threshold. The results can also be used to evaluate from first principles correlation functions for different laser beams. As an example the complex degree of mutual coherence of two laser beams is determined. It vanishes if one of the lasers is still below threshold and its value is close to unity well above threshold for observation times small compared to the inverse laser linewidth.     

Classical theory of laser noise

The classical theory of laser noise treats light in a classical manner, yet agrees with quantum theory for large particle numbers. The basic concept is that laser noise is caused by atomic jumps between lower and upper levels, and that atoms subjected to classically-prescribed optical fields are independent. The treatment of amplitude noise of single-mode cavities containing resonant three-level atoms is applicable to semiconductor lasers at moderate power. At high power one must account for the dependence of the gain on optical power and for state-occupancy fluctuations. The phasor theory that attributes noise to the beat between the oscillating field and the field spontaneously emitted in the mode by excited-state atoms cannot be understood consistently in semiclassical terms.     

Quantum theory of light and noise polarization in nonlinear optics

We present a consistent quantum theory of the electromagnetic field in nonlinearly responding causal media, with special emphasis on Chi(2) media. Starting from QED in linearly responding causal media, we develop a method to construct the cubic Hamiltonian expressed in terms of the complex nonlinear susceptibility in a quantum mechanically consistent way. In particular, we show that the method yields the nonlinear noise polarization, which together with the linear one is responsible for intrinsic quantum decoherence.     

Statistical and coherence effects in nonlinear optical parametric up conversion. II. Incoherent interaction

Parametric up conversion (PUC) by incoherent nonlinear optical mixing of second-order coherent strong pumping radiation and another weak chaotic input radiation with a finite spectral width is treated in the second part of this paper. The efficiency of PUC is calculated in the second approximation of the iterative method. It is shown that in the case of perfect phase matching the efficiency of PUC decreases with increase of both the spectral width of the input radiation and the dispersion of medium, whilst at considerable values of phase mismatch the reverse effect appears. The spectral distribution of the resulting sum-frequency radiation is calculated in the first approximation of the iterative method. There is a general tendency to narrowing of the spectral distribution of the generated radiation in the course of PUC. Moreover, when there is a phase mismatch present, a spectral shift of the central maximum of generated radiation towards the blue or red region, according to the signs of the phase mismatch and the typical dispersion coefficient, appears in the later phases of the PUC.     

A theory of optical coherence tomography

Analytical models of optical coherence tomography (OCT) of strongly turbid media of the type of biological tissues are developed on the basis of the theory of wave scattering in random inhomogeneous media. Similarity relations for signals of coherent and pulsed sounding are established, and general expressions for random realizations and statistical characteristics of tomograms are obtained. It is shown that after the appropriate modification the theory of image transfer in turbid media can be used for analysis of their informative properties. Simple formulas for estimating the visibility depth into the internal structure of biological tissues are proposed. Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41. No. 10, pp. 1258–1289, October 1998.     

Classical field theory methods in laser physics II. Statistical methods of laser physics

The second part of our work continues the analysis of the problem studied in the first par [J. Russ. Laser Res.,17, 205 (1996)]. We seek an answer to the following question: What is laser radiation from the viewpoint of the classical theory of wave fields? Here, we consider the statistical aspect of the problem. Moveover, we show how this aspect is connected with the formation of laser radiation from the level of spontaneous noise and how it governs the quality of laser radiation.     

Moment theory of self-trapped laser beams with nonlinear saturation

The moment theory of the quasi-optical equation is used to obtain simple analytical expressions for the relation between beam radius and power of stationary self-trapped light beams in passive nonlinear media with saturation of the index of refraction. In addition, the stability of these self-trapped beams is determined.     

The projection operator approach to the Fokker-Planck equation. II. Dichotomic and nonlinear Gaussian noise

Two models for the Freedericksz transition in a fluctuating magnetic field are considered: one is based on a dichotomic and the other on a nonlinear Gaussian noise. Both noises are characterized by a finite correlation time. It is shown that the linear response assumption leading to the best Fokker-Planck approximation in the dichotomic and nonlinear Gaussian cases can be trusted only up to the order ¹ and ⁰, respectively. The role of the corrections to the linear response approximation is discussed and it is shown how to replace the non-Fokker-Planck terms stemming from these corrections with equivalent terms of standard type. This technique is shown to produce perfect agreement with the exact analytical results (dichotomic noise) and to satisfactorily fit the results of analog simulation (nonlinear Gaussian noise).     

A microscopic theory of nucleation. II

Using the nonequilibrium statistical operator obtained in the preceding paper of the authors [1], equations describing the kinetics of nucleation in a nonequilibrium medium are derived. A Fokker-Planck equation is found for embryo distribution functions in the number of particles, energy, momentum, and c.m. coordinates with additional random forces due to non equilibrium processes in the medium. Hydrodynamic equations are obtained for the medium with account of thermodynamic forces due to discontinuities of thermodynamic parameters at the interphase boundary. The symmetry of cross (interphase) kinetic coefficients is considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 44–52, May, 1978.     

A bi-metric theory of gravitation. II

The bi-metric theory of gravitation proposed previously is simplified in that the auxiliary conditions are discarded, the two metric tensors being tied together only by means of the boundary conditions. Some of the properties of the field of a particle are investigated; there is no black hole, and it appears that no gravitational collapse can take place. Although the proposed theory and general relativity are at present observationally indistinguishable, some differences are pointed out which may some day be susceptible of observation. An alternative bimetric theory is considered which gives for the precession of the perihelion 5/6 of the value given by general relativity; it seems less satisfactory than the present theory from the aesthetic point of view.     

基于非线性光学理论的激光防护材料近况

按照防护原理,可将激光防护材料分为基于线性光学理论的防护材料和基于非线性光学理论的防护材料。综述了基于非线性光学理论的多种激光防护方法及其防护原理,介绍了基于非线性光学理论的多种新型防护材料及其研究进展,比较了四种基于非线性光学理论的激光防护方法的优、缺点,简短地分析了激光防护材料今后发展的基本方向。     

A new variant of nonlinear electrodynamics. II

The electrostatic solution of nonlinear electrodynamic equations is studied, the characteristic peculiarity of these equations being their consideration of the effect of electromagnetic spin on the space-time continuum. The energy of the nonlinear electromagnetic field proves to be finite for a quite wide spectrum of seed masses. Three-dimensional soliton solutions are constructed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 46–53, October, 1978.We express our deep indebtedness to N. N. Kolesnikov, V. R. Khalilov, and participants in the theoretical physics seminar led by Professor D. D. Ivanenko for evaluating the results of the study.     

Contribution to the theory of single-mode laser modulation II. Conduction modulation

The problem of conduction modulation of laser radiation is considered. It is shown that acoustic wave interacting with photon field in active medium causes a periodic modulation of the number of quanta in the single-mode gas laser. The problem of modulation is solved quantum mechanically and it is shown that dielectric and conduction modulation give similar results.     

A theory of patterns of passby noise

Some people say they are annoyed by traffic noise. There is rather a lot of evidence to show that where traffic noise is louder, more people say they are annoyed by it. On the basis of this sort of evidence, there is a consensus that road traffic noise causes annoyance, but some studies have detected unexplained peaks of annoyance in quieter places, or a plateau of annoyance in high noise. Such anomalies may especially affect those sensitive to noise. The pattern of alternation of passby noise and background traffic noise explains the positioning in soundspace of anomalies variously reported at 60 dB(A) L_eq, 4000 NV and 1800 NHV. Such anomalies occur where there are regular or rapidly alternating patterns of passby noise.     

A nonlinear theory of turbulent diffusion

It is shown that the well-known conservation laws for magnetic helicity and passive-scalar fluctuation intensity in the case of negligible molecular diffusion require that the hierarchy of nonlinear equations for the averaged Green function and the hierarchy of Bethe-Salpeter-type equations for fluctuation intensity be treated in a mutually consistent manner. These hierarchies are obtained to the sixth order in turbulent velocity correlators. Asymptotic formulas describing the evolution of scalar fluctuations and magnetic field are presented. A number of new effects are revealed that strongly affect diffusion, but are beyond the scope of the frequently used model of a delta-correlated turbulent field.     

A theory of 1/f noise

Letu() be an absolutely integrable function and define the random process where thet _i are Poisson arrivals and thes _i, are identically distributed nonnegative random variables. Under routine independence assumptions, one may then calculate a formula for the spectrum ofn(t), S _n(), in terms of the probability density ofs, p_s(). If any probability density p_s() having the property p_s() I for small is substituted into this formula, the calculated S_n() is such that S_n() 1 for small . However, this is not a spectrum of a well-defined random process; here, it is termed alimit spectrum. If a probability density having the property p_s() for small , where > 0, is substituted into the formula instead, a spectrum is calculated which is indeed the spectrum of a well-defined random process. Also, if the latter p_s is suitably close to the former p_s, then the spectrum in the second case approximates, to an arbitrary, degree of accuracy, the limit spectrum. It is shown how one may thereby have 1/f noise with low-frequency turnover, and also strict 1/f ^1– noise (the latter spectrum being integrable for > 0). Suitable examples are given. Actually, u() may be itself a random process, and the theory is developed on this basis.     

A nonlinear active noise control algorithm for virtual microphones controlling chaotic noise

In active noise control (ANC) systems, virtual microphones provide a means of projecting the zone of quiet away from the physical microphone to a remote location. To date, linear ANC algorithms, such as the filtered-x least mean square (FXLMS) algorithm, have been used with virtual sensing techniques. In this paper, a nonlinear ANC algorithm is developed for a virtual microphone by integrating the remote microphone technique with the filtered-s least mean square (FSLMS) algorithm. The proposed algorithm is evaluated experimentally in the cancellation of chaotic noise in a one-dimensional duct. The secondary paths evaluated experimentally exhibit non-minimum phase response and hence poor performance is obtained with the conventional FXLMS algorithm compared to the proposed FSLMS based algorithm. This is because the latter is capable of predicting the chaotic signal found in many physical processes responsible for noise. In addition, the proposed algorithm is shown to outperform the FXLMS based remote microphone technique under the causality constraint (when the propagation delay of the secondary path is greater than the primary path). A number of experimental results are presented in this paper to compare the performance of the FSLMS algorithm based virtual ANC algorithm with the FXLMS based virtual ANC algorithm.     

Prolongation theory. A new nonlinear Schrödinger equation

We discuss a new kind of nonlinear Schrödinger equation from the viewpoint of prolongation theory. It is shown that the equation possess a Lax pair with a 3 × 3 matrix structure. It is further demonstrated that by a multiple scale perturbation of Zakharovet al. it can be reduced to the usual KdV equation.