In nonlinear multimode optical systems, stochastic fluctuations of the intensities of the competing modes have signatures (characteristic moments, cumulants, and cross correlations) that depend on the number of modes and on any constraints on the total intensity. For the cases of maximum entropy fluctuations within a constant total intensity and within a fluctuating total intensity, we derive the intensity probability distribution functions and calculate the moments, cumulants, and cross correlation. The latter two are shown to be sensitive indicators of the number of interacting modes. When the modes are statistically equivalent, particular relations between the variance of one mode and the cross correlation of any two modes can indicate the type of mode coupling independent of their number. The predictions of these models are compared to recent experimental measurements on a multimode source of amplified spontaneous emission. Statistical results for a system that randomly jumps among single-mode states while maintaining constant total intensity are also calculated and are compared with the statistics of a (deterministic) two-mode bidirectional ring-laser simulation. The results are widely applicable to other coupled-mode systems including multimode lasers.

Based on an envelope-function approach developed in a previous publication [de Sterke and Sipe, Phys. Rev. A 38, 5149 (1988)], we give criteria for the experimental observation of gap solitons. Moreover, the theory is extended in several respects—it is shown that in the limit of harmonic time dependence, our treatment is equivalent to the effective-mass approximation in condensed-matter physics. Further, we derive equations of motion for the fields under more general conditions than before, including third-harmonic generation by the optical nonlinearity. By use of Hamiltonian methods it is shown that third-harmonic generation in periodic structures has quite different properties than in homogeneous materials.

We study the effect of slowly varying cavity losses on the bifurcation diagram of laser rate equations. Specifically, we consider class-B lasers, which include ruby, YAG, ${\mathrm{CO}}_2$, and semiconductor lasers. We show that the behavior of these lasers depends on two small parameters: v, the rate of change of the cavity decay, and γ, the loss rate for the population inversion. If v≪γ, the laser problem reduces to the study of a single nonautonomous equation previously studied by the authors [Phys. Rev. Lett. 53, 1818 (1984)]. However, if v=O(γ) or larger, the response of the laser is described by two strongly coupled nonautonomous equations. We solve these equations in the limit v=O(γ)→0 by using asymptotic methods. We show that (1) the laser admits a new, slowly varying, reference solution which is not a small perturbation of the nonzero-intensity steady-state solution when the cavity losses are constant and (2) large-amplitude slowly varying oscillations may develop. Our analytical approximations are compared with the exact numerical solution of the laser equations. Finally, we analyze numerically the effects of different initial conditions.

We develop a nonlinear theory of a two-photon correlated-spontaneous-emission laser (CEL) by using an effective interaction Hamiltonian for a two-level system coupled by a two-photon transition. Assuming that the active atoms are prepared initially in a coherent superposition of two atomic levels involved in the two-photon transition, we derive a master equation for the field-density operator by using our quantum theory for coherently pumped lasers. The steady-state properties of the two-photon CEL are studied by converting the field master equation into a Fokker-Planck equation for the antinormal-ordering Q representation of the field-density operator. Because of the injected atomic coherence, the drift and diffusion coefficients become phase sensitive. This leads to laser phase locking and an extra two-photon CEL gain. The laser field can build up from a vacuum in the no-population-inversion region, in contrast to an ordinary two-photon laser for which triggering is needed. We find an approximate steady-state solution of the Q representation for the laser field, which consists of two identical peaks of elliptical type. We calculate the phase variance and, for any given mean photon number, obtain the minimum variance in the phase quadrature as a function of the initial atomic variables. Squeezing of the quantum noise in the phase quadrature is found and it exhibits the following features: (1) it is possible only when the laser intensity is smaller than a certain value; (2) it becomes most significant for small mean photon number, which is achievable in the no-population-inversion region; and (3) a maximum of 50% squeezing can be asymptotically approached in the small laser intensity limit. As a by-product we also study the ordinary two-photon laser and find, e.g., photon-number variance and laser linewidth.

An optical ring cavity containing distributed nonlinear elements is proposed as a simple metaphorical model for investigating the dynamic properties of spatial chaos in a system far from thermal equilibrium. If the coupling between the elements is unidirectional, the stability of the disordered structure can be determined by the spatial Lyapunov exponent. This fact implies that spatial chaos is almost dynamically unstable and is replaced by spatiotemporal chaos. However, in the case of bidirectional coupling, the spatial chaos is self-induced over a wide range of the control parameter, which means that a memory function is formed cooperatively in the system. We predict several cooperative phenomena and describe their physical origin in terms of nonlinear dynamics. In particular, characteristics of spatial chaos applied to information storage are studied in detail. Applicability of predicted phenomena such as a cooperative all-optical switching, multivibrator operations, flip-flop operations, as well as spatial chaos memory, is discussed.

We propose and analyze experiments designed to probe the way in which the measurement process (the presence of a detector) influences the investigated system. These experiments are based on the fact that number states of the radiation field can be generated by the use of the micromaser and cavity quantum electrodynamics. It is shown that ‘‘which-path’’ (particle) information rules out interference (wave) effects due to the system-detector correlations and not due to a randomization of phase. Specific experiments based on neutron interferometry and quantum-beat techniques taken together with the micromaser are suggested and analyzed.

Using a new and sensitive technique, experiments on light-induced drift are performed on vibrationally excited ${\mathrm{CH}}_3$F in the buffer gases He, Ne, Ar, Kr, Xe, and ${\mathrm{CH}}_3$Cl. For the noble buffer gases, the difference in collision frequency between ground- and excited-state ${\mathrm{CH}}_3$F with respect to the buffer-gas molecules is found to range from 0.14% for He to 0.56% for Xe. For the polar buffer gas ${\mathrm{CH}}_3$Cl, this value is found to be 0.98%.

A formulation of fluid dynamics in terms of Lagrangian variables allows one to make direct use of the standard methods of statistical mechanics. However, most observations and empirical studies of large geophysical fluid systems are in terms of Eulerian variables, and this raises the issue of how to relate quantities given in terms of one set of variables to the corresponding quantities given in terms of the other set. In a completely general treatment of fluids, one must consider both oscillatory and translational modes of motion. The oscillatory modes are wavelike and lead to correlations which are analogous to those for phonons in the solid. The translational modes are particlelike and lead to correlations which correspond to diffusion in a weakly interacting gas. In this paper we consider the class of fluids for which the translational modes can be neglected. Based upon the assumption that the statistical distribution of the canonically conjugate Lagrangian variables is of a Gaussian form, we obtain a tractable expression for the Eulerian spectra in terms of the Lagrangian spectra and show that, in general, the two types of spectra are significantly different. In particular, it is shown that the Eulerian wave-number spectrum exhibits a large wave-number power-law decay which is similar to that often observed in geophysical systems and further is independent of the detailed nature of the Lagrangian wave-number spectrum. The large wave-number decay of the Eulerian spectrum is due to advection and is strictly a kinematic effect. This also implies that experiments which focus on the large wave-number advective tail cannot yield information about the true dynamics of the system. The application of this result to the problem of explaining the observed distribution of oceanic internal waves is discussed.

A systematic study of ion-dynamics effects in the first three lines of the Lyman and Balmer series of hydrogen has been carried out over the ranges ${10}^{22–}$${10}^{23}$ ${\mathrm{m}}^{\mathrm{-}3}$ and ${10}^{4–}$2×${10}^4$ K by means of a simulation technique that takes into account the joint effect of ions and electrons. The results obtained considering static ions match the results obtained through time ordering of the unified theory. The inclusion of ion movement in the simulations leads to excellent agreement with experimental data.

We propose a theoretical study of the nonlinear propagation of a Gaussian laser beam in a mixture of two liquids. Retaining electrostriction (dipole radiation forces), thermal expansion, and thermodiffusion (Soret effect) as the major mechanisms in the perturbation of the medium by the field, we analyze their relative importances in the nonlinear refraction and nonlinear beam power losses. Our attention is mostly focused on the case of mixtures close to critical (consolute) points, where the nonlinearities are very large. We show that one crosses over from a situation dominated by thermal processes to a situation dominated by dipole radiation forces by diminishing the beam diameter and/or the distance to the critical point. The second case is shown to be well within the reach of realistic experiments. Finally, we discuss the potentialities of electrostriction as an alternative to sedimentation in the gravity field to study critical mixtures and the potentialities of critical mixtures as model media in nonlinear optical engineering.

A general equation of mechanical equilibrium of fluid membranes subject to bending elasticity [reported in Phys. Rev. Lett. 59, 2486 (1987)] is derived in detail. The second variation of the shape energy, also obtained for arbitrary shapes, is used to analyze stability with respect to deformational modes for spherical and cylindrical vesicles. The former analysis is well known, while the latter is presented here for the first time. The theoretical results are shown to agree very well with previous numerical calculations. In addition, they provide the energies controlling the shape fluctuations and show that spontaneous curvature may transform cylinders into tapes or strings of beads. The study of the energy of infinitesimal deformations is finally extended to include the third variation. Applying the general result to the sphere, we obtain the critical value of spontaneous curvature below which oblate ellipsoids of a deformed sphere are more stable than prolate ones. It is shown to be the same regardless of whether volume or pressure is kept constant.

A previous treatment that holds for the Korteweg–de Vries equation is extended to cover the case of nonlinear integrable equations associated with the standard Zakharov-Shabat eigenvalue problem that has a complex discrete spectrum. Particularly, an analytical expression for the distribution function of solitons as a functional of the initial conditions is found. This distribution function gives the correct values of the infinite set of constants of motion and leads to a large number of conclusions that agree with previous numerical and analytical results. Special emphasis is given to a comparison of these results in the case of the derivative nonlinear Schrödinger equation. The distribution function is particularly useful for the statistical description of the nonlinear equations involved in the formalism (such as the nonlinear Schrödinger or the derivative nonlinear Schrödinger equations) when an ensemble of initial conditions is considered.

The effect of anisotropy, induced by a periodic modulation of the plate separation in circular Hele-Shaw geometry, on the viscous fingering patterns is studied. Experimentally the modulation of the separation is induced by etching a regular pattern of channels on one of the confining plates. We show that for etching patterns of sufficiently high symmetry, the bulk properties of the flow are isotropic on the length scale of investigations of the interfacial pattern. Hence anisotropic effects are due to the boundary conditions at the interface only. Such boundary conditions are modeled by considering the effect of an anisotropic, effective surface tension, and the macroscopic-interface equations are solved numerically to study the morphological phase diagram. We discuss our choice of the constant-flux operating mode, in which case the phase diagram is swept by adjusting one anisotropy parameter. Some qualitative arguments are offered for the nature of the phase diagram, which includes a transition region from radial growth characterized by a diffusion-limited-aggregation-like fractal dimension at low anisotropy to a dendritic region with evident sidebranching.

Experiments were conducted on planar slab targets of carbon, using a 20-J 5-nsec Nd:glass laser. We observed a much stronger contribution to linearly excited second-harmonic emissions due to the s-polarized beam than that due to the p-polarized beam. The scatter direction made an angle of 3π/4 with the forward direction of light propagation. Evidence for the saturation of parametric-decay instability is presented using s- as well as p-polarized light. The results are explained on the basis of a three-dimensional variation of the plasma density and existing theoretical formulations.

An experiment to detect the presence of 1/f fluctuations in the α-particle decay statistics of ${}_{}{}^{210}{}_{}{}^{}\mathrm{Po}$ has been conducted. No evidence for such fluctuations has been found, at least for frequencies in excess of ${10}^{\mathrm{-}6}$ Hz.

We obtain explicit predictions for velocity selection in two- and three-dimensional, symmetric, and one-sided models of dendritic solidification. We do this by using WKB techniques to derive a solvability condition, which is then solved numerically, for the existence of steady-state needle crystals. The comparison with available numerical results is reasonably good. We also discuss the comparison with experiments.

The Frenkel-Kontorova model of a chain of atoms in an external periodic potential is studied. The phase diagram of this model contains infinitely many tongues of commensurate phases separated by gaps of incommensurate structures. The period of these structures is described by a devil’s staircase (DS) function when the parameters which define the model are varied. The model exhibits a critical line above which the phason mode of the incommensurate phases is pinned and the DS is complete, while below the line the phason mode is unpinned and the DS is incomplete. We evaluate the critical line numerically and show that it has a fractal nature. The Hausdorff dimension $D_0$ and the spectrum of singularities f(α) of the gaps along the critical line are calculated. The analysis is performed for several forms of the periodic potential. The resulting $D_0$ and f(α) seem to be independent of the details of the potential with $D_0$=0.87±0.02. It is interesting to note that $D_0$ is equal, within the numerical uncertainty, to the Hausdorff dimension corresponding to the critical line of dissipative systems, although the f(α) of the two cases are found to be different.

A modified version of Rayleigh-Schrödinger (RS) perturbation theory, which has been developed previously, can dispense with the usual RS restriction that the unperturbed Hamiltonian should be diagonal in the chosen basis. This modified RS (MRS) perturbation scheme is discussed in some detail and extended to the degenerate case. The new degenerate MRS perturbation theory remains based upon a rearrangement of the unperturbed and perturbed parts of the Hamiltonian. Basically, an unperturbed Schrödinger equation is thereby produced which may be satisfied trivially and identically for an arbitrary Hamiltonian. The zeroth-order approximation to the energy remains a free parameter of the degenerate version of the MRS formalism. The explicit MRS expressions for the higher-order correction terms also preserve a strict analogy with their counterparts in the nondegenerate formalism. What is new is an enormous increase in the applicability and universality of the whole approach. Thus, a priori, the unperturbed Hamiltonian may now be chosen so as to have all of its larger off-diagonal elements much closer to those of the full original Hamiltonian than is permitted in the standard RS approach. This is a particular advantage of the whole MRS approach. By testing the method on two nontrivial model Hamiltonians, the method is demonstrated to have considerably improved convergence properties over the standard RS scheme, and at relatively little extra cost.

We present a new method for calculating electrostatic interactions in periodic systems with charged or dipolar particles. The method is appreciably faster than the Ewald summation method used until now in computer simulations of ionic and dipolar systems. The method may also be used for calculating the periodic crystal field in solids.

The calculation of the scaling properties of multifractal sets is presented as an eigenvalue problem. The eigenvalue equation unifies the treatment of sets that can be organized on regular trees—be they complete or incomplete (pruned) trees. In particular, this approach unifies the multifractal analyses of sets at the borderline of chaos with those of chaotic sets. Phase transitions in the thermodynamic formalism of multifractals are identified as a crossing of the largest discrete eigenvalue with the continuous part of the spectrum of the relevant operator. An analysis of the eigenfunctions is presented and several examples are solved in detail. Of particular interest is the analysis of intermittent maps, which shows the existence of an infinite-order phase transitions, and of (Smale) incomplete maps of the interval with finite and infinite rules of pruning of the multifractal trees.