An analytic nonperturbative treatment of degenerate four-wave mixing is presented to study Zeeman-coherence effects in phase conjugation. The counterpropagating pump waves are taken to be orthogonally circularly polarized and interact with different Zeeman sublevels. The nonlinear medium is modeled in terms of a homogeneously broadened, folded three-level system. Particular attention is paid to the phase-conjugate spectrum obtained by varying the Larmor frequency associated with the Zeeman splitting. A level-crossing resonance is found to occur in the saturation regime and manifests itself as a narrow peak or dip depending on various level decay rates. Even for a purely absorptive system, the Zeeman coupling of the ground-state degenerate sublevels is shown to produce significant phase-conjugate reflectivities at pump intensities much lower than those required for a two-level system.

Particular solutions of the logistic equation in nonlinear systems are sought. The study is related primarily to the two-mode operation of the laser and the competition or the predation of species in the population ecology. The method of solution is described in detail and the dependence on the model parameters is discussed for the above cases.

An analysis of the free-electron-laser (FEL) oscillator startup problem in the linear regime is presented. The model is spatially one dimensional, though many important three-dimensional effects are included heuristically. The electron beam consists of pulses of arbitrary shape separated by approximately twice the radiation transit time. The small gain per pass approximation is employed in deriving an energy rate equation, which describes the evolution of the radiation pulses within the resonator. The wiggler field is assumed to occupy a portion of the finite $Q$ resonator. In the energy rate equation, the spontaneous (incoherent) radiation term is represented by a source matrix, while the stimulated (coherent) radiation term is represented by a gain matrix. The effect of small variations in the mirror separation are investigated in the context of laser lethargy. Our analysis suggests possible methods which could substantially shorten the startup times in FEL oscillators. Finally, our results are compared with the FEL oscillator experiments performed at Stanford University.

A theory of the interaction of an atom with an intense nonlinear (non-Gaussian) noise field is developed, with emphasis on the connection with the underlying physics of laser coherence. We point out the possibility of obtaining exact solutions for the stochastically averaged atomic density matrix in terms of (matrix) continued fractions for a large class of nonlinear noise fields by generalizing the techniques developed by Risken and co-workers to solve nonlinear Fokker-Planck equations. As an example we discuss the absorption sepctrum of an atom strongly driven by noisy phase-locked radiation.

As a result of quantum fluctuations, a bistable system will undergo spontaneous transitions from one state to the other. For a bistable two-mode laser we present a theoretical analysis of the switching statistics by treating it as a one-dimensional first-passage—time (FPT) problem for the intensity of one of the modes. The methodology is also applicable to other bistable systems, such as in optical bistability. Several new asymptotic results for the small- and large-time behavior of various FPT distributions are given. The switching-time distribution is introduced as a weighted integral over FPT distributions and asymptotic expressions for this distribution are given. Several results of Monte Carlo computer simulations of the FPT problem are presented.

We consider a system consisting of an electron in a static magnetic field, interacting with the quantized electromagnetic field, within the nonrelativistic and electric dipole approximations (with a cutoff in momentum space). The Heisenberg equations of motion are solved exactly and the time evolution of the electric field is determined. The power spectrum of the scattered radiation is calculated when the electromagnetic field is initially in a coherent state. Our results for the line shape of the scattered radiation are shown to be valid for magnetic fields up to ${10}^{12}$ G. The quantization of the electromagnetic field allows one to consider effects of the natural linewidth and its dependence on the magnetic field. The renormalization of the electron mass is included in our treatment, and the results remain finite when the cutoff goes to infinity.

We show that in a gas laser the inhomogeneity in the distribution of amplifying atoms and the transverse Gaussian distribution of the saturating electromagnetic field give rise to different lens effects which are generally opposite to each other and influence the line shapes in different ways, shifting the Lamb dip either to the blue or to the red. For those laser phenomena which increase one or the other of the lens effects, the role of a diaphragm is of particular importance, since it determines the spatial distribution of the resonant field. We present here a new method for studying the resonant diffracted field in a laser. Also, we give a unified description of the various inhomogeneities. Doing away with the usual method of mean-field approximation, we numerically calculate, for a diaphragmed, monomode weak-gain gas laser, the intensity, variation of the beam diameter, and the radius of curvature as functions of the frequency.

The dynamic renormalization group yields a value for the critical exponent which we find to be too large. We believe that the true value is close to the decoupled mode value of $\frac{8}{15}{\pi }^2=0.054\mathrm{}$. The error in the renormalization-group calculation comes from the two-term truncation of the $\epsilon$ expansion. The ${\epsilon }^3$ term that is neglected is of opposite sign and as large as the ${\epsilon }^2$ term that is included. By avoiding the $\epsilon$ expansion entirely, we eliminate any truncation error and achieve greatly improved accuracy.

Onset behavior of diffusion generated by a chaotic motion of a one-particle system in a sinusoidal potential, driven by a time-dependent external field, is investigated, especially near the Pomeau-Manneville intermittency transition. It is numerically found that the diffusion coefficient diverges as the system approaches the intermittency transition. A possible interpretation of such anomalous enhancement of the diffusion coefficient is proposed in connection with the appearance of a temporally long coherence of the phase point.

We use the Rosenfeld-Ashcroft procedure of modeling the bridge function in the reference—hypernetted-chain integral equation with its hard-sphere values, and choose the sphere diameter so that the free energy of the system is minimized. The resulting integral equation is solved for both the long-range Coulomb potential and the short-range Lennard-Jones potential. The results are in excellent agreement with Monte Carlo data for the thermodynamics and structure of both systems. The method provides an entirely first-principles approach to the theory of the structure and thermodynamics of simple classical liquids.

A theoretical description is proposed for recently observed smectic-$A$ phases which possess antiferroelectric character. Order parameters are introduced that retain information on the head-tail asymmetry of the constituent molecules. Minimum ingredients are given for a mean-field model potential, with the additional use of the approximation of ideal orientational order. With reasonable estimates for the various types of interaction (dispersion, dipole—induced-dipole, and dipole-dipole) no ferroelectric smectic phase is found. Polarized layers can couple, however, to double layers, which leads to the antiferroelectric smectic-$A_2$ phase. In addition, an antiferroelectric smectic-$A_1$ phase is found, which has a single-layer periodicity. In that case there is a bilayer structure as in smectic-$A_2$, which is now not long ranged. For high orientational order the antiferroelectricity of these phases is shown to be the best compromise between the repulsion of the permanent dipole moments and the attractive induced interactions. The relative stability of the various phases is considered in detail. The predictions agree well with present experimental observations.

An Euler equation, consistent with Maxwell's equations, for describing the optically induced spatial reorientation of the director of a homeotropically oriented nematic liquid crystal is obtained for the case of normal incidence. The exact solution describing the orientation of the director is obtained. By examining the maximum deformation angle near the threshold, the threshold intensity and the criterion for the physical parameters that indicate whether the transition is first- or second-order at the threshold are obtained. The hysteresis accompanying the first-order Freedericksz transition is discussed. By choosing a compound (PAA, $p$-azoxyanisole) with suitable material parameters from known nematic liquid crystals, an experiment is proposed to observe, for the first time, a first-order Freedericksz transition in nematic liquid crystals. The Zel'dovich approach [Sov. Phys.-JETP 54, 32 (1981)] based on the geometrical-optics approximation is shown to be internally inconsistent and also inconsistent with the geometrical-optics approximation. The Euler equation using a self-consistent geometrical-optics approximation is also obtained, and turns out to be identical to our exact Euler equation, but different from the infinite-plane-wave approximation used by Durbin et al. [Phys. Rev. Lett. 47, 1411 (1981)]. Detailed comparisons between our approach and the Durbin approach are made. The dynamics of the transition are discussed and an approximate solution is given. The transient responses to the laser switch-on and switch-off are shown to have exponential time dependence. Finally, the effects of surface interactions on the transition are discussed and the exact solution is given. The procedure for determining the threshold, the saturation, and the parallel-state-maintenance intensities is given. We also discuss the first-order transition and propose experimental methods manifesting the effects of surface interaction. The criterion for the transition to be first order at any intensity is given.

The "inherent structures" which underlie the liquid state are those stable particle packings (potential minima) which can be reached by a steepest-descent quench on the potential-energy hypersurface. This paper explores the dynamics of transition between distinct inherent structures for a simple classical model of monatomic substances. Molecular-dynamics calculations with 32 and 108 particles have been carried out with running construction of the mapping to potential minima. This determines the distribution of stable packings according to their potential energy and shows how transition rates between alternative structures vary with total system energy. Melting and freezing events have been monitored in this manner. We observe occasional transitions in localized "two-state" (bistable) systems in strongly supercooled amorphous states. Transitions in fluid states display a peculiar intermittency that may have relevance to self-diffusion and viscous flow.

A new variational approach is proposed for the study of the many-body problem with the purpose of combining the advantages of perturbation-theoretic and variational approaches, while avoiding their disadvantages. A trial wave function is constructed so that a general expansion for the expectation value of an arbitrary physical quantity can be derived. Each term in the expansion can be calculated with the use of the techniques of Feynman diagrams, but in contrast with usual perturbation-theoretic approaches, every term in the expansion always converges. The procedure for applying the method to real systems is illustrated by a calculation of the ground-state energy of the electron gas at high densities.

High-resolution ac calorimetry was used to measure the heat capacity of $n$-hexyl-4′-$n$-pentyloxybiphenyl-4-carboxylate in the vicinity of the smectic-$A$ ($\mathrm{Sm}A$)—hexatic-$B$ and hexatic-$B-E$ transitions. The $\mathrm{Sm}A$—hexatic-$B$ transition was found to be continuous with critical exponents ${\alpha }^{-}={\alpha }^{+}=0.61\mathrm{}\pm 0.02$ and amplitude ratio $\frac{A^{+}}{A^{-}}=0.84\mathrm{}\pm 0.03$. The addition of a crystalline-$B$ material [N-(4-$n$-butyloxybenzylidene)-4′-octylaniline (4O.8), 10 wt.%] produced a weakly-first-order $\mathrm{Sm}A$—hexatic-$B$ phase transition. The data with $|T-T_c|\ge 100$ mK can be fitted by a power law with $\alpha =0.49\mathrm{}\pm 0.02$ and $\frac{A^{+}}{A^{-}}=0.82\mathrm{}\pm 0.03$. The large values for the critical exponents and the theoretical suggestion that the $\mathrm{Sm}A$—hexatic-$B$ transition may be near tricritical point (${\alpha }_t=0.5\mathrm{}$) are discussed. The hexatic-$B-E$ transition was found to be first order in nature.

We show that a recent prediction that the extent of reaction of a very dilute solute near the critical point of an almost pure solvent will show strong variation with temperature at fixed volume is in error. We give arguments to show that the behavior is at most weakly singular in the sense of Griffiths and Wheeler, point out the error in the earlier argument, and show that experiments adduced as evidence for the earlier argument are subject to alternative explanation. We also consider a prediction about the behavior near the critical point of a dissociating solvent.

A variational principle for relativistic magnetohydrodynamics is formulated. As an application, the properties of small-amplitude waves and their interaction with the mean flow are calculated. The generation of waves by an external current is incorporated into the formalism.

A mode-coupling theory is formulated for the spectrum of electric-field-gradient (EFG) fluctuations at the site of a probe particle in a classical liquid. The EFG-decay rate is approximated by taking into account coupling of the probe-particle motion to longitudinal and transverse current excitations of the liquid. With the introduction of some simplifying assumptions, the theory is applied to calculate the quadrupolar relaxation rate in liquid rubidium as a function of temperature. The effect of various probe-particle properties in the relaxation rate is discussed. The theory yields a $t^{-\frac{5}{2}}$ long-time tail for the EFG correlation function. In the electric potential responsible for the EFG $r$-dependent Sternheimer antishielding effects are included.

A Mori-type generalized Langevin equation is shown to be the only theoretical tool necessary for setting up a fast-computational method of evaluating EPR spectra. The advantages of this algorithm, concerning memory storage and time consumption, are clearly illustrated by explicitly evaluating the line shape of a nitroxide spin probe both in an isotropic liquid solution and in a liquid-crystal mesophase. This method makes theoretical EPR spectra of an orientated system in the slow-motion regime readily available. It is shown that the presence of an orientating potential renders the EPR spectrum much more sensitive to the details of molecular dynamics, thereby making this a potentially powerful tool for monitoring rotational dynamics in liquid mesophases.

A cluster-integral series is developed for the calculation of the droplet and crystallite partition functions involved in the classical theory of homogeneous nucleation. These formal series reduce the calculation of droplet free energies to the consideration of few-body integrals similar to those which determine the virial coefficients of an imperfect gas. Theories are proposed which correspond to infinite-order series resummations directly analogous to the Percus-Yevick and hypernetted-chain integral equations for simple liquids. The cluster-integral expansion coefficients through sixth order are evaluated for argon droplets over a wide range of temperatures by a very efficient specialized Monte Carlo sampling technique. These coefficients are used in two distinctly different ways: direct truncation of a suitable rearrangement of the series (star-tree approximation), and formation of Padé approximants for a generating function identified as being well suited for this purpose. By comparing the results obtained from these different procedures, it is concluded that the predicted droplet free energies are quite accurate in the range of droplet size of special interest in homogeneous nucleation theory, i.e., the inaccuracy appears to be less than 3% in the free energy per particle for droplets of 80 argon atoms.