We show that second-order nonlinear optical processes such as sum and difference generation can be observed in isotropic media, despite the fact that such processes are forbidden in the electric-dipole approximation. A theoretical treatment of quadrupole sum- and difference-frequency generation is given, which includes the effects of spin-orbit and Zeeman splittings. The quadrupole nonlinear susceptibility is evaluated and expressions for phase matching and output power are found. We have generally confirmed the theoretical predictions by observing quadrupole sum-frequency generation of ultraviolet light in sodium vapor. Deviations of the data from the simple theory can be satisfactorily accounted for by the effects of two-photon saturation, induced index of refraction changes, and single- and multi-photon absorption. Generalization of the dipole forbidden susceptibilities to the case of quadrupole-pumped processes is discussed, with difference-frequency generation in Cs vapor considered as an example.

Laser excitation of a two-level atom is analytically studied for arbitrary pulse shape, area, and energy. The ionization mechanism is modeled as a homogeneous width $\gamma$ of the atomic excited state. A transformation of the optical Bloch equations allows the dynamics to be completely described by a single equation for the area of the pulse at time $t$, $\Phi \mathrm{}\left(t\right)\mathrm{}$. Analytic solutions for the total bound population for a wide variety of pulse shapes can be found in two regions: $\Phi \mathrm{}\left(t\right)\mathrm{}\approx sin\Phi \mathrm{}\left(t\right)\mathrm{}$ and $\Omega \mathrm{}\left(t\right)>\gamma$, where $\Omega \mathrm{}\left(t\right)\mathrm{}$ is the resonance Rabi frequency. The generalization of the two-level atom results to the $N$-level atom is discussed.

We present in this paper a calculation of the static quadrupole polarizabilities and shielding factors of the atoms and a few ions of $3p$ open-shell systems. A fully coupled Hartree-Fock theory in the Roothaan formalism has been adopted. A variational approach is employed to obtain the first-order perturbed wave functions. Calculations have been performed for ground and some valence excited states of the systems. It is interesting to note that for all the systems under study the shielding factor is negative.

Stimulated emission from a relativistic beam of electrons passing through a spatially periodic right-hand circularly polarized magnetic field is considered. The amplification is found to be due to a ponderomotive bunching of the electrons. The effect is completely classical and for an infinite interaction distance a dispersion relation, which takes into account space-charge effects, describing the scattered field is derived. Conditions on the pump field amplitude, beam density, and momentum spread of the beam for emission from individual electrons to occur or for emission from plasma oscillations to occur are examined. Also, emission from individual electrons over a finite interaction distance is considered and gain is determined for distances less than an $e$-folding length.

A comprehensive theory of a new type of three-wave optical mixing (${\omega }_3=2\mathrm{}{\omega }_1-{\omega }_2$) is given for the spectroscopic purpose of determining ultra-short relaxation times (picoseconds or less) associated with excited states of condensed matter. Two incident light frequencies ${\omega }_1$ and ${\omega }_2$ are chosen so that they are both resonant with an inhomogeneously broadened optical transition and $|{\omega }_2-{\omega }_1|$ is in the vicinity of inverse relaxation times. The theory is developed for its major application to broad-band electronic transitions. The nonlinear susceptibility ${\chi }_R^{\mathrm{}\left(3\right)\mathrm{}}$ for this process (named resonant Rayleigh-type mixing) is calculated by a density-matrix formalism with the model of a two-level atomic system incorporating the distribution of resonance frequencies and longitudinal ($T_1$) and transverse ($T_2$) relaxation times. The result shows a frequency characteristic depending only on $T_1$, $T_2$, and ${\omega }_2-{\omega }_1$ in a broad-band limit, which serves for the determination of $T_1$ and $T_2$ in the frequency domain. The analysis is further extended to include the various effects as follows. The calculation of the saturation effect in the lowest order reveals that it modifies the shape of the frequency characteristic of the nonlinear susceptibility but does not affect seriously the determination of relaxation times. The effect of spectral cross relaxation within the inhomogeneous broadening is incorporated by a generalized density-matrix formalism. The resultant ${\chi }_R^{\mathrm{}\left(3\right)\mathrm{}}$ consists of two terms. The dominant term is the same as before except that $T_1$ is replaced by a combined relaxation time $T_1^{\prime }={(T_1^{-1\mathrm{}}+T_3^{-1\mathrm{}})}^{-1\mathrm{}}$, where $T_3$ is the cross-relaxation time. The other term originates from the inverse spectral-diffusion process. The effect of other energy levels located between the two levels under study is analyzed with a simple three-level model. The resultant ${\chi }_R^{\mathrm{}\left(3\right)\mathrm{}}$ also consists of two terms, the dominant one being the same as that for the two-level model except that $T_1^{-1\mathrm{}}$ should be interpreted as total population decay rate of the upper level. For accurate derivation of ${\chi }_R^{\mathrm{}\left(3\right)\mathrm{}}$ from experiment, we discuss the interference between the resonant and nonresonant susceptibility terms and the light-wave propagation effect. Finally, the close analogy between this type of optical mixing and the photon echo is discussed.

The three-wave optical mixing (${\omega }_3=2\mathrm{}{\omega }_1-{\omega }_2$) of the resonant Rayleigh type ($|{\omega }_1-{\omega }_2|\simeq \mathrm{inverse}\mathrm{relaxation}\mathrm{time}$, ${\omega }_1 \mathrm{and} {\omega }_2\simeq \mathrm{optical}\mathrm{transition}\mathrm{frequency}$) is applied to measure the subpicosecond relaxation times associated with the $S_0\to S_1$ electronic transition of 1,1'-diethyl-4,4'-quinocyanine iodide in ethanol and other dye solutions. Two independently tunable dye lasers excited by a nitrogen laser were used as light sources. The measured frequency responses of the nonlinear susceptibility $\left|{\chi }^{\mathrm{}\left(3\right)\mathrm{}}\right|$ as a function of ${\omega }_1-{\omega }_2$ were in good agreement with the theoretical formula based on an inhomogeneously broadened two-level model incorporating the phenomenological relaxation times. We also observed the interference effect between the resonant and nonresonant terms of the nonlinear susceptibility, which agree well with the theory. Thus we were able to determine the transverse relaxation time $T_2$ in the range of 0.02 ∼ 2 psec and the effective relaxation time $T_1^{\prime }$ (a combination of longitudinal- and cross-relaxation times) in the range of 0.1 ∼ 5 psec. These values were found to be largely dependent on the spectral position inside the broad absorption band and the sample preparation and deterioration, as well as on the kinds of dye materials and solvents. Details of the experimental procedure and a discussion of the interpretation of the experimental results are given.

A Fabry-Perot interferometer filled with a medium whose refractive index depends upon intensity has a multiple-valued transmission-vs-intensity characteristic. When the incident light is circularly polarized, and the nonlinearity is only cubic in the fields, Maxwell's equations may be solved exactly for the plane-parallel resonator field in terms of elliptic functions. An accurate approximate analysis is given for more general cases, including resonators with spherical mirrors and finite beams, where self-focusing is important. The theory is developed to yield design information for operation as an optical switch, "transistor," and power limiter.

The energy levels of two simple examples of screened Coulomb potentials have been analyzed using nonperturbative methods. The analysis indicates that the energy levels as a function of the perturbation parameter $\lambda$ have a branch cut along the negative real axis, starting from the origin. Furthermore, there are singularities on the second sheet, along $\left|\lambda \right|e^{\pm \frac{3\pi i}{2}}$ for $\left|\lambda \right|\to 0$. As a consequence of these singularities, the energy levels have an asymptotic series in $\lambda$, which means that one cannot use a power series in $\lambda$ to describe the energy levels to an arbitrary accuracy. The approximate but nonperturbative expression for the energy levels, which has been obtained by using dispersion relations, predicts energy levels which are in good agreement with those obtained from variational calculations.

The evolution of a small spherical system of many harmonic oscillators, initially in a coherent superradiant state, is considered. The dipole oscillators interact via their mutual fields, both longitudinal and transverse, including both radiation- and near-zone contributions. In principle, this interaction can dephase the vibration of oscillators and thereby decrease or eliminate the coherent superradiant contribution to the emitted radiation. It is shown, however, that if the system is sufficiently small, this dephasing makes only small corrections to the decay of the initial state. Thus the emission of a superradiant pulse and cooperative radiative decay are major processes in the system.

We have studied the temperature dependence of the anisotropy in the ultrasonic attenuation $\Delta \alpha$ in the nematic phase of MBBA at 1, 2, 4, and 10 MHz. $\Delta \alpha$ is analyzed in terms of a critical, a noncritical, and a nonrelaxing part. For the relaxation time of the noncritical component we find a reverse-barrier activation enthalpy $\Delta H_2=2.25\mathrm{}\pm 10\%$ kcal ${\mathrm{mole}}^{-1\mathrm{}}$. This value is consistent with rotational isomerism in the butyl end group of the molecule. The relaxation time of the critical component in the nematic phase is found to have a critical exponent $\gamma =1.06\mathrm{}\pm 10\%$. It is tentatively suggested that the critical part of $\Delta \alpha$ may be due to relaxation of fluctuations of the order parameter and not to relaxation of the order parameter itself.

We show that two strong but physically plausible assumptions allow all the measurable scaling exponents for very high-Reynolds-number incompressible turbulence to be expressed simply in terms of the exponent $\mu$ characterizing the dissipation fluctuations. The first assumption, introduced by Obukhov in 1962, relates the locally fluctuating dissipation to the locally fluctuating nonlinear energy transfer. The second assumption is that the same dissipation-length scale $\eta$ defines the crossover from the dissipation range to the inertial range for all of the velocity structure functions. The resulting exponent relations are the same as recently obtained from a simple geometrical model by Frisch, Sulem, and Nelkin, but the results appear less model dependent in the present context. In addition we introduce a locally defined energy transfer variable $T\left(\overrightarrow{\mathrm{x}}\right)={\eta }^2{\psi }^3\left(\overrightarrow{\mathrm{x}}\right)$, where $\psi \left(\overrightarrow{\mathrm{x}}\right)$ is any component of the velocity derivative tensor. We suggest that $T\left(\overrightarrow{\mathrm{x}}\right)$ has the same statistical properties as the locally defined viscous dissipation $\tilde{\epsilon }\left(\overrightarrow{\mathrm{x}}\right)=\nu {\psi }^2\left(\overrightarrow{\mathrm{x}}\right)$, where $\nu$ is the kinematic viscosity. This suggestion is compatible with our other results, and is capable of experimental test.

We employ the equivalence between Zwanzig's projection-operator formalism and perturbation theory to demonstrate that the approximate-solution technique of statistical linearization for nonlinear stochastic differential equations corresponds to the lowest-order $\beta$ truncation in both the consolidated perturbation expansions and in the "mass operator" of a renormalized Green's function equation. Other consolidated equations can be obtained by selectively modifying this mass operator. We particularize the results of this paper to the Duffing anharmonic oscillator equation.

The effect of the transition zone on the surface energy and the surface tension near the triple point is reported. Both quantities are calculated from the extended Kirkwood-Buff formulas for the cases of a linear, a cubic, and a Fermi density profile. The realistic Barker-Fisher-Watts (BFW) potential and the Parson, Siska, and Lee (MSVIII) potential are used. The experimental radial distribution function of Yarnell et al. is used for the bulk liquid and as an approximation for the transition zone. As a function of the zone width both realistic potentials lead to the same rapid monotonic increase for the surface energy, but a different mild monotonic decrease for the surface tension. The transition zone width can be determined approximately from the surface energy. If the three-body nonadditivity effects on both quantities are ignored, the experimental value 34.9 erg/${\mathrm{cm}}^2$ of the surface energy suggests a zone width of 6.2 to 8.5 Å, thus predicting a value of the tension of 12.1 dyn/cm for the BFW potential and 12.8 dyn/cm for the MSVIII potential as compared to the experimental value of 13.1 dyn/cm. Including the three-body nonadditivity effects estimated from the Axilrod-Teller triple-dipole potential, the zone width needed for a correct value of the surface energy is slightly larger (7.2 to 9.5 Å). However, the surface tension thus predicted is too small for all potentials.

A molecular theory of nematic liquid crystals is presented in which the short-range spatial correlations are accounted for by an orientation-averaged pair correlation function. This results in a two-fold self-consistency scheme whose spatial description resembles that in the conventional theory of classical liquids and whose orientational description resembles a mean-field theory. Numerical calculations are made for a simple potential chosen to describe para-azoxyanisole (PAA). Among the properties calculated are the two-phase coexistence curve, the order parameter along the coexistence curve, temperature dependence of the order parameter at constant pressure, and volume change and latent heat at the isotropic-nematic transition. Comparison with experimental data is made. There are encouraging signs to warrant further investigation.

Quadratic polarizabilities calculated from the three leading strongly coupled one-component-plasma theories are examined in the long-wavelength limit to see if they satisfy a recently established nonlinear compressibility sum rule. Only the Golden-Kalman polarizability calculated in the velocity-average-approximation exactly satisfies this rule to lowest order in the plasma parameter.

For any given velocity autocorrelation function, the incoherent scattering functions and the density self-correlation functions are uniquely determined in the Gaussian approximation. Since the natures of velocity autocorrelation functions have already been clarified, it is only necessary to evaluate the non-Gaussian correction terms. We have related these corrections to the deviations of partially-averaged mean-square displacements from the canonical average. Two time constants are involved: the collision time and the mixing time, the latter being much longer (by a factor of 50) than the former. The results are in qualitative agreement with molecular dynamics as well as neutron inelastic scattering data for liquid argon.

Thermodynamic properties of two-dimensional electron liquids are determined by numerical experiments over the domain of the plasma parameter $\epsilon =\frac{2\pi n{e}^4}{T^2}$, $0.05\le \epsilon \le 5\times {10}^3$, where $n$, $e$, and $T$ denote the number density, the unit charge, and the temperature in energy units, respectively. The correlation energy thus obtained is consistent with both the result of the plasma-parameter expansion calculation and that of earlier numerical experiments in the high-density domain. The pair correlation function $g\left(r\right)\mathrm{}$ agrees with the analytical result when $\epsilon$ is small. Short-range order appears at the critical plasma parameter ${\epsilon }_{\mathrm{sr}}$ in the range $10<\mathrm{}{\epsilon }_{\mathrm{sr}}<15\mathrm{}$. It is pointed out that the two-dimensional screening function $h\left(r\right)=\ln \mathrm{}\left[g\right(r)+1]+\frac{e^2}{\mathrm{Tr}}$ is approximately a linear function of the radius $r$ in the short-range domain, as in the case of three dimensions.

We describe a two-dimensional analytic calculation of the equilibrium diamagnetic field strength and shape, applicable in the regime of high magnetization to the high-energy low-angular-momentum ion orbits which characterize the distribution in a Migma fusion device.

Broadened Lyman-$\alpha$ and Lyman-$\beta$ x-ray profiles from hydrogenic, high-$Z$ radiators immersed in a hot, dense plasma are presented. Particular attention is given to broadened profiles originating in a plasma where there are two ionic species and in which ions and electrons have different temperatures. The applicability of these calculations to early laser-fusion experiments is discussed.

Within the formalism of Mori and Zwanzig a kinetic equation for the phase-space density of simple classical liquids is derived whose relaxation kernel is obtained as the solution of a second kinetic equation. The latter equation is closed by introducing a phenomenological relaxation rate. The solution of the resulting equations can be reduced to the evaluation of the self-correlation function for a tagged particle, studied earlier by the present authors. The calculated scattering functions $S(q, \omega )$ for liquid argon and rubidium are compared with neutron scattering experiments as well as molecular dynamics calculations. For liquid argon the earlier results of Duderstadt and Akcasu are improved, and it is furthermore shown that the present theory adequately accounts for the persistence of propagating density fluctuations in liquid rubidium.