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A generalization of the Navier-Stokes equation, valid for wavelengths and times of a molecular order of magnitude, is discussed on the basis of viscoelastic behavior of simple classical liquids. In this theory, transport coefficients are replaced by appropriate viscoelastic memory functions. The theory is verified by analyzing the data on current-correlation functions obtained from computer experiments.Three different models for the time dependence of the viscoelastic memory are investigated, namely, a single-exponential decay, a modified-exponential decay, and a Gaussian decay. It is observed that the memory functions are approximately Gaussian, at least for times of the order of one or two relaxation times. This is in agreement with a conjecture of Forster, Martin, and Yip. The wave-number dependence of the half-width of the Gaussian decay, and of the longitudinaland shear-viscosity coefficients, are found from computer experiments. The extrapolated values of these transport coefficients, in the limit k -0, are in good agreement with experiments on liquid argon.

High-frequency gratings with rectangular-groove profiles are used to generate high-efficiency beam splitters and beam deflectors. The effects of the grating design parameters, i.e., period, groove depth, duty cycle, number of phase levels, and polarization state (TE and TM) of the incoming signal, are considered. The case of the binary beam splitter grating is analyzed by using rigorous electromagnetic grating analysis. Fabrication techniques are presented in which three different lithographic techniques are considered (optical contact, deep-UV stepper reduction, and electron-beam direct write). Experimental results of 97% efficiency for the beam splitter grating and up to 80% for the beam deflector grating are reported.

expansion and evaluated leading terms for the phonon frequency and damping which were not available before.The rather sizable reduction in algebraic complexity using our method seen s to indicate that the next order, i. e. , the two-ring terms, is not out of reach. The study of two-ring diagrams will be much more than just a mathematical exercise. It will be a critical examination of the so-called "four-phonon process" and it will indicate the role of close-range interactions. The numerical results (5. 38)-(5.40) are not expected to be too useful in fitting the pho-non dispersion data of superfluid helium. Of course, useful numerical coefficients are not the objective of our calculation. We hope the above analysis has demonstrated a more novel method of calculation as well as given a clearer physical picture. Rev. 174, 227 (1968).The scattering length alone is not sufficient to deter-mine the interaction potential. Therefore, our model, which will assume a point interaction, will not illuminate effects pertaining to the detailed form of the potential. %'e shall keep the formalism here to a minimum. For a more detailed dielectric-constant formulation, see Ref. 5, Sec. II. P. C. Hohenberg and P. C. Martin, Ann. Phys. (N. Y. )~34 291 (1965).OThe Hugenholtz-Pines theorem is usually stated in terms of the full self-energy. However, it is trivial to show that only the irreducible part contributes.'~The rules of the standard finite-temperature perturbation theory can be found in many texts, for example, A. A. Abrikosov et al.Generalized hydrodynamics for classical fluids composed of structured molecules is discussed from a fundamental microscopic viewpoint. The analysis is concentrated on collective fluctuations of the intrinsic molecular angular momentum, and their coupling to the conserved densities of particle number, linear momentum, and energy. It is found from symmetry that the transverse components of linear and angular momentum are dynamically coupled, while the longitudinal angular momentum density moves independently of the other variables. By using the Zwanzig-Mori projection-operator technique, we derive closed and rigorous equations of motion for the set of fluctuations considered. For the case where the intrinsic angular momentum is not far from being separately conserved, approximations can be motivated which reduce these equations to simple relaxation equations, valid for small k and long times. General, Kubo-type expressions for the relaxation coefficients are given. Finally, sum-rule considerations are presented from which these coefficients can be approximately calculated.