Comparisons between variational, perturbational, and exact solutions for scattering from a random rough-surface model*

We present a method for constructing optimized pseudopotentials for general polyatomic systems which yield the “smoothest” orbitals localized in particular regions of space. (“Smoothest” orbitals are those for which the expectation value of the kinetic energy operator is a minimum.) We specifically consider the general case in which one can have several occupied orbitals localized in a particular region, rather than a single orbital. Recently, Anderson presented a pseudopotential formalism for constructing localized orbitals for a single isolated band in a polyatomic system. We have extended his formalism to the general multiorbital case, and show that for a specific choice of pseudopotential the Anderson equation yields optimal orbitals. We also show that the Hückel form of the Anderson pseudopotential equation yields identical eigenvalues (neglecting terms of order S2) to the optimal form, and orbitals differing from optimal orbitals by terms linear in S. The correction terms to the Hückel orbitals are obtained by first-order perturbation theory. Finally we examine the Adams–Gilbert equation, which is valid only for systems representable by a single Slater determinant, and indicate how one may solve this equation for the localized orbitals using perturbation theory with Anderson orbitals as the basis set.

A new approach to the problem of calculating the phonon component of the thermal conductivity of structurally disordered solids is presented. The basis of the method is the introduction of collective variables U(k) defined by where uα(j) denotes a Cartesian component of the displacement from equilibrium of the jth atom whose equilibrium site is at rj. For simplicity only monatomic solids are considered. The plane waves are chosen to satisfy periodic boundary conditions within a volume containing N atoms and the sum over k contains just N terms. In the case of amorphous solids the wave vectors are chosen to be within the Debye sphere. It is assumed that the waves are linearly independent. In the case of solids containing a large amount of crystalline order, for example when dislocations are present, it is not possible to give a precise definition of the upper limit in the sum over plane waves. For many practical purposes this is fairly unimportant, especially at low temperatures where only long-wavelength modes are excited. The merit of this approach is that it avoids describing a solid as a disordered lattice, which is very convenient in the case of amorphous solids and also when defects such as dislocations and interstitials are present in a lattice. The Boltzmann equation in this representation has precisely the same form as that of the standard theory of thermal conductivity. The collision terms have been calculated to first order in perturbation theory.

Diffractive corrections to the physical optics solution for the scattering of high-frequency sound by a random rough pressure release surface are derived. The correction terms are obtained by applying the composite-roughness theory to an expansion of the scattering integral. Results are presented for a three-dimensional rough surface and are compared with those obtained using a conventional composite-roughness approach.

We revisit the problem of one-dimensional Anderson localization, by providing\nperturbative expression for Lyapunov exponent of Anderson model with\nnext-nearest-neighbor (nnn) hopping. By comparison with exact numerical\nresults, we discuss the range of validity of the naive perturbation theory. The\nstability of band center anomaly is examined against the introduction of nnn\nhopping. New anomalies of Kappus-Wegner type emerge at nonuniversal values of\nwavelength when hopping to second neighbor is allowed. It is shown that\ncovariances in the first order of perturbation theory, develop singularities at\nthese resonant energies which enable us to locate them.\n

EXACT CONICAL-FLOW SOLUTIONS are available only for circular cones at zero angle of attack. For nonaxisymmetric cones or cones at angle of attack, only approximate methods exist. These methods are generally quite complicated and further limited to certain body shapes or certain Mach-number ranges.* 3 A great need was therefore felt for a simple approximate method applicable to any arbitrarily shaped conical body at zero incidence as well as at angle of attack. Such a method has been developed recently at Lockheed and is presented here in abbreviated form. The method is based on the theory. This theory determines the pressure on a conical body utilizing information for a symmetric cone at zero angle of attack with the same normal component of the free stream with respect to the surface as the local element of the body considered. This method works relatively well a t high Mach numbers. However, it is quite inconsistent at lower Mach numbers, especially for bodies which deviate considerably from circular cones. The equivalent-cone method does not give satisfactory results, mainly due to the fact that it considers only the local surface element on the body independent of the other body elements in the Newtonian-theory manner. These interference effects between the different surface elements have been taken into account in the newly developed method by adding to the equivalent-cone value of pressure an additional correction term for the pressure exchange along the surface of the body. This new method may be called the improved equivalent-cone method. I t covers the whole supersonic\Mach number range for which the conical flow conditions with attached shock at the apex of the body are fulfilled. The assumptions made for the correction term are that the pressure exchange along the surface is proportional to the pressure difference between the equivalent cone and circular cone. This reference circular cone is a cone having a pressure equal to the

First-order perturbation theory is employed to examine sound propagation in flowing media confined by a cylindrical waveguide. The use of perturbation theory allows examination of mode phase-speed changes due to any radially dependent flow w(r) as long as the flow magnitude is sufficiently small. The condition to be fulfilled is satisfied in the flow range: 0-0.3 m/s for the specific values of cylinder radius, ultrasound frequency, and sound speed analyzed in the present work [in the general case, however, the condition in Eq. (1) of the present work must be fulfilled]. This freedom of choice, i.e., the possibility to handle any radial flow profile, is used to analyze two flow profile cases: (1) where w(r) is a linear combination of a laminar flow profile and a flat profile corresponding to turbulent flow, and (2) where w(r) is a linear combination of a laminar flow profile and a more realistic logarithmic-dependent turbulent flow profile. In both cases, it is shown that large errors may result in ultrasound flow measurements if several modes are excited by the transmitting transducer, and that a logarithmic flow profile in the turbulent regime leads to somewhat larger measurement errors at high flow values as compared to assuming a simple flat profile in the turbulent regime.

A modification of the physical optics surface source density based on the Lynch variational approach is proposed for the case of scattering by one-dimensional pressure-release Gaussian random rough surfaces. Rather than the local radius of curvature, it involves the surface correlation length and introduces new criteria validating the physical optics evaluation of the coherent and incoherent scattering intensities. Closed expressions for the coherent reflection coefficient and the incoherent scattering cross section are derived without resorting to high-frequency approximation techniques. Numerical data based on these expressions and conventional geometrical optics shadowing theory are presented and shown to be in very good agreement with recently published exact Monte Carlo simulation results.

Silicon photonics is going trough a terrific expansion driven by several applications, from chip wiring to integrated sensors and telecommunications. Some applications, e.g. intra and inter chip connections and sensing, require long parallel waveguides for wiring or for connecting grating couplers (GCs) to devices situated in sensing micro-channels. In well packed photonics chips there are often long wiring waveguides parallel for several mm, so loss can be caused by light coupled back and forth between them (cross-talk), by scattering, wall roughness, mode mismatch, etc. This work aims to investigate cross-talk for long parallel waveguides, and to propose methods to reduce cross-talk loss when high integration density is required. We have designed and fabricated about 200 testing structures exploiting e-beam on silicon on insulator (SOI) chip, in order to test several parameters and to find out dominant loss mechanisms. All devices have been tested and measured using an automatic optical bench, in the wavelength range between 1500-1600 nm. Achieved results are promising, since they allow for comparing cross-talk for short as well as long interaction lengths (up to 5 mm), different waveguide width pairs, several separation distances, and for TE and TM polarization. For smaller gaps, having not symmetric pair of waveguides is very beneficial, since it results in a lower power coupling, e.g. about 20/14 dB of crosstalk reduction for TE/TM waveguides after 5 mm of propagation and gap of 0.5 μm. This can be very useful for the design of integrated photonics chips requiring high-density packaging of devices and waveguides.

SummaryHigh‐gain observers and sliding mode observers are two of the most common techniques to design observers (or differentiators) for lower triangular nonlinear dynamics. While sliding mode observers can handle globally bounded nonlinearities, high‐gain linear techniques can deal with globally Lipschitz nonlinearities. To gain in generality and avoid the usual assumption that the plant's solutions are bounded with known bound, we propose here to mix both designs in the more general case where the nonlinearities satisfy a global incremental affine bound. We inspire from the recently developed low‐power high‐gain observer technique, which relies on the interconnection of several second‐order high‐gain observers. Adding sliding‐mode correction terms into this low‐power structure enables to guarantee global convergence of the estimation error in finite‐time with gains depending only on the parameters of the incremental affine bound of the nonlinearities. The estimation error is also proved to be uniformly stable along solutions starting from any compact sets of initial conditions.

The diffusion equation in physical space–time for a Brownian particle driven by an external force field has been derived by Smoluchowski in the two particular cases where the external field is uniform or varies linearly with position (elastic force). In more general cases, correction terms must be added to the Smoluchowski equation. We show here how to use a multi-scale Chapman–Enskog expansion to obtain, in the hydrodynamic limit, the first corrective terms to the Smoluchowski equation, without any restriction on the friction coefficient, and for any sufficiently small position-dependent constant force field. We also compare our approach with the works of Wilemski, Titulaer, and van Kampen.

We report on a refined macroscopic model for slightly compressible gas slip flow in porous media developed by upscaling the pore-scale boundary value problem. The macroscopic model is validated by comparisons with an analytic solution on a two-dimensional (2-D) ordered model structure and with direct numerical simulations on random microscale structures. The symmetry properties of the apparent slip-corrected permeability tensor in the macroscale momentum equation are analysed. Slip correction at the macroscopic scale is more accurately described if an expansion in the Knudsen number, beyond the first order considered so far, is employed at the closure level. Corrective terms beyond the first order are a signature of the curvature of solid–fluid interfaces at the pore scale that is incompletely captured by the classical first-order correction at the macroscale. With this expansion, the apparent slip-corrected permeability is shown to be the sum of the classical intrinsic permeability tensor and tensorial slip corrections at the successive orders of the Knudsen number. All the tensorial effective coefficients can be determined from intrinsic and coupled but easy-to-solve closure problems. It is further shown that the complete form of the slip boundary condition at the microscale must be considered and an important general feature of this slip condition at the different orders in the Knudsen number is highlighted. It justifies the importance of slip-flow correction terms beyond the first order in the Knudsen number in the macroscopic model and sheds more light on the physics of slip flow in the general case, especially for large porosity values. Nevertheless, this new nonlinear dependence of the apparent permeability with the Knudsen number should be further verified experimentally.

SUMMARY We study properties of the energy-flux vector and other related energy quantities of homogeneous and inhomogeneous time-harmonic P and S plane waves, propagating in unbounded viscoelastic anisotropic media, both analytically and numerically. We propose an algorithm for the computation of the energy-flux vector, which can be used for media of unrestricted anisotropy and viscoelasticity, and for arbitrary homogeneous or inhomogeneous plane waves. Basic part of the algorithm is determination of the slowness vector of a homogeneous or inhomogeneous wave, which satisfies certain constraints following from the equation of motion. Approaches for determination of a slowness vector commonly used in viscoelastic isotropic media are usually difficult to use in viscoelastic anisotropic media. Sometimes they may even lead to non-physical solutions. To avoid these problems, we use the so-called mixed specification of the slowness vector, which requires, in a general case, solution of a complex-valued algebraic equation of the sixth degree. For simpler cases, as for SH waves propagating in symmetry planes, the algorithm yields simple analytic solutions. Once the slowness vector is known, determination of energy flux and of other energy quantities is easy. We present numerical examples illustrating the behaviour of the energy-flux vector and other energy quantities, for homogeneous and inhomogeneous plane P, SV and SH waves. 1 I NTR O DUCTION In this paper, we study the properties of the energy flux of homogeneous and inhomogeneous time-harmonic plane waves propagating in an unbounded viscoelastic anisotropic medium. The complexvalued energy-flux vector F (also called the Poynting or Umov‐ Poynting vector) is used as the quantity which fully characterizes the energy flux and the dissipation of energy of the wave. The real part of the vector F represents the time-averaged energy flux S. Both real, Re F = S, and imaginary, Im F, parts are closely related to the density of the time-averaged dissipated energy W d : W d equals twice the scalar product of −Im F and propagation vector P (real part of the slowness vector, perpendicular to the wave front) or twice the scalar product of Re F and attenuation vector A (imaginary part of the slowness vector, perpendicular to the plane of constant amplitude). The aim of this paper is to present a concise derivation of the expression for the complex-valued energy-flux vector F of homogeneous and inhomogeneous plane waves propagating in unbounded media of unrestricted anisotropy and viscoelasticity, to propose an algorithm for its calculation, and to perform analytical and numerical studies necessary for correct understanding of energy propagation and dissipation of the relevant plane waves. Considerable attention in seismology has been devoted to the energy flux of plane waves propagating in elastic anisotropic and in viscoelastic isotropic media. For elastic anisotropic media, see, for

There is a need for a solution to the problem of wave scattering from rough surfaces that is accurate when both the classical field perturbation and Kirchhoff approximations are not. In previous work it was shown numerically that the phase perturbation reflection and backscattering coefficients reduce to those of field perturbation theory and the Kirchhoff approximation in the appropriate limits; however, for cases when the classical solutions did not apply, the accuracy of the phase perturbation results could not be determined. In this paper we examine the validity of the phase perturbation technique for a region in parameter space when the two classical solutions fail. Numerical results for the bistatic scattering cross section are compared with exact numerical results for one-dimensional surfaces having a Gaussian roughness spectrum and satisfying a Dirichlet boundary condition. It is found that in the region considered the phase perturbation results agree with the exact results over all scattered angl...

Exact results on the critical function for the motion of an electron driven by two plane waves

Study of power flow mechanism of capacitive power transfer system based on Poynting vector analysis