Rough Surface Effects on the p-Wave Fermi Superfluids

Rough surface effects on p-wave Fermi superfluids are discussed on the basis of our recently proposed quasi-classical theory that can treat in a unified way the surface scattering ranging from the specular limit to the diffusive limit. We show how the transition temperature of the ABM state in a slab geometry depends on the surface roughness. In the diffusive limit, our result agrees with Kjäldmann et al. We consider the BW state, the ABM state and the polar state in a half-infinite geometry with a plane rough surface and discuss the self-consistent order parameter and the surface density of states at low temperatures. The rough surface effects on the Andreev scattering in the BW state are also discussed within the same framework.     

Boundary conditions for quasiclassical Green's Function for superfluid Fermi systems

We show that the quasiclassical Green's Function for Fermi liquids can be constructed from the solutions of the Bogoliubov-de Gennes equation within the Andreev approximation and derive self-consistent relations to be satisfied by the quasiclassical Green's function at the surfaces. The so-called normalization condition for the quasiclassical Green's function is obtained from this self-consistent relation. We consider a specularly reflecting wall, a randomly rippled wall, and a proximity boundary as model surfaces. Our boundary condition for the randomly rippled wall is different from that derived by Buchholtz and Rainer and Buchholtz.     

The superflow state of 3He-B at a diffusive wall. Quasiclassical calculations

We report on first computations considering effects of a rough wall on the counterflow state in superfluid ³He-B for high flow velocities. Using the quasiclassical Green's-function formalism supplemented by the boundary conditions for a diffusive wall, we calculate the order-parameter field and the supercurrent near a container wall for various pressures and temperatures. One of our results is that the current density at the wall as a function of the flow has a maximum at the velocity which is about half of the pair breaking velocity.     

Quasiclassical Green's function in slab geometry: Application to A-B transition of superfluid3He in a slab

We show that the quasiclassical Green's function method can be used in slab geometries when the separation is much longer than the Fermi wavelength; consequently, quantum interference effects due to the double walls can be averaged out. We apply the quasiclassical method to the A-B transition of superfluid³He in a slab with specular walls at arbitrary temperature. We consider the phase transition between the planar state and the BW state within the weak coupling theory. The phase transition is shown to be of second kind at all temperatures. We obtain the critical size at which the B phase becomes unstable as a function of temperature. AtT=0 K and at SVP, the critical size is estimated to be 0.78 µm.     

Boundary element modelling of flat plate floors under vertical loading

The present paper develops a boundary element model for flat plate floors. The floor slab is modelled using the shear‐deformable plate bending theory. Internal columns or walls are treated using internal collocation technique, where three interaction generalized forces are considered at the slab–column connection: two bending moments in two directions and shear force in the vertical direction. Such forces are considered to be constant over the column cross‐sections. The present technique takes into account the realistic geometric modelling of the column cross‐sections. The effect of considering such geometry in the analysis of the bending moments transferred from slab to columns is studied. Several examples, including solution of practical building slab, are presented. The results are compared to those obtained from other numerical methods to demonstrate the accuracy and the reliability of the present formulation. The present formulation can be considered as an accurate tool to predict moment transferred from the slab to the column. Copyright © 2004 John Wiley & Sons, Ltd.     

Transport in Fermi Liquids Confined by Rough Walls

I present theoretical calculations of the thermal conductivity of Fermi liquid \(^3\) He confined to a slab of thickness of order \(\sim \) 100 nm. The effect of the roughness of the confining surfaces is included directly in terms of the surface roughness power spectrum which may be determined experimentally. Transport at low temperatures is limited by scattering off rough surfaces and evolves into the known high-temperature limit in bulk through an anomalous regime in which both inelastic quasiparticle scattering and elastic scattering off the rough surface coexist. I show preliminary calculations for the coefficients of thermal conductivity. These studies are applicable in the context of electrical transport in metal nanowires as well as experiments that probe the superfluid phase diagram of liquid \(^3\) He in a slab geometry.     

Recovery of optical parameters in multiple-layered diffusive media: theory and experiments

Diffuse photon density waves have lately been used both to characterize diffusive media and to locate and characterize hidden objects, such as tumors, in soft tissue. In practice, most biological media of medical interest consist of various layers with different optical properties, such as the fat layer in the breast or the different layers present in the skin. Also, most experimental setups consist of a multilayered system, where the medium to be characterized (i.e., the patient's organ) is usually bounded by optically diffusive plates. Incorrect modeling of interfaces may induce errors comparable to the weak signals obtained from tumors embedded deep in highly heterogeneous tissue and lead to significant reconstruction artifacts. To provide a means to analyze the data acquired in these configurations, the basic expressions for the reflection and transmission coefficients for diffusive-diffusive and diffusive-nondiffusive interfaces are presented. A comparison is made between a diffusive slab and an ordinary dielectric slab, thus establishing the limiting distance between the two interfaces of the slab for multiple reflections between them to be considered important. A rigorous formulation for multiple-layered (M-layered) diffusive media is put forward, and a method for solving any M-layered medium is shown. The theory presented is used to characterize a two-layered medium from transmission measurements, showing that the coefficients of scattering, mu'(s) , and absorption, mu(a) , are retrieved with great accuracy. Finally, we demonstrate the simultaneous retrieval of both mu;(s) and mu(a).     

Effect of Rough Walls on Transport in Mesoscopic 3He Films

The interplay of bulk and boundary scattering is explored in a regime where quantum size effects modify mesoscopic transport in a degenerate Fermi liquid film of ³He on a rough surface. We discuss mass transport and the momentum relaxation time of the film in a torsional oscillator geometry within the framework of a quasiclassical theory that includes the experimentally determined power spectrum of the rough surface. The theory explains the anomalous temperature dependence of the relaxation rate observed experimentally. We model further studies on ³He confined in nanofluidic sample chambers with lithographically defined surface roughness. The improved understanding of surface roughness scattering can be extended to the analogous system of electrons in metals and suggests routes to improve the conductivity of thin metallic films.     

A hybrid finite element/method of moment formulation for singlefrequency eddy current inversion

Eddy-current inverse techniques using single-frequency currents have been applied with limited success to the reconstruction of crack width and thickness profiles primarily for one-dimensional and axisymmetric geometries. Because of the diffusive nature of the induced low-frequency eddy currents, the reconstruction process differs from high-frequency wave propagation methods. On the physical basis that both diffusive and wave phenomena can be described by the same Green's function with either a complex or real wave number, an integral formulation for the low-frequency magnetic vector potential is presented. By employing an iterative Born approximation algorithm and the method of moments, a reconstruction method for the conductivity profile in a metallic specimen is developed. To make this formulation amenable to complex geometries, finite-element analysis techniques are utilized to compute the integral kernel. The inversion process is tested with synthetic data generated by the numerical solution of a generic embedded flaw in a full-space and a surface-breaking defect     

Theory of Transverse Acoustic Impedance of Superfluid 3He

A microscopic theory of transverse acoustic impedance of superfluid ³He is presented. We calculate the stress tensor of superfluid ³He as a response to the oscillating rough wall. For that purpose, we extend a quasi-classical theory with the random S-matrix model for the rough wall to time dependent problems. By this formulation, we can take into account the pair breaking mechanism. Our numerical results show good agreement with the recent experiments in both the A-phase and the B-phase. We discuss that the frequency and the temperature dependence of the transverse acoustic impedance is dominated by the pair excitations involving the surface bound states.     

Hybrid finite-element-boundary integral algorithm to solve the problem of scattering from a finite and infinite array of cavities with stratified dielectric coating

This work presents a hybrid finite-element-boundary integral algorithm to solve the problem of scattering from a finite and infinite array of two-dimensional cavities engraved in a perfectly electric conducting screen covered with a stratified dielectric layer. The solution region is divided into interior regions containing the cavities and the region exterior to the cavities. The finite-element formulation is applied only inside the interior regions to derive a linear system of equations associated with unknown field values. Using a two-boundary formulation, the surface integral equation employing the grounded dielectric slab Green's function in the spatial domain is applied at the opening of the cavities as a boundary constraint to truncate the solution region. Placing the truncation boundary at the opening of the cavities and inside the dielectric layer results in a highly efficient solution in terms of computational resources, which makes the algorithm well suited for the optimization problems involving scattering from grating surfaces. The near fields are generated for an array of cavities with different dimensions and inhomogeneous fillings covered with dielectric layers.     

Stress singularities in a two-material wedge

A method for the determination of stresses in a two-material wedge-shaped region is presented. The method is applicable for plane strain or plane stress problems and treats the general case where each region is a wedge of arbitrary angle. The results are obtained by the use of the Mellin transform and the theory of residues.The characteristic equation is investigated to determine the stress singularity resulting from certain combination of geometry and material properties. A formal solution is then presented for the case where the loading is in the form of a point dislocation along the interface. This solution is the Green's function for the more general mismatch problems and therefore has applications in solving other problems with compatible boundary conditions. The results obtained show that for small values of r the dominant effect is due to geometry and the secondary effect is caused by the choice of elastic constants of the materials.This work was supported by the National Science Foundation under Grant GK-11977, and the Bell Telephone Laboratories Doctoral Support Program.     

Green's First Identity Method for Boundary-Only Solution of Self-Weight in BEM Formulation for Thick Slabs

The present paper develops a new technique for treatment of self-weight for building slabs in the boundary element method (BEM). Due to the use of BEM in the analysis, all defined variables are presented on the slab boundary (mesh is defined only along the slab boundary). Self-weight, however, is usually defined over slab domain, hence domain discretisation is required, which spoils the main advantage of the BEM. In this paper a new method is presented to transform self-weight domain integrals to the boundary for such slabs. The proposed method is based on using the so-called Green's first identity. All new kernels for generalized displacements, stress-resultants, and tractions are derived and listed explicitly. The present formulation is implemented into computer code and several examples are tested. Results are compared against results obtained from other numerical method to prove the accuracy and validity of the present formulation.     

Optical extinction by closely spaced parallel cylinders inside a finite dielectric slab

The formulation for the extinction and scattering cross sections of closely spaced parallel infinite cylinders in a dielectric medium of finite thickness is presented. We consider the general case of dissimilar refractive indices for the half-spaces on both sides of the slab, and the diameter and refractive index of each cylinder can be different. The formulation accounts for the coherent scattering between the cylinders and scattering of the multiply reflected internal waves inside the slab. Discontinuity in the refractive index across the dielectric slab interfaces results in boundary reflections that modify the angular distribution of the scattered intensity in both forward and backward directions. The extinction cross section, which is derived by a formal application of the optical theorem, is shown to consist of both a forward and a backward component. The general solution is applied to obtain the formulas for the cases of cylinders in front of a reflecting plane, cylinders inside a semi-infinite dielectric medium, and cylinders in free space.     

The application of the boundary integral equations method to subsonic flow with circulation past thin airfoils in a wind tunnel

Summary In this paper we apply the direct boundary integral equations method to subsonic flow with circulation past a thin airfoil in a wind tunnel. A set of Green's functions for the equations of the velocity perturbation is deduced. These functions together with the non-linear limit condition imposed just on the surface profile lead to a Fredholm integral equation of the second kind over the profile only. The integral formulation has the advantage not to truncate the flow domain and to save computing effort when numerical solving is performed, due to the lack of the tunnel walls discretisation. In the case of the incompressible fluid we use the exact equations of motion, which implies a valid solution for any shape of the profile, being not necessarily thin. In the case of the compressible fluid we use the linearized motion equations, which implies a valid solution only for thin airfoils. The integral equation obtained on the surface, together with the circulation integral formula are solved via a collocation method. The numerical tests made for the circular obstacle show a very good agreement with the theory. The numerical experiments on the NACA-4412 profile were made in order to determine the tunnel and the compressibility effects being compared to the unbounded flow.     

Light scattering from self-affine fractal silver surfaces with nanoscale cutoff: far-field and near-field calculations

We study the light scattered from randomly rough, one-dimensional, self-affine fractal silver surfaces with nanoscale lower cutoff illuminated by s- or p-polarized Gaussian beams a few micrometers wide. By means of rigorous numerical calculations based on the Green's theorem integral equation formulation (GTIEF), we obtain both the far- and near-field scattered intensities. The influence of diminishing the size of the fractal lower-scale irregularities (from approximately 50 nm to a few nanometers) is analyzed in the case of both single realization and ensemble-average magnitudes. For s polarization, variations are small in the far field, being significant only in the higher-spatial-frequency components of evanescent character in the near field. In the case of p polarization, however, the nanoscale cutoff has remarkable effects stemming from the roughness-induced excitation of surface-plasmon polaritons. In the far field, the effect is noticed both in the speckle pattern variation and in the decrease of the total reflected energy upon ensemble averaging, as a result of increased absorption. In the near field, more efficient excitation of localized optical modes is achieved with smaller cutoff, which in turn leads to huge surface electric field enhancements.     

Spatial coherence in strongly scattering media

We study the spatial coherence of an optical beam in a strongly scattering medium confined in a slab geometry. Using the radiative transfer equation, we study numerically the behavior of the transverse spatial coherence length in the different transport regimes. Transitions from the ballistic to the diffusive regimes are clearly identified.     

Green's function method in the radiative transfer problem. II. Spatially heterogeneous anisotropic surface

The most recent theoretical studies have shown that three-dimensional (3-D) radiation effects play an important role in the optical remote sensing of atmospheric aerosol and land surface reflectance. These effects may contribute notably to the error budget of retrievals in a broad range of sensor resolutions, introducing systematic biases in the land surface albedo data sets that emerge from the existing global observation systems. At the same time, 3-D effects are either inadequately addressed or completely ignored in data processing algorithms. Thus there is a need for further development of the radiative transfer theory that can rigorously treat both 3-D and surface anisotropy effects and yet be flexible enough to permit the development of fast forward and inversion algorithms. We describe a new theoretical solution to the 3-D radiative transfer problem with an arbitrary nonhomogeneous non-Lambertian surface. This solution is based on an exact semianalytical solution derived in operator form by the Green's function method. The numerical implementation is based on several parameterizations that accelerate the solution dramatically while keeping its accuracy within several percent under most general conditions.     

Second-order perturbation theory for scattering from heterogeneous rough surfaces

We propose a model to calculate scattering from inhomogeneous three-dimensional, rough surfaces on top of a stratified medium. The roughness is made up of an ensemble of deposits with various shapes and permittivities whose heights remain small with respect to the wavelength of the incident light. This geometry is encountered in the remote sensing of soil surfaces, or in optics wherever there are contaminated planar components. Starting from a volume-integral equation involving the Green's tensor of the stratified medium, we derive a height-perturbative expansion up to second-order. Our formulation, which depends explicitly on the profiles of each deposit and on the Fresnel coefficients of the layered substrate, accounts for double-scattering events and permits an evaluation of depolarization in the plane of incidence. Comparisons with rigorous calculations in the simplified case of two-dimensional geometries are presented. It is shown that the second-order scattering term can be much more important for heterogeneous surfaces than for their homogeneous counterparts.