Surface Profile Function Research Articles

Numerical study of vortex breaker optimization in a first stage oxygen tank

One of the crucial factors affecting the carrying capacity of the cryogenic liquid launch vehicle is the effective volume of the tank. Theoretical and experimental investigations on vortex breaker mechanisms have proposed promising schemes applied in the oxygen tank of the liquid-propellant launch vehicle to ensure the normal operation of the engine. In this paper, the liquid surface profile functions of the laminar core when the vortex generates were derived based on the Rankine vortex model. The dimensionless residual volume V/d3 and the Froude number were applied to compare the theoretical prediction of critical height with the actual simulation data of liquid oxygen. This comparison method can improve the model's accuracy. The efficiency of different basic shapes of vortex breakers was tested by conducting CFD modelling on a non-vertical outflow tank under a specific operating condition. Simulation results suggest negligible effects of heat transfer and surface tension. A circular plate is considered the optimal vortex breaker shape in traditional vertical outflow tanks, while a higher optimize efficiency was discovered in the half baffle basic shape in a non-vertical outflow tank by comparing the dimensionless residual volume and flow coefficient. A 34.26% reduction in flow resistance of half baffle breaker can be reached when applying a twenty-degree outlet pipe chamfering setting compared to a zero-degree chamfer. Considering practical operating limitations, it is concluded that a vortex breaker mechanism in a half baffle basic shape with a radius of 2.5d and a height of 4/d is the optimal scheme, which is suitable for all types of tanks. Its optimization efficiency of the residual volume reduction is about 56.68% compared to a no-breaker installation case. Lastly, a general equation based on CFD simulation for predicting the residual volume under a certain outflow velocity was proposed: V/d3≅αFr0.3, which trend is consistent with that of mathematical prediction V/d3≅αFr1/3. This consistency proves the accuracy and applicability of optimization strategy in this paper.

SOLVING FUNCTIONAL-TECHNOLOGICAL PROBLEMS USING A NON-PARAMETRIC APPROACH FOR CONTROL OF MICROGEOMETRY OF THE SURFACES OF DETAILS

In the article considers the effect of surface roughness on the functioning of a specific mechanism. In case of proven dependence of the studied functional property of surface microgeometry, a process of optimization of the sought parameter is carried out, in which the methods of mathematical analysis and theoretic foundations of technology of appliance and mechanical engineering are used for the basis of the theoretical studies. As control parameters of the specified optimal microgeometry, deciding the given task, the so-called non-parametric criteria are used, namely graphical of certain surface-profile functions.

X-ray beam-shaping via deformable mirrors: surface profile and point spread function computation for Gaussian beams using physical optics.

X-ray mirrors with high focusing performances are commonly used in different sectors of science, such as X-ray astronomy, medical imaging and synchrotron/free-electron laser beamlines. While deformations of the mirror profile may cause degradation of the focus sharpness, a deliberate deformation of the mirror can be made to endow the focus with a desired size and distribution, via piezo actuators. The resulting profile can be characterized with suitable metrology tools and correlated with the expected optical quality via a wavefront propagation code or, sometimes, predicted using geometric optics. In the latter case and for the special class of profile deformations with monotonically increasing derivative, i.e. concave upwards, the point spread function (PSF) can even be predicted analytically. Moreover, under these assumptions, the relation can also be reversed: from the desired PSF the required profile deformation can be computed analytically, avoiding the use of trial-and-error search codes. However, the computation has been so far limited to geometric optics, which entailed some limitations: for example, mirror diffraction effects and the size of the coherent X-ray source were not considered. In this paper, the beam-shaping formalism in the framework of physical optics is reviewed, in the limit of small light wavelengths and in the case of Gaussian intensity wavefronts. Some examples of shaped profiles are also shown, aiming at turning a Gaussian intensity distribution into a top-hat one, and checks of the shaping performances computing the at-wavelength PSF by means of the WISE code are made.

Influence of Uncompensated Solution Resistance on Diffusion Limited Chronocoulometric Response at Rough Electrode

Theory is developed for the charge transient under ohmic effect in reversible charge transfer process at disordered electrodes. Preliminary experimental results are also analyzed using this theory. Our results show that the deviation from Anson response is stemming from surface disorder and solution resistance. Detailed theoretical results are obtained and analyzed for fractal roughness with self-affine statistical properties. Expressions for concentration, charge density, and total charge have operator structure in Fourier transformed (deterministic) surface profile function. An elegant mathematical formula between the average charge transient and surface structure factor of roughness is obtained. Initial response is attributed to resistive effects of the electrolyte. The intermediate response is attributed to the interplay between various characteristics of the power spectrum and “ohmic equilibration layer thickness”. The theory marks that ohmic effects penetrate farther in time so much so to merge with time regimes which are outside the diffusion regimes giving nonideal behavior at electrode/electrolyte interface. Our results help with the quantitative understanding of generalized Anson response for Nernstian system affected by the uncompensated resistance at rough electrode/electrolyte interface.

Admittance of diffusion limited adsorption coupled to reversible charge transfer on rough and finite fractal electrodes

A theoretical model is developed to quantify the effect of diffusion controlled adsorption of electroactive species on (reversible) charge transfer admittance at a rough electrode. Results are obtained for the linearized isotherm and applied on various electrode roughness models, viz. (i) deterministic surface profile, (ii) roughness as random surface profile with known statistical properties, viz. the power spectrum of roughness and (iii) random profile function with finite self-affine fractal properties. The general expressions for admittance density and total admittance have a systematic operator structure in Fourier transformed surface profile function. Realistic model of roughness requires inclusion of randomness in surface morphology, hence the quantity of interest, the mean admittance is obtained (by taking ensemble average over all possible random configurations). Mathematical expression for the mean admittance of random morphology is the functional of the power spectrum of roughness. A special case of finite fractal roughness is characterized as self-affine scaling property over limited length scales and hence possess two lateral cutoff lengths along with the fractal dimension and a topothesy length (ℓτ). Our analysis explicitly relates the problem of rough electrode admittance with diffusion length (D/ω), isotherm (length) parameter (K) and fractal morphological characteristics. The three characteristic frequencies in this theory are: (a) lower adsorption crossover frequency –ωad≈D/K2, (b) adsorption–roughness topothesy dependent weak pseudo-quasireversibility characteristic frequency –ωq≈D/(ℓτK) and (c) high frequency small cutoff length (ℓ) dependent Warburg – anomalous Warburg crossover frequency –ωw≈D/ℓ2. Finally, theory shows emergence of pseudo-quasireversibility in intermediate frequency region due to electrode roughness in adsorption–diffusion system and curtailment of high frequency anomalous diffusion regime.

Erratum: Substrate influence on the plasmonic response of clusters of spherical nanoparticles [Phys. Rev. B83, 075426 (2011)

The reflection and scattering of light from surfaces is a topic that has been much discussed in the scientific literature. The application of optical methods for the characterization of surfaces and the understanding of natural phenomena is dependent on good models for the optical properties of surfaces and interfaces. In this work, we will examine three different aspects of the optical properties of surfaces and interfaces.The measurement of polarization effects of surfaces is dependent on wellperforming Mueller matrix ellipsometers. While designing Mueller matrix ellipsometers for a single wavelength is trivial, the design of broad-band imaging ellipsometers is a much more challenging problem. We discuss the use of genetic algorithms to optimize such ellipsometers in a very general fashion, and apply them to ellipsometers based on various liquid crystal technologies. The results include an ellipsometer design which was patented, and which outperforms previous designs both in terms of spectral bandwidth and noise propagation.Recent progress in the ability to manipulate matter on the nanoscale has led to great interest in the biological, chemical, and physical properties of nanoparticles and nano-sized structures. Depositing nanoparticles on top of a surface modifies the optical properties of both the particles as well as the substrate: any optical resonances that the particle(s) possess are modified by the substrate, and the reflectivity of the substrate is changed due to the presence of the particles. The electromagnetic interactions between nanoparticles and substrate are here investigated in the quasistatic regime, with discussion of both small clusters of nanoparticles as well as (infinite) periodic lattices of nanoparticles deposited on a substrate. The results include analysis of the interactions and resonances found in nanoparticle clusters deposited on a substrate.When light is incident on surfaces possessing random roughness or structure, the light is scattered according to the statistical properties of the surface. While there has been significant progress on surface scattering from surfaces with one-dimensional surface profile functions, less attention has been devoted to numerical models of surfaces possessing two-dimensional surface profiles. The main reason for this is that the solution of these models requires a large amount of computational power, even by today’s standards. We attempt to shed some light on optical effects observed in scattering from two-dimensional rough surfaces, with some focus on polarization effects, which are absent in scattering from one-dimensional surfaces. The Mueller matrix for a rough surface is also calculated, and describes the full polarization effects of rough surface scattering.

Second-order calculation of the local density of states above a nanostructured surface

We have numerically implemented a perturbation series for the scattered electromagnetic fields above rough surfaces, due to Greffet, allowing us to evaluate the local density of states to second order in the surface profile function. We present typical results for thermal near fields of surfaces with regular nanostructures, investigating the relative magnitude of the contributions appearing in successive orders. The method is then employed for estimating the resolution limit of an idealized near-field scanning thermal microscope.

Theory of Generalized Cottrellian Current at Rough Electrode with Solution Resistance Effects

A theory for effect of uncompensated solution resistance on Nernstian (reversible) charge transfer at an arbitrary rough electrode is developed. The significant deviation from the Cottrellian behavior is explained, as arising from the resistivity of the solution and geometric irregularity of the interface. Results are obtained for various electrode roughness models, viz., (i) deterministic surface profile, (ii) roughness as random surface profile with known statistical properties, and (iii) random functions with limited self-affine fractal properties. Expressions for concentration, current density, and total current transients have a systematic operator structure in Fourier transformed (deterministic) surface profile function. For a randomly rough electrode, the statistically averaged response is obtained by ensemble averaging over all possible surface configurations. An elegant mathematical formula between the average electrochemical current transient and surface structure factor of roughness is obtained. Realistic fractal roughness is characterized as self-affine scaling property over limited length scales. Limiting behavior in short time, is attributed to resistive effects of the electrolyte, roughness factor and surface curvatures. In the intermediate time, the anomalous power law behavior is attributed to the diffusion length weighted spatial frequency features of roughness. Our results help with the quantitative understanding of generalized Cottrellian response of moderately supported electrolytic solution at rough electrode/electrolyte interface.

Method for designing an arbitrary continuous refracting surface with specified property of wavefronts conversion

A method is proposed for designing an arbitrary continuous refracting surface that can realize conversion between two specified distributions of wavefront. Optimization technique is avoided by solving a partial differential equation governing the surface-profile function. This equation is obtained using the property of congruence and the law of refraction, and its solution in this special case is found to be the same as that obtained in wavefront conjugation. A method based on finding the local coefficients of the surface is developed to solve the equation numerically, and an example is given to illustrate its feasibility.

Inverse scattering with far-field intensity data: random surfaces that belong to a well-defined statistical class

We consider the inverse scattering problem of retrieving the surface profile function from far-field angle-resolved intensity data. The problem is approached as a nonlinear constrained optimization problem. The surface, assumed one-dimensional and perfectly conducting, is also assumed to be a realization of a Gaussian random process with a Gaussian correlation function with known standard deviation of heights (δ) and correlation length (a). Starting from rigorously calculated far-field angle-resolved scattered data, we search for the optimum profile using evolutionary strategies. Examples that illustrate the proposed scheme are presented. Aspects of the convergence and lack of uniqueness of the solution are discussed.

Surface plasmon polariton propagation near an index step

In this work, we study theoretically the scattering of p-polarized light of frequency ω from, and its transmission through, a system consisting of a dielectric medium (prism) characterized by a dielectric constant ϵ0 in the region x3>D; a metal film characterized by a complex, frequency-dependent dielectric function ϵ1(ω) in the region 0<x3<D; a dielectric film characterized by a dielectric constant ϵ2 in the region ζ(x1)<x3<0; and vacuum (ϵ3=1) in the region x3<ζ(x1). The light, whose plane of incidence is the x1x3-plane, is incident through the prism. For the surface profile function ζ(x1) we assume the form ζ(x1)=−dθ(x1)θ(L−x1), where θ(x1) is the Heaviside unit step function. Thus, we have a dielectric film of thickness d and dielectric constant ϵ2 covering the part of the lower surface (x3=0) of the metal film defined by 0<x1<L. The reduced Rayleigh equation for the amplitude of the light scattered back into the prism, R(q|k), is derived, as is the reduced Rayleigh equation for the amplitude of the light transmitted into the vacuum, T(q|k). These integral equations are solved numerically and the results are used to calculate the intensity of the scattered field in the near- and far-field regions, and the intensity of the transmitted light in the near-field region, for several values of L and of the wavelength of the incident light. The results provide information about the scattering of the surface plasmon polariton at the metal–vacuum interface, excited by the incident light, by index steps on that interface, which can be used to determine the thickness of the dielectric film d, its extent L, and its dielectric constant ϵ2.

Spectral densities of embedded interfaces in composite materials

The effective resistivity and conductivity of two media that meet at a randomly rough interface are computed in the quasistatic limit. The results are presented in the spectral density representations of the Bergman-Milton formulation for the properties of two-component composite materials. The spectral densities are extracted from computer simulations of resistor networks in which the random interface separates two regions containing different types of resistors. In the limit that the bond lengths in the resistor network are small compared to parameters characterizing the surface roughness, the resistor network simulation approximates the continuum limit of the two-component composite. The Bergman-Milton formulation is used to obtain a set of exact sum rules in the continuum limit for the spectral densities in terms of parameters describing the surface roughness and the simulation results are found to agree with these limiting forms. Perturbation theory results of the composite in the continuum limit for weakly rough random interfaces are also presented. An expansion of the spectral density is determined to second order in the surface profile function of the random interface and compared with the Bergman-Milton sum rules and computer simulation results. The formalism is applied to surface plasmons, electron energy loss, and light scattering from rough surfaces. Layered structures are discussed briefly.

The angular intensity correlation functions C (1) and C (10) for the scattering of light from randomly rough dielectric and metal surfaces

We study the statistical properties of the scattering matrix S(q|k) for the problem of the scattering of light of frequency ω from a randomly rough one-dimensional surface, defined by the equation x 3=ζ(x 1), where the surface profile function ζ(x 1) constitutes a zero-mean, stationary, Gaussian random process. This is done by studying the effects of S(q|k) on the angular intensity correlation function C(q,k|q′,k′)=⟨I(q|k)I(q′|k′)⟩−⟨I(q|k)⟩⟨I(q′|k′)⟩, where the intensity I(q|k) is defined in terms of S(q|k) by I(q|k)=L −1 1(ω/c)|S(q|k)|2, with L 1 the length of the x 1 axis covered by the random surface. We focus our attention on the C (1) and C (10) correlation functions, which are the contributions to C(q,k|q′,k′) proportional to δ(q−k−q′+k′) and δ(q−k+q′−k′), respectively. The existence of both of these correlation functions is consistent with the amplitude of the scattered field obeying complex Gaussian statistics in the limit of a long surface and in the presence of weak surface roughness. We show that the deviation of the statistics of the scattering matrix from complex circular Gaussian statistics and the C (10) correlation function are determined by exactly the same statistical moment of S(q|k). As the random surface becomes rougher, the amplitude of the scattered field no longer obeys complex Gaussian statistics but obeys complex circular Gaussian statistics instead. In this case the C (10) correlation function should therefore vanish. This result is confirmed by numerical simulation calculations.

Channel polaritons

By means of Green's second integral identity in the plane and a parametrization of the surface profile function, the dispersion relation for surface electromagnetic waves guided by a straight channel cut into the otherwise planar surface of a solid in contact with vacuum has been determined numerically. The solid can be either a metal or a dielectric medium and is characterized by an isotropic, real, frequency-dependent dielectric function $\ensuremath{\epsilon}(\ensuremath{\omega})$ that is negative in some range of frequencies. The resulting propagating bound electromagnetic modes are called channel polaritons.

Scattering of s-polarized electromagnetic plane waves from a film with a shallow random rough surface on a perfect conductor

Scattering of s-polarized electromagnetic planes waves from a film, with a shallow random rough one-dimensional surface, bounded by vacuum and a perfect conductor is calculated. An integral equation that relates the amplitude of the scattered field to the incident wave is found by use of the Rayleigh hypothesis. The integral equation is solved numerically and by use of the perturbation theory, up to the fourth order in the surface profile function. In the angular dependence of the incoherent part of the differential reflection coefficient, the backscattering peak and two additional satellite peaks are observed, owing to two guided waves supported by the film. Analysis of the perturbation solution reveals that the background scattering exhibits minima and maxima as functions of the thickness. By studying the behavior of the scattering as a function of the absorption index of the film, it is shown that the amplitudes of the peaks are low when k approximately 10(-2) and high when k approximately 10(-4).

Oblique-coordinate-system-based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings

A formulation of the Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic diffraction gratings is presented. The formulation is based on an oblique Cartesian coordinate system so that blazed surface-relief profiles and slanted volume permittivity variations can be treated simultaneously. The permittivity tensors of the anisotropic materials can be arbitrary, but their spatial variations are assumed to have the same period as the surface corrugations. Some of the previous authors’ works in the above individual aspects of the subject are improved on in regard to clarity, simplicity, and generality. Most important, the theory of Fourier factorization is applied throughout the analysis to ensure fast convergence of the numerical method when the permittivity tensors or the derivatives of the grating’s surface-profile functions have discontinuities. Numerical examples, including those of two types of multilayer magneto-optic gratings, are provided.

The Stokes matrix in conical scattering from a one‐dimensional perfectly conducting randomly rough surface

We calculate the elements of the Stokes matrix for the scattering of light from a one‐dimensional, perfectly conducting, randomly rough surface in the general case when the plane of incidence is not perpendicular to the generators of the surface. By using Green's second integral identity we obtain a pair of inhomogeneous integral equations for the surface values of the components of the total electric and magnetic fields in the system parallel to the generators of the surface. All components of the scattered field can be obtained from these two components, and from them the elements of the Stokes matrix can be calculated. These equations are solved numerically for each of 2000 realizations of the surface profile function, and the elements of the Stokes matrix are averaged over this ensemble of realizations of the surface profile function. We find that only four unique elements of the Stokes matrix are nonzero, as in the case in which the plane of incidence is normal to the generators of the surface (in‐plane scattering). Moreover, these nonzero elements in the conical scattering of light of wavelength λ can be obtained from the corresponding elements in the in‐plane scattering of light of an effective wavelength λeff = λ/ cos ф0, where ф0 is the conical angle. The results of the study enable complete information about the diffuse scattering properties of one‐dimensional perfectly conducting random surfaces to be obtained.

Impedance boundary conditions for a metal film with a rough surface

We have obtained local impedance boundary conditions for a metal film characterized by an isotropic, frequency-dependent, complex dielectric function epsilon(omega) that occupies the region zeta(x(1)) < x(3) < D. The surface-profile function zeta(x(1)) is assumed to be a single-valued function of x(1) that is differentiable as many times as is necessary. The electromagnetic field in the system is assumed to be p polarized with the plane of incidence, the x(1)x(3) plane. The results are used to study the scattering of p-polarized light from and its transmission through the metal film when the surface-profile function zeta(x(1)) in these calculations is assumed to be a stationary, zero-mean, Gaussian random process. These calculations are approximately four times faster than rigorous computer simulations, and their results are qualitatively and quantitatively in good agreement with those of the latter simulations.

Angular correlation functions of light scattered from weakly rough metal surfaces

The angular correlation functions of light diffusely scattered from weakly rough metal surfaces are studied. In experimental work, surfaces are well characterized, have highly one-dimensional roughness, and are studied in the case of p polarization. If there is significant plasmon polariton excitation, one type of intensity correlation function indicates that the diffuse intensity decorrelates rapidly as the angle of incidence is varied. It exhibits peaks that arise from an autocorrelation of identical intensities, or from the correlation of intensities related by the reciprocity principle. A second intensity correlation function expresses the symmetry of the diffuse intensity about the specular direction and is significant without plasmon polariton excitation. It is shown that the intensity correlation functions are directly related to two fundamental amplitude correlation functions. The latter are studied with perturbation theory, including all terms to fourth order in the surface profile function. The relation of the observed effects to backscattering enhancement is thus established, and favorable comparisons are made with the experimental results.

Surface electromagnetic waves in near-field optical scanning microscopy

Because their electromagnetic fields are localized to the surfaces that support them, surface electromagnetic waves are more sensitive to geometrical and dielectric perturbations in their propagation path than are volume electromagnetic waves incident on the same surface perturbations. This suggests that a near-field optical microscopy based on surface electromagnetic waves could reconstruct surface profiles with greater resolution than one based on the scattering of volume electromagnetic waves from surfaces with the same profiles. In this work we study the scattering of a surface plasmon polariton propagating along a metal surface and incident normally on an isolated surface defect. We calculate the intensity of the total field at constant height from the unperturbed surface to first order in the surface profile function. The result can be written in terms of the convolution of the surface profile function and a function that depends only on the properties of the metal surface. We invert this result by a Fourier transform method to obtain the surface profile function. As experimental data we use the results of a rigorous numerical solution of the corresponding reduced Rayleigh equation. Structures on the surface with lateral dimensions of the order of one-tenth the wavelength of the incident surface polariton have been reconstructed in this way.