Relative Reflectance Research Articles

On Dirichlet to Neumann and Robin to Neumann operators suitable for reflecting harmonic functions subject to a non-homogeneous condition on an arc

According to the Schwarz symmetry principle, every harmonic function vanishing on a real-analytic curve has an odd continuation, while a harmonic function satisfying homogeneous Neumann condition has an even continuation. Using a technique of Dirichlet to Neumann and Robin to Neumann operators, we derive reflection formulae for non-homogeneous Neumann and Robin conditions from a reflection formula subject to a non-homogeneous Dirichlet condition.

To the theory of determining the location of hydrate deposits in gas pipelines by acoustic sounding

Evolution of pressure perturbations propagating in pipeline filled with gas-and-drop medium representing “wet” methane at temperature below dew point and having damaged section, in form of extended narrowing of channel due to hydrate plug, is investigated. Hydrate formation is due to the presence of water (or its vapours) and gas, the components of which dissolve in water under certain conditions form a solid phase. Hydrate deposits help to reduce the cross-country capacity of gas pipelines and therefore their detection is a pressing task. It is proposed to solve the problem using acoustic methods. For this purpose mathematical model of propagation of acoustic waves in long-wave range in gas-and-droplet medium is considered. The horizontal pipeline appears semi-pointed, the solution is sought in the form of a harmonic wave. Wave is one-dimensional, having small amplitude of oscillations. Based on dispersion equations, dependence of phase velocity and attenuation coefficient on frequency of acoustic wave disturbance and on volume content of suspended phase (water droplets) are built. In the high frequency region, the attenuation coefficient increases with the volume content. The formulas for reflection and passage coefficients are derived taking into account pipeline narrowing due to hydrate deposits. The results of numerical calculations illustrating the dynamics of pulse signals depending on the thickness of the gas hydrate on the inner wall of the pipeline are presented. Calculations are based on forward and backward Fourier transformations and the use of software. It is established that the greater the hydrate deposit on the wall in thickness, the greater the amplitude of the returned reflected signal.

Atomistic S -matrix method for numerical simulation of phonon reflection, transmission, and boundary scattering

The control of phonon scattering by interfaces is critical to the manipulation of heat conduction in composite materials and semiconducting nanostructures. However, one of the factors limiting our understanding of elastic phonon scattering is the lack of a computationally efficient approach for describing the phenomenon in a manner that accounts for the atomistic configuration of the interface and the exact bulk phonon dispersion. Building on the atomistic Green's function (AGF) technique for ballistic phonon transport, we formulate an atomistic S-matrix method that treats bulk phonon modes as the scattering channels and can determine the numerically exact scattering amplitudes for individual two-phonon processes, enabling a highly detailed analysis of the phonon transmission and reflection spectrum as well as the directional dependence of the phonon scattering specularity. Explicit formulas for the individual phonon reflection, absorption and transmission coefficients are given in our formulation. This AGF-based S-matrix approach is illustrated through the investigation of: (1) phonon scattering at the junction between two isotopically different but structurally identical carbon nanotubes, and (2) phonon boundary scattering at the zigzag and armchair edges in graphene. In particular, we uncover the role of edge chirality on phonon scattering specularity and explain why specularity is reduced for the ideal armchair edge. The application of the method can shed new light on the relationship between phonon scattering and the atomistic structure of interfaces.

The broken string in Anti-de Sitter space

This paper describes an efficient method for solving the classical string equations of motion in (2+1)-dimensional anti-de Sitter spacetime. Exact string solutions are identified that are the analogs of piecewise linear strings in flat space. They can be used to approximate any smooth string motion to arbitrary accuracy. Cusps on the string move with the speed of light and their collisions are described by a reflection formula. Explicit examples are shown with the string ending on two boundary quarks. The technique is ideally suited for numerical simulations.

λ-Generalized Gamma Functions

The objective of present research is to introduce and analyze a new special function that occurs as a part of the kernel of the integral representation of the λ-generalized Hurwitz-Lerch zeta functions. One important aspect of the analysis of special functions is to study their properties. It is found that λ-generalized gamma function satisfies the log convex and derivative properties. Recurrence relation and reflection formula are achieved that are always important to study the behavior of new functions. As an application, a new series representation of the λ-generalized Hurwitz-Lerch zeta functions is established. These new results are validated by analyzing their important cases that agree with the known results. It is interesting to note that the coefficients in the series representation of the family of zeta functions are generalized from “1” to “gamma function” and then to “generalized gamma” and “λ-generalized gamma functions” in a simple and natural way.

Structural broadband absorbing metamaterial based on three-dimensional printing technology

In order to verify the feasibility of three-dimensional (3D) printing technology in preparing the metamaterial absorbers with complex structure, a three-layer broadband absorbing metamaterial is designed and fabricated by 3D printing technology. The surface layer and middle layer of the metamaterial are composed of periodic arrays with different unit dimensions and the bottom layer of a slab structure. The optimized thickness of the metamaterial is 4.7 mm. A composite absorbent which consists of carbonyl iron powder and nylon is used to fabricate the absorber. In experiment, the obtained absorber is vertically irradiated by an electromagnetic (EM) wave. Two strong absorption peaks at 5.3 GHz and 14.1 GHz are achieved, with the reflection losses of -15.1 dB and -12.5 dB, respectively. The superposition of the two absorption peaks results in a reflection loss below -10 dB in a range from 4 to 18 GHz. The effective EM parameters of the surface layer and the middle layer are calculated by the S parameter inversion method. An effective model of the three-layer structure absorber is proposed and its reflectivity is calculated by using a multilayer structure reflectivity formula. The calculated reflectivity agrees well with the measured one. The absorbing and resonance mechanisms of the two absorption peaks are investigated by analyzing the dynamic distributions of power density loss, electric field and magnetic field. It can be clearly confirmed that the reflection losses at 5.3 GHz and 14.1 GHz are primarily concentrated on the bottom layer and surface layer, and the broadband absorption performance can be derived from the superposition of broadband absorptions of the three absorbing layers. Meanwhile, the strong electric coupling effect between the adjacent units in the surface layer is demonstrated by analyzing the electric-field distributions, which indicates that the strong reflection loss at 14.1 GHz is mainly caused by the electric response. The multiple scattering effects among the three layers are also considered according to the magnetic field distributions at two resonance frequencies. It is shown that there are two magnetic responses at 5.3 GHz and 14.1 GHz, respectively, and the multiple scattering contributes to increasing the EM wave propagation distance and enhancing the power loss. The designed absorbing metamaterials in this paper achieve good broadband absorption performances, particularly in the low frequency band. Combined with 3D printing rapid technology, a promising route to constructing 3D absorbing metamaterials with complex structures is proposed, which would be of great significance and broad practical prospect.

Impact of the scattering phase function on the bulk reflectance of a turbid medium with large-scale inhomogeneities.

We study the total bulk reflectance of a turbid medium with large (as compared to the light wavelength) inhomogeneities at grazing angles of light incidence. To model highly forward scattering in the medium, we take advantage of the Reynolds-McCormick scattering phase function. Using the scaling analysis for the small-angle radiative transfer equation, we derive simple analytical formulas for the total reflectance. For grazing incidence angles, we find a range of values of the medium transport coefficients where the total reflectance proves to be a universal function of a single parameter, which is expressed in terms of the incidence angle, the absorption coefficient, and the transport scattering one. The explicit form of this function is governed by the specific angular profile of the scattering phase function. The results obtained are verified by comparison with results of a direct numerical integration of the radiative transfer equation.

Modeling and Simulation of Leaf Color Based on Virtual Rice

This paper presents a research object with the color changes of main stem leaves of two kinds of rice variety under different growth periods and nitrogen levels to improve the authenticity and practicability of visualization color model of virtual rice. The authors at first build an RGB-SPAD mathematic model of rice leaf; then obtain the leaf rendering equations via the BRDF diffuse reflection formulas; and at last apply the C++ language combined with the OpenGL graphics library programming to render and visualize appearances of the rice leaves. The leaf visualization color model constructed by this method accords with the law of rice growth, and thus it is expected to provide assistance in decision making to regulate and control the fertilizer level for rice cultivators according to the changes of leaf color.

Theoretical and numerical aspects of nonlinear reflection–transmission phenomena in acoustics

Equations of nonlinear acoustic wave motion in a non-classical lossy medium are used to derive generalised formulas describing the phenomena of reflection and transmission. Integral, non-local operators that are caused by the nonlinear effects in wave propagation and occur in reflection and transmission formulas are given in a form in which classical linear reflection and transmission coefficients are explicitly separated. Numerical calculations are performed for a simplified, one-dimensional wave travelling in a lossless medium. These simplifications reveal the pure effect of the impact of nonlinearities on the reflection and transmission phenomena. We consider adjacent media with different properties to illustrate various aspects of the problem. In particular, even if two media have the same linear impedance and the same material modules of the third order, we observe an explicit effect of the nonlinearity on the reflection phenomenon. The theoretical predictions are confirmed qualitatively by numerical calculations based on the finite difference time domain method.

The p-adic Arakawa–Kaneko–Hamahata zeta functions and poly-Euler polynomials

In this paper, we give a definition of the p-adic Arakawa–Kaneko–Hamahata zeta functions. These zeta functions interpolate Hamahata's poly-Euler polynomials at non-positive integers. We prove the derivative formula, the difference equation and the reflection formula of these zeta functions. Furthermore, we also prove a sums of products identity and a closed form of Hamahata's poly-Euler polynomials in terms of the Stirling numbers of the second kind.

Evaluating the performance of Lyzenga's water column correction in case-1 coral reef water using a simulated Wolrdview-2 imagery

This study identified the performance of Lyzenga water column correction under controlled condition of noise, water quality, and bottom type diversity. We build a simulated multispectral imagery, represent the subsurface reflectance of coral reef environment that receives by Worldview-2 satellite. The evaluation was performed to two different subsurface reflectance: Lee et al.’s and Maritonia et al.’s radiative transfer model. As a result, in the case of simple Maritonia et al.’s reflectance model, it shows that the effect of the water column is removable. However, if the actual just above surface reflectance is as more complex as Lee et al.’s model, then the Lyzenga's water column method could reduce the effect of water attenuation but not removed. In fact, based on the mathematical presentation of Lyzenga's model on simulation reflectance formula the effect of depth and water attenuation are in multiplication function. Thus, even after the correction, the water column attenuation effect stillremain.

Reflection formulae for ray tracing in uniaxial anisotropic media using Huygens's principle.

Ray tracing in uniaxial anisotropic materials is important because they are widely used for instrumentation, liquid-crystal displays, laser cavities, and quantum experiments. There are previous works regarding ray tracing refraction and reflection formulae using the common electromagnetic theory approach, but only the refraction formulae have been deduced using Huygens's principle. In this paper we obtain the reflection expressions using this unconventional approach with a specific coordinate system in which both refraction and reflection formulae are simplified as well as their deduction. We compute some numerical examples to compare them with the common expressions obtained using electromagnetic theory.

Theoretical and numerical aspects of nonlinear reflection–transmission phenomena in acoustics

Equations of nonlinear acoustic wave motion in a non-classical lossy medium are used to derive generalised formulas describing the phenomena of reflection and transmission. Integral, non-local operators that are caused by the nonlinear effects in wave propagation and occur in reflection and transmission formulas are given in a form in which classical linear reflection and transmission coefficients are explicitly separated. Numerical calculations are performed for a simplified, one-dimensional wave travelling in a lossless medium. These simplifications reveal the pure effect of the impact of nonlinearities on the reflection and transmission phenomena. We consider adjacent media with different properties to illustrate various aspects of the problem. In particular, even if two media have the same linear impedance and the same material modules of the third order, we observe an explicit effect of the nonlinearity on the reflection phenomenon. The theoretical predictions are confirmed qualitatively by numerical calculations based on the finite difference time domain method.

Boron Nitride and Sapphire Windows for 95-GHz Gaussian RF Beam

This paper presents the reflectivity and insertion loss characteristics of Boron Nitride (BN) and Sapphire RF windows for a 95-GHz Gaussian RF beam of a 100-kW gyrotron. The design of an RF window has been carried out using Fresnel’s formula for reflection and transmission of Gaussian electromagnetic wave at the boundary between two media. The return loss of the RF windows of 95 GHz was found to be less than −46.8 dB for BN and less than −57.8 dB for Sapphire, and the transmission loss of the RF windows was better than −0.06 dB for BN and better than −0.01 dB for Sapphire. Furthermore, the heat load due to RF power absorption over the disk surface has been calculated. The Sapphire RF windows have been found with lower heat load and deflection, and higher tensile stress than that of BN. The allowable stress level for Sapphire is higher than that of BN; therefore, using Sapphire is structurally safer to use.

Closed form valuation of American chained knock-in options

In this paper, we study pricing of American chained knock-in option. Chained barrier option is a new type of barrier option with two barrier levels. A knock-in American chained barrier option under a trigger clause is an option contract in which the option holder receives an American knock-in option conditional on the underlying asset price breaching a specified barrier level. We derive analytic valuation formulas for knock-in American chained options under the Black–Scholes pricing framework by using reflection principle and formulas for knock-in American options. Furthermore, we present some numerical solutions and plots of the value of knock-in American chained options.

Theoretical predictions and experimental suggestions for refraction behaviors occurring at lossy interfaces

According to the energy streamline method, refraction of energy flow of the transverse magnetic (electric) polarized electromagnetic waves occurring at the lossy/lossless interface may be either positive or negative, depending on the sign of the real part of permittivity (permeability) of the lossy medium. On the other hand, based on the reflection and refraction formulas developed by us, it is demonstrated that refraction behaviors of the electromagnetic wave occurring at the lossy/lossless interface relate to the difference between the electric damping angle and the magnetic angle of the lossy medium. Furthermore, the two-dimensional transmission lines with adjustable effective permittivity and permeability are proposed to be applied to experimentally test these predictions. This work systematically provides the basic refraction properties of electromagnetic waves occurring at lossy/lossless interfaces; it may be useful to further understand the observed refraction behaviors associated with the lossy interfaces and fully test the validity of reflection and refraction theories.

Anomalous reflection in the ultra-thin nano-strip antenna induced by incident field and displacement current phase matching

Recently, ultra-thin nano-strip antenna (thinner than skin depth) has attracted extensive attention as a key structure unit of metasurface. It can be used to realize anomalous enhanced reflection and absorption, as well as suppressed transmission, which has been used in photonic devices. However, there is still a lack of clear analysis to explain systematically these phenomena. In this paper, we present the concept of phase matching between the displacement current and the incident field, which initially leads to the anomalous scattering property of a single ultra-thin nano-strip antenna. Base on the property of a single nano-strip, we study the coupling of multi-nano-strip antennas and demonstrate that the randomized and periodic array of nano-strips have the same anomalous reflection spectrum. The phase matching is also reflected in the theoretic model, from which the reflection and transmission formulas for nano-strip array are derived. The formulas successfully predict the thickness influence on the reflection spectra of a periodic array. Our results can promote the development of ultra-thin nano-strip based metasurfaces.

Precise and fast computation of Fermi–Dirac integral of integer and half integer order by piecewise minimax rational approximation

Piecewise quadruple precision approximations of the Fermi–Dirac integral of integer order, 0(1)10, and half integer order, -9/2(1)21/2, are developed by combining (i) the optimally truncated Sommerfeld expansion, (ii) the piecewise truncated Chebyshev series expansion, and/or (iii) the reflection formula. They are used in constructing the double precision piecewise minimax rational approximations of the integral of the same orders. The relative errors of the new minimax approximations, which are all due to rounding off, are 3–13 machine epsilons at most and less than 4 machine epsilons typically while their CPU times are only 0.44–1.1 times that of the exponential function when -5⩽η⩽45. As a result, the new approximations run 16–31 times faster than the piecewise Chebyshev polynomial approximations (Macleod, 1998) for the physically important orders, -1/2(1)5/2, and 330–720 and 50–93 times faster than the combined series expansions (Goano, 1995) for the half integer orders, -1/2(1)21/2, and the integer orders, 1(1)9, respectively. A file of the Fortran codes of the obtained approximations and their test program and sample output is named xfdh.txt and located at: https://www.researchgate.net/profile/Toshio_Fukushima/.

Reflection and Transmission of Plane SH-Waves Through an Anisotropic Magnetoelastic Layer Sandwiched Between Two Semi-Infinite Inhomogeneous Viscoelastic Half-Spaces

Reflection and transmission coefficients due to incident plane SH-waves at an anisotropic magnetoelastic layer sandwiched between two inhomogeneous viscoelastic half-spaces are obtained. The inhomogeneity in viscoelastic coefficients of the lower half-space is considered as quadratic, while that of the upper half-space is considered as exponential. The solutions for the layer and half-spaces are obtained analytically. The formulae for reflection and transmission coefficients are obtained for viscoelastic half-spaces subjected to continuity conditions of interfaces of an anisotropic magnetoelastic layer with viscoelastic half-spaces and Snell’s law. These coefficients are found to be the function of phase velocity, wave number, inhomogeneity, thickness of layer, and angle of incidence. Two special cases are considered. In case I, the problem is reduced to reflection and transmission in two viscoelastic half-spaces, whereas case II considers two isotropic half-spaces. Numerical computations are carried out for a specific model. The graphs are plotted for reflection and transmission coefficients against the angle of incidence and non-dimensional wave number for inhomogeneity parameters, the angle at which the wave crosses the magnetic field, the thickness of the layer, and frequency. In general, as the inhomogeneity parameter increases, the reflection and transmission coefficients decrease, while the angle at which the wave crosses the magnetic field, the thickness of the layer and frequency have systematic effects. The effect of angle of incidence on the reflection and transmission coefficients is not monotonic, but converges for an angle of incidence for all the parameters. Also, graphs are plotted for phase shift versus angle of incidence. The effects of inhomogeneity, the angle at which the wave crosses the magnetic field, thickness of the layer and frequency are considered. The phase shift exhibits a similar nature for these parameters.

Equivalence of zeta function technique and Abel–Plana formula in regularizing the Casimir energy of hyper-rectangular cavities

Zeta function regularization is an effective method to extract physical significant quantities from infinite ones. It is regarded as mathematically simple and elegant but the isolation of the physical divergency is hidden in its analytic continuation. By contrast, Abel–Plana formula method permits explicit separation of divergent terms. In regularizing the Casimir energy for a massless scalar field in a D-dimensional rectangular box, we give the rigorous proof of the equivalence of the two methods by deriving the reflection formula of Epstein zeta function from repeatedly application of Abel–Plana formula and giving the physical interpretation of the infinite integrals. Our study may help with the confidence of choosing any regularization method at convenience among the frequently used ones, especially the zeta function method, without the doubts of physical meanings or mathematical consistency.