General Phase Function Research Articles

Numerical evaluation of oscillatory integrals via automated steepest descent contour deformation

Steepest descent methods combining complex contour deformation with numerical quadrature provide an efficient and accurate approach for the evaluation of highly oscillatory integrals. However, unless the phase function governing the oscillation is particularly simple, their application requires a significant amount of a priori analysis and expert user input, to determine the appropriate contour deformation, and to deal with the non-uniformity in the accuracy of standard quadrature techniques associated with the coalescence of stationary points (saddle points) with each other, or with the endpoints of the original integration contour. In this paper we present a novel algorithm for the numerical evaluation of oscillatory integrals with general polynomial phase functions, which automates the contour deformation process and avoids the difficulties typically encountered with coalescing stationary points and endpoints. The inputs to the algorithm are simply the phase and amplitude functions, the endpoints and orientation of the original integration contour, and a small number of numerical parameters. By a series of numerical experiments we demonstrate that the algorithm is accurate and efficient over a large range of frequencies, even for examples with a large number of coalescing stationary points and with endpoints at infinity. As a particular application, we use our algorithm to evaluate cuspoid canonical integrals from scattering theory. A Matlab implementation of the algorithm is made available and is called PathFinder.

Design method of imaging optical systems using confocal flat phase elements.

Imaging systems consisting of flat phase element such as diffractive optical element, holographic optical element, and metasurface have important applications in many fields. However, there is still a lack of generalized and efficient design methods of these systems, especially for systems with nonsymmetric configurations. We proposed a design method of imaging system consisting of flat phase elements based on confocal properties. The description of the generalized phase function for realizing point-to-point stigmatic imaging is derived. Given the focal length or magnification as well as the locations of the elements based on the design requirements, the phase functions can be calculated very fast and stigmatic imaging of the central field is realized. The systems can be taken as good starting points for further optimization, during which the rotationally symmetric or freeform phase terms can be added. Several design examples are demonstrated to validate the feasibility of the method. The proposed method increases design efficiency while decreasing the dependence on existing systems and skills significantly, and can be easily integrated into optical design software.

A convergent and asymptotic Laplace method for integrals

Watson’s lemma and Laplace’s method provide asymptotic expansions of Laplace integrals F(z)≔∫0∞e−zf(t)g(t)dt for large values of the parameter z. They are useful tools in the asymptotic approximation of special functions that have a Laplace integral representation. But in most of the important examples of special functions, the asymptotic expansion derived by means of Watson’s lemma or Laplace’s method is not convergent. A modification of Watson’s lemma was introduced in [Nielsen, 1906] where, by the use of inverse factorial series, a new asymptotic as well as convergent expansion of F(z), for the particular case f(t)=t, was derived. In this paper we go some steps further and investigate a modification of the Laplace’s method for F(z), with a general phase function f(t), to derive asymptotic expansions of F(z) that are also convergent, accompanied by error bounds. An analysis of the remainder of this new expansion shows that it is convergent under a mild condition for the functions f(t) and g(t), namely, these functions must be analytic in certain starlike complex regions that contain the positive axis [0,∞). In many practical situations (in many examples of special functions), the singularities of f(t) and g(t) are off this region and then this method provides asymptotic expansions that are also convergent. We illustrate this modification of the Laplace’s method with the parabolic cylinder function U(a,z), providing an asymptotic expansions of this function for large z that is also convergent.

Sampling Almost Periodic and Related Functions

We consider certain finite sets of circle-valued functions defined on intervals of real numbers and estimate how large the intervals must be for the values of these functions to be uniformly distributed in an approximate way. This is used to establish some general conditions under which a random construction introduced by Katznelson for the integers yields sets that are dense in the Bohr group. We obtain in this way very sparse sets of real numbers (and of integers) on which two different almost periodic functions cannot agree, which makes them amenable to be used in sampling theorems for these functions. These sets can be made as sparse as to have zero asymptotic density or as to be t-sets, i.e., to be sets that intersect any of their translates in a bounded set. Many of these results are proved not only for almost periodic functions but also for classes of functions generated by more general complex exponential functions, including chirps or polynomial phase functions.

Asymptotics of Gaussian integrals in infinite dimensions

We introduce an infinite-dimensional version of the classical Laplace method, in its original form, relative to a canonical Gaussian measure associated with a Hilbert space, and for a general phase function. Particular attention is given to the case of a phase function with finite-dimensional degeneracy. Explicit results on expansions in the form of power series in the relevant parameter, with estimates on remainders, are provided.

A Novel ISAR Imaging Method for Maneuvering Target Based on AM-QFM Model Under Low SNR Environment

Inverse synthetic aperture radar (ISAR) imaging for long range maneuvering small/stealth target plays an important role in the field of national defense surveillance. Due to low signal-to-noise (SNR) of received echo signal and the nonstationary Doppler frequency modulated (DFM) during the coherent processing intervals (CPI), the conventional range-Doppler (RD) algorithm is unsuitable to generate a well-focused ISAR image. Furthermore, the dihedral or trihedral components of the maneuvering target or range cell migration caused by the target's complex motion generate the time-varying amplitudes for the received signal, which brings a new challenge for ISAR imaging algorithm design. To overcome those issues, in this work, we propose a new method to obtain the well-focused ISAR image of long range maneuvering small/stealth target. This method consists of two main steps. First, the received signal in a range bin is characterized as multicomponent amplitude modulated-quadratic frequency modulated (AM-QFM) signal with time-varying amplitudes after the translational compensation. After that, a novel coherently integrated general cubic phase function (CIGCPF) is developed to estimate the parameters of multi-component AM-QFM signal. Compared with the existing ISAR imaging methods based on the constant amplitude signal model during the CPI, the well-focused ISAR image can be obtained by the proposed method with time-varying amplitude in a low SNR environment. Finally, the simulation results demonstrate the effectiveness of the proposed algorithm.

A well-conditioned Levin method for calculation of highly oscillatory integrals and its application

This paper is devoted to studying efficient calculation of generalized Fourier transform ∫ − 1 x f ( t ) e i ω g ( t ) d t . For the general phase function g ( t ) , we develop a modified Levin method by the spectral coefficient approach. A sparse and well-conditioned linear system is constructed to help accelerate calculation of highly oscillatory integrals. Numerical examples are included to show the convergence properties of the new method with respect to both quantities of collocation points and the frequency ω . Furthermore, we apply this approach to solving oscillatory Volterra integral equations.

Comments over homogenisation scales for interfacial emission and scattering by a divided medium: Beerian and non Beerian behaviours

In statistical methods of characterisation of porous media radiative properties, interfacial extinction cumulative distribution functions, scattering or absorption cumulative probabilities and general phase functions are generally determined from shots issued from random volume points instead of random interfacial points. Indeed, the first method is numerically much simpler and accurate than the second one. The validity of this approach is discussed and its limitations enlightened for both Beerian and non Beerian homogenised phases, and in the case of a diffuse reflection law or a general one.The explanation of the identity or difference between the results of the two previous types of extinction cumulative distribution functions comes from the comparison between the spatial scale at which these functions are determined and the own scales of the divided medium. The conditions for which a medium follows the Beer’s law are then defined in terms of spatial scales.Moreover, the modeling of interfacial emission for a non Beerian homogenised phase is in principle based on the reciprocity theorem and an integral formulation of the Generalised Radiative Transfer Equation. The validity of a simpler approach based on an effective absorption coefficient is also discussed, from the previous analysis.The validity of results of recent works published in IJHMT are finally discussed.

Estimation of a low-order Legendre expanded phase function of snow

The purpose of this paper is to estimate the scattering phase function of snow from angularly resolved measurements of light intensity in the plane of incidence. A solver is implemented that solves the scattering function for a semi-infinite geometry based on the radiative transfer equation (RTE). Two types of phase functions are considered. The first type is the general phase function based on a low-order series expansion of Legendre polynomials and the other type is the Henyey–Greenstein (HG) phase function. The measurements were performed at a wavelength of 1310nm and six different snow samples were analysed. It was found that a first order expansion provides sufficient approximation to the measurements. The fit from the first order phase function outperforms that of the HG phase function in terms of accuracy, ease of implementation and computation time. Furthermore, a correlation between the magnitude of the first order component and the age of the snow was found. We believe that these findings may complement present non-contact detection techniques used to determine snow properties.

Radiative transfer within strongly non homogeneous porous media: Application to a slab of packed particles

The original radiation transfer model is dedicated to strongly non homogeneous porous media with a transparent phase and an opaque one, characterised by a diffuse reflection law at local scale. It is only based on radiative statistical functions of extinction and scattering and a general phase function, which completely characterise the radiation properties of the medium. The model is a generalisation for these media, under integral formulation, of the Generalised Radiative Transfer Equation (GRTE). A rigorous expression of the emission source term from the radiation reciprocity principle is the key point. The model is developed and applied in the case of a slab of a packed bed of spherical particles, by using a statistical Monte Carlo method. It has been validated by comparison with results of the bibliography in the academic case of purely absorbing media and in the case of optically thick slabs. The model has finally been applied to realistic cases of scattering, absorbing and emitting porous media. The model also accurately accounts for local configurations resulting in non-monotonous temperature profiles.

Statistical characterization of near-wall radiative properties of a statistically non-homogeneous and anisotropic porous medium

The radiative properties statistical characterization method of Tancrez and Taine has been extended to statistically anisotropic and non-homogeneous porous media. When a homogenized phase of such a medium does not follow Beer’s law on extinction, extinction and scattering coefficients have no physical meaning: the phase is completely characterised by an extinction cumulative distribution function, a scattering cumulative probability, a general phase function and an effective optical index. A simple expression of the emission source term, in local thermal equilibrium conditions, is given for any non-homogeneous and anisotropic porous medium. The radiative transfer can be modelled by a Generalized Radiative Transfer Equation that is also extended to non-homogeneous porous media. This characterization method has been applied to the radiative properties of a packed bed of opaque and diffuse spherical particles bounded by a wall: the local porosity strongly varies near the wall and the homogenized phase is non Beerian. In the bulk of the medium, Beer’s law becomes valid and extinction, absorption and scattering coefficients have been determined.

A Closed‐Form Solution to Single Scattering for General Phase Functions and Light Distributions

AbstractDue to the intricate nature of the equation governing light transport in participating media, accurately and efficiently simulating radiative energy transfer remains very challenging in spite of its broad range of applications. As an alternative to traditional numerical estimation methods such as ray‐marching and volume‐slicing, a few analytical approaches to solving single scattering have been proposed but current techniques are limited to the assumption of isotropy, rely on simplifying approximations and/or require substantial numerical precomputation and storage. In this paper, we present the very first closed‐form solution to the air‐light integral in homogeneous media for general 1‐D anisotropic phase functions and punctual light sources. By addressing an open problem in the overall light transport literature, this novel theoretical result enables the analytical computation of exact solutions to complex scattering phenomena while achieving semi‐interactive performance on graphics hardware for several common scattering modes.

Encoding generalized phase functions on Dammann gratings

We extend the Dammann encoding concept to more general phase functions, i.e., to encode a phase pattern so that a number of diffracted orders have the same intensity. We demonstrate a Dammann vortex grating that generates different vortex patterns in the different diffracted orders. We also show a Fresnel lens that creates seven transverse focal spots in each of seven axial planes.

Complex geometrical optics solutions for anisotropic equations and applications

In this article, we construct complex geometrical optics solutions with general phase functions for second order elliptic equations in two dimensions. We then use these special solutions to treat the inverse problem of reconstructing embedded inclusions by boundary measurements.

All-frequency interactive relighting of translucent objects with single and multiple scattering

We present a technique, based on precomputed light transport, for interactive rendering of translucent objects under all-frequency environment maps. We consider the complete BSSRDF model proposed by Jensen et al. [2001]. which includes both single and diffuse multiple scattering components. The challenge is how to efficiently precompute all-frequency light transport functions due to subsurface scattering. We apply the two-pass hierarchical technique by Jensen et al. [2002] in the space of non-linearly approximated transport vectors, which allows us to efficiently evaluate transport vectors due to diffuse multiple scattering. We then include an approximated single scattering term in the precomputation, which previous interactive systems have ignored. For an isotropic phase function, this approximation produces a diffuse transport vector per vertex, and is combined with the multiple scattering component. For a general phase function, we introduce a technique from BRDF rendering to factor the phase function using a separable decomposition to allow for view-dependent rendering. We show that our rendering results qualitatively match the appearance of translucent objects, achieving a high level of realism at interactive rates.

Three-dimensional radiative transfer in an anisotropically scattering, plane-parallel medium: generalized reflection and transmission functions

The problem of spatially varying, collimated radiation incident on an anisotropically scattering, plane-parallel medium is considered. A very general phase function is allowed. An integral transform is used to reduce the three-dimensional radiative transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to derive nonlinear integral and integro-differential equations for the generalized reflection and transmission functions. The integration is over the polar and azimuthal angles—this formulation is referred to as the double-integral formulation. The integral equations are used to illustrate symmetry relationships and to obtain single- and double-scattering approximations. The generalized reflection and transmission functions are important in the construction of the solutions to many multidimensional problems. Coupled integral equations for the interior and emergent intensities are developed and, for the case of two identical homogeneous layers, used to formulate a doubling procedure. Results for an isotropic and Rayleigh scattering medium are presented to illustrate the computational characteristics of the formulation.

Asymptotic expansions for ornstein-uhlenbeck semigroups perturbed by potentials over banach spaces

An asymptotic expansion of Schilder-type integrals with general phase function on abstract Wiener spaces is given and good control on remainders is obtained. For Ornstein –Uhlenbeck semigroups perturbed by potentials on Banach spaces the asymptotic expansion is given in terms of explicitly discussed “classical orbits”, in the case of finitely many non-degenerate maxima of the phase function. A representation of the leading term by a solution of an infinite dimensional Sturm-Liouville problem is also provided

Statistically averaged rate equations for intense non-neutral beam propagation through a periodic solenoidal focusing field based on the nonlinear Vlasov–Maxwell equations

In this paper we present a detailed formulation and analysis of the rate equations for statistically averaged quantities for an intense non-neutral beam propagating through a periodic solenoidal focusing field Bsol(x) with axial periodicity length S=const. The analysis is based on the nonlinear Vlasov–Maxwell equations in the electrostatic approximation, assuming a thin beam with characteristic beam radius rb≪S, and small transverse momentum and axial momentum spread in comparison with the directed axial momentum pz=γbmβbc. The global rate equation is derived for the self-consistent nonlinear evolution of the statistical average 〈χ〉=Nb−1∫dXdYdX′dY′χFb, where χ(X,Y,X′,Y′,s) is a general phase function, and Fb(X,Y,X′,Y′,s) is the distribution function of the beam particles in the transverse phase space (X,Y,X′,Y′) appropriate to the Larmor frame. The results are applied to investigate the nonlinear evolution of the generalized entropy, mean canonical angular momentum 〈Pθ〉, center-of-mass motion for 〈X〉 and 〈Y〉, mean kinetic energy (1/2)〈X′2+Y′2〉, mean-square beam radius 〈X2+Y2〉, and coupled rate equations for the unnormalized transverse emittance ε(s) and root-mean-square beam radius Rb(s)=〈X2+Y2〉1/2. Most importantly, the present derivation of nonlinear rate equations for various statistical averages 〈χ〉 allows for general azimuthal variation (∂/∂θ≠0) of the distribution function and self-field potential, and therefore represents a major generalization of earlier calculations carried out for the case of axisymmetric beam propagation.

Solution of the equation of transfer with general phase function by a modified spherical-harmonic method

The equation of transfer with general phase function has been solved by a modified form of spherical-harmonic method. The solutions in case of certain particular phase functions are then derived from the general one.

Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures

Radiative heat transfer in two-dimensional rectangular enclosures is studied using the S- N discrete ordinates method. The medium in the enclosure is gray and absorbs, emits, and anisotropically scatters radiative energy. General Mie-anisotropic phase functions are treated by Legendre polynomial expansions. The average incident radiations and the radiative heat fluxes are presented in graphical and tabular forms. The phase function anisotropy plays a significant role in the radiation heat transfer when the boundary condition is nonsymmetric, but it is not important for symmetric environments. Side wall heat losses are significant for back scattering phase functions, moderate optical thicknesses, large scattering albedos, and small reflectivities.