Renormalization group interpretation of the Born and Rytov approximations

In this paper the method of renormalization group (RG) [Phys. Rev. E54, 376 (1996)] is related to the well-known approximations of Rytov and Born used in wave propagation in deterministic and random media. Certain problems in linear and nonlinear media are examined from the viewpoint of RG and compared with the literature on Born and Rytov approximations. It is found that the Rytov approximation forms a special case of the asymptotic expansion generated by the RG, and as such it gives a superior approximation to the exact solution compared with its Born counterpart. Analogous conclusions are reached for nonlinear equations with an intensity-dependent index of refraction where the RG recovers the exact solution.     

Validity of diffraction tomography based on the first born and the first rytov approximations

Using computer simulations we examine the ranges of validity of the first Born and first Rytov approximations employed in diffraction tomography. To that end we apply the filtered backpropagation(FBP) algorithm in conjunction with the first Born approximation and the hybrid FBP algorithm in conjunction with the first Rytov approximation. We find that the range of validity of the first Born approximation is approximately 3 times smaller than that of the first Rytov approximation and that the range of validity of each approximation can be expressed in terms of the product of the refractive-index difference between the object and the background and the size of the object. Also, we establish precise criteria for the validity of diffraction tomography within each of these two approximations. For the first Rytov approximation the validity of the hybrid FBP algorithm is found to be limited by phase-unwrapping problems.     

ABCD matrix for reflection and refraction of Gaussian light beams at surfaces of hyperboloid of revolution and efficiency computation for laser diode to single-mode fiber coupling by way of a hyperbolic lens on the fiber tip

We report the formulation of an ABCD matrix for reflection and refraction of Gaussian light beams at the surfaces of the hyperboloid of revolution that separate media of different refractive indices. The analysis includes an arbitrary angle of incidence and is based on matching the optical phase at the interface. Finally, we deduce expressions for spot sizes and wave-front radii and use them to obtain the ABCD matrix. Based on the formulated ABCD matrix for refraction under paraxial approximation, we also report a simple theoretical investigation of the coupling efficiency of a laser diode to a single-mode fiber with a hyperbolic lens formed on its tip.     

Nonlinearized approach to profile inversion

A method for reconstructing the index of refraction of a bounded inhomogeneous object of known geometric configuration from measured far-field scattering data is presented. This work is an extension of recent results on the direct scattering problem wherein the governing domain integral equation was solved iteratively by a successive relaxation technique. The relaxation parameters were chosen to minimize the residual error at each step. Convergence of this process was established for indices of refraction much larger than required for convergence of the Born approximation. For the inverse problem, the same technique is applied, except is this case both the index of refraction and the field are unknown. Iterative solutions for both unknowns are postulated with two relaxation parameters at each step. They are determined by simultaneously minimizing the residual errors in satisfying the domain integral equation and matching the measured data. This procedure retains the nonlinear relation between the two unknowns. Numerical results are presented for the dielectric slab. The algorithm is shown to be effective in cases where the iterative solution of the direct problem is rapidly convergent and outperforms the Born-based approaches.     

An iterative solution of the two-dimensional electromagnetic inverse scattering problem

A new method, based on an iterative procedure, for solving the two-dimensional inverse scattering problem is presented. This method employs an equivalent Neumann series solution in each iteration step. The purpose of the algorithm is to provide a general method to solve the two-dimensional imaging problem when the Born and the Rytov approximations break down. Numerical simulations were calculated for several cases where the conditions for the first order Born approximation were not satisfied. The results show that in both high and low frequency cases, good reconstructed profiles and smoothed versions of the original profiles can be obtained for smoothly varying permittivity profiles (lossless) and discontinuous profiles (lossless), respectively. A limited number of measurements around the object at a single frequency with four to eight plane incident waves from different directions are used. The method proposed in this article could easily be applied to the three-dimensional inverse scattering problem, if computational resources are available.     

Photon density waves scattered from cylindrical inhomogeneities: theory and experiments

We present an analytical solution for the scattering of diffuse photon density waves from an infinite circular, cylindrical inhomogeneity embedded in a homogeneous highly scattering turbid medium. The analytical solution, based on the diffusion approximation of the Boltzmann transport equation, represents the contribution of the cylindrical inhomogeneity as a series of modified Bessel functions integrated from zero to infinity and weighted by different angular dependencies. This series is truncated at the desired precision, similar to the Mie theory. We introduce new boundary conditions that account for specular reflections at the interface between the background medium and the cylindrical inhomogeneity. These new boundary conditions allow the separate recovery of the index of refraction of an object from its absorption and reduced scattering coefficients. The analytical solution is compared with data obtained experimentally to evaluate the predictive capability of the model. Optical properties of known cylindrical objects are recovered accurately. However, as the radius of the cylinder decreases, the required measurement signal-to-noise ratiorapidly increases. Because of the new boundary conditions, an upperlimit can be placed on the recovered size of cylindrical objects with radii below 0.3 cm if they have a substantially different index of refraction from that of the background medium.     

Solution of convergence problem in ultrasound inverse scattering tomography

When the product of contrast and size of an object, which is to be reconstructed by using the ultrasound inverse scattering tomography algorithm, is large, it is well known that those algorithms fail to converge to a unique global minimum. In order to solve this well known and difficult convergence problem, in this paper we present a new method, which converges to the true solution, for obtaining the scattering potential without using the Born or Rytov approximation. This method converts the nonlinear nature of the problem into a linear one. Through computer simulations we will show the validity of the new approach for high contrast two-dimensional scattering objects which are insonified by an incident ultrasound plane wave. Numerical results show that the reconstruction error is very small for circularly symmetric two-dimensional cylindrical objects whose refractive indices range from small to even sufficiently large values for which the previous inverse scattering algorithms fail to converge.     

A Modified Born Approximation for Scattering in Isotropic and Anisotropic Elastic Solids

A new modified Born approximation (MBA) is presented that significantly extends the range of validity of the Born approximation to include the pulse-echo responses of strongly scattering inclusions in an elastic solid. The MBA also improves on the doubly distorted Born approximation (DDBA), a similar modification of the Born approximation that has been recently developed. These improvements are demonstrated by comparing the MBA, the Born approximation and the DDBA with the exact separation of variables solution for spherical inclusions in isotropic media. Furthermore, it is shown that the form of the MBA remains valid even for the pulse-echo scattering of an anisotropic inclusion in a general anisotropic elastic medium so that it is potentially applicable to a wide class of flaws and materials.     

Skewed distribution of irradiance predicted by the second-order Rytov approximation.

The first-order Rytov approximation predicts that the probability-density function for intensity fluctuations in a random medium should be a log normal distribution. Here the corrections to this prediction that arise from the second term in the Rytov series are explored. The primary effect is to skew the distribution so as to favor values that are less than the average. This effect is controlled by the Rytov variance beta0(2) alone, and the predicted distribution contains no adjustable parameters. The theoretical result is compared with numerical simulations for weak scattering of plane and spherical waves. The agreement is quite good unless the intensity fluctuations are very large or very small relative to the mean irradiance. In those ranges, the predictions require additional terms in the Rytov expansion.     

Image estimation from scattered field data

The purpose of this paper is to reappraise the linearizing methods frequently used to solve inverse scattering problems. We describe inversion algorithms based on the Born and the Rytov approximations and the nature of the distortions obtained in the reconstructions when using them. We present extensions of these methods, namely, the distorted-wave Born and the distorted-wave Rytov approximations, which incorporate prior knowledge about part of the scattering structure. A method for inverting scattered field data using these distorted-wave approximations is described, which retains the computational simplicity of the Born and the Rytov techniques. Some examples of their use with simulated and real data are given. A further extension of our distorted-wave formalism, which leads to improvements of the reconstructed image, is suggested. This entails a spectral estimation procedure also based on the incorporation of prior knowledge about the scatterer. This spectral estimation procedure can be useful for interpolation of scattered field data as well as resolution enhancement.     

Interference Figures of Biaxial Crystal Plates (Optic Axes Figures)

Abstract

The intensity distribution in the interference field produced by a plane parallel biaxial crystal plate between crossed polarizers and transmitted by convergent light (optic axes figures) is presented as an explicit formula. This has been achieved by a suitable procedure of approximation which consists of treating the crystal as an isotropic medium and using Fresnel's formulae for the transmitted wave amplitudes at both surfaces. The anisotropy of the crystal is taken into account only in the determination of the orthogonal vibration directions and of the refractive indices pertaining to the two waves originating from the double refraction at the first surface of the crystal. The interference figures calculated by this method are in close agreement with the exact results from Rath and Pohl.     

A Comparison of Extinction Efficiencies in the Eikonal and the Anomalous Diffraction Approximations for Spheres

Abstract

Extinction curves in the eikonal approximation and the anomalous diffraction approximation have been compared with exact extinction curves for homogeneous spheres of various refractive indices. Our results show the anomalous diffraction approximation to be more suitable for extinction calculations.     

径向速度变化地层井孔声场的快速模拟

在声波测井过程中,为了精细刻画井孔附近岩石和流体特性所引起的速度径向变化规律,常用的方法是对井孔到原状地层的区域进行径向分层,然后利用传播矩阵的方法来模拟丼周的速度变化,然而,这种方法用于波形数据反演的过程中十分耗时。提出了一种利用Born近似快速模拟纵波速度径向变化的方法。推导了速度扰动时散射场积分表达式,并据此计算径向速度扰动模型下的散射声场,得到其近似解析解。数值模拟结果表明,与传统的方法相比,新的近似解能给出合理的模拟结果,提供了一种利用声波波形数据反演速度径向变化的新思路。     

Phenomenological Rashba model for calculating the electron energy spectrum on a cylinder

The energy spectrum of an electron on the surface of a cylinder is calculated using the Pauli equation with an additional term that takes into account the spin-orbit interaction. This term is taken in the approximation of a phenomenological Rashba model, which provides exact expressions for the wave functions and the electron energy spectrum on the cylinder surface in a static magnetic field.     

Impedance of single-turn coil surrounding a conducting cylinder with a flaw

The impedance of a single-turn coil which surrounds a conducting cylinder with a flaw is calculated employing Green's function technique. The Born approximation is used in order to know how the impedance change due to the presence of a flaw depends on the conductivity and size of a flaw.     

Optical phase retrieval by use of first Born- and Rytov-type approximations

The first Born and Rytov approximations of scattering theory are introduced in their less familiar near-field versions. Two algorithms for phase retrieval based on these approximations are then described. It is shown theoretically and by numerical simulations that, despite the differences in their formulation, the two algorithms deliver fairly similar results when used for optical phase retrieval in the near and intermediate fields. The algorithms are applied to derive explicit solutions to four phase-retrieval problems of practical relevance to quantitative phase-contrast imaging and tomography. An example of successful phase reconstruction by use of the Born-type algorithm with an experimental x-ray image is presented.     

Investigation of Domains of Validity of Approximation Methods in Forward Light Scattering from Long Cylinders

Domains of validity of approximation methods in light scattering from non-absorbing transparent long cylinders of diameter h 5 w m have been clearly identified. Numerical comparisons between these methods and the exact theory for various values of the refractive index and size parameter of the cylinder are presented. The utility of the eikonal approximation and the corrections to this approximation is emphasized.     

Semi-empirical inelastic cross sections for electron transport in liquid water

Electron inelastic cross sections for water in the liquid phase are important for developing Monte Carlo codes that simulate the full degradation of any radiation beam in biological matter. The limited experimental information for condensed targets and the complexity of the background theory has led to largely heuristic semi-empirical models. The present work makes use of the dielectric formalism under the first Born approximation to develop inelastic cross sections for low-energy electron transport in liquid water. A Drude model was used to describe the energy-loss distribution at the dipole limit on the basis of optical data, while the impulse approximation and an empirical generalised-oscillator-strength provided the extension to finite momentum transfer. Born corrections established earlier for water vapour were applied at low impact energies. Core-electron transitions were treated by a binary model with exchange terms. Sum-rules were satisfied to within 1-2% while an 1-value of about 80 eV was obtained. A comparison with other studies is provided.     

Quantitative characterization of flaws near a surface using inverse born approximation

Quantitative reconstruction of volumetric flaws near a surface of an elastic solid has been carried out experimentally by analyzing the scattered ultrasonic waves. The inverse Born approximation (developed for flaws in bulk materials) was tested for the first time in the determination of the size, shape, and orientation of near-surface flaws. We have studied spherical solid inclusions at various depths below the surface. In addition we examined an approximately 2:1 prolate spheroidal inclusion which was located one major axis below the surface. The determination of the flaw's size, shape, and orientation in terms of an equivalent ellipsoid is realized by performing nonlinear least-squares iteration of the one-dimensional Born inversion results obtained at various scattering directions within a finite aperture. The reconstruction is in good agreement with the actual parameters of the flaw.     

ABCD matrix for reflection and refraction of Gaussian beams at the surface of a parabola of revolution

We report the formulation of an ABCD matrix for reflection and refraction of Gaussian light beams at the surface of a parabola of revolution that separate media of different refractive indices based on optical phase matching. The equations for the spot sizes and wave-front radii of the beams are also obtained by using theABCD matrix. With these matrices, we can more conveniently design and evaluate some special optical systems, including these kinds of elements.