Tilt-invariant theory of rough-surface scattering: I

The small-slope approximation (SSA) in rough-surface scattering theory uses the surface slope as a small parameter of expansion. But, from the physical point of view, the slope may not be a restrictive parameter because we can change the slope of a surface simply by tilting the coordinate system. We present the theory of rough-surface scattering in a coordinate-invariant form. The new method, tilt-invariant approximation (TIA), leads to a different expansion that does not require that the slope of a surface be small. For a small Rayleigh parameter this approximate solution provides the correct perturbation theory, for a large Rayleigh parameter it provides the Kirchhoff approximation with several correcting terms.     

Tilt-invariant theory of rough-surface scattering: I

Abstract

The small-slope approximation (SSA) in rough-surface scattering theory uses the surface slope as a small parameter of expansion. But, from the physical point of view, the slope may not be a restrictive parameter because we can change the slope of a surface simply by tilting the coordinate system. We present the theory of rough-surface scattering in a coordinate-invariant form. The new method, tilt-invariant approximation (TIA), leads to a different expansion that does not require that the slope of a surface be small. For a small Rayleigh parameter this approximate solution provides the correct perturbation theory, for a large Rayleigh parameter it provides the Kirchhoff approximation with several correcting terms.     

Convergence of solutions to the rough-surface scattering problem

Abstract

A new formulation of the rough-surface scattering problem is obtained that is closely linked to the Kirchhoff approximation. The governing equation is cast into a form amenable to solution by the method of successive approximations. The domain of convergence of this solution is established and is shown to apply also to the odd-ordered operator expansion, small-slope approximation and perturbation theory provided that the slope of the scattering surface is everywhere less than unity. The analysis is performed for scattering from one-dimensional pressure-release surfaces. Numerical examples are presented for sinusoidal and echelette gratings.     

Convergence of solutions to the rough-surface scattering problem

A new formulation of the rough-surface scattering problem is obtained that is closely linked to the Kirchhoff approximation. The governing equation is cast into a form amenable to solution by the method of successive approximations. The domain of convergence of this solution is established and is shown to apply also to the odd-ordered operator expansion, small-slope approximation and perturbation theory provided that the slope of the scattering surface is everywhere less than unity. The analysis is performed for scattering from one-dimensional pressure-release surfaces. Numerical examples are presented for sinusoidal and echelette gratings.     

Surface-integral formulation of scattering theory

We formulate scattering theory in the framework of a surface-integral approach utilizing analytically known asymptotic forms of the two-body and three-body scattering wavefunctions. This formulation is valid for both short-range and long-range Coulombic interactions. New general definitions for the potential scattering amplitude are presented. For the Coulombic potentials, the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure. New post and prior forms for the Coulomb three-body breakup amplitude are derived. This resolves the problem of the inability of the conventional scattering theory to define the post form of the breakup amplitude for charged particles. The new definitions can be written as surface-integrals convenient for practical calculations. The surface-integral representations are extended to amplitudes of direct and rearrangement scattering processes taking place in an arbitrary three-body system. General definitions for the wave operators are given that unify the currently used channel-dependent definitions.     

High-order solutions of three-dimensional rough-surface scattering problems at high frequencies. I: the scalar case

We present a new high-order numerical method for the solution of high-frequency scattering problems from rough surfaces in three dimensions. The method is based on the asymptotic solution of appropriate integral equations in the high-frequency regime, in a manner that bypasses the need to resolve the fields on the scale of the wavelength of radiation. Indeed, inspired by prior work in two dimensions, we seek a solution of the integral equation in the form of a slow modulation of the incoming radiation, and we choose a series expansion in inverse powers of the wavenumber to represent the unknown slowly varying envelope. As we show, this framework can be made to yield an efficiently computable recursion for the terms in the series to any arbitrary order. The resulting algorithms generally provide a very significant improvement over classical (e.g. Kirchhoff) approximations in both accuracy and applicability and they can, in fact, effectively produce results with full double-precision accuracy for configurations of practical interest and up to the resonance regime.     

NN scattering in higher-derivative formulation of baryon chiral perturbation theory

We consider a new approach to the nucleon-nucleon scattering problem in the framework of the higher-derivative formulation of baryon chiral perturbation theory. Starting with a Lorentz-invariant form of the effective Lagrangian we work out a new symmetry-preserving framework where the leading-order amplitude is calculated by solving renormalizable equations and corrections are taken into account perturbatively. Analogously to the KSW approach, the (leading) renormalization scale dependence to any finite order is absorbed in the redefinition of a finite number of parameters of the effective potential at given order. On the other hand, analogously to Weinberg’s power counting, the one-pion-exchange potential is of leading order and is treated non-perturbatively.     

The Haag-Ruelle formulation of scattering in hyperfunction quantum field theory

The Källén-Lehmann representation for two-point Wightman Fourier hyperfunctions and the cluster property for truncated vacuum expectation values are established in the framework of hyperfunction quantum field theory. With some additional assumptions these properties allow one to verify the Haag-Ruelle asymptotic conditions.     

A dynamical formulation of one-dimensional scattering theory and its applications in optics

We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrödinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, antilasing, and unidirectional invisibility.     

High-order solutions of three-dimensional rough-surface scattering problems at high-frequencies. Part II: The vector electromagnetic case

We introduce a new numerical scheme for three-dimensional electromagnetic rough-surface scattering simulations with the capability of delivering very accurate results from low to high frequencies at a cost that is independent of the wavelength of radiation. The method is an extension of the ideas and techniques introduced in the first paper of this series (Waves in Random and Complex Media, 15 (2005), pp. 1-16) to the vector electromagnetic case, and it is based on the solution of an integral equation formulation of the scattering problem. As in the scalar case, the solution of the integral equation (i.e. the current) is expressed as a slow modulation of an oscillatory exponential of known phase, and explicit recursive formulae are derived for the successive terms in a series expansion of the slow envelope in inverse powers of the wavenumber. As we show, and in spite of the considerably more involved nature of the derivations and resulting formulae, the performance of the method retains the favourable characteristics that were demonstrated in the treatment of acoustic scattering problems. In particular, results with full double-precision accuracy are presented for various geometries, incidences and polarizations.     

Supersymmetric formulation of information theory in the presence of a phase transition

The Channon entropy is represented as a path integral over superfield components. This allows one to describe, in addition to information growth, the memory of a system undergoing a chain of bifurcations. Institute for Physics and Technology, Sumy. Research and Development Enterprise of the Plant for Repair of Electrotechnical Equipment. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 89–93, February, 1993.     

Radiation pressure of a string on a ring: A solution in terms of the scattering theory

The possibility of reducing the 1D problem of calculating the radiation pressure on a ring to a linear scattering problem solved in terms of partial scattering amplitudes is demonstrated. The latter approach is analogous to that previously applied to the 3D case??in particular, to the case of a spherically symmetric inclusion in an ideal liquid. The ring is represented as an inclusion with a monopole scattering amplitude. To achieve full qualitative coincidence of the expressions obtained for the 1D and 3D cases, the corresponding scattering amplitudes are considered as factors multiplying the fundamental solutions to the 1D and 3D Helmholtz equations. The results of the study enable a unified approach to the two aforementioned problems differing in dimension.     

An approximate phase shift formula applied to elastic scattering of electrons by mercury atoms

A recently published, approximate formula for the scattering phase shifts of electrons in a screened Coulomb field of the Hulthén type is investigated in more detail and shown to be applicable to the fast computation of the elastic differential scattering cross section and the Mott polarization and other functions describing polarization phenomena. Although these phase shifts show a much less pathological behaviour than the pure Coulomb phases, the partial wave expansion of the scattering amplitudes still has a weak divergence in the forward direction corresponding to a singularity which is, however, much less severe than the singularity of the Rutherford scattering amplitude. The examples chosen for demonstration in tabular form are concerned with 46 keV or 204 keV electrons elastically scattered by mercury atoms, a case which has received attention in other publications. The tables include exact values for a three-term Yukawa potential which is to represent, in analytical form, the realistic potential, exact values for a suitable Hulthén potential, and values computed by means of the approximate phase shift formula. As to the choice of the screening exponent of the Hulthén potential which enters the approximate phase shift formula, it is found that this exponent should be adjusted so as to approximate the realistic potential uniformly. The value so obtained is smaller, in the case of the mercury atoms here considered by about 11%, than the value which would be needed for fitting the realistic potential shift, due to screening, at the origin.     

Nucleon scattering to the continuum in terms of the two-fermion theory of multistep direct reactions

Cross sections of the (p,xp) reaction on ⁹³Nb were measured at an incident energy of 26.5 MeV and analysed together with the existing data for nucleon scattering between 18 and 26 MeV. The (p,p′) and (n,n′) emission spectra and angular distributions have been described consistently in the framework of the two-fermion multistep direct theory of Feshbach, Kerman and Koonin. We have confirmed the predictions following our earlier analyses that the strength of the effective nucleon-nucleon interaction does not depend on the reaction channel provided that all pre-equilibrium reactions are included. The latter requires allowing for the giant resonances in continuum from the energy-weighted sum rules. The method applied avoids double-counting of collective and onestep direct excitations.