Rational Reflection Coefficients Research Articles

Discrete Dirac systems on the semiaxis: rational reflection coefficients and Weyl functions

ABSTRACTWe consider the cases of the self-adjoint and skew-self-adjoint discrete Dirac systems, obtain explicit expressions for reflection coefficients and show that rational reflection coefficients and Weyl functions coincide.

Scattering for general-type Dirac systems on the semi-axis: reflection coefficients and Weyl functions

We show that for general-type self-adjoint and skew-self-adjoint Dirac systems on the semi-axis Weyl functions are unique analytic extensions of the reflection coefficients. New results on the extension of the Weyl functions to the real axis and on the existence (in the skew-self-adjoint case) of the Weyl functions follow. Important procedures to recover general-type Dirac systems from the Weyl functions are applied to the recovery of Dirac systems from the reflection coefficients. We explicitly recover Dirac systems from the rational reflection coefficients as well.

Inverse scattering designs of dispersion-engineered planar waveguides.

We have introduced a semi-analytical IS technique suitable for multipole, rational function reflection coefficients, and used it for the design of dispersion-engineered planar waveguides. The technique is used to derive extensive dispersion maps, including higher dispersion coefficients, corresponding to three-, five- and seven-pole reflection coefficients. It is shown that common features of dispersion-engineered waveguides such as refractive-index trenches, rings and oscillations come naturally from this approach when the magnitude of leaky poles in increased. Increasing the number of poles is shown to offer a small but measureable change in higher order dispersion with designs dominated by a three pole design with a leaky pole pair of the smallest modulus.

Inverse scattering for an AKNS problem with rational reflection coefficients

We study the AKNS (Ablowitz–Kaup–Newell–Segur) inverse scattering problem for rational reflection coefficients on a semi-infinite interval. We demonstrate that the Marchenko integral equations for the AKNS version on such an interval can be solved in a direct and straightforward way by algebraic methods for any set of rational reflection coefficients, vanishing at infinity. The general AKNS scattering problem for this case, as well as the usual symmetry reductions, are discussed. The connection to an alternative procedure by Rourke and Morris (1992 Phys. Rev. A 46 3631) is pointed out. Our procedure is built around a constant matrix, M, which can be constructed from the poles and residues of the rational reflection coefficients. Under certain conditions, which we define as minimal symmetry and which then suitably constrains the distribution of eigenvalues, we show that it is always possible to represent the resulting potentials as truncated N-soliton potentials. This procedure is of interest for solving initial-boundary value problems of integrable hyperbolic systems by the inverse scattering transform (IST) applied to a semi-infinite or finite interval.

Explicit solutions to the Korteweg–de Vries equation on the half line

Certain explicit solutions to the Korteweg–de Vries equation in the first quadrant of the xt-plane are presented. Such solutions involve algebraic combinations of truly elementary functions, and their initial values correspond to rational reflection coefficients in the associated Schrödinger equation. In the reflectionless case such solutions reduce to pure N-soliton solutions. An illustrative example is provided.

The Korteweg-de Vries hierarchy: structure and solution

We attempt to realize the structure of the Korteweg-de Vries (KdV) hierarchy by using a simple dimensional analysis. The specific results presented refer to equations of the hierarchy and conserved Hamiltonian densities associated with them. Based on the Gel’fand-Levitan-Marchenko equation we construct a series expansion for the unstable solution of the KdV-like equations by using the continuous part of the spectrum for the Schrodinger operator. Our result is in exact agreement with that obtained from the Sabatier’s formulation of the inverse problem for rational reflection coefficients.

Rational reflection coefficients in inverse scattering for a Dirac system

The inverse scattering problem (ISP) on the whole line for a Dirac system is considered. The reflection coefficient (FC) is represented as a rational function with an arbitrary number of poles. The method of solving for the Gel'fand-Levitan-Marchenko (GLM) equation generated by a rational reflection coefficient (RFC) is extended to n poles, when a spectral gap is present. The explicit solution in the case of three poles is presented. Graphs of the potential as a function of distance are displayed for cases having up to four poles.

Inverse problem for Sturm-Liouville operators with rational reflection coefficient

In this paper we give exact formulas for the potential associated to a Sturm-Liouville equation when the reflection coefficient function is rational. The solution is given in terms of a minimal realization of the reflection coefficient function.

Inverse scattering theory for optical waveguides and devices: Synthesis from rational and nonrational reflection coefficients

Electromagnetic inverse scattering theory is applied to the synthesis of optical waveguides and devices from specified reflection and transmission characteristics. The permittivity profiles in the inhomogeneous core regions of the devices are reconstructed from their transverse reflection coefficients. Two applications of this theory are demonstrated with examples using specified reflection coefficients: design of a single‐mode inhomogeneous optical waveguide and design of an optical logic gate. The previous inverse scattering theory which used reflection coefficients that are rational functions of the transverse wavenumber has been supplemented with an inverse scattering theory that uses nonrational reflection coefficients. This inverse scattering theory uses an iterative discretized space‐time finite‐difference method with the “leapfrog” algorithm. The initial values for the iterative method are obtained from the inverse theory that uses rational function reflection coefficients. The inverse scattering calculations have been verified by finite‐difference frequency domain direct scattering techniques.

The stability of one-dimensional inverse scattering

The stability of one-dimensional inverse scattering is discussed. The analysis is performed analytically using perturbation theory for reflection coefficients that are rational functions of the wavenumber and numerically for realistic reflection coefficients. It is shown that the same instability features occur for real reflections as for rational reflection coefficients. There are three types of error considered which are common in the application of the data. Explicitly examined are an erroneously chosen DC level, an amplitude error and a time error. It is shown that the physical reason for the instabilities of the one-dimensional inverse problem is the lack of sufficient energy penetration in the potential that is examined.

Half-solitons as solutions to the Zakharov-Shabat eigenvalue problem for rational reflection coefficient. II. Potentials on infinite support

It is shown how to invert the Zakharov-Shabat eigenvalue problem for rational scattering coefficients, in order to produce potentials defined over the whole real line. The method reduces the problem to finding two semi-infinite potentials—which can be efficiently calculated using the soliton-lattice algorithm described in a previous paper. The inversion is usually unique for given rational scattering coefficients, since they usually specify the scattering data of the system uniquely. In the case of nonunique inversion, it is possible to obtain a family of potentials from the inversion. Each member of the family has the same scattering data, except for differing residues. Examples of the inversion include an ‘adiabatic’ inversion pulse for use in nuclear magnetic resonance, and a demonstration of how the cubic nonlinear Schrödinger equation may be integrated.

Half solitons as solutions to the Zakharov-Shabat eigenvalue problem for rational reflection coefficient with application in the design of selective pulses in nuclear magnetic resonance.

It is shown how the Zakharov-Shabat (ZS) eigenvalue problem for rational reflection coefficient may be reduced to the ZS problem with zero reflection coefficient. The soliton solutions to this reduced problem are obtained using the B\acklund transform. Hence the solutions to the original problem are shown to be half solitons. It is demonstrated how selective pulses in nuclear magnetic resonance may be calculated using this technique. In particular, almost perfect 90\ifmmode^\circ\else\textdegree\fi{} self-refocused and 180\ifmmode^\circ\else\textdegree\fi{} refocusing selective pulses are demonstrated.

Inverse-scattering view of modal structures in inhomogeneous optical waveguides

To understand the physical meaning of rational reflection coefficients in inverse-scattering theory for optical waveguide design [ J. Opt. Soc. Am. A6, 1206 ( 1989)], we studied the relationship between the poles of the transverse reflection coefficient and the modes in inhomogeneous dielectrics. By using a stratified-medium formulation we showed that these poles of the spectral reflection coefficient satisfy the same equation as the guidance condition in inhomogeneous waveguides. Therefore, in terms of wave numbers, the poles are the same as the discrete modes in the waveguide. The radiation modes have continuous real values of transverse wave numbers and are represented by the branch cut on the complex plane. Based on these results, applications of the Gel’fand–Levitan–Marchenko theory to optical waveguide synthesis with the rational function representation of the transverse reflection coefficient are discussed.

Optical waveguides for imaging applications: Design by an inverse scattering approach

AbstractAn inverse scattering theory is used to design optical waveguides capable of transmitting spatial images without degradation. The propagation characteristics of N modes each carrying a pixel of the image are specified by a transverse rational reflection coefficient. The Gel'fand‐Levitan‐Marchenko inverse scattering theory is used to obtain the unique solution of the permittivity profile of the waveguide from the reflection coefficient

Discontinuous soliton-like solution to the self-induced-transparency equations

By using the ∂ -approach for rational reflection coefficients and dispersion relations with moving singularities we build a solution to SIT equations which belongs to a new class of discontinuous soliton-like solutions of some nonlinear evolution equations.

A Kalman filter solution of the inverse scattering problem with a rational reflection coefficient

This paper presents a new inverse scattering method for reconstructing the reflectivity function of symmetric two-component wave equations, or the potential of a Schrödinger equation, when the reflection coefficient is rational. This method relies on the so-called Chandrasekhar equations which implement the Kalman filter associated to a stationary state-space model. These equations are derived by using first a general layer stripping principle to obtain some differential equations for reconstructing a general scattering medium, and by specializing these recursions to the case when the probing waves have a state-space model.

On the solutions of the Gel’fand–Levitan equation

The solutions of the Gel’fand–Levitan equation are obtained in a closed form for general rational reflection coefficients, and as a convergent limit of closed forms for reflection coefficients sufficiently close to being rational.

Eigenvalues and eigenfunctions associated with the Gel’fand–Levitan equation

It is shown here that the solutions of the Gel’fand–Levitan equation for inverse potential scattering on the line may be expressed in terms of the eigenvalues and eigenfunctions of certain associated operators of trace class. The details are sketched for the case of rational reflection coefficients, and carried out for the simplest class of examples.

Rational reflection coefficients and inverse scattering on the line

Inverse scattering for the Schrodinger equation on the line is studied for reflection and transmission coefficients that satisfy the usual regularity conditions and are rational functions ofk. The origin is still a particular point, but the potentials do not need to be cut at this point like in previous studies. Giving up this restriction corresponds to the existence of poles for both reflection coefficients in both upper and lower halfk-planes. It is shown that the problem reduces to solving a linear algebraic system. A different algorithm, made of a sequence of Darboux-Backlund transforms, gives also the solution in closed form and enables to study separately modifications of both sides of the potential due to the introduction of poles. Thus it paves the way for approximation studies. Generalizations and particular problems will be studied in forthcoming papers.

The applicability of an inverse method for reconstruction of electron-density profiles

Inverse scattering theory is used to reconstruct profiles of electron density from the analytic representation of the reflection coefficient. The complex reflection coefficient <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(k)</tex> is represented as a rational function of the wavenumber <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> . Using a three-pole approximation for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r(k)</tex> , the one-dimensional inverse scattering theory is applied to obtain a closed-form expression for the electron-density profile function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q(x)</tex> . The integral equation of the inverse scattering theory (Gelfand-Levitan equation) is solved by a differential-operator technique, and several numerical examples are given. Three-pole reflection coefficients are found to be applicable to the reconstruction of relatively thin electron layers which might be generated in the laboratory. Rational reflection coefficients with an increased number of poles are found to be necessary to simulate other electron layers of physical interest. This is demonstrated by comparison of multipole reflection coefficients in the Butterworth approximation with reflection coefficients calculated from Epstein's direct scattering theory for electron layers. A parameter in the Epstein theory, which characterizes the total electron content of the layer, is related to the number of poles needed to reconstruct that layer. Estimates are thus obtained of the number of poles needed to reconstruct ionospheric layers and other plasmas of physical interest.