Extinction Theorem Research Articles

On the completeness of the natural modes for quantum mechanical potential scattering

The set of natural modes, associated with quantum mechanical scattering from a central potential of finite-range is shown to be complete. The natural modes satisfy a non-Hermitian homogeneous integral equation, or alternatively, are solutions of the time independent Schrödinger equation subject to a recently formulated nonlocal boundary condition (the quantum mechanical extinction theorem). An expansion theorem similar to that of Hilbert–Schmidt is formulated, valid for values of the solution of the scattering integral equation inside the range of the potential. The boundary conditions generated by the quantum mechanical extinction theorem are shown to be closely connected with the Jost function.

The Birth and Death Chain in a Random Environment: Instability and Extinction Theorems

Let $(Y_n)$ be a recurrent Markov chain with discrete or continuous state space. A model of a birth and death chain $(Z_n)$ controlled by a random environment $(Y_n)$ is formulated wherein the bivariate process $(Y_n, Z_n)$ is taken to be Markovian and the marginal process $(Z_n)$ is a birth and death chain on the nonnegative integers with absorbing state $z = 0$ when a fixed sequence of environmental states of $(Y_n)$ is specified. In this paper, the property of uniform $\phi$-recurrence of $(Y_n)$ is used to prove that with probability one the sequence $(Z_n)$ does not remain positive or bounded. An example is given to show that uniform $\phi$-recurrence of $(Y_n)$ is required to insure this instability property of $(Z_n)$. Conditions are given for the extinction of the process $(Z_n)$ when (i) $(Z_n)$ possesses homogeneous transition probabilities and $(Y_n)$ possesses an invariant measure on discrete state space, and (ii) $(Z_n)$ possesses nonhomogeneous transition probabilities and $(Y_n)$ has general state space.

Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces

In this paper we present an analysis of the conditioning character of the matrices obtained in the different methods for solving the integral equations that appear in the scattering of waves from a hard corrugated surface. It is found that the matrices derived from the Rayleigh hypothesis become ill conditioned for large corrugations when the approach is not valid at any rate. The method based on the extinction theorem developed by Masel, Merrill, and Miller is analytically correct, but it becomes ill conditioned and no practical numerical solutions can be obtained for large values of the corrugation strength, depending on the precision of the computer used. Finally the self-consistent method with the numerical procedure developed by Garcia and Cabrera is well conditioned and leads always to a good practical numerical solution for all kinds of corrugations, no matter how large.

Internal field resonance structure: Implications for optical absorption and scattering by microscopic particles

Mie scattering theory is used to calculate the efficiency factors for absorption by microscopic dielectric spheres. Resonances in the efficiency factors for absorption and resonances in the amplitudes of the electric and magnetic multipoles which occur in an expansion of the fields inside the dielectric sphere are discussed. Several trends in the strengths and width of the various resonances as functions of the absorption coefficient of the sphere, size parameter, multipole order, and multipole resonance number are given. A formal solution for elastic (Mie) and inelastic (Raman) scattering by microscopic particles is derived from the extinction theorem. With this background, some implications of the resonances in the interpretation of absorption, fluorescence, and Raman scattering by microscopic particles are discussed.

The extinction theorem in atom-surface scattering

We present a generalization of the extinction theorem of optics to treat atom-surface scattering. The method, which is valid for a wide class of atom-surface potentials, is not limited to the case of small roughness. Its connections with Rayleigh approximation are discussed.

The Ewald Dynamical Theory of Reflection and Transmission of Light by a Dielectric Slab. The Extinction Theorem in Spatially Dispersive Media

The dielectric slab is considered as a system of classical dipoles fixed on the lattice points of N + 1 parallel planes of a simple cubic lattice. The dipoles interact with retarded electromagnetic and non-retarded mechanical forces. Thus the media with spatial dispersion are taken into account, too. The problem of reflection and transmission of light by this system is solved in two steps. First, the amplitudes of the dipole moments are evaluated as a function of the incident electromagnetic wave. Then the electromagnetic field generated by the vibrating dipoles of the entire slab is calculated. The resulting formulae for the angle of refraction and for the intensity of the reflected and transmitted waves coincide with the classical Snell law and Fresnel formulae, if the wavelength of the incident light is much greater than the lattice constant of the slab. A generalized proof of the Ewald extinction theorem valid for all wavelengths and for spatially dispersive media is given.

Equivalence of the Ewald-Oseen extinction theorem as a nonlocal boundary-value problem with Maxwell’s equations and boundary conditions*

It has been shown in the past that the Ewald-Oseen extinction theorem can be generalized into a nonlocal boundary-value problem and that the scattering of an electromagnetic wave on a medium of arbitrary response can be described in terms of this boundary-value problem. The scattered field inside the medium (the interior-scattering problem) is determined, without any reference to the scattered field outside the medium, by solving a set of coupled partial differential equations, subject to certain nonlocal boundary conditions which are generalizations of the Ewald-Oseen extinction theorem. Once the field on the boundary of the medium is known, the exterior field (the exterior-scattering problem) is evaluated by direct integration. Moreover, a hypothesis was put forward according to which these nonlocal boundary conditions completely and uniquely specify the solution of the interior-scattering problem. In this paper it is shown that this hypothesis is true. The equivalence of the interior- and exterior-scattering problems to Maxwell’s equations and boundary conditions is proven. For simplicity, the proof is confined to nonmagnetic media. Thus, if Maxwell’s equations and boundary conditions have a unique solution, the interior-scattering problem will also have a unique solution, namely, that obtained from the former. Some illustrative examples are presented in the special case of a linear medium.

On the applicability of the Lippmann-Schwinger approach to the scattering of particles by semiinfinite crystals

We present a formal approach, using integral equations of the Lippmann-Schwinger type, to the scattering of particles by semiinfinite periodic potentials. The two possible physical situations, scattering of free plane waves and scattering of Bloch waves, are treated with two different integral equations. The validity of this formalism to describe the particle wavefunction in the whole space is established, and its connection with the matching method is studied. The corresponding extinction theorems, one for each situation, are derived. Interesting relations among the probability amplitudes for both scattering problems are found.

Null field approach to scalar diffraction I. General method

Invoking the optical extinction theorem (extended boundary condition) the conventional singular integral equation (for the density of reradiating sources existing in the surface of a totally reflecting body scattering monochromatic waves) is transformed into infinite sets of non-singular integral equations, called the null field equations. There is a set corresponding to each separable coordinate system (we say that we are using the ‘elliptic’, ‘spheroidal’, etc., null field method when we employ 'elliptic cylindrical', ‘spheroidal’, etc., coordinates). Each set can be used to compute the scattering from bodies of arbitrary shape, but each set is most appropriate for particular types of body shape as our computational results confirm. We assert that when the improvements (reported here) are incorporated into it, Waterman’s adaptation of the extinction theorem becomes a globally efficient computational approach. Shafai’s use of conformal transformation for automatically accomodating singularities of the surface source density is incorporated into the cylindrical null field methods. Our approach permits us to use multipole expansions in a computationally convenient manner, for arbitrary numbers of separated, interacting bodies of arbitrary shape. We present examples of computed surface source densities induced on pairs of elliptical and square cylinders.

The interaction of light with a semiinfinite dielectric as a phonon problem; the generalized Snellius law and Fresnel formulae

Using Ewald's formulae it is possible to evaluate the electromagnetic field generated by a mechanical plane wave propagating through a semiinfinite dipole crystal. The problem of refraction and reflection of an external electromagnetic wave on a semiinfinite dipole crystal is thus reduced to the evaluation of the forced vibrations of this crystal excited by the incident electromagnetic wave. Applying this approach, the generalized Snellius law and generalized Fresnel formulae can be deduced. In the limit of long waves these general formulae change into the usual classical expressions. The Ewald extinction theorem follows directly from the lattice dynamics of our system.

Optical properties of rough surfaces: General theory and the small roughness limit

The diffraction of electromagnetic waves from the rough surface of a material of finite permittivity is examined for the case where the wavelength of the incident radiation is comparable to the dimensions of the surface roughness. Two methods of calculation studied are the Rayleigh method and the extinction-theorem integral-equation method. The latter is shown to have a unique exact solution. This property is, in turn, used to show how the Rayleigh method can be modified to give convergent results. The extinction theorem is also used to reduce the Rayleigh equations to a simpler form. These reduced equations, which are extremely convenient to use in the case of small roughness, are applied in this case to find perturbative expressions for the reflected field and for the surface-plasmon dispersion relation.

Interaction of electromagnetic waves at rough dielectric surfaces

A systematic perturbation theory is developed to study the interaction of electromagnetic waves at rough dielectric surfaces. The Ewald-Oseen extinction theorem is used as the basis of the perturbation theory. Explicit expressions for both first-order and second-order fields (in terms of the surface roughness parameter) at all points in space are presented. The incident field is treated very generally and thus it can be in any state of polarization. The surface is characterized by a structure function and hence the results are valid both for periodic and statistical surfaces. The first-order fields are used to calculate various types of scattering cross sections for arbitrary incident fields characterized by coherence matrices. The coherence matrix for scattered fields is given. The equivalent electric and magnetic surface currents on a flat surface which would lead to the same expressions for first-order fields as those obtained for the rough surface, are calculated using the extinction theorem. The extinction cross section is calculated using an appropriately formulated optical theorem. This leads in a straightforward manner to a change in the value of the reflectivity as compared to the value on a flat surface. The experiment of Teng and Stern is discussed briefly in the light of the expressions for extinction cross sections. Finally, first-order and second-order fields are used to discuss Smith-Purcell radiation, i.e., the radiation emitted by an electron moving parallel to a grating surface. This case corresponds to the conversion of evanescent waves into homogeneous waves due to surface roughness. The relation of some of our results to those of Crowell, Elson, Ritchie, Juranek, Lalor, Marvin et al., Maradudin, and Mills is also discussed.

The extinction theorem in nonlinear optics

The apparatus of integral equations in linear optics suggested by Ewald and Oseen has been developed for nonlinear media. The relations between micro- and macroscopic values have been found for anisotropic and nonlinear media. The formulation of the extinction theorem applied to nonlinear optics is given.

Relation between Waterman's extended boundary condition and the generalized extinction theorem

The relation of Waterman's extended boundary condition to the generalized Ewald-Oseen extinction theorem is established. This unifies different formulations of the electromagnetic scattering. Our proof also establishes the validity of Waterman's extended boundary condition for an arbitrary type of material medium.

The generalized Ewald-Oseen extinction theorem and the integral equation in electromagnetic scattering

Four different types of response functions are used to derive the generalized Ewald-Oseen extinction theorem and the integral equation for the treatment of scattering of electromagnetic waves from a material medium. No surface terms appear explicitly in the integral equation. The present formulation is specially suited for perturbation calculations since the response functions, which appear, are not necessarily the ones corresponding to free space.

Integral equation treatment of scattering from rough surfaces

A self-consistent integral equation describing scattering from rough surfaces is obtained. This integral equation automatically takes into account the discontinuous behavior of the zeroth-order fields and Green's functions and leads to results for various scattering and extinction cross sections in agreement with the results obtained by other methods such as those based on the extinction theorem and the Rayleigh-Fano method.

Electrodynamics of nonlocal bounded dielectrics

We investigated the electrodynamics of a bounded nonlocal medium, characterized by the non-translationally invariant susceptibility introduced by Zeyher, Birman, and Brenig. We obtained dispersion equations for the propagating and surface modes, as well as the extinction theorems, and additional boundary conditions appropriate to this susceptibility. The calculation was carried out for various angles of incidence, and polarizations leading to different reflectivities than in other work using a translationally invariant susceptibility. We also derived a new surface polariton dispersion equation. An analysis of an attenuated-total-reflection experiment shows that the new dispersion equation predicts altered reflectivities from those predicted by competing theories.

A simple model to explain the slowing down of light in a crystalline medium

Two main questions arise when one seeks to explain the propagation of light in a dielectric medium: (i) Why does the wave slow down? (ii) Why is there apparently only a forward wave? There is a subtle answer to these questions provided by the Ewald–Oseen extinction theorem, but the physical mechanism involved is not very easily visualized. This article sets forth a simple one-dimensional model of parallel polarizable slabs that makes the mechanism clear and links the two questions in an interesting way.

Scattering states and bound states as solutions of the schrödinger equation with nonlocal boundary conditions

The problem of determining the Schr\"odinger wave function of a nonrelativistic particle that is either scattered by a potential of a finite range or that is bound to it is reformulated in a novel way. It is shown that in either case the wave function must satisfy a certain boundary condition on the surface that delimits the effective range of the potential. For scattering states the boundary condition is analogous to the mathematical formulation of the Ewald-Oseen extinction theorem of classical electromagnetic theory. The new formulation is illustrated by determining the scattering states and the bound states for a central potential. It is also shown that a boundary condition that is used in band-structure calculations in solids is an immediate consequence of our quantum-mechanical extinction theorem for bound states.

Generalization of classical rigorous theory of dispersion and Helmholtz-Kirchhoff integral theorem

New forms of the Ewald-Oseen extinction theorem and a diffraction formula are obtained by the generalization of the classical rigorous theory of dispersion. These theorem and formula reduce to the Helmholtz-Kirchhoff integral theorem in vector form under certain condition.