Symmetric Scatterers Research Articles

Unfolded optical glory of spheroids: backscattering of laser light from freely rising spheroidal air bubbles in water

Enhancement in backscattering known as glory scattering results from geometric and material properties of spherically symmetric scatterers. The wave-front shape near the spherical scatterer is locally a circular torus. Radiation from a toroidal wave front is axially focused on the backward-directed axis. It is shown that the axial point caustic unfolds to an astroid caustic as the scatterer's shape changes from spherical to slightly spheroidal. The wave front pertinent for slightly spheroidal scatterers was modeled as a toroidal wave front with a superimposed harmonic angular perturbation. Experimental observations are displayed for cross-polarized backscattering by freely vertical rising, slightly oblate spheroidal air bubbles in water illuminated by a horizontally propagating laser beam. These patterns were recorded with a camera for two different incident-beam polarization directions relative to the axis of rotational symmetry of the bubble. Angular scattering patterns were also computed using a perturbation analysis based on use of the harmonically perturbed toroidal wave front and physical optics. Bubble oblateness was estimated from features of the angular scattering pattern and from hydrodynamic relations.

Necessary and sufficient condition for a scattering amplitude to correspond to a spherically symmetric scatterer

It is proved that A( θ′, θ, k) = A( θ′· θ, k) is a necessary and sufficient condition for a scattering amplitude A( θ′, θ, k) to correspond to a spherically symmetric potential q( x) = q(| x|).

Transfer matrix theory of leaky guided waves

We present a general transfer matrix approach for the propagation of guided waves in presence of inhomogeneities or «scatterers». We particularly address the problem of the coupling with radiation modes leading to a leakage of the guided wave to the surrounding bulk medium at each scattering. Examples are surface acoustic waves, electromagnetic or acoustic excitations or evanescent electromagnetic waves near a boundary, guided waves in optical fibers… From symmetry and conservation laws, we obtain, in the case of symmetric scatterers, the general form of the transfer matrix in terms of four independent real parameters. For ID-periodic lattices of identical scatterers, we show that leakage vanishes at the band edge: this coherent effect stems from the complete destructive interference between the converted radiations at each scatterer. This result demonstrates that coherent leakage is deeply different from a usual «dissipation» effect. Finally, we discuss the competition between Anderson localization and coherent leakage in the presence of disorder. Near the band edges and in the presence of disorder, the «attenuation length» due to the leakage is much larger than the localization length. We suggest an experimental situation where these effects could be observed Etude generale de la propagation d'ondes guidees en presence d'inhomogeneites et de diffuseurs. Exemples: ondes acoustiques de surface, excitations electromagnetiques et acoustiques, onde evanescentes proches d'une frontiere, ondes guidees dans les fibres optiques

The density matrix of an infinite muffin‐tin system and mutually avoiding scattering paths

AbstractFor independent quantum particles moving in an infinite ordered or disordered array of spherically symmetric scatterers the density matrix and related quantities for the incident field are represented via the superposition of all scattered waves. The corresponding formulae contain products of two propagators which are defined for mutually avoiding scattering paths. Therefore the usual vertex effects disappear. The new formulation forms a sound mathematical base for earlier trials. It is valid also for stationary non‐equilibrium situations and can therefore be employed to construct a non‐classical transport theory.

Structure of resonances for symmetric scatterers

This paper classifies the resonances and root vectors in the problem of scattering by tetrahedral and cubic clusters of zero-range potentials. It is shown that the symmetry of scatterer leads to the following properties of the S matrix: the resonances (roots of the S matrix) and root vectors can be classified in accordance with special subgroups of the symmetry groups of the scatterers. It is also proved that the series of root vectors are complete in some subspaces of L/sub 2/ (S/sup 2/), using for this some facts from the theory of entire functions of exponential type.

Scattering from a fluid sphere revisited via a transition matrix approach

The Anderson model for scattering of a sound wave from a fluid sphere, used widely in biological applications, is rederived by using a transition matrix approach. The variant used is MOOT (method of optimized truncation), in which the amount of dissatisfaction of boundary conditions due to a finite basis set is minimized [W. M. Visscher, J. Appl. Phys. 51, 825 (1980)]. The method is applicable to nonspherical scatterers and to layered structures of scatterers, and the computation is greatly simplified for axially symmetric scatterers. The reproductibility of the Anderson model and the properties of the transition matrix render this method very promising for acoustic scattering from nonspherical marine organisms with thin elastic shells and swimbladders. [Work supported by NASA.]

Irreducible analysis of Raman spectra for all crystal point groups

Techniques for the optimal and maximal extraction of information by Raman scattering experiments are investigated. All optical interactions are assumed to be electric dipole. For each crystal point group symmetry, a minimal set of choices of experimental arrangement (i.e. choice of polarisation vectors for incoming and outgoing light beams, scattering geometry etc.) is given, together with the transformations necessary to determine the irreducible parts of the spectrum in each of several bases. The more detailed tables are presented as a supplementary publication, SUP 70031. While valid for all crystal point groups, they are restricted to the case of time-reversal-invariant scattering interactions, and also to symmetric scatterers. The latter restriction includes systems which may have an antisymmetric part to the scattering amplitude, but in which the scattering intensity is unchanged by interchanging the incoming and outgoing polarisation vectors. Circular polarisation studies are necessary for a complete analysis of any system. However, the conventional 90 degrees and 180 degrees scattering geometries are adequate. Related symmetries of the Mueller matrix of the scatterer are also analysed.

Eigenfunctions of expected value operators in the Wishart distribution, II

Let V = ( v ij ) denote the k × k symmetric scatter matrix following the Wishart distribution W( k, n, Σ). The problem posed is to characterize the eigenfunctions of the expectation operators of the Wishart distribution, i.e., those scalar-valued functions f( V) such that ( E n f)( V) = λ n, k f( V). A finite sequence of polynomial eigenspaces, EP spaces, exists whose direct sum is the space of all homogeneous polynomials. These EP subspaces are invariant and irreducible under the action of the congruence transformation V → T′ VT. Each of these EP subspaces contains an orthogonally invariant subspace of dimension one. The number of EP subspaces is determined and eigenvalues are computed. Bi-linear expansions of |I + VA| −n 2 and (tr VA) r into eigenfunctions are given. When f( V) is an EP polynomial, then f( V −1) is an EP function. These EP subspaces are identical to the more abstractly defined polynomial subspaces studied by James.

Multiple Scattering in Disordered Materials. I. Theory of the Effective Medium for Homogeneous One‐Component Systems of Correlated Scatterers

AbstractMultiple scattering of scalar waves propagating in a macroscopic system composed of identic spherically symmetric nonoverlapping scatterers is treated theoretically for the general case of anisotropic scattering at correlated scatterers. A basic approximation is involved in the treatment of the back‐scattering of the wave going out from a fixed scatterer. The incident field within the homogeneous medium obeys a local wave equation with medium wave number k. The theory yields a system of equations for the relative strength coefficients of the angular momentum components of the incident field. The wave number has to be determined from this system of equations too. Back‐scattering itself cannot be treated correctly using a naive medium‐like approach.

Eigenfunctions of Expected Value Operators in the Wishart Distribution

Let $(X_{1,m}, X_{2,m}, \cdots X_{k,m}), 1 \leqslant m \leqslant n,$ be a sample of size $n$ from the $k$ dimensional normal distribution with mean vector $\mu$ and covariance matrix $\Sigma.$ Let $V = (\nu_{ij}), 1 \leqslant i, j \leqslant k,$ denote the symmetric scatter matrix where $v_{ij} = \sum_m(X_{i,m} - \mu_i)(X_{j,m} - \mu_j).$ The problem posed is to characterize the eigenfunctions of the expectation operators of the Wishart distribution, i.e., those scalar valued functions, $f(V),$ such that $E(f(V)) = \lambda_{n,k}f(\Sigma).$ If $f$ is an eigenfunction then (a) for nonsingular $T,f(T'VT)$ is an eigenfunction and (b) for integral $p, |V|^{p/2}f(V)$ is an eigenfunction. For $k \leqslant 2,$ a complete solution of the problem is given. For $k = 1$ the functions $f(v) = cv^\alpha$ are the only eigenfunctions. For $k = 2,$ a function $f$ is an eigenfunction if and only if (i) $f$ is homogeneous and (ii) $4 \frac{\partial^2 f}{\partial v_{22}\partial v_{11}} - \frac{\partial^2 f}{\partial^2v_{21}} = C|V|^{-1}f.$ A representation of eigenfunctions is given in terms of sums of associated Legendre functions. Relationships between eigenfunctions and harmonic functions are indicated. Any homogeneous polynomial is proved to be a linear combination of polynomial eigenfunctions.

A new space-time integral equation for two-dimensional transient scattering

In this letter a general time-domain integral equation for solving 2-dimensional scattering problems is presented. This new approach improves the integrodifferential equations commonly employed for the transient scattering of plane waves on cylindrical perfectly conducting obstacles. Moreover, this approach solves the problem of circular symmetric scatterers illuminated by transient incident waves that have the same symmetry.

Coherent electromagnetic waves in pair-correlated random distributions of aligned scatterers

Recent results for the corresponding scalar problem are generalized to coherent electromagnetic waves in random distribution of pair-correlated obstacles (aligned or averaged over alignment). Proceeding essentially as before, we obtain dispersion equations by averaging the vector–dyadic functional equation relating the multiple and single scattered amplitudes of the obstacles. In general, for aligned nonradially symmetric scatterers, the resulting bulk indices of refraction specify anisotropic media; the anisotropy arises either from the scatterers’ properties (physical parameters or shape, or both) or from their distribution, or from both. The illustrations include both isotropic and anisotropic cases, and the explicit results generalize earlier ones.

Coherent scalar field in pair-correlated random distributions of aligned scatterers

Dispersion equations for coherent propagation of scalar waves in random distributions of pair-correlated obstacles (aligned or averaged over alignment), are obtained by averaging the functional equations relating the multiple and single scattered amplitudes of the obstacles. The resulting bulk indices of refraction and bulk parameters, for aligned nonradially symmetric scatterers, specify anisotropic media; the anisotropy arises either from the scatterers’ properties (physical parameters or shape, or both) or from their distribution, or from both. The illustrations include both isotropic and anisotropic cases (in one to three dimensions), and the explicit results generalize earlier ones.

Propagation through a precipitation medium: Theory and measurement

The theory of propagation through a precipitation medium is based here on the differential propagation constant and two additional complex constants. The latter reduce to a single real parameter, the "canting angle," for certain simple models, for example, for a population of equioriented symmetrical scatterers. In a uniform medium the constants are closely related to the elements of an equivalent permittivity tensor. It is shown that the parameters can be determined from a sequence of depolarization measurements.

Multiple scattering of waves by the doubly periodic planar array of obstacles

We analyze the three−dimensional problem of multiple scattering by a doubly periodic planar array of bounded obstacles, and compare the results with those for the grating of parallel cylinders in two dimensions and for the periodic line in three. Plane wave integral forms of the scattered field and of the multiple scattered amplitude lead directly to the array−mode functional representation in terms of the single scattered amplitude and to simple approximations corresponding to array resonances for near−grazing evanescent modes. For the grating and for the periodic line, a grazing mode corresponds to reinforcement of the excitations an obstacle receives from the waves scattered by its neighbors; for the doubly periodic array, to reinforcement of the waves of the lattice lines perpendicular to the mode’s direction. We also derive spherical wave (and conical−cylindrical wave) representations of the solution, and exhibit the results for spherically symmetric scatterers as a special case.

Diagonal operators in disordered systems

This considers a single electron moving in a disordered array of spherically symmetric scatterers, the whole immersed in an external, static, uniform magnetic field. It is shown in a straightforward manner under very broad assumptions about the distribution of scatterers that the averaged scattering matrix and other related operators are diagonal with respect to a complete set of eigenstates describing a free electron in a magnetic field.

Multiple scattering of sound by a periodic line of obstacles

Equations derived earlier for multiple scattering by arbitrary three-dimensional configurations are applied to a line of equally spaced identical obstacles. We develop several different representations and approximations for the field and derive the appropriate energy and scattering theorems. Spherically symmetric scatterers are considered as a special case. In general, the results are analogous to those obtained before for the two-dimensional problem of scattering by a planar grating of identical cylinders but differ essentially in that now the scattered modes are conical instead of planar. A major difference is that resonance maxima (analogous to the Wood's anomalies for the planar grating) may occur not only for wavelengths that are slightly larger than the grazing (Rayleigh) wavelengths, but also for wavelengths that are slightly smaller. We also derive closed-form approximations for small scatterers with arbitrary spacing, and then specialize the results to small spacing.

Light Scattering from Coated Spheres: Model for Biological Cells

Efficient methods for the calculation of light scatteringintensityfunctions for concentrically coated spheres (~10-micro diam) are discussed. This model represents many types of biological cells whose nuclei have a low refractive index (~1.1) and cytoplasms with a slightly lower refractive index. Studies are made on the relationships between the scattering coefficients for nonabsorbing, spherically symmetric scatterers. The physical origin of these coefficients is examined for absorbing scatterers. A comparison of the angular half-width of the scattered intensity functions for the coated sphere and an equivalent homogeneous sphere shows that diffraction dominates the small angle scattering in both cases. At larger angles, the coated sphere scattering pattern is more structured and quite sensitive to core sphere size, suggesting a possible method of distinguishing types of biological cells that are similar in gross size but different in internal detail.

Relativistic Scattering of Electromagnetic Waves by Moving Obstacles

The excitation for an obstacle moving with constant velocity in free space (Σ) is transformed to the scatterer's rest system (Σ′), and the corresponding far-field scattering U′∼Ua′(g) is transformed to Σ to obtain U ∼ Ua(G). A simple expression is derived for the scattering amplitude G in terms of the conventional g, and known functional forms U[G] give the scattered field in Σ as a complex integral and its inverse-distance expansion. In dyadic form, G̃ = p̃r·g̃·p̃, where p̃ corresponds to transforming the excitation from Σ to Σ′, and p̃r (a planar reciprocal) to transforming the scattering from Σ′ to Σ. Then we transform the Green's function surface and volume integral representations, and the multipole series for U′, and apply the results to spherically symmetric scatterers, arbitrary small scatterers, large tenuous scatterers, and to cylinders and slabs.

Pressure Shifts of Highly Excited States of Atoms

A reinvestigation is given of Fermi's work on the shift of highly excited states of atoms embedded in a gaseous medium due to the interaction of the active electron with the medium particles. For the case treated by Fermi, spherically symmetric medium particles, it is found that effects neglected by him cancel and that his results are essentially correct. The method is extended to the case where the medium is composed of diatomic molecules, and it is shown that when rotational excitation is not important, only low-energy elastic scattering contributes in a manner similar to the spherically symmetric scatterer. A simple two-rotational-state model is also analyzed in the case where rotational excitation may be important. It is concluded from the model that when the method used is applicable, real rotational coupling is never important.