Showing papers on "Plane wave published in 1969"

Dielectric rectangular waveguide and directional coupler for integrated optics

Abstract: We study the transmission properties of a guide consisting of a dielectric rod with rectangular cross section, surrounded by several dielectrics of smaller refractive indices. This guide is suitable for integrated optical circuitry because of its size, single-mode operation, mechanical stability, simplicity, and precise construction. After making some simplifying assumptions, we solve Maxwell's equations in closed form and find, that, because of total internal reflection, the guide supports two types of hybrid modes which are essentially of the TEM kind polarized at right angles. Their attenuations are comparable to that of a plane wave traveling in the material of which the rod is made. If the refractive indexes are chosen properly, the guide can support only the fundamental modes of each family with any aspect ratio of the guide cross section. By adding thin lossy layers, the guide presents higher loss to one of those modes. As an alternative, the guide can be made to support only one of the modes if part of the surrounding dielectrics is made a low impedance medium. Finally, we determine the coupling between parallel guiding rods of slightly different sizes and dielectrics; at wavelengths around one micron, 3-dB directional couplers, a few hundred microns long, can be achieved with separations of the guides about the same as their widths (a few microns).

Continuous and Discrete Inverse‐Scattering Problems in a Stratified Elastic Medium. I. Plane Waves at Normal Incidence

Abstract: This paper investigates the problem of obtaining an analytic solution and practical computational procedures for recovering the properties of an unknown elastic medium from waves that have been reflected by or transmitted through the medium. The medium consists of two homogeneous half‐spaces in contact with a heterogeneous region. The analytic solution is obtained by transforming the equation of motion for the propagation of plane waves at normal incidence in a stratified elastic medium into a one‐dimensional Schrodinger equation for which the inverse‐scattering problem has already been solved. The practical computational procedures are obtained by solving the corresponding discrete inverse‐scattering problem resulting from approximating the heterogeneous region with a sequence of homogeneous layers such that the travel time through each layer is the same. In both the continuous and discrete inverse scattering problems, the impedance of the medium as a function of travel time is recovered from the impulse response of the medium. A discrete analogy of the continuous solution is also developed. Similar results are obtained for a stratified elastic half space bounded by a free surface.

High-Frequency Scattering by a Transparent Sphere. II. Theory of the Rainbow and the Glory

Abstract: The treatment, initiated in Paper I [J. Math. Phys. 10, 82 (1969)], of the high‐frequency scattering of a scalar plane wave by a transparent sphere is continued. The main results here are an improved theory of the rainbow and a theory of the glory. The modified Watson transformation is applied to the third term of the Debye expansion of the scattering amplitude in terms of multiple reflections. Only the range 1

Reflection of Plane Waves from the Flat Boundary of a Micropolar Elastic Half‐Space

Abstract: Reflection of plane waves from stress free flat surface of micropolar elastic half space, presenting reflection laws and amplitude ratios

Fluctuations of a Beam Wave Propagating Through a Locally Homogeneous Medium

Abstract: : General formulations for the fluctuations of a beam wave propagating through a homogeneous or locally homogeneous medium are given in terms of the spectral density of the index of refraction. The amplitude and phase correlation functions and the mean square fluctuations are derived for a homogeneous medium showing the dependence on the radial distance in the transverse plane of the beam. The amplitude and phase structure functions are derived for a locally inhomogeneous medium. The correlation functions and the structure functions do not depend only on the difference coordinate, but they are functions of the radial coordinates in the beam cross section. This particular inhomogeneity, however, is shown to be an analytic continuation of the homogeneous or locally homogeneous case. The mean square amplitude fluctuation for the Kolmogorov's locally homogeneous medium is shown to behave as a plane wave for short distance and then becomes less than that of a spherical wave, and its spectrum is shown to behave as 1/K for large K in contrast with the plane and spherical waves. The spread of the beam radius is shown to be approximately the 8/3 powers of the distance L for small distance and its increase depends on the magnitude of the index of refraction fluctuation. (Author)

Self-Consistent Orthogonalized-Plane-Wave and Empirically Refined Orthogonalized-Plane-Wave Energy-Band Models for Cubic ZnS, ZnSe, CdS, and CdSe

Abstract: First-principles orthogonalized-plane-wave energy-band calculations have been carried out for cubic ZnS, ZnSe, CdS, and CdSe, using a nonrelativistic formalism and Slater's free-electron exchange approximation. The calculations were first carried out in terms of a physically realistic trial crystal potential, and then iterated to obtain a self-consistent solution. So far as we are aware, these are the first fully convergent, fully self-consistent energy-band solutions reported for cubic II-VI semiconducting compounds. In spite of the simplified treatment of exchange effects, and the neglect of relativistic and correlation effects, the first-principles solutions are in qualitative and semi-quantitative agreement with experiment in all cases. It is shown briefly how improved solutions can be obtained by introducing small, carefully chosen empirical corrections. The adequacy of various energy-band models was tested by calculating the optical spectrum (actually ${\ensuremath{\epsilon}}_{2}$) and comparing this with the experimental spectrum. Actually, this comparison checks only certain features of these energy-band models. It would be helpful to have additional experimental information so that other features, such as the energy separation between principal and subsidiary conduction-band minima, could also be checked.

An explicit determination of the empty space-times on which the wave equation satisfies Huygens' principle†

Abstract: The validity of Huygens' principle in the sense of Hadamard's ‘minor premise’ is investigated for scalar wave equations on curved space-time. A new necessary condition for its validity in empty space-time is derived from Hadamard's necessary and sufficient condition using a covariant Taylor expansion in normal coordinates. A two component spinor calculus is then employed to show that this necessary condition implies that the plane wave space-times and Minkowski space are the only empty space-times on which the scalar wave equation satisfies Huygens' principle.

Diffracted Wave Fields Expressible by Plane-Wave Expansions Containing only Homogeneous Waves

Abstract: The angular-spectrum representation of wave fields is shown to yield scalar Helmholtz-equation solutions (in a half-space) that tend in the L2 limit of the mean to prescribed boundary values on an infinite plane boundary for all square-integrable boundary values. This result is employed to obtain several important properties of source-free wave fields, i.e., wave fields (satisfying square-integrable boundary values) that contain only homogeneous plane waves in their angular-spectrum representations: (1) the function describing a source-free wave field can be extended into the region behind the plane boundary to give a bounded continuous solution of the scalar Helmholtz equation in all space; (2) the function describing a wave field can be extended to the whole space of three complex variables as an entire function satisfying a certain inequality if, and only if, the wave field is source free; (3) the two-dimensional autocorrelation function of the wave field on planes parallel to the boundary plane is independent of the distance of the plane from the boundary if, and only if, the field is source free; and (4) a boundary value exists that produces a three-dimensional, pseudoscopic, real image of a wave field if, and only if, the wave field is source free. A series-mode expansion for source-free fields in terms of the boundary value is derived and is shown to be absolutely and uniformly convergent. The series is transformed into a two-dimensional Taylor series (with coefficients determined in terms of the boundary value) and another series that displays explicitly the contribution to the wave field due to each partial derivative of boundary value. The series are valid representations of wave fields in a broad class that is the natural extension of the source-free class of fields when the restriction that the boundary value be square integrable is removed. The series are used to derive two angular-spectrum representations for wave fields in the extended class and to discuss the properties of those wave fields.

Calculations of the X-ray emission bands of copper using augmented plane wave Bloch functions

Abstract: The L and M X-ray emission bands of metallic copper are calculated in the one-electron approximation using augmented plane wave Bloch functions analysed into s, p and d symmetry components. The effect on the calculated intensity distributions of lifetime broadening due to Auger processes is estimated in a simple manner, and appropriately broadened curves are compared with the available experimental data.

On the Transmission of Plane Waves through Layered Linear Viscoelastic Media

Abstract: The propagation of two‐dimensional time harmonic waves through a plane layered viscoelastic medium is considered in terms of the equivalent elastic plane strain problem with modified Lame constants that are complex and frequency dependent. The problem is formulated directly in terms of stresses and displacements rather than potentials and is solved by matrix methods. If the incident wave is not attenuated in the direction parallel to the layering, as would occur if the incident wave were to travel through a semi‐infinite elastic half‐space before striking the viscoelastic layers, interface waves could be generated only if one of the layers is “pseudoelastic,” i.e., has at least one real wave speed. In this case, interface waves may be generated in the same manner as in the purely elastic case. If the incident wave is attenuated in this direction, however, interface waves could be generated for specific angles of incidence and material properties.

Combined Plane-Wave Tight-Binding Method for Energy-Band Calculations with Application to Sodium Iodide and Lithium Iodide

Abstract: A method of calculating energy bands, which we name the mixed-basis method, is developed. This method uses a wave-function expansion which includes free-atom or ion wave functions and plane waves, and it may incorporate relativistic effects. It is, in general, not equivalent to the orthogonalized-plane-wave (OPW) method, although the cases for which the two methods are equivalent are developed. A sample calculation is made for LiI and NaI, for which x-ray, valence, and conduction states are calculated. The agreement between theory and x-ray and optical-absorption data is excellent. The importance of relativistic effects is explored, and comparisons are made with conventional OPW results, which include relativity by first-order perturbation theory.

Non-linear wave propagation in a relaxing gas

Abstract: We consider the propagation of waves of small finite amplitude e in a gas whose internal energy is characterized by two temperatures T (translational) and Ti (internal) in the form e = CvfT + CvfTi, and Ti is governed by a rate equation dTi/dt = (T − Ti)/τ. By means of approximations appropriate for a wave advancing into an undisturbed region x > 0, we show that to order eδ, the equation satisfied by velocity takes the non-linear form \[ \bigg(\tau\frac{\partial}{\partial t}+1\bigg)\bigg\{\frac{\partial u}{\partial t}+\bigg(a_1+\frac{\gamma + 1}{2}u\bigg)\frac{\partial u}{\partial x}-{\textstyle\frac{1}{2}}\lambda\frac{\partial^2u}{\partial x^2}\bigg\}=(a_1-a_0)\frac{\partial u}{\partial x}, \] where a1, a0 are the frozen and equilibrium speeds of sound in the undisturbed region, δ = ½(1 − (a20/a21)), and λ is the diffusivity of sound due to viscosity and heat conduction (λ may be neglected except when discussing the fine structure of a discontinuity). Some numerical solutions of this model equation are given.When e is small compared with δ, it is also possible to construct a solution for the flow produced by a piston moving with a constant velocity by means of a sequence of matched asymptotic expansions. The limit reached for large times for either compressive or expansive pistons is the expected non-linear solution of the exact equations. For a certain range of advancing piston speeds, this is a fully dispersed wave with velocity U in the range a0 a1 the solution is discontinuous, and indeterminate in the absence of viscosity; a singular perturbation technique based on λ is then used to determine the structure of the wave head.

Effect of a Shear Layer on Plane Waves of Sound in a Fluid

Abstract: Available theory indicates that plane waves of sound are reflected and refracted at an interface of relative motion (a velocity discontinuity) between two regions of fluid. If the relative velocity is sufficiently great, three types of reflection occur, ordinary, total, and amplified, depending on the incident wave angle. In the amplification regime, theory predicts resonances. Here the velocity discontinuity replaced by a transition layer of finite thickness separating the two fluid regions. This layer is approximated by two equal velocity discontinuities (Model I) and by a linear velocity profile shear layer (Model II). For one example in the regime of ordinary reflection, the effects of thickness are negligible for thicknesses up to 1/10 of the incident wavelength. For the chosen examples in amplified reflection, extreme reductions in transmission and reflection coefficients occur for a thickness as little as 1/50 of a wavelength. The two models approach (at unequal rates) total reflection and zero transmission at larger thicknesses. These effects are produced by layers of fluid traveling at or near the apparent wave speed parallel to the shear layer. Such layers tend to “insulate” the two fluid regions from each other.

Electronic structure of terbium using the relativistic augmented-plane-wave method.

Abstract: The energy bands for Tb have been calculated by the relativistic augmented-plane-wave method. The resulting Fermi surface is presented, and agreement is found with the characteristic wave vector describing the spiral phase of Tb.

Scattering of a Plane Wave at a Plane Interface Separating Two Moving Media

Abstract: The problem of reflection and transmission of a plane electromagnetic wave at an interface separating two moving media is discussed. Contrary to previous studies, the model does not excludo normal components of the velocities with respect to the interface. The general formalism yields dyadic reflection and transmission coefficients. The results are applied to the Cerenkov radiation phenomenon, the Fizeau experiment, clear-air scattering (when the two media have the same constitutive parameters in their respective rest frames), and ray optics in moving media.

Reflection and Transmission by a Random Medium

Abstract: The mean field is considered for a bounded medium with a refractive index having a real random part. It has been shown previously that for suitable ranges of the amplitude and correlation length of the refractive index fluctuations, this field satisfies a certain integrodifferential equation. This equation is solved for a plane wave incident from either side on the plane boundary of a semi-infinite random medium, for the Green's function of a semi-infinite random medium, and for a plane wave incident on a slab of random medium, provided that the background refractive index is homogeneous throughout. The cases of both one- and three-dimensional fluctuations are considered, and explicit expressions are given for the reflection,transmission, and coupling coefficients for a medium with an exponential correlation function.A Wiener-Hopf factorization required for other correlation functions is described, as are methods for treating reflection and transmission at a curved boundary of a random medium. A principal finding is the inadequacy of treating the mean wave in a bounded random medium by using just the refractive index for an unbounded random medium, for in addition we must include a transition layer near the boundary.

Rejection of a Coherent Arrival at an Array

Abstract: Analysis is presented for the enhancement of a plane‐wave signal at an acoustic array relative to noise of which a part is also a plane wave from another direction. The process considered (described in the preceding paper) employs conventional beam forming to estimate the interfering plane‐wave form, which is then removed at each element by subtraction. Removal of interference by more than one plane wave, and the relation to optimum beam forming, are briefly considered.

Three-Dimensional Electromagnetic Scattering from a Circular Tube of Finite Length

Abstract: An electromagnetic plane wave in either E or H polarization is normally incident on a cylinder which is assumed to be an infinitely thin‐walled, perfectly conducting, circular tube of finite length with complete generality both in radius and length. For either polarization both axial and transverse currents are induced. The problem is transformed into determining an infinity of Fourier components by expanding each function of θ into a Fourier series. For each Fourier component a pair of integral equations for the total currents is obtained. These are decoupled in a special sense permitting the transverse total current to be obtained first, followed by the axial total current. The diflerence current is then defined and expressed in terms of the total current, making possible the determination of the inside and the outside total current. These are experimentally measurab e quantities. The theoretical formulas for the scattered fields, the far field patterns, and the scattering cross sections are also derived in terms of the total current. Finally, sample results, both theoretical and experimental, are included.

Scattering obliquely incident microwaves by a moving plasma column.

Abstract: The problem of the interaction of obliquely incident microwaves with a plasma column which is moving uniformly in the axial direction is treated analytically. An arbitrary polarization for the incident plane wave is assumed. Two methods in solving this problem are presented. Extensive numerical results for the scattered energy in the backward and broadside directions and the angular distribution of the scattered energy are obtained for various interesting ranges of the parameters involved. It is found that cross‐polarized field components are induced even at normal incidence when the plasma medium is moving with respect to the observer and that cross‐polarized field components disappear at an incident angle θ0=sin−1(vz/c), where vz is the velocity of the moving plasma and c is the speed of light in vacuum when the plasma column is imbedded in free space.

An inverse scattering technique for electromagnetic bistatic scattering

Abstract: It is shown that the total field produced by a plane wave incident upon a scattering body can be expressed at all points in space as the sum of the incident field and the Fourier transform of a qua...

Dielectric Function of a Model Insulator.

Abstract: A wave-number- and frequency-dependent dielectric function has been calculated for a model insulator using the random-phase approximation. The model insulator consists of a free-electron conduction band represented by a single orthogonalized plane wave (OPW) and a valence band represented by linear combinations of ionic wave functions. All energy bands are assumed parabolic and isotropic. Numerical results are obtained for Ar, KCl, CsCl, MnS, and Si, in reasonable agreement with previous calculations. Screened exchange potentials have been obtained by using an interpolation formula, and evaluated for the chlorine ion in CsCl.

Electric-field distribution in the focal region of an offset paraboloid

Abstract: The electric field in the vicinity of the focus of a finite offset-paraboloid reflector, illuminated by a linearly polarised plane wave, is investigated. The paper begins with a general formulation of the reflected field calculation procedure for an arbitrary reflecting surface. The formulation is then specialised to apply to an offset paraboloid. Extensive computed data of the field in the focal region of an offset paraboloid are presented and compared with the field distribution in the focal region of a symmetrical paraboloid. It is shown that the electric-field distribution in the focal region of an offset paraboloid is almost the same as for a symmetrical paraboloid with the equivalent focal length.

Backscattering from an impedance loaded slotted cylinder

Abstract: When a thick infinite cylinder is illuminated by a plane wave with a perpendicular electric field at normal incidence, its backscattering can be controlled by implementing an impedanced-backed longitudinal slot on its surface. An exact theory is developed to account for the scattered field from an impedance loaded slotted cylinder. Optimum slot loadings for minimizing and maximizing the backscattering from the cylinder are determined. A series of experiments has been conducted to verify the theory. Excellent agreement between theory and experiment is obtained. The present study indicates that a very effective reduction in the backscattering from the cylinder can be achieved by a proper design of the slot loading.

Electronic Energy Bands, Excitons, and Plasmons in Lithium Fluoride Crystal

Abstract: The electronic band structure of lithium fluoride has been worked out using the mixed basis (MB) method recently proposed by one of the authors. Due to the effect of exchange, F− wave functions are rather compact, and mixing of plane waves in the valence band wave functions is unimportant. The minima of the conduction band occur at equivalent L-points in the Brillouin zone. Recent ultraviolet, soft X-ray, and electron energy loss spectra are analyzed in the light of the present calculation.