Showing papers on "Superposition principle published in 1983"

Magnetic excitations in layered media: Spin waves and the light-scattering spectrum

Abstract: We present an analysis of the spin-wave spectrum of a semi-infinite stack of ferromagnetic films, each of which is separated by a gap filled by a nonmagnetic medium. This is done within a formalism which includes the Zeeman and dipolar contributions to the spin-wave energy, with exchange omitted. We then calculate the spin-wave contribution to the Brillouin spectrum of such a system, in the backscattering geometry. The aim is to compare the spectrum for scattering from a sample with this geometry, with that from an isolated film. Two features unique to the stack appear in the spectrum. Each film, in isolation, possesses surface spin waves on its boundaries (Damon-Eshbach waves). In the layered geometry these interact to form a band of excitations of the array, which has nonvanishing component of wave vector normal to the stack. We find a feature in the spectrum associated with scattering from this band of modes; the position of the peak is controlled by dispersion introduced by interfilm interactions. Under certain conditions, the semi-infinite stack possesses a surface spin wave, whose eigenfunction is a linear superposition of individual film states, with amplitude that decays to zero as one moves down into the stack interior. This mode also produces a distinct feature in the light-scattering spectrum. These points are illustrated with a series of calculations of the spectrum, for parameters characteristic of layered ultrathin coherent structures.

The control of boundary-layer transition using a wave-superposition principle

Abstract: An experimental study has been made of the concept of controlling boundary-layer transition by superimposing in the flow Tollmien–Schlichting waves that are of equal amplitude and antiphased to the disturbances that grow and lead to transition. The cases that have been considered are transition arising from a single-frequency two-dimensional disturbance and transition arising from a nonlinear interaction between two waves of different frequency. A feedback system for controlling transition has also been studied. In each case, both hot-wire surveys and flow visualization have shown that it is possible to delay transition but that the flow cannot be restored completely to its undisturbed state. This appears to be a consequence of interactions between the very weak three-dimensional background disturbances in the flow and the primary two-dimensional waves. The implications of these findings in an implementation of the concept are discussed.

Time-Dependent Superposition of Spinors

Abstract: Inverting the spin state of one of the two coherent waves propagating within a neutron interferometer by means of a radio-frequency spin-flip device leads to a nonstationary interference pattern. By using stroboscopic neutron detection one can resolve this to demonstrate the nonclassical behavior of spinor superposition.

Incoherent Superposition of Multiple Self-imaging Lau Effect and Moiré Fringe Explanation

Abstract: Characteristics of the diffraction field of two spatially separated linear diffraction gratings under incoherent illumination are studied. The analytical model is based on the spatial superposition of many mutually incoherent Talbot (self-imaging) effects. It permits a simple explanation of the basic parameters of the ‘incoherent’ diffraction images: axial localization, spatial period and lateral displacement. The moire fringe formation in space, the imaging of a grating by a second grating, and the Lau effect are considered; the results are compared with the former studies in the literature.

Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects

Abstract: We present the first rigorous electromagnetic theory of diffraction in nonlinear optics. This theory allows the study of any type of nonlinear grating: bare or coated, whatever the groove depth and the profile of grating and coatings may be. The formalism developed here is derived from Maxwell's equations. The existence of the excitation and its nonlinear feature on the one hand, and the diffraction of the pump beams and of the signal on the other hand, are fully taken into account. The calculation reported here is valid for all cases of polarization (TM or TE) of the pump beams and of the signal. Two expressions of the nonlinear polarization at the signal frequency are derived. One is valid below the modulated region; the other one, inside this region. These two expressions take into account all the diffracted orders at the pump frequencies: propagating and evanescent. We then get the expression of the electromagnetic field at the signal frequency everywhere: not only outside the modulated region, but also inside it. The results thus obtained show that this electromagnetic field is a superposition of a diffracted field, with radiated and evanescent orders, and an infinite number of elementary driven waves. We also derive the nonlinear grating equation which allows the determination of the directions of propagation of the radiated diffracted orders. This is achieved using a new geometrical construction. It is shown that the evanescent diffracted orders at the signal frequency and at the pump frequencies can be resonantly excited. The regorous feature of the electromagnetic theory developed here allows us to get the following new and important result: There exists an optimal groove depth for which the electromagnetic resonance contribution to the enhancement of the nonlinear optical process is the strongest. These results can be applied to the study of different nonlinear optical processes, such as enhanced second-harmonic generation, surface-enhanced Raman scattering, Pockels effect, and optical rectification.

On the basis set superposition error in potential surface investigations. I. Hydrogen‐bonded complexes with standard basis set functions

Abstract: The basis set superposition errors on the energy of the complexes FH ⋅⋅⋅ OH2 and FH ⋅⋅⋅ NCH are investigated at the SCF level. The two‐dimension energy‐potential surfaces, corresponding to the slow and fast frequencies of the hydrogen bridge, are calculated without and with the counterpoise correction. The corresponding quadratic, cubic, and quartic force constants are computed. Similar calculations are reported for the electric dipole moment and its derivatives. These investigations show that the basis set superposition error: (i) does not decrease, in the vicinity of the equilibrium configuration of the complex, when standard high‐quality basis sets are used; (ii) shifts the minimum of the potential surface, distorts this surface, and changes the force constants in a significant but not drastic manner.

Theoretical and experimental investigation of a microstrip radiator with multiple lumped linear loads

Abstract: A simple theory is developed for the analysis of microstrip patch elements which are loaded at one or more points with lumped linear load impedances. The analysis is based on a cavity model in which the shape of the field distribution between the patch and ground plane is assumed to be well approximated by that of the resonant modes of a corresponding magnetic- and electric-walled cavity. The resonant mode of the loaded cavity is represented as an appropriate superposition of the modes of the corresponding unloaded cavity. The characteristic equation for the resonant frequencies of the loaded cavity is obtained in terms of the load impedances and the unloaded cavity multiport open-circuit parameters. An analysis of the input impedance of a rectangular microstrip element shorted at an arbitrary point has been implemented and the results show good agreement with experiment. An ancilliary result showing the equivalence between a thin strip and a circular cylinder model of a feed current distribution...

Dispersion relation for propagation of light in cholesteric liquid crystals

Abstract: A general theory of the light propagation in periodic structures characterized by a uniform rotation of the dielectric tensor about a given axis is presented. Starting from a fundamental approach of Dreher and Meier, which is mostly numerical, an analytical solution of the characteristic equation has been found which can be used to calculate the wave vectors as a function of $\ensuremath{\omega}$ and of the incidence angle ${\ensuremath{\theta}}_{i}$. The electromagnetic wave is described as a superposition of elementary modes having the form of Bloch waves. Each elementary mode is represented by a sum of plane waves elliptically polarized, whose wave vectors are the roots of the characteristic equation. The analysis of the solutions of such an equation allows us to draw a more complete map of the stability and instability regions for light propagation in helical structures than the ones currently available in the literature. The coexistence of two distinct modes, with different polarization states, determines the shape of the stability map. Each mode presents a series of Bragg instabilities. Between the two Bragg instabilities of the same order a further instability exists which is common to both modes and does not satisfy the Bragg conditions. All instability bands, with the exception of only one of the first order, vanish at normal incidence. This occurs for any value of the optical anisotropy and is a peculiarity of perfectly ordered helical structures. The bandwidth increases with ${\ensuremath{\theta}}_{i}$, and overlapping may occur. Typical plots of dispersion curves and attenuation constants are reported. Finally, we compute the intensity and the polarization state of the light reflected from a thin film, in order to clarify the controversial point about the structure\char22{}doublet or triplet\char22{}of the higher-order reflection bands.

Translation, rotation and superposition of linear quadtrees

Abstract: In Gargantini (1982a) it has been shown that storing black nodes of a quadtree is sufficient to retrieve any basic property associated with quadtrees. To achieve this, each black node must be represented as a quaternary integer whose digits (from left to right) describe the path from the root to that node. The sorted sequence of quaternary integers representing a given region is called the linear quadtree associated with that region. Such a structure has been shown to save more than two-thirds of the memory locations used by regular quadtrees. In this paper we present procedures for translating and rotating a region and consider the superposition of binary images with different characteristics (such as different resolution parameter, different pixel size and/or different center). Translation, rotation, and superposition are shown to be O(N log N) operations; for translation N is the number of black pixels; for rotation N is the number of black nodes; for superposition N is the sum of black nodes or black pixels of the two images, depending on whether or not the two regions are centered on the same raster.

Superposition Method for Analysis of Free-Edge Stresses

Abstract: Superposition techniques were used to transform the edge stress problem for composite laminates into a more lucid form. By eliminating loads and stresses not contributing to interlaminar stresses, the essential aspects of the edge stress problem are easily recognized. Transformed problem statements were developed for both mechanical and thermal loads. Also, a technique for approximate analysis using a two dimensional plane strain analysis was developed. Conventional quasi-three dimensional analysis was used to evaluate the accuracy of the transformed problems and the approximate two dimensional analysis. The transformed problems were shown to be exactly equivalent to the original problems. The approximate two dimensional analysis was found to predict the interlaminar normal and shear stresses reasonably well. Previously announced in STAR as N83-32846

The harmonic and transient electromagnetic response of a thin dipping dike

Abstract: An exact solution is given for the electromagnetic induction in a dipping dike of finite conductivity, represented as a thin half‐sheet in a nonconducting surrounding. The problem is formulated for arbitrary dipole or circular loop Tx-Rx configurations. The formal solution obtained by the Wiener‐Hopf technique is cast into a rapidly convergent triple integral suitable for an effective numerical treatment. A good agreement is found between numerical results and analog measurements available for harmonic excitation. The transient response is obtained as a superposition of the half‐sheet free‐decay modes and is illustrated by some numerical examples for coincident loops, including a diagram for the approximate determination of conductance and depth of a vertical dike.

A contribution to eddy current calculations in plane and axisymmetric multiconductor systems

Abstract: For steady-state eddy current problems in Plane and axisymmetric multiconductor systems calculations by means of three different methods are described. The first method is a finite element approach using the superposition principle resulting in the admittance matrix of the total multiconductor system. The second method described is the integrodifferential approach providing a direct implementation of the total current condition. In case of multiconductor systems consisting of sub-systems with series connected conductors and given voltages the integrodifferential method has to be used combined with the superposition principle. The third method is based on the numerical solution of the underlying Fredholm integral equation of second kind. Some examples are given to show the particular advantages and disadvantages of the various methods with respect to computation time and accuracy of solution. Furthermore, the application of different higher-order isoparametric elements for each of the methods is investigated.

Scattering by Weakly Nonlinear Objects

Abstract: An attempt is made to construct a consistent scattering theory for systems involving nonlinear objects. Weak nonlinearity is assumed, such that harmonic generation is present, but shock wave formation is excluded. Mathematically this is described by constitutive relations in the form of Volterra series. For periodic (as opposed to monochromatic) waves, this procedure facilitates algebraic constitutive relations and dispersion equations in the transform space. Weak nonlinearity, as defined here, implies phase matching, i.e., all harmonics of the fundamental wave possess the same phase velocity (this is the reason that shock waves cannot be formed). Due to this stipulation superposition is allowed in a restricted sense, facilitating the construction of arbitrary wave solutions, e.g., cylindrical and spherical waves, by using sums (integrals) of plane waves.Using wave solutions from the linear theory, various boundary value problems can be discussed. Plane interfaces are considered, displaying well known pro...

The Effect of the Interference of Traveling and Stationary Waves on Time Variations of the Large-Scale Circulation.

Abstract: It is hypothesized that the interference of stationary and traveling waves of the same longitudinal can cause some of the observed time variations in the large-scale circulation. To explore this hypothesis the eight-winter average structure of a regularly occurring, westward propagating disturbance which we earlier called the “16-day wave” is further documented. Energy quantities are calculated as this 16-day wave moves in and out of phase with the stationary or time-mean wave. The resulting time variations are similar to some already reported in the literature. Eddy heat momentum transport associated with energy conversions have phase relationships between pressure levels that can be approximately predicted by a simple linear superposition of the observed stationary waves and traveling external Rossby waves. In further support of the hypothesis, cross-spectral results determined from independent data show a reasonable agreement with these predictions.

A fast recursive algorithm for binary-valued two-dimensional filters

Abstract: An efficient algorithm for computing the response of a linear spatially varying digital image filter to an arbitrary digital input image is described. This response is the superposition summation of the input image with a digital point spread function (PSF). It is assumed here that the PSF is binary valued. The approach to this computation is based on the Principle of Inclusion and Exclusion. This approach leads to a new efficient algorithm. This algorithm is also efficient when the PSF is spatially invariant.

Optical Properties of Cholesteric Liquid Crystals at Oblique Incidence

Abstract: The optical properties of a periodic structure characterized by a uniform rotation of the dielectric tensor about a given axis are theoretically analyzed. The electromagnetic wave is described as a superposition of elementary modes having the form of Bloch waves. Each elementary mode is represented by a sum of plane waves elliptically polarized, whose wavevectors are the solutions of a characteristic equation. This equation, presented in a preceding paper, is furtherly analyzed, in order to obtain the wave vectors in terms of a power series of a small parameter δ, representing the anisotropy of the dielectric tensor. The coefficients of the series up to terms containing δ6 are explicitly given, and the corresponding truncation errors computed. The spectral composition and the polarization states of the Bloch waves are also analyzed and discussed for different values of the incidence angle in the frequency range containing the lower reflection bands. In particular it is shown that in the regions b...

New Bäcklund Transformations and Superposition Principle for Gravitational Fields with Symmetries

Abstract: Vector Backlund transformations which relate solutions of the vacuum Einstein equations having two commuting Killing fields are introduced. Such transformations generalize those found by Pohlmeyer in connection with the nonlinear δ model. A simple algebraic superposition principle, which permits the combination of Backlund transforms in order to get new solutions, is given. The superposition preserves the asymptotic flatness condition, and the whole scheme is manisfestly O(2, 1) invariant.

Analysis of observables in diagonally polarized light scattering experiments

Abstract: We analyze the polarization of light scattered from an ensemble of stationary, randomly oriented molecules of arbitrary size, for ‘‘diagonally’’ polarized incident light (linearly polarized at ±45° to the scattering plane). The results are expressed in terms of the ‘‘diagonal modulation,’’ the change in scattered intensity when the incident polarization switches from +45° to −45°. We use the exact nonrelativistic operators for the interaction of light and matter, and we use no wave function approximations. The diagonal modulation is expressed as a superposition of four independent observables, which may be separated by polarization filters in the observation channel. Three of these are found to posses interesting characteristics after the classical orientation average is taken: (a) they vanish when the damping constants of the molecule are set to zero; and (b) they vanish in the limit of a dipolar scattering tensor. This analysis therefore suggests experiments in which combinations of damping constants an...

Paley-Wiener criterion for relaxation functions

Abstract: It is shown how the Paley-Wiener theorem in Fourier-transform theory can provide the bound for physically acceptable relaxation functions for long times. In principle the linear exponential decay function, and hence also a superposition of linear exponential decay functions, does not provide an acceptable description of relaxation phenomenon although the Paley-Wiener bound can be made to approach arbitrarily close to linear exponential. A class of relaxation functions proposed recently obeys the Paley-Wiener bound. The general necessity for time-dependent relaxation rates is emphasized and discussed.

Relation Between Certain Quasi-Vortex Solutions and Solitons of the Sine-Gordon Equation and Other Nonlinear Equations

Abstract: It is shown that the quasi-vortex type solutions recently studied by Hudak of the sine-Gordon equation, u x x + u y y =sin u , can be derived from the known multiple soliton solutions by the proper procedure. This shows in principle the existence of the multiple quasi-vorte solutions. It also shows that the superposition of usual solitons and quasi-vortex solutions are possible for this equation. Implication of the present results to other soliton equations is briefly discussed.

The kinematics of a stochastic field of internal waves modified by a mean shear current

Abstract: Modes and dispersion curves of high-frequency internal waves are calculated from observed profiles of the Brunt-Vaisala frequency and of the mean shear current. Cases treated are from the tropical Atlantic (GATE) and from the North Atlantic (JASIN). The dispersion characteristics are anisotropic. The feature increases in importance with increasing frequency, wavenumber, and mode number. The mean shear strongly deforms the eigenfunctions for modes 2 and higher at upstream wave directions, and separated wave guides occur at different depths. The higher the mode number, the more effective is the critical-layer absorption. By means of the stochastic superposition of shear modes a model of the kinematics of a field of free, linear internal waves is constructed for the GATE situation. The first mode strongly dominates and the waves constitute a narrow band process in frequency—wavenumber space. Thus, most of the energy is in regions where the anisotropy of the system of dispersion curves and shear modes is weak, and the mean flow introduces only a moderate kinematic distortion in the wave field. It is concluded that the principal features of the GATE spectra, a shoulder in the energy spectra between 1.5 and 4 cph and a strong directionality within this frequency band, are related to dynamic processes. A detailed modeling of the various cross spectra is impossible without taking the mean current into account.