Showing papers on "Superposition principle published in 1980"

Integrable nonlinear Klein-Gordon equations and Toda lattices

Abstract: We present a class of nonlinear Klein-Gordon systems which are soluble by means of a scattering transform. More specifically, for eachN≧2 we present a system of (N−1) nonlinear Klein-Gordon equations, together with the correspondingN ×N matrix scattering problem which can be used to solve it. We illustrate these with some special examples. The general system is shown to be closely related to the equations of the periodic Toda lattice. We present a Backlund transformation and superposition formula for the general system.

Limitation of rotation sensing by scattering.

Abstract: There appear to be limitations in the operation of optical-fiber Sagnac gyro rotation sensors that have imposed a minimum measurable rotation rate that is much higher than that caused by quantum noise. We show that one source of limitation, namely, the superposition of a nonreciprocal pair of waves generated by backward scattering from the incident waves, can result in significant error but can be mitigated by appropriate system design and signal modulation.

Inverse scattering problems in optics

Abstract: 1. Progress in Inverse Optical Problems.- 1.1 Inverse Problems in Optics and Elsewhere.- 1.2 Survey of Recent Results.- 1.2.1 Phase, Uniqueness, and Estimation.- 1.2.2 Radiometry and Coherence.- 1.2.3 A Moment Problem.- 1.3 Construction of Lambertian Scatterers.- 1.3.1 Lambertian Source Correlation.- 1.3.2 Random Scatterer Models.- 1.4 Organization of this Volume.- References.- 2. The Inverse Scattering Problem in Structural Determinations.- 2.1 Philosophical Background.- 2.2 The Direct Scattering Problem.- 2.2.1 Description of the Medium.- 2.2.2 The Scattered Fields.- 2.2.3 Expression for the Intensity.- 2.3 Analytic Description and Properties of Scattered Fields.- 2.3.1 Entire Functions of the Exponential Type.- 2.3.2 Distributions of Zeros for Functions of Class E.- 2.3.3 Encoding of Information by Zeros.- 2.4 The Deterministic Problem.- 2.4.1 Limitations of Measurements.- 2.4.2 The Phase Problem.- 2.4.3 Solutions to the Zero Problem.- 2.4.4 Zero Location.- 2.4.5 Extensions of the Method.- 2.5 The Statistical Problem.- 2.5.1 Overall Characterization of the Medium.- 2.5.2 Analytical Properties of Overall Descriptors.- 2.5.3 Determination of Overall Descriptors from Finite Records.- 2.6 Conclusions.- References.- 3. Photon-Counting Statistics of Optical Scintillation.- 3.1 Introductory Remarks.- 3.2 Photon-Counting Statistics.- 3.2.1 Single-Interval Statistics.- 3.2.2 Photon-Correlation Spectroscopy.- 3.2.3 Instrumental Effects.- 3.2.4 Noise and Statistical Accuracy.- 3.3 Scattering Theory.- 3.3.1 Mechanisms and Theories for Strong Scattering.- 3.3.2 The "Discrete-Scatterer" Model.- 3.3.3 K Distributions.- 3.3.4 Correlation Functions.- 3.4 Limit Distributions in the Random Walk Problem.- 3.4.1 The Gaussian Limit.- 3.4.2 Negative Binomial Number Fluctuations.- 3.4.3 A Population Model.- 3.5 Experiments.- 3.5.1 Dynamic Scattering by Nematic Liquid Crystals.- 3.5.2 Hot-Air Phase Screen.- 3.5.3 Extended Atmospheric Turbulence.- 3.5.4 Other Experiments.- 3.6 Concluding Remarks.- References.- 4. Microscopic Models of Photodetection.- 4.1 Photoelectron and Photon Statistics.- 4.1.1 Definition of the Problem.- 4.1.2 Ideal and Real Detection.- 4.2 Models for Ideal Detection - a Review.- 4.2.1 Mandel's Formula.- 4.2.2 Perturbation Approach.- 4.2.3 Field Attenuation.- 4.2.4 Inversion Problem.- 4.3 Open-System Detection Scheme.- 4.3.1 Detector Model.- 4.3.2 Relation Between Atomic and Field Dynamics.- Field Dynamics.- Dynamics of the Atomic Moments.- 4.3.3 Photocounting Probability.- 4.4 Disturbing Effects.- 4.4.1 Dark Currents and Noise.- Photodetectors.- Noise in Photoconductive Detectors.- Noise in Photomultipliers.- PMT Statistics.- 4.4.2 Dead Time Effects.- 4.4.3 Coherence and Sampling Effects.- Time Effects.- Spatial Effects.- Sampling Effects.- Other Counting Experiments.- 4.5 Temperature Effects in Photodetection.- 4.5.1 Langevin Equations of Motion.- The Field Equation.- Connection Between Atomic and Field Dynamics.- 4.5.2 Photocounting Probability.- 4.5.3 Applications.- Numerical Examples and Discussion.- 4.6 Summary of Statistical Methods.- 4.6.1 Random Variables.- Examples.- 4.6.2 Stochastic Processes.- 4.6.3 The Statistical Description of the Radiation Field.- 4.7 The Statistical Description of Open Systems.- 4.7.1 Equation of Motion of the Reduced Density Matrix.- 4.7.2 Langevin Equations.- References.- 5. The Stability of Inverse Problems.- 5.1 Ill-Posedness in Inverse Problems.- 5.1.1 Well-Posed and Ill-Posed Problems.- 5.1.2 Ill-Posedness and Numerical Instability.- 5.1.3 General Formulation of Linear Inverse Problems.- 5.1.4 Prior Knowledge as a Remedy to Ill-Posedness.- 5.1.5 Holder and Logarithmic Continuity.- 5.2 Regularization Theory.- 5.2.1 An Outline of Miller's Theory.- 5.2.2 Eigenfunction Expansions and Numerical Filtering.- 5.2.3 Tikhonov Regularization Method.- 5.2.4 Stability Estimates.- 5.3 Optimum Filtering.- 5.3.1 Random Variables in a Hilbert Space.- 5.3.2 Best Linear Estimates.- 5.3.3 Mean-Square Errors.- 5.3.4 Comparison with Miller's Regularization Method.- 5.4 Linear Inverse Problems in Optics.- 5.4.1 Inverse Problems in Fourier Optics.- Prolate Spheroidal Wave Functions (PSWF).- Perfect Lowpass Filter.- Bandwidth Extrapolation.- 5.4.2 Inverse Diffraction.- Inverse Diffraction from Plane to Plane.- Inverse Diffraction for Cylindrical Waves.- Inverse Diffraction from Far-Field Data.- 5.4.3 An Inverse Scattering Problem for Perfectly Conducting Bodies.- 5.4.4 Inverse Scattering Problems in the Born Approximation.- 5.4.5 Object Reconstruction from Projections and Abel Equation.- 5.4.6 Concluding Remarks and Open Problems.- References.- 6. Combustion Diagnostics by Multiangular Absorption.- 6.1 Absorption in Homogeneous Media.- 6.2 Multiangular Scanning.- 6.2.1 Basic Equation.- 6.2.2 Two-Dimensional Fourier Transform.- 6.2.3 Linear Superposition Techniques.- 6.2.4 Algebraic Reconstruction Techniques (ART).- 6.2.5 Applications and Results.- 6.3 The Reconstruction Procedure.- 6.3.1 Reconstruction Errors.- 6.3.2 An Observation of the Oversampling Requirement of Reconstruction.- 6.3.3 Number of Measurements M x N in Combustion Application.- 6.3.4 The Convolution Algorithm.- 6.3.5 Simulated Test Functions and Results.- 6.3.6 Algebraic Reconstruction.- 6.3.7 Benefits of Additional Digital Signal Processing.- 6.3.8 Conclusion.- 6.4 Experimental Aspects.- References.- 7. Polarization Utilization in Electromagnetic Inverse Scattering.- 7.1 Scope.- 7.1.1 Definitions of the Electromagnetic Inverse Problem.- 7.1.2 Definitions of Exact, Unique, and Approximate Methods.- 7.1.3 Incompleteness and A Priori Knowledge, Data Limitedness and Self-Consistency.- 7.2 The Vector Diffraction Integral, Its Far-Field Approximations, and Some Tauberian Relations.- 7.2.1 Basic Scattering Phenomena, Nomenclature, and Radar Definitions.- 7.2.2 The Stratton-Chu Vector Diffraction Integral and the Vector-Current Integral Equations.- 7.2.3 Far Scattered Fields in the Physical Optics Limit and Their Vector Corrections.- 7.2.4 Time-Domain Target Modeling: Utilization of Some Tauberian Theorems.- 7.3 The Radar Scattering and Target Polarization Matrices.- 7.3.1 Basic Electromagnetic Polarization Descriptors.- 7.3.2 Radar Scattering Matrices and Radar Measurables.- 7.3.3 Kennaugh's Optimum Polarization Pairs.- 7.3.4 Radar Target and Clutter Characteristic Operators.- Single Radar Target Classification.- The Time-Varying Distributed Target.- Synthetic Aperture Imagery.- 7.4 Inverse Scattering Theories in Various Electromagnetic Frequency Regimes.- 7.4.1 The Low Frequency Regime: Rayleigh-Gans Theory.- 7.4.2 The Resonant Frequency Regime: Natural Frequency Expansion.- 7.4.3 Physical Optics Far-Field Inverse Scattering Theories: Broad-Band Approach.- Fourier Transform Method of Physical Optics.- POFFIS in Time, Frequency, and Projection Domain.- The Limited Aperture Problem.- Polarizational Correction.- 7.4.4 Geometrical Optics Inverse Scattering Asymptotic Theories.- GOIS and the Minkowski Problem.- Vector Extension of GO Equivalent Curvature Inverse Method.- Scattering Center Discrimination: Kell's Monostatic-Bistatic Equivalence Theorem.- 7.5 Vector Holography and Polarization Utilization.- 7.5.1 Vector Wavefront Reconstruction and Interferometry.- 7.5.2 Polarization Dependence in Millimeter and Microwave Holography.- 7.5.3 The Postulate of Inverse Boundary Conditions.- 7.5.4 Near-Field Approach to Vector Inverse Scattering.- 7.6 Conclusions.- 7.6.1 Summary.- 7.6.2 Unresolved Vector Inverse Problems.- 7.6.3 Limitations and Omissions.- 7.6.4 Recommendations.- References.- Additional References with Titles.

Normal and Shear Impact of Layered Composite With a Crack: Dynamic Stress Intensification

Abstract: The dynamic response of a layered composite under normal and shear impact is analyzed by assuming that the composite contains an initial flaw in the matrix material. One of the objectives was to develop an analytical method for determining dynamic stress solutions which should lead to a numerical method which utilizes Fourier transform for the space variable and Laplace transform for the time variable. The time-dependent angle loading is separated into two parts: a symmetric and a skew-symmetric with reference to the crack plane. By superposition, the transient boundary conditions consist of applying normal and shear tractions to a crack embedded in a layered composite; one phase of the composite could represent the fiber while the other could be the matrix. Mathematically, these conditions reduce the problem to a system of dual integral equations solved in the transform plane for the transform of the dynamic stress-intensity factor.

The superposition principle for homojunction solar cells

Abstract: The superposition principle for solar cells states that the current flowing in an illuminated cell subject to a forward bias V is given by the algebraic sum of the short-circuit photocurrent and the current which would flow at bias V in the dark. Several authors have published arguments establishing the validity of this principle for homojunction cells operated so that the minority-carrier concentrations in the quasi-neutral regions do not exceed low injection levels. All these arguments depend on the assumption that the quasi-Fermi levels are constant across the depletion region of a forward-biased, illuminated cell. The accuracy of this assumption is examined in detail in the present paper. It is found that the quasi-Fermi levels do, in fact, vary significantly across the depletion region of an illuminated cell operated at short-circuit or low forward bias. However, it is shown that if the carrier mobilities are reasonably high and the carrier lifetimes reasonably long, the superposition principle still provides an excellent description of the cell characteristics at all bias levels. The superposition principle may seriously overestimate the efficiency of cells fabricated on poor-quality substrates with very short lifetimes and low mobilities.

A model with non‐reflecting boundaries for use in explicit soil—structure interaction analyses

Abstract: The purpose of this paper is to present an accurate and efficient model for use in explicit soil-structure interaction analyses for seismic excitation. The main body of the paper is concerned with the implementation of non-reflecting boundaries at the base and vertical faces of a two-dimensional finite element or finite difference soil mesh. The paper proposes a scheme for implementing viscous dashpots as energy absorbers at the base of the model where the seismic excitation is applied. To achieve this, the form of the seismic input is modified from the conventional accelerogram to a force time history that can be obtained from a standard deconvolution program with very minor modifications. The proposed scheme is also applicable to frequency-domain solutions. For the lateral boundaries, a superposition non-reflecting boundary formulation is recommended. The paper shows how easily the standard formulation can be modified to accommodate a seismic excitation that is assumed to be vertically propagating from the base of the model. An example is presented to demonstrate the accuracy of the proposed model.

Elliptical and hyperbolic zone plates

Abstract: Elliptical and hyperbolic zone plates have been constructed by optical methods recording on a photosensitive material the model produced by coherent superposition of two wave fronts, one spherical and the other cylindrical. When the foci of the wave fronts coincide, we have linear zone plates. In the reconstruction, the effects of linear and nonlinear recording produce the appearance of linear foci situated symmetrically on both sides of the zone plates. Experimental results are presented.

Chapter vii – elements of the theory of interference and interferometers

Abstract: Introduction IN Chapter III a geometrical model of the propagation of light was derived from the basic equations of electromagnetic theory, and it was shown that, with certain approximations, variations of intensity in a beam of light can be described in terms of changes in the cross-sectional area of a tube of rays. When two or more light beams are superposed, the distribution of intensity can no longer in general be described in such a simple manner. Thus if light from a source is divided by suitable apparatus into two beams which are then superposed, the intensity in the region of superposition is found to vary from point to point between maxima which exceed the sum of the intensities in the beams, and minima which may be zero. This phenomenon is called interference. We shall see shortly that the superposition of beams of strictly monochromatic light always gives rise to interference. However, light produced by a real physical source is never strictly monochromatic but, as we learn from atomistic theory, the amplitude and phase undergo irregular fluctuations much too rapid for the eye or an ordinary physical detector to follow. If the two beams originate in the same source, the fluctuations in the two beams are in general correlated, and the beams are said to be completely or partially coherent depending on whether the correlation is complete or partial. In beams from different sources, the fluctuations are completely uncorre-lated, and the beams are said to be mutually incoherent. When such beams from different sources are superposed, no interference is observed under ordinary experimental conditions, the total intensity being everywhere the sum of the intensities of the individual beams.

False Paradoxes of Superposition in Electric and Acoustic Waves.

Abstract: When acoustic or electromagnetic waves cancel by destructive interference, the wave impedance reflected to the sources of the wave energy changes so that the input power is reduced correspondingly Because this rather subtle point is not discussed in most instructional treatments of wave physics, misconceptions concerning the apparently ’’missing’’ energy are widespread, as reflected in semipopular science writing This paper presents an analysis of the cases of destructive and constructive wave interference, and analogous problems for lumped electrical and mechanical systems

Superposition fringe shear interferometer

Abstract: A new white light shear interferometer based on the use of superposition fringes is described (superposition fringe shear interferometer, SFSI). The SFSI enables one to test for plane waves; chromatic as well as all other aberrations can be measured. Test examples and the secondary spectrum of a microscope objective are given. By spectroscopically dispersing a slit section of the shear interferogram the chromatic aberrations can be displayed. The mean phase difference between the two interfering waves can be adjusted by tilting one of the two interferometer etalons. The whole setup is mechanically stable. Shear interferograms can be obtained by inserting an interference filter in the ray path.

A model theory for the fibrous absorber. Part 1: Regular fibre arrangements

Abstract: The axial propagation of sound waves in a model consisting of parallel fibers is calculated. The viscous forces and the thermal conduction are taken into account. This leads to viscous waves and to thermal waves besides the usual acoustic compression wave. The potential function for the total field near a fiber is treated as the superposition of the radiated field from the fiber itself and of the scattered fields from all the other fibers. The explicit field equations for a regular square fiber arrangement is derived and the influence of the order of symmetry of the arrangement is discussed. This leads to simplifications in the field equations and to field equations for the case of a homogeneous fiber distribution.

Linear modes of convection in the solar envelope

Abstract: The structure of the solar convective envelope calculated according to mixing-length theory is not consistent with the transport of the convective heat flux by any superposition of the fundamental linear convective modes. If the amplitudes of these modes can be estimated by consideration of the most important nonlinear terms in the equations of motion, it becomes possible to find solar models which are consistent with mode transport of the heat through the convective zone. These models may be characterized by a mixing length which varies from point to point through the envelope. The linear modes in such models show peaks in velocity at certain preferred length scales, but these scales do not necessarily correspond to those of the solar motions, such as the supergranulation.

Resolution of the Coulomb field into evanescent modes

Abstract: We show that the field of a charge at rest can always be considered as a superposition of evanescent waves of zero frequency. By proposing a two-dimensional Fourier expansion (instead of the three-dimensional one imposed by Landau and Lifshitz), we obtain a development of the Coulomb field in plane evanescent waves of zero frequency. This development is not valid in an arbitrary plane that contains the charge. By proposing a one-dimensional Fourier expansion we obtain a development in cylindrical evanescent waves of zero frequency. This last development is not valid in an arbitrary axis that contains the charge. These expansions enable us to analyze electrostatic boundary-value problems in a novel way.

Quantum dynamics and a semiclassical description of the photon

Abstract: Nonstationary, superposition wave functions are used with computer graphics to reveal the dynamic quantum trajectories of several molecular and electronic transitions. These methods are then coupled with classical electromagnetic theory to provide a conceptually clear picture of the emission process and emitted radiation localized in time and space.

Dirac's superposition of pure states extended to the statistical operators

Abstract: A definition of superposition relation for the statistical operators is proposed which is equivalent to the one of Varadarajan. It is found that the superposition relation is preserved under a general linear dynamics and under tensor product.