Complex Plane Wave Research Articles

Generation of a polarized optical field from a given scalar field for wide-viewing-angle holographic displays

Although the scalar wave theory does not fully describe the nature of the light, which is a vector wave field, it is common to characterize the optical field propagated from an object as a scalar wave in 3DTV research. A given scalar wave field is usually directly mapped to the transverse field components of a simply polarized optical field during the generation of the optical field by a holographic 3D display; this procedure gives accurate results in terms of the optical intensity if the field is paraxial. However, due to the nonnegligibly large longitudinal component of the vector field, this method fails in wide-angle fields. In order to generate a polarized optical field which has the same resultant effect as prescribed by a wide-angle scalar field, we extend this simple polarization approach to tilted and rotated planes. That is, we first put a constraint on the generated vector wave field such that each plane wave component has a simple polarization relation at its locally transverse plane. Then, we map the complex plane wave amplitudes of a given scalar field to the plane wave amplitudes of locally transverse field components of the polarization-constrained optical field. As a consequence of the developed mapping, for an observer which is located at a tilted and rotated plane and captures a locally paraxial segment of the incoming wave field, the intended scalar intensity results can also be obtained in wide-viewing-angle 3D holographic displays. The developed model is valid if the chosen local coordinate frames of the locally transverse planes of the propagating plane waves vary slowly as the propagation direction changes. For a 3DTV setup where the display is located at a z-plane and the observers are located away from the z-axis, an analytical representation for appropriate coordinate frames is developed. For the same setup, computer simulation results showed that the excessive amplification of the longitudinal component due to the conventional mapping does not arise in the proposed method and the desired scalar results are observed. Moreover, since the determined polarization state is preserved at each tilted and rotated plane, the new model can also be used in an optical setup where the polarization information is important. As a result, the proposed mapping enables the scalar wave theory to be used in wide-viewing angle holographic display applications under the given constraints.

Complex ray in anisotropic solids: Extended Fermat's principle

In contrast to homogeneous plane waves, solutions of the Chris-toffel equation for anisotropic media, for which a determined number of can be observed in a fixed direction of observation, inhomogeneous plane waves provide a continuum of rays that propagate in this direction. From this continuum, some complex plane waves can be extracted for verifying a definition of quasi-arrivals, based on the condition that the time of flight would vary the less in extension to the Fermat's principle that stipulates a stationary time of flight for wave arrivals. The dynamic response in some angular zones contain prominent, although not singular, features whose arrivals cannot be described by the classical ray theory. These wave packet's arrivals can be described by quasi-fronts associated to specific inhomogeneous plane waves. The extent of the phenomena depends on the degree of anisotropy. For weak anisotropy, such quasi-fronts can be observed. For strong anisotropy, the use of inhomogeneous plane waves, due to their complex slowness vector, permits a simple description of quasi-arrivals that refer to the internal diffraction phenomenon. Some examples are given for different wave surfaces, showing how the wave fronts can be extended beyond the cuspidal edges for forming closed wave surfaces.

Geometry and scaling of tangled vortex lines in three-dimensional random wave fields

The short- and long-scale behaviour of tangled wave vortices (nodal lines) in random three-dimensional (3D) wave fields is studied via computer experiment. The zero lines are tracked in numerical simulations of periodic superpositions of 3D complex plane waves. The probability distribution of local geometric quantities such as curvature and torsion are compared to previous analytical and new Monte Carlo results from the isotropic Gaussian random wave model. We further examine the scaling and self-similarity of tangled wave vortex lines individually and in the bulk, drawing comparisons with other physical systems of tangled filaments.

Strongly degenerate homogeneous pseudo-Kähler structures of linear type and complex plane waves

We study the class K2+K4 of homogeneous pseudo-Kähler structures in the strongly degenerate case. The local form and the holonomy of a pseudo-Kähler manifold admitting such a structure are obtained, leading to a possible complex generalization of homogeneous plane waves. The same question is tackled in the case of pseudo-hyper-Kähler and pseudo-quaternion Kähler manifolds.

Model Independent MRE Data Analysis

For the diagnosing modality called MRE (magnetic resonance elastography), the displacement vector of a wave propagating in a human tissue can be measured. The average of the local wavelength from this measured data could be an index for the diagnosing, because the local wave length becomes larger when the tissue is stiffer. By assuming that the local form of the wave is given approximately as multiple complex plane waves, we identify the real part of the complex linear phase of the strongest plane wave of this multiple complex plane waves, by first applying the FBI transform (Fourier-Bros-Iagolnitzer transform) with an appropriate size of Gaussian window and then taking the maximum of the modulus of the transform with respect to the Fourier variable. The real part of the linear phase is nothing but the real inner product of the wave vector and the position vector. Similarly the imaginary part of the linear phase describes the attenuation of the wave and it is given as a real inner product of a real vector and the position vector. This vector can also be recovered by our method. We also apply these methods to design some denoising and filtering for noisy MRE data.

Detection and extraction of orientation-and-scale-dependent information from two-dimensional GPR data with tuneable directional wavelet filters

Abstract The Ground Probing Radar (GPR) is a valuable tool for near surface geological, geotechnical, engineering, environmental, archaeological and other work. GPR images of the subsurface frequently contain geometric information (constant or variable-dip reflections) from various structures such as bedding, cracks, fractures, etc. Such features are frequently the target of the survey; however, they are usually not good reflectors and they are highly localized in time and in space. Their scale is therefore a factor significantly affecting their detectability. At the same time, the GPR method is very sensitive to broadband noise from buried small objects, electromagnetic anthropogenic activity and systemic factors, which frequently blurs the reflections from such targets. This paper introduces a method to de-noise GPR data and extract geometric information from scale-and-dip dependent structural features, based on one-dimensional B-Spline Wavelets, two-dimensional directional B-Spline Wavelet (BSW) Filters and two-dimensional Gabor Filters. A directional BSW Filter is built by sidewise arranging s identical one-dimensional wavelets of length L , tapering the s -parallel direction (span) with a suitable window function and rotating the resulting matrix to the desired orientation. The length L of the wavelet defines the temporal and spatial scale to be isolated and the span determines the length over which to smooth (spatial resolution). The Gabor Filter is generated by multiplying an elliptical Gaussian by a complex plane wave; at any orientation the temporal or spatial scale(s) to be isolated are determined by the wavelength. λ of the plane wave and the spatial resolution by the spatial aspect ratio γ , which specifies the ellipticity of the support of the Gabor function. At any orientation, both types of filter may be tuned at any frequency or spatial wavenumber by varying the length or the wavelength respectively. The filters can be applied directly to two-dimensional radargrams, in which case they abstract information about given scales at given orientations. Alternatively, they can be rotated to different orientations under adaptive control, so that they remain tuned at a given frequency or wavenumber and the resulting images can be stacked in the LS sense, so as to obtain a complete representation of the input data at a given temporal or spatial scale. In addition to isolating geometrical information for further scrutiny, the proposed filtering methods can be used to enhance the S/N ratio in a manner particularly suitable for GPR data, because the frequency response of the filters mimics the frequency characteristics of the source wavelet. Finally, signal attenuation and temporal localization are closely associated: low attenuation interfaces tend to produce reflections rich in high frequencies and fine-scale localization as a function of time. Conversely, high attenuation interfaces will produce reflections rich in low frequencies and broad localization. Accordingly, the temporal localization characteristics of the filters may be exploited to investigate the characteristics of signal propagation (hence material properties). The method is shown to be very effective in extracting fine to coarse scale information from noisy data and is demonstrated with applications to noisy GPR data from archaeometric and geotechnical surveys.

The properties of a dynamo wave in the case of intense meridional circulation

The solution of a Parker dynamo was constructed for the case of intensive meridional circulation using a method similar to WKB. We show how to build a solution in the transition from the traveling-wave regime to the regime of stationary magnetic field configuration. A solution of the Hamilton-Jacobi equations was discovered for the problem of a dynamo containing a triple point in the complex plane wave vector.

Comparison of methods for computing interference elastic waves in thin-layered media. 2

The paper is an immediate continuation of the paper where the solution of the problem on the propagation of low-frequency waves in thin-layered media by the dispersion equation method was considered in detail. In the present article, the solution of a similar problem is given for an elastic layer and a half-space, which are in rigid contact, by the method of superposition of complex plane waves. Bibliography: 17 titles.

Geoacoustic inversion of short range source data using a plane wave reflection coefficient approach

Acoustic time series data were collected in a shallow, hard bottom lake environment located in central Texas using both short range (2 m) implosive data, obtained with the source and a single hydrophone located near mid-depth in the waveguide, along with longer range implosive and explosive data from a near surface source to a bottom mounted hydrophone. Matched field inversions using simulated annealing were performed with a ray trace plus complex plane wave reflection coefficient forward propagation model that was validated in previous work. Isolating bottom interacting paths to perform the inversions is shown to be essential to reduce parameter uncertainties in the hard bottom environment and enables a systematic approach to the inversions which establishes the number of layers needed to represent the lake environment. Measured transmission loss data from a towed source were compared through a RMS error analysis to modeled transmission loss, constructed with the parameters from inversions of data from several source types, to further establish the validity of the inversion approach for this environment. Geoacoustic parameters obtained by inversions of short range, low frequency impulsive data are used to predict transmission loss at longer ranges and higher frequencies. The range dependence of the global minimum is discussed.

Numerical study of geoacoustic inversion in range-dependent ocean environments

Simulated acoustic data for two-dimensional waveguides were used to estimate the properties of a sea bed over which the depth of the water column varies with range. The simulated data were produced by a parabolic equation method. No approximate knowledge of the local sea-bed representation or parameter bounds was given prior to the inversions. In addition, little approximate information was given on the nature of the range dependence of the sea bed. The inversion methodology used a simulated annealing approach and a variation of the algorithm that generated the original acoustic data. For the purpose of consistency, a second method was employed, which used the same simulated annealing approach but with a range-dependent ray approach in which the sea-bed interaction is described by a complex plane wave reflection coefficient. Various combinations of frequencies and source-receiver positions commonly used in real acoustic measurements at sea were selected for the inversions. The goal of the inversion was to efficiently achieve a coarse description of the environment by several inversions; short-range data at low frequencies were employed for the purpose of efficiency. In the later stages of the inversion process, data from longer ranges and higher frequencies were included to provide additional details of the sea bed. Sea-bed parameters were assumed to vary linearly with range between geoacoustic profiles defined at specific ranges. Inversion results obtained prior to the reporting of the actual solutions are presented.

A method of solving plane initial and boundary-value problems of the dynamic theory of elasticity

A method for solving plane initial and boundary-value problems (1BVP) in the dynamic theory of elasticity (DTE), in which the solutions are represented in the form of a continuous planar superposition of arbitrary analytic functions which are complex plane waves of arbitrary shape is described. By representing the solutions in this form it is possible to use the Radon transformation, by means of which plant: lBVP of the DTE reduce to boundary-value problems of the theory of functions of a complex variable: in the simplest cases of those of Dirichlet or Keldysh-Sedov, and in more complicated cases to Riemann-Hilbert or Riemann ( matching) problems. The method is illustrated on some basic IBVP of the DTE in a half-plane. An analytic solution for the basic mixed problem of the DTE, in which there is no boundary condition that applies over the entire infinite interval of the half-plane boundary, is :found for the fast time.

The reflectivity method with polar sampling

AbstractA modification to the reflectivity method that results in more accurate and efficiently computed theoretical seismograms is introduced. The reflectivity method involves an explosive point source and a radially symmetric earth, and in the temporal frequency domain the displacement field and the complex plane wave reflectivity are, to within a phase factor, Hankel transform pairs in the polar coordinate variables radial offset and wavenumber. This motivates formulation of the time-transformed displacement field into a Fourier-Bessel series over the reflectivity function, and replacement of the traditional method for sampling the plane-wave reflectivity with the proper polar sampling scheme, based on the finite Fourier transform representation for a band-limited signal. Several numerical examples are discussed.