Calculation of Diffraction Field Using Incremental Theory of Diffraction for Observation Points Outside the Keller Cone

Fast and accurate calculation of the diffraction field is a challenging problem in computer-generated holography. In this work, we present the performances of three fast algorithms that can be used in accurate computation of the diffraction field from a point cloud object on a display device which has pixelated structure. Performances of the algorithms are evaluated according to computation speed and error on the reconstructed object. First algorithm is based on scaling of a pre-computed one-dimensional (1D) kernel, the second one utilizes a look-up-table (LUT) which is formed by 496 pre-computed 1D kernels, and third one is a hybrid algorithm which uses LUT of the second algorithm with the scaling function employed in the first algorithm. 1D kernels are used to keep the allocated memory space by the LUT in manageable sizes. Third algorithm yields the best error performance compared to other two, but its computation time performance is not as good as the others. Even if, the second algorithm excels the first and the third ones in computation time, it gives the worst error performance among the presented algorithms. The computation times of all three algorithms are improved by having parallel computing.

In this paper, a fast algorithm is proposed for accurate calculation of the scalar optical diffraction on a pixelated optical device used in the reconstruction process from a three-dimensional object that is formed by scattered sample points over the space. In computer-generated holography, fast and accurate calculation of the diffraction field is an important and a challenging problem. Therefore, several fast algorithms can be found in the literature. The accuracy of the calculations can be determined by the signal processing techniques and the numerical methods used in the calculation of diffraction fields. Furthermore, the quality of reconstructed objects can be affected by the properties of optical devices employed in the reconstruction process. For instance, the pixelated structure of those devices has a significant effect on the reconstruction process. Therefore, the pixelated structure of the display device has to be taken into account. Furthermore, fast calculation of the diffraction pattern can be a bottleneck in dynamic holographic content generation. As a solution to the problems, we propose a fast and accurate algorithm based on a precomputed one-dimensional kernel and scaling of that kernel for the computation of the diffraction pattern for a pixelated display.

In his landmark paper, dated February 1962, Prof. Joseph Keller detailed the notion of Keller's cone. He stated, “Geometrical optics, the oldest and most widely used theory of light propagation, fails to account for certain optical phenomena called diffraction....” I had a personal encounter with Keller's cone at a hotel in Florida! In the early morning on October 14, 2007, I witnessed Keller's cone on the door of my hotel room, resulting from edge diffraction from a TV stand due to the sun's rays coming through an opening of a window curtain. By now we know how to construct the local plane-wave behavior of the diffracted field along the diffracted rays, by invoking the fact that an edge forms one of the caustics of the diffracted wavefront. We also know that the diffracted field is of the order of k-1/2, in comparison to the geometrical-optics field, which is of the order of k0. In many antenna and scattering problems, this added diffracted term immensely enhances the accuracy of the total field. Due to the boundary-layer properties of the diffracted ray field along the incident and reflected shadow boundaries and also caustics, the original form of Keller's construction fails. These shadow-boundary shortcomings have been overcome through construction of the Uniform Theory of Diffraction (UTD, by Kouyoumjian and Pathak), the Uniform Asymptotic Theory (UAT, by Ahluwalia, Boersma, Lewis, Lee, and Deschamps), and the Spectral Theory of Diffraction (STD, by Rahmat-Samii, Ko, and Mittra). An overview and the salient features of these theories are revisited in a novel and unified manner, and a representative reflector-antenna example is highlighted.

In his landmark paper dated February 1962, Prof. Keller introduced the notion of Keller's cone and stated, "Geometrical optics, the oldest and most widely used theory of light propagation, fails to account for certain optical phenomena called diffraction...". I had a personal encounter with Keller's cone at a hotel in Florida! In an early morning on October 14, 2007, I witnessed Keller's cone on the door of my hotel room resulting from edge diffraction from a TV stand due to the sun rays coming through an opening of a window curtain. By now we know how to construct the local plane wave behavior of the diffracted field along the diffracted rays by invoking the fact that an edge forms one of the caustics of the diffracted wave front. We also know that the diffracted field is of the order of k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/2</sup> in comparison to the geometrical optics field which is of the order of k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> . In many antenna and scattering problems this added diffracted term enhances the accuracy of the total field immensely. Due to the boundary layer properties of the diffracted ray field, along the incident and shadow boundaries and also caustics the original form of Keller's construction fails. This shortcoming has been overcome through construction of the Uniform Theory of Diffraction (UTD by Kouyoujian and Pathak), Uniform Asymptotic Theory (UAT by Ahluwalia, Beorsma, Lewis, Lee and Deschamps) and the Spectral Theory of Diffraction (STD by Rahmat-Samii and Mittra). An overview and salient features of these theories are revisited in a novel and unified manner.

Fast calculation of diffraction field from three-dimensional objects should be achieved to obtain three-dimensional holographic television. Furthermore by taking into account the pixellated structure of the display device in diffraction field calculation, we can obtian more successful results. Proposed method not only paves the way to compute diffraction field in real-time but also takes into account the pixellated structure of the display device. Both numerical and optical experiments give similar results.

Electromagnetic scattering characteristics change significantly from breaking waves, which is considered to be one reason for sea spike phenomenon（HH polarization scattering intensity close to or even greater than VV polarization scattering intensity). Spiky sea clutter is often treated falsely as targets, which affects radar performance in target detection in the sea surface background. Thus the investigation on the physical mechanism of the sea spike phenomenon can help mitigate false alarms. In this paper, the authors investigate the microwave backscattering from the wedge-shaped breaking waves, which is simulated with the dihedral impedance wedge of finite length. The physical optical field of the breaking waves is calculated with the Kirchhoff approximation. Based on the Maliuzhinets method with using the precise impedance boundary condition, the impedance wedge scattering solution in spectral integral representation is presented. The spectral function is derived by the perturbation method with respect to the oblique incident angle based on the incidence normal to or grazing to the edge. After obtaining the spectral function, the asymptotic theory is used to determine the diffraction field of impedance wedge at an arbitrary skew incidence. The equivalent edge currents are derived from the uniform diffraction of impedance wedge by combining the physical optical coefficients and diffracted coefficients. Backscattering radar cross-sections（RCSs) of the diffracted field from 120 impedance wedge are calculated in both HH and VV polarizations, and the effects of frequency and permittivity on the wedge diffraction are discussed as well. The physical optical field backscattering from 135 impedance wedge is compared with the total field with considering the diffraction effects. Further calculations and analyses for backscattering from the three-dimensional extension breaking waves are presented by using the contribution of edge diffraction field to correct the physical optics field. Numerical results show that the backscattering RCS of impedance diffracted field in HH polarization is greater than that in VV polarization in the Keller cone. Therefore, the diffraction effects will make the backscattering RCS of the total field in HH polarization greater than that in VV polarization when the breaking wave grows to near-collapse stage at a small grazing angle with upwind observation. This indicates that the wedge diffraction is one of the causes of sea spike phenomenon.

Calculation of a three-dimensional (3D) diffraction field from arbitrarily distributed field samples over 3D space is an important problem in holographic 3D television systems. Straightforward superposition of diffracted fields from the samples may not provide accurate calculation of the diffraction field because of the possible mutual couplings between those samples. We define an inverse problem to overcome that deviation caused by mutual couplings. First, the diffraction field on a reference plane is estimated accurately from the known field values at the sampling points. Then, the diffraction field over the entire space can be calculated from the field on the reference plane. Sparse representation of the diffraction field on the reference plane may provide a suitable framework in terms of L1-norm minimization. Once the diffraction field over the reference plane is obtained, the harmonics that form the diffraction field can be calculated. After that, the field over the entire 3D space can be found by using those harmonics. In this work, we proposed a method based on the SPGL1 algorithm that solves the inverse problem of accurate calculation of the diffraction field on the reference plane by using fewer samples in calculation compared to the methods based on L2-norm minimization. Furthermore, the proposed method requires less memory allocation as well.

On the elemination of infinities in the PO component of equivalent edge currents

A new formulation of a time domain incremental theory is introduced. This approach is applied to the scattering of a pulsed plane wave incident on a circular disk. It is shown that the scattered field is free from singularities at caustics and exhibits a notable wave structure outside Keller's cone.

A new formulation of a time domain incremental theory is introduced. This approach is applied to the scattering of a pulsed plane wave incident on a circular disk. It is shown that the scattered field is free from singularities at caustics and exhibits a notable wave structure outside Keller's cone.

Ray tracing (RT) techniques are successful deterministic approaches used in predicting radio propagations. One of the best known RT techniques is Shooting and Bouncing Rays (SBR). SBR can give effective results in especially indoor environments containing obstacles. In this study, a radio propagation prediction in a tunnel with narrow cross-section and obstacles is carried out using SBR. In the SBR technique used, Snell's law and Keller's Cone are used for the geometrical calculations. The electromagnetic loss calculations are carried out with the help of Fresnel's law and UTD (Uniform Theory of Diffraction). The SBR technique is used in a revised 3D simulator environment, called RayLA developed in Java and JOGL. In the study, the signal levels from a source running at 2.4 GHz have been first measured in a real tunnel with narrow-section and obstacles, and the tunnel environment has been modeled in the 3D simulator. Then, the results obtained from the SBR technique have been compared with both the real measurements and the results taken from OMNET++/Castalia which is another simulation platform.

An incremental length diffraction coefficient (ILDC) formulation is presented for the canonical problem of a locally tangent wedge with surface impedance boundary conditions on its faces. The resulting expressions are deduced in a rigorous fashion from a Sommerfeld spectral integral representation of the exact solution for the canonical wedge problem. The ILDC solution is cast into a convenient matrix form which is very simply related to the familiar geometrical theory of diffraction (GTD) expressions for the field on the Keller cone. The scattered field is decomposed into physical optics, surface wave, and fringe contributions. Most of the analysis is concerned with the fringe components; however, the particular features of the various contributions are discussed in detail.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Low-frequency limit is the lowest operating frequency of the compact antenna test range (CATR), which is determined by the electrical size of the specifically designed reflector. Due to the existence of edge diffraction, the size of the reflector must be large enough compared to the wavelength to ensure the quality of the quiet zone (QZ), especially at low frequencies. In this article, a concave serrated-edge reflector (CSER) is presented, which can achieve a new low-frequency limit of 15 wavelengths. Keller cone is utilized to analyze the edge-diffracted rays. The lower level of the diffracted field is achieved in the concave serrated design because these undesired fields are controlled away from the QZ. All results show that the performance of the CSER is superior to other serrated ones.

Holographic near-eye display (H-NED) is one of the most promising technologies in three-dimensional (3D) augmented reality and virtual reality displays due to its ability to provide all depth cues. The numerical calculation of diffraction fields is the basis and key step in the design flow of H-NEDs, and the sampling is the core. In this study, from the perspective of phase space optics, a systematical analysis on the sampling of the wavefield and intensity in H-NEDs is presented. The space-bandwidth product evolution of the wavefield and intensity are given via Wigner distribution function. The sampling criteria, minimum number of samples and conservation law of nonuniform sampling points of the intensity are provided. Such a comprehensive sampling analysis provides guidelines for the correct numerical calculation of diffraction fields in H-NEDs.

Fast calculation of diffraction field from a three-dimensional object is an important problem in holographic three-dimensional television. As a result of this, several fast algorithms can be found in the literature, but most them omit pixelated structure of the display device used in the reconstruction process. We propose a fast algorithm for diffraction field calculation from a three-dimensional object for a pixelated display device. Real-time calculations can also be achieved when the proposed algorithm is run on a graphical processing unit. Performance assessment of the algorithm is obtained by the computation time of the diffraction field and the error on the reconstructed object. The proposed algorithm employs a precomputed look-up-table which is formed by one-dimensional kernels. Each kernel in the look-up-table denotes a diffraction field on the display device from a point light source at a specific depth along longitudinal axis. Allocated memory space by the precomputed look-up-table is optimized according to the sampling policy of the depth parameter. Look-up-table formed by uniformly sampled depth parameter provides the same error on the reconstructed object with less number of kernels. Also, optical experiments are conducted and successful results are obtained.