Incomplete Cylindrical Functions Research Articles

Fresnel diffraction from sector apertures

The derivation of the analytic representation of the diffraction field from the sector aperture in the Fresnel zone is presented. The new solution is expressed in the form of a convergent series on incomplete cylindrical functions in the Poisson form that makes it possible to calculate and obtain three-dimensional patterns of the field amplitude in the whole image plane. The obtained patterns correspond to the Fresnel approximation and are true for arbitrary values of the opening angle and for the radius of sectoral aperture.

Analytic computation of line-drawn objects in computer generated holography

Digital holography is a promising display technology that can account for all human visual cues, with many potential applications i.a. in AR and VR. However, one of the main challenges in computer generated holography (CGH) needed for driving these displays are the high computational requirements. In this work, we propose a new CGH technique for the efficient analytical computation of lines and arc primitives. We express the solutions analytically by means of incomplete cylindrical functions, and devise an efficiently computable approximation suitable for massively parallel computing architectures. We implement the algorithm on a GPU (with CUDA), provide an error analysis and report real-time frame rates for CGH of complex 3D scenes of line-drawn objects, and validate the algorithm in an optical setup.

On closed-form expressions for the approximate electromagnetic response of a sphere interacting with a thin sheet — Part 1: Theory in the frequency and time domain

In mineral exploration and geologic mapping of igneous and metamorphic terranes, the background is often dominantly resistive. The most important electromagnetic interaction is between a discrete conductor and an overlying sheet of conductive overburden (e.g., glacial clays or weathering products of the basement rocks). To enable the electromagnetic modeling of these common situations, here I provide closed-form expressions for the approximate electromagnetic response of a sphere embedded in highly resistive rocks and interacting with an overlying thin sheet. The sphere is assumed to be dipolar and excited by a locally uniform field. The expressions in the time and frequency domains are represented as sums of complete and incomplete cylindrical functions. New asymptotic approximations are provided for the efficient evaluation of the required incomplete cylindrical functions. The frequency-domain formulas are validated by numerical transformation to the time domain and comparison to the time-domain solution.

Physical Optics Curved-Boundary Dielectric Plate Scattering Formulas for an Accurate and Efficient Electromagnetic Characterization of a Class of Natural Targets

Closed-form representations of the physical optics (PO) field scattered in the far zone by plane penetrable dielectric angular sectors of arbitrary opening angle featuring a conical-section boundary are derived in terms of incomplete cylindrical functions (ICFs). The proposed expressions, possibly in combination with PO formulas for the scattering from polygonal plates, allow one to evaluate the scattering from flat dielectric plates with both convex and concave curved edges in a very efficient manner and constitute a useful tool to improve the accuracy of the geometrical characterization of a class of natural scatterers. In order to check the correctness and computational effectiveness of the proposed scattered-field solutions, comparisons with results obtained by accurate completely numerical PO calculations are provided. Simulation data by the method of moments are also presented to assess the applicability of the PO approximation to the considered sample scattering geometries.

Closed‐form physical optics expressions for the radar cross section of perfectly conducting plane angular sectors

AbstractClosed‐form expressions for the physical optics (PO) field scattered in the far zone by perfectly electrically conducting (PEC) plane angular sectors with a conical‐section boundary are derived in terms of incomplete cylindrical functions (ICFs). The developed representations are very efficient from a computational point of view and constitute a useful tool to analytically predict the radar cross section (RCS) of metal plates with both convex and concave curved edges. The correctness of the proposed scattered field formulas is demonstrated through comparisons with the results obtained by an accurate completely numerical PO approach. A sample analysis of the scattering from a more elaborate geometry, which can be seen as a juxtaposition of angular sectors, is also presented. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 50: 160–165, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22989

Exact Closed-Form Expression of a Sommerfeld Integral for the Impedance Plane Problem

A Sommerfeld integral for an impedance half-plane is one of the classical problems in electromagnetic theory. In this paper, the integral is evaluated into two series representations, which are expressed in terms of the exponential integral and the Lommel function, respectively. It is shown that the series expansions absolutely converge for any parameters, such as distance or surface impedance. Then based on the Lommel function expansion, an exact closed-form expression of the integral is formulated. This expression is written in terms of incomplete Weber integrals, which are directly related to incomplete cylindrical functions such as incomplete Lipschitz-Hankel integrals. Additionally, a complete asymptotic series of the integral is obtained based on the exponential integral expansion. With conventional methods such as steepest descent method, it is cumbersome to derive the divergent series. The validity of all the formulations derived in this paper is demonstrated through a comparison with a numerical integration of the integral for various situations

Series expansions for the incomplete Lipschitz‐Hankel integral Je0(a, z)

Bessel series expansions are derived for the incomplete Lipschitz‐Hankel integralJe0(a, z). These expansions are obtained by using contour integration techniques to evaluate the inverse Laplace transform representation for Je0(a, z). It is shown that one of the expansions can be used as a convergent series expansion for one definition of the branch cut and as an asymptotic expansion if the branch cut is chosen differently. The effects of the branch cuts are demonstrated by plotting the terms in the series for interesting special cases. The Laplace transform technique used in this paper simplifies the derivation of the series expansions, provides information about the resulting branch cuts, yields integral representations for Je0(a, z), and allows the series expansions to be extended to complex values of z. These series expansions can be used together with the expansions for Ye0(a, z), which are obtained in a separate paper, to compute numerous other special functions, encountered in electromagnetic applications. These include: incomplete Lipschitz‐Hankel integrals of the Hankel and modified Bessel form, incomplete cylindrical functions of Poisson form (incomplete Bessel, Struve, Hankel, and Macdonald functions), and incomplete Weber integrals (Lommel functions of two variables).

Uniform asymptotic expansions of a class of incomplete cylindrical functions

For the uniform asymptotic expansio of Incomplete Cylindrical Functions of Bessel form a modification of Bleistein's general procedure is given. It offers the convenience of completely separating the processes of determining the expansion coefficients and of repeatedly integrating by parts. This is of practical relevance with a view to delegating the otherwise very tedious calculation of higher expansion coefficients to one of the available computer codes for the algebraic-analytic manipulation of given expressions.

Theory of Incomplete Cylindrical Functions and Their Applications

Theory of Incomplete Cylindrical Functions and Their Applications

Evaluation of incomplete cylindrical functions

Evaluation of incomplete cylindrical functions

Tables of Incomplete Cylindrical Functions

Tables of Incomplete Cylindrical Functions