Comments on "New plane wave spectrum formulations for the near-fields of circular and strip apertures"

The above paper (see ibid., vol.24, p.438-449, July (1976)) applies a plane wave spectrum (PWS) formulation to diffraction problems involving circular and strip apertures and gives new results in terms of Fresnel integrals for the electric field near the aperture. In this note, a discussion of those new results is presented; conclusions are: As a technique for solving electromagnetic aperture diffraction problems, the particular PWS described gives inadequate results, especially for near-fields, and by using the standard Keller's formula, a geometrical theory of diffraction (GTD) solution for the diffracted field from a circular aperture is obtained, but the solution does not in general agree with the one given     

On the aperture and pattern space factors for rectangular and circular apertures

A consequence of a recently discovered edge condition for planar apertures is that all planar aperture distributions are separable physically into a product of an edge factor and an aperture space factor, analogous to the way in which the radiation pattern separates into a product of an element factor and a pattern space factor. An exact relationship between these aperture and pattern space factors for physically realizable vector fields is derived here for rectangular and for circular apertures. For rectangular apertures it leads to a two-dimensional set of doubly orthogonal functions that are characteristic of the aperture geometry. Characteristic functions for circular apertures, however, are shown to exist only if the vector fields are circularly symmetric, although for scalar fields they exist for completely arbitrary aperture distributions with arbitrary edge taper. For rectangular apertures the characteristic functions consist of products of spheroidal functions and for circular apertures they are obtained from a generalization of the spheroidal functions. Some of the properties of these generalized spheroidal functions are developed here.     

高斯光束圆环衍射与光学系统质量评价

根据任意波型的光波的小孔衍射公式计算了高斯光束圆环衍射的光强分布与环围功率分布;分析了以平面波的圆孔衍射分析激光发射光学系统的光学质量所造成的误差。指出:正确地评价光学系统的质量,必须考虑入射波型和光学系统的具体结构。     

Electric and Magnetic Coupling through Small Apertures in Shield Walls of Any Thickness

A method is presented for evaluating the coupling between two identical resonant cavities coupled by a small aperture in a plane common wall of arbitrary thickness. The coupling is related to the frequencies of the symmetric and asymmetric modes of oscillation of the coupled cavity structure, and a variational technique is used to determine those frequencies. The method is applied to circular and rectangular apertures, and it is shown that the coupling is separable into electric and magnetic terms. The results enable theoretical solutions to be obtained for the electric and magnetic polarizabilities of circular and rectangular apertures in walls of zero thickness, and equivalent polarizabilities to be obtained when the wall thickness is nonzero. Curves of numerical values are given for circular and rectangular apertures. With zero wall thickness, the results obtained are the same as those of Bethe for a circular aperture and give good agreement with Cohn's experimental results for rectangular apertures.     

The design of classical offset Dragonian reflector antennas with circular apertures

The vector aperture field of classical offset Dragonian dual-reflector antennas is derived using geometrical-optics concepts. This field then yields the equivalent paraboloid of the geometry. From these results, the conditions for an axially symmetric equivalent paraboloid, when a circular aperture is assumed, are obtained. A complete step-by-step geometrical-optics-based design procedure for optimum classical offset Dragonian antennas with circular apertures is then presented (i.e., zero geometrical-optics cross-polarization and minimum spillover). This procedure is demonstrated by two design examples.     

Electromagnetic coupling to a cable through apertures

Electromagnetic field penetration through apertures is investigated numerically and experimentally. The coupling to a cable located behind an aperture is determined in terms of the open circuit voltage of the cable. A square and a circular aperture are taken as examples, and the open circuit voltage is calculated and verified by an experiment. The apertures are modeled by employing Babinet's principle and solving the complementary flat plate problem using wire-grid modeling. Comparisons with radar cross section measurements were also made to verify the wire-grid modeling.     

Polynomial Approximations for the Transverse Magnetic Polarizabilities of Some Small Apertures

Polynomial approximations are given for the magnetic polarizabilities of some small apertures of various shapes, as functions of the aperture width to length ratios, for the case where the applied magnetic field is across the narrower dimension of the aperture. The shapes considered are the rectangle, diamond, rounded end slot, and ellipse, of which only the last has an exact solution.     

The extended aperture method for the determination of the shadow region radiation of parabolic reflectors

In this paper the problem of the shadow region radiation of reflectors with circular apertures is examined. The approach is based on an extension of the aperture method of calculating antenna patterns. Using this extended aperture method and assuming that the reflector is an absorbing screen, the shadow region pattern within small angles from the axis of the reflector is determined. These methods are applicable only to cases where the wavelength is small with respect to the dimensions of the reflector. The main conclusions are the following: the shadow region field is primarily dependent on the illuminating field which is present on the outer edges of the reflector. For a uniform illumination the front to back ratio is of the order of[(2pi/lambda)rho]^{2}(wherelambdais the wavelength andrhothe radius of the aperture).     

Effect of near field radiators on the radiation leakage through perforated shields

The electromagnetic coupling through an infinite conducting sheet perforated with a finite array of apertures excited by a metal radiator placed in its near field is investigated. The coupling is analyzed numerically by the method of moments (MoM) with an emphasis on understanding the effect of the interactions between the array of apertures and the metal radiator placed in its proximity. The radiation leakage through the array of apertures is found to be significant due to the proximity to the antenna, even if the apertures are electrically small. The effect of a number of other factors like antenna size, antenna type, aperture spacing, and aperture sizes on the extent of radiation leakage occurring due to coupling between the array of apertures and the radiator is also investigated. Numerical results confirm that plane wave theory underestimates the electromagnetic coupling through small apertures by antennas placed in their near field.     

Closed-form low frequency solutions for electromagnetic waves through a frequency selective surface

A simple closed-form approximate solution is given to the problem of transmission of a low-frequency electromagnetic wave through a frequency selective surface (FSS). FSS are periodic metal plates (or their complimentary apertures) sandwiched between dielectric slabs. At low frequencies, the induced currents on the metal plates may be approximated by a known function with a constant coefficient to be determined by the boundary conditions. Based on such a "one-mode" approximation, we derive a closed-form solution for the scattered field for FSS with multiple narrow rectangular slots, with a single wide rectangular aperture, and a circular aperture. When compared with the available exact solutions, we find that the one-mode approximation is valid when the periodaof the FSS is such thata/lambdais small enough that no grating lobe appears, e.g.,a < lambdafor normal incidence.     

Aperture placement effects in oxide-defined vertical-cavitysurface-emitting lasers

A study of the effects of aperture placement on the properties of vertical-cavity surface-emitting lasers (VCSELs) is presented. When thin apertures are placed at the peak of the electric field standing wave optical losses are very high for small apertures. The threshold current increases with decreasing aperture size and two-dimensional diffraction like patterns are evident in the far field. For apertures placed at a node, optical losses appear to be negligible, and loss of optical confinement is apparent for apertures below 2 πm     

Asymptotic behaviour of TE11 mode coupling in circular apertures

An asymptotic formula for mode coupling in a finite planar array of circular apertures, which was derived previously, is compared with the exact solution for the case of TE11 mode coupling in two circular waveguides. It is shown that for dominant mode operation of a circular waveguide the asymptotic formula is accurate enough for most practical purposes, even for aperture spacings of the order of a wavelength.     

New plane wave spectrum formulations for the near-fields of circular and strip apertures

The plane wave spectrum (PWS) method has previously been applied to analyze the near-field of planar apertures. The main goal of this paper is to present new PWS formulations for the near-fields of strip and circular apertures. Only special cases are developed in detail. For example, the uniform and parabolic aperture distributions are developed for the circular aperture. These new formulations are expressed in terms of either elementary functions or Fresnel integrals. Consequently, they permit considerably more rapid and efficient calculations than previous near-field formulations, by either the PWS or the aperture integration approach. The new formulations are especially advantageous for large circularly symmetric apertures (on the order of100lambdaand larger) in that computational efficiencies are improved by an order of magnitude or two over the original PWS formulation. The improvement over aperture integration techniques is more than a factor of 1000 for the100lambdaaperture.     

The transmission coefficient of elliptical and rectangular apertures for electromagnetic waves

A simple scalar method using Kirchhoff's boundary values is applied to the diffraction problems of circular, elliptical, and rectangular apertures for normally incident electromagnetic waves. As far as circular apertures are concerned, a simple formula can be derived not only for the diffraction pattern but also for the transmission coefficient. This formula yields good results for apertures greater than0.8lambda. Even in the ease of elliptical apertures a simple formula can he derived for the diffraction pattern. For the elliptical aperture, as well as the rectangular one, the transmission coefficient was found in the form of an integral. Relief models and diagrams are given for the transmission coefficients of the elliptical and the rectangular apertures as a function of the two aperture parameters. Diagrams are given which explain the dependence of the transmission coefficient on the aperture parameters. A comparison with other more complicated methods of approximation and with measurements shows both good agreement and the great practical value of the simple method of approximation used.     

Electromagnetic transmission through multiple circular apertures in a thick conducting plane

The boundary-value problem of electromagnetic wave scattering from multiple circular apertures in a thick conducting plane is rigorously solved. The eigenfunction expansion, integral transform, and superposition principle are utilized to represent the scattered field in the discrete and continuous modes. The boundary conditions are enforced to obtain a set of simultaneous equations for the discrete modal coefficients. The transmission coefficient is represented in a fast convergent series. Computation is performed to illustrate the behavior of transmission and coupling through multiple circular apertures in terms of the aperture geometry.     

Radiation patterns of circular apertures with Gaussian illumination

We analyze the radiation pattern from circular apertures with Gaussian electric field distribution. Patterns are calculated using the Fraunhofer diffraction integral and we obtain a convenient expression for the FWHM beamwidth as a function of the edge taper. The aperture efficiency can be calculated for blocked and unblocked apertures in analytic form. The beam efficiency must be computed numerically, but can be obtained in a straightforward way from the beamwidth.     

Mutual coupling between two small circular apertures in aconducting screen

With the use of the reaction integral and two-dimensional Fourier transform, an analytical expression for mutual coupling between two small circular apertures in a conducting screen, excited by a normally incident plane electromagnetic wave, is obtained. Numerical examples for two different polarizations of the plane wave are investigated. The expression for the mutual admittance gives a correct value of the self-admittance of a small aperture when the distance between the holes is equal to zero     

Time Domain Analysis of the Near-Field Radiation of Shaped Electrically Large Apertures

The near-field radiation of several electrically large apertures has been computed based on the aperture field convolution method, including a square aperture, a circular aperture, and a serrated edge aperture. The frequency range is 8.0–12.0 GHz. The aperture diameters are limited to 5.00 m and the corresponding electrical size is 133–200 wavelengths. The time domain spectrum has been obtained by chirp z-transform with a Hamming window and the mechanism for the generation of the aperture near-field radiation in time domain is analyzed. The calculation results demonstrate that the near-field radiation of the aperture can be approximately seen as the synthesis of a plane wave from the aperture, cylindrical waves from the edges, and spherical waves from the corners. By this method, the direction, position, and magnitude of incoming waves in the near-field region can be estimated, and the aperture design can be modified to meet the requirements of the near-field radiation. The near field of the three apertures mentioned above has been compared, and it is shown that the serrated edge aperture has more uniform direct wave and lower diffraction waves, making it a candidate for compact range (CR) aperture design.      

Hybrid thin-slot algorithm for the analysis of narrow apertures infinite-difference time-domain calculations

A technique to incorporate a half-space aperture integral equation into a finite-difference time-domain (FDTD) code based on the offset Yee mesh (see K.S. Yee, ibid., vol. AP-14, p.302-7, 1966) is presented. To introduce the technique, linear apertures that are electrically narrow in both width and depth are discussed. The method incorporates an independent time-marching solution for the aperture problem into the FDTD code so that the aperture formally does not exist within the main FDTD mesh. A feedback scheme is introduced so that full exterior and interior coupling is included in the aperture solution. The technique is particularly useful for the analysis of apertures that are narrow both in width and depth with regard to the FDTD spatial cell. Previous thin-slot methods are shown to significantly underestimate the transverse gap electric field for this case, and an explanation for this is provided with the aid of the hybrid algorithm     

On using a sense wire to quantitate the magnetic flux leakagethrough an aperture in an electromagnetic shield

A formulation is developed for determining the sensitivity of positioning a sense wire behind an aperture in an electromagnetic shield to quantitate the flux leakage through the aperture. Both hardened and unhardened circular apertures are considered. Comparisons with measured data are made to verify the analysis