Circular Reflector Research Articles

Stable and unstable cylindrical-laser-resonator theory in an angular spectrum representation

A new theory is formulated which describes the electromagnetic field inside of a laser resonator through an expansion of the field into an angular spectrum of plane waves. The resonator is composed of two identical, circular reflectors of arbitrary size, focal length, and axial separation. This theory is not limited by the usual paraxial approximations, and provides integral expressions for the spatial distribution of the field over the interior of the resonator in three dimensions. The integral expression for the transverse electric component of the field is evaluated numerically for several of the lower loss modes in confocal, spherical, and unstable resonators with linear dimensions the order of ten wavelengths. These results indicate that some of the modes in confocal and spherical resonators behave in some respects like modes that are unstable if the resonator has dimensions the order of a few wavelengths. The effect on the spatial distribution of the resonator field due to saturation of the amplifying medium is found to be slight if spatial hole burning is neglected.

Fourier Analysis of Spherical Aberrations

Aberration in imaging systems due to the presence of spherical surfaces is usually treated by the expansion of the aberration function in terms of a complete set of polynomials that are orthogonal over the interior of a circle of unit radius. This method involves a set of complicated programs, and the interpretation of the components in the expansion in terms of physical quantities is not possible. To avoid such complications, Fourier transform techniques are used. This method is applied to determine the diffraction pattern of a circular acoustical reflector. It is shown that the amplitude distribution function using this technique and the language of J. W. Goodman's Introduction to Fourier Optics is consistent with experimental results obtained in the laboratory. In particular, the effects of varying the radius of the spherical surface, index of refraction, and wavelength in the reduction of spherical aberration are obvious in the resulting equation. [This work supported by ARPA and monitored by ONR.]

Advantages of a Two-Transducer Frequency Analysis Technique to Characterize Discontinuities

In an earlier paper [J. Acoust. Soc. Amer. 48, 102(A) (1970)], we have shown that the frequency analysis of a broad-band ultrasonic echo may be used to determine the size and orientation of flat circular reflectors. A single transducer was used as both source and receiver. The one-transducer technique is not completely satisfactory for the case of discontinuities having both random shape and orientation. Recently, we have used a second transducer to receive ultransonic pulses scattered by a reflector. Defects in materials were simulated by irregularly shaped metal shims suspended with random orientation in water. The received frequency spectrum when analyzed shows characteristic maxima and minima, which depend on the shape, size, and orientation of the reflectors. A model—which assumes an interference pattern formation of the scattered spherical wavelets at the receiver—relates the spacing between frequency maxima and the positions of the two transducers to the size and orientation of the “discontinuities.” The measurement and the analysis is greatly simplified with the use of the second transducer, and size determination of the discontinuities can usually be made within 10%–20%. [Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation.]

Correction to "Synthesis of the primary fields of a transmitting feed for a circular reflector"

Correction to "Synthesis of the primary fields of a transmitting feed for a circular reflector"

Properties of cylindrical boreholes as reference defects in ultrasonic inspection

The properties of cylindrical boreholes, used as reference defects, lying perpendicular to an ultrasonic beam were examined. The influence of geometrical parameters on the echo amplitude are discussed and the echo amplitude of cylindrical boreholes are compared with the DGS-diagram for circular reflectors.

The efficiency of twin circular flashtube reflectors for laser pumping

An analytical comparison is made between elliptical and circular reflectors used in a twin flashtube cavity for solid state laser pumping. Results show that the transfer efficiency is reasonably independent of cavity radius. Calculations for a constructed laser using the proposed circular reflectors given the drop in efficiency to be only 5% which agrees with measurements to within the experimental error. This easily constructed design also gives good uniformity of pumping.

Synthesis of the primary fields of a transmitting feed for a circular reflector

A two-dimensional electromagnetic boundary-value problem consisting in the synthesis of the primary fields of a transmitting feed antenna for a circular-cylindrical reflector is considered. A primary field that corrects for the phase aberration of the reflector is synthesized as a sum of circular waves whose common origin is displaced from the center of curvature of the reflector. Limitations on the gain of the feed-reflector combination as a function of the number of waves and the displacement are discussed. The synthesized fields are compared with those obtained from geometrical optics.

Fields in the image space of symmetrical focusing reflectors

The fields scattered by circular symmetric reflector illuminated by a linearly polarised wave incident normally on the aperture are calculated from the induced surface currents. It is shown that the fields in the axial region can be represented by a spectrum of near-spherical hybrid waves propagating along the axis. For large microwave focusing reflectors, the wavefronts are effectively plane in the significant part of the image space. The axial wave fields are linear combinations of the TEln and TMln fields appropriate for circular metal pipes, but can be bounded only by anisotropic-reactance surfaces. Axial-wave theory is used to investigate the characteristics of the fields in the focal region of a paraboloidal reflector, when the incident wave is uniform and plane. For radiotelescope focal ratios, the image structure differs significantly from the classical Airy pattern, deduced by scalar analysis, of optical focusing systems. Energy vortexes circulating about the dark rings influence the efficiency obtainable from aperture-type feeds in the focal plane. Application of axial-wave analysis to spherical reflectors, and the synthesis of high-efficiency low-noise feeds, using hybrid-waves in corrugated guides, are described briefly.

Fresnel region approximations for wide angles and large fresnel numbers

A general approximation is developed which permits the accurate computation of radiation patterns from circular apertures and reflectors by single integration both at wide angles and for large Fresnel numbers. Both the far-field and small angle Fresnel region approximations are obtained as special cases. The approximation is applicable to regions where the ratio of the aperture diameter to the distance to the point of observation is small and where this distance is many wavelengths. A comparison is made with the values determined by precise numerical integration for a certain range of parameters and shows good agreement for both the amplitudes and phases of the integrals within the expected range of validity of the approximation.

The efficiency of reflector systems for solid-state lasers

A technique for determining the efficiency of reflector systems is discussed. The efficiency and distribution of the light used for pumping in elliptical and circular cylindrical reflector systems are investigated experimentally. The experimental results are compared with calculations.

Angular Spectra of Optic Cavities*†

The resonant modes of optic interferometer cavities are investigated in the angular spectrum domain. The investigation is based on an integral equation (governing the relation between the normal modes and cavity geometry) which is derived by using the self-consistent Rayleigh formulation for solving diffraction problems. For plane-parallel cavities this integral equation can be solved by means of a series expansion of orthogonal functions characteristic of the cavity geometry without making any assumption about the relative magnitudes of end reflector dimensions and separation. The solution also provides, in addition to a comparison with the solutions of the approximated Huygens’ integral equation as found by other investigators, a direct way for obtaining the angular plane-wave spectrum (or the radiation pattern) of the beam emerging from the cavity. The particular cases of plane-parallel cavities with infinite-strip and circular end reflectors are considered in this paper.

Optical Resonator Modes—Circular Reflectors of Spherical Curvature*

Optical Resonator Modes—Circular Reflectors of Spherical Curvature*

Resonant Modes of Optic Interferometer Cavities I Plane-Parallel End Reflectors*

A study is made of the resonant or normal modes of optic and quasioptic interferometer cavities with plane–parallel end reflectors. The solution of the integral equation governing the relation between the normal modes and the geometry of the cavity is found by means of a series expansion of orthogonal functions. The terms of the series for the normal modes can be interpreted as Fraunhofer diffraction patterns characteristic of the geometry of the end reflectors. Various geometries, such as the infinite-strip, rectangular, and circular end reflector cavities, are considered and the results plotted and interpreted.

Spatial Coverage of Radar Reflectors

Passive reflectors may be employed to enhance the radar return of a space vehicle for some phases of tracking, as in orbital rendezvous. Contour charts prepared on a suitable base grid provide an objective means of evaluating the spatial coverage of different designs for this application. The superiority of a circular corner reflector over square or triangular designs is demonstrated by plotting their backscattering areas in this manner. Comparison can also be made of altogether different types of reflectors, as illustrated by a jack, which provides very poor return relative to the three types of corners.

Near-Field Investigation of a Shaded Transducer

An investigation has been made of the near-field phase and pressure amplitude distribution across the aperture of a shaded line-in-cone transducer. The transducer consists of a line of 16 cylindrical elements mounted within and along the axis of a right circular conical reflector. Data were taken for various numbers and combinations of the 16 cylindrical elements. Computed far-field patterns for these various combinations are compared with their respective patterns measured in the far field. (This work was supported by the Bureau of Ships.)

Aperture-to-medium coupling on line-of-sight paths: Fresnel scattering

An electromagnetic signal propagated through a medium of randomly varying dielectric exhibits random-phase fluctuations. The phase of the total signal is the average of all rays which strike the reflector and is therefore smoothed by a receiver-reflector combination of finite aperture. This paper presents a theoretical analysis and numerical results for this phase-smoothing for small phase perturbations. The receiver is assumed to be a circular parabolic reflector with a collecting feed at the focus. The propagation is described by Fresnel scattering and the one dimensional ray theoretical expressions employed. Closed form results are obtained for three separate space correlation models of the random dielectric medium. These results are valid so long as the scattering parameter L\lambda l_{0}^{-2} is small.

Diffraction by cylindrical reflectors

The electromagnetic diffraction problem is formulated in terms of the determination of the currents induced on a refiector-scatterer by a primary source. All field quantities are readily determined therefrom.An approximate solution ignores the interaction of currents on the refiector-scatterer in that it relates the surface current to the incident magnetic field. This is the geometrical-optics current. For the strip and circular reflector with line source at its centre, corrections to the geometrical-optics currents are obtained by an approximate analytic technique. In each case this yields an equivalent line current at the edges.The field of the line current is non-isotropic; its directional gain is determined by appealing to the Sommerfeld half-plane solution. The line currents do not greatly affect the total pattern; this indicates that the main character of the diffraction can be obtained from the geometrical-optics currents.Measurements of the diffracted field of a circular cylindrical reflector with line feed at its centre were made in a parallel-plane device. A description of some of the features of the equipment is given. The results confirm that geometrical-optics currents are themselves satisfactory, but are inconclusive with respect to the correction line currents.

Radio Equipment for Cross Channel Air Services

IN March 1931 a demonstration was given by the International Telephone and Telegraph Laboratories of radio telegraphic communication across the English Channel on a wave-length of about 17 cm. (see NATURE of April 11, 1931, p. 564.) According to a note in the Electrician and the Electrical Review for November 18, a somewhat similar equipment to that used in the above demonstration has been ordered by the Air Ministry for use in connexion with cross-Channel flying services. This equipment will be manufactured by Messrs. Standard Telephones and Cables Ltd. in their Hendon factory, and it will operate on a wave-length in the neighbourhood of 15 am. The oscillations corresponding to this wave length will be generated by special valves and will be led to the transmitting aerial, which is less than one inch long, situated at the focus of a circular reflector about 10 ft. in diameter. This reflector will be focused on to a similar reflector at the receiving station. The equipment ordered by the Air Ministry will be located at the Lympne air-port, near Hythe, and will operate in conjunction with a similar equipment ordered by the French Air Ministry to be situated at St. Inglevert aerodrome, nearly seven miles south-west of Calais. It will be used for announcing the arrival and departure of aeroplanes that are not fitted with radio, and for routine service messages. An interesting feature of this new service will be the use of teleprinters for both receiving and transmitting messages. In this way typewritten messages will actually be sent across the Channel by radio, thus providing a permanent record at each end. It is expected that the station will be in opera tion early next spring and its use will relieve the volume of traffic at Croydon and Lympne radio stations very considerably.