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In this paper the geometrical theory of diffraction is extended to treat the radiation from apertures or slots in convex perfectly conducting surfaces. It is assumed that the tangential electric field in the aperture is known so that an equivalent infinitesimal source can be defined at each point in the aperture. Surface rays emanate from this source which is a caustic of the ray system. A launching coefficient is introduced to describe the excitation of the surface ray modes. If the field radiated from the surface is desired, the ordinary diffraction coefficients are used to determine the field of the rays shed tangentially from the surface rays. The field of the surface ray modes is not the field on the surface; hence if the mutual coupling between slots is of interest, a second coefficient related to the launching coefficient must be employed. In the region adjacent to the shadow boundary, the component of the field directly radiated from the source is represented by Fock-type functions. In the illuminated region the incident radiation from the source (this does not include the diffracted field components) is treated by geometrical optics. This extension of the geometrical theory of diffraction is applied to calculate the radiation from slots on elliptic cylinders, spheres, and spheroids.

A servo controlled Fabry-Perot interferometer using capacitance micrometers to sense departures from parallelism and variations in the mean spacing of the Fabry-Perot plates is described. The parallelism and mean spacing of the plates is maintained over the full 75 mm aperture of the instrument to an accuracy better than the surface quality of the plates ( approximately lambda /150 at 500 nm).

A multielement high power monopulse feed was investigated for excitation of a phased array. This feed has the advantages of high antenna efficiency, effective independent sidelobe control for the sum and difference patterns, and high power handling capability. A 32-element device has been designed which yields theoretical sidelobe levels less than -31 dB for both the sum and difference patterns over the 2800 to 3200 MHz frequency band. The aperture efficiency is 69.5 percent which includes 0.1 dB spillover loss and 1.40 dB tapering loss. The difference slope efficiency defined as the actual slope gain over the maximum possible value is 50 percent at midband. An experimental feed was constructed to verify the predicted performance characteristics. Measured data on both the radiation patterns and the antenna gain agreed very Well with the analytical results. The measured sidelobe level is below -30 dB for all sum and difference beams over the 2800 to 3200 MHz. In addition, this feed can be designed for low power intensity at the feed aperture so that very high power operation can be attained.

A model is proposed for the gravity flow of particles that is particularly applicable to cohesive powders. The flow criterion is first estimated by describing the distribution of solids pressure within a vessel and by calculating the ‘flowability’ of a cylindrical block above the aperture. A flow rate equation is derived from a dynamic force balance on the block which gives an acceleration and a velocity profile in the discharging block. The seemingly steady flow is in fact a rapidly fluctuating one as the loading switches from one state to another. Computed results from the model explain several well known phenomena in particle discharge from hoppers and the method described gives a clear basis for design.

SUMMARY An annoying phenomenon in many low power electron microscope negatives is a centrally located overexposed spot. It is shown that its occurrence is associated with the use of thin-foil objective apertures, and that the spot most probably is produced by electrons permeating the thin-foil material. The overexposed spot can be avoided either by using a concentrated nearly focused illuminating beam, in practical terms by using a short exposure time, or by using a thick, disc-shaped objective aperture.

A distributed RC circuit model is used to describe an interdigital electrode transducer (IDT) having finite conduction in the electrode stripes. The distributed circuit is described by a set of differential equations whose solutions yield the current and voltage distributions along the aperture of the IDT. It is found that the electrode resistance causes a distortion of the excited wave amplitude and phase due to the nonuniform voltage and current distribution. An equivalent circuit for the terminal properties is also derived which illustrates the effects of conduction loss. The theory is also used to predict electrode efficiency, effective aperture weighting, and phase shift in weighted arrays.

The quality of the reflecting surface and the mechanical perfection of the bending mechanism are factors which affect the performance of curved glass reflectors in optically focusing small-angle X-ray scattering cameras. The intensity of the reflected beam can be enhanced by gold-coating the reflector, both because of the higher reflectivity of gold and because larger grazing incidence angles are possible, thus increasing the optical aperture. Coating imperfections and the use of increased incidence angles are factors which increase the parasitic scattering. A new bending mechanism is described in which two 200 mm long reflectors are arranged in series to form a reflector of effective length of nearly 400 mm.

Calculations have been performed on the contrast available in thick specimens. Ten modes of operation have been considered, six in the conventional electron microscope (CEM) and four in the scanning transmission electron microscope (STEM). Electrons passing through the specimen fall into four categories, elastically scattered, inelastically scattered, unscattered, and those scattered both elastically and inelastically. For the various operation modes these groups are taken in combinations to form practical signal intensities. In the calculation of practical intensities, the angular distribution of plurally scattered electrons is considered. The fraction of each scattered group passing through the microscope aperture has been calculated for various types of illumination. The practical intensities are used to calculate the contrast available in stained and unstained specimens. For very thick specimens bright field offers the best combination of contrast and signal intensity. Other contrast modes show a reve...

The simultaneous correction of third order aperture aberration and axial chromatic aberration is studied theoretically to determine practical upper limits of the accelerating voltage. Aberrations of a round objective lens are compensated by a quadrupole-octopole corrector similar in design to the Archard, Deltrap and Hardy systems. Severe tolerances on lens stability, large electrostatic fields in the quadrupoles, and chromatic aberration of magnification are the primary factors that adversely affect the performance of these systems at high voltage. The variational calculus is used to compute correctors with low electrostatic fields in the quad­rupoles. When lens stability requirements are also considered, the general configuration of high performance systems is deduced. In addition to internal corrector structure, five parameters including voltage and magnification are used to define a system. Optical and field properties are calculated for a wide variety of these parameters. It is concluded that chromatic correction is feasible for accelerating voltages up to 500 kV. Lens stability requirements become very severe for beam energies greater than the electron rest energy, 511 keV. Tabular results are given for systems designed for resolutions of 1, 0.5 and 0.1 nm. In these examples lens stability requirements are one part in 5 x 105, 106and 5 x 106re­spectively. Configuration changes and their effects on stability and field of view are discussed.

A new interdigital electrode transducer geometry has been conceived and its theoretical performance verified experimentally. The transducer is composed of sets of electrodes which are connected in series through an offset or "dog-leg" electrode, thereby significantly increasing the electrical impedance of the transducer over that of a conventional one having the same aperture. This transducer radiates and receives a uniform straight-crested wavefront. The transducer is shown to have unique impedance properties suited for wide-aperture surface-wave devices or those requiring some form of wave-amplitude weighting. Experiments have been conducted on YZ LiNbO/sub 3/ which verify that the terminal impedance of the transducer is proportional to the square of the number of equal-aperture constant-amplitude sections.

A system is proposed for improved sensitivity, resolution and depth delineation in gamma -ray cameras. An array of apertures is used to provide different views of the radiating source. This aperture plate is between the source and the gamma -ray detector. Each aperture is mechanically modulated at a different frequency so that its resultant image can be separated from the others. On reconstruction the individual images are demodulated by using the appropriate frequency. The individual images are translated and combined to view any section of the radiating source. The noise problem and the requirements for orthogonality of the modulating signals are considered.

Expressions have been derived for the transfer function of a circular aperture suffering from astigmatism and operating in non-uniform illumination. The intensity distribution considered is given as t(x, y) = 1 - A(x 2 +y 2). The modifications in the optical transfer function by changing A, the apodization parameter, has been shown graphically.

Interferometers based on the amplitude division principle, the primary example being the Michelson, with their multiplex and throughput advantages, are now the classical instruments for infrared spectroscopy. In seeking designs to optimize the optical efficiency of astronomical telescopes in feeding amplitude and phase modulated Michelson interferometers, wave optical solutions for shaped three-mirror systems have been found which offer mechanical compactness, while preserving plane wavefronts and maximizing the energy falling coherently on the final detector. For the sub-mm region, where the ratio λ/D of wavelength to aperture is not small compared to unity, our solutions are valuable in providing shaped surfaces which minimize diffraction losses by providing a 20 dB Gaussian taper on the plane fronts through each interferometer. The numerical methods described incorporate an original treatment of the reflection of radiation at a 45° off-axis paraboloid where the beam divergence is such that Fresnel diffraction cannot be neglected.

The Sparrow criterion of resolution is used to obtain an expression for the limit of resolution of a circular aperture, with non-uniform illumination by partially coherent light. Also the image intensity distributions for the two partially coherent points with different separations are obtained.

In a recent paper the author proposed a self-focusing theory based on linear stability analysis of the quasi-optical equation [1]. Strictly, the theory applies only to a plane wave of infinite aperture. In this paper the theory is extended to beams of finite aperture, which may be initially convergent or divergent.

Theoretical consideration is given to the error of the optical transfer function caused by noise and truncation of the measured line spread function. The error due to noise is shown to be an increasing function of the scanning range of the line spread function. An optimum scanning range is found to exist that minimizes the total error resulting from noise and truncation. Numerical calculation is presented for the case of an aberration-free circular aperture and the change in total error with varying scanning range and noise level is illustrated.

Characteristics of the cross slot coupling aperture applicable to circularly polarized equalizer networks is presented. This coupling mechanism is analyzed and experimental results indicate good agreement between theory and practice. Extension of the single-cavity unit to multiple direct-coupled dual-mode equalizer networks is also discussed.

A pulsed electron paramagnetic resonance spectrometer is described, and applications of the instrument to the determination of longitudinal relaxation times of dilute solutions of free radicals and of spin‐labeled proteins using the saturation‐recovery method are discussed. The spectrometer employs a bimodal cavity; the pumping and observing microwave powers are coherent; and all modulating frequencies, delays, and aperture widths are derived from a master clock. Free‐induction‐decay signals may be observed in such equipment, and special techniques are introduced to avoid interference of these signals with the saturation‐recovery signals.

This paper describes recent advances in the development of high-thrust tubular colloid sources for satellite applications. Single emitter technology has advanced to the point where steady reliable performance has been demonstrated at greater than 111 /zN (25 yJb) per unit, 75% time-of-flight efficiency, and 1300 sec specific impulse for periods on the order of 100 hr. The most significant findings were the importance of orienting the emitter vertically in order to equalize gravitational effects around the emitter rim, the need to enlarge the extractor aperture, and the development of special tooling techniques for manufacturing emitters. A four-source module was subjected to extensive performance testing in addition to runs of 100 hr and 360 hr. Long-term steady performance was demonstrated at several levels up to 547 /*N at 75% time-of-flight efficiency and 1325 sec specific impulse. The ability to neutralize the beam without interfering with thruster performance was also demonstrated. Beam probe measurements showed that 90% of the beam was contained within a 15° half angle. The thruster's ability to provide constant performance to within ±3% was also demonstrated. During short-term testing at high-feed pressures, the thruster produced 1339 /iN at 70% efficiency and 1029 sec specific impulse.

A study of the field distribution in a square aperture on a plane conducting screen illuminated by a plane electromagnetic wave at normal incidence is presented. In solving the problem, the square aperture is first formulated into a complementary square plate scattering problem. The distribution of current on the plate is then found by solving the corresponding wire mesh model with the aid of a computer program for linear wire antennas. A simple averaging process is then introduced in a straightforward manner to convert the currents flowing along the wires into continuously distributed equivalent surface currents on the square plate. The surface current density is, in turn, equal to the value of magnetic field intensity by the boundary condition for a perfect conductor. Finally, a direct application of the electromagnetic form of Babinet's principle gives the electric field distribution of the aperture. The far zone field in terms of a radar cross section of a square plate compares well with an experiment. A discussion is also included comparing the electric field distribution of a square aperture and circular apertures.

Summary Instrumental factors affecting the precision and accuracy of measurements with the Vickers M86 scanning and integrating microinterferometer were investigated. ‘Spot’ measurements of OPD (optical path difference) could be made to a precision better than 0.001 Λ (wavelengths); the coefficient of variation of scanning measurements of IOPD (integrated OPD) depended on the object, typical values being approximately ± 5% for a human erythrocyte in water, and ± 10% for a mouse sperm head in water. Repeated measurements enabled the dry mass of a specimen to be estimated to any desired precision. Instrumental calibration to give results in absolute units is described. Early prototypes required frequent calibration, but with production instruments the calibration varies little from day to day, and very similar results are obtained if a specimen is measured in different parts of the scanned field. A slight, approximately sinusoidal deviation from linearity was sometimes observed when the apparent OPD or IOPD was plotted against expected values. The error, which probably occurs with all instruments employing a Jamin double-refracting interference system, was maximally about ± 0.02 Λ for a specimen of true OPD 0.25 Λ or 0.75 Λ, but was zero for a specimen of true OPD 0.5 Λ. The error could be minimized or eliminated by the correct choice of an interference fringe for ‘spreading’, careful adjustment of the condenser and background OPD, and especially by reduction of the condenser aperture. Scanning measurements of IOPD were shown experimentally and theoretically to be insensitive to changes in the measuring-spot diameter or errors in focus. Valid measurements of dry mass can therefore be made even if the three-dimensional specimen (e.g. a living cell or isolated nucleus) is considerably thicker than the depth of focus of the objective used.

The first order parameters and the second order aberrations are derived for double focusing mass spectrometers with a toroidal electrostatic field and Wien filter. The useful field area to accommodate a given aperture beam and the Q values are calculated. Some numerical examples reveal the possibilities of the combinations of diverging, converging and dispersive elements.

A description is given of experimental equipment developed especially to enable the usefulness of scanned ultrasonic holography in ophthalmic diagnostics to be evaluated. Its useful aperture is 40 mm which it can scan with either 160 lines in 23 seconds or 80 lines in 12 seconds using pulses of 10 MHz ultrasound carrier 0.5 μs or longer. The resolution is approximately 0.5 mm and a single scan line is completed within 150 ms. It works in a direct pulse-echo mode, the ultrasonic images being those obtained by direct reflection. Facilities include simultaneous A and B-scan displays for which the carrier pulse length is usually 0.5 μs. Initial work indicates that although the application of ultrasonic holography will depend critically upon the development of satisfactory clinical techniques, the usefulness of a fast almost real-time B-scan is beyond doubt. The whole system is now undergoing evaluation at Moorfields Eye Hospital and a report on its performance will appear in the next issue.

An apodization scheme for increasing the Sparrow resolution limit under the partially coherent illumination is formulated via the calculus of variations. The required pupil functionsa re obtained by solving an homogeneous Fredholm integral equation and evaluated as functions of the coherence condition of illumination and the total light flux passing through the aperture.

Systems of a catadioptric form generally have central obscurations that give rise to annular-type entrance pupils. These are found in most cases to be well approximated to the region contained within two ellipses which are generally of different eccentricities. The O.T.F. (optical transfer function) for an annular aperture at a given image point may be expressed as the combination of four integrals, one for each of the possible overlap areas which together describe the form of the overlap region between the two ellipses. Each integral is of a similar nature to the one giving the O.T.F. associated with an elliptical aperture. A computer programme has been written to calculate the O.T.F. using Hopkin's algorithm (1957). Because only one analytical example (aberration-free case) could be obtained for a comparison with this programme, O.T.F. measurements were carried out experimentally on a simple doublet having a central obscuration in order that a more substantial check could be made. The maximum error enco...

The divergent-lens effect of the electrode aperture (slit or hole) on high-intensity ion beams, where the space charge cannot be neglected, are calculated to a first approximation with the help of basic physical laws. Simple relations are derived for the effective lens power resulting from these phenomena and for the location of the final virtual object to be imaged in the collector region. Results are given both for concentric-cylinder (slit) and radially symmetric (hole) configurations.

SUMMARY The Foucault in-focus method for viewing magnetic domains in a conventional electron microscope has been modified. The main feature of our modification is to introduce an aperture below the intermediate lens and to use it for stopping out one part of the split central spot instead of the objective aperture. This arrangement substantially reduces the axial astigmatism due to the objective aperture and makes it possible to reach the point resolution 5–6 nm in both light and dark domains. The image resolution is limited by the small image magnification achieved using the three-stage optical system of a Tesla BS 413 microscope.

The use of complementary semicircular or half-plane apertures in the objective lens back focal plane of an optical system to determine from image intensity measurements the complex image wavefunction is discussed. This method for the solution of the phase problem in optics is restricted to bright-field optics, where the primary wave is allowed to interfere with the scattered wave. It is shown that in this modified bright-field situation, using semicircular (half-plane) apertures rather than the normal circular objective aperture, it is possible to determine for complementary positions of the apertures a Hilbert relation between the real and imaginary parts of the image wavefunction. The initial estimate for the complex wavefunction may be systematically corrected for nonlinear effects occurring in the equations for the image intensity.

Behavior of the ion sheath near an aperture of ion source is investigated via a nonlinear Poisson equation for an electric potential function in a cylindrical coordinate system. On the assumptions of free electrons with Maxwellian distribution and ion flow against the aperture, the sheath condition and the condition under which the plasma surface is concave are derived. These conditions are shown to depend on the aperture radius, potential of electrode U, electron temperature kT/sub e/ and initial ion velocity. The ion current through the aperture with maximum radius is found to increase significantly when U is greater than or equal to 20kT/sub e/. Behavior of the potential distribution and plasma boundary under various situations are also demonstrated by means of numerical calculations. (auth)

An analytical and experimental investigation of the radiation properties of wide-flare corrugated E-plane sectoral horns (10 < 0 < 70, where ao is the half-flare angle) excited in the HE11 mode [1] with improved E-plane radiation patterns and suppressed side-lobe levels over a broad bandwidth is described in this paper. Given the flare length and half-flare angle of the horn, the far-field radiation pattern is determined by two methods, one of which involves aperture-field integration, based on vector diffraction formula over a constant phase surface at the mouth of the horn and the other of which employs an expansion of the aperture field in terms of free-space cylindrical TE and TM wave functions. Good agreement is obtained between the two methods with measured radiation patterns for several values of 0. Given the flare length, variation of the on-axis gain with 0of the horn is analytically investigated, and the design procedure for determining optimum 0, in order to obtain maximum on-axis gain, is also indicated. Further-more, design and experimental studies on wide-flare horns under consideration are also summarized. Copyright ? 1974 by The Institute of Electrical and Electronics Engineers, Inc.

An approach to image processing is made by means of image scanning with an aperture array. The restoration of one-dimensional linear blur is taken as an example, and methods of designing the aperture array and evaluating its performance are discussed. The quality criterion of restored images with regard to resolution and detection is introduced and applied to the practical system.

Speckle Interferometry has originated as an extension of methods previously used by Fizeau, Michelson and Pease for observing stellar details in the presence of atmospheric seeing. The « fine structure » of stellar images long reported by some observers of double stars is in fact a speckle pattern similar to that observable in diffused laser beams. As could be predicted from this interpretation, 1 000 speckles were found in the image given by the 5-meter Palomar telescope when observing in monochromatic light with short exposures. The fringes observed by Michelson and Pease, using a Fizeau screen on the aperture, are a special case of speckle which can be analyzed visually because of its simpler structure. The convolution of the specklesby details of the stellar object may be detected by time-averaging the autocorrelation function (or the power spectrum) of the instantaneous image, giving the autocorrelation function (or the power spectrum) of the object. Observations were carried out using a television sensor and optical processing techniques. The diameters of several supergiant stars were measured and limb darkening evidenced in α Orionis and o Ceti. A dozen stars were found to be binary, including Algol for which the predicted third body is thus evidenced. Future developments include observations at very low levels in the photon counting mode, and real time electronic processing. Use of several large telescopes in the synthetic aperture mode is also becoming feasible.

Introduction. The SIMAC [1], [2] is constituted by the association of a Fabry-Perot Interferometer and a large aperture spectrograph [F/2]. This allows it to achieve a very high resolution (8.105 to 106) with a very short time of exposure. Its constitution leads to a semi-sequential recording process so that a linear dispersion is obtained on the photographic plate. Moreover it allows one to increase the signal to noise ratio without changing camera lens as necessary with a classical spectrograph. In this paper we describe the technical aspects of the instrument and we give some experimental results. A complete theoretical and experimental discussion has been given previously in [3]. - Theoretical discussion. In an other paper [4] we establish that the three high resolution spectrographs (grating, Fabry-Perot, SIMAC) accept, in the same time, the same amount of energy for the same resolution. It follow that the fondamental difference is the shortest value of the focal length that the instruments are able to receive because of the graininess of the photographic emulsion. Discussion about the signal to noise ratio indicates what is the most appropriated instrument suitable for a specific problem. - Technical issue. A) Interferometer: It is a classical Fabry-Perot interferometer. An important improvement in the linearity of the scanning has been obtained by use of a flow controller (constant differential type). B) Spectrograph: Grating is manufactured by Bauch and Lomb (250 X 125 mm, 300 Lines/mm, blazed at 63° 27). As the present time the camera lens is a catadioptric system at F/2 and the focal length is the shortest theoretically possible (250 mm). But recent works have shown that this camera has to be modified: because of the large aperture, the receiver is a photographic film and experimental works proved that it was not convenient for position measurements (displacement of gelatin, deformation of the support can reach 10 to 40 μ). Secondly, with a catadioptric system, the plate holder, which occult the incident beam, must be very small, so we lose the advantage described in part I. The new camera will have the shortest focus length consistent with a photographic plate (glass support). C) Optical mounting: The arrangement is classical. The highest signal to noise ratio is obtained by use of the whole diameter of the Fabry-Perot. A soustractive dispersion double monochromator is used to get absolute symetric images in view of the recording with a photoelectric spectrum plate comparator [3]. - Results. Three reproductions show the possibilities of the SIMAC. The first one is an isotope displacement of Samarium. The second and third ones are molecular spectra of Iodic absorption and Lanthane-Oxyd emission. In these last cases, the Fortrat-Deslandres parabols are clearly perceived owing to the linearity of the scanning. - Conclusion. If the SIMAC accept finally the same amount of energy by time unit that the others high resolution spectrographs, it presents the advantages of a great flexibility. Without any mounting modifications, it can be a high speed spectrograph or a good signal to noise ratio spectrograph; even, in this last case, it keeps the advantages of the great aperture instruments.

The use of matrix methods to calculate ion displacements in quadrupole fields is applied to a detailed simulation of mass filter operation . Ion transmission is calculated for various scan lines and various apertures both with and without fringing fields. The experimentally observed tailing on the low mass side due to insufficient cycles of the r.f. field is explained and its avoidance investigated. Study of ion transmission as a function of initial phase shows that the optimum initial phase for miximising instrument aperture is associated with the necessity for a longer instrument. Conversely, the optimum phase for minimising instrument length has the poorest useful aperture.

A plane wave is symmetrically incident on a spherical shell with a circular aperture. The fields inside and outside the cavity are expanded in terms of spherical vector wave functions and the modal coefficients are found by application of the least squares method to the boundary conditions. Computed data are obtained and the results exhibited in the form of amplitude curves of the interior and aperture fields as functions of position for a variety of cavity and aperture sizes. Within the cavity it appears that the field variations are primarily determined by the cavity size and that the aperture size serves only to scale them.

The application of Hilbert transforms to the evaluation of electron microscope images is examined. In particular, the use of complementary semicircular apertures in the objective lens back focal plane to determine from intensity measurements the complex image wavefunction is discussed. Bright-field optics is considered where the primary electron beam is allowed to interfere with the scattered electron beam. It is shown that in this modified bright-field situation, using semicircular apertures rather than the normal circular objective aperture, it is possible to determine for complementary positions of the apertures a Hilbert transform relation between the real and imaginary parts of the image wavefunction. The initial estimate for the complex wavefunction may be corrected for nonlinear terms occurring in the equations for the image intensity. In this way it is possible to correct the image wavefunction for nonlinear terms contributing as much as 50% to the modulation of the image intensity; and an image contrast as high as 70%, well above that encountered in the conventional microscopy of biological materials, does not seem to invalidate the solution for the image wavefunction. The application of the Hilbert transform to electron microscopy is illustrated with computed images comprising an array of single atoms, and the effects of objective lens aberrations are included. The solutions for the complex wavefunction are valid in all cases examined and the correction for nonlinear terms in the image intensity may be made in 1-5 iterations, the lower figure corresponding to the contrast normally achieved in the transmission electron microscopy of biological materials. Additionally, the stability of the solution for the image wavefunction is examined when the effects of photographic noise, objective aperture misalignment and inelastic background are included in the image intensity profiles.

A scanning transmission energy analyzing microscope (STEAM) and an automated microprobe are described and their use explained. In the former instrument the image is scanned across an aperture and quantitative information recorded. In the latter the movement of the beam can be programmed and the x-ray data taken into storage. ImagesFIGURE 1.FIGURE 4.FIGURE 5.FIGURE 6.FIGURE 8.FIGURE 10.

The beam-mode expansion method used in the discussions of the diffraction of a Gaussian wave beam through an aperture is applied to a system of two circular or square aperture stops, and the analytical expressions of the power transmission and conversion coefficients of a fundamental mode through the system are obtained. By using these expressions, the optimum incidence conditions that maximize the power transmission coefficient of the fundamental mode can be known. These conditions coincide formally with those obtained by Kogelnik and Yariv for an incident wave having a prolate spheroidal-wave function distribution. Both circular and square geometries can be analyzed in the same way.