Publisher Summary This chapter discusses applications, specifically dealing with the performance characteristics of optical systems. The linear prolate functions are a set of band limited functions, which like the trigonometric functions, are orthogonal and complete over a finite interval. However, unlike the trig functions, they are also complete and orthogonal over the infinite interval. Fourier transform of a linear prolate function is proportional to the same prolate function. The mathematical properties of the circular prolates, which make them readily applicable to the analysis of optical imagery, are summarized in the chapter. The optical applications arise from the modern use of the prolates as a convenient set of one-dimensional, orthogonal functions. These applications have been on the general subject of optical systems analysis. Performance characteristics of the laser have been established, along with the ultimate ability of lens and lensless systems to form high quality images. There is one physical phenomenon that unifies these applications—diffraction at a finite aperture—and it can be said that this phenomenon is optimally analyzed by use of the prolate functions.

VIII Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions

Apodization theory is concerned with the determination of the distribution of light over the exit pupil of an optical system required in order to achieve a suppression of the side lobes of the diffraction pattern. Here analytic solutions are given to the problem of determining the distribution of light in the exit pupil to concentrate maximally the illuminance in a geometrically similar region of the image plane. Both slit and circular apertures are treated.

Analytic Solution of Two Apodization Problems

Upholding the diffraction limit in the focusing of light and sound

Filters of the type exp(inθ)sgn Jn(zor) with θ the azimuthal angle and r the radial coordinate in the filter plane, are shown to maximize the energy content in a narrow annular image of radius zo with respect to incident energy. The simplest optimal filter, sgnJo(zor), is well approximated by the binary circular phase grating sgn cos(zor − π/4). The single lobe of the first order image of this filter contains 46% of the incident energy within the half-width 0.4λf/a, centered around the image radius Nλf/(2a), where N ≃ zo/π is the number of filter sections.

Optimal filter design for annular imaging.

The distortion of the image of an object, produced by diffraction, can be reduced by filtering. A class of filters is developed for minimizing the first term in the moment expansion of the error. The resulting spread functions have high attenuation rate, negligible side lobes, and are optimum for imaging smooth objects. One- and two-dimensional systems are considered; the analysis includes coherent and incoherent illumination.

Apodization for Optimum Imaging of Smooth Objects

It is recognized that the most favorable distribution of radiant energy in a diffraction pattern is that which corresponds to the best concentration around the center O. This hypothesis is expressed by an extremal condition on the factor of encircled energy E(W), that is, the ratio of the energy inside a circle of radius W and centered on the diffraction pattern, to the total energy in the same.A study of the effects of spherical aberration on this factor of encircled energy has shown that aberration always tends to decrease the factor from the value obtained with an Airy pattern. However, this factor may be increased by the use of an amplitude filter at the pupil of the optical system.In treating the case of amplitude filters one may use a rigorous analysis in terms of Taylor's series in (1−x2)p−1 or a polynomial Tn(x) of degree n−1 in terms of (1−x2).The corresponding amplitudes in the diffraction pattern are Г(W) and Гn(W); the maximum factor of encircled energy E(Wm) and En(Wm). The following converge...

Maximum of the factor of encircled energy

An optical system which is symmetrical about some axis forms a diffraction pattern around its focus F in an observation plane II.The factor of encircled energy E(W) is, apart from a normalization factor, the energy spread over the II plane inside a circle of radius W, centered on F.Although it is a good test for the quality of an optical system, the factor of encircled energy has not been used a great deal because of the lengthy calculations that its numerical determination requires. Two rapid computing methods are exposed here, with tables allowing a complete determination of E(W).

Guy Lansraux

Papers

Maximum of the factor of encircled energy

Numerical determination of the factor of encircled energy relative to a diffraction pattern of revolution

Une représentation de la diffraction instrumentale

DIFFRACTION ITÉRÉE DE LA FIGURE (sin X/X) PAR UN RÉSEAU INFINI