QC 351 A7 no. 04 / This technical report was prepared from the specifications document used for the bids for the University of Arizona telescope. This telescope was funded by the National Science Foundation under the Science Development Program. This document is being included in the Optical Sciences report series since it may have some relevance to the programs in this area.

QC 351 A7 no. 06 / A 6-inch aperture 3-mirror optical system has been designed that yields high resolution over a flat field of 6° at a speed of f/1.8. All surfaces are strongly aspheric. A proposed application of this new type of camera is for sky survey and stellar photographic photometry in the far ultraviolet under a current NASA Apollo AAP program. A general discussion of 3-mirror reflective systems is given.

QC 351 A7 no. 09 / Meinel and Shack recently devised a three-element reflective optical system for a specific astronomical application that appears to offer rather general uses for fast, wide-field cameras. A three-element system is mathematically elegant in that this is the minimum number of surfaces necessary to provide zero third-order spherical aberration, coma, and astigmatism for any distribution of spacings and powers.. Two-element configurations of surfaces, such as the Schwarzschild and Couder, in general, leave one of the above aberrations uncorrected. The maximum speed and field of good definition of two- mirror designs are also quite limited. The Meinel-Shack three-mirror system provides much superior performance since the degrees of freedom afforded through the use of aspheric deformations on all surfaces permits the exact solution for the fifth -order aberrations astigmatism 5, coma 5 and spherical 5 to be eliminated along with astigmatism 3, coma 3 and spherical 3. Moreover, it is possible to design practical systems with zero Petzval sum yielding a flat focal surface, and achieve an excellent balance of the remaining fifth -order aberrations, elliptical coma and oblique spherical aberration. The ultimate optical performance is governed by the resulting balance of any remaining higher - order aberrations. A preliminary report of the design approach used by Meinel and Shack is given in Optical Sciences Technical Report No. 6, appended to this report.

QC 351 A7 no. 12 / The technique of coherent detection has been used to explore the problems involved in detecting extremely low power levels. An input signal power level of 5 x 10^-19 watts of 3.39u radiation was detected with voltage S/N of 2, in good agreement with theory. The major experimental problem was elimination of feedback from the local oscillator into the laser source. Narrowness of bandwidth was limited by instability in detector bias. Neither of these difficulties presents a fundamental limitation to a well designed receiver of light from a distant source.

QC 351 A7 no. 14 / Lens designers who use automatic programs and large computers are no longer considered to be pioneers but are now an integral part of the world's rapidly-expanding optical industry. Various techniques of automatic correction, though still in the developmental stage, are finding daily application. A designer armed with a powerful lens design program and a capable computer may now design, in a matter of hours, a lens system which would have taken months to perfect only a few years ago; the results are consistently better than those obtained by classical methods, using log tables and desk calculator. Realizing that much has been learned about automatic design in the past decade, and realizing that the time was ripe for this knowledge to be shared, the Institute of Optics (University of Rochester) sponsored an international conference on Lens Design with Large Computers. Following is an agenda and list of speakers at this conference, held July 5-8, 1966, in Rochester, New York.

QC 351 A7 no. 17 / The following report was originally prepared by P. N. Slater as a progress report on a Multi-Spectral Tracking Telescope for the Apollo Applications Program under NASA contract NSR 03-002-066. Since the work is an outgrowth from investigations supported by other contract funding at the Optical Sciences Laboratory, the report has been adapted as a Technical Report for the information of the other sponsors. The basic design, following the Meinel-Shack three-mirror arrangement, represents a particularly excellent refinement by R. V. Shack. The extreme compactness of the design is of benefit to this application, and even though the central obscuration is large, the modulation transfer function remains satisfactory for the widely-used reconnaissance films. --A. B. Meinel

QC 351 A7 no. 21 / It is found that, despite the finite extent of any real pupil, an absorption coating exists which, when applied to the pupil of an optical system, results in nearly perfect formation of the coherent image. It is suspected that this coating causes a significant loss of total illumination. On the other hand, an "active" pupil (one for which there is a gain in light flux) would allow the quasi-perfect imagery without any loss in total illumination. When active pupils become a technological reality, this will be a good use for one.

QC 351 A7 no. 22 / The statistical approach must be used to describe the image when object brightness, optical pupil characteristics, and image detection are all subject to random fluctuations. Use of the statistical characteristic W (the Fourier transform of probability density) is found to clarify the phenomenon and to result in a linear theory. There is a characteristic function W, corresponding to the joint statistics of the log modulus and the phase, for: the object spectrum, optical transfer function, detection transfer function and image spectrum. These four W-functions are the statistical analogy to the four Fourier spectra themselves, if the object radiation is either perfectly coherent or perfectly incoherent. Thus, in analogy to the ordinary Fourier theory of image formation, there is (1) a statistical transfer theorem linking object and image fluctuations, (2) a statistical transfer function for the optics, which may be computed from the optical pupil statistics, (3) sampling theorems, and other analogous results. The W-functions are found to determine all moments of each spectral distribution, and to imply that the moments themselves obey a transfer theorem. Also, the optical characteristic W-function seems to be useful as a quality criterion of optical system stability. In the deterministic limit, the statistical theory goes over into the ordinary Fourier theory of image formation. Random detection noise is a natural parameter of the theory, so that application to practical problems seems eminently possible.

QC 351 A7 no. 23 / It has been traditional to constrain image processing to linear operations upon the image. This is a realistic limitation of analog processing. In this paper, the calculus of variations is used to find the optimum, generally non - linear, processor of a noisy image. In general, such processing requires the use of an electronic computer. The criterion of optimization is that expectation (10. - (5j1K) be a minimum. Subscript j denotes the spatial frequency w. at which the unknown object spectrum 0 is to be re- stored, 0 denotes the optimum restoration by this criterion, and K is an even power at the user's discretion. A further generality is to allow the image- forming phenomenon to obey an arbitrary law I. = L(T., O., N.). Here, J J J T. denotes the intrinsic system characteristic (usually the optical transfer function), and N. represents a noise function. The optimum Oj is found to be the root of a finite polynomial. When the particular value K = 2 is used, the root O. is known analytically, along with the expected, mean -square, minimal error due to its use. When K = 2, processor O. has the added significance of minimizing the total mean -square restoration error over the spatial object. This error may be further minimized by choice of an optimal processing bandwidth. Particular processors O. are found for the "image recognition" problem and for the case of a "white" object region. The latter case is numerically simulated.

QC 351 A7 no. 24 / A lens system may be judged by its ability to relay information from object to image. A pertinent criterion of optical quality is h, the change in entropy between corresponding sampling points in the object and image planes. Since h is a unique function of the optical pupil, for a given bandpass 20 of the object, through the proper choice of a pupil function it is possible to maximize h at a given Q. Physically, the optimum pupil function is an absorption coating applied to a diffraction - limited lens system. A numerical procedure is established for determining, with arbitrary accuracy, the optimum absorption coating, the resulting transfer function, and the maximum h, all at a given 0. These quantities are determined, both for the one -dimensional pupil and the circular pupil, in the approximation that the optimum pupil function may be represented as a Fourier - ( Bessel) series of five terms. The computed values of hmax, at a sequence of 52 values, are estimated to be correct to 0.2% for the 1 -D pupil, and to 0.5% for the circular pupil. The optimum pupil functions are apodizers at small S2 and superresolvers at large 0. Finally, we use the computed curve of hmax to relate the concept of "information transfer" to that of "classical resolving power ": we show that a binary object (as defined) cannot radiate information to the image when the spacing between object sampling points is less than 0.87 times the Rayleigh resolution length.