QC 351 A7 no. 26 / The tensor equations of elasticity in nonorthogonal curvilinear coordinates are presented in a form suitable for the method of dynamic relaxation. This method is described briefly and then is applied to the solution of the problem of elastic deformation of curved mirrors.

QC 351 A7 no. 29 / Most optical systems have rotational symmetry. For such systems, we establish a method of finding (a) the maximum attainable modulation transfer function (MTF) at arbitrary frequency wo; and (b) the required pupil function U(wo;p). Physically, the latter comprises absorbing films in the pupil of diffraction -limited optics. The method of solution is numerical and iterative, based on the Newton -Raphson algorithm. Solutions (a) and (b) are established at frequencies wo = 0.1, 0.2, ..., 0.9 (x optical cutoff). The computed (a) are correct to ±0.0001 over all wo indicated. Quantities (b) have an average error over each pupil of ±0.002 for frequencies wo <- 0.5. With 0.6 s wo <- 0.9, the error is 0.01. The curve of maximum MTF(wo) seems smooth enough to allow for accurate interpolation. Solutions (a) and (b) were also found over the finer subdivision wo = 0.05, 0.1, 0.15, ..., 0.8 with slightly less accuracy than above, in order to allow for interpolation of pupils U(wo;p) over values wo. This seems possible for 0.05 < wo < 0.40. The maximum MTF(wo) shows appreciable gain (e.g., 8% at wo = 0.2) over the MTF for uncoated, diffraction -limited optics at all wo except in the inter- mediate region 0.4 <- wo <- 0.6. In the high - frequency band 0.5 <- wo <- 1.0, however, the maximum MTF(wo) shows little gain over the MTF due to an uncoated, diffraction -limited pupil with the proper central obscuration. The light loss due to each U(wo;p) may be measured by the total energy transmission and the Strehl flux ratio. These are plotted against wo, .and indicate moderate light loss.

QC 351 A7 no. 31 v1 / "The Physics of Quantum Electronics," á two -week, non-credit course sponsored by the Optical Sciences Center, was held from June 17 through June 28, 1968, on the campus of Northern Arizona University in Flagstaff. The course was directed by Professors S. F. Jacobs (University of Arizona) and M. O. Scully (Massachusetts Institute of Technology) and was patterned after the tutorial symposiums sponsored in 1966 and 1967 by Colorado State University. Designed primarily for advanced students, research scientists, and technical administrators working in the general area of quantum electronics and coherence physics, the course attracted 90 participants from all over the world. A list of attendees appears at the end of this report.

QC 351 A7 no. 31 v2 / The quantum theory of noise plays a very important role in the laser and optical parametric oscillator. In these lectures we shall consider in some detail a damped harmonic oscillator as well as a rotating-wave van der Pol oscillator which describes a laser. We begin with a brief reivew of the quantization of the radiation field and the coherent state representation.1,2 This representation is especially appropriate for oscillators far above threshold since it shows the very close correspondence between the quantum and classical theory. The theory of damping will be presented from both the Langevin and reduced density operator viewpoints. When the system is a mode of the radiation field, the density operator in the coherent state representation is a Fokker-Planck type equation. We shall present an elementary discussion of the optical parametric oscillator.

QC 351 A7 no. 32 / A random perturbation in the wavefront transmitted through an optical system results in diffraction effects in the image which are describable in terms of the statistical measures of the wavefront perturbations, whereas the geometrical effects are determined only by the statistics of the gradient of the wavefront perturbation. By using a model in which the gradient statistics are held constant while the depth of perturbations is varied, differences between the geometrical and diffraction images are made apparent. For an rms depth Qw /A > 0.5, the diffraction and geometric images are statistically indistinguishable, but for smaller depths the diffraction image departs from the geometric image, separating in effect into an undisturbed core and a diffracted halo, between which the partition of the image power depends on the depth of the wavefront perturbations. Various proper- ties of the diffraction image as a function of the rms depth of the perturbations are discussed.

QC 351 A7 no. 33 / Image formation is a major area of Optical Sciences Center research. Parallel studies are progressing on analog and digital image processing techniques. This report describes the current progress in the analog apparatus and research. By February, 1968, the mechanical assembly of the system had been completed. The optics were prepared and coated; beam splitters were fabricated and assembled. The electronics package was debugged and all elements were readied for initial tests. These tests were designed to provide systematic checkout and image restoration indications. The approach taken thus far is to optimize the system and study the broad aspects of signal -to -noise ratios that can be achieved with the analog method. It can be seen that, while analog may be faster and less expensive, it is less flexible than a digital approach. The description of mask making leads one to realize that refinements of the analog method are extensive. The balanced system must be achieved for proper gains in image processing. No firm conclusions are drawn in this report. Activity in image processing will continue throughout FY 69. The indication from these experiments, however, is that there is insufficient signal -to -noise ratio in the material shown in Fig. 5 to permit much objective improvement in image quality before excessive grain modulation occurs. The question of whether the subjective improvement is useful in a practical sense has not yet been addressed.

QC 351 A7 no. 34 / Despite its necessarily finite aperture, an optical system can theoretically be coated to produce arbitrarily perfect imagery over a limited field. When the object is of limited extent, this field can be made the optical conjugate to the object, so that the whole object is imaged with arbitrary precision. The required pupil coating approximates low- contrast cosine fringes over its central region; toward the aperture edge the frequency and amplitude rapidly accelerate. The maximum occurs as a narrow spike. The frequency near the central region varies directly with the total extent of the conjugate field and inversely with the required central core width A in the point amplitude response. As t is made arbitrarily narrow, the point amplitude response approaches the form of a sinc function over the field of view. This function is precisely the point amplitude for a diffraction-limited pupil with a magnified aperture of 1/A times the given pupil aperture! The only image property that is not in compliance with this effective aperture magnification is that of total illumination. This is severely reduced from that of the original, uncoated aperture, and is the major restriction on practical use of the derived pupil. Applications to microscopy and telescopy are discussed.

QC 351 A7 no. 35 / A thin plate of crystalline quartz has been fabricated for use as a quarterwave plate with He -Ne 3.39 -um lasers and has been used to determine the birefringence of the quartz at that wavelength. With a single thin plate, first -order relative retardation is achieved without recourse to an air - spaced pair of thicker plates having opposing retardation. Compared with such thick, air-spaced plates, thin quartz plates are much less costly to make, and they allow use of this excellent optical material much farther out into the infrared region. Because the birefringence at 3.39 um had not yet been measured, an extrapolated value was used to determine an approximate thickness for the plate that was to be fabricated. Using the plate thus fabricated (0.128 mm thick), two methods were followed to independently determine the birefringence: First, with the plate normal to a linearly polarized 3.39 -um laser beam, the state of polarization of the transmitted beam was measured, yielding the relative retardation, and hence the birefringence, after determination of the plate thickness. The second independent determination of birefringence was obtained by measuring the plate tilt necessary to produce exactly circularly polarized light. The average of the 3.39 -um birefringence values obtained from the two methods was .0065 t .0001, corresponding to a quarterwave plate thickness of .1304 mm. The possibility of using thin crystalline quartz for infrared wave plates is attractive. However, one must consider both anisotropic absorption and anisotropic Fresnel reflection, which vary with wavelength. Only if these anisotropic losses can be balanced or made negligible (as by anti - reflection coating) can a perfect waveplate be made.

QC 351 A7 no. 41 / A function PvN(ß) exists whose finite Fourier transform over a specified range of its argument is asymptotic (with N) to an Airy distribution with arbitrary scale compression. Consequently, when the function is applied as a passive coating to a diffraction -limited lens of fixed aperture,the point amplitude response collapses inward as if the lens were physically replaced by a diffraction-limited lens of greater aperture. Investigating the implications of coating PN(ß) to image theory, we find the following: (1) The scalar wave equation has intrinsically a particlelike solution. (2) A modification of PN(ß) causes an arbitrarily narrow depth of focus. (3) An arbitrary point amplitude response may be optically produced. (Suppose g(x) to be a required, and arbitrary, point response function with G(ß) its finite Fourier transform. Then pupil PN(ß)G(ß) produces g(x), asymptotic with N.) (4) When applied onto any band -limited pupil G(ß), coating PN(ß) effectively extrapolates G(ß) arbitrarily beyond the bounds of the aperture. Some amusing analog devices, based on the extrapolating property (No. 4 above), are next developed. These are an optical analog signal extrapolator, a picture extrapolator, and an analog method of band -unlimited image processing. We also suggest the existence of a laser "superposition mode" whose out- put would be arbitrarily directive, and the possibility of using an acoustical pupil NO) to resolve these long wavelengths with near-optical quality. The ultimate limitations on the practical use and fabrication of pupil PN(ß) are discussed.

QC 351 A7 no. 43 / The optical elements of a cassegrain telescope are commonly tested individually, with their axes in a horizontal position. When these optical elements are inserted in the telescope, the resulting imagery is often dis- appointing. The quality of the imagery in the telescope may be predicted more accurately if the primary and secondary mirrors, with their axes in the vertical position, are tested against each other with the aid of null compensating reimaging optics. An example is given to illustrate the application of the technique.