This symposium is a result of the author's observation over the better part of the last decade that workers in each of the fields of optics, electro-optics, acoustics, and electromagnetics have been solving very similar problems with little awareness of what is being done outside their field. Many of the tools and quite a few of the important results have been reinvented independently in each of these fields. There are excellent and fundamental reasons for this. The purpose of this introductory paper is to attempt to convey a bit of philosophy wrapped around some very fundamental Physics about the field of synthesis and its interdisciplinary aspects. This paper presents two basic ideas: 1. Synthesis is a problem that is different and distinct from analysis, and should be treated as such. 2. Most electromagnetic and acoustic synthesis problems ultimately deal with physics that can be expressed mathematically in the same form.

Several years ago we published some results concerning the structure of geometrical wavefronts in a homogeneous medium. A description of a wavefront was obtained as a general solution of the eikonal equation. By means of some standard techniques in differential geometry this led to the description of the general caustic. This in turn led us to the study of the geometric aberrations of an optical system from a completely different point of view. These results also suggest an approach to the physics of. the Propagation of light in a homogeneous medium which ought to lead to a proper vector diffraction theory. Recall Hertz' approach to the problem of the spherical wavefront in which he transformed the Maxwell equations into wave equations for the vector and scalar potential functions. To apply these to the spherical wavefront he transformed the arguments of the aradient, divergence and curl operators to a spherical coordinate system. The near-field solution of the wave equation predicted the existence of the dipole oscillator. The farfield solution yielded the now well-known description of the polarization and energy distribution on a spherical wavefront. Present work involves applying these same techniques to the general wavefront obtained as a solution of the eikonal equation. The vector differential operators have been transformed to an appropriate generalized coordinate system. The wave equation for the Potential functions have been obtained and an intermediate integral has been found. As is stands at the present time the electric and magnetic vectors can be expressed in terms of this intermediate integral.

Numerical propagation of scalar fields within thin annuli over short distances demands very high sampling density when employing techniques which evaluate the Fresnel-Kirchhoff integral. Previous work on using asymptotic expansion methods has been limited to field distributions which display a high degree of azimuthal symmetry. This paper describes an extension of the approach to more asymmetric field distributions. The resolution requirements for this technique are discussed and some numerical results are presented. The problem of propagating thin annular scalar field distributions over short distances lends itself to very high fresnel number configurations and consequently requires high resolution sample density. The fresnel number is defined as

Significant recent developments in elastic wave scattering theory are discussed in terms of their effect on the use of ultrasonics for quantitative flaw characterization in structural materials. The discussion begins with an introduction on nondestructive evaluation (NDE) describing the need for a quantitative ultrasonics technology that includes a knowledge of the fundamental ultrasound-flaw interaction and the ability to apply that knowledge in practice. Our present level of understanding is then illustrated using several examples of theoretical results for both direct and inverse scattering problems. Very little mathematical detail is employed in the discussion with the emphasis placed, rather, on graphically displaying results and showing differing effects for different scatterers. Concluding the discussion is a description of a new, sophisticated ultrasonic test instrument, an ultrasonic test bed, under construction at the Lawrence Livermore National Laboratory. This facility, when completed, will allow for the testing and practical implementation of existing and emerging advanced ultrasonic concepts such as described here.

The normalized scalar wave equation is solved for propagation parameters which depend on only the normalized frequency V and an arbitrary refractive-index profile. Analytic expressions relate these parameters to the optical fiber diameter, materials, and wavelengths of interest. These expressions are used to find dispersion and bandwidth. In this way it is possible to determine fiber parameters so that minimum dispersion occurs at preferred operating wavelengths. Then refractive-index profiles may be modified to improve bandwidth over desirable ranges.

As optical fiber technology matures, complexity of optical waveguides and waveguide components also grows. Traditional techniques which may be used to analyze circular fibers are no longer adequate. An assessment of several modern analytical/numerical techniques which have been used successfully to treat problems of practical interests is given. Il-lustrative numerical examples are also presented.

The so-called pseudo-Maxwell are a set of partial differential eauations that strongly resemble the Maxwell equations, yet are based only on Fermat's principle, the idea of an orthotomic system of rays, and certain theorems from differential gecmetry. From Fermat's principle, applying the Euler equation from the variational calculus, one obtains the ray equation whose solutions describe ray paths in an inhomogeneous medium. We define an aggregate of such rays as an orthotomic system if it is possible to find a sur-face orthogonal to all rays in the aggregate. Making use of the Frenet equations from differential geometry, one may derive relationships between certain geometrical vectors and their derivatives. These are the pseudo-Maxwell equations. Their existence is' paradoxical. Are they merely a mathematical artifact, an accidental quirk of the notation we are accustomed to use? Or do they indicate that there is more geometry lurking in the physics of electricity and magnetism than we ever dreamed of in our philosophies?

Radiant power from the images of adjacent point sources will spill over into each other even though their separation is greater than the system resolution element. This spillover or system optical crosstalk can be troublesome if a point source of interest is in the vicinity of a much brighter point source. A method is presented for computation of the constituent of system optical crosstalk due to optical surface scatter. This analysis provides the scatter function at the image based on the scatter function for each surface and the surface position along the optical path.

Optical systems that perform non-projective transformations are rarely synthesized by intent. Most systems familiar in practice are designed to provide the closest approximation to a projective transformation that is allowed by physics, technology, or economy. The advent of the laser brings many new applications for optical transformations - the non-projective variety being a late-comer. Requirements in the fields of laser materials processing, optical data processing, high energy lasers, and laser fusion, just to name those areas already penetrated, lead one to consideration for unconventional grooming of wavefront irradiance profiles. Transformations such as changing a wavefront irradiance distribution from flat-like to gaussian-like, or vice versa, or changing the wavefront area obscuration while maintaining its focusability, are typical examples of applications gaining an increasing interest. Following the laws of geometrical optics, yet violating certain fundamental rules of imaging, the present paper develops principles of design and analysis of non-projective transformations in optics, and explores one possible application.

Microwave holography is a useful experimental technique for imaging remote or inaccessible objects and for diagnostics of antennas, radomes, and scatterers; however, diffraction restricts image resolution. This paper describes a method for improving resolution in microwave holography. The holograms are either spherical or circular. Porter's scalar theory of curved holograms is extended to vector fields by using rectangular components to treat the effects of wave polarization. The mathematical formulation is a Helmholtz diffraction integral. This integral is written as a convolution for currents on line segments. The convolution is applied to the spatial frequency spectra of images. The spectra of dipole antennas are analytically continued, and the current distributions are exactly reconstructed. An experimental example is described; it is diffraction of a half-wavelength wide slit in a conducting screen. The analytic continuation of the holographically reconstructed nearfield produced images with resolution approximately 1/4 wavelength. Before continuation, resolution was 0.6 wavelength. In addition, the boundary condition of vanishing tangential over the metal screen was better satisfied in the image produced by continuation.

A technique for designing antenna arrays whose sources are located along a line is presented. Unlike most approaches to linear array synthesis, this technique optimizes source locations as well as source strengths. The technique is based on a modification of Prony's method.

In the synthesis problem a designer specifies the field that is to be incident on a system, and the field that it is desired that the system produce from this incident field by refraction, reflection, diffraction, scatterng, and/or reradiation. Mathematically and physically, this is an inverse scattering problem. In an inverse scattering problem, the fields in the inhomogeneous wave equation are known, and it is desired to solve for the source term. N. N. Bojarski has derived an Exact Inverse Scattering Theory for such "inverse source" problems. The problem of determining the generalized refractive index (i.e., the complex permeability and dielectric constant for an electromagnetic problem, or the velocity and absorption for an acoustic problem) distribution of an inhomogeneous medium from measurements of the fields scattered by the medium can also be treated using this theory. This solution is applicable to all remote probing problems, including radar, sonar, "profiling" of inhomogeneous propagation media, nondestructive evaluation, and seismic exploration.

The dual-shaped reflector synthesis problem was first solved by Kinber and Galindo in the early 1960's for the circularly symmetric-shaped reflectors. Given an arbitrary feed pattern, it was shown that the surfaces required to transform this feed pattern by geometrical optics into any specified phase and amplitude pattern in the specified output aperture are found by the integration of two simultaneous nonlinear ordinary differential equations. For the offset noncoaxial geometry, however, it is shown that the equations found by this method are partial differential equations which, in general, do not form a total differential. Hence the exact solution to this problem is believed to be generally not possible. It is also shown, however, that for many important problems the partial differential equations form a nearly total differential. It thus becomes possible to generate a smooth subreflector by integration of the differential equations and then synthesize a main reflector which gives an exact solution for the specified aperture phase distribution. The resultant energy (or amplitude) distribution in the output aperture as well as the output aperture periphery are then approximately the specified values. A representative group of important solutions are presented which illustrate the very good quality that frequently results by this synthesis method. This includes high gain, low sidelobe, near-field Cassegrain, and different (f/D) ratio reflector systems. While the above approach generates the solutions for the surfaces by numerically solving a line integral along the surface, more recent approaches develop the entire area from either an 'initial' line (usually of symmetry) or a point. The latter development is particularly useful in addressing the question of the existence of solutions - a currently controversial and very significant question. In this regard it is most important to pay careful attention to the required constraints posed in the problem.

A planar aperture synthesis procedure which predicts measured radiation pattern data accurately and which exploits known physical parameters of the actual antenna is described for use in computer-aided radome analysis. The plane wave spectrum (PWS) representation, with the geometrical optics approximation, is used to characterize radiation from the array of four identical, discrete, conical horn elements that was studied. Solutions for the PWS of each element from measured array patterns over the visible region are presented, and a digital signal processing algorithm is described for extrapolating the aperture-limited PWS into the evanescent region as required to determine the near field of each element. The element near fields, having bounded support, are combined to produce a near field for the complete array. The array near field is used in a computer-aided radome analysis to demonstrate the accuracy to which the measured antenna patterns are predicted for the case of a free space radome. For completeness, comparisons of measured and computed patterns for a tangent ogive Rexolite (Σ_r = 2.54) radome are presented.

In designing multi-element phased arrays, whether for radar antennas or ultrasonic imaging systems, it is always desirable to keep the number of sensor elements in the array to a minimum. This results in economic savings in the construction of the array and in computational savings in the processing of the collected data for the reconstruction of images. With this reduction in the number of elements comes an immediate trade-off in the array's performance between image resolution and the introduction of ambiguities into the image through grating lobes. This paper presents results on the optimization of phased arrays for a specified number of elements with respect to grating lobes, side lobes and image resolution. This is accomplished by adjusting the spacings as well as the individual gains of the elements. The results are applied to one and two dimensional Cartesian arrays and also to Fresnel lens arrays.

This paper describes the need for non-raytracing schemes in the optical design and analysis of large carbon-dioxide lasers like the Gigawatt,¹ Gemini, ² and Helios³ lasers currently operational at Los Alamos, and the Antares 4 laser fusion system under construction. The scheme currently used at Los Alamos involves characterizing the various optical components with a Zernike polynomial sets obtained by the digitization6 of experimentally produced interferograms of the components. A Fast Fourier Transform code then propagates the complex amplitude and phase of the beam through the whole system and computes the optical parameters of interest. The analysis scheme is illustrated through examples of the Gigawatt, Gemini, and Helios systems. A possible way of using the Zernike polynomials in optical design problems of this type is discussed. Comparisons between the computed values and experimentally obtained results are made and it is concluded that this appears to be a valid approach. As this is a review article, some previously published results are also used where relevant.

The application of coded aperture imaging to the measurement of fuel location in nuclear fuel pins undergoing simulated reactor accident conditions is aescribed. Modifications to normal coded aperture techniques, including the use of a one-dimensional, thici, (1.5 cm) aperture, are described. The predicted and experimentally measured performance of the aperture is compared, and results of steady state imaging tests using the Coded Aperture Imaging Jystem (CAIS) are shown.

The position location accuracy of a point source in the fields of view of two telescopes with the same focal plane is considered. It is shown that for detectors sized to equal the distance to the first zero of the coherently added system, there is little practical difference in accuracy between a system that adds the flux coherently as opposed to a system that adds it incoherently.

A development of the Aldis equations is presented which permits exact computation of the contribution of each surface to the aberration of a ray at the focal plane. This development allows for the analysis of the general, rotationally symmetric asphere. A surface-by-surface analysis with the Aldis equations is compared with a third-order analysis.

We have derived an analytic procedure for recording a flat, volume phase, transmission holographic optical element at a wavelength different from that at which it is to be used. The procedure guarantees that the resulting element will have diffraction limited aberration performance. Furthermore, it guarantees, to a first order, that the Bragg condition for high diffraction efficiency will be satisfied. The technique gives simple analytic expressions for the required object and reference construction beam phases at the element. In general, the object and reference construction beams must be realized using computer generated holograms in conjunction with conventional refractive or reflective optics.

The design of pulsed, high power excimer lasers incorporating unstable resonators is an area of major importance. The successful recording of UV holograms using an e-beam pumped XeF laser with unstable optics opens up the possibility of laser medium characterization by this procedure. This paper describes the optical configuration used in the recording of such holograms, and the measurement of the excimer laser's temporal coherence length by a holographic procedure. As a conclusion, several reasons for the proposed employment of UV holographic techniques in the design of high power excimer lasers are noted.