This paper deals with the effects of non-uniform amplitude aperture illumination on the three-dimensional intensity distribution in the neighborhood of focus of an optical system which has a circular aperture with and without primary and secondary spherical aberrations expressed by the Zernike polynomials. A large number of intensity distributions are calculated numerically by the 24-point Gaussian quadrature method with an electronic computer. Diagrams of the contour lines of intensity (the isophotos) are shown and are compared with the result of the previously published papers on uniform amplitude aperture illumination. When the non-uniformity of positive amplitude aperture illumination becomes large, the bright central nucleus of the three-dimensional image becomes longer and narrower similar to the case of an annular aperture. Consequently both the focal depth and resolving power are increased. When in the negative case the non-uniformity becomes large, the three-dimensional intensity distributions are spread over a broad region of the geometrical cone of rays, whereby the higher orders of diffraction patterns are attenuated.
The problem of automatic lens design is generally shown to be equivalent to minimize a merit function Φ (x1••••xn) whose magnitude depends upon the residual aberrations of the system. In this report, the merit function Φ is computed at n-dimensional lattice points (μ1, μ2 …… μn) distributed over the region of independent variables where a set of positive integers (μ1, μ2 •••• μn) represents each lattice point. Φ (μ1•••• μn) can be expanded to a series of “sub-functions” Φk (μ1, μ2••••μk, μk+1.0••••μn0) each of which is the intersection of Φ with the k-dimensional sub-space containing a starting point μ0(μ10, μ20••••μn0). When the minimum of the sub-function Φk is known, the procedure of finding the minimum of Φk+1 is derived and this procedure of “sub-minimization” is applied successively to a series of sub-function Φ1, Φ2•••• until the final minimum of Φn(=Φ) in the n-dimensional space is reached. (successive minimization method) Both the extremely slow convergence and complicated numerical computations of the steepest descent method are avoided by the present method. In case of repeated applications of the successive minimization method corresponding to the continuous variation of external parameters included in Φ, both the starting point and the lattice distance are improved at each minimization process, the improvement being based upon the results of the previous minimization. (variable metric method) By this technique, the accuracy of solutions is remarkably increased. The whole methods described above have definite advantages when applied to the problem of automatic correction of first- and third-order aberrations. In this case, the numerical computing process becomes considerably simple by means of lattice-points expression of the present method. As numerical examples, the whole computing processes of triplet design are presented. The variation of residual third-order spherical aberration, when all the other first- and third-order aberrations are corrected to target values, is also computed against various values of thin-lens system parameters.
The development of electronic computers has brought much possibility in evaluating methods of lens systems, and the theory of transfer function is now being applied to practical lens design. In this paper the evaluation by spot diagram is discussed. A method of plotting spot diagram is proposed by utilizing the line printer of computer as the X-Y plotter. As the geometrical optical intensity distribution Ig (x, y) given by spot diagram is expressed in the form of Ig (x, y)=N-1_??_δ(x-xi, y-yi) the convolution t (x, y) with the turbidity r (x, y) of the image receiver-the total impulse response-is given by t(x, y)=N-1_??_r(x-xi, y-yi) Calculated intensity distributions by computer are compared with the measured ones for an actual lens and the agreement is satisfactory. These things can also be applied for the case of transfer function. The Strehl definition t (O, O) is calculated for several cases of the primary and secondary spherical aberrations, and it is confirmed that there are two extremum positions of S. D. in these cases.
Diffraction images by polarizing microscope with crossed Nicols were extensively studied by Kubota and Saito. Their work is mainly on the diffraction image of a point source through this optical system. In this paper, the study is furthered to diffraction images of various types of extended coherent source such as rectangular, circular and triangular types. These images are of unusual patterns different from the Airy-type ones formed by an ordinary optical system, and the patterns are theoretically calculated on an electronic computer and shown in contour lines of intensity. Photographs taken with a ruby optical laser are shown for comparison. The relation between the diffraction images and the size of the various extended sources is discussed.
n automatic plotter has been developed. With this device, a number of spots on the cathode-ray tubes can be plotted rapidly, automatically and accurately. This device can be utilized as an. X-Y recorder as well as an indicator of intensity or density distribution in the field of optics or statistics. And also, if the viewing storage tube is used as an indicator, this device may be used. as the light source of the image synthesizer in the field of optics. This device consists of three parts: a D-A converter, a photo-reader and a cathode ray oscillo-scope or memoriscope. The D-A converter is of all-transistor circuit. The input data of this device are those that are punched out on paper tape with binary cord directly obtained through the electronic computer. The speed for treating the data is about 60_??_80 msec per one point. The accuracy of this device is better than ±0.5% of full scale; it is enough for practical use with satisfactory linearity and reproducibity.
For evaluation of the point image of a lens, the distance in functional space, d(R, R0)=[∫(R-R0)2dN]1/2 is introduced. Cases of out-of-focused aberration free lens and secondary spherical aberration are shown as samples. By this method, difference between the optical transfer function curves can be scrutinized.
The spot diagram of an optical system is obtained by interpolation method and skew ray tracing method. For unique evaluation of the optical image, the image plane which minimizes or maximizes one of the following merit functions is searched. The adopted merits are (a) ∑ri2, the squared summation of the distance of each spot from the origin and (b) ∑|ri|, ∑ri2, ∑1/(|ri+Δr) and ∑1/(|ri|+Δr)2. The origin for (a) is the point of intersection of the principal ray with the image plane and that for (b) is the center of gravity. IBM 7090 is used for computation. The obtained results are in good agreement with those of Kuwabara's.