Since optical systems have distinctive features as compared to electical communication systems, some formulation should be prepared for the optical image in order to use it in information theory of optical systems. In this paper the following formula for the inten-sity distribution of the image by an optical system having a given aperture constant a in the absence of both aberration and defect in focussing is obtained by considering the nature of illumination, namely, coherence, partial coherence and incoherence; _??_ where I(y) is the intensity of the image at a point of coordinate y, T12 the phase coherence factor introduced by H. H. Hopkins etc., E(x) the complex transmission coefficient of the object and A(x) the complex amplitude of the incident waves at the object, and the integra-tion is taken over the object plane. The above expression has some interesting features; namely, the “intensity matrix” composed of the element anm mentioned above is a positve-definite Hermitian matrix, and the diagonal elements are given by the intensities sampled at every point of the image plane separated by the distance λ/2a, and the trace of the matrix or the sum of diagonal elements is equal to the total intensity integrated over the image plane. Since an Hermitian matrix can be reduced to diagonal form by a unitary transformation, the intensity distribution of the image can be expressed as _??_ where λ1, λ2……λn, ……are non-negative eigenvalues of the intensity matrix. In case of coherent illumination, only the first term of the above equation remains and all the other terms are zero, because the rank of the coherent intensity matrix is one, and its only non-vanishing eigenvalue is equal to the total intensity of the image. On the other hand, the rank of the incoherent intensity matrix is larger than the rank of any other coherent or partially coherent cases. The term of the largest eigenvalue in the above formulation may be especially important, because it will correspond to the coherent part of the image in case of partially coherent illumination. From the intensity matrix of the image obtained by uniform illumination of the object having uniform transmission coefficient, we may derive an interesting quantity, namely _??_ where λn is the n-th eigenvalue of the intensity matrix and I0 is the trace of the matrix. d is zero for the coherent illumination and becomes log N for the incoherent illumination, where N is the “degree of freedom” of the image of the area S, namely, N=4a2S/λ2. The value of d for partially coherent illumination is a posititve quantity smaller than log N. A quantity δ=(d0-d)/d0 may be regarded as a measure of the “degree of coherence” of the illumination, where d0=log N and δ is unity for the coherent case and zero for perfectly incoherent case. The sampling theorem for the intensity distribution is derived, and the relation between elements of intensity matrix and intensities sampled at every point separated by the dis-tance λ/4a is shown.
Studies on information volume and resolving power of ideal lens, emulsion and their combination are given. (I) Chart contrast versus resolving power characteristics of ideal lens and photograhic emulsion, (II) resolving power of emulsion-emulsion combination and (III) resolving power of ideal lens-emulsion combination are considered theoretically and the results of calculation are compared with hitherto published data.
Calculation and considerations on the response function of photographic lens (where aberration beyonds 2λ) are given. Response function for some types of aberration and different focus positions are calculated numerically from geometro-optical response. Results are represented by contour lines in focus position versus spatial frequency at the following three cases. 1. Primary spherical aberration alone. 2. Primary and secondary aberrations, and 3. Primary, secondary and tertiary aberrations. Response function are evaluated as their integrant with weighting function (receptor's response function), consideration on position of the best focus and the best correction of spherical aberration are described.
As a method of measuring the response function of photographic lenses, the image scanning method is investigated. A photoelectric photometer suitable for this purpose is constructed and a photographic lens is tested with a square wave line patterned object and is evaluated by a square wave response function. The change in wave form of the image with frequencies is also explained by using the response function.
In optical computations on skew rays, there are various methods besides the trigonometrical ray tracing method which is generally used, but they are for special cases and more or less complicated. With respect to an optical system composed of spherical surfaces and planes, a trial is made in this paper to establish a systematic method of computations that is suited for skew ray tracing and calculations of aberration. The attempt seems successful to some extent by choosing three independent variables for the entrance ray enabling the classification of aberration terms by their orders and thereby deriving adequate variables and simple formulas for the ray tracing. The results are available to differential calculations, the method of which is left for future publication.