Recently D. Gabor in England and the writer have independently reached the same conclusion, namely, that the intensity distribution of images can be described by using positive-definite Hermitian matrices under various conditions of illumination. According to the writer's results, the intensity distribution of an image obtained by a pupil of numerical aperture α can be described as the following quadrature form, I(x; α)=(Ø(x; α), AØ (x; α)), (1) where A is the intensity matrix having n, m-element Anm=∫∫Γ12 (X1, X12) E* (X1) E(X2) u* (X1-nπ/kα)u (X2-mπ/kα) dX1dX2, (2) and ø (x; α) is the vector, n-th component of which is given by the n-th sampling function for the image obtained by the pupil, that is, øn(x; α)=sin(kαx-nπ)/(kαx-nπ). (3) It is shown in this paper that the given intensity matrix A is transformed into another matrix B when the wave in an image by the pupil is transmitted through another pupil of numerical aperture β, namely, B=T'*AT, (4) where T is the transmission matrix concerning the second pupil and T'* is the matrix, m, n-element of which is equal to the complex conjugate of n, m-element of the above matrix T. The equation (4) can be derived as follows; since the transmission function u of the combined system of the two pupils mentioned above is given by the well-known convolution integral of transmission functions u and u' of the first and second pupils respectively, namely, u(X-x)=∫+∞-∞u(X-ξ)u'(ξ-x)dξ, (5) and the first transmission function u(X-ξ) can be expressed in series by using the sampling theorem for the image by the pupil of numerical aperture α, the transmission factor u(X-nπ/kβ) in the matrix element Bnm is expressed as u(X-nπ/kβ)=+∞Σn=-∞u(X-nπ/kα)Tnn', (6) and Tnn'=∫un(ξα)u'(ξ-n'π/kβ)dξ, (7) where un(ξ; α) is the n-th sampling function for the image by the said pupil. Inserting the equation (6) into the n, m-element of intensity matrix B, which is of the form similar to the equation (2), the equation (4) can be obtained, where the transmission matrix T is given by the element Tnn' obtained above. In the case of α≥β, the element of transmission matrix can be expressed as Tnn'=(π/kα)u'(nπ/kα-n'π/kβ), (8) that is, the n, n'-element of the transmission matrix indicates how much complex amplitude of wave can be produced at the n'-th sampling point in the second image plane by the wave having unit amplitude at the n-th sampling point in the first image plane.
The relation between wave optics and geometrical optics is studied by using the Fourier analysis. The geometric optical intensity distribution of light Ig(x, y) in a point image is expressed by Ig(x, y)=_??_ where φ(u, v) is the aberration function and (x, y), (u, v) are the normalized coordinates on the image plane and exit pupil respectively, A being the area of (u, v) region. The geometric optical response function Rg (s, t) is given by the Fourier transform of Ig(x, y), that is, Rg (s, t)=_??_ dudv, where (s, t) is the normalized line frequency. The above formula of Rg (s, t) is then compared with the wave optical response function deduced by Hopkins, and it is shown that, when the magnitude of the aberration function φ(u, v) is small compared with the wave length, the wave optical intensity distribution Iw(x, y) can be expressed approximately by the convolution of Ig(x, y) and the intensity distribution of the aberration-free diffraction image, and when the magnitude of φ(u, v) becomes large compared with the wave length, Iw(x, y) is given approximately by Ig(x, y), if Ig(x, y) does not become infinite at any point. In the case in which Ig(x, y) becomes infinite at certain points, if we neglect the high frequency Fourier component of their intensity distribution, Iw(x, y) becomes nearly equal to Ig(x, y).
Experiments on “relative information volume” (VIr, ) I, IV) and considerations on their results are described. In the evaluation of photographic lens, it is necessary to consider it in connection withh the characteristics of receptor, as shown in the auther's previous papers1). VIr, is obtained by a simple apparatus (Fig. 4). Area type response slit (Fig. 3: receptor's response) is convolutioned with the line image made by the lens under test (Fig. 1). By this method, integrated value of the response function for lens-receptor combination can be obtained directly. Experimental results obtained by using several lenses show clearly the absolute value of VIr, depth of focus and displacement of best focus by stop down etc. (Fig. 6 and Fig. 7) These results coincide well with the theoretical results which is described in report IV.
Routine colorimetric work is by spectrophotometric measurement and calculation of tristimulus value for which hand worked colorimetric computors have gradually come into general use. While the comparison of accuracy of various methods of colorimetric calculation has been made by a number of investigators (Nickerson, Azuma, Okada and others), no report on the reading accuracy of colorimetric computor has so far been brought to notice. Investigation has been conducted on the personal errors of three female assistants of the age 20_??_30 and of over one year's experience using a thirty selected ordinate colorimetric computor and samples 7.5R 4/8, 7.5R 3/4, 7.5G 4/6, 7.5G 3/4, 7.5PB 5/6 and 7.5B 3/2. The computor and spectral characteristic curves of the samples are shown in figures (Fig. 1 and Fig. 2). Observed tristimulus, their means and standard errors are summarized in tables and figure (Tables 1 and 2, Fig. 3). The personal error by experienced observers appears to be around ±0.0009. Hence, for colors darker than value 3/, the error may surpass 1 NBS unit unless good care is taken. The reading error and the chromaticity error are presented by simple formulations (8.1 and 8.2).
Approximate relations between tristimulus values (X, Y and Z) of color and color temperature (T) of Planckian Radiation in the C. I. E. trichromatic system are investigated with numerous colors. Following relations X=ax+bx (1/T×106)+cx(1/T×106)2 Y=ay+by (1/T×106)+cy(1/T×106)2 Z=az+bz (1/T×106)+cz(1/T×106)2 when T≈3000_??_7000°K are found. For these empirical equations, some theoretical considerations are made. These quadratic forms can be adopted for the calculation of tristimulus values of color at any color temperature of Planckian Radiation.
For an obliquely incident light, a silver interference filter has two transmission bands of different wavelengths, one of which corresponds to the polarization, parallel and the other perpendicular to the plane of incidence. So the optical train of a polarizer, an oblique silver interference filter and a rotating analyser driven by a motor transmits the lights of two wavelengths alternately. The authors constructed a light comparator for two wavelengths composed of this optical train, a photocell, an amplifier and a phase detector. If the interference filter is embedded obliquely in a transparent block of high index medium, the angle of refraction in the dielectric layer of the filter becomes so large that the gaps of the two transmission bands are widely separated. A phase compensating plate is inserted into this optical train to make the transmitted light circularly polarized in the wavelength region in. which the two transmission bands overlap one another. This instrument can be applied both to monitoring the thickness of every layer of dielectric multi-layer interference filters in course of evaporation in vacuum and to measuring color temperatures of light sources with high precision. For the latter application, discussion is made how to determine the two wavelengths to obtain the highest sensitivity and, moreover, the highest accuracy according to the definition of color temperature.
Properties of thin oxide films of Cd on glass are studied. Thickness of the films is determined by Strong's method. It was found neccessary to evaporate cadmium over a previously evaporated thin film of chromium to ensure even deposit. Thin Cd films deposited on glass are found to have exceptionally high electrical conductivity and optical transmission. Heating enhances these properties, irreversively. As a typical example, Cd film of 5501 in thickness, heated at 300°C, has the specific resistance of 36×10-4Ω. cm and the white-light transmission of 78%. A current of 45 mA through this glass plate (width 2.25cm) raised the temperature of the plate by 60°C (from 5°C). The film has a possible use as transparent conducting coatings for the purpose of de-icing windows by direct electrical heating.
A new distinctness-of-image gloss meter has been constructed with which the distinctnessof-imge gloss of samples is determined by measuring the ratio r of the intensity of specularly reflected light to that of diffusely reflected in the direction deviated by a small angle s from the former. The relation between r and s is first explored in the range from 36' to 72' of s for the incident angle α of 60° (Fig. 3), and then the relation between r and α in the range from 40° to 80° of α for s=45' (Fig. 4).
In appraising the quality of polished plate glass surface, fine roughness down to 200300 Å is called to account, and it can be actually detected by the unaided eye. This is because of the schlieren phenomenon which takes place when the human eye looks at the concerned glass surface. A general wave-optical theory on the schlieren method is developed, taking the resolving power of the viewing system into consideration, and the contrast caused by regular and irregular roughness of the glass surface is estimated. The results shows that the optical instrument such as human eye could easily detect very fine roughness with enough contrast regardless of its resolving power, and that, for inspection of polished plate glass surface, the distance between the surface of the sample and the light-screening plate in the schlieren apparatus has to be near the range of clear vision of the eye. The schlieren patterns of various polished glass surfaces photographed by such an apparatus are compared with the multiple-beam interferograms of the same samples. Requirement for the degree of smoothness of polished glass surface appears to be more exacting than expected from the Rayleigh's λ/4 criterion.
The peculiar properties of a train of three linear polarizers, theoretically treated by R. C. Jones, are experimentally investigated. A combination of three sheet polarizers and a wave plate of adequate relative retardation is used to examine the birefringence of the polarizer. Estimated value of relative retardation of the polarizer (Polaroid Corp. product, H type, t=0.3 mm) is 0.24 λ for a monochromatic light of 440 mμ. Calculated and observed values of the change in the shape of transmittance curve by the principal axes being out of alignment are in fair agreement. Adjustment of the alignment was possible down to 1/2° which was sufficient for the required accuracy.
A goniophotometric spectrometer, consisting of a goniophotometer and a monochrometer of Littrow type, has been constructed to measure detailed reflection characteristics of textile fabrics. -(The characteristics of materials other than textile fabrics, glass for example, are determined with the aid of a monochrometer). With this apparatus, measurement of the intensity of monochromatic beam reflected from the textile fabrics of frosted rayon staple (white colour) is carried out at various angles of incidence as well as reflection. The results obtained are as follows: (1) With the wavelength longer than 500 mμ, maximum intensity of the reflected beam is observed in the direction almost parallel to the plane of the reflecting surface. (2) With the wavelength shorter than 420 mμ, on the other hand, the maximum is observed in the direction of the incident beam. (3) With the increase of wavelength, the reflectivity increases between 420 mμ and ca. 630 mμ and decreases between ca. 630 mμ and 640 mμ. From these observations it may be concluded that the reflection characteristics of textile fabrics of frosted rayon staple are mainly due to the scattering of light by fibres constituting the textile fabrics and by the particles of pigments (TiO2) contained in their surface layer. (4) Denoting the reflectivity in the direction of the angle of specular reflection by Rs, and that in the direction of the normal to the specimen by Rn, the curve of Rs-Rn against wavelength is very similar to that of the contrast gloss at any angle of incidence. Hence, the contrast gloss is a useful measure of the so-called gloss, so far as the textile fabrics of frosted rayon staple of achromatic color is concerned.