The matrix formulation for the intensity distribution of images obtained by a circular aperture is derived by using the circle sampling theorem obtained by the writer. By assuming that the amplitude distribution of the wave on a plane is limited within a circle and its Fourier-transform is also limited within another circle, it can be shown that the amplitude distribution mentioned above can be described by both Fourier-Bessel and circle sampling expansions. Then, by using these expansions two types of intensity matrices are derived for the intensity distribution of the above wave. The intensity matrix for the illumination upon the object plane is calculated by using H. H. Hopkins' phase-coherence factor, and intensity matrices for waves on the object plane after transmission of the object, on the planes of entrance and exit pupils and on the image plane are derived by the successive matrix-transformation from the above intensity matrix for the illumination. The elements of transformation matrices are generally obtained by considering the relation between a circle-sampling value (or Fourier-Bessel coefficient) of the waves on the first plane and the circle-sampling values (or Fourier-Bessel coefficients) of the waves on the second plane. The experimental methods determining the elements of an intensity matrix of a given image are discussed. Namely, concerning the circle-sampling type of intensity matrix, a matrix element Ams: nt can be obtained by measuring the intensity at the origin of the Fraunhofer diffraction pattern of waves produced after passing waves of a given image through the “circle sampling filter”, which is transparent on the circumferences of sampling circles having radii λms/ka and λnt/ka and has phase factors exp (-imθ) and exp (-inθ) respectively on these circumferences and intercepts the light outside of these circumferences. The Fourier-Bessel type of intensity matrix can be obtained in the same way-by inserting the circle-sampling filter mentioned above on the exit pupil of the image-forming system.
Spot diagrams give much information about the image formation of an optical system. In order to evaluate the quality of image formed by a lens, it is shown that the geometric optical response function and the intensity distribution can be calculated from the co-ordinates of the spots on the image plane, and the image evaluation by the geometric optical “Strehl definition” is discussed. Furthermore, the relation between wave optical image and geometric optical image is considered in the case of partially coherent illumination, and it is deduced that when the wave aberration is large compared with the wavelength of light (more than 2λ), the degree of coherence on the object plane scarcely influences the intensity distribution on the image plane.
Spectral transmittance of evaporated glasses of several indices is measured with the purpose of applying achromatic coating to photographic lenses. The transmittance is found much higher than that with the conventional monolayer; it is inclined to be filter-cutting for ultraviolet region. From the determined spectral characteristics, coefficients relating to color correction, lens flare and transmittance for the use of color film are obtained. The achromatic coating is especially effective for color photography when used on low index glasses in the lens system in combination with the monolayer reflection-reducing film.
In designing the lenses for projection etc., many designers calculate the aberrations in the directions which make transverse magnifications smaller. Recently, in the measurement of response function, the lens performance is frequently regarded in the direction which makes transverse magnification larger. In both cases the lenses are investigated in the states, in which the object-image correspondences are reversed to the actual conditions. Expressions for the aberrations of the system, in which the object-image correspondence is reversed, can be derived from the ordinary fifth order aberration theory. The results, induced from these expressions, are discussed.
Using the apparatus and methods previously reported by the author1)8), reflection characteristics of several kinds of paper are measured with white and monochromatic beams of light. The results obtained are as follows: 1. From the comparative measurements of printing papers, white and dyed by red ink, it is found that the magnitude of specular component of coated paper scarecely changes by dying and, consequently, experimentally decomposed specular and diffuse components are thought to be theoretically also valid. For uncoated paper, the polar curve of the diffuse component is somewhat deformed from a circle expected by Lambert's law and the specular component has some color. 2. The fact that the specular component does not follow Fresnel's formula for coated papers may be the result of the interference effect of fine clay coated on them. 3. The process of calendering, as regards to the reflection characteristics, corresponds to making the specular component large and making the diffuse component small to nearly the same amount.
Concerning a transparent colored solution, the change of concentration causes the change of luminance and excitation purity. The experiments are intended to make it clear which effect of variation of luminance or that of excitation purity is predominant in color discrimination in such cases. From the results of experiments using colorimeter, it is inferred that, at low concentrations, luminance discrimination predominantes, but the effect of purity is not negligible at concentrations beyond a certain value which depends on colors. Another experiment on the effects of field brightness shows similar results.
It has been believed that in the polarizing microscope no light is passed through the optical system when the polarizers are crossed. But, in practical optical system, the transmittances of the refracting surfaces are different for the components parallel and perpendicular to the plane of incidence and the plane of polarization is rotated during the passage of the system. Accordingly, a little amount of light passes through the crossed Nicols. The diffraction image formed by this leaked light is quite different from the regular Airy disc. It is expressed with [sin 2θ. J3(γ)/r]2, where θ and r are the polar coordinates in the image plane and J3(r) is the third order Bessel function. There is a dark cross across the image and the diffraction pattern i a four-leaf clover. Diffraction image of a coherently illuminated slit with this optical system also shows unusual pattern. Theoretical derivation of these patterns are shown with photographic illustrations.
The following three types of interference filters are devised to obtain monochromatic filters of high transmission and narrow pass band. 1. Higher order interference filter By making use of the second order interference, the transmission band is reduced to less than half of that with the conventional first order interference and the performance of the filter is improved. Using higher order interference, a monochromatic filter of a very narrow pass band is made by superposing a first order interference filter for the same wavelength to suppress unwanted transmission band that would appear in the longer wavelength side. 2. Multilayer transparent interference filter A 15-layer filter with 84% in maximum transmission and 5.2mμ in half band width is made. By increasing the number of layers to 19 and 23 and the order of interference, still narrower band width is obtained. 3. Interference filter made of metallic and transparent films combined Using the merits of metallic and transparent films at the same time, an interference filter of good performance with a small number of layers is made. A 7-layer symmetrical second order interference filter had 38.6% in maximum transmission and 4.2nμ in half band width.
Imaging characteristics of lens with random fluctuation of complex transmittance in the pupil are studied by response function and noise theory. Random fluctuation of light (optical noise) is classified as follows: optical noise according to the condition of illumination. Callier Q-factor is investigated with the idea of amplitude noise. The relation between Q-factor and granularity is discussed. Phase noise is mainly treated in the applications because of its distinctive features on no loss in the amount of light and on the capability of perfect cut-off characteristics in response function. Response function of the combined system with phase noise filter and aberration-free lens could be designed by some factors concerning the random fluctuation of phase just as one wishes. Applications of the phase noise, on the filtering of periodic structure as screen dots or as scanning lines in TV, on the removing of moire in the rescreening and on the soft focus effect in portrait photography, are illustrated.