This is a historical translation of the seminal paper by H. Gamo, originally published in Oyo Buturi (Applied Physics, a journal of The Japan Society of Applied Physics) Vol. 25, pp. 431–443, 1956. English translation by Kenji Yamazoe, with further editing by the translator and Anthony Yen.

Since optical systems have distinctive features as compared to electrical 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 intensity distribution of the image by an optical system having a given aperture constant α in the absence of both aberration and defect in focusing is obtained by considering the nature of illumination, namely coherent, partially coherent, and incoherent: I(y)=∑n∑manmun(y)um*(y),where u_n(y)  =  sin 2πα/λ (y  −  nλ/2α) / 2πα/λ (y  −  nλ/2α) and a_nm  =  (2α/λ)²  ∬  Γ₁₂(x₁  −  x₂) E(x₁) E^* (x₂)  |  A(x₁)  ||  A^* (x₂)  |  u_n(x₁)u_m(x₂)dx₁ dx₂. I(y) is the intensity of the image at a point of coordinate y, Γ₁₂ the phase coherence factor introduced by H. H. Hopkins et al., E  (  x  )   the complex transmission coefficient of the object and A  (  x  )   the complex amplitude of the incident waves at the object, and the integration is taken over the object plane. The above expression has some interesting features, namely the “intensity matrix” composed of the element a_nm mentioned above is a positive-definite Hermitian matrix, and the diagonal elements are given by the intensities sampled at every point of the image plane separated by a distance λ  /  2α, and the trace of the matrix or the sum of diagonal elements is equal to the total intensity integrated over the image plane. Since a Hermitian matrix can be reduced to diagonal form by a unitary transformation, the intensity distribution of the image can be expressed as I(y)=λ1|∑S