Mathematical analysis of intensity distribution of the optical image in various degrees of coherence of illumination (representation of intensity by Hermitian matrices)

Abstract. 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 un(y)  =  sin 2πα/λ (y  −  nλ/2α) / 2πα/λ (y  −  nλ/2α) and anm  =  (2α/λ)2  ∬  Γ12(x1  −  x2) E(x1) E* (x2)  |  A(x1)  ||  A* (x2)  |  un(x1)um(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 anm 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|∑Si1ui|2+λ2|∑Si2ui|2+⋯+λn|∑Sinui|2+⋯,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 d=−∑n(λn/I0)log(λn/I0),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  =  4α2S  /  λ2. The value of d for partially coherent illumination is a positive 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 the 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 distance λ  /  4α is shown.