照明のコヒーレンス性を考慮した光学像の強度分布に関する理論

強度のエルミート行列による表示

抄録

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 a_nm 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……λ_n, ……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 λ_n 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.