Abstract

In this paper the relation between wave optics and geometrical optics is studied by using Fourier analysis. The geometric optical intensity distribution of light in the point image Ig(x,y) is expressed by

Ig(x,y)=a1|2ϕu22ϕυ2(2ϕuυ)2|1,
where ϕ(u,υ) is the aberration function and (x,y), (u,υ) are the normalized coordinates in the image plane and the exit pupil, respectively, a being the area of the (u,υ) region. Hereupon the geometric optical response function Rg(s,t) is given by the Fourier transform of Ig(x,y), that is;
Rg(s,t)=a1Aexp[2πi(su+tυ)ϕ]uυ,
where (s,t) are the normalized line frequencies.

Then the foregoing formula for Rg(s,t) is compared with the wave optical response function which H. H. Hopkins has deduced, and it is shown that, when the magnitude of the aberration function ϕ(u,ϕ) is small compared with the wavelength, 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,ϕ) becomes large compared with the wavelength, Iw(x,y) is given approximately by Ig(x,y), if Ig(x,y) does not become infinite at any point. In the case where Ig(x,y) is infinite at certain points, if we neglect the high-frequency Fourier component of the intensity distribution, Iw(x,y) is nearly equal to Ig(x,y).

© 1958 Optical Society of America

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