A lens system may be judged by its ability to relay entropy from object to image. The pertinent criterion of optical quality is h, the loss of entropy between corresponding sampling points in the object and image planes. Since h is a unique function of the optical pupil, depending on the object only through its cutoff frequency Ω, by the proper choice of a pupil function it is possible to maximize h at each given Ω. Physically, the optimum pupil function is an absorbing film applied to a diffraction-limited lens system. A numerical procedure is established for determining, with arbitrary accuracy, the optimum pupil function, the resulting transfer function, and the maximum h, all at a given Ω. These quantities are determined, both for the one-dimensional pupil and the circular pupil, in the approximation that the optimum pupil function may be represented as a Fourier (Bessel) series of five terms. The computed values of hmax, at a sequence of Ω values, are estimated to be correct to 0.2% for the 1-D pupil, and to 0.5% for the circular pupil. The optimum pupil functions are apodizers at small Ω and superresolvers at large Ω.
© 1968 Optical Society of America
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