Abstract

Multidimensional Fourier transforms, Parseval’s theorem, and Schwarz’s inequality, used in concert, show how to apodize a lens to minimize the <i>n</i>th moment of the irradiance in the diffraction pattern for a given central irradiance. Replacing Schwarz’s inequality with the calculus of variations, we can find the pupils that minimize quotients of certain different moments. In particular, the mean-square miss distance of a photon gun is minimized if the waves it launches have amplitudes that decrease as the <i>J</i><sub>o</sub> Bessel function moves from the center of the projecting lens and reach zero at the rim.

© 1969 Optical Society of America

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  1. I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Co., New York, 1951), p. 24+.
  2. D. M. Y. Somerville, An Introduction to the Geometry of N Dimensions (Dover Publications, Inc., New York, 1958), p. 135.
  3. Reference 1, p. 44+. I have used three-dimensional Fourier transforms of "generalized" apertures [C. W. McCutchen, J. Opt. Soc. Am. 54, 240 (1964).] and sources [C. W. McCutchen, J. Opt. Soc. Am. 56, 727 (1966).] in studying diffraction images and coherence in three dimensions. The present paper is unconnected with this earlier work. Generalized apertures and sources have a kind of reality in the three dimensions of the real world. The present work concerns only the image in the two dimensions of the focal plane (or at infinity in the limit of ever-increasing focal length), and dimensions beyond the second exist only in the imagination.
  4. G. Lansraux and G. Boivin, Can. J. Phys. 39, 158 (1961).
  5. P. Jacquinot and B. Roizen-Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964), p. 62.

Boivin, G.

G. Lansraux and G. Boivin, Can. J. Phys. 39, 158 (1961).

Jacquinot, P.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964), p. 62.

Lansraux, G.

G. Lansraux and G. Boivin, Can. J. Phys. 39, 158 (1961).

Roizen-Dossier, B.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964), p. 62.

Sneddon, I. N.

I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Co., New York, 1951), p. 24+.

Somerville, D. M. Y.

D. M. Y. Somerville, An Introduction to the Geometry of N Dimensions (Dover Publications, Inc., New York, 1958), p. 135.

Other (5)

I. N. Sneddon, Fourier Transforms (McGraw-Hill Book Co., New York, 1951), p. 24+.

D. M. Y. Somerville, An Introduction to the Geometry of N Dimensions (Dover Publications, Inc., New York, 1958), p. 135.

Reference 1, p. 44+. I have used three-dimensional Fourier transforms of "generalized" apertures [C. W. McCutchen, J. Opt. Soc. Am. 54, 240 (1964).] and sources [C. W. McCutchen, J. Opt. Soc. Am. 56, 727 (1966).] in studying diffraction images and coherence in three dimensions. The present paper is unconnected with this earlier work. Generalized apertures and sources have a kind of reality in the three dimensions of the real world. The present work concerns only the image in the two dimensions of the focal plane (or at infinity in the limit of ever-increasing focal length), and dimensions beyond the second exist only in the imagination.

G. Lansraux and G. Boivin, Can. J. Phys. 39, 158 (1961).

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics III, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1964), p. 62.

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