Abstract

Bounds on the intensity and the variation of the far field <i>F</i>(<i>u,v</i>) of a given object ƒ(<i>x,y</i>) are developed in terms of the energy and the area of the object. An amplitude function ƒ(<i>x,y</i>) is determined for maximizing <i>F</i>(<i>u</i><sub>0</sub><i>v</i><sub>0</sub>). The results are extended to objects with circular symmetry. The analysis is applied to the following apodization problem: Given a pupil of specified boundary <i>R</i>, a transmission function ƒ(<i>x,y</i>) of energy <i>E</i> is sought such that the energy of its far field in a region <i>S</i> of the <i>u</i>-<i>v</i> plane is maximum.

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959).
  2. A. Papoulis, MRI Symposium on Generalized Networks (Polytechnic Inst. of Brooklyn Press, 1967).
  3. A. Papoulis, The Fourier integral and its Applicatins (McGraw—Hill Book Co., New York, 1962).
  4. I. N. Sneddon, Fourier Transforms (McGraw—Hill Book Co., New York, 1951).
  5. Lord Rayleigh, Phil. Mag. (5), 11, 214 (1881). See also Ref. 1, p. 397.
  6. G. Petian, La Théorie des Fonctions de Bessel (Centre National de la Recherche Scientifique, Paris, 1955).
  7. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill Book Co., New York, 1951).
  8. A. Papoulis, IEEE Trans. Information Theory IT-11, 4 (1965).
  9. D. Slepian, Bell System Tech. J. 43, 3009 (1964).
  10. p. Jacquinot and B. Roizen-Dossier in Progress in Optics III, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1964).
  11. R. Barakat, J. Opt. Soc. Am., 52, 264 (1962).
  12. The function 2πaD(w,w0) is called the directrix by Lansraux and Boivin.13
  13. G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).
  14. D. Slepian, H. O. Pollak, and H. J. Landau, Bell System Tech. J. 40, 43 (1961).

Barakat, R.

R. Barakat, J. Opt. Soc. Am., 52, 264 (1962).

Boivin, G.

G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959).

Jacquinot, p.

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

Landau, H. J.

D. Slepian, H. O. Pollak, and H. J. Landau, Bell System Tech. J. 40, 43 (1961).

Lansraux, G.

G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill Book Co., New York, 1951).

A. Papoulis, The Fourier integral and its Applicatins (McGraw—Hill Book Co., New York, 1962).

A. Papoulis, IEEE Trans. Information Theory IT-11, 4 (1965).

A. Papoulis, MRI Symposium on Generalized Networks (Polytechnic Inst. of Brooklyn Press, 1967).

Petian, G.

G. Petian, La Théorie des Fonctions de Bessel (Centre National de la Recherche Scientifique, Paris, 1955).

Pollak, H. O.

D. Slepian, H. O. Pollak, and H. J. Landau, Bell System Tech. J. 40, 43 (1961).

Rayleigh, Lord

Lord Rayleigh, Phil. Mag. (5), 11, 214 (1881). See also Ref. 1, p. 397.

Roizen-Dossier, B.

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

Slepian, D.

D. Slepian, H. O. Pollak, and H. J. Landau, Bell System Tech. J. 40, 43 (1961).

D. Slepian, Bell System Tech. J. 43, 3009 (1964).

Sneddon, I. N.

I. N. Sneddon, Fourier Transforms (McGraw—Hill Book Co., New York, 1951).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959).

Other

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959).

A. Papoulis, MRI Symposium on Generalized Networks (Polytechnic Inst. of Brooklyn Press, 1967).

A. Papoulis, The Fourier integral and its Applicatins (McGraw—Hill Book Co., New York, 1962).

I. N. Sneddon, Fourier Transforms (McGraw—Hill Book Co., New York, 1951).

Lord Rayleigh, Phil. Mag. (5), 11, 214 (1881). See also Ref. 1, p. 397.

G. Petian, La Théorie des Fonctions de Bessel (Centre National de la Recherche Scientifique, Paris, 1955).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill Book Co., New York, 1951).

A. Papoulis, IEEE Trans. Information Theory IT-11, 4 (1965).

D. Slepian, Bell System Tech. J. 43, 3009 (1964).

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

R. Barakat, J. Opt. Soc. Am., 52, 264 (1962).

The function 2πaD(w,w0) is called the directrix by Lansraux and Boivin.13

G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).

D. Slepian, H. O. Pollak, and H. J. Landau, Bell System Tech. J. 40, 43 (1961).

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