Abstract

If a field g (x ,z) satisfies the diffusion equation ∂<sup>2</sup>g /∂x<sup>2</sup>+2<i>j k</i> (∂g /∂z)=0, then its ambiguity function <i>X</i>(x ,v ,z)=∫<sup>∞</sup>g (η+x/2,z)g <sup>*</sup> (η-x/2,z)e <sup>-jrη</sup><i>d</i>η satisfies the wave equation <i>v</i><sup>2</sup>(∂<sup>2</sup><i>X</i>/∂<i>x</i><sup>2</sup>)-<i>k</i><sup>2</sup>(∂<sup>2</sup><i>X</i>/∂<i>z</i><sup>2</sup>)=0. A theory of Fresnel diffraction and Fourier optics results, involving merely coordinate transformations of the independent variables of the aperture ambiguity function. As an application, a simple expression for the width of the diffracted beam is derived in terms of certain moments of the amplitude of the incident wave. The analysis is extended to signals crossing a layer of a random medium. At the exit plane, the field is partially coherent and it spreads as it propagates. The broadening of beam width due to the loss in coherence is related to the statistical properties of the layer.

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 382.
  2. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 316.
  3. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), p. 26.
  4. Reference 2, p. 358.
  5. P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, New York, 1953), p. 49.
  6. R. O. Harger, Synthetic Aperture Radar Systems; Theory and Design (Academic, New York, 1970).
  7. Reference 2, p. 346.
  8. D. E. Vackman, Sophisticated Signals and the Uncertainty Principle, translated from Russian (Springer, New York, 1958), p. 49.
  9. A. W. Rihaczek, Principles of High-ResolutionR adar (McGraw-Hill, New York, 1969), p. 118.
  10. Reference 2, p. 416.
  11. J. W. Goodman, INtroduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 83.
  12. A. Papoulis, J. Opt. Soc. Am. 58, 653 (1968).
  13. Reference 8, p. 53.
  14. A. Papoulis, J. Opt. Soc. Am. 62, 1423 (1972).
  15. C. W. McCutchen, J. Opt. Soc. Am. 99, 1163 (1969).
  16. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 282.
  17. Reference 16, p. 377.
  18. Reference 16, p. 476.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 382.

Goodman, J. W.

J. W. Goodman, INtroduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 83.

Harger, R. O.

R. O. Harger, Synthetic Aperture Radar Systems; Theory and Design (Academic, New York, 1970).

McCutchen, C. W.

C. W. McCutchen, J. Opt. Soc. Am. 99, 1163 (1969).

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), p. 26.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 282.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 316.

A. Papoulis, J. Opt. Soc. Am. 62, 1423 (1972).

A. Papoulis, J. Opt. Soc. Am. 58, 653 (1968).

Rihaczek, A. W.

A. W. Rihaczek, Principles of High-ResolutionR adar (McGraw-Hill, New York, 1969), p. 118.

Vackman, D. E.

D. E. Vackman, Sophisticated Signals and the Uncertainty Principle, translated from Russian (Springer, New York, 1958), p. 49.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 382.

Woodward, P. M.

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, New York, 1953), p. 49.

Other (18)

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 382.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), p. 316.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), p. 26.

Reference 2, p. 358.

P. M. Woodward, Probability and Information Theory with Applications to Radar (Pergamon, New York, 1953), p. 49.

R. O. Harger, Synthetic Aperture Radar Systems; Theory and Design (Academic, New York, 1970).

Reference 2, p. 346.

D. E. Vackman, Sophisticated Signals and the Uncertainty Principle, translated from Russian (Springer, New York, 1958), p. 49.

A. W. Rihaczek, Principles of High-ResolutionR adar (McGraw-Hill, New York, 1969), p. 118.

Reference 2, p. 416.

J. W. Goodman, INtroduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 83.

A. Papoulis, J. Opt. Soc. Am. 58, 653 (1968).

Reference 8, p. 53.

A. Papoulis, J. Opt. Soc. Am. 62, 1423 (1972).

C. W. McCutchen, J. Opt. Soc. Am. 99, 1163 (1969).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), p. 282.

Reference 16, p. 377.

Reference 16, p. 476.

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