Abstract

A generalization of the Fresnel approximation in diffraction theory is proposed. The phase term in the diffraction integral is approximated by a parabolic variation, not by a binomial expansion but rather by matching up at the critical points for asymptotic evaluation of the integral. The method provides a correction to the optical coordinates of the Fresnel diffraction theory that extends its region of validity. It is applied to diffraction of an inclined plane wave by a half-plane.

© 1991 Optical Society of America

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References

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  1. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).
  2. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  3. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  4. B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed. (Clarendon, Oxford, 1950).
  5. C. J. R. Sheppard, M. Hyrnevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” submitted to J. Opt. Soc. Am. A.

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Baker, B. B.

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed. (Clarendon, Oxford, 1950).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Copson, E. T.

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed. (Clarendon, Oxford, 1950).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

Hyrnevych, M.

C. J. R. Sheppard, M. Hyrnevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” submitted to J. Opt. Soc. Am. A.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

Sheppard, C. J. R.

C. J. R. Sheppard, M. Hyrnevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” submitted to J. Opt. Soc. Am. A.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Other (5)

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed. (Clarendon, Oxford, 1950).

C. J. R. Sheppard, M. Hyrnevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” submitted to J. Opt. Soc. Am. A.

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Figures (2)

Fig. 1
Fig. 1

Geometry of diffraction of a plane wave by a half-plane.

Fig. 2
Fig. 2

Matching of the phase variation in the diffraction integral by a parabola.

Equations (13)

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U ( x , y , z ) = i k 2 π S exp ( i k R ) R d x d z .
U ( x , y , 0 ) = i k 2 π × S exp ( i k { x cos α + [ ( x x ) 2 + y 2 + z 2 ] 1 / 2 } ) [ ( x x ) 2 + y 2 + z 2 ] 1 / 2 × d x d z .
I = + exp ( i k d 2 + z 2 ) d z d 2 + z 2 .
z = d sinh θ ,
I = + exp ( i k d cosh θ ) d θ = i π H 0 ( 1 ) ( k d ) ,
I = 2 π i k d exp ( i k d ) ,
U ( x , y ) = k 2 π i 0 exp ( i k { x cos α + [ ( x x ) 2 + y 2 ] 1 / 2 } ) [ ( x x ) 2 + y 2 ] 1 / 4 d x .
x = r cos θ , y = r sin θ ,
U ( r , θ ) = k 2 π i 0 exp { i k [ x cos α + ( x 2 2 x r cos θ + r 2 ) 1 / 2 ] } [ ( x 2 2 x r cos θ + r 2 ) 1 / 2 ] 1 / 2 d x .
x = r sin ( θ α ) / sin α
U ( r , θ ) = k π r i 1 1 + sin θ sin α × 0 exp { i k [ x 2 sin 2 α 2 r cos 2 ( θ α 2 ) 2 x tan ( θ α 2 ) + r ] } d x .
F ( p ) = p exp ( i q 2 ) d q ,
U ( r , θ ) = exp ( i π / 4 ) π 1 + cos ( θ α ) sin α ( sin θ + sin α ) × exp [ i k r cos ( θ α ) ] F [ ( 2 k r ) 1 / 2 sin ( θ α 2 ) ] .

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