Abstract

Sufficient conditions for the application of the Fresnel approximation of the two-dimensional Kirchhoff diffraction formula are derived for spatially band-limited input signals diffracted by a slit. The method used is an extension of the one given by Papoulis, who derives conditions only for input signals that are sufficiently smooth. For band-limited input signals, the Fresnel region can have an upper limit for the axial extent, depending on the maximum frequency in the input signal.

© 1975 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968).
  2. A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956).
  3. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Erdélyi, A.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968).

Papoulis, A.

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

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968).

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956).

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

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Figures (4)

FIG. 1
FIG. 1

Diffraction by a slit.

FIG. 2
FIG. 2

Fresnel region (shaded) for two different values of the aperture L and the spatial cut-off frequency F of the input signal. (a) λF = 10−3, L = 2 × 104λ; (b) λF = 10−2, L = 10 × 104λ. The regions to the right of the curves A are the Fresnel regions, taking into account the physical aperture L only. The rest of the Fresnel region is based on the fictive aperture. The regions between the corresponding curves B and C are the transition regions where only a part of the fictive aperture is within the physical aperture.

FIG. 3
FIG. 3

Stationary point xs and the contributing aperture interval 4(λz0)1/2 for a single-frequency input signal.

FIG. 4
FIG. 4

Fresnel region (shaded) on the z0 axis for a band-limited signal with a maximum frequency F and no aperture.

Equations (16)

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υ ( x 0 , z 0 ) = L / 2 + L / 2 g ( ρ ) u ( x ) exp ( j k ρ ) d x ,
g ( ρ ) = 1 2 ( λ ρ ) 1 / 2 ( 1 + z 0 / ρ ) exp ( j 1 4 π ) .
u ( x ) = F + F U ( f ) exp ( 2 π j x f ) d f ,
υ ( x 0 , z 0 ) = F + F U ( f ) L / 2 + L / 2 g ( ρ ) exp ( j k ρ + 2 π j x f ) d x d f .
k ρ = k z 0 + k ( x x 0 ) 2 2 z 0 k ( x x 0 ) 4 8 z 0 3 + .
z 0 / λ > 25 1 / 3 ( | x 0 | + 1 2 L λ ) 4 / 3 ,
x = x s = x 0 λ f z 0 { 1 λ 2 f 2 } 1 / 2 .
x s x 0 λ f z 0 .
2 λ p 2 [ ( 1 + z 0 p / λ ) 1 / 2 λ f ] < x x s < 2 λ p 2 [ ( 1 + z 0 p / λ ) 1 / 2 + λ f ] ,
2 ( λ z 0 ) 1 / 2 < x x s < + 2 ( λ z 0 ) 1 / 2 .
| x x 0 | < λ F z 0 + 2 ( λ z 0 ) 1 / 2 .
1 400 ( λ F ) 4 { 1 ( 1 40 λ F ) 1 / 2 } 4 < z 0 / λ < 1 400 ( λ F ) 4 { 1 + ( 1 40 λ F ) 1 / 2 } 4 .
| x 0 | 1 2 L < λ z 0 F + 2 ( λ z 0 ) 1 / 2 .
υ ( x 0 , z 0 ) = g ( z 0 ) exp ( j k z 0 ) × L / 2 + L / 2 u ( x ) exp ( j k ( x x 0 ) 2 2 z 0 ) d x .
2 λ F z 0 + 4 ( λ z 0 ) 1 / 2 < 1 / ( 10 F ) ,
4 ( λ z 0 ) 1 / 2 < 1 / ( 10 F ) .