Abstract

The predicted sharp cutoff of the transfer function of a coherent imaging system is not seen experimentally. The experiment suggests that this is due to the neglect of the diffraction, caused by the finite lens, of the individual plane-wave components of the angular spectrum of the object wave. It is shown that this omission is equivalent to dropping a quadratic phase term in the description of the image obtained with the Fresnel–Kirchhoff diffraction formula.

© 1974 Optical Society of America

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References

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  1. See for example, J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968).
  2. W. T. Cathey, J. Opt. Soc. Am. 60, 738A (1970).
  3. W. T. Cathey, J. Opt. Soc. Am. 60, 1552A (1970).
  4. J. A. Ratcliffe, in Reports on Progress in Physics, Vol. 19, edited by A. C. Strickland (Physical Society, London, 1956), pp. 188–267.
    [Crossref]
  5. S. Herman, J. Opt. Soc. Am. 61, 1428 (1971).
    [Crossref]
  6. W. T. Cathey, Optical Information Processing and Holography, (Wiley, New York, 1974).

1971 (1)

1970 (2)

W. T. Cathey, J. Opt. Soc. Am. 60, 738A (1970).

W. T. Cathey, J. Opt. Soc. Am. 60, 1552A (1970).

Cathey, W. T.

W. T. Cathey, J. Opt. Soc. Am. 60, 738A (1970).

W. T. Cathey, J. Opt. Soc. Am. 60, 1552A (1970).

W. T. Cathey, Optical Information Processing and Holography, (Wiley, New York, 1974).

Goodman, J. W.

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

Herman, S.

Ratcliffe, J. A.

J. A. Ratcliffe, in Reports on Progress in Physics, Vol. 19, edited by A. C. Strickland (Physical Society, London, 1956), pp. 188–267.
[Crossref]

J. Opt. Soc. Am. (3)

W. T. Cathey, J. Opt. Soc. Am. 60, 738A (1970).

W. T. Cathey, J. Opt. Soc. Am. 60, 1552A (1970).

S. Herman, J. Opt. Soc. Am. 61, 1428 (1971).
[Crossref]

Other (3)

W. T. Cathey, Optical Information Processing and Holography, (Wiley, New York, 1974).

J. A. Ratcliffe, in Reports on Progress in Physics, Vol. 19, edited by A. C. Strickland (Physical Society, London, 1956), pp. 188–267.
[Crossref]

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

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Figures (5)

Fig. 1
Fig. 1

One pair of waves passing through a lens.

Fig. 2
Fig. 2

Tracing the wave pair to find the cutoff frequency.

Fig. 3
Fig. 3

Coherent image of a grating formed by large lens diameter.

Fig. 4
Fig. 4

Coherent image of same grating, formed by lens with smaller diameter than in Fig. 3.

Fig. 5
Fig. 5

Coherent image of same grating, formed by lens with smaller diameter than in Fig. 4.

Equations (19)

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A ( u , v ) = U ( x , y ) exp [ - i 2 π ( u x + v y ) ] d x d y ,
U ( x , y ) = A ( u , v ) exp [ i 2 π ( u x + v y ) ] d u d v ,
w d i - f = D f
w - l d i tan 2 χ .
d i - f f = d i d 0
l = d i d 0 ( D - d 0 tan 2 χ ) .
tan 2 χ = 2 λ u ,
l = d i d 0 ( D - 2 λ d 0 u ) .
u co = D 2 λ d 0 .
u ci = D 2 λ d i .
u co u ci = d i d 0 = M ,
U 3 ( x 3 , y 3 ) = 1 λ 2 d i d 0 h ( x 3 , y 3 ; d i ) U 1 ( x 1 , y 1 ) h ( x 1 , y 1 ; d 0 ) · P ( x 2 , y 2 ) exp { - i 2 π [ x 2 ( x 1 λ d 0 + x 3 λ d i ) + y 2 ( y 1 λ d 0 + y 3 λ d i ) ] } d x 2 d y 2 d x 1 d y 1 ,
h ( x , y ; d ) exp [ i k ( x 2 + y 2 ) 2 d ] ,
U 3 ( x 3 , y 3 ) = d 0 d i h ( x 3 , y 3 ; d i ) P ˜ ( x 1 + d 0 d i x 3 , y 1 + d 0 d i y 3 ) · U 1 ( x 1 , y 1 ) h ( x 1 , y 1 ; d 0 ) d x 1 d y 1 ,
P ˜ ( x 1 λ d 0 + x 3 λ d i , y 1 λ d 0 + y 3 λ d i )
λ 2 d 0 2 P ˜ ( x 1 + d 0 d i x 3 , y 1 + d 0 d i y 3 ) .
U 3 ( x 3 , y 3 ) = 1 M h ( x 3 , y 3 ; d i ) { P ˜ ( x 3 M , y 3 M ) [ U 1 ( - x 3 M , - y 3 M ) h ( x 3 M , y 3 M ; d 0 ) ] } .
A 3 ( u , v ) = - M 3 λ 2 d i d 0 h ( u , v ; - 1 λ 2 d i ) { P ( - M u , - M v ) [ A 1 ( - M u , - M v ) h ( u , v ; - 1 λ 2 d 0 ) ] } .
U 2 ( x 2 , y 2 ) = U 1 ( x 1 , y 1 ) × exp { i k [ ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 2 d ] } d x 1 d y 1 , = U 1 ( x 2 , y 2 ) h ( x 2 , y 2 ; d )