Abstract

Formulas are developed that permit the calculation of the diffraction images of truncated, one-dimensional, periodic objects in the presence of incoherent illumination using transfer function theory. Special attention is paid to the important problem of determining the minimum number of cycles that such a target must possess in order for it to act effectively as an infinite periodic target. Illustrative calculations are carried out for aberration-free, defocused systems.

© 1967 Optical Society of America

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References

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  1. J. Pastor, Opt. Acta 9, 237 (1962).
    [CrossRef]
  2. R. Barakat, A. Houston, J. Opt. Soc. Am. 53, 1371 (1963).
    [CrossRef]
  3. W. Charman, Phot. Sci. Eng. 8, 253 (1964).
  4. R. Barakat, M. Morello, J. Opt. Soc. Am. 52, 992 (1962).
    [CrossRef]
  5. R. Barakat, A. Houston, J. Opt. Soc. Am. 55, 1142 (1965).
    [CrossRef]

1965

1964

W. Charman, Phot. Sci. Eng. 8, 253 (1964).

1963

1962

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Figures (8)

Fig. 1
Fig. 1

Three-cycle square wave and sine wave objects.

Fig. 2
Fig. 2

Diffraction images of truncated sine wave objects (L = 1) as seen through the aberration-free system in the Fraunhofer receiving plane W2 = 0: – · – three-cycle target, — five-cycle target, - - - seven-cycle target, – · · – nine-cycle target.

Fig. 3
Fig. 3

Diffraction images of truncated square wave objects (L = 1) as seen through the aberration-free system in the Fraunhofer receiving plane W2 = 0: – · – three-cycle target, — five-cycle target, - - - seven-cycle target, – · · – nine-cycle target.

Fig. 4
Fig. 4

Diffraction images of truncated sine wave objects (L = 1) as seen through the defocused aberration-free system in the receiving plane W2 = 0.5 λ: – · – three-cycle target, — five-cycle target, - - - seven-cycle target, – · · – nine-cycle target.

Fig. 5
Fig. 5

Diffraction images of truncated square wave objects (L = 1) as seen through the defocused aberration-free system in the receiving plane W2 = 0.5 λ: – · – three-cycle target, — five-cycle target, - - - seven-cycle target, – · · – nine-cycle target.

Fig. 6
Fig. 6

Diffraction images of both truncated square wave and sine wave objects (L = 1) as seen through the defocused aberration-free system in the receiving plane W2 = 1.0 λ: – · – three-cycle targets, — five-cycle targets, - - - seven-cycle targets. The scale on which the graph is drawn renders impossible the distinction between square and sine wave objects.

Fig. 7
Fig. 7

Comparison of diffraction images of three-, five-, and seven-cycle sine wave and square wave targets (L = 1) for the defocused aberration-free system in the receiving plane W2 = 0.5 λ; — square wave, - - - sine wave.

Fig. 8
Fig. 8

Comparison of diffraction images of three-, five-, and seven-cycle sine wave and square wave targets (L = 1.5) for the defocused aberration-free system in the receiving plane W2 = 0.5 λ: – square wave, - - - sine wave.

Tables (2)

Tables Icon

Table I Modulation of Truncated Sine and Square Wave Targets (L = 1) for the Aberration-Free System in Fraunhofer Receiving Plane, W2 = 0

Tables Icon

Table II Modulation of Truncated Sine and Square Wave Targets (L = 1) for the Defocused (W2 = 0.5 λ) Aberration-Free System

Equations (18)

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o ( v ) = 0 ( - < v < - 5 L ) 1 ( - 5 L < v < - 3 L ) 0 ( - 3 L < v < - L ) 1 ( - L < v < L ) 0 ( L < v < 3 L ) 1 ( 3 L < v < 5 L ) 0 ( 5 L < v < ) .
O ( ω ) = 1 2 π - O ( v ) exp ( - i v ω ) d v .
O ( ω ) = ( π ω ) - 1 ( sin ω L - sin 3 ω L + sin 5 ω L ) .
O ( ω ) = ( - 1 ) N - 1 π ω l = 1 N ( - 1 ) l - 1 sin ( 2 l - 1 ) ω L .
O ( ω ) = ( 2 π ω ) - 1 [ ( sin 2 N ω L ) / cos ω L ] .
O ( v ) = 1 2 [ 1 + ( - 1 ) N + 1 cos π v 2 L ] ( v 2 N L ) 0 ( v > 2 N L )
O ( ω ) = π 2 ω ( sin 2 N ω L ) / ( π 2 - 4 ω 2 L 2 ) .
I ( ω ) = T ( ω ) O ( ω ) ,
i ( v ) = - 2 2 T ( ω ) O ( ω ) exp ( i v ω ) d ω .
i ( v ) = 1 π 0 2 cos v ω sin ( 2 N ω L ) T ( ω ) ω cos ω L d ω ,
i ( v ) = π 0 2 cos v ω sin ( 2 N ω L ) T ( ω ) ω ( π 2 - 4 ω 2 L 2 ) d ω .
i ( v ) 1 π 0 2 cos v ω sin ( 2 N ω L ) T ( ω ) ω d ω .
( cos ω L ) - 1 1 + ( 1 / 2 ) ( ω L ) 2 + ,
( π 2 - 4 ω 2 L 2 ) - 1 1 / π 2 [ 1 + ( 4 / π 2 ) ( ω L ) 2 + ] .
i sq 1 π 0 2 cos v ω sin ( 2 N ω L ) ω T ( ω ) d ω + L 2 2 π 0 2 ω cos v ω sin ( 2 N ω L ) T ( ω ) d ω +
i si 1 π 0 2 cos v ω sin ( 2 N ω L ) ω T ( ω ) d ω + 4 L 2 π 3 0 2 ω cos v ω sin ( 2 N ω L ) T ( ω ) d ω + .
i sq - i si = π 2 - 8 2 π 3 L 2 0 2 ω cos v ω sin ( 2 N ω L ) T ( ω ) d ω +
modulation = I ˜ max - I ˜ min I ˜ max + I ˜ min ,

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