Abstract

The images formed under incoherent illumination by two gratings in tandem are analyzed using the theory of grating imaging. The images are categorized on the basis of various integer parameters that arise in the analysis. A transfer function is introduced for describing the imaging process. The image positions are related to the foci of a Fresnel zone plate.

© 1985 Optical Society of America

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References

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  1. E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
    [CrossRef]
  2. J. Jahns, A. W. Lohmann, “The Lau effect: a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  3. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
    [CrossRef]
  4. R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
    [CrossRef]
  5. R. Sudol, B. J. Thompson, “Lau effect, theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
    [CrossRef] [PubMed]
  6. G. J. Swanson, E. N. Leith, “Lau effect and grating imaging,” J. Opt. Soc. Am. 72, 552–555 (1982).
    [CrossRef]
  7. G. J. Swanson, “Partially coherent imaging and interferometry based on diffraction gratings,” Ph.D. dissertation (University of Michigan, Ann Arbor, Michigan, 1983;available from University Microfilms, Ann Arbor, Michigan).
  8. K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
    [CrossRef]
  9. B. J. Chang, “Grating based interferometer,” Ph.D. dissertation (University of Michigan, Ann Arbor, Michigan, 1974;available from University Microfilms, Ann Arbor, Michigan).
  10. Ref. 7, p. 37.
  11. K.-H. Brenner, A. Lohmann, J. Ojeda-Casteneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–16 (1983).
    [CrossRef]

1983 (2)

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

K.-H. Brenner, A. Lohmann, J. Ojeda-Casteneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–16 (1983).
[CrossRef]

1982 (1)

1981 (1)

1979 (3)

J. Jahns, A. W. Lohmann, “The Lau effect: a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

1948 (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Brenner, K.-H.

K.-H. Brenner, A. Lohmann, J. Ojeda-Casteneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–16 (1983).
[CrossRef]

Chang, B. J.

B. J. Chang, “Grating based interferometer,” Ph.D. dissertation (University of Michigan, Ann Arbor, Michigan, 1974;available from University Microfilms, Ann Arbor, Michigan).

Gori, F.

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Jahns, J.

J. Jahns, A. W. Lohmann, “The Lau effect: a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Lau, E.

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Leith, E. N.

Lohmann, A.

K.-H. Brenner, A. Lohmann, J. Ojeda-Casteneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–16 (1983).
[CrossRef]

Lohmann, A. W.

J. Jahns, A. W. Lohmann, “The Lau effect: a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Ojeda-Casteneda, J.

K.-H. Brenner, A. Lohmann, J. Ojeda-Casteneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–16 (1983).
[CrossRef]

Patorski, K.

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

Sudol, R.

R. Sudol, B. J. Thompson, “Lau effect, theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
[CrossRef] [PubMed]

R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

Swanson, G. J.

G. J. Swanson, E. N. Leith, “Lau effect and grating imaging,” J. Opt. Soc. Am. 72, 552–555 (1982).
[CrossRef]

G. J. Swanson, “Partially coherent imaging and interferometry based on diffraction gratings,” Ph.D. dissertation (University of Michigan, Ann Arbor, Michigan, 1983;available from University Microfilms, Ann Arbor, Michigan).

Thompson, B. J.

R. Sudol, B. J. Thompson, “Lau effect, theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
[CrossRef] [PubMed]

R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

Ann. Phys. (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Acta (1)

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–758 (1983).
[CrossRef]

Opt. Commun. (4)

J. Jahns, A. W. Lohmann, “The Lau effect: a diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

K.-H. Brenner, A. Lohmann, J. Ojeda-Casteneda, “Lau effect: OTF theory,” Opt. Commun. 46, 14–16 (1983).
[CrossRef]

Other (3)

G. J. Swanson, “Partially coherent imaging and interferometry based on diffraction gratings,” Ph.D. dissertation (University of Michigan, Ann Arbor, Michigan, 1983;available from University Microfilms, Ann Arbor, Michigan).

B. J. Chang, “Grating based interferometer,” Ph.D. dissertation (University of Michigan, Ann Arbor, Michigan, 1974;available from University Microfilms, Ann Arbor, Michigan).

Ref. 7, p. 37.

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Figures (4)

Fig. 1
Fig. 1

General grating imaging. G1 and G2 are arbitrary periodic structures of spatial frequencies f1 and f2 respectively. Each source point is characterized by an incident spatial frequency f0. Observation is at plane P.

Fig. 2
Fig. 2

Images formed for the case r = 2, l = 2.

Fig. 3
Fig. 3

Plot of the transmittance functions for the object t1 and the imaging element t2.

Fig. 4
Fig. 4

Plot of the image irradiance produced by the object and imaging element of Fig. 3.

Equations (52)

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t 1 = n a n exp ( i 2 π n f 1 x )
t 2 = m b m exp ( i 2 π m f 2 x ) ,
u p = n m a n b m exp [ i 2 π ( n f 1 + m f 2 ) x ] × exp { i 2 π λ [ ( z 0 + z 1 ) n f 1 + z 1 m f 2 ] f 0 } × exp { i π λ [ z 0 n 2 f 1 2 + z 1 ( n f 1 + m f 2 ) 2 ] }
I p = | u p | 2 = n n , m m , a n a n * b m b m * × exp { i 2 π [ ( n n ) f 1 + ( m m ) f 2 ] x } × exp { i 2 π λ [ ( n n ) ( z 0 + z 1 ) f 1 + ( m m ) z 1 f 2 ] f 0 } exp { i π λ [ ( n 2 n 2 ) z 0 f 1 2 + z 1 ( n f 1 + m f 2 ) 2 z 1 ( n f 1 + m f 2 ) 2 ] } ,
z 0 = l 2 λ f 1 2 , z 1 = p 2 λ f 1 2 , f 2 = r f 1 ,
I p = n n m m , a n a n * b m b m * × exp { i 2 π [ ( n n ) + r ( m m ) ] f 1 x } × exp { i π ( f 0 / f 1 ) [ ( l + p ) ( n n ) + r p ( m m ) ] } × exp { i ( π / 2 ) [ ( l + p ) ( n 2 n 2 ) + p r 2 ( m 2 m 2 ) + 2 r p ( n m n m ) ] } .
ϕ ( n , n , m , m ) = [ ( l + p ) ( n n ) + r p ( m m ) ] .
ϕ ( m , m ) = 2 ( m m ) .
f 0 = f 0 + f 1 ϕ ,
ϕ ( n , n , m , m ) = 0 .
( l + p ) ( n n ) + r p ( m m ) = 0 .
z 1 = z 0 1 + ( f 2 / f 1 ) ( m m n n ) ,
I p = n n m m a n a n * b m b m * × exp { 2 π [ ( n n ) + r ( m m ) ] f 1 x } × exp { i ( π / 2 ) [ ( l + p ) ( n 2 n 2 ) + p r 2 ( m 2 m 2 ) + 2 r p ( n m n m ) × δ [ ( l + p ) ( n n ) + r p ( m m ) ] ,
I p = n n m m , a n a n * b m b m * exp [ i 2 π ( l / p ) ( n n ) f 1 x ] × exp { i ( π / 2 ) r l [ r p / ( l + p ) ( m 2 m 2 ) } × δ [ ( l + p ) ( n n ) + r p ( m m ) ] .
k = ( m m n n ) ,
I p = n n m m a n a n * b m b m * exp [ i 2 π ( l / p ) ( n n ) f 1 x ] × exp { i ( π / 2 ) [ ( r l + k ) ( m 2 m 2 ) ] × δ [ k ( n n ) + ( m m ) ] .
I 1 = | t 1 | 2 = n n a n a n * exp [ i 2 π ( n n ) f 1 x ] = K A K exp [ i 2 π K f 1 x ] ,
A K = n a n a K n * .
I p = K A K { m m b m b m * exp [ i ( π / 2 ) ( r l / k ) ( m 2 m 2 ) ] × δ [ k K + ( m m ) ] } exp [ i 2 π ( l / p ) K f 1 x ] .
OTF = m m b m b m * exp [ i ( π / 2 ) ( r l / k ) ( m 2 m 2 ) ] × δ [ k K + ( m m ) ] .
| r l / k | = { q 1 = 0 , 4 , 8 , q 2 = 2 , 6 , 10 , q 3 = 1 , 3 , 5 , .
OTF 1 = K b K δ { k K + K } ,
OTF = K b K ( 1 ) K δ ( k K + K ) ,
OTF = K b 2 K δ { k K + K } .
k = α / β ,
I p = K A β K b α K exp [ i 2 π ( l / p ) β K f 1 x ] for q 1 ,
I p = K A β K b α K ( 1 ) α K exp [ i 2 π ( l / p ) β K f 1 x ] for q 2 ,
I p = K A β K b α K exp [ i 2 π ( l / p ) β K f 1 x ] for q 3 .
I p = K A K b k K exp [ i 2 π ( l / p ) K f 1 x ] for q 1 ,
I p = K A K b k K ( 1 ) k K exp [ i 2 π ( l / p ) K f 1 x ] for q 2 ,
I p = K A K b 2 k K exp [ i 2 π ( l / p ) K f 1 x ] for q 3 .
k = ( l + p ) r p ,
r l / | k | = q ,
1 l + 1 p = r ± q ,
F = ± q / r 2 .
l = q k / r .
p ± = l 1 ± r | k | .
l = 4 / r .
q k = 1.4 = 4.1 = 2.2 ,
p ± = 4 r ( 1 ± r ) ,
p ± = 4 r ( 1 ± 2 r ) ,
p ± = 4 f ( 1 ± 4 r ) ,
l = q .
I p = K A K b K exp [ 2 π ( l / F ) K f 1 x ] for q 1 ,
I p = K A K b K ( 1 ) K exp [ i 2 π ( l / F ) K f 1 x ] for q 2 ,
I p = K A K b 2 K exp [ i 2 π ( l / F ) K f 1 x ] for q 3 .
r | k | = 1 .
t 2 = m b exp ( i 2 π m f 2 x ) = b m δ ( x m f 2 ) .
t 1 = m a n exp ( i 2 π n f 1 x ) , a n = sin ( 2 π f 1 n c 1 ) π n
t 2 = m b m exp ( i 2 π m f 2 x ) , b m = sin ( 2 π f 2 m c 2 ) π n .
k r c 2 1 2 f 1 c 1 .
k 1 DC 1 DC 2 .

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