Abstract

Lau imaging is shown to be a special case of generalized two-grating interference and is explained in terms of the phenomenon of grating imaging, wherein a grating is imaged by a second grating.

© 1982 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417–423 (1948).
    [Crossref]
  2. J. Jahns and A. W. Lohmann, “The Lau effect: A diffraction experiment with incoherent illumination,” Opt. Commun. 28, 263–267 (1979).
    [Crossref]
  3. R. Sudol and B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
    [Crossref]
  4. R. Sudol and B. J. Thompson, “Lau effect, theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
    [Crossref] [PubMed]
  5. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
    [Crossref]
  6. F. O. Weinberg and N. B. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36, 227–230 (1959).
    [Crossref]
  7. E. Leith and G. Swanson, “Achromatic interferometers for white light optical processing and holography,” Appl. Opt. 19, 638–644 (1980).
    [Crossref] [PubMed]
  8. B. J. Chang, “Grating based interferometer,” Ph.D. thesis (The University of Michigan, Ann Arbor, Mich., 1974), University Microfilms, Ann Arbor, Mich., order no. 74-25-170.

1981 (1)

1980 (1)

1979 (3)

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

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

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

1959 (1)

F. O. Weinberg and N. B. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36, 227–230 (1959).
[Crossref]

1948 (1)

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

Chang, B. J.

B. J. Chang, “Grating based interferometer,” Ph.D. thesis (The University of Michigan, Ann Arbor, Mich., 1974), University Microfilms, Ann Arbor, Mich., order no. 74-25-170.

Gori, F.

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

Jahns, J.

J. Jahns and 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.

Lohmann, A. W.

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

Sudol, R.

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

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

Swanson, G.

Thompson, B. J.

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

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

Weinberg, F. O.

F. O. Weinberg and N. B. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36, 227–230 (1959).
[Crossref]

Wood, N. B.

F. O. Weinberg and N. B. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36, 227–230 (1959).
[Crossref]

Ann. Phys. (1)

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

Appl. Opt. (2)

J. Sci. Instrum. (1)

F. O. Weinberg and N. B. Wood, “Interferometer based on four diffraction gratings,” J. Sci. Instrum. 36, 227–230 (1959).
[Crossref]

Opt. Commun. (3)

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

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

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

Other (1)

B. J. Chang, “Grating based interferometer,” Ph.D. thesis (The University of Michigan, Ann Arbor, Mich., 1974), University Microfilms, Ann Arbor, Mich., order no. 74-25-170.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Lau effect. An extended source S illuminates two gratings, G1 and G2, that are separated by a distance z0. Observation of the fringes is made at P, the back focal plane of lens L2.

Fig. 2
Fig. 2

Grating interferometer. Two orders from the first grating, G1, are selected and propagate to grating G2. Two diffracted orders, one from each of the two incident beams, are selected from G2 and combine at the observation plane P.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

u n m = a n b m exp [ i 2 π ( f 0 + n f 1 + m f 2 ) x ] × exp [ - i π λ z 0 ( n f 1 + f 0 ) 2 exp [ - i π λ z 1 ( n f 1 + m f 2 + f 0 ) 2 ] ,
I = u n m + u n m 2 = a n 2 b m 2 + a n 2 b m 2 + 2 a n b m a n b m cos { 2 π [ ( n - n ) f 1 + ( m - m ) f 2 ] x - 2 π λ [ ( n - n ) z 0 f 1 + ( n - n ) z 1 f 1 + ( m - m ) z 1 f 2 ] f 0 - π λ [ ( n 2 - n 2 ) z 0 f 1 2 + z 1 ( f 1 n + f 2 m ) 2 - z 1 ( f 1 n + f 2 m ) 2 ] } .
z 1 = - z 0 / [ 1 + ( f 2 f 1 ) ( m - m ) ( n - n ) ] .
I = a n 2 b m 2 + a n 2 b m 2 + 2 a n b m a n b m cos { 2 π [ ( n - n ) f 1 + ( m - m ) f 2 ] x + π λ z 0 f 1 f 2 ( n - n ) ( m + m ) } .
z 1 = - z 0 / [ 1 + ( m 2 - m 2 ) ( n 2 - n 2 ) ( f 2 f 1 ) 2 + 2 ( n m - n m ) ( n 2 - n 2 ) ( f 2 f 1 ) ] .
1 z 0 + 1 z 1 = 1 F g ,
F g = - z 0 ( f 1 f 2 ) ( n - n m - m ) .
M = ( m - m ) f 1 ( n - n ) f 1 + ( m - m ) f 2 = z 1 z 0 .
t 1 ( x ) = n a n exp ( i 2 π n f 1 x ) ,
t 2 ( x ) = m b m exp ( i 2 π m f 2 x ) ,
u p = n m u n m ,
I p = n m n n a n b m a n * 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 ] } .
I p = a 1 2 b - 1 2 + a - 1 2 b 1 2 + 2 a 1 b 1 a - 1 b - 1 cos { 2 π ( 2 f 1 - 2 f 2 ) x - 2 π λ [ 2 f 1 ( z 1 + z 0 ) - 2 f 2 z 1 ] f 0 } .
I p = n m n m a n a n * b m b m * exp [ i 2 π ( n - n + m - m ) f 1 x ] exp { - i 2 π λ [ ( n - n ) ( z 0 + z 1 ) + ( m - m ) z 1 ] f 1 f 0 } exp - i π λ { ( n 2 - n 2 ) z 0 + z 1 [ ( n + m ) 2 - ( n + m ) 2 ] } f 1 2 δ n - n + m - m ,
I p = n m n m a n b m a n * b m * exp [ i 2 π ( n - n + m - m ) f 1 x ] × exp { i π λ z 0 f 1 2 [ ( n - n ) 2 + 2 m ( n - n ) ] } δ n - n + m - m .
F z 1 = - F ( n - n + m - m ) z 0 ( n - n ) .
I p = n m n m a n b m a n * b m * exp [ - i 2 π ( n - n ) f 1 z 0 x F ] × exp { i π λ z 0 f 1 2 [ ( n - n ) 2 + 2 m ( n - n ) ] } δ n - n + m - m .
n n a n a n * exp [ - i 2 π f 1 z 0 F ( n - n ) x ] δ n - n + m - m = k a k exp ( - i 2 π f 1 z 0 F k x ) δ k + m - m ,
I p = k a k { m m b m b m exp [ - i π λ z 0 f 1 2 ( m 2 - m 2 ) ] δ k + m - m } × exp ( - i 2 π f 1 z 0 F k x ) .
I p = k a k { m m b m b m exp [ - i π 2 α ( m 2 - m 2 ) ] δ k + m - m } × exp ( - i 2 π f 1 z 0 F k x ) .
I p = k m a k b m b m + k cos [ 2 π f 1 z 0 F k x + π 2 α ( 2 m + k ) k ] .
I p = k a k b k ( - 1 ) k cos ( 2 π k z 0 F f 1 x ) .
I p = k a k b k cos ( 2 π k z 0 F f 1 x ) .
I p = k even a k b k cos ( 2 π k z 0 F f 1 x ) = k a 2 k b 2 k cos ( 4 π k z 0 F f 1 x ) .