Abstract

Using a matrix formulation of Fresnel diffraction, we describe discrete gratings that exhibit self-images at fractions of the Talbot distance. Such structures are obtained as solutions of an eigenvalue equation.

© 1996 Optical Society of America

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References

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  1. J. T. Winthrop, C. R. Worthington, J. Opt. Soc. Am. 55, 373 (1965).
    [CrossRef]
  2. W. D. Montgomery, J. Opt. Soc. Am. 57, 772 (1967).
    [CrossRef]
  3. V. Arrizón, J. Ojeda-Castaneda, Opt. Lett. 20, 118 (1995).
    [CrossRef] [PubMed]
  4. V. Arrizón, J. Ojeda-Castaneda, Appl. Opt. 33, 5925 (1994).
    [CrossRef] [PubMed]
  5. H. Cohn, Advanced Number Theory (Dover, New York, 1962), p. 10.
  6. J. P. Guigay, Opt. Acta 18, 677 (1971).
    [CrossRef]
  7. G. E. Shilov, Linear Algebra (Dover, New York, 1977) p. 263.
  8. J. Ibarra, J. Ojeda-Castaneda, Opt. Commun. 96, 294 (1993).
    [CrossRef]
  9. J. C. Barreiro, P. Andrés, J. Ojeda-Castaneda, Opt. Commun. 73, 106 (1989).
    [CrossRef]

1995 (1)

1994 (1)

1993 (1)

J. Ibarra, J. Ojeda-Castaneda, Opt. Commun. 96, 294 (1993).
[CrossRef]

1989 (1)

J. C. Barreiro, P. Andrés, J. Ojeda-Castaneda, Opt. Commun. 73, 106 (1989).
[CrossRef]

1971 (1)

J. P. Guigay, Opt. Acta 18, 677 (1971).
[CrossRef]

1967 (1)

1965 (1)

Andrés, P.

J. C. Barreiro, P. Andrés, J. Ojeda-Castaneda, Opt. Commun. 73, 106 (1989).
[CrossRef]

Arrizón, V.

Barreiro, J. C.

J. C. Barreiro, P. Andrés, J. Ojeda-Castaneda, Opt. Commun. 73, 106 (1989).
[CrossRef]

Cohn, H.

H. Cohn, Advanced Number Theory (Dover, New York, 1962), p. 10.

Guigay, J. P.

J. P. Guigay, Opt. Acta 18, 677 (1971).
[CrossRef]

Ibarra, J.

J. Ibarra, J. Ojeda-Castaneda, Opt. Commun. 96, 294 (1993).
[CrossRef]

Montgomery, W. D.

Ojeda-Castaneda, J.

V. Arrizón, J. Ojeda-Castaneda, Opt. Lett. 20, 118 (1995).
[CrossRef] [PubMed]

V. Arrizón, J. Ojeda-Castaneda, Appl. Opt. 33, 5925 (1994).
[CrossRef] [PubMed]

J. Ibarra, J. Ojeda-Castaneda, Opt. Commun. 96, 294 (1993).
[CrossRef]

J. C. Barreiro, P. Andrés, J. Ojeda-Castaneda, Opt. Commun. 73, 106 (1989).
[CrossRef]

Shilov, G. E.

G. E. Shilov, Linear Algebra (Dover, New York, 1977) p. 263.

Winthrop, J. T.

Worthington, C. R.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Opt. Acta (1)

J. P. Guigay, Opt. Acta 18, 677 (1971).
[CrossRef]

Opt. Commun. (2)

J. Ibarra, J. Ojeda-Castaneda, Opt. Commun. 96, 294 (1993).
[CrossRef]

J. C. Barreiro, P. Andrés, J. Ojeda-Castaneda, Opt. Commun. 73, 106 (1989).
[CrossRef]

Opt. Lett. (1)

Other (2)

G. E. Shilov, Linear Algebra (Dover, New York, 1977) p. 263.

H. Cohn, Advanced Number Theory (Dover, New York, 1962), p. 10.

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Figures (2)

Fig. 1
Fig. 1

Transmittance in one cell of a discrete grating G(x) formed by Q equally spaced subintervals.

Fig. 2
Fig. 2

Transmittance in one cell of a discrete version of a Blazé grating formed by Q uniformly distributed phase levels between 0 and 2π. This phase grating exhibits a self-image at z = Zt/2Q.

Tables (1)

Tables Icon

Table 1 Eigenvalues and Orthogonal Eigenvectors of Matrix D for the Distance z = Zt/8

Equations (15)

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G ( x ) = L = 0 Q - 1 a L B ( x - L d / Q ) ,
B ( x ) = n = - rect ( x - n d d / Q ) .
u [ x , z = Z t / N ] = L = 0 Q - 1 c L G ( x - L d / Q ) ,
c L = 1 / Q exp [ i π ( L 2 / Q - 1 / 4 ) ] .
u [ x , z = Z t / N ] = L = 0 Q - 1 b L B ( x - L d / Q ) ,
β = D α ,
α = [ a 0 a 1 a Q - 1 ] ,             β = [ b 0 b 1 b Q - 1 ] , D = [ c 0 c Q - 1 c 2 c 1 c 1 c 0 c 3 c 2 c Q - 1 c Q - 2 c 1 c 0 ] .
D N = I .
u [ x , z = M Z t / N ] = L = 0 Q - 1 d L B ( x - L d / Q ) ,
γ = D M α .
D M α = K α .
D α = K α ,
α · α p = m = 0 Q - 1 a * a m = a * m = 0 Q - 1 a m = 0 .
1 2 [ exp ( - i π / 4 ) 1 exp ( i 3 π / 4 ) 1 1 exp ( - i π / 4 ) 1 exp ( i 3 π / 4 ) exp ( i 3 π / 4 ) 1 exp ( - i π / 4 ) 1 1 exp ( i 3 π / 4 ) 1 exp ( - i π / 4 ) ] [ a 0 a 1 a 2 a 3 ] = K [ a 0 a 1 a 2 a 3 ] .
- 2 [ - 1 / 2 0 1 / 2 0 ] + i 2 [ 0 1 / 2 0 - 1 / 2 ] = [ 1 i - 1 - i ] ,

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