Abstract

Construction of one-dimensional multilevel phase gratings is described that, when illuminated by a coherent plane wave, lead to the formation of amplitude binary gratings with an arbitrary value of the opening ratio. The gratings are proposed as array illuminators that can provide a significantly high compression factor together with a large number of uniformly illuminated points.

© 1993 Optical Society of America

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References

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  1. J. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
    [CrossRef]
  2. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).
  3. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  4. K. Patorski, P. Szwaykowski, “Light intensity distribution in the Fresnel diffraction region of a non-sinusoidal phase grating,” Opt. Appl. 11, 627–631 (1981).
  5. G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
    [CrossRef]
  6. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–381 (1965)
    [CrossRef]

1990 (1)

1989 (1)

J. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

1988 (1)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).

1984 (1)

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
[CrossRef]

1981 (1)

K. Patorski, P. Szwaykowski, “Light intensity distribution in the Fresnel diffraction region of a non-sinusoidal phase grating,” Opt. Appl. 11, 627–631 (1981).

1965 (1)

Indebetouw, G.

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
[CrossRef] [PubMed]

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).

Patorski, K.

K. Patorski, P. Szwaykowski, “Light intensity distribution in the Fresnel diffraction region of a non-sinusoidal phase grating,” Opt. Appl. 11, 627–631 (1981).

Streibl, J.

J. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

Szwaykowski, P.

K. Patorski, P. Szwaykowski, “Light intensity distribution in the Fresnel diffraction region of a non-sinusoidal phase grating,” Opt. Appl. 11, 627–631 (1981).

Thomas, J. A.

Winthrop, J. T.

Worthington, C. R.

Appl. Opt. (1)

J. Mod. Opt. (1)

J. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

G. Indebetouw, “Propagation of spatially periodic wavefields,” Opt. Acta 31, 531–539 (1984).
[CrossRef]

Opt. Appl. (1)

K. Patorski, P. Szwaykowski, “Light intensity distribution in the Fresnel diffraction region of a non-sinusoidal phase grating,” Opt. Appl. 11, 627–631 (1981).

Optik (1)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).

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Figures (4)

Fig. 1
Fig. 1

Geometry of the optical system.

Fig. 2
Fig. 2

Transmittance function t0(x) of a unitary cell.

Fig. 3
Fig. 3

Intensity distribution at the distance z = Zt/n behind the binary amplitude grating for (a) n odd and α = 1/n; (b) for both n and n/2 even and α = 2/n; (c) for n even but n/2 odd and α = 2/n.

Fig. 4
Fig. 4

Profiles of the phase gratings plotted for the values of parameters (a) n = 5; (b) n = 7; (c) n = 8; (d) n = 9.

Tables (2)

Tables Icon

Table 1 Basic Properties of the Coefficients C(L, n) and Values of Opening Ratio Required for Uniform Intensity Distribution at the Observation Plane z = Zt/n

Tables Icon

Table 2 Phase Profiles of Difraction Gratings Calculated for n = 5, 7, 8, and 9 by Extracting the Imaginary Part of C(L, n)

Equations (27)

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z = Z t / n ,             n = 2 , 3 , ,
t ( x ) = t 0 ( x ) m = - δ ( x - m d ) .
t 0 ( x ) = { 1 , for x w / 2 0 , for x > w / 2 ,
f ( x , z ) = exp ( i k z ) i λ z t ( x ) h ( x , z ) = exp ( i k z ) i λ z t 0 ( x ) Δ ( x , z ) ,
Δ ( x , z ) = m = - δ ( x - m d ) h ( x , z ) .
Δ ( x , z ) = 1 d m = - exp [ - i π λ z ( m / d ) 2 ] exp ( i 2 π m x / d ) .
Δ ( x , z = Z t / n ) = 1 d m = - exp ( - i 2 π m 2 / n ) exp ( i 2 π m x / d ) .
m = n L + q ,
Δ ( x , z = Z t / n ) = 1 d L = - q = 0 n - 1 exp [ - i 2 π n ( n L + q ) 2 ] × exp [ i 2 π ( n L + q ) x / d ] = 1 d L = - q = 0 n - 1 exp ( - i 2 π n q 2 ) exp ( i 2 π q x / d ) × exp ( i 2 π n L x / d ) .
L = - exp ( i 2 π n L x / d ) = d n L = - δ ( x - L d / n ) .
Δ ( x , z = Z t / n ) = L = - C ( L , n ) δ ( x - L d / n ) ,
C ( L , n ) = 1 n q = 0 n - 1 exp [ i 2 π n q ( L - q ) ] .
f ( x , z = Z t / n ) = L = - C ( L , n ) t 0 ( x - L d / n ) ,
C ( L + K n , n ) = C ( L , n ) .
C ( L , n ) = 1 n q = - p n - 1 - p exp [ i 2 π n q ( L - q ) ] ,
C ( L = 2 K , n ) = C ( L = 0 , n ) ,
C ( L = 2 K + 1 , n ) = C ( L = 1 , n ) .
C ( 1 , n ) = C ( 0 , n ) 0.
C ( L = 2 K , n = 4 M + 2 ) = 0 ,
C ( L = 2 K , n = 4 M ) = C ( L = 0 , n = 4 M ) 0 ,
C ( L = 2 K + 1 , n = 4 M ) = 0 ,
C ( L = 2 K + 1 , n = 4 M + 2 ) = C ( L = 1 , n = 4 M + 2 ) 0.
C ( L = 2 K , n ) = 1 n q = 0 n - 1 exp [ i 2 π n q ( 2 K - q ) ] = 1 n q = 0 n - 1 exp { - i 2 π n [ ( q - K ) 2 - K 2 ] } = 1 n exp ( i 2 π n K 2 ) q = 0 n - 1 exp [ - i 2 π n ( q - K ) 2 ] = 1 n exp ( i 2 π n K 2 ) q = - K n - 1 - K exp ( - i 2 π n q 2 ) = exp ( i 2 π n K 2 ) C ( L = 0 , n ) ,
C ( L = 2 K + 1 , n ) = 1 n q = 0 n - 1 exp [ i 2 π n q ( 2 K + 1 - q ) ] = 1 n exp [ i 2 π n K ( 1 + K ) ] × q = 0 n - 1 exp [ i 2 π n ( q - K ) ( 1 - q + K ) ] = 1 n exp [ i 2 π n K ( 1 + K ) ] × q = - K n - 1 - K exp [ i 2 π n q ( 1 - q ) ] = exp [ i 2 π n K ( 1 + K ) ] C ( L = 1 , n ) .
C ( L = 1 , n ) = 1 n q = 0 2 p exp [ i 2 π 2 p + 1 q ( 1 - q ) ] = 1 n q = 0 2 p exp { - i 2 π 2 p + 1 [ - q + q 2 + ( 2 p + 1 ) q + p 2 - p 2 ] } = exp ( i 2 π 2 p + 1 p 2 ) q = 0 2 p exp [ - i 2 π 2 p + 1 ( q - p ) 2 ] = exp ( i 2 π 2 p + 1 p 2 ) q = - p p exp ( - i 2 π 2 p + 1 q 2 ) = exp ( i 2 π 2 p + 1 p 2 ) C ( L = 0 , n = 2 p + 1 ) .
C ( L = 0 , n = 4 M + 2 ) = 1 n q = 0 4 M + 1 exp ( - i π 2 M + 1 q 2 ) = 1 n [ q = 0 2 M exp ( - i π 2 M + 1 q 2 ) + q = 2 M + 1 4 M + 1 exp ( - i π 2 M + 1 q 2 ) ] = 1 n { q = 0 2 M exp ( - i π 2 M + 1 q 2 ) + q = 0 2 M exp [ - i π 2 M + 1 ( q + 2 M + 1 ) 2 ] } = 1 n [ q = 0 2 M exp ( - i π 2 M + 1 q 2 ) - q = 0 2 M exp ( - i π 2 M + 1 q 2 ) ] = 0.
C ( L = 1 , n = 4 M ) = 1 n q = 0 4 M - 1 exp [ i π 2 M q ( 1 - q ) ] = 1 n { q = 0 2 M - 1 exp [ i π 2 M q ( 1 - q ) ] + q = 2 M 4 M - 1 exp [ i π 2 M q ( 1 - q ) ] } = 1 n { q = 0 2 M - 1 exp [ i π 2 M q ( 1 - q ) ] + q = 0 2 M - 1 exp [ i π 2 M ( q + 2 M ) ( 1 - q - 2 M ) ] } = 1 n { q = 0 2 M - 1 exp [ i π 2 M q ( 1 - q ) ] - q = 0 2 M - 1 exp [ i π 2 M q ( 1 - q ) ] } = 0.

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