Abstract

A general method is described that permits demonstration that only three different array illuminators based on the Talbot effect can be produced.

© 1992 Optical Society of America

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References

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  1. H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
    [CrossRef]
  2. N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
    [CrossRef]
  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. W. H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).
  5. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images: I,” J. Opt. Soc. Am. 55, 373–381 (1965).
    [CrossRef]
  6. A. W. Lohmann, Optical Information Processing (Uttenreuth, Berlin, Germany, 1986).
  7. K. Patorski, “The Self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 1–110.
    [CrossRef]
  8. E. Keren, O. Kafri, “Diffraction effects in moiré deflectometry,” J. Opt. Soc. Am. A 2, 111–120 (1985).
    [CrossRef]
  9. W. D. Montgomery, “Algebraic formulation of diffraction applied to self-imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1986).
    [CrossRef]

1990 (1)

1989 (1)

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

1986 (1)

1985 (1)

1977 (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

1965 (1)

1836 (1)

W. H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).

Dammann, H.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Kafri, O.

Keren, E.

Klotz, E.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Lohmann, A. W.

Montgomery, W. D.

Patorski, K.

K. Patorski, “The Self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 1–110.
[CrossRef]

Streibl, N.

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

Talbot, W. H. F.

W. H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).

Thomas, J. A.

Winthrop, J. T.

Worthington, C. R.

Appl. Opt. (1)

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Philos. Mag. (1)

W. H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9, 401–407 (1836).

Other (2)

A. W. Lohmann, Optical Information Processing (Uttenreuth, Berlin, Germany, 1986).

K. Patorski, “The Self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 1–110.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Binary phase grating.

Fig. 2
Fig. 2

Phasor diagrams for the array illuminators listed in Table I.

Tables (1)

Tables Icon

Table I Array Illumination Based on the Talbot Effect

Equations (34)

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u 0 ( x ) = ( m ) c m exp ( j 2 π x m / d )             ( m = 0 , ± 1 , ± 2 , ± 3 , ) .
u z ( x ) = ( m ) c m exp [ - j 2 π z λ m 2 / ( 2 d 2 ) ] exp ( j 2 π x m / d ) .
z λ / ( 2 d 2 ) = 1 ;
z T = 2 d 2 / λ ,
u z ( x ) = ( m ) c m exp ( - j 2 π m 2 z / z T ) exp ( j 2 π x m / d ) .
u 0 ( x ) = ( m ) c m exp [ j 2 π x ( cos α λ + m d ) ] exp ( j 2 π y cos β λ ) .
f x = cos α λ + m d ,             f y = cos β λ .
u z T ( x , y ) = exp [ j 2 π λ ( x cos α + y cos β + z T cos ν ) ] × ( m ) c m exp [ j 2 π d m ( x - z T cos α ) ] .
u 0 ( x ) = { 1 + [ exp ( j ϕ ) - 1 ] rect ( x / w ) } * 1 d comb ( x / d ) ,
u 0 ( x ) = { exp ( j ϕ ) x - m d < w / 2 1 x - m d elsewhere .
U 0 ( f x ) = 1 δ ( f x ) + [ exp ( j ϕ ) - 1 ] × w d sinc ( f x w ) ( m ) δ ( f x - m / d ) .
c 0 = 1 + w d [ exp ( j ϕ ) - 1 ] ,
c n = w d [ exp ( j ϕ ) - 1 ] sinc ( n w d )             ( n = ± 1 , ± 2 , ± 3 , ) .
u 0 ( x ) = c 0 + ( n ) c n exp ( j 2 π x n d ) = c 0 + Δ u 0 .
Δ u 0 = { Δ u 01 = 1 - c 0 = w d [ 1 - exp ( j ϕ ) ] , Δ u 02 = exp ( j ) - c 0 = - ( 1 - w d ) [ 1 - exp ( j ϕ ) ] .
u z ( x ) = c 0 + ( n ) c n exp ( - j 2 π n 2 z z T ) exp ( j 2 π x n d ) .
u z ( x ) = c 0 + exp ( j θ ) ( n ) c n exp ( j 2 π x n / d ) = c 0 + exp ( j θ ) Δ u 0
u z ( x ) = { c 0 + exp ( j θ ) Δ 01 , c 0 + exp ( j θ ) Δ u 02 .
c 0 2 = u 01 2 ,
{ 1 + w d [ exp ( j ϕ ) - 1 ] } { 1 + w d [ exp ( - j ϕ ) - 1 ] } = w 2 d 2 [ 1 - exp ( j ϕ ) ] [ 1 - exp ( - j ϕ ) ] .
cos ϕ = 1 - 1 2 w / d ,
c 0 2 = u 02 2 ,
{ 1 + w d [ exp ( j ϕ ) - 1 ] } { 1 + w d [ exp ( - j ϕ ) - 1 ] } = ( 1 - w d ) 2 ( 1 - exp ( j ϕ ) ] [ 1 - exp ( - j ϕ ) ] .
cos ϕ = 1 - 1 2 ( 1 - w / d ) .
n 2 / T = integer + 1 / T             ( for every n ) ,
exp ( j θ ) = exp ( - j 2 π t / T ) .
n 2 = ( 2 h - 1 ) 2 = 4 h ( h - 1 ) + 1             ( h = 0 , ± 1 , ± 2 , ± 3 , ) .
exp ( j θ ) = exp ( - j 2 π t / 2 ) ,             exp ( - j 2 π t / 4 ) .
n 2 = ( 3 h ± 1 ) 2 = 3 h ( 3 h ± 2 ) + 1.
exp ( j θ ) = exp ( - j 2 π t / 3 ) .
c 0 + exp ( j θ ) Δ u 02 = 0.
cos ϕ = 1 - 1 2 ( 1 - 1 / 2 ) = 0 ,             c 0 = ( 1 + j ) / 2 , Δ u 01 = ( 1 - j ) / 2 ,             Δ u 02 = - ( 1 - j ) / 2.
c 0 + exp ( - j 3 π / 2 ) Δ u 01 = 1 + j .
0 : 1 + j 2 = 0 : 2 ,

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