Abstract

We describe a variety of multilevel phase structures that can be used to generate Lohmann's array illuminators. We report several experimental verifications of the synthesis of such multilevel phase structures by using simple binary curves in a conventional optical processor.

© 1994 Optical Society of America

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References

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  1. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik 79, 41–45 (1988).
  2. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  3. V. Arrizón, J. Ojeda-Castañeda, “Talbot array illuminator with binary phase gratings,” Opt. Lett. 18, 1–3 (1993).
    [CrossRef] [PubMed]
  4. D. Xiao-Yi, “Talbot effect and the array illuminators, that are based on it,” Appl. Opt. 31, 2983–2986 (1992).
    [CrossRef]
  5. J. R. Leger, G. J. Swanson, “Efficient array illuminator using binary-optics phase plates as fractional Talbot planes,” Opt. Lett. 15, 288–290 (1990).
    [CrossRef] [PubMed]
  6. P. Szwaykowsky, V. Arrizón, “Talbot array illuminators with multilevel phase gratings,” Appl. Opt. 32, 1109–1114 (1993).
    [CrossRef]
  7. 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]
  8. J. Ojeda-Castañeda, V. Arrizón, “Synthesis of 1-D phase profiles with variable optical path,” Microwave Opt. Technol. Lett. 5, 429–432 (1992).
    [CrossRef]

1993 (2)

1992 (2)

J. Ojeda-Castañeda, V. Arrizón, “Synthesis of 1-D phase profiles with variable optical path,” Microwave Opt. Technol. Lett. 5, 429–432 (1992).
[CrossRef]

D. Xiao-Yi, “Talbot effect and the array illuminators, that are based on it,” Appl. Opt. 31, 2983–2986 (1992).
[CrossRef]

1990 (2)

1988 (1)

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

1965 (1)

Arrizón, V.

Leger, J. R.

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).

Ojeda-Castañeda, J.

V. Arrizón, J. Ojeda-Castañeda, “Talbot array illuminator with binary phase gratings,” Opt. Lett. 18, 1–3 (1993).
[CrossRef] [PubMed]

J. Ojeda-Castañeda, V. Arrizón, “Synthesis of 1-D phase profiles with variable optical path,” Microwave Opt. Technol. Lett. 5, 429–432 (1992).
[CrossRef]

Swanson, G. J.

Szwaykowsky, P.

Thomas, J. A.

Winthrop, J. T.

Worthington, C. R.

Xiao-Yi, D.

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Figures (7)

Fig. 1
Fig. 1

Schematic representation of a binary amplitude grating at plane z = 0 and its self-image at plane z = Zt. A uniform irradiance distribution appears at plane z = (M/N)Zt; x and y are lateral coordinates.

Fig. 2
Fig. 2

Two periods of the transmittance of the binary amplitude grating G(x).

Fig. 3
Fig. 3

Constant phase intervals inside one period of the multilevel phase distribution at the distance z = (M/N)Zt behind a binary amplitude grating of opening ratio: (a) α = 1/5 (for N = 5), (b) α = 1/4 (for N = 8), (c) α = 1/5 (for N = 10).

Fig. 4
Fig. 4

Multilevel phase profiles generated for a binary amplitude grating with opening ratio α = 1/5 at the fractional Talbot distance: (a) z = (1/5)Zt, (b) z = (2/5)Zt, (c) z = (3/5)Zt, (d) z = (4/5)Zt,.

Fig. 5
Fig. 5

Schematic representation of the optical setup employed for synthesizing one-dimensional phase distributions.

Fig. 6
Fig. 6

(a) Input object for generating the multilevel phase distribution in Fig. 4(d). (b) Frequency spectrum of the input object.

Fig. 7
Fig. 7

(a) Uniform irradiance distribution at the output plane z = 0 of the processor and (b) the binary irradiance distribution with opening ratio α = 1/5 observed at a distance z = (1/5)Zt behind the plane z = 0.

Tables (2)

Tables Icon

Table 1 Properties of |C(L, M, N)|2 and the Values of Required α for Uniform Irradiance at the Observation Plane z = (M/N)Zt

Tables Icon

Table 2 Phase Values Corresponding to N = 5 for M = 1, …, 4

Equations (27)

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G ( x ) = m = t ( x md ) = t ( x ) m = δ ( x md ) ,
u ( x , z ) = t ( x ) p ( x , z ) ,
p ( x , z ) = 1 d m = exp [ i π λ z ( m / d ) 2 ] exp ( i 2 π mx / d ) .
p [ x , z = ( M / N ) Z t ] = 1 d m = exp [ i 2 π ( M / N ) m 2 ] exp ( i 2 π mx / d ) .
p [ x , z = ( M / N ) Z t ] = 1 d L = n = 0 N 1 exp [ i 2 π ( M / N ) n 2 ] × exp ( i 2 π nx / d ) exp ( i 2 π NLx / d ) .
L = exp ( i 2 π NLx / d ) = d N L = δ ( x Ld / N ) .
p [ x , z = ( M / N ) Z t ] = L = C ( L , M , N ) δ ( x Ld / N ) ,
C ( L , M , N ) = 1 N m = 0 N 1 exp [ i 2 π N m ( L Mm ) ] .
u [ x , z = ( M / N ) Z t ] = L = C ( L , M , N ) t ( x Ld / N ) = L = 0 N 1 C ( L , M , N ) G ( x Ld / N ) .
C ( L , 0 , N ) = { 1 for L multiple of N , 0 otherwise .
| C ( L , M , N ) | 2 = 1 N 2 q = 0 N 1 p = 0 N 1 exp [ i 2 π N L ( q p ) ] × exp [ i 2 π N ( q 2 p 2 ) M ] .
| C ( L , M , N ) | 2 = 1 N 2 p = 0 N 1 n = 0 N 1 exp ( i 2 π N ln ) × exp [ i 2 π N n ( n + 2 p ) M ] = 1 N n = 0 N 1 exp [ i 2 π N n ( L Mn ) ] × C ( 2 Mn , 0 , N ) .
C ( 2 Mn , 0 , N ) = { 1 only if n = 0 , N / 2 with N even 1 only if n = 0 N odd 0 otherwise .
| C ( L , M , N ) | 2 = 1 N { 1 for odd N 1 + exp [ i π ( L MN / 2 ) ] for even N .
F ( L ) = C ( L , M , N ) | C ( L , M , N ) | ,
Δ z = [ 1 ( M / N ) ] Z t .
C ( L , M , N ) = { exp [ i ϕ ( L , M , N ) ] C ( 0 , M , N ) for N = 2 K + 1 or N = 4 K exp [ i ϕ ( L , M , N ) ] C ( 1 , M , N ) for N = 4 K + 2 ,
exp [ i 2 π N m ( L Mm ) ]
C ( L , M , N ) = exp [ i 2 π N ( rL + M r 2 ) ] × 1 N m = 0 N 1 exp [ i 2 π N ( mL m 2 M + 2 mMr ) ] .
ϕ ( L , M , N ) = 2 π N ( rL + M r 2 ) ,
r ( L , M , N ) = { QN L 2 M for N = 4 K or N = 2 K + 1 QN L + 1 2 M for N = 4 K + 2 ,
r ( L , 1 , N ) = { L / 2 N = 4 K or 2 K + 1 ( even L ) ( N L ) / 2 N = 2 K + 1 ( odd L ) ( 1 L ) / 2 N = 4 K + 2 ( odd L ) .
r ( L , M , N ) = r ( L , 1 , N ) ,
L = { QN + ML N = 4 K or 2 K + 1 ( even L ) QN + ML MN N = 2 K + 1 ( odd L ) QN + ML M + 1 N = 4 K + 2 ( odd L ) .
T ( x , y ) = m = K K δ [ y ϕ ( x ) mY ] .
u ( x , z = 0 ) = exp ( i 2 π μ 0 y ) exp [ i 2 π μ 0 ϕ ( x ) ] .
OPD = 2 π μ 0 ( ϕ max ϕ min ) .

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