Abstract

An approach for designing Talbot array illuminators that produce quasi-arbitrary numbers of spots and quasi-arbitrary distributions of these spots in the output plane is proposed. The approach consists in combining elementary Talbot array illuminators. In addition to the quasi-arbitrary topology of the output spots, the resulting compound array illuminator shows several advantages such as its ability to be used for both coherent and incoherent imaging. Compared with elementary array illuminators, the compound illuminator does not add any specific technological complication.

© 2006 Optical Society of America

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References

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  1. N. Streibl, "Beam shaping with optical array generators," J. Mod. Opt. 36, 1559-1573 (1989).
    [CrossRef]
  2. M. Barge, H. Hamam, Y. Defosse, R. Chevallier, and J. L. de Bougrenet, "Array illuminators using diffractive optical elements," J. Opt. 27, 151-170 (1996).
    [CrossRef]
  3. H. Dammann and E. Klotz, "Coherent optical generation and inspection of two-dimensionnal periodic structures," Opt. Acta 24, 505-515 (1977).
    [CrossRef]
  4. A. W. Lohmann and J. A. Thomas, "Making an array illuminator based on the Talbot effect," Appl. Opt. 29, 4337-4340 (1990).
    [CrossRef] [PubMed]
  5. H. Hamam, "Design of Talbot array illuminators," Opt. Commun. 131, 359-370 (1996).
    [CrossRef]
  6. H. Hamam, "Talbot imaging and unification," Appl. Opt. 42, 7052-7059 (2003).
    [CrossRef] [PubMed]
  7. J. T. Winthrop and C. R. Worthington, "Theory of Fresnel images. I. Plane periodic objects in monochromatic light," J. Opt. Soc. Am. 55, 373-381 (1965).
    [CrossRef]
  8. H. Hamam, "Simplified linear formulation of Fresnel diffraction," Opt. Commun. 144, 89-98 (1996).
    [CrossRef]
  9. K. Patorski, "The self-imaging phenomenon and its applications" inProgress in Optics, E. Wolf, ed. (North-Holland, 1989), Vol. 27, pp. 1-110.
    [CrossRef]
  10. J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1968).
  11. V. Arrizon and J. Ojeda-Castaneda, "Fresnel diffraction of substructured gratings: matrix description," Opt. Lett. 20, 118-120 (1995).
    [CrossRef] [PubMed]
  12. H. Hamam and J. L. de Bougrenet de la Tocnaye, "Multilayer array illuminators with binary phase plates at fractional Talbot distances," Appl. Opt. 35, 1820-1826 (1996).
    [CrossRef] [PubMed]
  13. H. Hamam, "Lau array illuminator," Appl. Opt. 43, 2888-2894 (2004).
    [CrossRef] [PubMed]
  14. X. Y. Da, "Multiple-image formation by Fresnel-Dirac sampling," Appl. Opt. 34, 299-302 (1995).
    [CrossRef] [PubMed]
  15. H. Hamam, "Applet for various types of TAILs." Links for other TAILs and applications associated with them are at www.umoncton.ca/genie/electrique/Cours/Hamam/Optics/Talbot/Tail.htm.
  16. H. Hamam, "Applet for designing TAIL with arbitrary fan-outs and topologies." Other links for TAIL designs and applications are also given at www.umoncton.ca/genie/electrique/Cours/Hamam/Optics/Talbot/ArbiTail.htm.
  17. E. Ioseliani, "Fresnel diffraction by two-dimensional periodic structures," Opt. Spectrosc. 55, 544-547 (1983).

2004 (1)

2003 (1)

1996 (4)

H. Hamam, "Simplified linear formulation of Fresnel diffraction," Opt. Commun. 144, 89-98 (1996).
[CrossRef]

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, and J. L. de Bougrenet, "Array illuminators using diffractive optical elements," J. Opt. 27, 151-170 (1996).
[CrossRef]

H. Hamam, "Design of Talbot array illuminators," Opt. Commun. 131, 359-370 (1996).
[CrossRef]

H. Hamam and J. L. de Bougrenet de la Tocnaye, "Multilayer array illuminators with binary phase plates at fractional Talbot distances," Appl. Opt. 35, 1820-1826 (1996).
[CrossRef] [PubMed]

1995 (2)

1990 (1)

1989 (1)

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

1983 (1)

E. Ioseliani, "Fresnel diffraction by two-dimensional periodic structures," Opt. Spectrosc. 55, 544-547 (1983).

1977 (1)

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

1965 (1)

Arrizon, V.

Barge, M.

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, and J. L. de Bougrenet, "Array illuminators using diffractive optical elements," J. Opt. 27, 151-170 (1996).
[CrossRef]

Chevallier, R.

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, and J. L. de Bougrenet, "Array illuminators using diffractive optical elements," J. Opt. 27, 151-170 (1996).
[CrossRef]

Da, X. Y.

Dammann, H.

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

de Bougrenet, J. L.

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, and J. L. de Bougrenet, "Array illuminators using diffractive optical elements," J. Opt. 27, 151-170 (1996).
[CrossRef]

de la Tocnaye, J. L. de Bougrenet

Defosse, Y.

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, and J. L. de Bougrenet, "Array illuminators using diffractive optical elements," J. Opt. 27, 151-170 (1996).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1968).

Hamam, H.

H. Hamam, "Lau array illuminator," Appl. Opt. 43, 2888-2894 (2004).
[CrossRef] [PubMed]

H. Hamam, "Talbot imaging and unification," Appl. Opt. 42, 7052-7059 (2003).
[CrossRef] [PubMed]

H. Hamam, "Design of Talbot array illuminators," Opt. Commun. 131, 359-370 (1996).
[CrossRef]

H. Hamam, "Simplified linear formulation of Fresnel diffraction," Opt. Commun. 144, 89-98 (1996).
[CrossRef]

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, and J. L. de Bougrenet, "Array illuminators using diffractive optical elements," J. Opt. 27, 151-170 (1996).
[CrossRef]

H. Hamam and J. L. de Bougrenet de la Tocnaye, "Multilayer array illuminators with binary phase plates at fractional Talbot distances," Appl. Opt. 35, 1820-1826 (1996).
[CrossRef] [PubMed]

Ioseliani, E.

E. Ioseliani, "Fresnel diffraction by two-dimensional periodic structures," Opt. Spectrosc. 55, 544-547 (1983).

Klotz, E.

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

Lohmann, A. W.

Ojeda-Castaneda, J.

Patorski, K.

K. Patorski, "The self-imaging phenomenon and its applications" inProgress in Optics, E. Wolf, ed. (North-Holland, 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]

Thomas, J. A.

Winthrop, J. T.

Worthington, C. R.

Appl. Opt. (5)

J. Mod. Opt. (1)

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

J. Opt. (1)

M. Barge, H. Hamam, Y. Defosse, R. Chevallier, and J. L. de Bougrenet, "Array illuminators using diffractive optical elements," J. Opt. 27, 151-170 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

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

Opt. Commun. (2)

H. Hamam, "Design of Talbot array illuminators," Opt. Commun. 131, 359-370 (1996).
[CrossRef]

H. Hamam, "Simplified linear formulation of Fresnel diffraction," Opt. Commun. 144, 89-98 (1996).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (1)

E. Ioseliani, "Fresnel diffraction by two-dimensional periodic structures," Opt. Spectrosc. 55, 544-547 (1983).

Other (4)

H. Hamam, "Applet for various types of TAILs." Links for other TAILs and applications associated with them are at www.umoncton.ca/genie/electrique/Cours/Hamam/Optics/Talbot/Tail.htm.

H. Hamam, "Applet for designing TAIL with arbitrary fan-outs and topologies." Other links for TAIL designs and applications are also given at www.umoncton.ca/genie/electrique/Cours/Hamam/Optics/Talbot/ArbiTail.htm.

K. Patorski, "The self-imaging phenomenon and its applications" inProgress in Optics, E. Wolf, ed. (North-Holland, 1989), Vol. 27, pp. 1-110.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1968).

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Figures (5)

Fig. 1
Fig. 1

Two different TAILs: (a) one with a complex profile having Dirac pulses with different complex weights separated by an interval of 2 d / q and (b) one that can be implemented by a multilevel phase plate.

Fig. 2
Fig. 2

Compound TAIL producing arbitrary output with four spots per period: (a) one period of the replay field, (b) one elementary cell in one period of the compound TAIL, (c) one period of the compound TAIL.

Fig. 3
Fig. 3

Setup for concentrating energy in one period: h(x) is the transmittance of the compound TAIL. In the output plane, one obtains one period of h(x, z), namely, t(x). t 1 ( x ) is the wave field just behind the lens and the phase element. t 1 ( x , z 2 ) is the diffraction field of t 1 ( x ) at distance z 2 . p h ( x ) is the transmittance of the phase element.

Fig. 4
Fig. 4

Compound TAIL imaging setup. The object is reproduced in several replicas at distance z 2 . Two elementary TAILs are included. Only five periods (vertically) of the replay field are shown. In reality, the replay field extends horizontally and vertically as a two-dimensional periodic amplitude distribution.

Fig. 5
Fig. 5

Compound TAILs and their respective replay fields observed at z = 1 / 3 Z T . Top: phase distribution of one period of each compound TAIL. Bottom: respective amplitude distribution of one period of each replay field.

Equations (39)

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h ( x , z ) = FR z { h ( x , 0 ) } = exp ( i 2 π z / λ ) exp ( i π / 4 ) λ z × h ( x ) f k ( x , z ) ,
f k ( x , z ) = exp ( i π x 2 λ z ) .
h ( x , z ) = m = 0 q / 2 1 T ( m , p , q ) h ( x d 2 + 2 d q m ) .
T ( m p , p , q ) = cst ( - 1 ) m exp ( i 2 π p q m 2 ) .
h ( x , z ) = k T ( x , p , q ) h ( x ) .
k T ( x , p , q ) = m = 0 m = q / 2 1 T ( m , p , q ) δ ( x d 2 2 m d q ) .
h ( x , 0 ) = k T ( x , p , q ) h ( x , z ) .
h ( x , z ) = cst   n = n = + δ ( x n d ) .
h ( x , 0 ) = cst [ m =0 m = q / 2 1 T ( m , p , q ) × δ ( x d 2 2 m d q ) ] n = n = + δ ( x n d )
h ( x , 0 ) = cst   n = n =+ m = 0 m = q / 2 1 T ( m , p , q ) × δ ( x n d d 2 2 m d q ) .
T ( m + q 2 , p , q ) = T ( m , p , q ) ,
h ( x , 0 ) = m = m = + T ( m , p , q ) δ ( x d 2 2 m d q ) .
h ( x , 0 ) = cst [ m =0 m = q / 2 1 ( 1 ) m exp ( i 2 π p q m 2 ) × δ ( x d 2 2 m p d q ) ] n = n = + δ ( x n d ) .
h ( x , 0 ) = cst    m = m = + ( 1 ) m exp ( i 2 π m 2 q ) × δ ( x d 2 2 m d q ) .
h ( x , 0 ) = cst [ m = 0 m = q / 2 1 ( 1 ) m exp ( i 2 π p q m 2 ) × δ ( x d 2 2 m p d q ) ] n = n = + rect [ q 2 d ( x n d ) ] ,
h ( x , 0 ) = cst { m = 0 m = q / 2 1 ( 1 ) m exp ( i 2 π p q m 2 ) × rect [ q 2 d ( x d 2 2 m d q ) ] } n = n = + δ ( x n d ) .
R C = d Δ x = q 2 .
h ( x , 0 ) = cst { m = 0 m = q / 2 1 ( 1 ) m exp ( i 2 π p q m 2 ) × rect [ q d ( x d 2 2 m d q ) ] } n = n = + δ ( x n d ) .
h ( x , z ) = t ( x ) = n = n = + t ( x n d ) ,
t ( x ) = cst   rect ( q 2 d x ) m = 1 m = M δ ( x x m ) .
t ( x ) = m = 1 m = M δ ( x x m ) .
1 z 2 + 1 z 1 = 1 z ,
l ( x ) = exp ( i π x 2 λ f L )
p h ( x ) = n = n = + exp ( i π 2 q p n 2 ) rect ( x n d )
d = z 2 z d = z 1 + z 2 z 1 d .
h ( x , z ) = t ( x ) = n = n = + t [ - z 1 z 2 ( x n d ) ] .
1 z 2 + 1 z 1 = 1 z ,
ima ( x ) = exp [ i π x 2 λ ( z 2 z ) ] n = n = + exp ( - i π 2 q p n 2 ) × s [ z 1 z 2 ( x n d ) ] ,
s ( x ) = obj ( x ) t ( x ) = m = 1 m = M obj ( x x m ) .
z 1 z 2     2 + 1 z 2 = 1 z 2 z
t ( x , z 1 ) = FR z 1 { t ( x ) } = exp ( i π x 2 λ z 1 ) + t ( x 1 ) exp ( i π x 1     2 λ z 1 ) × exp ( i 2 π x 1 x λ z 1 ) d x 1 .
t ( x , z 1 ) = t 2 ( x ) = exp ( i π x 2 λ z 1 ) × FT { t ( u ) exp ( i π u 2 λ z 1 ) } | u = x / ( λ z 1 ) .
t 1 ( x , z 2 ) = t 2 ( x ) n = n = + exp [ j π ( x n d ) 2 λ z ] .
t 1 ( x , z 2 ) = exp [ i π x 2 λ ( 1 z 1 z 1 ) ] × FT { t ( u ) exp ( i π u 2 λ z 1 ) } | u = x / ( λ z 1 ) × n = n = + exp ( i π n 2 d 2 λ z ) exp ( i π 2 x n d λ z ) .
t 1 ( x ) = exp ( i π x 2 λ z 2 ) × FT { FT { t ( u ) exp ( i π u 2 λ z 1 ) } | u = x / ( λ z 1 ) × n = n = + exp ( i π n 2 d 2 λ z ) × exp ( i π 2 x n d λ z ) } | u = x / ( λ z 2 ) .
t 1 ( x ) = exp [ i π x 2 λ ( z 1 z 2 2 + 1 z 2 ) ] n = n = + exp ( i π n 2 d 2 λ z ) × t ( z 1 z 2 x + n z 1 z d ) .
t ( x ) = exp [ i π x 2 λ ( z 1 z 2     2 + 1 z 2 1 f L ) ] × n = n = + exp [ i π n 2 ( d 2 λ z q 2 p ) ] t [ z 1 z 2 ( x n d ) ] .
z 1 z 2     2 + 1 z 2 1 f L
h ( x , z ) = t ( x ) = n = n = + t [ z 1 z 2 ( x n d ) ] .

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