Abstract

The fractional Talbot effect brings into play a superimposition of shifted and complex weighted replicas of the original object. This phenomenon can be used to replicate images of nonperiodic objects by means of Talbot array illuminators. These diffractive elements can also be used to concentrate replicas into a single image. These techniques are useful for several applications such as beam shaping.

© 2003 Optical Society of America

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References

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  1. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  2. 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]
  3. H. Hamam, “Simplified linear formulation of diffraction,” Opt. Commun. 144, 89–98 (1996).
    [CrossRef]
  4. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  5. S. Zhao, C. Zhou, P. Xi, H. Wang, L. Liu, “Number of phase levels in a two-dimensional separable Talbot array illuminator,” J. Opt. Soc. Am. A 18, 103–1070 (2001).
    [CrossRef]
  6. H. Hamam, 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]
  7. H. Hamam, “Design of Talbot array illuminators,” Opt. Commun. 131, 359–370 (1996).
    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  9. B. Smolinska, A. Smolinska Kalestynski, “Autoidolon of quasi-periodic optical objects,” Opt. Acta 25, 257–263 (1978).
    [CrossRef]
  10. A. Smirnov, A. D. Galpern, “Effect of errors of a periodic transparency,” Opt. Spectrosc. 48, 324–326 (1980).
  11. E. Loseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. 55, 544–547 (1983).
  12. J. Goodman, A. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
    [CrossRef]
  13. H. Hamam, “Applet for Talbot imaging,” http://www.umoncton.ca/genie/electrique/Cours/Hamamh/Optics/Talbot/TalbImagFrac.htm .
  14. F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 29, 119–126 (1992).
    [CrossRef]
  15. H. Hamam, “Programmable multilayer diffractive optical elements,” J. Opt. Soc. Am. A 14, 2223–2230 (1997).
    [CrossRef]

2001

1997

1996

H. Hamam, “Simplified linear formulation of diffraction,” Opt. Commun. 144, 89–98 (1996).
[CrossRef]

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

H. Hamam, 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]

1992

F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 29, 119–126 (1992).
[CrossRef]

1990

1983

E. Loseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. 55, 544–547 (1983).

1980

A. Smirnov, A. D. Galpern, “Effect of errors of a periodic transparency,” Opt. Spectrosc. 48, 324–326 (1980).

1978

B. Smolinska, A. Smolinska Kalestynski, “Autoidolon of quasi-periodic optical objects,” Opt. Acta 25, 257–263 (1978).
[CrossRef]

1970

J. Goodman, A. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

1967

de Bougrenet de la Tocnaye, J. L.

Galpern, A. D.

A. Smirnov, A. D. Galpern, “Effect of errors of a periodic transparency,” Opt. Spectrosc. 48, 324–326 (1980).

Goodman, J.

J. Goodman, A. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Goodman, J. W.

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

Hamam, H.

Liu, L.

Lohmann, A. W.

Loseliani, E.

E. Loseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. 55, 544–547 (1983).

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]

Silvestri, A.

J. Goodman, A. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

Smirnov, A.

A. Smirnov, A. D. Galpern, “Effect of errors of a periodic transparency,” Opt. Spectrosc. 48, 324–326 (1980).

Smolinska, B.

B. Smolinska, A. Smolinska Kalestynski, “Autoidolon of quasi-periodic optical objects,” Opt. Acta 25, 257–263 (1978).
[CrossRef]

Smolinska Kalestynski, A.

B. Smolinska, A. Smolinska Kalestynski, “Autoidolon of quasi-periodic optical objects,” Opt. Acta 25, 257–263 (1978).
[CrossRef]

Thomas, J. A.

Wang, H.

Wyrowski, F.

F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 29, 119–126 (1992).
[CrossRef]

Xi, P.

Zhao, S.

Zhou, C.

Appl. Opt.

IBM J. Res. Dev.

J. Goodman, A. Silvestri, “Some effects of Fourier-domain phase quantization,” IBM J. Res. Dev. 14, 478–484 (1970).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

B. Smolinska, A. Smolinska Kalestynski, “Autoidolon of quasi-periodic optical objects,” Opt. Acta 25, 257–263 (1978).
[CrossRef]

Opt. Commun.

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

F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 29, 119–126 (1992).
[CrossRef]

H. Hamam, “Simplified linear formulation of diffraction,” Opt. Commun. 144, 89–98 (1996).
[CrossRef]

Opt. Spectrosc.

A. Smirnov, A. D. Galpern, “Effect of errors of a periodic transparency,” Opt. Spectrosc. 48, 324–326 (1980).

E. Loseliani, “Fresnel diffraction by two-dimensional periodic structures,” Opt. Spectrosc. 55, 544–547 (1983).

Other

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

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]

H. Hamam, “Applet for Talbot imaging,” http://www.umoncton.ca/genie/electrique/Cours/Hamamh/Optics/Talbot/TalbImagFrac.htm .

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Figures (7)

Fig. 1
Fig. 1

Fractional Talbot effect: (a) Talbot kernel k T (x, p, q), (b) initial periodic field, (c) diffraction pattern h(x, z) at fractional Talbot distance z = p/ qZ T . h(x, z) = h(x) * k T (x, p, q).

Fig. 2
Fig. 2

Talbot imaging setup: The object is reproduced in several replicas at distance z 2. The phase plate and the lens in the dashed rectangular box transform the divergent beam into a plane wave. Δx, permitted lateral shift.

Fig. 3
Fig. 3

Talbot unification setup: t(x), periodic structure. In the image plane we obtain an image of one period of t(x). t 1(x) is the wave field just behind structure t(x). t 1(x, z 2) is the diffraction field of t 1(x) at distance z 2. ph(x) is the transmittance of the phase element.

Fig. 4
Fig. 4

System that uses Talbot imaging and unification. Whereas neither at the input nor at the output is a periodic structure imposed, the system profits from the fractional Talbot effect. The input, a single pattern, is made periodic by the Talbot unification subsystem (S 1). Then any operation that uses the fractional Talbot effect, such as logic operations (S 2), can be performed. The periodic structure is passed through a Talbot unification subsystem (S 3) to produce a single pattern at the output.

Fig. 5
Fig. 5

Talbot unification system used for beam shaping: (a) a nonuniform beam is entered; (b) this beam can be considered a quasi-periodic two-dimensional array of small cells (periods). When this beam goes through the Talbot unification subsystem, all cells are added constructively to the central cell and we obtain the square cell at the output (magnified in this figure).

Fig. 6
Fig. 6

Input object composed of four small segments spread in a large dark zone.

Fig. 7
Fig. 7

At the one-eighth Talbot plane, the four segments of Fig. 6 are replicated and fill all dark zones of the original to form an Escher-like figure. Only four periods are shown.

Equations (33)

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hx, z=FRzhx, 0= expi2πz/λexp-iπ/4λz hx * fkx, z,
fkx, z=expjπ x2λz.
hx, z=a=0q/2-1 Ta, p, qhx- d2+ 2adq,
Ta, p, q= 2qb=0q/2-1 exp-iπ2 b2q p+b×expiπ 4abq,
hx, z=kTx, p, q * h x,
kTx, p, q=m=0m=q/2-1 Tm, p, qδx- d2- 2mdq.
Tmp, p, q=cst-1m exp-i2π pq m2.
hx, 0=kTx, -p, q * h x, z.
kTx, -p, q=kTx, q-p, q.
Ix, z=rectq2d x * n=-n=+ δx-nd.
Ix=Ix, 0=rectq2d x * n=-n=+ kTx-nd, -p, q
Ix=n=-n=+m=0m=q/2-1 Tm, -p, qrect×q2dx- d2- 2mdq-nd.
Ta, p, q=T a+q/2, p, q,
Ix=m=-m=+ Tm, -p, qrectq2dx- d2- 2mdq.
Ix=FR-zrectq2d x * n=-n=+ δx-nd.
Ix=rectq2d x * n=-n=+ fkx-nd, -z
Ix=n=-n=+exp-jπ x-nd2λz * rectq2d x.
Ix=n=-n=+ exp-jπ x-nd2λz.
1z2+ 1z1= 1z,
sx=expiπ x2λz2-zn=-n=+ exp-j π2qp n2×s- z1z2 x+n z2z d.
z2z1 Gd= z2z d= z1+z2z1 d
G z1z2 d= z1+z2z2 d= z1z d.
phx=n=-n=+ expj π2qp n2 rect - z1z2 x+n z2z d.
fL=z2-z.
expiπ x2λz2-zexp-iπ x-dx2λz2-zshifted lens=expiπ dx2λz2-zcons tan texp-i2π xdxλz2-zprism.
hx, z=FRzhx =expiπ x2λz-+ hx1expiπ x12λz ×exp-i2π x1xλzdx1.
hx, z=expiπ x2λzFThuexpiπ u2λzu=x/λz.
sx, z=expiπ x2λz1FTsuexpiπ u2λz1u=x/λz1.
s1x=sx, zn=-n=+ exp-jπ x-nd2λz.
s1x=exp-iπ x2λz2n=-n=+ exp-jπ n2d2λz× expjπ 2xndλzFTsu×expiπ u2λz1u=x/λz1.
s1x, z2=expiπ x2λz2FTn=-n=+ exp-jπ n2d2λz ×expjπ 2xndλzFTsu×expiπ u2λz1u=x/λz1u=x/λz2.
sx=s1x, z2=expiπ x2λz2-zn=-n=+ exp-jπ n2d2λz×s- z1z2 x+n z2z d.
sx=expiπ x2λz2-zn=-n=+×exp-j π2qp n2s- z1z2 x+n z2z d.

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