Abstract

We show that the transmittance of a finite Talbot array illuminator (TAI) can be expressed by the phase distribution of a pixelated lens, modulated by a discrete phase grating (G). Thus the TAI reconstruction field is given by the convolution of the grating’s Fourier transform, with the point-spread function of the pixelated lens. On the basis of this approach we propose a method to improve the performance of a finite TAI by modifying the basic cell of the grating factor G.

© 2000 Optical Society of America

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References

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  1. 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]
  2. J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
    [CrossRef]
  3. P. Szwaykowski, V. Arrizón, “Talbot array illuminators with multilevel phase gratings,” Appl. Opt. 32, 1109–1114 (1993).
    [CrossRef] [PubMed]
  4. V. Arrizón, J. Ojeda-Castañeda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 (1994).
    [CrossRef] [PubMed]
  5. C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115, 40–44 (1995).
    [CrossRef]
  6. H. Hamam, J. L. B. de la Tocnaye, “Multilayer array illuminators with binary phase plates at fractional Talbot distances,” Appl. Opt. 35, 1820–1826 (1996).
    [CrossRef] [PubMed]
  7. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  8. W. Klaus, Y. Arimoto, K. Kodate, “Talbot array illuminators providing spatial intensity and phase modulation,” J. Opt. Soc. Am. 14, 1092–1102 (1997).
    [CrossRef]
  9. T. J. Suleski, “Generation of Lohmann images from binary-phase Talbot array illuminators,” Appl. Opt. 36, 4686–4691 (1997).
    [CrossRef] [PubMed]
  10. W. Klaus, Y. Arimoto, K. Kodate, “High-performance Talbot array illuminators,” Appl. Opt. 37, 4357–4365 (1998).
    [CrossRef]
  11. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  12. V. Arrizón, G. Rojo-Velázquez, “Fractional Talbot effect: compact description,” Opt. Rev. 7, 129–131 (2000).
    [CrossRef]
  13. N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
    [CrossRef]
  14. E. Carcolé, J. Campos, S. Bosch, “Diffraction theory of Fresnel lenses encoded in low-resolution devices,” Appl. Opt. 33, 162–174 (1994).
    [CrossRef] [PubMed]
  15. V. Arrizón, J. G. Ibarra, A. Serrano-Heredia, “Split Talbot array illuminators,” Opt. Commun. 123, 63–70 (1996).
    [CrossRef]
  16. V. Arrizón, E. Lopez-Olazagasti, “Binary phase gratings for array generation at 1/16 of Talbot length,” J. Opt. Soc. Am. A 12, 801–804 (1995).
    [CrossRef]
  17. Ch. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminations,” Appl. Opt. 38, 284–290 (1999).
    [CrossRef]
  18. V. Arrizón, E. Carreón, L. A. González, “Self-apodization of low-resolution pixelated lenses,” Appl. Opt. 38, 5073–5077 (1999).
    [CrossRef]

2000 (1)

V. Arrizón, G. Rojo-Velázquez, “Fractional Talbot effect: compact description,” Opt. Rev. 7, 129–131 (2000).
[CrossRef]

1999 (2)

1998 (1)

1997 (2)

T. J. Suleski, “Generation of Lohmann images from binary-phase Talbot array illuminators,” Appl. Opt. 36, 4686–4691 (1997).
[CrossRef] [PubMed]

W. Klaus, Y. Arimoto, K. Kodate, “Talbot array illuminators providing spatial intensity and phase modulation,” J. Opt. Soc. Am. 14, 1092–1102 (1997).
[CrossRef]

1996 (2)

1995 (2)

C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115, 40–44 (1995).
[CrossRef]

V. Arrizón, E. Lopez-Olazagasti, “Binary phase gratings for array generation at 1/16 of Talbot length,” J. Opt. Soc. Am. A 12, 801–804 (1995).
[CrossRef]

1994 (2)

1993 (1)

1990 (1)

1989 (1)

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

1982 (1)

1971 (1)

J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

1965 (1)

Arimoto, Y.

W. Klaus, Y. Arimoto, K. Kodate, “High-performance Talbot array illuminators,” Appl. Opt. 37, 4357–4365 (1998).
[CrossRef]

W. Klaus, Y. Arimoto, K. Kodate, “Talbot array illuminators providing spatial intensity and phase modulation,” J. Opt. Soc. Am. 14, 1092–1102 (1997).
[CrossRef]

Arrizón, V.

Bosch, S.

Campos, J.

Carcolé, E.

Carreón, E.

de la Tocnaye, J. L. B.

Fienup, J. R.

González, L. A.

Guigay, J. P.

J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

Hamam, H.

Ibarra, J. G.

V. Arrizón, J. G. Ibarra, A. Serrano-Heredia, “Split Talbot array illuminators,” Opt. Commun. 123, 63–70 (1996).
[CrossRef]

Klaus, W.

W. Klaus, Y. Arimoto, K. Kodate, “High-performance Talbot array illuminators,” Appl. Opt. 37, 4357–4365 (1998).
[CrossRef]

W. Klaus, Y. Arimoto, K. Kodate, “Talbot array illuminators providing spatial intensity and phase modulation,” J. Opt. Soc. Am. 14, 1092–1102 (1997).
[CrossRef]

Kodate, K.

W. Klaus, Y. Arimoto, K. Kodate, “High-performance Talbot array illuminators,” Appl. Opt. 37, 4357–4365 (1998).
[CrossRef]

W. Klaus, Y. Arimoto, K. Kodate, “Talbot array illuminators providing spatial intensity and phase modulation,” J. Opt. Soc. Am. 14, 1092–1102 (1997).
[CrossRef]

Liu, L.

C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115, 40–44 (1995).
[CrossRef]

Lohmann, A. W.

Lopez-Olazagasti, E.

Ojeda-Castañeda, J.

Rojo-Velázquez, G.

V. Arrizón, G. Rojo-Velázquez, “Fractional Talbot effect: compact description,” Opt. Rev. 7, 129–131 (2000).
[CrossRef]

Serrano-Heredia, A.

V. Arrizón, J. G. Ibarra, A. Serrano-Heredia, “Split Talbot array illuminators,” Opt. Commun. 123, 63–70 (1996).
[CrossRef]

Stankovic, S.

Streibl, N.

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

Suleski, T. J.

Szwaykowski, P.

Thomas, J. A.

Tschudi, T.

Winthrop, J. T.

Worthington, C. R.

Zhou, C.

C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115, 40–44 (1995).
[CrossRef]

Zhou, Ch.

Appl. Opt. (10)

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)

W. Klaus, Y. Arimoto, K. Kodate, “Talbot array illuminators providing spatial intensity and phase modulation,” J. Opt. Soc. Am. 14, 1092–1102 (1997).
[CrossRef]

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]

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

Opt. Acta (1)

J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

Opt. Commun. (2)

C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115, 40–44 (1995).
[CrossRef]

V. Arrizón, J. G. Ibarra, A. Serrano-Heredia, “Split Talbot array illuminators,” Opt. Commun. 123, 63–70 (1996).
[CrossRef]

Opt. Rev. (1)

V. Arrizón, G. Rojo-Velázquez, “Fractional Talbot effect: compact description,” Opt. Rev. 7, 129–131 (2000).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

(a) Phase distribution of a TAI with reconstruction plane z = 16Z T /17, (b) reconstruction field computed with FTE formulation [Eq. (2)]. Only two periods are shown.

Fig. 2
Fig. 2

Finite TAI with nine periods and reconstruction plane z = 16Z T /17. Phase distributions (for x ≥ 0) of (a) pixelated lens L(x), (b) grating factor G(x), (c) TAI.

Fig. 3
Fig. 3

Similar to Fig. 2, for M = 1 and identical values for the other parameters.

Fig. 4
Fig. 4

Corresponding to TAI in Fig. 2 we obtained intensity distributions for (a) PSF of the pixelated lens L(x), (b) train of Dirac’s deltas of (xz), and (c) TAI reconstruction field.

Fig. 5
Fig. 5

Similar to Fig. 4, for TAI in Fig. 3.

Fig. 6
Fig. 6

First modification of the grating factor in Fig. 2, for generation of 11 spots: (a) basic cell of optimized grating factor G(x), (b) train of impulses of (xz), (c) reconstruction field of modified TAI.

Fig. 7
Fig. 7

Second optimization (on the grating factor in Fig. 6): (a) basic cell of optimized grating factor G(x), (b) train of Dirac’s deltas of (xz), and (c) reconstruction field of modified TAI.

Fig. 8
Fig. 8

Optimization of TAI in Fig. 2, constrained to 14 phase levels: phase distributions for (a) PL and (b) first modified grating factor G.

Fig. 9
Fig. 9

Second (phase-constrained) optimization, on grating factor in Fig. 8: (a) basic cell of optimized grating factor G(x), (b) train of Dirac’s deltas of (xz), (c) reconstruction field of modified TAI.

Tables (1)

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Table 1 Phase Distributions of the Initial Grating Factor (G)a

Equations (20)

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tx=fcx  t0x,
ψx=L=0N-1 CL, M, Ntx-Ld/N,
CL, M, N=expi πL22MNTL, M, N,
TL, M, N=12MN1/2m=02M-1expi πN2M m2×exp-i πM mL.
CL, M, N=0,  even N/2,  odd L,
CL, M, N=0,  odd N/2,  even L,
CL, M, N0,  otherwise.
expiψL=exp-iπL2/2MNexp-iϕTL,
tx=LxGx,
Lx=L=-QQexp-i πL22MNrectx-Lpp,
Gx=L=-exp-iϕTLrectx-Lpp.
FRfx, z=1λz- fβexpiπβ2λz×exp-i2π βxλzdβ,
Ψx=1λzFRLx, z  G˜xλz.
fcx=m=02M-1 fdx-md,
ψdx=1dN2M1/2 expi πNx22Md2n=-exp-i2πnNx/d=12MN1/2n=-expi πn22MNδx-ndN.
ψcx=m=02M-1 ψdx-md=L=- CL, M, Nδx-Ld/N,
tx=L=-QQexpiψLrectx-Lpp,
Ψx=1p2MN1/2L=-QQexp-iϕTL×exp-iπLxMNpIx, L,
Ix, L=-rectξpexpiπξ2λz×exp-i2π2MNpxp-Lξdξ.
Ψx=12MN1/2L=-QQexp-iϕTL×exp-iπLxMNpsinc12MNxp-L.

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