Abstract

The number of phase levels of a Talbot array illuminator is an important factor in the estimation of practical fabrication complexity and cost. We show that the number (L) of phase levels of a Talbot array illuminator has a simple relationship to the prime number. When there is an alternative π-phase modulation in the output array, the relations are similar.

© 2001 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 (Stuttgart) 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. J. R. Leger, G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional Talbot planes,” Opt. Lett. 15, 288–290 (1990).
    [Crossref] [PubMed]
  4. L. Liu, “Lau cavity and phase locking of laser arrays,” Opt. Lett. 14, 1312–1314 (1989).
    [Crossref] [PubMed]
  5. V. Arrizón, J. Ojeda-Castañeda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 (1994).
    [Crossref] [PubMed]
  6. P. Szwaykowski, V. Arrizón, “Talbot array illuminator with multilevel phase gratings,” Appl. Opt. 32, 1109–1114 (1993).
    [Crossref] [PubMed]
  7. V. Arrizon, J. Ojeda-Castañeda, “Fresnel diffraction of substructured gratings: matrix description,” Opt. Lett. 20, 118–120 (1995).
    [Crossref] [PubMed]
  8. M. Testorf, J. Ojeda-Castañeda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am A 13, 119–125 (1996).
    [Crossref]
  9. J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
    [Crossref]
  10. 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]
  11. C. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminations,” Appl. Opt. 38, 284–290 (1999).
    [Crossref]
  12. V. Arrizón, E. López-Olazagasti, “Binary phase grating for array generation at 1/16 of Talbot length,” J. Opt. Soc. Am. A 12, 801–804 (1995).
    [Crossref]
  13. W. Klaus, Y. Arimoto, K. Kodate, “Talbot array illuminators providing spatial intensity and phase modulation,” J. Opt. Soc. Am. A 14, 1092–1102 (1997).
    [Crossref]
  14. W. Klaus, Y. Arimoto, K. Kodate, “High-performance Talbot array illuminators,” Appl. Opt. 37, 4357–4365 (1998).
    [Crossref]
  15. T. J. Suleski, “Generation of Lohmann images from binary-phase Talbot array illuminators,” Appl. Opt. 36, 4686–4691 (1997).
    [Crossref] [PubMed]
  16. J. R. Leger, G. Mowry, “External diode-laser array cavity with mode-selecting mirror,” Appl. Phy. Lett. 63, 2884–2886 (1993).
    [Crossref]
  17. J. R. Leger, G. Mowry, D. Chen, “Model analysis of Talbot cavity,” Appl. Phy. Lett. 64, 2937–2939 (1994).
    [Crossref]
  18. M. Testorf, V. Arrizón, J. Ojeda-Castañeda, “Numerical optimization of phase-only elements based on the fractional Talbot effect,” J. Opt. Soc. Am. A 16, 97–105 (1999).
    [Crossref]
  19. H. Wang, C. Zhou, L. Liu, “Simple Fresnel diffraction equations of a grating for Talbot array illumination,” Opt. Commun. 173, 17–22 (2000).
    [Crossref]
  20. S. Nowak, C. Kurtsiefer, T. Pfau, C. David, “High-order Talbot fringes for atomic matter waves,” Opt. Lett. 22, 1430–1432 (1997).
    [Crossref]
  21. C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
    [Crossref]
  22. H. Wang, C. Zhou, L. Jilang, L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
    [Crossref]

2000 (2)

H. Wang, C. Zhou, L. Liu, “Simple Fresnel diffraction equations of a grating for Talbot array illumination,” Opt. Commun. 173, 17–22 (2000).
[Crossref]

H. Wang, C. Zhou, L. Jilang, L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
[Crossref]

1999 (3)

1998 (1)

1997 (3)

1996 (1)

M. Testorf, J. Ojeda-Castañeda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am A 13, 119–125 (1996).
[Crossref]

1995 (3)

1994 (2)

J. R. Leger, G. Mowry, D. Chen, “Model analysis of Talbot cavity,” Appl. Phy. Lett. 64, 2937–2939 (1994).
[Crossref]

V. Arrizón, J. Ojeda-Castañeda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 (1994).
[Crossref] [PubMed]

1993 (2)

P. Szwaykowski, V. Arrizón, “Talbot array illuminator with multilevel phase gratings,” Appl. Opt. 32, 1109–1114 (1993).
[Crossref] [PubMed]

J. R. Leger, G. Mowry, “External diode-laser array cavity with mode-selecting mirror,” Appl. Phy. Lett. 63, 2884–2886 (1993).
[Crossref]

1990 (2)

1989 (1)

1988 (1)

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

1971 (1)

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

Arimoto, Y.

Arrizon, V.

Arrizón, V.

Chen, D.

J. R. Leger, G. Mowry, D. Chen, “Model analysis of Talbot cavity,” Appl. Phy. Lett. 64, 2937–2939 (1994).
[Crossref]

David, C.

Denz, C.

C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[Crossref]

Guigay, J. P.

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

Jilang, L.

H. Wang, C. Zhou, L. Jilang, L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
[Crossref]

Klaus, W.

Kodate, K.

Kurtsiefer, C.

Leger, J. R.

J. R. Leger, G. Mowry, D. Chen, “Model analysis of Talbot cavity,” Appl. Phy. Lett. 64, 2937–2939 (1994).
[Crossref]

J. R. Leger, G. Mowry, “External diode-laser array cavity with mode-selecting mirror,” Appl. Phy. Lett. 63, 2884–2886 (1993).
[Crossref]

J. R. Leger, G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional Talbot planes,” Opt. Lett. 15, 288–290 (1990).
[Crossref] [PubMed]

Liu, L.

H. Wang, C. Zhou, L. Liu, “Simple Fresnel diffraction equations of a grating for Talbot array illumination,” Opt. Commun. 173, 17–22 (2000).
[Crossref]

H. Wang, C. Zhou, L. Jilang, L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
[Crossref]

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]

L. Liu, “Lau cavity and phase locking of laser arrays,” Opt. Lett. 14, 1312–1314 (1989).
[Crossref] [PubMed]

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 (Stuttgart) 79, 41–45 (1988).

López-Olazagasti, E.

Mowry, G.

J. R. Leger, G. Mowry, D. Chen, “Model analysis of Talbot cavity,” Appl. Phy. Lett. 64, 2937–2939 (1994).
[Crossref]

J. R. Leger, G. Mowry, “External diode-laser array cavity with mode-selecting mirror,” Appl. Phy. Lett. 63, 2884–2886 (1993).
[Crossref]

Nowak, S.

Ojeda-Castañeda, J.

Pfau, T.

Stankovic, S.

C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[Crossref]

C. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminations,” Appl. Opt. 38, 284–290 (1999).
[Crossref]

Suleski, T. J.

Swanson, G. J.

Szwaykowski, P.

Testorf, M.

M. Testorf, V. Arrizón, J. Ojeda-Castañeda, “Numerical optimization of phase-only elements based on the fractional Talbot effect,” J. Opt. Soc. Am. A 16, 97–105 (1999).
[Crossref]

M. Testorf, J. Ojeda-Castañeda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am A 13, 119–125 (1996).
[Crossref]

Thomas, J. A.

Tschudi, T.

C. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminations,” Appl. Opt. 38, 284–290 (1999).
[Crossref]

C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[Crossref]

Wang, H.

H. Wang, C. Zhou, L. Jilang, L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
[Crossref]

H. Wang, C. Zhou, L. Liu, “Simple Fresnel diffraction equations of a grating for Talbot array illumination,” Opt. Commun. 173, 17–22 (2000).
[Crossref]

Zhou, C.

H. Wang, C. Zhou, L. Liu, “Simple Fresnel diffraction equations of a grating for Talbot array illumination,” Opt. Commun. 173, 17–22 (2000).
[Crossref]

H. Wang, C. Zhou, L. Jilang, L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
[Crossref]

C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[Crossref]

C. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminations,” Appl. Opt. 38, 284–290 (1999).
[Crossref]

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]

Appl. Opt. (6)

Appl. Phy. Lett. (2)

J. R. Leger, G. Mowry, “External diode-laser array cavity with mode-selecting mirror,” Appl. Phy. Lett. 63, 2884–2886 (1993).
[Crossref]

J. R. Leger, G. Mowry, D. Chen, “Model analysis of Talbot cavity,” Appl. Phy. Lett. 64, 2937–2939 (1994).
[Crossref]

J. Opt. Soc. Am A (1)

M. Testorf, J. Ojeda-Castañeda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am A 13, 119–125 (1996).
[Crossref]

J. Opt. Soc. Am. A (3)

Microwave Opt. Technol. Lett. (1)

H. Wang, C. Zhou, L. Jilang, L. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
[Crossref]

Opt. Acta (1)

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

Opt. Commun. (3)

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]

C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[Crossref]

H. Wang, C. Zhou, L. Liu, “Simple Fresnel diffraction equations of a grating for Talbot array illumination,” Opt. Commun. 173, 17–22 (2000).
[Crossref]

Opt. Lett. (4)

Optik (Stuttgart) (1)

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

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Figures (4)

Fig. 1
Fig. 1

(a) and (b) Symmetric structures of phase factors within one period for even M and odd M, respectively. Because of the symmetries, the number of phase levels of (a) and (b) are at most M/2 + 1 and (M - 1)/2 + 1, respectively.

Fig. 2
Fig. 2

(a) Symmetric structure in the case of M = 4t, and (b) the equal phases in the case of M = t 2.

Fig. 3
Fig. 3

(a) and (b) Symmetrical structures within one period for even M I and odd M I , respectively. The number of phase levels of (a) and (b) are at most M I and M I + 1, respectively.

Fig. 4
Fig. 4

Symmetric structure in the case of M I = 2t: (a) t = 4q + 1, and (b) t = 4q + 3. (c) The phase structure in the case of M = t 2.

Tables (4)

Tables Icon

Table 1 Simple Relations between the Number (L) of Phase Levels of Talbot Array Illuminators and the Opening Ratio (1/M) of the Generated Arraya

Tables Icon

Table 2 Numerical Examples of the Number (L) of Phase Levels of Talbot Array Illuminators and the Opening Ratio (1/M) of the Illumination Array up to M = 16

Tables Icon

Table 3 Simple Relations among the Number (L) of Phase Levels of π-Phase-Modulated Talbot Array Illuminators and the Intensity-Opening Ratio (1/M I ) of the Generated Arraya

Tables Icon

Table 4 Numerical Examples of the Number (L) of Phase Levels of the Talbot Array Illuminator and the Intensity-Opening Ratio (1/M I ) of the Illumination Array in the π-Phase Modulated Case up to M I = 16

Equations (42)

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ϕ k = rk 2 M   π
ϕ k = rk k - 1 M   π
pr = k r M + 1 ,
L = M 2 + 1 - δ L
L = M + 1 2 - δ L
ϕ k 2 - ϕ k 1 = r k 2 2 - k 1 2 M   π .
ϕ k 2 - ϕ k 1 = 2 π c ,
k 2 2 - k 1 2 = 2 M c / r .
k 2 - k 1 k 2 + k 1 = c 2 M .
k 2 - k 1 k 2 + k 1 = q 1 q 2 ,
k 2 - k 1 = q 1 ,
k 2 + k 1 = q 2 ,
k 1 = q 2 - q 1 2 ,
k 2 = q 2 + q 1 2 .
k 2 - k 1 k 2 + k 1 = 2 2 c 2 t ,
k 1 = t - 2 c ,
k 2 = 2 t - k 1 .
ϕ k 2 - ϕ k 1 = r k 2 - k 1 k 1 + k 2 - 1 M   π .
k 2 - k 1 k 2 + k 1 - 1 = c 2 M ,
k 2 - k 1 k 2 + k 1 - 1 = t - n nt 2 = c 2 M ,
M + 1 2 - t - 1 2 = t t - 1 2 + 1 .
δ L n = 2 δ L n - 1 + 1 - - 1 n 2 .
ϕ k = rk k - 1 M I   π
ϕ k = rk 2 M I   π
pr = k r M I + 1 ,
L = M I - δ L
L = M I + 1 - δ L
ϕ k 2 - ϕ k 1 = r k 2 - k 1 k 1 + k 2 - 1 M I   π ;
k 2 - k 1 k 2 + k 1 - 1 = c 2 M I ,
k 2 - k 1 = q 1 ,
k 2 + k 1 - 1 = q 2 ;
k 1 = q 2 - q 1 + 1 2 ,
k 2 = q 2 + q 1 + 1 2 .
k 1 = t + 1 2 - 2 c ,
k 2 = t + 1 2 + 2 c .
ϕ k + t = r k + t k + t - 1 M I   π = ϕ k + rk π + r t - 1 2   π ,
ϕ 2 q + 1 t = r 2 q + 1 t 2 t 2   π = π ,
ϕ 2 q t = r 2 q t 2 t 2   π = 0 ,
ϕ k 1 = rk 1 2 M I   π ,
ϕ k 2 = rk 2 2 M I   π .
k 2 2 - k 1 2 = c 2 M I = 2 c t 1 t 2 .
t 1 t 2 - t 1 - t 2 + 1 2 + t 1 + t 2 - 2 + 2 = t 1 + 1 t 2 + 1 2 .

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