Abstract

Under specific circumstances the fractional Talbot effect can be described by simplified equations. We have obtained simplified analytic phase-factor equations to describe the relation between the pure-phase factors and their fractional Talbot distances behind a binary amplitude grating with an opening ratio of (1/M). We explain how these simple equations are obtained from the regularly rearranged neighboring phase differences. We point out that any intensity distribution with an irreducible opening ratio (M N/M) (M N < M, where M N and M are positive integers) generated by such an amplitude grating can be described by similar phase-factor equations. It is interesting to note that an amplitude grating with additional arbitrary phase modulation can also generate pure-phase distributions at the fractional Talbot distance. We have applied these analytic phase-factor equations to neighboring (0, π) phase-modulated amplitude gratings and have analytically derived a new set of simple phase-factor equations for Talbot array illumination in this case. Experimental verification of our theoretical results is given.

© 1999 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. L. Liu, “Lau cavity and phase locking of laser arrays,” Opt. Lett. 14, 1312–1314 (1989).
    [CrossRef] [PubMed]
  4. S. Nowak, C. Kurtsiefer, T. Pfau, C. David, “High-order Talbot fringes for atomic matter waves,” Opt. Lett. 22, 1430–1432 (1997).
    [CrossRef]
  5. 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]
  6. L. Liu, X. Liu, L. Ye, “Joint Talbot effect and logic operated moiré patterns,” J. Opt. Soc. Am A 7, 970–976 (1990).
    [CrossRef]
  7. V. Arrizón, J. Ojeda-Castaneda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 (1994).
    [CrossRef] [PubMed]
  8. M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
    [CrossRef]
  9. P. Szwaykowski, V. Arrizón, “Talbot array illuminator with multilevel phase gratings,” Appl. Opt. 32, 1109–1114 (1993).
    [CrossRef] [PubMed]
  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, L. Wang, T. Tschudi, “Solutions and analyses of the fractional-Talbot array illuminations,” Opt. Commun. 147, 224–228 (1998).
    [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]

1998 (1)

C. Zhou, L. Wang, T. Tschudi, “Solutions and analyses of the fractional-Talbot array illuminations,” Opt. Commun. 147, 224–228 (1998).
[CrossRef]

1997 (2)

1996 (1)

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

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

1994 (1)

1993 (1)

1990 (3)

1989 (1)

1988 (1)

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

Arimoto, Y.

Arrizón, V.

Berry, M. V.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

David, C.

Klaus, W.

Klein, S.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Kodate, K.

Kurtsiefer, C.

Leger, J. R.

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]

L. Liu, X. Liu, L. Ye, “Joint Talbot effect and logic operated moiré patterns,” J. Opt. Soc. Am A 7, 970–976 (1990).
[CrossRef]

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

Liu, X.

L. Liu, X. Liu, L. Ye, “Joint Talbot effect and logic operated moiré patterns,” J. Opt. Soc. Am A 7, 970–976 (1990).
[CrossRef]

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.

Nowak, S.

Ojeda-Castaneda, J.

Pfau, T.

Swanson, G. J.

Szwaykowski, P.

Thomas, J. A.

Tschudi, T.

C. Zhou, L. Wang, T. Tschudi, “Solutions and analyses of the fractional-Talbot array illuminations,” Opt. Commun. 147, 224–228 (1998).
[CrossRef]

Wang, L.

C. Zhou, L. Wang, T. Tschudi, “Solutions and analyses of the fractional-Talbot array illuminations,” Opt. Commun. 147, 224–228 (1998).
[CrossRef]

Ye, L.

L. Liu, X. Liu, L. Ye, “Joint Talbot effect and logic operated moiré patterns,” J. Opt. Soc. Am A 7, 970–976 (1990).
[CrossRef]

Zhou, C.

C. Zhou, L. Wang, T. Tschudi, “Solutions and analyses of the fractional-Talbot array illuminations,” Opt. Commun. 147, 224–228 (1998).
[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. (3)

J. Mod. Opt. (1)

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

J. Opt. Soc. Am A (1)

L. Liu, X. Liu, L. Ye, “Joint Talbot effect and logic operated moiré patterns,” J. Opt. Soc. Am A 7, 970–976 (1990).
[CrossRef]

J. Opt. Soc. Am. A (2)

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]

C. Zhou, L. Wang, T. Tschudi, “Solutions and analyses of the fractional-Talbot array illuminations,” Opt. Commun. 147, 224–228 (1998).
[CrossRef]

Opt. Lett. (3)

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 (7)

Fig. 1
Fig. 1

One-dimensional (1/M) opening-ratio grating illuminated with monochromatic light at fractional Talbot distances (p/2M)Z T , where p and M have no common divisor; there will be pure-phase distributions. In reverse it can be used for array illumination.

Fig. 2
Fig. 2

(a) At the distance (p/2M)Z T the neighboring phase differences are rearranged regularly according to (k p = pk 1 mod M) from the differences at the distance (1/2M)Z T . (b) Reverse relation (k 1 = rk p mod M), i.e., the neighboring δϕ p (k p + 1) and δϕ p (k p ) should be the same as at (1/2M)Z T , with a lateral shift of r.

Fig. 3
Fig. 3

Distributions at the distance (p/2M)Z T generated by an amplitude grating with an opening ratio of (1/M), as shown in Fig. 1, where p and M have a common divisor. The M S pulses within one period have different phases θ k , where k = 1, 2, … , M S , and can be calculated with Eqs. (1), (2), and (24).

Fig. 4
Fig. 4

Intensity opening ratio of the field distributions at the distance (p/2M N )Z T generated by an amplitude grating with an opening ratio of (1/M) but increased M N times, i.e., M N /M, where p and M N have no common divisor, M N M, M N and M are positive integers, and c (=M/ M N ) might not be an integer.

Fig. 5
Fig. 5

Arbitrary phase-modulated input amplitude grating (one period is composed of M elements). This grating could generate a pure-phase distribution at the distance (p/2M)Z T , where p and M have a common divisor and M c is the maximum common divisor between p and M. The θ k representations are simplified representations of exp(iθ k ), where k = 1, 2, … , M c and the zero representations denote opaque.

Fig. 6
Fig. 6

(a) Characteristics of the pure-phase distribution at the distance (p/2M)Z T generated by an input amplitude grating with an opening ratio of M = 8q, where q is a positive integer. (b) Neighboring π-phase-modulated input amplitude grating, where M = 8q or M = 8q + 4.

Fig. 7
Fig. 7

(a) Schematic illustration of the experimental arrangement with d = 410 µm. (b) Our theoretical distribution (— ao-38-2-284-i001 —) and the measured distribution (-- ❋ --). See text for details.

Tables (1)

Tables Icon

Table 1 Relations among M, p, and r and the Corresponding Pure-Phase Factors in the Case of M = 7

Equations (33)

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ϕk=rk2M π,  k=1, 2,, M,
ϕk=rkk-1M π,  k=1, 2,, M,
pr=krM+1,
ϕk=ϕk+1-ϕk.
kp=pk1 mod M,
ϕk=k2M π,  k=1, 2,, M.
ϕ1k1=ϕk1+1-ϕk1=πM2k1+1,
ϕpkp=ϕ1k1.
ϕpkp+1=ϕ1k1+r, ϕpkp+2=ϕ1k1+2r, ϕpkp+l=ϕ1k1+lr=π2k1+lr+1M,
ϕpkp+1=ϕpkp=ϕ1k1, ϕpkp+2=ϕpkp+1+ϕpkp+1=ϕ1k1+ϕ1k1+r,
ϕpkp+m=l=0m-1 ϕ1k1+lr=l=0m-1π2k1+lr+1M=πMm2r+2k1-r+1m.
k1=r-12,
ϕpm=rm2M π,
ϕpkp=ϕ1k1.
k1=r2kp+1-12.
k1=r2kp+1-12+krM.
ϕ1k1+1=ϕpkp+p.
ϕM-k=ϕk,  k=1, 2,, M/2.
ϕM-k+1=ϕk,  k=1, 2,, M-1/2.
1Mk=1Mcosϕk+m-ϕk=0if m=1, 2,, M-11if m=0, M.
Mc=maxcomdivp, M.
Ms=M/Mc.
ps=p/Mc.
psrs=ksMs+1,
ϕk+M2=ϕk+-1kπ=ϕk+1--1kπ2,
E2k=expiϕk+expiϕk+M2+π+expiϕk+1+expiϕk+1+M2+π=expiϕk1+expikπ+π+expiϕk+1(1+expik+1π+π)=2×expiϕ2s-1,
ϕs=r2s-12M π=rss-1M1π,
ϕk+M2=ϕk+-1k+1π,
E2k=2 expiϕ2s,
ϕs=rs2MI π,
prI=kIMI+1,
ϕs+MI=ϕs-π,
12MIs=12Mcosϕs+m-ϕs=1if m=0, 2MI-1if m=MI0otherwise,

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