Abstract

A study of the stochastic description of the Talbot effect in the temporal domain under random timing jitter is presented. The relevant statistical quantity is the variance. The variance of a train of pulses, each one affected by random timing jitter, shows peaks in the edges of the pulses. When this train is Talbot-imaged, the variance becomes flattened along the unit interval corresponding to each pulse as a result of the dispersion of the individual pulses of the train. Fractional Talbot devices are also analyzed. In particular, it is shown that this smoothing effect also occurs in Talbot devices leading to N× repetition rates of the original train.

© 2004 Optical Society of America

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References

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  1. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), pp. 1–108.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. L. E. Jiang, M. E. Grein, H. A. Haus, E. P. Ippen, and H. Yokohama, “Timing jitter eater for optical pulse trains,” Opt. Lett. 28, 78–80 (2003).
    [CrossRef] [PubMed]
  13. A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).
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  15. C. Gómez-Reino, M. V. Pérez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer-Verlag, Berlin, 2002), Chap. 7.

2003 (2)

2002 (1)

2000 (3)

1999 (1)

1996 (1)

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

1994 (1)

1981 (1)

1975 (1)

R. Ulrich, “Image formation by phase coincidences in optical waveguides,” Opt. Commun. 13, 259–264 (1975).
[CrossRef]

Agogliati, B.

Arcangeli, L.

Atkins, S.

Azaña, J.

Bao, C.

Bekker, A.

Belmonte, M.

Berry, M. V.

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

Fernández-Pousa, C. R.

Fischer, B.

Flores-Arias, M. T.

Grein, M. E.

Haus, H. A.

Ibsen, M.

Ippen, E. P.

Jannson, J.

Jannson, T.

Jiang, L. E.

Klein, S.

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

Laporta, P.

Longhi, S.

Marano, M.

Muriel, M. A.

Papoulis, A.

Pérez, M. V.

Prunei, V.

Svelto, O.

Ulrich, R.

R. Ulrich, “Image formation by phase coincidences in optical waveguides,” Opt. Commun. 13, 259–264 (1975).
[CrossRef]

Vodonos, B.

Yokohama, H.

Zervas, M. N.

Appl. Opt. (1)

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. (1)

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

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

Opt. Commun. (1)

R. Ulrich, “Image formation by phase coincidences in optical waveguides,” Opt. Commun. 13, 259–264 (1975).
[CrossRef]

Opt. Lett. (5)

Other (4)

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol. XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), pp. 1–108.

A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984).

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, Boston, 2001).

C. Gómez-Reino, M. V. Pérez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer-Verlag, Berlin, 2002), Chap. 7.

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Figures (8)

Fig. 1
Fig. 1

Contributions of type |f|2 (solid curve) and |f|2 (dashed curve) to the input (top graph) and output (bottom graph) variances due to a single pulse.

Fig. 2
Fig. 2

Ten central unit intervals of the input (top) and output trains of pulses (bottom) in the integer Talbot device characterized by the index γ/α=2. The rms width of the input pulses is tp=0.100 and the standard deviation of random timing jitter is tj=0.015.

Fig. 3
Fig. 3

Superposition of ten input pulses in a unit interval in the integer Talbot device characterized by the index γ/α=2. Parameters as in Fig. 2.

Fig. 4
Fig. 4

Superposition of ten output pulses in a unit interval in the integer Talbot device characterized by the index γ/α=2. Parameters as in Fig. 2.

Fig. 5
Fig. 5

Input variance in two unit intervals obtained from Eq. (22) (dashed curve) and through a numerical estimate over 100 sample trains (dots); output variance for γ/α=2 in two unit intervals obtained from Eq. (24) (solid line) and through a numerical estimation (dots). Parameters as in Fig. 2.

Fig. 6
Fig. 6

Superposition of ten sequences of output pulses in a unit interval in the fractional Talbot device characterized by the index γ/α=1/4, leading to a 4× repetition rate. The rms width of the input pulses is tp=0.050 and the standard deviation of random timing jitter is tj=0.020.

Fig. 7
Fig. 7

Input variance in two unit intervals obtained from Eq. (22) (dashed curve) and through a numerical estimate over 100 sample trains (dots); output variance for γ/α=1/4 in two unit intervals obtained from Eq. (24) (solid curve) and through a numerical estimation (dots). Parameters as in Fig. 6.

Fig. 8
Fig. 8

Superposition of ten sequences of output pulses in a unit interval in the fractional Talbot device characterized by the index γ/α=1/2, leading to a 2× repetition rate. The rms width of the input pulses is tp=0.040 and the standard deviation of random timing jitter is tj=0.007.

Equations (30)

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xz=-i β22 2xt2,
hξ(t-t)=(-2πiξ)-1/2 exp[-i(t-t)2/2ξ],
x0(t)=k=-+f0(t-kt0-ak).
f0(t)=(2πtp2)-1/2 exp(-t2/2tp2).
p(ak)=(2πtj2)-1/2 exp(-ak2/2tj2).
xξ(t)=-+hξ(t-t)x0(t)dt.
xξ(t)=k=-+fξ(t-kt0-ak)=k=-+-+hξ(t-t)f0(t-kt0-ak)dt,
fξ(t-kt0-ak)=(2πρ)-1/2 exp[-(t-kt0-ak)2/2ρ],
xξ(t)=k=-+fξ(t-kt0-ak)=-+hξ(t-t)x0(t)dt.
fξ(t-kt0-ak)=-+dakp(ak)fξ(t-kt0-ak)=(2πρ˜)-1/2 exp[-(t-kt0)2/2ρ˜],
x0(t)=k=-+f¯0(t-kt0),
f¯0(t)=(2πt¯2)-1/2 exp(-t2/2t¯2).
ξ=t022π γα
Vξ(t)=|xξ(t)-xξ(t)|2=xξ(t)xξ(t)*-xξ(t)xξ(t)*,
Vξ(t)=k,m=-+fξ(t-kt0-ak)fξ(t-mt0-ap)*-k=-+fξ(t-kt0-ak)2.
Vξ(t)=k=-+[|fξ(t-kt0-ak)|2-|fξ(t-kt0-ak)|2],
Vξ(t)=1tp4π m=-+ 1η1(ξ)2π exp-(t-mt0)22η12(ξ)-1t¯4π s=-+ 1η2(ξ)2π exp-(t-st0)22η22(ξ),
η12(ξ)=ξ2+tp2tˆ22tp2,η22(ξ)=ξ2+t¯42t¯2,
Vξ(t)=m=-+gm exp(2πimt/t0)=g0+2g±1 cos(2πt/t0)+
gm=1t0tp4π exp[-2π2m2η12(ξ)/t02]-1t0t¯4π exp[-2π2m2η22(ξ)/t02].
1t0 -t0/2t0/2Vξ(t)dt=g0=1t04π 1tp-1t¯,
V0(t)|f(t-a0)|2-|f(t-a0)|2=18π2 1tpη1(ξ) exp-t22η12(ξ)-1tη¯2(ξ) exp-t22η22(ξ).
η12(ξ)=t028π2 t02tp2 γ2α2+tˆ22.
Vξ(t)g0.
tpt0=12N2 ln 10.
2g±1=exp(-2π2η12(ξ)/t02)t0tpπ-exp(-2π2η22(ξ)/t02)t0t¯π.
2π2η22(ξ)t02=14N2 t02tp2+tj2+π2 tp2t02+π2 tj2t02
>14N2 t02tp2+tj2>18N2 t02tp2
=ln 10,
Apeakσξ=((1/2παt¯2)+σξ2)1/2σξ1σξt¯2πα2π1/4 1α tpt01/2 t0tj,

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