Abstract

Integer and fractional temporal self-imaging phenomena occur when an ideal (infinite-duration) periodic optical pulse sequence propagates through a suitable dispersive medium under the first-order dispersion approximation. I analytically and numerically investigate the impact of nonidealities in the input periodic pulse sequence, especially finite duration of the sequences as well as intensity and phase fluctuations between pulses, on the temporal self-imaging phenomena. I derive conditions for which effects associated with these nonidealities can be neglected. Under these conditions, the intensity of the input nonideal finite sequence can also be self-imaged—integer and fractional self-imaging are also possible—by propagation through a suitable dispersive medium. The resulting self-images of the input signal not only maintain the temporal features of the original individual pulses (temporal shape and duration) but also the total temporal duration of the finite input sequence and the original intensity fluctuations between pulses. The analytical results are confirmed by means of numerical simulations.

© 2003 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 XXVII, E. Wolf, ed. (Elsevier Science, Amsterdam, The Netherlands, 1989), pp. 1–108.
  2. M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
    [CrossRef]
  3. M. Mansuripur, “The Talbot effect,” Opt. Photon. News 8(4), 42–47 (1997).
  4. T. Jannson and J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373–1376 (1981).
  5. F. Mitschke and U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
    [CrossRef]
  6. J. Azaña and M. A. Muriel, “Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. 24, 1672–1674 (1999).
    [CrossRef]
  7. J. Azaña and M. A. Muriel, “Temporal Talbot effect in fiber gratings and its applications,” Appl. Opt. 38, 6700–6704 (1999).
    [CrossRef]
  8. S. Longhi, M. Marano, P. Laporta, O. Svelto, M. Belmonte, B. Agogliati, L. Arcangeli, V. Pruneri, M. N. Zervas, and M. Ibsen, “40-GHz pulse-train generation at 1.5 μm with a chirped fiber grating as a frequency multiplier,” Opt. Lett. 25, 1481–1483 (2000).
    [CrossRef]
  9. V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Lebedev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
    [CrossRef]
  10. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001).
    [CrossRef]
  11. Ch. Deckers, “Etude de l’influence de la dimension finie d’un reseau sur la formation des images de Fresnel,” Nouv. Rev. Opt. 7(2), 113–119 (1976).
    [CrossRef]
  12. A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. 44(2), 208–212 (1978).
  13. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
    [CrossRef]
  14. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1987).

2001

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Lebedev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001).
[CrossRef]

2000

1999

1998

F. Mitschke and U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
[CrossRef]

1997

M. Mansuripur, “The Talbot effect,” Opt. Photon. News 8(4), 42–47 (1997).

1996

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

1994

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

1981

1978

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. 44(2), 208–212 (1978).

1976

Ch. Deckers, “Etude de l’influence de la dimension finie d’un reseau sur la formation des images de Fresnel,” Nouv. Rev. Opt. 7(2), 113–119 (1976).
[CrossRef]

Agogliati, B.

Arcangeli, L.

Azaña, J.

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]

Borisov, V. I.

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Lebedev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

Deckers, Ch.

Ch. Deckers, “Etude de l’influence de la dimension finie d’un reseau sur la formation des images de Fresnel,” Nouv. Rev. Opt. 7(2), 113–119 (1976).
[CrossRef]

Ibsen, M.

Jannson, J.

Jannson, T.

Klein, S.

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

Kolner, B. H.

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

Laporta, P.

Lebedev, V. I.

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Lebedev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

Longhi, S.

Mansuripur, M.

M. Mansuripur, “The Talbot effect,” Opt. Photon. News 8(4), 42–47 (1997).

Marano, M.

Minkovich, V. P.

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Lebedev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

Mitschke, F.

F. Mitschke and U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
[CrossRef]

Morgner, U.

F. Mitschke and U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
[CrossRef]

Muriel, M. A.

Perepechko, S. N.

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Lebedev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

Pruneri, V.

Smirnov, A. P.

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. 44(2), 208–212 (1978).

Starodumov, A. N.

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Lebedev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

Svelto, O.

Zervas, M. N.

Appl. Opt.

IEEE J. Quantum Electron.

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7, 728–744 (2001).
[CrossRef]

J. Mod. Opt.

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

J. Opt. Soc. Am.

Nouv. Rev. Opt.

Ch. Deckers, “Etude de l’influence de la dimension finie d’un reseau sur la formation des images de Fresnel,” Nouv. Rev. Opt. 7(2), 113–119 (1976).
[CrossRef]

Opt. Commun.

V. P. Minkovich, A. N. Starodumov, V. I. Borisov, V. I. Lebedev, and S. N. Perepechko, “Temporal interference of coherent laser pulses in optical fibers,” Opt. Commun. 192, 231–235 (2001).
[CrossRef]

Opt. Lett.

Opt. Photon. News

F. Mitschke and U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
[CrossRef]

M. Mansuripur, “The Talbot effect,” Opt. Photon. News 8(4), 42–47 (1997).

Opt. Spectrosc.

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. 44(2), 208–212 (1978).

Other

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1987).

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

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Figures (5)

Fig. 1
Fig. 1

Results from simulation of propagation of a nonideal finite periodic optical pulse sequence (soliton-shape pulses under a Gaussian envelope) through a first-order dispersive medium. The medium’s dispersion coefficient is fixed to achieve a fractional self-imaging effect with a repetition rate multiplication factor of m=2. The input sequence has a sufficient number of input pulses to ensure undistorted self-imaging [according to inequality (19)]. (a) Input sequence to the dispersive medium. (b) Output sequence from the dispersive medium (solid curve) and input temporal envelope (dashed curve). Inset: output individual pulse (solid curve) and input individual pulse (dashed curve) (n.u. refers to normalized units).

Fig. 2
Fig. 2

Same definitions as for Fig. 1, but the input sequence does not have a sufficient number of input pulses to ensure undistorted self-imaging [according to inequality (19)].

Fig. 3
Fig. 3

Same definitions as for Fig. 2.

Fig. 4
Fig. 4

Results from simulation of propagation of a finite periodic optical pulse sequence, composed of 20 identical Gaussian-shape pulses (square envelope; T1=100 ps; T1/Δt02) through a first-order dispersive medium. Two-dimensional image: normalized average optical intensity of the output signal from the dispersive medium for different values of the dispersion coefficient (input signal for dispersion equals 0). One-dimensional plots: normalized average optical intensity of the output signal for the dispersion values corresponding to (I) first-order (s=1) fractional self-imaging effect with a repetition rate multiplication factor of m=2, (II) first-order (s=1) integer self-imaging effect, and (III) second-order (s=2) integer self-imaging effect (n.u. refers to normalized units).

Fig. 5
Fig. 5

Same definitions as for Fig. 4, but with a higher duty cycle for the input sequence, T1/Δt04.

Equations (20)

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a1(t)=p=-+a0(t-pT1).
|Φ¨0|=smT122π,
a2(t)p=-+exp(jϕp)a0[t-p(T1/m)],
x(t)=p=1Npulsesepa0(t-pT1),
a1(t)=p=-+a0(t-pT1)
x(t)=p=-+e(t=pT1)a0(t-pT1)=e(t)p=-+a0(t-pT1)=e(t)a1(t).
y(t)=x(t) *  h(t)=[e(t)a1(t)] *  h(t)-+e(τ)a1(τ)exp-j (t-τ)22Φ¨0dτ=exp-j t22Φ¨0-+e(τ)a1(τ)×exp-j τ22Φ¨0expj tτΦ¨0dτ,
y(t) exp-j t22Φ¨0 E(ω=-t/Φ¨0) *  -+a1(τ)exp-j τ22Φ¨0expj tτΦ¨0dτ,
-+a1(τ)exp-j τ22Φ¨0expj tτΦ¨0dτ=expj t22Φ¨0-+a1(τ)exp-j (t-τ)22Φ¨0dτ expj t22Φ¨0[a1(t) *  h(t)]=expj t22Φ¨0a2(t),
y(t) exp-j t22Φ¨0E(ω=-t/Φ¨0) * expj t22Φ¨0a2(t)=exp-j t22Φ¨0E(ω=-t/Φ¨0) * expj t22Φ¨0×p=-+exp(jϕp)a0[t-p(T1/m)].
y(t) exp-j t22Φ¨0E(ω=-t/Φ¨0) * p=-+exp(jωipt)×exp(jϕp)a0[t-p(T1/m)],
ωip=ωi(t=pT1/m)=pT1mΦ¨0.
Y(ω)=I[y(t)]e(t=Φ¨0ω)p=-+×exp[j(pT1/m)(ω-ωip)]×exp(jϕp)A0(ω-ωip),
e(Φ¨0ω)e(Φ¨0ωip)et=p T1m
Y(ω)p=-+e(t=pT1/m)×exp[j(pT1/m)(ω-ωip)]×exp(jϕp)A0(ω-ωip),
y(t)p=-+e[t=p(T1/m)]×exp(jωipt)exp(jϕp)a0[t-p(T1/m)].
Iy(t)p=-+|e[t=p(T1/m)]|2I0[t-p(T1/m)],
Φ¨0ΔωenvΔt0.
ΔtenvK2πsmT12Δt0,
NpulsesK2πsmT1Δt0.

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