Abstract

The letter presents a technique for Nth-order differentiation of periodic pulse train, which can simultaneously multiply the input repetition rate. This approach uses a single linearly chirped apodized fiber Bragg grating, which grating profile is designed to map the spectral response of the Nth-order differentiator, and the chirp introduces a dispersion that, besides space-to-frequency mapping, it also causes a temporal Talbot effect.

© 2007 Optical Society of America

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References

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  1. H. J. A. da Silva and J. J. O'Reilly, "Optical pulse modeling with Hermite - Gaussian functions," Opt. Lett. 14, 526- (1989).
    [CrossRef] [PubMed]
  2. R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, "Ultrafast all-optical differentiators, " Opt. Express 14, 10699-10707 (2006).
    [CrossRef] [PubMed]
  3. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, "Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating," Opt. Express 15, 371-381 (2007).
    [CrossRef] [PubMed]
  4. M. Kulishov and J. Azaña, "Design of high-order all-optical temporal differentiators based on multiple-phase-shifted fiber Bragg gratings," Opt. Express 15, 6152-6166 (2007).
    [CrossRef] [PubMed]
  5. Y. Park, R. Slavik, J. Azaña "Ultrafast all-optical first and higher-order differentiators based on interferometers" Opt. Lett. 32, 710-712 (2007).
    [CrossRef] [PubMed]
  6. M. A. Preciado, V. García-Muñoz, and M. A. Muriel "Ultrafast all-optical Nth-order differentiator based on chirped fiber Bragg gratings," Opt. Express 15, 7196-7201 (2007).
    [CrossRef] [PubMed]
  7. A. G. Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, "Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA," Electron. Lett. 35, 1096-1097 (1999).
    [CrossRef]
  8. M. A. Preciado, V. García-Muñoz, and M. A. Muriel "Grating design of oppositely chirped FBGs for pulse shaping," IEEE Photon. Technol. Lett. 19, 435-437 (2007).
    [CrossRef]
  9. J. Azaña and L. R. Chen, "Synthesis of temporal optical waveforms by fiber Bragg gratings: a new approach based on space-to-frequency-to-time mapping, " J. Opt. Soc. Am. B 19, 2758-2769 (2002).
    [CrossRef]
  10. S. Longhi, M. Marano, P. Laporta, and V. Pruneri, "Multiplication and reshaping of high-repetition-rate optical pulse trains using highly dispersive fiber Bragg gratings," IEEE Photon. Technol. Lett. 12, 1498-1500 (2000).
    [CrossRef]
  11. S. Longhi, M. Marano, P. Laporta, O. Svelto, "Propagation, manipulation, and control of picosecond optical pulses at 1.5 μm in fiber Bragg gratings, J. Opt. Soc. Am. B 19, 2742-2757 (2002).
    [CrossRef]
  12. J. Azaña and M. A. Muriel, "Temporal Talbot effect in fiber gratings and its applications," Appl. Opt. 38, 6700-6704 (1999).
    [CrossRef]
  13. J. Azaña and M. A. Muriel, ‘‘Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,’’ IEEE J. Quantum Electron. 36, 517-527 (2000).
    [CrossRef]
  14. J. T. Mok and B. J. Eggleton, "Impact of group delay ripple on repetition-rate multiplication through Talbot self-imaging effect," Opt. Commun. 232, 167-178 (2004).
    [CrossRef]

2007 (5)

2006 (1)

R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, "Ultrafast all-optical differentiators, " Opt. Express 14, 10699-10707 (2006).
[CrossRef] [PubMed]

2004 (1)

J. T. Mok and B. J. Eggleton, "Impact of group delay ripple on repetition-rate multiplication through Talbot self-imaging effect," Opt. Commun. 232, 167-178 (2004).
[CrossRef]

2002 (2)

2000 (2)

S. Longhi, M. Marano, P. Laporta, and V. Pruneri, "Multiplication and reshaping of high-repetition-rate optical pulse trains using highly dispersive fiber Bragg gratings," IEEE Photon. Technol. Lett. 12, 1498-1500 (2000).
[CrossRef]

J. Azaña and M. A. Muriel, ‘‘Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,’’ IEEE J. Quantum Electron. 36, 517-527 (2000).
[CrossRef]

1999 (2)

J. Azaña and M. A. Muriel, "Temporal Talbot effect in fiber gratings and its applications," Appl. Opt. 38, 6700-6704 (1999).
[CrossRef]

A. G. Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, "Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA," Electron. Lett. 35, 1096-1097 (1999).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

A. G. Jepsen, A. E. Johnson, E. S. Maniloff, T. W. Mossberg, M. J. Munroe, and J. N. Sweetser, "Fibre Bragg grating based spectral encoder/decoder for lightwave CDMA," Electron. Lett. 35, 1096-1097 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Azaña and M. A. Muriel, ‘‘Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,’’ IEEE J. Quantum Electron. 36, 517-527 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

S. Longhi, M. Marano, P. Laporta, and V. Pruneri, "Multiplication and reshaping of high-repetition-rate optical pulse trains using highly dispersive fiber Bragg gratings," IEEE Photon. Technol. Lett. 12, 1498-1500 (2000).
[CrossRef]

M. A. Preciado, V. García-Muñoz, and M. A. Muriel "Grating design of oppositely chirped FBGs for pulse shaping," IEEE Photon. Technol. Lett. 19, 435-437 (2007).
[CrossRef]

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

Opt. Commun. (1)

J. T. Mok and B. J. Eggleton, "Impact of group delay ripple on repetition-rate multiplication through Talbot self-imaging effect," Opt. Commun. 232, 167-178 (2004).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Other (1)

H. J. A. da Silva and J. J. O'Reilly, "Optical pulse modeling with Hermite - Gaussian functions," Opt. Lett. 14, 526- (1989).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1.
Fig. 1.

Architecture of the system. Periodic pulse train is processed by an apodized linearly chirped FBG.

Fig. 2.
Fig. 2.

Plots (a) and (d) show the amplitude of the spectral response corresponding to the FBG (solid), and to an ideal differentiator (dashed) for first and second examples, respectively. The temporal waveforms are showed in plots (b) and (c) for first example, and in plots (e) and (f) for second. Plots (b) and (e) correspond to a Gaussian pulse as input, and plots(c) and (f) correspond to an antisymmetric Hermite-Gaussian pulse as input. In plots (b), (c), (e) and (f) we show the input pulse in dashed line, the ouput pulse for the designed system in solid line, and the output pulse for the ideal differentiator in dotted line (indistinguishable from solid line in (b) and (c), and hardly distinguishable from solid line in (e) and (f)).

Equations (9)

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R ( ω ) H N , w ( ω ) 2 = ω N W ( ω ) 2
n ( z ) = n av ( z ) + Δ n max 2 A ( z ) cos [ 2 π Λ 0 z + φ ( z ) ]
C K = 4 n av 2 ( c 2 ϕ ̈ r )
L = ϕ ̈ r c Δ ω g ( 2 n av )
ϕ ̈ r = s m T 2 2 π s = 1 , 2 , 3 , , m = 1 , 2 , 3 , ,
ϕ ̈ r > > ( Δ t g ) 2 8 π
A ( z ) = [ ln ( 1 R ( ω ) ( ω = sign ( C K ) Δ ω g L z ) ) 32 n av 2 π ω 0 2 ϕ ̈ r Δ n max 2 ] 1 2
R ( ω ) = { C R ( ω Δ ω g ) [ 1 + tanh ( 4 16 ω Δ ω g ) ] } 2
A ( z ) = C A ( ln { 1 C R 2 z L a 2 N [ 1 + tanh ( 4 16 z L a ) ] 2 } ) 1 2

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