## Ultrafast all-optical Nth-order differentiator and simultaneous repetition-rate multiplier of periodic pulse train

Optics Express, Vol. 15, Issue 19, pp. 12102-12107 (2007)

http://dx.doi.org/10.1364/OE.15.012102

Acrobat PDF (241 KB)

### 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

## 1. Introduction

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, and 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]

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]

11. S. Longhi, M. Marano, P. Laporta, and 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]

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]

12. J. Azaña and M. A. Muriel, “Temporal Talbot effect in fiber gratings and its applications,” Appl. Opt. **38**, 6700–6704 (1999). [CrossRef]

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]

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]

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]

**15**, 7196–7201 (2007). [CrossRef] [PubMed]

## 2. Theory

*f*)=

_{out}(t*d*, where

^{N}_{fin}(t)/dtN*f*and

_{in}(t)*f*are the complex envelopes of the input and output of the system respectively, and

_{out}(t)*t*is the time variable. In frequency domain,

*F*=(

_{out}(*ω*)*jω*)

^{N}*F*(

_{in}*ω*), where

*F*(

_{in}*ω*) and

*F*are the spectral functions of

_{out}(ω)*f*and

_{in}(t)*f*, respectively (ω is the base-band frequency, i.e.,

_{out}(t)*ω*=

*ω*, where

_{opt}-ω_{0}*ω*is the optical frequency, and

_{opt}*ω*is the central optical frequency of the signals). Thus, an Nth-order differentiator is essentially a linear filtering device providing a spectral transfer function of the form

_{0}*H*(

_{N}*ω*)=

*F*(

_{out}*ω*)/

*F*(

_{in}*ω*)=(

*jω*)

*. We are interested in obtaining an analytic expression of a feasible spectral response, so the ideal spectral response function must be windowed,*

^{N}*H*(

_{N,w}*ω*)=

*H*(

_{N}*ω*)

*W*(ω)=(

*jω)*, where

^{N}W(ω)*W(ω)*is a window function.

*H*(

_{r}*ω*)=(

*R*(

*ω*))

^{1/2}exp(

*jϕ*(ω))∝|

_{r}*H*(ω)|exp(

_{Nw}*jϕ*(

_{N}*ω*)+

*jω*

^{2}

*/2) where*

_{r}*H*(

_{r}*ω*),

*R(ω)*, and

*ϕ*are the spectral response in reflection, reflectivity and phase of the FBG,

_{r}(ω)*ϕ*=phase(

_{N}(ω)*H*(

_{N}*ω*))=phase(

*H*(

_{N,w}*ω*)) and

*ϕ*(

_{r}*ω*)/

*∂ω*

^{2}is the first order dispersion coefficient of the FBG, which is a constant value for linearly chirped FBGs. Regarding the reflectivity, we have:

*N*is odd, the differentiator spectral response presents a π-phase shift at

*ω*=0. In our approach, this condition is attained by introducing a

*π*-phase shift in the grating at z=0.

*C*, and the length of the grating

_{K}=∂^{2}φ(z)/∂z^{2}*L*, can be calculated from [13

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]

*Δω*is the grating bandwidth.

_{g}12. J. Azaña and M. A. Muriel, “Temporal Talbot effect in fiber gratings and its applications,” Appl. Opt. **38**, 6700–6704 (1999). [CrossRef]

*s/m*must be an irreducible rational fraction. Repetition rate multiplication can be achieved for

*m*>1. As a result the reflected signal has a repetition rate

*m*times that of the input signal.

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]

*Δt*can be calculated from the temporal length of ℑ

_{g}^{-1}[

*H*], and ℑ

_{N,w}(ω)^{-1}denotes inverse Fourier transform.

*T*

^{2}≫(Δ

*t*)

_{g}^{2}, and it is probable that (5) not only satisfies (6), but greatly exceeds it, so from (4) we can deduce that it is necessary to use a longer FBG than strictly required for space-to-frequency mapping.

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]

*ω*/2π) of 193 THz, an effective refractive index

_{0}*n*=1.45, a band of interest (Δ

_{eff}*ω*/2π) of 5 THz centred at

*ω*

_{0}(

*ω*

_{0}-Δ

*ω*/2≤

*ω*

_{opt}≤

*ω*

_{0}+Δ

*ω*/2), a grating bandwidth Δ

*ω*=Δ

_{g}*ω*, a maximum reflectivity of 50 %, and a pulse train period

*T*=40 ps.

*H*. We choose a function based on a hyperbolic tangent as window,

_{1}(ω)=jω*W(ω)=W*=(1/2)[1+tanh(4-|16

_{th}(ω)*ω*/Δ

*ω*|)], and we have

_{g}*H*=

_{1,w}(ω)*H*. The desired reflectivity is obtained from (1):

_{1}(ω)W_{th}(ω)*C*=2.1238 is a normalization constant to get a maximum reflectivity of 50 %.

_{R}^{-1}[

*H*] we obtain Δ

_{1,w}(ω)*t*≈2 ps. Using expressions (5)and (6) we have

_{g}_{r}|≫1.5915×10

^{-25}

*s*

^{2}/rad.

_{r}=-1.2732×10

^{-22}

*s*

^{2}/

*rad*, where

*s*=1 and

*m*=2 have been selected. This implies that the input repetition rate is multiplied by two, so we have an output period of repetition

*T*=

_{out}*T*/2=20 ps. The odd order of 1st differentiator implies that π-phase shift must be introduced in the grating at

_{in}*z*=0.

*C*=1.2011 is a normalization constant selected to get a normalized apodization profile function 0≤

_{A}*A(z)*≤1, and

*N*=1.

*t*≈2 ps, and we design the same technological parameters as in the first example (so repetition rate is also doubled). The apodization profile is given by (9), where

_{g}*C*=181.02, and

_{R}*N*=4 (same

*L*and

*C*as for first example).

_{A}*f*∝exp(-

_{in,1}(t)*t*

^{2}/(2

*σ*

^{2})) with σ=800 fs, are showed in Fig. 2(b), and 2(e) for first and second example, respectively. The output pulse corresponding to an antisymmetric HG pulse described by

*f*(

_{in,2}*t*)∝∂

*f*∝

_{in,1}(t)/∂t*t*·exp(-

*t*

^{2}/(2

*σ*

^{2})) are showed in Fig. 2(c), and 2(f) for first and second example, respectively.

*L*≫0.051 cm for space-to- frequency, which can be satisfied from

*L*>5 cm. The length of the grating designed (41.346 cm) is much longer, but is within the accuracy of currently available fabrication techniques, as it can be seen in [11

11. S. Longhi, M. Marano, P. Laporta, and 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]

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]

## 4. Conclusion

**15**, 7196–7201 (2007). [CrossRef] [PubMed]

**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]

11. S. Longhi, M. Marano, P. Laporta, and 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]

## Acknowledgements

## References and Links

1. | H. J. A. da Silva and J. J. O’Reilly, “Optical pulse modeling with Hermite - Gaussian functions,” Opt. Lett. |

2. | R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express |

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 |

4. | M. Kulishov and J. Azaña, “Design of high-order all-optical temporal differentiators based on multiple-phase-shifted fiber Bragg gratings,” Opt. Express |

5. | Y. Park, R. Slavik, and J. Azaña “Ultrafast all-optical first and higher-order differentiators based on interferometers” Opt. Lett. |

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 |

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. |

8. | M. A. Preciado, V. García-Muñoz, and M. A. Muriel , “Grating design of oppositely chirped FBGs for pulse shaping,” IEEE Photon. Technol. Lett. |

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 |

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. |

11. | S. Longhi, M. Marano, P. Laporta, and O. Svelto, “Propagation, manipulation, and control of picosecond optical pulses at 1.5 µm in fiber Bragg gratings, J. Opt. Soc. Am. B |

12. | J. Azaña and M. A. Muriel, “Temporal Talbot effect in fiber gratings and its applications,” Appl. Opt. |

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. |

14. | J. T. Mok and B. J. Eggleton, “Impact of group delay ripple on repetition-rate multiplication through Talbot self-imaging effect,” Opt. Commun. |

**OCIS Codes**

(060.2340) Fiber optics and optical communications : Fiber optics components

(230.1150) Optical devices : All-optical devices

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 6, 2007

Revised Manuscript: September 5, 2007

Manuscript Accepted: September 6, 2007

Published: September 7, 2007

**Citation**

Miguel A. Preciado and Miguel A. Muriel, "Ultrafast all-optical Nth-order differentiator and simultaneous repetition-rate multiplier of periodic pulse train," Opt. Express **15**, 12102-12107 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12102

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### References

- H. J. A. da Silva and J. J. O'Reilly, "Optical pulse modeling with Hermite - Gaussian functions," Opt. Lett. 14, 526- (1989). [CrossRef] [PubMed]
- R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, "Ultrafast all-optical differentiators, " Opt. Express 14, 10699-10707 (2006). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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. 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]
- 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]
- 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]
- J. Azaña and M. A. Muriel, "Temporal Talbot effect in fiber gratings and its applications," Appl. Opt. 38, 6700-6704 (1999). [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]
- 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]

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