## Ultrafast all-optical Nth-order differentiator based on chirped fiber Bragg gratings

Optics Express, Vol. 15, Issue 12, pp. 7196-7201 (2007)

http://dx.doi.org/10.1364/OE.15.007196

Acrobat PDF (456 KB)

### Abstract

In this letter we present a technique for the implementation of Nth-order ultrafast temporal differentiators. This technique is based on two oppositely chirped fiber Bragg gratings in which the grating profile maps the spectral response of the Nth-order differentiator. Examples of 1^{st}, 2^{nd}, and 4^{th} order differentiators are designed and numerically simulated.

© 2007 Optical Society of America

## 1. Introduction

1. N. Q. Ngo, S. F. Yu, S. C. Tjin, and C. H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. **230**, 115–129, (2004). [CrossRef]

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

1. N. Q. Ngo, S. F. Yu, S. C. Tjin, and C. H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. **230**, 115–129, (2004). [CrossRef]

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

4. 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]

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 “Grating design of oppositely chirped FBGs for pulse shaping,” IEEE Photon. Technol. Lett. **10**, 435–437 (2007). [CrossRef]

_{a}, is the spectral shaper, and provides the spectral response for pulse shaping. The second, FBG

_{b}, cancels the dispersion introduced by the first grating. Obviously, the order of the FBGs can be arbitrarily selected.

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. I. Littler, M. Rochette, and B. Eggleton, “Adjustable bandwidth dispersionless bandpass FBG optical filter,” Opt. Express **13**, 3397–3407 (2005). [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. I. Littler, M. Rochette, and B. Eggleton, “Adjustable bandwidth dispersionless bandpass FBG optical filter,” Opt. Express **13**, 3397–3407 (2005). [CrossRef] [PubMed]

8. I. Littler, M. Rochette, and B. Eggleton, “Adjustable bandwidth dispersionless bandpass FBG optical filter,” Opt. Express **13**, 3397–3407 (2005). [CrossRef] [PubMed]

## 2. Theory

*f*(

_{out}*t*)=

*d*(

^{N}f_{in}*t*)/

*dt*, where

^{N}*f*(

_{in}*t*) and

*f*(

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

*t*is the time variable. We can also express this in frequency domain as,

*F*(

_{in}*ω*)=(j

*ω*)

*(*

^{N}F_{out}*ω*) where

*F*(

_{in}*ω*) and

*F*(

_{out}*ω*) are the spectral functions of

*f*(

_{in}*t*) and

*f*(

_{out}*t*), respectively (

*ω*is the base-band frequency, i.e.,

*ω*=

*ω*-

_{opt}*ω*, where

_{0}*ω*is the optical frequency, and

_{opt}*ω*is the central optical frequency of the signals). Thus, the spectral response of the ideal Nth-order differentiator is:

_{0}*W*(

*ω*) is a window function, which must be selected to meet:

*trans*(

*ω*) is a transient function which must have low amplitude values at the edges of the band of interest in order to avoid an abrupt discontinuity. Notice that not any window function verifies this condition on

*trans*(

*ω*), even in the case of a window function presenting low values at the edges of the band of interest.

*H*(

_{syst}*ω*), proportional to the differentiator spectral response:

*H*(

_{a}*ω*),

*H*(

_{b}*ω*),

*R*(

_{a}*ω*),

*R*(

_{b}*ω*),

*ϕ*(

_{a}*ω*),

*ϕ*(

_{b}*ω*) are the spectral response in reflection, reflectivity and phase of the FBGs. In this approach we assume that FBG

_{b}is a dispersion compensator, so we can consider that

*R*(

_{b}*ω*) presents an ideal flat-top response in the band of interest, so the shape of the reflectivity is influenced by FBG

_{a}solely. Thus, we have:

*(*ϕ ¨

_{a}*ω*) = -

*(*ϕ ¨

_{b}*ω*) =

*, where*ϕ ¨

_{a}*(*ϕ ¨

*ω*) denotes ∂

^{2}

*ϕ*(

*ω*)/∂

*ω*

^{2}, and

*is a constant value, which is obtained from the FBG*ϕ ¨

_{a}_{a}design.

_{a}. The refractive index of FBG

_{a}can be written as:

*ω*=0. In our approach, this condition is attained by introducing a π-phase shift in the grating of FBG

_{a}at

*z*=0 The chirp factor of FBG

_{a}, which is defined as

*C*=∂

_{K,a}^{2}

*φ*(

_{a}*z*)/∂

*z*

^{2}, and

*L*can be calculated from [9

_{a}9. 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]

*c*is the light vacuum speed, and Δ

*ω*is the FBG

_{g,a}_{a}bandwidth. It is well known that when a chirped FBG introduces an enough high dispersion, the spectral response of the grating is a scaled version of its corresponding apodization profile [10

10. 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]

*t*is the temporal length of the inverse Fourier transform of the FBG

_{a}_{a}spectral response without the dispersive term, which can be calculated from the temporal length of ℑ

^{-1}[

*H*(

_{N,w}*ω*)], where ℑ

^{-1}denotes inverse Fourier transform. It is worth noting that the broader (narrower) bandwidth, the shorter (longer) minimum length of the grating required for FBG

_{a}to map properly the spatial profile on the spectral response [6

6. M. A. Preciado, V. García-Muñoz, and M. A. Muriel “Grating design of oppositely chirped FBGs for pulse shaping,” IEEE Photon. Technol. Lett. **10**, 435–437 (2007). [CrossRef]

*ω*) [11]. In the case of high reflectivity an approximate function [12] must be applied over

*R*(

_{a}*ω*). In particular, a logarithmic based function is used in our approach, and we obtain an expression which is valid for both weak and strong gratings:

## 3. Examples and results

^{st}, 2

^{nd}and 4

^{th}order differentiators, which are numerically simulated. For all the examples we assume a carrier frequency (

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

_{0}*n*=1.45 for FBG

_{eff,a}_{a}, a band of interest (Δ

*ω*/2π) of 5 THz centred at

*ω*(

_{0}*ω*-Δ

_{0}*ω*/2 ≤

*ω*≤

_{opt}*ω*+Δ

_{0}*ω*/2), a FBG

_{a}bandwidth Δ

*ω*=Δ

_{g,a}*ω*, and a maximum reflectivity for FBG

_{a}of 90 %.

^{st}-order differentiator. The corresponding ideal spectral response is

*H*(

_{1}*ω*)=j

*ω*, and we choose a function based on a hyperbolic tangent as window,

*W*(

_{th}*ω*)=(1/2)[1+tanh(4-|16

*ω*/

*Δω*|)]:

_{a}) must be designed to properly map the desired spectral response. From the temporal length of ℑ

^{-1}[

*H*

_{1,w}(

*ω*)] we obtain Δ

*t*≈2 ps. Using expression (9) we have |

_{a}*|>>1.5915×10*ϕ ¨

_{a}^{-25}

*s*

^{2}/

*rad*, and choose

*=-1.6×10*ϕ ¨

_{a}^{-23}

*s*

^{2}/

*rad*. Moreover, the odd order of 1

^{st}differentiator implies that π-phase shift must be introduced in FBG

_{a}at

*z*=0. The desired reflectivity for FBG

_{a}in the band of interest is obtained from (5):

*C*=0.659 is a normalization constant selected to get a normalized apodization profile function 0≤

_{A}*A*(

_{a}*z*)≤1, and

*N*=1.

_{b}) is

*(*ϕ ¨

_{b}*ω*) = -

*= -1.6×10*ϕ ¨

_{a}^{23}

*s*

^{2}/

*rad*, which must present a flat top spectral response in the band of interest.

^{nd}order differentiator using the same methodology. We obtain again Δ

*t*≈2 ps, so we have the same technological parameters as in the first example. The apodization profile which is given by (13), where

_{g,a}*C*= 13.568, and

_{R}*N*=2 (same

*L*and

_{a}*C*as for first example).

_{A}^{th}order differentiator. We have again Δ

*t*≈2 ps, and the same technological parameters, with an apodization profile described by (13), where

_{g,a}*C*= 243.1, and

_{R}*N*=4 (same

*L*and

_{a}*C*as for first example).

_{A}_{a}), the dispersion compensator (FBG

_{b}), and the whole system, for the first, second and third example, respectively. Figures 2(d), 2(e), and 2(f) compare the spectral responses of the spectral shaper (FBG

_{a}) and the ideal differentiator, for the first, second and third example, respectively. Figures 2(g), 2(h), and 2(i) show the temporal waveform of the input pulse and the output pulses of the designed system, and the ideal differentiator, for the first, second and third example, respectively. We have applied an input gaussian envelope pulse, described by

*f*(

_{in}*t*) ∝ exp(-

*t*

^{2}/(2σ

^{2})), with σ = 500 fs (FWHM= 1.177 ps).

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]

**13**, 3397–3407 (2005). [CrossRef] [PubMed]

**13**, 3397–3407 (2005). [CrossRef] [PubMed]

14. I. C. M. Littler, L. Fu, and B. J. Eggleton, “Effect of group delay ripple on picosecond pulse compression schemes,” Appl. Opt. **44**, 4702–4711 (2005). [CrossRef] [PubMed]

## 4. Conclusion

^{st}, 2

^{nd}, and 4

^{th}order differentiators.

**13**, 3397–3407 (2005). [CrossRef] [PubMed]

## Acknowledgments

*Ministerio de Educacion y Ciencia*under Project “Plan Nacional de I+D+I TEC2004-04754-C03-02”.

## References and Links

1. | N. Q. Ngo, S. F. Yu, S. C. Tjin, and C. H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. |

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

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

4. | 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 |

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 “Grating design of oppositely chirped FBGs for pulse shaping,” IEEE Photon. Technol. Lett. |

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. | I. Littler, M. Rochette, and B. Eggleton, “Adjustable bandwidth dispersionless bandpass FBG optical filter,” Opt. Express |

9. | 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. |

10. | 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 |

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. | B. Bovard, “Derivation of a matrix describing a rugate dielectric thin film,” Appl. Opt. |

13. | James F. Brennan III and Dwayne L. LaBrake, “Fabrication of chirped fiber bragg gratings of any desired bandwidth using frequency modulation,” US patent 6728444 (April 2004). |

14. | I. C. M. Littler, L. Fu, and B. J. Eggleton, “Effect of group delay ripple on picosecond pulse compression schemes,” Appl. Opt. |

**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: February 20, 2007

Revised Manuscript: May 16, 2007

Manuscript Accepted: May 23, 2007

Published: May 29, 2007

**Citation**

Miguel A. Preciado, Victor Garcia-Muñoz, and Miguel A. Muriel, "Ultrafast all-optical Nth-order differentiator
based on chirped fiber Bragg gratings," Opt. Express **15**, 7196-7201 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7196

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### References

- N. Q. Ngo, S. F. Yu, S. C. Tjin, and C. H. Kam, "A new theoretical basis of higher-derivative optical differentiators," Opt. Commun. 230, 115−129 (2004). [CrossRef]
- 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]
- 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 "Grating design of oppositely chirped FBGs for pulse shaping," IEEE Photon. Technol. Lett. 10, 435-437 (2007). [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]
- I. Littler, M. Rochette, and B. Eggleton, "Adjustable bandwidth dispersionless bandpass FBG optical filter," Opt. Express 13, 3397-3407 (2005). [CrossRef] [PubMed]
- 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. 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, 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]
- B. Bovard, "Derivation of a matrix describing a rugate dielectric thin film," Appl. Opt. 27, 1998-2004 (1988). [CrossRef] [PubMed]
- J. F. BrennanIII and D. L. LaBrake, "Fabrication of chirped fiber bragg gratings of any desired bandwidth using frequency modulation," US patent 6728444 (April 2004).
- I. C. M. Littler, L. Fu, and B. J. Eggleton, "Effect of group delay ripple on picosecond pulse compression schemes," Appl. Opt. 44, 4702-4711 (2005). [CrossRef] [PubMed]

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