## Simultaneous ultrafast optical pulse train bursts generation and shaping based on Fourier series developments using superimposed fiber Bragg gratings

Optics Express, Vol. 15, Issue 17, pp. 10878-10889 (2007)

http://dx.doi.org/10.1364/OE.15.010878

Acrobat PDF (259 KB)

### Abstract

We propose an all-fiber method for the generation of ultrafast shaped pulse train bursts from a single pulse based on Fourier Series Developments (FDSs). The implementation of the FSD based filter only requires the use of a very simple non apodized Superimposed Fiber Bragg Grating (S-FBG) for the generation of the Shaped Output Pulse Train Burst (SOPTB). In this approach, the shape, the period and the temporal length of the generated SOPTB have no dependency on the input pulse rate.

© 2007 Optical Society of America

## 1. Introduction

1. A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. **19**, 161–235 (1995). [CrossRef]

*et al*. [2

2. J. Azaña, R. Slavík, P. Kockaert, L. R. Chen, and S. LaRochelle, “Generation of customized ultrahigh repetition rate pulse sequences using superimposed fiber Bragg gratings,” J. Lightwave. Technol. **21**, 1490–1498 (2003). [CrossRef]

3. J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L. R. Chen, B. J. Eggleton, and J. Azaña, “4x100 GHz pulse train generation from a single-wavelength 10 GHz mode-locked laser using superimposed fiber Bragg gratings and nonlinear conversion,” IEEE/OSA J. Lightwave. Technol. **24**, 2091–2099 (2006). [CrossRef]

4. 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 Bragg gratings,” Opt. Lett. **24**, 1672–1764 (1999). [CrossRef]

5. M.P. Petropoulos, A. D. Ibsen, D. J. Ellis, and Richardson, “Rectangular pulse generation based on pulse reshaping using a superstructured fiber Bragg grating,” IEEE/OSA J. Lightwave Technol. **19**, 746–752 (2001). [CrossRef]

6. N. K. Berger, B. Levit, and B. Fischer, “Reshaping periodic light pulses using cascaded uniform fiber Bragg gratings,” J. Lightwave Technol. **24**, 2746–2751 (2006). [CrossRef]

7. 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]

9. 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]

*et al*. [10

10. 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]

2. J. Azaña, R. Slavík, P. Kockaert, L. R. Chen, and S. LaRochelle, “Generation of customized ultrahigh repetition rate pulse sequences using superimposed fiber Bragg gratings,” J. Lightwave. Technol. **21**, 1490–1498 (2003). [CrossRef]

11. J. A. Bolger, I. C. M. Littler, and B. J. Eggleton, “Optimisation of superimposed chirped fibre Bragg gratings for the generation of ultra-high speed optical pulse bursts,” Opt. Commun. **271**, 524–531 (2007). [CrossRef]

## 2. Theory

*c*. The frequency response of the whole filter can be expressed as

_{p}*H*is the

_{p}(ω)*p*individual filter and

_{th}*H*is defined as the generator filter with bandwidth

_{0}(ω)*δω*. The frequency separation between the individual filters is set by

*Δω*. The sum ranges from -

*p*to

_{min}*p*where

_{max}*p*and

_{min}*p*are integer numbers. The number of replicated filters is min max

_{max}*M=p*

_{min}+

*p*

_{max}+1. The frequency

*ω*corresponds to the base band frequency, i.e.

*ω=ω*

_{opt}-ω_{0}, where

*ω*is the optical frequency and

_{opt}*ω*is the central optical frequency of the whole filter. The corresponding complex envelope impulse response is:

_{0}*h*is the impulse response of the generator filter

_{0}(t)*H*and acts as overall temporal envelope, hence the SOPTB duration

_{0}(ω)*t*is set by the duration of

_{h}*h*. The shape of

_{0}(t)*h*has to be squared in order to avoid pulse to pulse amplitude variations. The second term consists of an

_{0}(t)*M*elements sum of complex numbers multiplied by frequency equispaced complex exponentials. This second term acts as pulse shaper of the individual pulses forming the output signal.

*y(t)*of the whole filter to a temporal input pulse with complex envelope

*x(t)*is given by the convolution of the impulse response of the whole filter with the input pulse.

*Y(ω)*is equal to the product of the input pulse spectrum

*X(ω)*with the spectral response of the filter

*H(ω)*.

*y(t)*can be approximated to the impulse response when the input pulse spectrum

*X(ω)*is considered flat all over the filter bandwidth, that is,

*s*is a complex constant of proportionality. Under this assumption, provided that the shape of

*h*is flat and the set of complex numbers

_{0}(t)*c*coincide with the coefficients of a FSD with fundamental harmonic set by

_{p}*Δω*, the input signal can be reshaped to any desired form with repetition period

*T*=2

*π*/Δ

*ω*. Equation (5) only applies when the input pulse spectrum

*X(ω)*can be considered flat all over the whole filter bandwidth. This consideration limits the filter bandwidth to the frequency range where the input spectrum can be considered as flat. A limited filter bandwidth reduces the number of individual filters that can be implemented. In other words, a limited filter bandwidth limits the number of harmonics of the FSD. A reduction on the number of harmonics leads to a less accurate shaping of the individual pulses that compose the SOPTB.

*δω*of the generator filter

*H*. That is, if the input pulse spectrum

_{0}(ω)*X(ω)*can be considered flat just over the bandwidth of each individual filter

*H*that form the whole filter, then the output frequency response

_{p}(ω)*Y(ω)*can be expressed as

*X*can be chosen as samples of

_{p}*X(ω), X*or by energetic reasons in a way that the approximated function has the same total energy that the function to be approximated. In the temporal domain, the complex envelope of the temporal response has the form

_{p}=X(pΔω)*d*with the coefficients of the FSD with fundamental harmonic Δω of the desired SOPTB (

_{p}=X_{p}c_{p}*y(t)*).

*pth*individual filter

*R*that forms the whole filter is defined as

_{p,max}*c*is normalized to the unity, the maximum reflectivity of the whole filter corresponds to the maximum reflectivity of the generator filter.

_{p}*φ*of the

_{p}*p*individual filter spectral response is determined by the phase of the complex number

_{th}*c*.

_{p}=d_{p}/X_{p}7. 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]

5. M.P. Petropoulos, A. D. Ibsen, D. J. Ellis, and Richardson, “Rectangular pulse generation based on pulse reshaping using a superstructured fiber Bragg grating,” IEEE/OSA J. Lightwave Technol. **19**, 746–752 (2001). [CrossRef]

9. 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]

10. 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]

## 3. Grating design for FSD filters based on Uniform S-FBGs

5. M.P. Petropoulos, A. D. Ibsen, D. J. Ellis, and Richardson, “Rectangular pulse generation based on pulse reshaping using a superstructured fiber Bragg grating,” IEEE/OSA J. Lightwave Technol. **19**, 746–752 (2001). [CrossRef]

## 4. Grating design for FSD filters based on S-LCFBGs

7. 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]

*H*is the reflectivity, the global phase and the central frequency of each filter. Consequently all the individual filters should have the same bandwidth and the same dispersion

_{p}(ω)*ϕ*̈. For a given length, the filter bandwidth sets the chirp factor [9

9. 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]

10. 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]

*t*≫1/

_{h}*δω*. The frequency separation

*Δω*is lower than the bandwidth of the generator filter

*δω*to avoid frequency overlapping between the individual filters. The frequency overlapping causes ripples in the amplitude and in the group delay of the frequency response. These ripples lead to a decrease of the filter energetic efficiency and to an increase of the pulse to pulse amplitude variations in the SOPTB temporal amplitude envelope [4

4. 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 Bragg gratings,” Opt. Lett. **24**, 1672–1764 (1999). [CrossRef]

*δω*should be equal to the frequency separation

*Δω*. Then, for the case of maximal energetic efficiency, a condition which relates the SOPTB duration with the period of the SOPTB is obtained

**19**, 435–437 (2007). [CrossRef]

*ω*is the central optical frequency of the whole filter and

_{0}*R*is calculated using Eq. (9).

_{p,max}## 5. Examples and results

*t*of 0.5 ps and a carrier frequency

_{FHWH}*ω*

_{0}/(2

*π*) of 193 THz. The temporal length of the SOTPB and its period are set to 40 ps and 4 ps respectively. The frequency separation of the individual filters is 250 GHz, but in the regions where the FSD coefficients are null the separation is greater. The length of all the gratings is 4.12 mm and the effective refractive index of the optical fiber is 1.452.

### 5.1 FSD filters based on Uniform S-FBGs

^{-4}, which does not achieve RI saturation. The minimal RI modulation for this example is 1.3×10

^{-5}which is greater than the minimum RI achievable. The energetic efficiency, defined as the quotient between the input and the output signal energy, is about 0.31%; hence in most cases an amplificator should be placed after the pulse shaper.

**19**, 746–752 (2001). [CrossRef]

### 5.2 FSD filters based on S-LCFBGs

*X(ω)*can be considered constant over each individual filter bandwidth and the approximation presented in Eq. (6) applies.

^{-3}which does not achieve RI saturation. Additionally, the total amount of RI modulation can be reduced simply by reducing the reflectivity of the generator filter. This reduction of reflectivity avoids index saturation and produces a flatter overall impulse response. The only drawback of the reduction of the reflectivity is that the energetic efficiency decreases, but it continues to be much greater than for the case with dc term.

### 5.4 Tolerance Analysis

*et al*. [5

**19**, 746–752 (2001). [CrossRef]

*t*=0.5 ps) and for the extreme case where the FWHM input pulse width is 0.7 ps, (40% greater than the design width) for the three examples presented in the previous sections. The main effect of this pulse width mismatching is an increase of the 10–90% rise/fall time. The 10–90% rise/fall time increases from 0.3 ps for the ideal case to 0.7 ps for the non-ideal case in the first example (uniform S-FBG) and from 0.25 ps to 0.7 ps in the two other examples (S-LCFBGs) Moreover, no increase of the amplitude variation across the top of each pulse is observed.

_{FWHM}## 6. Conclusion

## Acknowledgments

*Ministerio de Educación y Ciencia*under the projects “Plan Nacional de I+D+I TEC2004-04754-C03-02” and “Plan Nacional de I+D+I TEC2007-68065-C03-02”. Víctor García-Muñoz also thanks the support of the

*Telecommunications Department, Faculté Polytechnique de Mons*, where he has been a visiting researcher with a grant of the

*Consejo Social Universidad Politécnica de Madrid*.

## References and links

1. | A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. |

2. | J. Azaña, R. Slavík, P. Kockaert, L. R. Chen, and S. LaRochelle, “Generation of customized ultrahigh repetition rate pulse sequences using superimposed fiber Bragg gratings,” J. Lightwave. Technol. |

3. | J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L. R. Chen, B. J. Eggleton, and J. Azaña, “4x100 GHz pulse train generation from a single-wavelength 10 GHz mode-locked laser using superimposed fiber Bragg gratings and nonlinear conversion,” IEEE/OSA J. Lightwave. Technol. |

4. | 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 Bragg gratings,” Opt. Lett. |

5. | M.P. Petropoulos, A. D. Ibsen, D. J. Ellis, and Richardson, “Rectangular pulse generation based on pulse reshaping using a superstructured fiber Bragg grating,” IEEE/OSA J. Lightwave Technol. |

6. | N. K. Berger, B. Levit, and B. Fischer, “Reshaping periodic light pulses using cascaded uniform fiber Bragg gratings,” J. Lightwave Technol. |

7. | 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 |

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

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

10. | 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 |

11. | J. A. Bolger, I. C. M. Littler, and B. J. Eggleton, “Optimisation of superimposed chirped fibre Bragg gratings for the generation of ultra-high speed optical pulse bursts,” Opt. Commun. |

12. | T. Erdogan, “Fiber Grating Spectra,” IEEE/OSA J. Lightwave Technol. |

**OCIS Codes**

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

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(230.1150) Optical devices : All-optical devices

(320.5540) Ultrafast optics : Pulse shaping

(320.7080) Ultrafast optics : Ultrafast devices

**ToC Category:**

Optical Devices

**History**

Original Manuscript: July 6, 2007

Revised Manuscript: August 13, 2007

Manuscript Accepted: August 13, 2007

Published: August 14, 2007

**Citation**

Víctor García-Muñoz, Miguel A. Preciado, and Miguel A. Muriel, "Simultaneous ultrafast optical pulse train bursts generation and shaping based on Fourier series developments using superimposed fiber Bragg gratings," Opt. Express **15**, 10878-10889 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-17-10878

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### References

- A. M. Weiner, "Femtosecond optical pulse shaping and processing," Prog. Quantum Electron. 19, 161-235 (1995). [CrossRef]
- J. Azaña, R. Slavík, P. Kockaert, L. R. Chen, and S. LaRochelle, "Generation of customized ultrahigh repetition rate pulse sequences using superimposed fiber Bragg gratings," J. Lightwave. Technol. 21, 1490-1498 (2003). [CrossRef]
- J. Magné, J. Bolger, M. Rochette, S. LaRochelle, L. R. Chen, B. J. Eggleton, and J. Azaña, "4x100 GHz pulse train generation from a single-wavelength 10 GHz mode-locked laser using superimposed fiber Bragg gratings and nonlinear conversion," J. Lightwave. Technol. 24, 2091-2099 (2006). [CrossRef]
- 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 Bragg gratings," Opt. Lett. 24, 1672-1764 (1999). [CrossRef]
- P. Petropoulos, M. Ibsen, A. D. Ellis, D. J. Richardson, "Rectangular pulse generation based on pulse reshaping using a superstructured fiber Bragg grating," J. Lightwave Technol. 19, 746-752 (2001). [CrossRef]
- N. K. Berger, B. Levit, aand B. Fischer, "Reshaping periodic light pulses using cascaded uniform fiber Bragg gratings," J. Lightwave Technol. 24, 2746- 2751 (2006). [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]
- I. Littler, M. Rochette, and B. Eggleton, "Adjustable bandwidth dispersionless bandpass FBG optical filter," Opt. Express 13, 3397-3407 (2005). [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. 19, 435-437 (2007). [CrossRef]
- S. Longhi, M. Marano, P. Laporta, O. Svelto, "Propagation, manipulation, and control of picosecond optical pulses at 1.5 ?m in fiber Bragg gratings," J. Opt. Soc. Am. B 19, 2742-2757 (2002). [CrossRef]
- J. A. Bolger, I. C. M. Littler and B. J. Eggleton, "Optimisation of superimposed chirped fibre Bragg gratings for the generation of ultra-high speed optical pulse bursts," Opt. Commun. 271, 524-531 (2007). [CrossRef]
- T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 15, 1277-1294 (1997). [CrossRef]

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