## Transform-limited picosecond pulse shaping based on temporal coherence synthesization

Optics Express, Vol. 15, Issue 15, pp. 9584-9599 (2007)

http://dx.doi.org/10.1364/OE.15.009584

Acrobat PDF (353 KB)

### Abstract

A simple and efficient optical pulse re-shaper based on the concept of temporal coherence synthesization is proposed and analyzed in detail. Specifically, we demonstrate that an arbitrary chirp-free (transform-limited) optical pulse waveform can be synthesized from a given transform-limited Gaussian-like input optical pulse by coherently superposing a set of properly delayed replicas of this input pulse, e.g. using a conventional multi-arm interferometer. A practical implementation of this general concept based on the use of conventional concatenated two-arm interferometers is also suggested and demonstrated. This specific implementation allows the synthesis of any desired temporally-symmetric optical waveform with time features only limited by the input pulse bandwidth. A general optimization algorithm has been developed and applied for designing the system specifications (number of interferometers and relative time delays in these interferometers) that are required to achieve a desired optical pulse re-shaping operation. The required tolerances in this system have been also estimated and confirmed by numerical simulations. The proposed technique has been experimentally demonstrated by reshaping an ≈1-ps Gaussian-like optical pulse into various temporal shapes of practical interest, i.e. picosecond transform-limited flat-top, parabolic and triangular pulses (all centered at a wavelength of ≈ 1550nm), using a simple two-stage interferometer setup. A remarkable synthesis accuracy and high energetic efficiency have been achieved for all these pulse re-shaping operations.

© 2007 Optical Society of America

## 1. Introduction

3. T. Otani, T. Miyajaki, and S. Yamamoto, “Optical 3R Regenerator using wavelength converters based on electroabsorption modulator for all-optical network applications,” IEEE Photon. Technol. Lett. **12**, 431–433 (2000). [CrossRef]

4. F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. Roelens, P. Petropoulos, and D. J. Richardson, “Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express **14**, 7617–7622 (2006). [CrossRef] [PubMed]

3. T. Otani, T. Miyajaki, and S. Yamamoto, “Optical 3R Regenerator using wavelength converters based on electroabsorption modulator for all-optical network applications,” IEEE Photon. Technol. Lett. **12**, 431–433 (2000). [CrossRef]

4. F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. Roelens, P. Petropoulos, and D. J. Richardson, “Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express **14**, 7617–7622 (2006). [CrossRef] [PubMed]

5. A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. **19**, 161–237 (1995). [CrossRef]

6. T. Kurokawa, H. Tsuda, K. Okamoto, K. Naganuma, H. Takenouchi, Y. Inoue, and M. Ishii, “Time-space conversion optical signal processing using arrayed-waveguide grating,” Electron. Lett. **33**, 1890–1891 (1997). [CrossRef]

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

8. Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber gratings,” Opt. Express **14**, 12670–12678 (2006). [CrossRef] [PubMed]

6. T. Kurokawa, H. Tsuda, K. Okamoto, K. Naganuma, H. Takenouchi, Y. Inoue, and M. Ishii, “Time-space conversion optical signal processing using arrayed-waveguide grating,” Electron. Lett. **33**, 1890–1891 (1997). [CrossRef]

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

8. Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber gratings,” Opt. Express **14**, 12670–12678 (2006). [CrossRef] [PubMed]

3. T. Otani, T. Miyajaki, and S. Yamamoto, “Optical 3R Regenerator using wavelength converters based on electroabsorption modulator for all-optical network applications,” IEEE Photon. Technol. Lett. **12**, 431–433 (2000). [CrossRef]

*two-stage*interferometric setup.

## 2. Operation principle

### 2.1 Temporal coherence synthesization

10. V. Narayan and D. L. MacFarlane, “Bursts and codes of ultrashort pulses,” IEEE Photon. Technol. Lett. **5**, 1465–1467 (1993). [CrossRef]

11. M. Shen and R. A. Minasian, “Toward a high-speed arbitrary waveform generation by a novel photonic processing structure,” IEEE Photon. Technol. Lett. **16**, 1155–1157 (2004). [CrossRef]

## 2.2 Practical implementation using concatenated two-arm interferometers

## 3. General design algorithm

*M*concatenated two-arm interferometers. As detailed below,

*M*actually represents the maximum number of available interferometers to be used in the pulse shaping system. The parameters to be optimized are the

*M*relative time delays, i.e.

*τ*

*with*

_{j}*j*=1 …

*M*; each one corresponding to each of the concatenated interferometers. Assuming also that an input pulse

*e*(

*t*) (corresponding optical spectrum

*E*(

*ω*)) is launched at the system input, the output pulse spectrum

*A*(

*ω*) can be expressed as follows [12]:

*ω*=

*ω*

^{′}+

*ω*

*, where*

_{c}*ω*

^{′}is the base-band frequency, and

*ω*

*is the central (carrier) optical frequency of the input pulse. The corresponding relationship in the temporal domain can be written as:*

_{c}*a*(

*t*) and

*e*(

*t*) are the complex envelopes of the output and input optical pulses, respectively. In what follows, we assume that

*e*(

*t*) is a real function, representing a transform-limited pulse. The design is further simplified if the relative time delays in the interferometers are fixed to satisfy the following ‘phase-matching’ condition:

*τ*

*=2*

_{j}*n*

_{j}*π*/

*ω*

*, where*

_{c}*n*

*is an arbitrary integer. In this case, Eq. (2) can be reduced to:*

_{j}*a*(

*t*) is the result of overlapping various (2

*) delayed replicas of the input optical pulse envelope*

^{M}*e*(

*t*). This overlapping process can be properly designed to construct the desired output envelope shape. Specifically, in order to determine the system parameters, namely the relative time delays, that are required to obtain a target output pulse shape

*a*

*(*

_{targ}*t*), we use a weighted root-mean-square error function, defined as follows:

*I*

*=|*

_{targ}*a*

*|*

_{targ}^{2}and

*I*

*=|*

_{calc}*a*

*|*

_{calc}^{2}are the target intensity (desired output pulse shape) and calculated (actually obtained) intensity, respectively,

*a*

*(*

_{calc}*t*)=

*a*(

*t*) (which is obtained from Eq. (3)),

*t*

*is the discrete set of times where the output pulse envelope is evaluated,*

_{n}*N*being the total number of samples over a predefined time window, and

*W*

*=*

_{n}*W*(

*t*

*) is the weight for each sampling point. We use a general algorithm to search for the optimum set of parameters (relative time delays) that minimizes this error function. As stated before, the purpose of the weighting function is to emphasize the local regions of the output waveform over which a higher re-shaping accuracy is desired (as an example, this procedure can be used to minimize the oscillations in the flat-top region of a rectangular-like pulse waveform by simply allocating a higher weighting value over this more critical region).*

_{n}## Procedure make(shape, weight, M, T,D)

*shape*⇒

*I*

*(*

_{targ}*t*

_{1}…

*t*

*)*

_{N}*weight*⇒

*W*(

*t*

_{1}…

*t*

*)*

_{N}*set*:

*Err*=∞

*for*

*τ*

_{1}=

*τ*

_{1.1}…

*τ*

_{1,S}

*for*

*τ*

*=*

_{M}*τ*

_{M,1}…

*τ*

_{M,S}

*calculate the output intensity based on the*

*relativetime delays vector*(

*τ*

_{1},…

*τ*

*)⇒*

_{M}*I*

*(*

_{calc}*t*

_{1}…

*t*

*);*

_{N}*calculate the absolute error*:

*Error*=

*if*(

*Error*<

*D*×

*Err*)

*Err*←

*Error*;

*Optimum*(

*τ*

_{1},…

*τ*

*)←(*

_{M}*τ*

_{1},…

*τ*

*);*

_{M}*τ*

*(to be optimized), is assigned S different possible values uniformly increasing from*

_{j}*τ*

_{j,1}=0 to a pre-defined maximum value

*τ*

_{j,S}(ignoring the parameters permutation). Thus, the total number of possible combinations that are evaluated for searching the optimal set of time delays (with minimum error) would be equal to

*M*×

*S*, where

*M*is the maximum number of available interferometers. Notice that in our algorithm we have assumed that permutations between the different interferometers are not important and we can allocate the relative time delays to arbitrarily placed interferometers. For each set of time delays, the resultant output pulse waveform,

*a*

*(*

_{calc}*t*) is calculated according to Eq. (3) and subsequently, the new weighted mean squared error is obtained from Eq. (4). Notice that since all the time delays,

*τ*

*(1, 2,…,*

_{j}*M*) are initialized at ‘0’, (which is equivalent to assume that there is no interferometer), the number of “required” interferometers in the system increases to (

*j*+1) after every

*j*×

*S*loops (

*j*=1, 2,…,

*M*). In principle, the new set of design parameters will be accepted as the optimal one (

*Optimum*) only if their associated weighted squared error (

*Error*) is lower than the present minimum weighted squared error (

*Err*) multiplied by a parameter

*D*(<1). The parameter

*D*(margin factor) defines an additional mechanism to determine the minimum number of interferometers that are required to achieve the target pulse re-shaping operation with the desired accuracy. Basically, a new set of parameters will be accepted as the optimal solution

*only*if they provide a certain improvement over the presently selected solution; the required “improvement” is quantitatively evaluated in terms of the associated mean squared error and is controlled through the parameter

*D*. To be more concrete, the margin factor

*D*sets the minimum relative error difference (

*δ*=1-

*D*) that is necessary to accept the new solution as the optimal one, i.e. the solution under evaluation (with associated mean squared error

*Error*) will be accepted as the optimal solution only if (

*Err*-

*Error*)/

*Err*>

*δ*(where we reiterate that

*Err*is the mean squared error of the presently selected solution). Thus, if the use of more than

*L*(<

*M*) interferometers does not offer an improvement over the already selected solution, according to the criterion fixed through the parameter

*D*, then the rest of the time delays, i.e. variables from

*τ*

_{1}~

*τ*

_{M-L}will be set to zero in the optimal solution. In this way, the minimum number of required interferometers (

*L*) will be automatically provided by the algorithm, according to the minimum relative error improvement fixed “a priori” by an experienced designer.

*τ*

*=*

_{j}*2πn*

*/*

_{j}*ω*

*. In this way, the time sampling that is used in the design algorithm for each time-delay variable can be made much longer than the ultrashort time resolution associated with the used carrier frequency. In practice, the above condition can be achieved by very fine tuning of each of the relative time delays around its nominal design value (see more discussions on this point in Section 5).*

_{c}## 4. Design examples

*α*(0≤

*α*≤1), and the Full-Width Half-Maximum (FWHM) pulse time-width

*T*. The corresponding frequency-domain representation of the raised cosine function in Eq. (5) is:

*α*. The figure evidences the role of the two parameters (roll-off factor

*α*and pulse time width

*T*) in the general function. As the parameter

*α*varies from zero towards one, the shape of the raised cosine function is softened, changing from an ideal square shape (

*α*=0) to a “minimally smoothed” triangular shape (

*α*=1).

*λ*=3.2 nm). The considered input pulse time shape [intensity shown in the inset of Fig. 4(a)] was obtained by taking the inverse Fourier transform of the square root of the measured input pulse spectrum, assuming a constant spectral phase (see descriptions in Section 6 below). For our numerical designs, the time variable has been quantized in sampling steps of 0.01 ps.

## 4.1 Pulse shaping based on two cascaded interferometers

*τ*

_{1},

*τ*

_{2}and

*τ*

_{3}=

*τ*

_{1}+

*τ*

_{2}). Specifically, we have targeted the synthesis of flat-top, triangular and parabolic optical pulse shapes, all of them with the same FWHM time duration of 2 ps. The corresponding numerically simulated results are shown in Figs. 4–6. Additionally, this same system has been designed to synthesize a longer flat-top pulse, see results in Fig. 4(b).

## 4.1.1 Flat-top optical pulses

*α*=0.75 and

*T*=2 ps. A proper weighting function was used to minimize the error along the flat-top region (weight factor of 1000) and along the waist region (weight factor of 100). The used weighting function (normalized units) is plotted in Fig. 4(a) with a black dotted line. The use of this weighting function allowed us to minimize the fluctuations in the flat-top region while obtaining the desired time width and suitable rising and falling slopes. As anticipated, the desired pulse re-shaping algorithm could be accurately implemented using only two concatenated interferometers; the optimum relative delays for this design were 0.37 ps and 1.22 ps, respectively. Figure 4(a) (blue, dotted curve) shows the intensity of the numerically simulated output pulse after propagation through the optimized pulse shaper. The numerically obtained output pulse shape was almost identical to the target square-like time waveform (not shown here).

*α*=0.75 and

*T*=3.1 ps. A similar weighting function to that of the previous flat-top pulse was employed (the normalized weighting function is shown in Fig. 4(b) with a black dotted curve). The optimum relative time delays in this case were 0.93 ps and 1.90 ps. The numerically obtained output pulse shape (shown in Fig. 4(b) with a blue, dotted curve) was almost identical to the target square-like shape.

## 4.1.2 Triangular optical pulse

*α*=1 and

*T*=2 ps. No weighting function is required in this case. As anticipated, this pulse re-shaping operation can be accurately achieved using only two concatenated interferometers; the optimum relative time delays in these interferometers are 0.76 ps and 1.40 ps. Simulation results are shown in Fig. 5, where the blue, dotted line is the numerically simulated output pulse waveform. We emphasize again that the numerically obtained pulse waveform was almost undistinguishable from the target waveform.

## 4.1.3 Parabolic optical pulse

4. F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. Roelens, P. Petropoulos, and D. J. Richardson, “Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express **14**, 7617–7622 (2006). [CrossRef] [PubMed]

*P*

*is the peak power of the pulse, and*

_{p}*T*

*is the FWHM temporal duration. The envelope shape (intensity) of an ideal 2-ps (FWHM) parabolic pulse is shown in Fig. 6 (orange, solid curve). As for an ideal square or triangular waveform, an ideal parabolic pulse exhibits an infinite bandwidth and consequently, cannot be synthesized in practice. A practical, bandwidth-limited version of this waveform could be obtained by simply convolving the ideal parabolic function [given by Eq. (7)] with a bandwidth-limited (e.g. Gaussian) pulse. In our design, we have chosen a 2-ps ideal parabolic pulse as the target time waveform but have used a proper weighting function (shown in Fig. 6 with a black, dotted curve) to fully optimize the synthesization process over the central region of the optical pulse. Specifically, this was achieved by fixing the weighting factor to 1,000 over the region of interest. Notice that the central part of a parabolic waveform is indeed the most important pulse region from a practical viewpoint [4*

_{P}**14**, 7617–7622 (2006). [CrossRef] [PubMed]

## 4.2 Design example with more than two concatenated interferometers

*D*in the proposed optimization algorithm can be used to determine the minimum number of interferometers that are required to achieve a desired pulse shaping operation with a sufficiently high accuracy. Specifically, we assumed the same input optical pulse as in the examples shown above (0.95-ps Gaussian-like pulse) and we targeted the generation of a triangular pulse waveform with a FWHM time duration of 3 ps (i.e. the target function was the raised cosine with parameters

*α*=1 and

*T*=3 ps). The target pulse waveform is shown in Fig. 7 (blue, dotted curves). A suitable weighting function was used to optimize the temporal response along the waveform slopes (a weight of 1,000 was employed over these critical regions). Figure 7(a) (solid, red curve) shows the optimized output pulse waveform when only two interferometers were considered in the design (

*M*=2); in this case, the optimal relative delays were 1.17 ps and 2.11 ps. A significant deviation is observed between the numerically calculated output waveform and the target shape, particularly in the slopes of the triangular pulse. Following this optimization process, the relative deviation between the mean squared errors of the two best solutions (solutions with the two lowest mean squared errors) was estimated to be 1.5 × 10

^{-15}. In a second trial, the number of interferometers was left undetermined (

*M*> 3) and the parameter

*D*was fixed to 1-1.5×10

^{-16}. This value was set to ensure that the algorithm could escape from the relative minimum-error solution found for the case of two interferometers (

*δ*=1-

*D*<1.5×10

^{-15}). The optimization algorithm provided a solution consisting in 3 concatenated interferometers (

*L*=3) with relative time delays fixed to 0.9 ps, 1.2 ps, and 2 ps, respectively. In this case, we estimated a relative deviation between the two lowest mean squared errors of 1.4×10

^{-16}(as expected, this relative deviation is lower than the minimum acceptable relative error difference fixed in the algorithm,

*δ*=1-

*D*=1.5×10

^{-16}). Figure 7(b) (solid, red line) shows the numerically simulated output pulse shape obtained with this optimal configuration; the synthesized waveform is visibly much closer to the target pulse than in the case of two interferometers [Fig. 8(a)], having synthesized the pulse slopes with a very high precision.

## 5. Error analysis

*τ*

*=2*

_{j}*n*

_{j}*π*/

*ω*

*, where*

_{c}*n*

*is an arbitrary integer an*

_{j}*ω*

*is the central (carrier) optical frequency of the input pulse. This condition guarantees that the delayed replicas of the input optical pulses are summed up with no phase variations among them. In practice, the time delays may deviate from this condition and this is expected to affect critically the pulse shaping result. One should bear in mind that given the high value of the central optical frequency, even a small change in each relative time delay may translate into a significant phase variation in the interfering pulses. Mathematically, it can be inferred from Eq. (2) and Eq. (3) above that deviations in the relative time delays (Δ*

_{c}*τ*

*) with respect to its optimal values (*

_{j}*τ*

*) will result into the following distorted output waveform:*

_{j}*φ*

*=*

_{j}*ω*

*Δ*

_{c}*τ*

*≪2π), these delay variations should satisfy the following condition: Δ*

_{j}*τ*

*≪2π/*

_{j}*ω*

*. Assuming a typical central optical frequency*

_{c}*ω*

*=2π×192THz (corresponding wavelength around 1.55µm), this translates into very tight acceptable tolerances for the relative time delays in the system, i.e. Δ*

_{c}*τ*

*≪5.2-fs; in other words, the path-length differences in the used interferometers need to be controlled with sub-micron resolutions. Obviously, this condition is much more critical than the one resulting from deviations in the time delays among the overlapped pulse replicas (assuming no phase variations), since the tolerances associated with these deviations are given by the following condition Δ*

_{j}*τ*

*≪*

_{j}*τ*

*, typically resulting in tolerances in the sub-picosecond range (assuming (sub-)picosecond pulse re-shaping operations such as those investigated in this paper).*

_{j}*ω*

*=2π×192THz. We remind the reader that the optimum relative time delays are 0.369 ps and 1.218 ps. Notice that in order to evaluate the effect of phase variations, which was obviated in the above system design procedure, the sampling time is now fixed to 1-fs, corresponding to phase changes of 1.2 rad. Two different cases are separately evaluated, namely the effect of deviations in the longer time delay [results shown in Fig. 8(a)] and the effect of deviations in the shorter time delay [results shown in,. 8(b)], assuming in each case that the other time delay is set to its optimal value. In both cases, the time delay under analysis was deviated between -1fs to 1fs with respect to its optimal value which corresponds to ±0.3 µm optical path-length instability in air. As expected, these small time delay variations were sufficient to induce a visible distortion in the synthesized output waveforms. Concerning the variations in the longer time delay [Fig. 8(a)], we observed that as we deviated further from the optimum relative time delay, the synthesized pulse was broadened and a deeper undershoot was introduced over the flat-top region. The system appeared to be more sensitive to a decrease in the time delay (with respect to its optimal value) rather than to a delay increasing. In the case of variations in the shorter time delay [Fig. 8(b)], the synthesized pulse was slightly broadened with a significant central undershoot when the delay was increased with respect to its optimal value; in contrast, a delay decreasing translated into a narrowing of the pulse width.*

_{c}## 6. Experimental results and discussions

*τ*

*=*

_{j}*2πn*

*/*

_{j}*ω*

*) was precisely adjusted by monitoring the output pulse spectrum (using an OSA); specifically, the path-length difference in each interferometer needed to be tuned with sub-micrometer resolution to ensure that the periodic spectrum modulation corresponding to the assigned time delay,*

_{c}*τ*

*, was properly shifted to be exactly symmetric with respect to the pulse center wavelength. Examples of output spectra obtained after these fine adjustments are shown in the inset of Fig. 4(b) (results corresponding to the two flat-top pulse synthesis experiments described above).*

_{j}14. L. Lepetit, G. Chériaux, and M. Joffre, “Linear technique of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B **12**, 2467–2474 (1995). [CrossRef]

15. Y. Park, F. Li, and J. Azaña, “Characterization and optimization of optical pulse differentiation using spectral interferometry,” IEEE Photon. Technol. Lett. **18**, 1798–1800 (2006). [CrossRef]

14. L. Lepetit, G. Chériaux, and M. Joffre, “Linear technique of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B **12**, 2467–2474 (1995). [CrossRef]

5. A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. **19**, 161–237 (1995). [CrossRef]

## 7. Conclusions

## Acknowledgments

## References and links

1. | F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, “All-optical pulse reshaping and retiming systems incorporating pulse shaping fiber Bragg grating,” J. Lightwave Technol. |

2. | L. K. Oxenløwe, M. Galili, H. C. H. Mulvad, R. Slavík, Y. Park, J. Azaña, and P. Jeppesen, “Flat-top pulse enabling 640 Gb/s OTDM demultiplexing,” |

3. | T. Otani, T. Miyajaki, and S. Yamamoto, “Optical 3R Regenerator using wavelength converters based on electroabsorption modulator for all-optical network applications,” IEEE Photon. Technol. Lett. |

4. | F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. Roelens, P. Petropoulos, and D. J. Richardson, “Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express |

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

6. | T. Kurokawa, H. Tsuda, K. Okamoto, K. Naganuma, H. Takenouchi, Y. Inoue, and M. Ishii, “Time-space conversion optical signal processing using arrayed-waveguide grating,” Electron. Lett. |

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

8. | Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber gratings,” Opt. Express |

9. | Y. Park and J. Azaña, “Optical pulse shaping technique based on a simple interferometry setup,” in Proc. of IEEE LEOS 2006 Annual Meeting. Paper TuN2, pp. 274–275. |

10. | V. Narayan and D. L. MacFarlane, “Bursts and codes of ultrashort pulses,” IEEE Photon. Technol. Lett. |

11. | M. Shen and R. A. Minasian, “Toward a high-speed arbitrary waveform generation by a novel photonic processing structure,” IEEE Photon. Technol. Lett. |

12. | C. K. Madsen and J. H. Zhao, |

13. | I. Glover and P. Grant, |

14. | L. Lepetit, G. Chériaux, and M. Joffre, “Linear technique of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B |

15. | Y. Park, F. Li, and J. Azaña, “Characterization and optimization of optical pulse differentiation using spectral interferometry,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

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

(120.2440) Instrumentation, measurement, and metrology : Filters

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: May 21, 2007

Revised Manuscript: July 13, 2007

Manuscript Accepted: July 13, 2007

Published: July 18, 2007

**Citation**

Yongwoo Park, Mohammad H. Asghari, Tae-Jung Ahn, and José Azaña, "Transform-limited picosecond pulse shaping based on temporal coherence synthesization," Opt. Express **15**, 9584-9599 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9584

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### References

- F. Parmigiani, P. Petropoulos, M. Ibsen, D. J. Richardson, “All-optical pulse reshaping and retiming systems incorporating pulse shaping fiber Bragg grating,” J. Lightwave Technol. 19, 746–752 (2001).
- L. K. Oxenløwe, M. Galili, H. C. H. Mulvad, R. Slavík, Y. Park, J. Azaña, P. Jeppesen, “Flat-top pulse enabling 640 Gb/s OTDM demultiplexing,” Conference on Lasers and Electro-Optics Europe (CLEO-Europe) Munich, Germany, June 2007, Paper CI8-1.
- T. Otani, T. Miyajaki, S. Yamamoto, “Optical 3R Regenerator using wavelength converters based on electroabsorption modulator for all-optical network applications,” IEEE Photon. Technol. Lett. 12, 431–433 (2000). [CrossRef]
- F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. Roelens, P. Petropoulos, D. J. Richardson, “Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express 14, 7617–7622 (2006). [CrossRef] [PubMed]
- A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–237 (1995). [CrossRef]
- T. Kurokawa, H. Tsuda, K. Okamoto, K. Naganuma, H. Takenouchi, Y. Inoue, M. Ishii, “Time-space conversion optical signal processing using arrayed-waveguide grating,” Electron. Lett. 33, 1890–1891 (1997). [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]
- Y. Park, M. Kulishov, R. Slavík, J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber gratings,” Opt. Express 14, 12670–12678 (2006). [CrossRef] [PubMed]
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