We propose a novel linear filtering scheme based on ultrafast all-optical differentiation for re-shaping of ultrashort pulses generated from a mode-locked laser into flat-top pulses. The technique is demonstrated using simple all-fiber optical filters, more specifically uniform long period fiber gratings (LPGs) operated in transmission. The large bandwidth typical for these fiber filters allows scaling the technique to the sub-picosecond regime. In the experiments reported here, 600-fs and 1.8-ps Gaussian-like optical pulses (@ 1535 nm) have been re-shaped into 1-ps and 3.2-ps flat-top pulses, respectively, using a single 9-cm long uniform LPG.
©2006 Optical Society of America
The use of long-period fiber gratings (LPGs) for ultrafast optical pulse manipulation has been subject of considerable recent research work –. In particular, LPGs have been applied for optical pulse repetition rate multiplication, code generation and recognition  as well as for more general all-optical signal processing functionalities, such as ultrafast optical differentiation , . In this paper, we focus on the potential of LPG devices for optical pulse shaping applications , and in particular, we investigate the use of LPGs for a classic pulse shaping task, namely generation of flat-top pulses –.
The generation of ultrashort flat-top temporal intensity profiles is highly desired for a range of non-linear optical switching and frequency conversion applications , –. From a practical viewpoint, the two following features are highly desired: (i) Linear pulse re-shaping techniques are attractive because they generally require a simpler implementation and are independent of the input optical pulse power –; (ii) An all-fiber implementation allows for low insertion loss, robust, and low cost realization with full compatibility with fiber-based systems , . The main drawback of a linear re-shaping method for flat-top pulse generation is that the steepness of the rise/decay edges in the synthesized flat-top waveform is limited by the input optical bandwidth. Concerning linear all-fiber flat-top temporal pulse shapers, previously published work has mainly dealt with the use of customized in-fiber Bragg gratings (FBGs) ,  operating in reflection. In this approach, the FBGs are used as linear filters operating over input Gaussian-like (or soliton-like) optical pulses and they are designed to achieve the desired sinc-like spectral transfer function. This pulse shaping strategy is thus based on the proper manipulation of the spectral-domain features of the input optical pulse in order to obtain the spectral profile that corresponds to the desired temporal profile. This so-called Fourier-domain approach has been extensively used in conventional optical pulse shapers based on bulk diffraction gratings . The main limitations of FBG-based optical pulse shapers are that (i) an extremely complex amplitude and phase grating profile is usually required and (ii) due to the FBG limited bandwidth, temporal waveforms faster than a few tens of picoseconds cannot be easily synthesized. It has been previously anticipated that all these limitations could be overcome using LPGs instead of FBGs. Two LPG-based techniques have been reported so far. The first one  is difficult to implement in practice as the pulse re-shaping operation needs to be realized via a ‘core mode to cladding mode’ coupling or vice versa (i.e. the input and output signals are in different fiber propagation modes). We recently reported another alternative method  - it operates in transmission (the light pulse propagates exclusively in the fiber core mode) and is based on the use of a single uniform LPG designed to operate as an optical differentiator . In particular, we found out that a flat-top waveform is generated when a Gaussian-like optical pulse is linearly filtered by the LPG with its resonance frequency slightly detuned from the pulse carrier frequency. Moreover, it was shown that pulses with an ideal flat-top can be obtained by accurately adjusting the LPG resonance strength. This flat-top pulse shaper also appeared to be capable of operation down to the sub-picosecond regime , promising a broad range of applications. However, the principle of operation of this new flat-top pulse shaper is not evident as it is not based on the conventional Fourier-domain approach . Moreover, we anticipate that this same operation principle could be practically implemented using other linear filtering platforms (i.e. it is not restricted to application using LPG filters).
In this paper, we provide a comprehensive study of this novel flat-top optical pulse re-shaping scheme, including a formal and general explanation of the same. We show that the flat-top pulse re-shaping can be understood as a simple linear filtering of a transform-limited (chirp-free) symmetric optical pulse by a properly wavelength-detuned all-optical differentiator. As we have recently realized and reported the design and experimental demonstration of all-optical pulse time differentiation using LPGs , , we used the developed LPG devices to demonstrate our new approach of flat-top generation. We remind the reader that in our previous works on LPG-based time differentiation, sub-picosecond Gaussian-like optical pulses were differentiated to directly obtain first-order odd-symmetry Hermite-Gaussian time waveforms, which are very different waveforms to those presented here. From the principle of operation we reveal here, it follows that flat-top pulses of different duration can be generated using a single LPG device (similarly to the optical differentiator ). Here, we also demonstrate this unprecedented design flexibility experimentally by re-shaping of 600-fs and 1.8-ps Gaussian-like optical pulses into 1-ps and 3.2-ps flat-top pulses, respectively, using a single 9-cm long uniform LPG. To the best of the authors’ knowledge, these waveforms are the fastest flat-top temporal pulses ever synthesized with an all-fiber device (about 10-20 times faster than in previous works , ).
2. Operation principle
A uniform LPG operating in full-coupling condition (e.g. when κL=π/2, where κ is the coupling coefficient and L is the grating length) has been shown  to provide a linear filtering function around the LPG resonance frequency, ω 0, described by V(ω)∝-jωU(ω), where U(ω) and V(ω) are the spectra of the input and output complex envelopes u(t) and v(t), respectively (ω is the base-band frequency variable defined as ω=ω’-ωcar, where ω’ is the optical frequency variable and ωcar is the carrier frequency of the input and output optical signals). Notice that in the previous notation it is assumed that the signals’ carrier frequency ω car coincides with the LPG resonance frequency ω 0, ωcar=ω 0. In this case, the LPG linear filtering function is that of the first-order temporal differentiator, i.e. in the temporal domain, v(t)∝∂u(t)/∂t. In what follows, we assume an input Gaussian pulse, u(t)∝exp(-at 2 ), where a is a constant of no relevance for our purposes. The corresponding differentiated pulse is an odd-symmetry Hermite-Gaussian (HG) waveform , , ∂u(t)/∂t∝t·exp(-at 2), which consists of two concatenated pulses with identical amplitude profiles inversed in time that have a relative phase difference of π. Let us now consider a finite detuning of the input pu\lse carrier frequency ωcar with respect to the filter’s resonance frequency ω 0, Δω=ωcar-ω 0. The LPG filtering function is now given by V(ω)∝-jωU(ω)+jΔωU(ω), which in the temporal domain corresponds with the following operation: v(t)∝∂u(t)/∂t+jΔωu(t). Thus, the temporal intensity profile of the signal at the LPG output can be written as
which is a superposition of the differentiated waveform (HG waveform with a double-pulse intensity profile) and the original Gaussian pulse intensity profile, where the Gaussian pulse intensity peak coincides with the energy valley in the double-pulse waveform. The amount of energy of the original Gaussian pulse that is present in the generated pulse is proportional to the frequency detuning Δω. As the frequency detuning Δω increases, the differentiated pulse is gradually re-shaped in such a way that the central valley of the temporal double-pulse intensity is filled by the residual original Gaussian profile, leading to the formation of a single flat-top pulse for an optimal frequency detuning. Generally, such pulse is not transform-limited (e.g. the pulse temporal profile exhibits a certain non linear phase variation). This should pose no problem with many applications in which only the intensity is of interest , .
It has been anticipated that any desired ultrafast temporal waveform could be synthesized as a linear superposition of a Gaussian pulse and its successive time derivatives . Our proposal is based on this general property and in particular, we show that a flat-top intensity waveform can be well approximated by only two terms of the general series, i.e. by a proper combination of a Gaussian pulse and its first time derivative. Moreover, we also propose and demonstrate an extremely simple mechanism to implement this idea; this mechanism is based on the use of an optical differentiator where the relative weight between the two required temporal terms can be easily adjusted by the pulse-differentiator frequency detuning Δω according to the expression in Eqn. (1).
We have proved by numerical simulations that an optical pulse with a completely flat-top can be synthesized by an optimal combination of a Gaussian pulse and its first time derivative. In the proposed practical implementation of this idea, one should expect certain deviations from the ideal filtering characteristics affecting the pulse shape. Specifically, the flatness of the synthesized waveform will be affected mainly by a slight difference between the spectral transmission of an ideal differentiator and that of a real LPG filter. Obviously, these deviations becomes more significant over broader bandwidths, i.e. for shorter input pulses (see for instance our results below). Moreover, in practice, it may be difficult to set the LPG resonance - pulse wavelength detuning to the exact optimal operation point. We have however observed that the flat-top pulse re-shaping process is relatively robust to detuning from the optimal conditions. To give an idea of the required precision, assuming an ideal 8-cm LPG, a wavelength detuning of 0.1-nm from the optimal operation point would induce a valley in the pulse’s flat top with a relative depth of 0.04dB. The relative depth of this valley would become as large as 0.24dB for a wavelength detuning from optimal conditions of 0.5 nm. Finally, we have also observed, that the flatness and symmetry in the synthesized pulse can be optimized by adjusting the LPG coupling strength. Numerical results showing this interesting effect were reported in . A deeper understanding of this phenomenon would require further investigation but this is outside the scope of the present work.
The described pulse re-shaping process is illustrated by the numerical results shown in Fig. 1. Specifically, Fig. 1(a) shows the simulated intensity and phase temporal profiles of the waveforms generated at the output of a uniform LPG (parameters given below) when an ideal Gaussian optical pulse with a FWHM time width of 1.8 ps (also shown in the figure) is launched at the LPG input, assuming three different values for the input pulse - LPG wavelength detuning, Δλ=0 (dotted, red curves, corresponding to the optical differentiation process), -0.4 nm (dashed, green curves), and -1.3 nm (solid, blue curves). A nearly flat-top optical pulse is generated when the wavelength detuning is fixed to Δλ=-1.3 nm. We emphasize that as expected for a linear pulse re-shaping process, the rising and decaying times in the synthesized flat-top pulse are determined by those of the bandwidth-limited input pulse. To appreciate this feature, the input pulse in Fig. 1 is shifted in time to get its rise edge close to the rise edge of the generated flattop waveform.
3. Experiments and discussions
The LPG used in our proof-of-concept experiments was made in a standard telecom fiber (SMF-28, Corning Inc.) using the point-by-point technique with a CO2 laser . Specifically, the used fiber LPG was 8.9 cm long, with a period of 415 µm, which corresponds to coupling into the 5-th odd cladding mode at a resonance wavelength of 1534.9 nm. During fabrication, the LPG parameters were adjusted to achieve operation close to the full-coupling condition .
Figure 2 shows the measured (solid lines) amplitude and phase spectral characteristics of the fiber LPG; these characteristics were very close to those theoretically predicted (also shown in Fig. 2 with dashed lines), except for an intensity insertion loss of approximately 3 dB. In particular, the LPG exhibited the desired linear amplitude transfer function around the grating resonance wavelength with complex zero transmission at this wavelength. We believe that the insertion loss of the used LPG is most probably caused by a higher-order coupling and could be minimized by optimizing the grating index variation (to approach the ideal sine-like function).
The input pulses in our experiments were nearly transform-limited Gaussian-like optical pulses generated from a passively mode-locked wavelength-tunable fiber laser (Pritel Inc.) with Full-Width-Half-Maximum (FWHM) pulse widths ranging from 600 fs to 1.8 ps. The pulses from the laser were first launched into a polarization controller, as the fiber LPG was slightly birefringent, and were subsequently propagated through the LPG-based pulse shaper. The output spectrum was monitored using an optical spectrum analyzer (OSA). We used a fiber-based Fourier-transformed spectral interferometry (FTSI) setup to retrieve the complex temporal waveform of the output pulse (in this setup, the input pulse was used as the reference pulse) . We have previously applied this same technique for fully characterizing (in magnitude and phase) and optimizing ultrafast odd-symmetry Hermite-Gaussian optical pulses obtained by differentiation of sub-picosecond Gaussian-like input pulses using LPGs [3, 16]. Similarly to our previous reports, in this present work, FTSI allowed us to monitor the obtained temporal waveform and optimize the experiment conditions, namely wavelength detuning (in our experiment, this was achieved via fiber laser tuning) and slight adjustment of the LPG coupling coefficient (via LPG straining ), so that to achieve the desired flat-top temporal waveform. Fig. 3(a) and 3(b) show the amplitude and phase temporal profiles recorded at the LPG output when a 1.8 ps optical pulse was launched at the fiber input for different values of the pulse - LPG wavelength detuning, Δλ=0 (dotted, red curves), -0.4 nm (dashed, green curves), and -1.3 nm (solid, blue curves). The measured spectra corresponding to the three shown temporal waveforms are shown in Fig. 4. There is an excellent agreement between the experimental results (Fig. 3) and those expected from numerical simulations (Fig. 1). The estimated FWHM time width of the experimentally generated flat-top waveform (for Δλ=-1.3 nm) is ≈ 3.2 ps. The energetic efficiency of the performed filtering process (ratio of the output power to the input power) was measured to be 5.5%.
A remarkable property of the proposed technique is that with a specific LPG structure, flat-top waveforms of various durations can be synthesized by use of input pulses with different time widths. In other words, the duration of the generated flat-top waveform can be changed by simply tuning the time-width of the input pulse (after proper re-adjustment of the experiment conditions, e.g. pulse - LPG frequency detuning). Following with our proof-of-concept experiments, Fig. 5(b) shows the numerically simulated and experimentally generated flat-top temporal intensity profiles using a 600 fs (FWHM) Gaussian-like input pulse. In this case, the generated flat-top waveform has a FWHM time-width of ≈ 1 ps. For completeness, the results corresponding to flat-top pulse generation using a 1.8-ps input pulse are also shown in Fig. 5(a). As anticipated, the rising/decaying times in each synthesized waveform are determined by those of the corresponding bandwidth-limited optical pulse. For instance, the shown results evidences that the 1-ps flat-top waveform exhibits steeper edges than the 3.2-ps flat top pulse; obviously, this is associated with the fact that a broader input pulse bandwidth was used for synthesizing the shorter flat-top waveform. As mentioned above, due to its broader bandwidth, the shorter pulse covers larger part of the LPG amplitude spectral response, which has characteristics slightly deviating from that of an ideal differentiator; this translates into the observed slightly more significant asymmetry and fluctuations in the synthesized flat-top pulse. Using numerical simulations, we have estimated that flat-top waveforms with time widths down to ≈ 550 fs (from ≈ 290-fs input Gaussian-like pulses) could be efficiently synthesized using the specific LPG reported here. Shorter input pulses were found theoretically to degrade the steepness of the fall edge of the flat-top pulse. However, shorter flat-top pulses could be generated using a LPG with a broader resonance bandwidth. For verification purposes, we also measured the autocorrelations of the generated flat-top waveforms, see Fig. 5(c); as expected, the measured autocorrelation traces were nearly triangular.
An important aspect to consider in a pulse re-shaping process is the energetic efficiency. It has been previously shown that processing of a long (narrow bandwidth) pulse with a high-bandwidth differentiator deteriorates the energetic efficiency of the process as a larger portion of the input pulse bandwidth is close to the filter’s zero transmission (i.e. a larger portion of the input pulse spectrum is filtered out by the LPG) . Thus, for optimum performance in terms of energetic efficiency, the input pulse bandwidth should approach that of the differentiator. In our examples, an energetic efficiency of ≈ 20% has been measured for the 1-ps flat-top pulse re-shaping process. This is a value considerably higher than that obtained in the 3.2-ps flat-top pulse generation experiment (measured energetic efficiency of ≈ 5.5%), in good agreement the predictions made above. To obtain a higher energetic efficiency for the longer pulses (e.g., 1.8 ps), an LPG with narrower bandwidth (realizable using a longer LPG or a LPG that couples into a different cladding mode) should be used.
A new, simple and efficient all-fiber method for generating flat-top optical pulses down to the sub-picosecond regime has been proposed and experimentally demonstrated. This technique is based on re-shaping an input Gaussian-like optical pulse with a slightly wavelength-detuned optical differentiator, which in our implementation was realized using a suitable uniform LPG filter. The technique allows synthesizing flat-top waveforms of different durations with the same LPG device. The demonstrated approach could be readily applied for upgrading previously reported pulse reshaping, retiming and temporal switching systems based on the use of ultrashort flat-top temporal waveforms – for operation over significantly higher data rates. In a more general framework, our results highlight the capabilities of LPG technology for pulse re-shaping operations in the picosecond/sub-picosecond range, a temporal regime which cannot be easily reached with FBG technology.
This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) through its Strategic Grants Program and by the Grant Agency of AS, Czech Republic, contract No. B200670601.
References and links
1. S. J. Kim, T. J. Eom, B. H. Lee, and C. S. Park, “Optical temporal encoding/decoding of short pulses using cascaded long-period fiber gratings,” Opt. Express 11, 3034–3040 (2003). http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-23-3034 [CrossRef] [PubMed]
3. R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators, ” Opt. Express 14, 10699–10707 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-22-10699 [CrossRef] [PubMed]
4. M. Kulishov and J. Azaña, “Ultrashort pulse propagation in uniform and nonuniform waveguide long-period gratings,” J. Opt. Soc. Am. A 22, 1319–1332 (2005). [CrossRef]
5. see for instance, A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19, 161–237 (1995). [CrossRef]
6. 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]
7. L. Qian, A. M. H. Wong, S. A. Neata, and X. Gu, “Simple and efficient optical pulse shaping: new algorithm and experimental demonstration,” Conference on Lasers and Electro-Optics (CLEO)2006, Long Beach, CA, USA. Paper JWB-33.
8. J. H. Lee, P. C. The, P. Petropoulos, M. Ibsen, and D. J. Richardson, “All-optical modulation and demultiplexing systems with significant timing jitter tolerance through incorporation of pulse shaping fiber Bragg gratings,” IEEE Photon. Technol. Lett. 14, 203–205 (2002). [CrossRef]
9. 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. 19, 746–752 (2001).
10. L. K. Oxenlowe, M. Galili, A. T. Clausen, and P. Jeppesen, “Generating a square switching window for timing jitter tolerant 160Gb/s demutiplexing by the optical Fourier transform technique,” Proc. of the 32nd European Conference on Optical Communication (ECOC 2006), Cannes, France, September 2006. Paper We2.3.4.
11. M. Kulishov, Y. Park, J. Azaña, and R. Slavík, “(Sub-)Picosecond Flat-Top Waveform Generation using a Single Uniform Long-Period Fiber Grating,” in Proc. of European Conference on Optical Communications (ECOC 2006), Cannes, France, September 2006. Paper We2.3.7
13. I. Bralwish, B. L. Bachim, and T.K. Gaylord, “Prototype CO2 laser-induced long-period fiber grating variable optical attenuators and optical tunable filters,” Appl. Opt. 43, 1789–1793 (2004). [CrossRef]
14. R. Slavík, “Extremely deep long-period fiber grating made with CO2 laser,” IEEE Photon. Technol. Lett. 18, 1705–1707 (2006). [CrossRef]
15. 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]
16. 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]