1. Introduction

There has been much recent interest in parabolic pulse generation in optical fiber amplifiers with normal group velocity dispersion (GVD) [1–9]. Such parabolic pulses are of wide-ranging practical significance since they are generated asymptotically in the fiber amplifier independent of the shape or the noise properties on the input pulse, and possess linear chirp which leads to efficient pulse compression [3]. They are also of fundamental interest as they represent a particular class of solution of the nonlinear Schrödinger equation (NLSE) with gain that is associated with self-similar evolution of the pulse intensity and chirp in such a way that the intensity profile retains its parabolic shape and resists the deleterious effects of optical wavebreaking. These self-similar parabolic pulses (similaritons) also possess interesting intermediate asymptotic behaviour in the low amplitude wings, and this has also been the subject of some recent theoretical analysis [5, 6]. To date, experimental studies of parabolic pulse generation have been restricted to fiber amplifiers where the gain and normal GVD are associated with the addition of dopants such as Er³⁺ [2] or Yb³⁺ [3,7,8] in the fiber core. Numerical studies of such doped fiber amplifiers have been largely restricted to the case where the pulse propagation is accurately modelled by the addition of a simple constant gain term to the standard NLSE, although some work has also been carried out to investigate the effects of a longitudinally varying gain profile [4], as well as the limiting effects of the finite transition linewidth [9].

In addition to these studies of doped fiber amplifiers, a recent numerical study has also shown that fiber Raman amplifiers based on microstructure or photonic crystal fiber (PCF) can support the generation of parabolic pulses when pumped with high power 20 W nanosecond pulses typical of those obtained from diode pumped high power erbium doped fiber amplifier chains [10]. In this paper, we demonstrate both numerically and experimentally that the use of PCF is not a necessary requirement for parabolic pulse Raman amplification, and that parabolic pulses at 1550 nm can be obtained in a Raman amplifier based on standard non-zero dispersion shifted fibre (NZ-DSF) using a commercial watt-level CW pump source at 1455 nm. An important aspect of our experiments is the use of frequency-resolved optical gating (FROG) to measure the intensity and chirp of the amplifier output pulses, thus directly confirming their expected parabolic intensity profile and linear chirp.

2. Numerical simulations

We begin by presenting the results of numerical simulations demonstrating that parabolic pulse generation in NZ-DSF is possible at 1550 nm under typical experimental conditions. Our simulations are based on the generalized NLSE: $\frac{\partial E(z,t)}{\partial z}=-\frac{\alpha }{2}E-i\frac{{\beta }_2}{2}\frac{{\partial }^{2}E}{\partial t^2}+\frac{{\beta }_3}{6}\frac{{\partial }^{3}E}{\partial t^3}+i\gamma \left(1+\frac{i}{{\omega }_0}\frac{\partial }{\partial t}\right)\left[E(z,t)\left({\int }_0^{\infty }R\left(t\text{'}\right){\mid E(z,t-t\text{'})\mid }^{2}dt\text{'}\right)\right]$ Here, E(z,t) is the pulse envelope in a co-moving frame, and the function R(t)=(1-f _R)δ(t)+f _R h _R(t) includes instantaneous and delayed Raman contributions with the fractional Raman contribution f _R=0.18. For h _R, we used the measured Raman response of silica. We consider a Raman amplifier based on NZ-DSF with dispersion parameters at 1550 nm given by β₂=4.89×10^-3 ps²m^-1 and β₃=1.09×10^-4 ps³m^-1 so that that the fiber is normally dispersive at the wavelength of the input pulses which are being amplified. The nonlinearity coefficient used was γ=2.23×10^-3 W^-1m^-1. The simulations assume a noise-free continuous wave Raman pump with a wavelength of 1455 nm such that near-optimal Raman gain is obtained at 1550 nm. The (wavelength-dependent) linear fiber loss α was also included. These fiber parameters were chosen to correspond to the NZ-DSF which was used in our experiments that are described below in Section 3. In addition, the input pulse initial conditions to the simulations corresponded to the intensity and chirp of our 1550 nm pulse source which were measured experimentally using second harmonic generation (SHG)-FROG.

For input pulses of 0.75 pJ energy and duration (FWHM) of 10 ps, Fig. 1 shows the simulation results obtained for a CW Raman pump power of 0.8 W. The evolution plot in Fig. 1(a) shows that both the amplitude and temporal width of the input pulses increase with propagation, and that the amplified pulse intensity assumes a parabolic profile at the amplifier output after 5.3 km of propagation. The energy of the simulation output pulses under these conditions was 37 pJ, corresponding to an amplifier gain of 17 dB. The parabolic nature of the output pulses is shown more explicitly in Fig. 1(b). Here, in the top figure, the solid lines show the output pulse intensity profile (on a linear scale) as well as the pulse chirp, and these are respectively compared with least-squares parabolic and linear fits which are shown as the circles. The good fits that are obtained illustrate clearly the linearly-chirped parabolic nature of the output pulses from the Raman amplifier under these conditions. The solid line in the bottom figure reproduces the output intensity profile on a logarithmic scale to show how the intensity profile from simulations exhibits the rapid falloff in the wings which is characteristic of a parabolic pulse. Moreover, the comparison between the simulation output profile and the parabolic fit (circles) on a logarithmic scale highlights the close agreement between simulation and the parabolic fit over several orders of magnitude. For comparison, this figure also includes gaussian (long dashes) and sech² (short dashes) fits to illustrate the much improved goodness of fit that is obtained using a parabolic pulse profile.

 

Fig.1. (a) Parabolic pulse evolution over 5.3 km in a NZ-DSF Raman amplifier. (b) The top figure shows the simulation output pulse intensity and chirp (solid lines) together with parabolic and linear fits respectively (circles). The bottom figure plots the simulation output (solid line) and parabolic fit (circles) on a logarithmic scale, and also includes gaussian (long dashes) and sech2 (short dashes) fits to illustrate the comparatively poor fits obtained using these pulse shapes compared to a parabolic pulse.

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3. Experimental results

As discussed in detail in Ref. [5], the observation of parabolic pulse characteristics in a fiber amplifier depends on the judicious choice of both the input pulse and normal GVD fiber parameters that are used. The simulation results above are significant as they indicate that the particular parameter range required for parabolic pulse generation in Raman amplifiers around 1550 nm can be achieved with readily available 1550 nm sources and current-generation Raman amplifier technology.

These simulations have been directly confirmed using the experimental set-up shown in Fig. 2. In these experiments, we used a Raman amplifier based on 5.3 km of NZ-DSF copumped by a CW Keopsys 2 W source at 1455 nm. To ensure that the amplifier characteristics could be studied over the full range of the pump source, connectors and WDM couplers with high power ratings were used. The 1550 nm input pulses were obtained from a Pritel FFL passively modelocked fibre laser with a repetition rate of 22 MHz. Second harmonic generation FROG (SHG-FROG) was then used to characterize the pulses before and after amplification in the NZ-DSF. The fidelity of the FROG measurements were checked using standard techniques based on comparisons of the independently-measured autocorrelation and spectrum with the FROG trace marginals. Pulse retrieval was performed using the generalized projections algorithm.

 

Fig 2. Schematic diagram of experimental set-up.

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We first present experimental results obtained using input pulses of 0.75 pJ energy. SHG-FROG was used to characterize the input pulses before injection in the amplifier, with the pulses typically exhibiting a near transform-limited sech² intensity profile of 10 ps FWHM. As discussed above, it was the retrieved electric field from these measurements that was used as the initial injected pulse in the simulation results shown above in Fig. 1. The output pulses obtained with an amplifier gain of 17 dB were characterized using SHG-FROG with Fig. 3(a) showing the measured FROG trace obtained. The circles in Fig. 3(b) shows the intensity and chirp retrieved from this measurement, where the FROG retrieval error was 0.005 on a 256×256 grid. The retrieved intensity and chirp are compared with those obtained from the simulations (solid lines), and we note good agreement between the intensity profiles over more than 2 orders of magnitude, and both the linearity and the slope of the output pulse chirp. These results confirm the numerical predictions that parabolic pulses generation is possible in a normal-GVD fiber Raman amplifier.

 

Fig. 3. (a) Measured FROG trace of the amplified output pulses obtained with 0.75 pJ input pulses and 17 dB gain. (b) The retrieved intensity and chirp (circles) compared with the expected results from the numerical simulations described in Section 2 (solid lines).

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The results above have been complemented by additional experiments using 10 ps input pulses with a higher input pulse energy of 2.4 pJ. In particular, for this fixed input pulse energy, experiments were carried out to study the dependence of the output pulse characteristics on the Raman amplifier gain by varying the 1455 nm pump power over the range 0.8–1.6 W. The results of these experiments for three particular values of gain are shown in Fig. 4. Here, we plot the measured SHG-FROG traces as well as the associated retrieved intensity profiles and pulse chirp (solid lines). The circles in the figure show the corresponding least-squares parabolic and linear fits to the experimentally-retrieved intensity and chirp. The FROG errors for these data were (a) 0.005 (on a 256×256 grid), (b) 0.008 (on a 512×512 grid) and (c) 0.013 (on a 512×512 grid). We note in this context that retrieval errors for highly chirped pulses must be interpreted in light of the significant non-zero data fraction of the trace, which is defined in terms of the fraction of the data having an intensity greater than 1% of the trace maximum [11]. For the FROG traces in Figs. 4(b) and (c), the non-zero data fraction is around 20% so that these retrieval errors are acceptably low.

 

Fig. 4. Measured FROG traces and retrieved intensity and chirp obtained at the amplifier output for gains of (a) 11.4 dB, (b) 17.7 dB and (c) 20.8 dB. The retrieved intensity and chirp (solid lines) are compared respectively with least-squares parabolic and linear fits (circles), clearly showing the evolution of the amplifier output to a parabolic profile with increasing gain.

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From this figure, the visual inspection of the FROG traces reveals how the increase in the amplifier gain is associated with the simultaneous increase in both the temporal and spectral width of the amplified pulse. Moreover, the comparison between the measured intensity and the parabolic least-squares fit shows that, even though some characteristic features of parabolic pulses are apparent at a low gain of 11.4 dB, the output intensity profiles clearly become increasingly parabolic with increasing gain. This is particularly reflected in the rapidity of the falloff of the pulse wings. Figure 5(a) presents a summary of these experiments, plotting the temporal and spectral width of the amplified pulses as a function of Raman gain. Note that the temporal width is taken here as the separation between the zero crossings of the least squares parabolic pulse fit to the experimental data.

The linearity of the retrieved chirp on the output pulse would be expected to lead to efficient compression using only linear chirp compensation provided by, for example, a simple grating pair compressor. In this regard, an important feature of the FROG characterization is that it permits the numerical computation of the expected compressed pulse characteristics based on the retrieved intensity and phase. For example, for the data in Fig. 4(c), numerical linear chirp compensation yields the compressed pulse characteristics shown in Fig. 5(b), which show negligible chirp variation across the pulse center. Here, the compressed pulse peak power is 172 W and the duration is 1.4 ps which, when compared to the original input pulses from the fiber laser, represents a factor of 7 temporal compression and a factor of 750 increase in peak power.

 

Fig. 5 (a) Pulse duration and spectral width as a function of amplifier gain. (b) Intensity and chirp of compressed pulse obtained after linear chirp compensation of the output parabolic pulse obtained with 20.8 dB gain.

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4. Conclusions

The results in this paper represent the first experimental demonstration of parabolic pulse generation in a Raman amplifier. The use of SHG-FROG for pulse characterization allows the measurement of such characteristic parabolic pulse features such as the linear chirp and the rapid falloff in the wings of the intensity profile. Numerical compression of the measured output pulse predict that near chirp-free compressed pulses can be obtained.

This work has been supported by a Fonds National pour la Science contract ACI-Photonique PH43, a Centre National de la Recherche Scientifique contract Equipe-Projet EPML3 and the Conseil Régional de Bourgogne.