1. Introduction

It is now a known fact that a high power light pulse with a parabolic intensity profile and a linear frequency chirp can execute nonlinear propagation in a passive fiber with normal dispersion while maintaining its parabolic shape and linear chirp, and therefore by avoiding the dramatic effects of optical wave-breaking [1]. Such a wave-breaking-free pulse, that propagates in a self-similar manner, is called “similariton”. Similaritons can be generated in rare-earth-doped fiber amplifiers [2–5] and in Raman fiber amplifiers [6,7] with uniform normal dispersion. Moreover, similaritons can be generated from a laser resonator [8] or by use of a dispersion-decreasing fiber [9]. It is noteworthy that the characteristics of similaritons created in a constant longitudinal gain amplifier are determined only by the incident pulse energy and amplifier parameters [4,10]. Consequently, any arbitrary input pulse with a constant energy can transform into a unique output similariton [10]. Furthermore, the similariton chirp is independent of the input pulse energy and depends only on the gain and dispersion of the amplifier [4].

In this paper, by use of the remarkable features of similaritons, we achieve temporal synthesis of a chirp-free optical pulse of arbitrary temporal shape. The technique presents the unique feature that the synthesized pulse is independent of the intensity and chirp profiles, peak power, and wavelength of the original input pulse. Our method for achieving the synthesis of optical pulses lies in a two-step procedure. In the first step, the original input pulse is transformed into a similariton by a self-similar propagation through a normal-dispersion Raman amplifier. Next, spectral filtering and chirp compensation of the similariton are simultaneously achieved by means of a linearly chirped fiber Bragg grating (FBG) [11,12] The FBG transfer function determines the specific shape of the synthesized pulse. Our experiments have used different techniques, including cross-correlation and spectrum measurements, and the PICASO technique [13], to directly characterize the intensity and phase profiles of the initial, parabolic and synthesized pulses.

2. Similaritons

Evolution of the slowly varying envelope E(z,t) of an initial pulse during its propagation in a normal-dispersion Raman fiber amplifier can be modeled by the nonlinear Schrödinger equation (NLSE) with a constant longitudinal gain g :

where β₂ and γ are the second-order dispersion and Kerr nonlinear coefficients, respectively. Note that g represents an effective gain coefficient which includes both the distributed gain of the Raman amplifier and the fiber losses. The similariton, which corresponds to the self-similar asymptotic solutions of Eq. (1) in the limit z→∞ is characterized by a parabolic intensity profile and a positive linear chirp [4]. In the spectral domain the similariton solution Ẽ(z,ω) is also a parabolic function, whose analytical expression is given by :

with Ẽ(z,ω) = 0 for ∣ω∣ > ω _p(z). In Eq. (2) ψ_o is a constant, C = g/(3β ₂) and $A_o=\frac{1}{2}{\left(\frac{g{U}_{\mathit{in}}}{\sqrt{\frac{\gamma {\beta }_2}{2}}}\right)}^{\frac{1}{3}}$, where U_in is the initial pulse energy. The pulse spectral width ω _p(z) increases exponentially with propagation distance: ${\omega }_p\left(z\right)=A_o\sqrt{\frac{2\gamma }{{\beta }_2}}\exp \left(\frac{\mathrm{gz}}{3}\right)$. These parabolic pulses have a positive linear chirp given by t_c(ω) = ω/C. Equation (2) shows that the similariton characteristics are determined only by the energy of the initial pulse, and not by its specific shape (duration and functional form). Moreover, the chirp parameter C is independent of the initial pulse energy. The linear positive similariton chirp can be completely compensated by means of a linearly chirped FBG providing large amounts of linear negative dispersion. Chirp-free pulses of arbitrary temporal shapes and durations can then be generated by spectral filtering of the dechirped similariton. As dechirping and spectral filtering are both linear effects, they can be performed simultaneously by means of a properly designed linearly chirped FBG.

3 . Experimental setup

The experimental setup shown in Fig. 1 was used to perform the pulse synthesis from generation of optical similaritons. The initial pulses were produced by a picosecond fiber laser (Pritel PFL) that offered the possibility of varying the pulse duration, wavelength and profile. The PFL output was subsequently split into two beams with different intensities. The beam with the weakest intensity was used as the initial pulses (to be synthesized), whereas the beam with the strongest intensity was used as reference pulses for cross-correlation measurements as explained below. The initial pulses were transformed into similaritons by means of a normal-dispersion Raman amplifier made up of a 7.3-km standard non-zero dispersion-shifted fiber (NZ-DSF), with the following parameters at 1550 nm: dispersion (β₂ = 2.0 × 10^-3 ps² m^-1, dispersion slope of 0.081 ps/nm/km, nonlinear coefficient γ = 2.0 × 10^-3 W^-1 m^-1 and losses of 0.244 dB/km. The fiber was pumped via a wavelength-division-multiplexing (WDM) coupler with a launched power of 1.25 W from a cw Raman laser (Keopsys) operating at 1455 nm. The effective Raman amplification gain was g = 5.47 × 10^-4 m^-1 (2.38 dB/km). The similaritons generated at the amplifier output were coupled into a linearly chirped FBG (Teraxion), which is connected to an optical circulator for chirp compensation and spectral filtering in a reflection configuration. The FBG had a dispersion of 6.1 ps/nm and its frequency transfer function was a Gaussian with full-width at half-maximum (FWHM) bandwidth of 2.13 nm centered around 1550 nm.

 

Fig. 1. Pulse synthesis and characterization experimental setup.

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The characteristics of the initial, parabolic and synthesized pulses were monitored both in the frequency and in the time domain, by using an optical spectrum analyzer (OSA) and a based second-harmonic generation intensity correlator, respectively. Full characterization of the intensity and phase of the initial and synthesized pulses was achieved by means of the PICASO (phase and intensity from correlation and spectrum only) technique [13]. On the other hand, in order to precisely measure the intensity and phase of the similaritons, we have used a cross-correlation between the similariton and a high-power reference pulse (see Fig. 1), which was completely characterized in intensity and phase from frequency-resolved optical gating (FROG) measurements [14]. Note that we have checked that the FROG characterization of the reference pulse, which is a replica of the initial pulse with higher energy, leads to intensity and phase profiles identical to those determined from PICASO retrieval of the initial pulse, confirming the reliability of the PICASO technique. Let us also note that the sensitivity of our FROG setup was not sufficient to precisely characterize the parabolic similariton and synthesized pulses.

 

Fig. 2. (a) Intensity from PICASO retrievals of the initial pulse. (b) Intensity and chirp profiles from cross-correlation measurements of the similariton pulse (circles). Parabolic and linear fits (solid lines) of respectively intensity and chirp profiles. (c) Intensity and phase profiles from PICASO retrievals of the synthesized pulse (circles). Gaussian fit (solid line).

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4. Experimental results and discussion

We have considered amplification of a transform-limited initial pulse with an energy of U_in = 4.95 pJ. Figure 2(a) shows the intensity profile of this initial pulse retrieved from the PICASO technique. The pulse duration and spectral bandwidth (FWHM) are 5.8 ps and 53 GHz, respectively. Figure 2(b) shows the intensity and chirp profiles of the amplified pulse retrieved from cross-correlation with the reference pulse and direct spectrum measurement. The experimental output pulse characteristics (circles) are compared with least-squares parabolic and linear fits of the intensity and chirp profiles, respectively (solid lines). The good agreement between experimental results and corresponding fits illustrates clearly the linearly-chirped parabolic nature of the similariton at the Raman amplifier output. Let us note that linear low-intensity wings appear on the experimental pulse intensity profile. These low-intensity wings reveal that linearly chirped parabolic pulses are approximate self-similar solutions of the NLSE in the high-intensity limit [15]. The chirp slope at the amplifier output was found to be C = 12.6 GHz/ps, whereas the corresponding value obtained from the theoretical asymptotic model is C = 14.5 GHz/ps. The similariton spectral bandwidth (FWHM) was found from measurements to be 0.64 THz, which is also in good agreement with the analytical value ω_p/√2 = 0.72 THz. The intensity and phase profiles for the synthesized pulse retrieved from the PICASO technique are shown in Fig. 2(c). The phase across the pulse is constant, which implies that the synthesized pulse is transform-limited. The experimental intensity profile (circles) is compared with a Gaussian fit (solid line). The pulse duration and spectral bandwidth (FWHM) are found to be 1.95 ps and 0.25 THz, respectively, which leads to a time bandwidth product of 0.49 that is quite close to the value of 0.44 corresponding to a transform-limited Gaussian pulse.

 

Fig. 3. Intensity from PICASO retrievals of three different initial pulses with identical energy of 4.95 pJ (a) and corresponding synthesized pulses (b). (c) Experimental evolution of the temporal width of the synthesized pulse autocorrelation as a function of the input pulse energy.

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We have also used PICASO retrievals to investigate the invariance of the synthesized pulse with respect to the initial pulse characteristics at constant input energy. Figure 3(a) shows the intensity of three initial pulses with quite different shapes but having the same energy U_in = 4.95 pJ. More precisely, the input pulse parameters are the following: FWHM temporal widths of 8, 5.8 and 2.5 ps, and peak powers of 0.47, 0.73 and 1.25 W (crosses, circles and solid line, respectively). Figure 3(b) shows the intensity of the synthesized pulses associated to the three initial pulses. As can be seen in Fig. 3(b), all the input pulses lead essentially to the same output pulse characteristics. In another series of experiments we have studied the dependence of the synthesized pulse with respect to the input pulse energy at a given initial pulse shape. Figure 3(c) exhibits the evolution of the FWHM temporal width of the synthesized pulse autocorrelation as a function of input pulse energy. Remarkably we can clearly observe that variation of input energy over a wide range (1.5 to 30 pJ) leads to only very slight variations of the autocorrelation width (from 2.6 to 3 ps), confirming that in the case of a constant gain profile the similariton chirp is independent of the input energy. The slight continuous increase of the pulse width with input energy is due to depletion of the Raman pump wave, that induces variation of the effective gain and modification of the similariton chirp, which is therefore not completely compensated by the FBG. On the other hand, a careful inspection of Fig. 3(c) reveals a small jump in the variation of the synthesized pulse width at very small input energy. We attribute this behavior to the fact that the output amplified pulse has not yet reach the self-similar asymptotic solution, and therefore, the amplified pulse bandwidth is not sufficiently large in comparison with the FBG bandwidth. Finally, we have observed (but not shown here) that for a fixed energy and constant profile, initial pulses with different wavelengths (ranging from 1547 to 1553 nm) all evolve to a unique synthesized pulse at 1550 nm.

5. Conclusions

In this paper we have presented a novel technique for pulse synthesis based on similariton generation, which exhibits remarkable properties: the synthesized pulse characteristics are independent of the initial pulse duration, profile and energy and present a certain tolerance with respect to the initial pulse wavelength. We anticipate that pulse synthesis by use of similaritons will find wide application in many areas of optics such as ultrafast optics, optical communication and optical information processing.