We present direct real-time experimental measurements and numerical modeling of temporal and statistical properties for the Ytterbium-doped fiber laser with spectral bandwidth of ~2 GHz. The obtained results demonstrate nearly exponential probability density function for intensity fluctuations. A significant decrease below the Gaussian probability has been experimentally observed for intensity fluctuations having value more than 2.5 of average intensity that may be treated as indication of some mode correlations.
© 2013 OSA
Recently, generation dynamics and statistical properties of CW fiber lasers have attracted a great attention. In particular, extreme optical rogue waves  could be generated in a Raman fiber laser (RFL) [2, 3] and lasers of other types [4–6] having mechanisms different from that for rogue wave generation in single-pass systems such as supercontinuum generators, Raman amplifiers etc [7–12]. Well-developed numerical approaches based on nonlinear Schrödinger equation (NLSE) describe well an interplay between numerous nonlinear interacting longitudinal modes in a long fiber resonator  and allow detailed investigation of both the spectral and temporal/statistical properties for CW Raman fiber lasers [14–16]. Since the multimode generation is strongly fluctuating at short time scale, it is more correct to call this regime as “quasi-CW”. Experimental investigation of the RFL generation dynamics is challenging, since its full optical bandwidth (typically several hundreds GHz) is much bigger than real-time bandwidth of oscilloscopes (up to 60 GHz for the newest models). So, direct experimental measurements of the dynamics are usually strongly averaged and do not provide precise information about overall statistical properties. To overpass this, one can use spectral filtering techniques to study temporal and statistical properties within optical bandwidth comparable with measurement bandwidth , or perform indirect measurements .
An Ytterbium-doped fiber laser (YDFL) is another quasi-CW laser source that is even more prevalent device than RFLs due to its high power and efficiency. Typical YDFL spectral width at moderate power is around 0.1 nm , equivalent to 30 GHz optical bandwidth. However, a narrowband YDFL generation has been demonstrated recently  with 10-50 pm width variation in 1-12 W power range. So, the total optical bandwidth of this laser is surely within the real-time bandwidth of oscilloscopes, thus making feasible measurements of real-time intensity dynamics and statistical properties of total laser radiation. Such narrow-band laser is still highly multimode generating simultaneously a number of interacting longitudinal modes and having thus stochastic time dynamics. Therefore it offers for the first time to our knowledge an opportunity to measure generation dynamics and its statistical properties in real time in full optical bandwidth.
In this work we experimentally study temporal and statistical properties of a quasi-CW fiber laser within full optical bandwidth using specially designed ultra-narrowband linearly polarized YDFL. Both temporal and statistical properties have been measured revealing almost thoroughly stochastic nature of radiation at time scale down to 400 ps. In addition, numerical simulations of YDFL’s temporal and statistical properties have been performed, also for the first time to our knowledge. Experimental and numerical data appear to be in good qualitative agreement.
Experimental setup is shown in Fig. 1(a) . The 4-m long YDFL cavity consists of 3 m Yb-doped double-clad polarization maintaining fiber (Liekki YB1200-6/125DC-PM) and 1 m passive fiber. The Yb-ions concentration in active fiber equals to 9.7·1025 m−3. The FBG with 98% reflectivity at the left end of the cavity has ~0.2 nm wide reflection spectrum. At the other end, the laser resonator was formed by the narrowband FBG with ~70 pm spectral width and 25% reflectivity at 1064 nm. To avoid power-induced wavelength shifts of FBG`s spectral profile, the output FBG center wavelength was temperature-controlled to keep maximum power. Pumping of the laser was provided by two laser diodes, operating at 970-976 nm depending on pump current and temperature. More details on design of the narrowband linearly-polarized YDFL can be found in .
The system lases at 1064 nm in quasi-CW regime with the output power in 0.5 - 3.7 W range. Note that at low power near the threshold, a pulsed regime is observed similar to that one described in  with self-pulsations and self-sweeping of the generating modes, see also [21, 22] for details of the mechanism. In the quasi-CW domain the generated power depends almost linearly on the coupled multimode pump power (Fig. 2 ). The generation slope efficiency at optimum is about 60% being typical for YDFLs.
The generated optical spectrum is extremely narrow having full width at half maximum of only 7 pm (~2 GHz) at maximum generated power, Fig. 3 . The spectrum is measured by a scanning Fabry–Perot interferometer with free spectral range of 14 GHz and resolution of 0.2 GHz . Despite narrow spectrum, the laser still should be considered as multimode since it maintains the generation of about 100 different longitudinal modes, as the modes separation in cavity with length L = 4 m is c/2Ln~25 MHz. In the short-cavity YDFL the spectrum broadens nearly linearly with increasing generation power owing to the self-phase modulation at negligible dispersion of the narrowband multimode radiation . Note that in RFLs having much longer cavities (and, therefore, having much bigger number of generation modes), the main broadening mechanism is the combined effect of the dispersion and four-wave mixing between different longitudinal modes that results in square-root broadening law both in stationary regime [23, 24] and during the radiation build-up . Both broadening mechanisms lead to the same shape of the spectral profile described by the hyperbolic secant function [19, 23, 24]. We have checked that a hyperbolic secant function provides also a good fit for our spectral profiles, see Fig. 3(a).
Though the total radiation is quasi-CW, relative dephasing of modes should result in strong intensity fluctuations on sub-nanosecond scale (being inversely proportional to the spectral width). Indeed, the 7-pm wide spectrum corresponds to typical fluctuation time of 500 ps and 2 GHz bandwidth. This allows us to measure the dynamics of intensity fluctuations in real time for full optical bandwidth and to study statistical properties of the YDFL radiation. To do this we use calibrated LeCroy WavePro 725Zi-A real-time oscilloscope with bandwidth of 2.5 GHz and LeCroy OE455 photodetector with bandwidth of 3.5 GHz. In measurements, we pay a special attention on saturation of the photodiode response function as it could change the measured intensity pdf sufficiently for high peak intensities. On the other hand, too low average power results in high background noise that could affects the radiation statistics at low intensity values. The intensity statistics is measured using a large number of real-time traces. Total probability is normalized to unity. Autocorrelation function (ACF) for the measured intensity was calculated by a standard algorithm: ACF equals to unity at zero time offset.
Experimentally measured intensity evolution reveals high contrast intensity fluctuations being as high as several mean intensity value. The typical time trace and corresponding autocorrelation function are shown in Fig. 4 . The measured intensity pdf (probability density function) is shown in Fig. 5 at different powers. Since the sensitivity of the photodiode is limited by the background noise, the measured pdf has a narrow dip at low intensities. However, the data outside this region seems to be correct, as we have carefully checked a linearity of the photodiode response up to maximum peak intensity.
3. Numerical modeling and comparison with experiment
In addition, we numerically simulated the generation dynamics and investigated statistical properties of the generated radiation using a NLSE-based model previously reported to be efficient for modeling of RFLs [3, 26]. To model YDFL generation here, we use exactly the same model as presented in , namely NLSE which includes second-order dispersion, Kerr nonlinearity and gain combined with rate equations for effective 2-level Ytterbium system to properly describe its gain saturation, see  for all details. The ytterbium emission and absorption cross-sections, Fig. 1(b), were provided by the manufacturer. We use iterative procedure similar to that one used for modeling of Brillouin fiber lasers  and Stokes-anti-Stokes generation in RFLs . Power is averaged over numerical time window T that in our simulation is equal to the round-trip time. A short length of the laser cavity allows us to model time dynamics over the entire round trip. Indeed, the roundtrip time is 38.7 ns (TR = 2nL/c, where L = 4m). We have checked that decreasing the time steps, changing the grid size etc. do not affect the results.
First, numerical model describes well the power balance in the system predicting the value of generation power ant its qualitative behavior at increasing pumping, Fig. 2. However, to achieve quantitative agreement some minor experimental factors have to be taken into account. For example, pump laser diode wavelength shifts from 970 to 976 nm while pump power increases because of pump diode heating. As the pump absorption cross-section depends strongly on wavelength, Fig. 1(b), the pump wavelength shift leads to the sufficient changes in generation efficiency. This effect is responsible for the fine variations in the slope of power curve, Fig. 2. Accurate measurement of the pump diode temperature-induced shift and including the measured dependence into a numerical model allows us to achieve a good quantitative agreement between numerical and experimental data and accurately describe the laser output power and efficiency.
The calculated and measured optical spectra are in good qualitative agreement in the all power range of quasi-CW laser operation, Fig. 3. The spectrum is bell-shaped being close to the hyperbolic secant function. Fluctuations in the simulated spectral profile are attributed to amplitude fluctuations of the longitudinal modes . As it was already mentioned, the generated spectrum is broadened with increasing power due to nonlinear effects, namely due to self-phase modulation of the fluctuating intensity induced by modes dephasing . Note that we have not included in our model the effect of spatial hole burning (sufficient in a short linear cavity) that leads to a constant addition to the spectral width . Taking this effect into account a more accurate quantitative prediction of the spectral width value and its dependence on the pump power may be achieved.
Numerically simulated time dynamics of the output radiation demonstrates that the total intensity fluctuate strongly, Figs. 1(a) and 1(b), having typical fluctuation time of 500 ps that is closed to the experimental one. The ACF function in numerical modeling is similar to experimental one having the same temporal width, Fig. 4(c). This additionally proves that our real-time measurements are made for the total optical bandwidth. However, there are some deviations at large time detunings in ACF.
Finally, we compare calculated intensity pdfs with the measured ones in the available intensity domain, see Fig. 5. In numerical simulations, the intensity pdf is almost purely exponential showing just a small increase of events with zero intensity. At the same time, experimentally measured pdfs have a dip near zero intensities which could be an indication of background noise influence and averaging due to the limited electrical bandwidth of the oscilloscope (2.5 GHz) as some part of the laser light goes beyond it at high power (see Fig. 3), both effects preventing to reach zero intensity level. Outside the low-intensity region the experimental curves nearly coincide with the calculated ones up to normalized intensity level of 2.5, above which a sufficient decrease below the Gaussian probability is experimentally observed for both power data corresponding to spectral widths of 1.5 GHz (Fig. 5(a)) and 2 GHz (Fig. 5(b)). At that the deviation is stronger for 2 GHz spectral width being closer to the oscilloscope bandwidth, that may also indicate some influence of the averaging effect on high-intensity fluctuations. Nevertheless, non-exponential intensity pdfs observed at different powers could be treated as an indirect experimental indication that some mode correlations exist in YDFL generation similar to RFLs . Moreover, the suppression of high-intensity rare events in generation dynamics of the narrowband linearly-polarized YDFL has been also confirmed in the experiments on its second harmonics generation .
By presenting Fig. 4 (c) and Fig. 5 that show deviations between experimental results and numerical modeling in autocorrelation function and intensity pdf, we would like also to stress the following point. Despite very good agreement between numerical modeling and experimental results that have been demonstrated at Figs. 2, 3 for the key laser performance characteristics (generation power and spectrum), the fine comparison that involves statistical properties of laser behavior is still an open problem. The deviations in comparison of statistical properties can be attributed either to peculiarities of measurement techniques or to some physical effects that affect generated radiation statistics without affecting key average laser characteristics.
Thus, we present an experimental and numerical investigation of spectral and temporal properties of the narrowband Ytterbium-doped fiber laser. The real time dynamics of quasi-CW radiation is measured and simulated for the first time to our knowledge. Results of numerical modeling and experiments are in good quantitative agreement for the laser generation power and generation spectrum. The measured and calculated statistical properties are in qualitative agreement. Intensity pdf is close to exponential, but reveals decrease of the experimentally measured values below the Gaussian probability at normalized intensities above 2.5 that may be treated as indication of some mode correlations.
Authors would like to acknowledge the support of the European Research Council, SB RAS partner integration project N43, grants of the Russian Ministry of Education and Science, Russian Foundation for Basic Research, Department of General Physics of the Russian Academy of Sciences, Dynasty Foundation, and thank E. V. Podivilov, S. K. Turitsyn for helpful discussions.
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