April 15, 2015 / Vol. 40, No. 8 / OPTICS LETTERS

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Intensity dynamics and statistical properties of random distributed feedback fiber laser Oleg A. Gorbunov,1,2,* Srikanth Sugavanam,3 and Dmitry V. Churkin1,2,3 1

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Institute of Automation and Electrometry, SB RAS, Novosibirsk 630090, Russia 2 Novosibirsk State University, Novosibirsk 630090, Russia

Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, UK *Corresponding author: [email protected] Received January 21, 2015; accepted March 1, 2015; posted March 18, 2015 (Doc. ID 232967); published April 9, 2015

We present first experimental investigation of fast-intensity dynamics of random distributed feedback (DFB) fiber lasers. We found that the laser dynamics are stochastic on a short time scale and exhibit pronounced fluctuations including generation of extreme events. We also experimentally characterize statistical properties of radiation of random DFB fiber lasers. We found that statistical properties deviate from Gaussian and depend on the pump power. © 2015 Optical Society of America OCIS codes: (140.3490) Lasers, distributed-feedback; (140.3430) Laser theory; (030.6600) Statistical optics. http://dx.doi.org/10.1364/OL.40.001783

Fiber lasers with random distributed feedback (DFB) owing to Rayleigh scattering are versatile recently emerged sources of radiation [1,2] with a number of applications in sensing and telecommunications [3–6]. Random DFB fiber lasers of various generation properties have been demonstrated including tunable [7], multi-wavelength [8,9], narrow-band [10], cascaded lasers [11,12], diodepumped lasers [13], and lasers with polarized pumping [14]. The generation efficiency of a random DFB fiber laser can be superior to those of lasers with conventional cavity design [15–17] due to the specific power distribution along the cavity [18]. Very recently, the processes of how the generation spectrum of random fiber laser is formed have been described [19]. To do that, the new type of wave kinetic systems—active cyclic wave kinetic systems—are considered, and nonlinear kinetic theory of laser’s spectrum formation is developed. In wave kinetics, it is usually assumed that the radiation has purely Gaussian statistics. This allows it to split and to average correlations functions. To date, there were no any experimental measurements of statistical properties of random DFB fiber lasers. Moreover, the temporal properties of such lasers have not been investigated so far as well. In general, temporal and statistical properties of quasiCW fiber lasers to which class the random DFB fiber laser belongs are of practical and fundamental importance and interest. For instance, generation of extreme and rogue events in highly stochastic generation of quasiCW fiber lasers has been reported [20–22]. More generally, quasi-CW lasers allow for addressing classical questions of wave turbulence in versatile fiber optic experiments, see, for example, recent works [23–26] and review [27]. In the present Letter, we present for a first time experimental measurements of temporal and statistical properties of a random distributed-feedback fiber laser. For experimental studies of temporal and statistical properties of the random DFB fiber laser, we used forward-pumped configuration [2], Fig. 1(a). The cavity is formed using a 40-km span of Corning SMF 28 fiber. A continuous wave Raman fiber laser operating at 1455 nm is used as the pump source, which is splice-coupled to 0146-9592/15/081783-04$15.00/0

the fiber span using a WDM. In our experiments, we used narrow-band fiber Bragg grating (FBG) to achieve a narrow-band generation of a random fiber laser [10]. The FBG used has a Gaussian profile, with central wavelength at 1550.5 nm, FWHM of 50 pm, and reflectivity ∼98%. It is spliced to the random fiber laser configuration near the pump end, so half-open configuration is implemented [2]. Output from the laser is measured at the far end of the 40-km fiber span. A WDM is used at the output to remove any undepleted pump power. A two-stage isolator (return losses >60 dB) is also used to isolate the laser from back reflections arising from the measurement apparatus. A 33-GHz real time oscilloscope together with a 50-GHz bandwidth photodetector is used for acquisition of the intensity dynamics of the laser. The typical generation spectrum of the laser has width around 1 nm, Fig. 1(b). Despite us having used narrow FBG in the cavity, the spectrum becomes sufficiently broader than FBG bandwidth because of nonlinear kinetic spectral broadening [19]: the spectrum width increases from 0.05 nm near the generation threshold up to 1.4 nm at high power. This implies that the optical bandwidth of the generated radiation is higher than the electrical bandwidth of measurement setup almost in all power range. Strong fluctuations are observed in generated intensity at sub-nanosecond time scale, Fig. 2. The observed stochastic intensity dynamics are similar to stochastic turbulent-like intensity dynamics observed in other types of quasi-CW fiber lasers with conventional cavities based

Fig. 1. (a) Experimental setup. (b) Spectrum width dependence over pump power. At insert—generation spectrum at 3 W. © 2015 Optical Society of America

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on point-based mirrors [20–22]. The higher the power, the shorter a typical time of intensity fluctuations. At the same time, the probability of intense events seems to be higher at lower power. Note that it has been numerically reported that a random fiber laser should have stochastic intensity dynamics [28], but there were no any experimental confirmations of that. Origin of stochastic nature of intensity dynamics in quasi-CW Raman fiber lasers with conventional (pointbased) cavities is multiple four-wave mixing processes between different distinct longitudinal modes, which could be described in terms of weak wave turbulence [29]. In lasers with random DFB, there are no stationary cavity modes, so generation spectrum can be treated as continuous spectrum. Nonlinear interactions between different frequency components in such spectrum lead as well to the irregular intensity dynamics and could be described within nonlinear kinetic theory of laser’s spectrum [19] within an assumption that the overall statistics of fluctuations is Gaussian for amplitudes. Further, we directly measure the intensity statistics of a random fiber laser at different powers. We do measurements of extremely long time traces (109 samples in each trace) and calculate from the traces intensity probability density functions (PDFs) at each power level, Fig. 3(a). In central area, PDFs for all powers overlay. The lower the power, the more probable events of extreme intensities. We measured events with intensities up to 20 times higher than average value. At higher powers, intensity PDFs’ far wings are almost exponential. Note that despite intensity, PDFs’ wings can be well approximated by exponential function, and a decay factor (or slope in logarithmic scale) is not equal to universal slope of minus 0

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Fig. 3. Intensity PDFs measured at (a) different pump powers: 1.6 W (black line), 2.2 W (gray line), and 3 W (light gray line), and (b) different oscilloscope bandwidths: 30 GHz (black line), 10 GHz (gray line), and 3 GHz (light gray line) at pump power 1.6 W.

1, which should be observed for completely Gaussian statistics [30]. PDF narrowing in similar conditions were observed experimentally and numerically in [31,32] performs analytical calculation of PDF of a Raman Stokes wave. In general, intensity PDFs should provide exact information on statistical properties of the generated radiation, but this is not usually the case for radiation of quasi-CW lasers. The reasons for that is broad optical spectrum of optical bandwidth being larger than electrical bandwidth of measurement setup. Indeed, there is an area of intensity PDF at “negative” normalized intensities, Fig. 3(a). This is an indication that the laser radiation has frequencies higher than that could be measured by used oscilloscope (i.e., the optical bandwidth of the measured signal is higher than the electrical bandwidth of the measurement’s setup), and instrumental noise could be important [33]. These technical limitations could affect the conclusion on the statistical properties of the radiation, as the position of the most probable intensity and, potentially, the slope of the PDF could be altered. However, despite these limitations, one could still conclude whether the radiation of the studied random laser has Gaussian statistics or not. Indeed, in the case of bandwidth-limited measurements, i.e., when measurement’s bandwidth is lower than real optical bandwidth, one should make a series of measurements by decreasing the ratio between electrical measurement bandwidth, E, and optical bandwidth of the signal, S, i.e., ratio E∕S should be changed [33]. The plot of the intensity PDFs over E∕S ratio will reveal whether the statistics is Gaussian or not, as the Gaussian statistics has universal behavior of its slope in such representations. Measurements of the intensity PDF presented at Fig. 3(a) are already measurements at different E∕S ratios, as the spectrum width increases with pump power, so E∕S ratio decreases with power if the measurement bandwidth is fixed. However, the statistical properties of the radiation could be different at different power levels. So we implement another procedure to change E∕S ratio, namely we fix the generation power and directly decrease the electrical measurement bandwidth using settings of the oscilloscope, Fig. 3(b). We repeat such measurements at different powers. Note that strong changes in the shape of intensity PDFs are observed while E∕S ratio is changed. Further we approximate the intensity PDF wing at large intensities by exponential function and plot its slope (in logarithmic scale), i.e., decay factor versus E∕S ratio, Fig. 4(a). We also plot the dependence of the intensity PDF’s slope calculated within the model of independent frequency components having purely Gaussian statistics of amplitudes, i.e., for the radiation having completely exponential statistics for total intensity, see [33] for details. We found that the radiation statistics of studied random DFB fiber laser deviates from Gaussian, as dependence of PDF slope does not coincide with those predicted for Gaussian statistics both at low (1.6 W) and high (3 W) power. At high powers, the intensity PDF of the random DFB fiber laser is narrower than predicted for Gaussian statistics as the PDF’s slope has lower levels than predicted. At low power, the intensity PDF is broader than for Gaussian statistics.

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Note that there is a different behavior of measured PDF slope over E∕S ratio at different power levels: compare squares and triangles at Fig. 4(a). This means that statistical properties of the radiation are different at different power levels. Our measurements of PDF slope for different pump power at fixed electrical bandwidth E provides additional information, Fig. 4(a), circles. Both data sets (at varied power and varied electrical bandwidth) meets at E∕S  0.5 (that is equivalent to 1.6 W at full electrical bandwidth) and at E∕S  0.25 (that is equivalent to 3 W at full electrical bandwidth), so one can track how the statistical properties are gradually changed over power increase: namely the distribution becomes gradually narrower with pump power increase. Note that it could be difficult to approximate the intensity PDF by exponential function, as measured probability density function could have nonexponential wing at large intensities. To characterize radiation in a more precise way, we calculate the probability of intensity to have a value larger than some set value, chosen here to be 4 times larger than average intensity as an example, PI > 4hIi, Fig. 4(b). Such probability does not depend on whether the measured intensity PDF approximates well by exponential function or not, so more precise information about statistical properties could be extracted. Again, for a radiation having Gaussian statistics, there is a universal behavior of probability of extreme events with intensity higher than 4 average values over E∕S ratio. Figure 4(b) directly shows that the probability of extreme events is higher at low powers than at high powers [compare squares and triangles on Fig. 4(b)]. At the same time, the probability of extreme events monotonously decreases while pump power increases, Fig. 4(b), circles. There is a good correspondence between probability measured while power varied at fixed bandwidth and bandwidth varied at fixed power. Finally we define from the experimentally measured intensity dynamics the intensity autocorrelation function (ACF) as Kτ  hIt · It  τit ∕hIt2 it , see insert on Fig. 4(c). It’s background level at large time detunings also indicate in a clear way if the statistic is Gaussian or not. For a completely Gaussian statistics, the ACF background level is equal to 0.5 [15]. Deviation of measured values of ACF background level from 0.5 gives

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