In present paper correlations between different parts of spectrum of a fiber laser with randomly distributed feedback (RDFL) were experimentally measured directly. Implemented statistical analysis demonstrate weak cross-correlations between different lines in generation spectrum. These correlations were vizualized by plotting 2-D probability density functions. Linear correlation coefficient (Pearson coefficient) was calculated for each pair of spectrum lines

Churkin, Dmitry V.

Gorbunov, Oleg

Sugavanam, Srikanth

Vatnik, Ilya D.

English

Oleg Gorbunov

Dmitry V. Churkin

Srikanth Sugavanam

Ilya D. Vatnik

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Revealing spectral cross-correlations in radiation of multiwavelength fiber laser with randomly distributed feedback

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PROCEEDINGS OF SPIESPIEDigitalLibrary.org/conference-proceedings-of-spieRevealing spectral cross-correlationsin radiation of multiwavelength fiberlaser with randomly distributedfeedbackI.  Vatnik, O.  Gorbunov, S.  Sugavanam, D.  ChurkinI.  Vatnik, O.  Gorbunov, S.  Sugavanam, D.  Churkin, "Revealing spectralcross-correlations in radiation of multiwavelength fiber laser with randomlydistributed feedback," Proc. SPIE 10683, Fiber Lasers and Glass Photonics:Materials through Applications, 1068335 (17 May 2018); doi:10.1117/12.2307311Event: SPIE Photonics Europe, 2018, Strasbourg, FranceDownloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 5/29/2018 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use  Revealing spectral cross-correlations in radiation of multiwavelength fiber laser with randomly distributed feedback  I. Vatnika, O. Gorbunova,b, S. Sugavanamc, D. Churkina aNovosibirsk State University, Pirogova str. 2, Novosibirsk 630090, Russia; bInstitute of Automation and Electrometry SB RAS, Academician Koptyug ave. 1, Novosibirsk 630090, Russia; cAston Institute of Photonic Technologies, Aston University, Birmingham, B4 7ET, United Kingdom; ABSTRACT In present paper correlations between different parts of spectrum of a fiber laser with randomly distributed feedback (RDFL) were experimentally measured directly. Implemented statistical analysis demonstrate weak cross-correlations between different lines in generation spectrum. These correlations were vizualized by plotting 2-D probability density functions. Linear correlation coefficient (Pearson coefficient) was calculated for each pair of spectrum lines.  Keywords: laser radiation statistics, fiber laser with randomly distributed feedback 1. INTRODUCTION Statistics of fiber lasers' radiation is continuously studied during the last decade. General approach, used for description of statistical properties of conventional single-mode lasers cannot be directly used for multimode fiber radiation [1], generated in fiber lasers. Nonlinear interaction of numerous modes lead to complicated temporal dynamics, for which statistical description is only possible. Non-trivial properties of radiation dynamics in fiber lasers with fiber Bragg gratings was observed in numerical simulations [2] and a number of experiments in different fiber laser types [3, 4]. It was shown, that probability density function (PDF) has non-exponential shape, what indicates presence of spectral correlations. More general approach was implemented in [5] to predict general properties of radiation statistics of turbulent radiation of a fiber laser. Fiber lasers with randomly distributed feedback (RDFLs) have additional differ as there are no separate spectral modes due to absence of localized mirrors, so their radiation statistical properties are even more mysterious. Indirect observations [6] show that PDF of generated radiation has non-exponential shape, so correlations between different spectral components should exist. In present paper an attempt to measure these correlations directly was performed. Weak correlations were observed and analyzed. 2. SIMULTANEOUS MEASUREMENT OF DIFFERENT SPECTRAL LINES IN RDFL As one of key problems for experimental investigation of statistical propertied of fiber laser radiation is fast decay of spectral power density during shift from spectrum center to spectrum edge a scheme of RDFL with Lyot filer (LF) was chosen. In this single armed, forward pumped configuration a LF is placed at one end of long fiber span, what results in a flat modulated spectrum, convenient for direct measurement of correlations. For more details see [7]. Experimental setup is schematically shown on fig. 1a. Fiber Lasers and Glass Photonics:  Materials through Applications, edited byStefano Taccheo, Jacob I. Mackenzie, Maurizio Ferrari, Proc. of SPIE Vol. 106831068335 · © 2018 SPIE · CCC code: 0277-786X/18/$18 · doi: 10.1117/12.2307311Proc. of SPIE Vol. 10683  1068335-1Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 5/29/2018 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use  Figure 1.The laser cavfiber laser op(WDM) as thin transmissimeasured at tWDM and twLasing with pthe thresholdconsist of sincomparable aPDF of singlposition in geand in dip betduring formaseparate linesFigure 2.RFLLyotfilter(a) a) Experimentity is formederating at 14e pump sourcon, a 50:50 che far end of o stage isolatower of hund are shown ogle lines. Wmplitude – die lines were neration specween major ption of the e cannot be tre Technique of WDM40 kmal setup b) Cha using a 40 k55 nm is splie. The Lyot foupler basedthe 40 km fibor (return lossreds mW is on fig. 1b; chidth of a sucfference is lesexperimentalltrum: lines ineaks PDF hantire spectruated as statistsimultaneous m SMF28WDracteristic genem span of Cce-coupled toilter (LF) is s fiber mirrorer span. Undes > 60 dB) rbserved at puaracteristic sph line is equs than 6 dB, y measured i center of twve significantm, which haically indepeneasurement of M X:xISOration spectrumorning SMF  the ultralonpliced to the R was used toepleted pumpespectively. mp higher thectrum widthal by order o– so all lines hn [8]. It was o major peakly longer tailss strongly nodent and shotwo lines' temp of RDFL with28 standard tg fiber span uFL-FL confi reflect the  power and san 1.1 W (see is 13 nm. Df magnitudeave enough shown that Ps have PDF c. Performed hn-Gaussian uld correlate woral dynamics LF elecommunicsing a waveguration nearfiltered radiapurious back  details in [6]ue to LF inf to oscilloscoenergy to be rDF shape oflose to Gaussalf-qualitativstatistics, havith each oth. ations fiber. Alength divisio the pump ention back. Lareflections ar). Laser spectluence it is ppe bandwidthegistered by p single lines ian while at se analysis of e allowed toer.    CW Ramann multiplexerd. As it worksser output ise removed byra well aboveeriodical and. Lines havehotodetectordepend on itspectrum edgetheir interplay assume tha        .    t Proc. of SPIE Vol. 10683  1068335-2Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 5/29/2018 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use  Process of simultaneous measurements of different lines of generated laser spectrum is schematically illustrated at fig. 2. A coupler at laser output divides radiation into two channels. Each of them is passed through tunable spectral filter to cut desirable spectral line, which then are detected by two fast (50 GHz) photodetectors, transmitting electrical signals to two channels of oscilloscope operating with bandwidth 16 GHz. In experiments position and width of one spectral filter was fixed at the edge of spectrum, while second was swept through other spectral lines of left major peak. Length of measured time dynamics was 109 points with discretization step 12.5 ps. Totally 7 pairs of lines were measured.  Figure 3. Characteristic temporal dynamics at two channels Fig. 3 illustrates two simultaneous temporal dynamics from two oscilloscope channels. It is hard to notice some correlation visually, what indicates that it can be expected to appear weak, if any. But calculation of cross-correlation function (CCF)  clearly shows cross-correlation peak between lines, however its amplitude is small. CCF for the closest pair of lines is shown at fig. 4.   ( ) ( ) ( )21211IIdttItITCCF∫ −= ττProc. of SPIE Vol. 10683  1068335-3Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 5/29/2018 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use   Figure 4. Cross-correlation function for nearest lines (1555,82 nm and 1556.27 nm). Time shift between channels possibly arises due to different fiber length used after splitting signal at laser output. Well-pronounced peak indicates presence of correlations between two channels. Peak is significantly broadened, reason is not clear by now. 3. CORRELATION ANALYSIS General characterization of two correlated traces can be provided with two-dimensional probability density function (2d PDF), which characterizes probability of simultaneous observation of intensities I1 and I2 in channels 1 and 2 respectively. If correlation between traces is nonzero than 2d PDF cannot be factorized to multiplication of PDFs of both traces:  To approximate 2d PDF a two-dimensional histogram was built: all range of intensity fluctuation was divided into 1024 bins for both channels, and all simultaneously measured intensity pairs were distributed over this 2d matrix. As our buffer was 109 points for 1024·1024 = 1.048576·106 bins approximation was expected to be accurate. 2d PDF for the nearest pair of lines is shown on fig. 5  ( ) ( ) ( )221121, IPIPIIP ⋅≠Proc. of SPIE Vol. 10683  1068335-4Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 5/29/2018 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use   Figure 5. Two-dimensional probability for pair of closest lines (1555,82 nm and 1556.27 nm). Visually noticeable asymmetry relevant to line I1 = I2 is the result of correlation of traces and arises from the fact that 2d PDF cannot be factorized into production of one-dimensional probabilities. Knowledge of 2d PDF allows calculation of distribution two-dimensional moments and cumulants:   Correlation between two traces is often characterized by linear correlation coefficient (Pearson coefficient), varying between 1 and -1:  Pearson coefficient dependence on wavelength of second channel line is shown on fig. 6.  ( )∫= 212121 , dIdIIIPII knnkα 2010202210202001101111 ,, αακαακααακ −=−=−=022011κκκ=rProc. of SPIE Vol. 10683  1068335-5Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 5/29/2018 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use   Figure 6. Pearson coefficient as function of spectral detuning. Pearson coefficient is quite small – less than 0.1 – and decays with increase of separation between lines in spectrum. Decrease of cross-correlation for distant lines looks consistent, but note that even for highest spectral distance in our experiment –  more than 6 nm – coefficient is the same order of magnitude as for the closest pair (spectral distance 0.9 nm). Thus we can conclude that spectral correlations exist in radiation of RDFL with LF and can be detected by direct observation. Small qualitative values of correlation can be probably explained by big enough spectral width of lines; further search for cross-correlations is more convenient to perform with narrow spectral lines. 4. CONCLUSION A direct simultaneous measurement of long time traces (109 points) representing two lines of spectra of fiber laser with randomly distributed feedback with Lyot filter was performed. Statistical correlation of the edge line and any other lines in spectrum, measured in experiment, are clearly visualized by peaks of cross-correlation function. Two-dimensional probability and conditional probability functions were calculated as histograms to visualize correlations. It was shown that Pearson correlation coefficient is quite small by value – about 0.03-0.04 – but is non-zero for all, even distat lines in spectrum.  5. ACKNOWLEDGMENTS This work is supported by the Russian Foundation for Basic Research (16-32-00772). 6. REFERENCES [1] R. Loudon, Quantum Theory of Light (OUP Oxford, 2000) [2] D. V. Churkin, and S. V. Smirnov, “Numerical modelling of spectral, temporal and statistical properties of Raman fiber lasers,” Opt. Commun. 285, 2154–2160 (2012). [3] S. Randoux, and P. Suret, "Experimental evidence of extreme value statistics in Raman fiber lasers." Opt. Lett. 37, 500-502 (2012). [4] A. E. Bednyakova, O. A. Gorbunov, M. O. Politko, S. I. Kablukov, S. V. Smirnov, D. V. Churkin, M. P. Fedoruk, and S. A. Babin, “Generation dynamics of the narrowband  Yb-doped fiber laser,” Opt. Express 21, 8177-8182 (2013). [5] P. Walczak, S. Randoux, and P. Suret, “Statistics of a turbulent raman fiber laser,” Opt. Lett. 40, 3101–3104 (2015). Proc. of SPIE Vol. 10683  1068335-6Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 5/29/2018 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use  [6] O. A. Gorbunov, S. Sugavanam, and D. V. Churkin, “Intensity dynamics and statistical properties of random distributed feedback fiber laser,” Opt. Lett. 40, 1783–1786 (2015) [7] S. Sugavanam, Z. Yan, V. Kamynin, A. S. Kurkov, L. Zhang, and D. V. Churkin, “Multiwavelength generation in a random distributed feedback fiber laser using an all fiber lyot filter,” Opt. Express 22, 2839–2844 (2014). [8] Gorbunov O.A., Sugavanam S., Vatnik I.D., Churkin D.V., "Statistical properties of radiation of multiwavelength random DFB fiber laser", Optics Express, 24, 16, 19417-19423 (2016) Proc. of SPIE Vol. 10683  1068335-7Downloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 5/29/2018 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use

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Revealing spectral cross-correlations in radiation of multiwavelength fiber laser with randomly distributed feedback

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