Stability analysis of second order pulsed Raman laser in dispersion managed systems

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1 Stability analysis of second order ulsed Raman laser in disersion managed systems Salih K. Kalyoncu, Shiming Gao, En-Kuang Tien, Yuewang Huang, Dogukan Yildirim, Enver Adas, Stefano Wabnitz and Ozdal Boyraz, EECS Deartment, University of California, Irvine, 9697, USA Diartimento di Elettronica er l Automazione, Università di Brescia, 5 Brescia, Italy EE Deartment, founding member, Istanbul Sehir University, Istanbul, Turkey oboyraz@uci.edu ABSTRACT Wavelength tunable synchronous ulse sources are highly desirable for sectroscoy and otical diagnostics. The common method to generate short ulses in the fiber is the use of nonlinear induced sectral broadening which result in soliton shaing in anomalous disersion regime. However, to generate ultra-short ulses, broadband gain mechanism is also required. In recent years, Raman fiber lasers have retrieved strong interest due to their caability of serving as um sources in gain-flattened amlifiers for otical communication systems. The fixed-wavelength Raman lasers have been widely studied in the last years, but recently, much focus has been on the multi wavelength tunable Raman fiber lasers which generate outut Stokes ulses in a broad wavelength range by so called cascaded stimulated Raman scattering. In this aer we investigate synchronous st and nd order ulsed Raman lasers that can achieve frequency sacing of u to cm - that is highly desired for CARS microscoy. In articular, analytical and numerical analysis of ulsed stability derived for Raman lasers by using disersion managed telecom fibers and umed by 5nm fiber lasers. We show the evolution of the st and nd order Stokes signals at the outut for different um ower and SMF length (determines the net anomalous disersion) combinations. We investigated the stability of disersion managed synchronous Raman laser u to second order both analytically and numerically. The results show that the stable nd order Raman Stokes ulses with.4w to.w eak ower and s to.5s ulse width can be achieved in disersion managed system. Keywords: Second order ulsed Raman laser, disersion managed system, stability analysis, variational analysis.. INTRODUCTION For sectroscoy and otical diagnostics, hase coherent and wavelength tunable synchronous ulse source is a great demand. The nonlinear induced sectral broadening in an anomalous disersion regime is the widely used method to generate short ulses in the fiber by so called soliton comression []. Stimulated Raman scattering with a wide gain bandwidth of ~6THz has an ability to generate femtosecond ulses. Additionally, Raman gain can be generated at any wavelength in a nonlinear medium. U to date, the Fiber Raman Lasers (FRL) have attracted much attention for their wide sread alications such as um sources in gain-flattened amlifiers and tunable sources for characterization of otical devices [-]. In articular, multi wavelength tunable fiber Raman lasers are highly desirable to generate Stokes ulses at a broad wavelength range. In chemistry and biological alications, with higher order cascaded ulsed Raman lasers with a broad frequency sacing which is highly desirable for CARS microscoy are generated [4]. In this aer, we studied the stability analysis of the st and the nd order ulsed Raman lasers with frequency tuning range u to cm -. Particularly, detailed numerical analysis is conducted to investigate the stability regime of the Stokes ulses u to second order in a disersion managed system umed by 5nm fiber laser. Stable Stokes ulses are generated with resect to different um ower and single mode fiber (SMF) length that consists of disersion managed (DM) system to form the cavity. Variational analysis is also derived to analytically analyze the steady state dynamics of the Stokes ulses inside the cavity and to redict the outut ulse arameter by solving ordinary differential equations with a eriodic boundary conditions which seems an alternative to time consuming numerical simulations. The analytical results are comared with simulation results to show good agreement with the derived equations. The results illustrate that the stable nd order Raman Stokes ulses with.4mw to.w eak ower and s to.5s ulse width can be achieved in DM system. Nonlinear Otics and Alications V, edited by Mario Bertolotti, Proc. of SPIE Vol SPIE CCC code: X//$8 doi:.7/ Proc. of SPIE Vol Downloaded From: htt://siedigitallibrary.org/ on /5/ Terms of Use: htt://siedl.org/terms

2 . SIMULATION SETUP and ANALYTICAL EQUATIONS To generate stable ulsed lasers by Raman rocess, disersion management of um, first order and second order ulses is the utmostt imortant arameter. Most of the commercial DSFs and SMFs have high disersion values and sloes, and hence it is not ossible to maintain a reasonable walk-off (overlaing of ulses) through the roagation. To eliminate walk off and erform stability analysis following exerimental setu is being investigated, Figure a. Disersion (s / km-nm ) Disersion Profile of DM System DM for st Stokes DM for nd Stokes L SMF L=8m Figure. Designed set u (a) and the disersion managed system for each Stokes ulses (b) Length (m) 8 System is umed by 5nm Mode Locked Laser with s ulse width. The laser cavity is designed as roosed in [7] that mainly consists of two cascaded sections, the disersion managed (DM) system as the gain mechanismm and the single mode fiber (SMF) for ulse shaing. Disersion management is formed by cascading highly nonlinear fiber (HNLF) ieces which has a nearly zero disersion and disersion sloe in a eriodic configuration. In articular, we restrict our analysis for commercially available fibers from different vendors. The characteristics of the HNLF s and the single mode fiber (SMF) used in the simulations are summarized in Table. Table. The characteristics of the fibers used in the simulations [6]. HNLF (+) sloe HNLF (-) sloe SMF Fiber loss (α).9 db/km.9 db/km. db/km Nonlinearity coefficient (γ).5 (/W-km).5 (/W-km) (/W-km) A eff.7 (μm ).7 (μm ) 5 (μm ) Raman coefficient (g R ).4 (/W-km).4 (/W-km).8(/W-km) Zero disersion ( ) / sloe 5nm /.(s/nm km) Length L=8m (4 section) 5nm / -.(s/nm km) L= =8m (4 section) nm /.76(s/nm km) L=m-mm In order to raise the net disersion to anomalous regime, SMF is cascaded with the disersion managed system to generate stable soliton like ulses. Thus, each round tri can be assumed as a eriod of the modified disersion managed system having net anomalous disersion. Figure b illustrates the disersion rofile for each Stokes order in fibers that satisfies the described design criteria. The overlaing of Stokes ulses at the beginning is satisfied by inserting a stretcher (delay line) in one of the ring cavities to match the um reetition rate. Proc. of SPIE Vol Downloaded From: htt://siedigitallibrary.org/ on /5/ Terms of Use: htt://siedl.org/terms

3 The roagation of Stokes ulses at steady state can be modeled by using variational analysis method which assumes the signal maintains its shae even though its ower, ulse width and chir changes in a continuous manner and remains same eriodically after each round tri. The well known equations describing the evolution of single ulse arameters in variational method [5] can be modified by taking signal to signal interactions into consideration such as Raman amlification, cross hase modulation (XPM) and walk off as a erturbation in the system. In articular, the hysical effects of XPM and the walk off on the chir evolution, um deletion, and gain reduction become more dominant [8]. The modified analytical equations for the ulse arameters, ower (P), ulse width (T), chir(c) and the temoral osition (t) assuming all the ulses having Gaussian shae are: dp d t t s d t ex R + T T T = αp g P P t = T w dp t t P dt t s d t = α P + g P ex P g P ex P R R + T T s T T T dt β C g T P R s = T T s T P () () () (4) dc T T T P s P P + P + P g T R C T T T s T β γ = ( + C ) + d t P s P = w + grt T T s T dp t P dt t d t = α P + g P ex P R + T T T T dt β C g T P R = (8) T T dc β γ T T P = ( + C ) + P + P + P g T R C T T T T d t P = gt R T T where w ( D + D ) ( D + D ) ( ) and ( ) = D( ) d w = D( ) d. Here the subscrits j=,, corresond to um, st Stokes and nd Stokes signals resectively. In addition, α is the fiber loss (5) (6) (7) (9) () Proc. of SPIE Vol Downloaded From: htt://siedigitallibrary.org/ on /5/ Terms of Use: htt://siedl.org/terms

4 (km - ), g R is the Raman gain coefficient (W - km - ), γ is the nonlinearity coefficient (W - km - ), D is the disersion value (s/nm-km), β is the grou velocity disersion arameter (s /km) and w, stands for walk of times between um- st Stokes and st - nd Stokes ulses.. SIMULATION RESULTS and STABILITY ANALYSIS The main objective of the analysis is to determine if the outut Stokes ulses converge to disersion managed solitons at steady state. Slit Ste Fourier Method (SSFM) is being used to solve the couled NLS equations. The st and nd order Stokes signals are obtained at the outut for different um ower and SMF length (determines the net anomalous disersion) combinations. In Figure, the outut Stokes ulses and their eak ower evolution are shown as an examle of transient analysis. The enveloe of the outut Stokes ulses is also characterized by comaring with a Gaussian ulse with the same ulse arameters. Power (W) st Order Stokes Pulse SSFM ulse Gaussian fit Peak Power (W).5.5 Peak Power evolution of st Stokes Power (W) Time (s).4... nd Order Stokes Pulse SSFM ulse Gaussian fit Time (s) Peak Power (W) Normalizied Time ( t / T ) R Peak Power evolution of nd Stokes Normalizied Time ( t / T ) R Figure. The outut Stokes ulses (a) with Gaussian fitting and the evolution of their eak owers (b) at L SMF =5m and P um =.7W. T RT is the round tri time of the cavity. The intra cavity ulse dynamics of the Stokes ulses are also investigated by extracting the evolution of ulse arameters through the single ass. Figure illustrates the evolution of eak ower and the ulse width (FWHM) of the Stokes ulses at steady state. The dynamics of the ulse arameters of the Stokes ulses has similar characteristics. Through the disersion managed section of the cavity where all the Raman interactions occur, the gain and the nonlinear effects such as single hase modulation (SPM) and the cross hase modulation (XPM) are the dominant terms. Thus, the Stokes ulse are amlified and ositively chired though the disersion managed (DM) fibers. Because the disersion length (L D =T / β ) through the individual highly nonlinear fibers (HNLF) is much more than the amlification (L A = (g R A ) - ) and nonlinear lengths (L NL = (γ A ) - ) and due to net zero disersion of the DM system, linear disersive effects become negligible as seen by the ulse width. However, in the second section of the cavity (standard SMF), since the ulses become searate due the walk off, there is no ulse to ulse interaction. The grou velocity disersion (GVD) is the dominant effect though the SMF. Since the ulses becomes highly ositively chired in the DM section of the cavity due to nonlinear hase, the Stokes ulses first comress and then broaden as seen by the dynamic of eak ower and ulse width. Proc. of SPIE Vol Downloaded From: htt://siedigitallibrary.org/ on /5/ Terms of Use: htt://siedl.org/terms

5 st Stokes Peak Power (W) Proagation dstance (m) st Stokes Pulse width (s) Proagation dstance (m) nd Stokes Peak Power (W) Proagation dstance (m) nd Stokes Pulse width (s) Proagation dstance (m) Figure. The intra-cavity dynamics of the steady state ulse arameters, eak ower and ulse width of st order Stokes (a) and nd order Stokes (b) at L SMF =5m and P um =.7W are shown. The analytical and simulation results are comared. st Stokes Peak Power (W) Solid = results Dashed = results Pum Power (W) nd Stokes Peak Power (W) Pulse width (s).5.5 nd Stokes signal st Stokes signal Solid = results Dashed = results Pum Power (W) Figure 4. Steady state eak ower (4a) and ulse width (4b) of the Stokes ulses with resect to um ower are shown. The correctness of analytical derivations is also illustrated in the figure by comaring with the simulation results. Proc. of SPIE Vol Downloaded From: htt://siedigitallibrary.org/ on /5/ Terms of Use: htt://siedl.org/terms

6 ly derived ulse arameters such as eak ower and ulse width of the st and the nd order Stokes at steady state with resect to different um owers are also illustrated in Figure 4 with numerical results. The intra cavity dynamics of these arameters are analyzed by assuming the steady state values found by solving the differential equations at any oint inside the cavity as the initial condition and integrating the equations for one round tri [9]. The analytical and numerical results are shown Figure for secific SMF length and um ower (L SMF =5m and P um =.7W). The agreement with the numerical results obtained by SSFM verifies the validity of our variational analysis. To investigate the stable regime of the outut Stokes ulses, system is deely analyzed by changing both the length of the SMF from m to m and for each case tuning the um eak from.7w to.76w. The results are shown in the following contour lots indicating the eak ower and ulse widths of st and nd order Stokes ulses. Pum Power (W) Pum Power (W) st Stokes Peak Power (W) nd Stokes Peak Power (W) Pum Power (W) Pum Power (W) st Stokes Pulse width (s) nd Stokes Pulse width (s) Figure 5. Contour lots for the eak ower and ulse width of the st and nd order Stokes ulses due to the SMF length and um ower. Figure 5 illustrates the eak owers and ulse widths at which the outut Stokes ulses converge at steady state. At low um owers which are very close to threshold only the st order Stokes can be generated. When the um ower is increased, st order Stokes signal starts to build u in the system and converge. If the final eak ower of the st Stokes signal reaches the threshold for nd Stokes signal, it generates the nd Stokes ulse. At high um owers due to imbalance between strong nonlinearity and weak disersion, st Stokes signal starts to break u and system becomes unstable. The robustness to nonlinearity increases roortionally with net disersion. 4. CONCLUSION We investigated the stability of disersion managed synchronous Raman laser u to second order both analytically and numerically. The variational analysis is conducted to derive the differential equations which exlain the intra cavity dynamics of Stokes ulses in terms of their ulse arameters. The couled differential equations with a eriodic boundary conditions are solved to obtain the outut Stokes ulse arameters at steady state. The analytically and numerically derived ulse arameters of the Stokes ulses are comared to illustrate the correctness of the analytical analysis. The results show that the stable nd order Raman Stokes ulses with.4w to.w eak ower and s to Proc. of SPIE Vol Downloaded From: htt://siedigitallibrary.org/ on /5/ Terms of Use: htt://siedl.org/terms

7 .5s ulse width can be achieved in DM system. Results can be further imroved by otimizing the arameters of the disersion management. 5. ACKNOWLEDGEMENT This work is suorted by DARPA RADER Program and EU grant PIRG7-GA REFERENCES [] C.J.S. Matos, S.V. Poov, and J.R. Taylor, Short-ulse, all-fiber, Raman laser with disersion comensation in a holey fiber Ot. Lett. Vol. 8, No. (). [] G H. Renner, S. Cierullies, M. Krause, Scaling rules for Raman fiber lasers OFC, Atlanta, USA (Paer MF5), (). [] M. Krause, H. Renner, Theory and design of double-cavity Raman fiber lasers J. Lightwave Technology, Vol., No. 8 (5). [4] S. Cierullies, H. Renner, E. Brinkmeyer, Numerical otimization of multi-wavelength and cascaded Raman fiber lasers Ot. Commun. 7(-6),. -8 (). [5] G.P. Agrawal, [Nonlinear Fiber Otics 4 th edition], Academic Press (9). [6] OFS Secialty Photonics Division, htt://ofscatalog.secialtyhotonics.com [7] H.A. Haus and M. Nakazawa, Theory of the fiber Raman soliton laser, J. Ot. Soc. Am. B/Vol. 4, No.5, (May/987). [8] A. Hook, D. Anderson, and M. Lisak, Effects of cross-hase modulation and um deletion on Stokes ulse dynamics in stimulated Raman scattering, J. Ot. Soc. Am. B, Vol. 6, No., (Oct/989). [9] Cristian Antonelli, Jeff Chen, Franz X Kartner, Intracavity ulse dynamics and stability for assively mode-locked lasers, Otics Exress, Vol. 5, No., (May/7). Proc. of SPIE Vol Downloaded From: htt://siedigitallibrary.org/ on /5/ Terms of Use: htt://siedl.org/terms