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Communications in Physics, Vol. 23, No. 2 (2013), pp. 171-177 INFLUENCE OF THE FREQUENCY-CHIRP ON PULSE IN PASSIVELY MODE-LOCKING OPTICAL FIBER RING LASER BUI XUAN KIEN Electric Power University 235 Hoang Quoc Viet Str, Tu Liem Dist., Hanoi, Vietnam Email: kiendhv2000@yahoo.com TRAN HAI HUNG Nghe An Pedagogical College TRINH DINH CHIEN Hanoi University of Science, Vietnam National University, Hanoi 334 Nguyen Trai Str., Thanh Xuan Dist., Hanoi, Vietnam Received 05 March 2013; revised manuscript received 31 March 2013 Abstract. We consider a model of a passively mode-locking fiber ring laser bult using a saturable absorber and a chirped fiber Bragg grating to balance dispersion and nonlinearity. The evolution of the slowly envelope of the optical field in a loop fiber subject to dispersion , Kerr nonlinearity, frequency- chirp and nonlinear absorption is given by the generalized complex Ginzburg-Landau equation. The influence of the frequency-chirp on the pulse is simulated and discussed, and the stationary conditions concerning the chirp parameter are found out for our laser. I. INTRODUCTION In recent years there has been an extensive growth in research activities focused on developing ultrashort (< 10 ps) pulse sources based on rare-earth-doped-fiber [1–4]. The short pulse fiber-lasers have been passively mode-locked by using the chirp-fiberBragg-grating (C-FBG) to control dispersion, the polarization controller, the semiconductor saturable absorber mirror (SESAM) or the multiple quantum well saturable absorber (MQW-AS) [5]. The theory of mode-locking is based on the master equation derived under the condition that nonlinear changes to the intra-cavity pulse must be small per round-trip and fast saturable absorber action [6]. The carrier dynamics is adiabatically eliminated due to the sub-picosecond material response time and pulse widths of order ten picosecond. To obtain analytic results, the form of a complex Ginzburg-Landau equation is used, where the nonlinear contributions of the saturable absorber are approximated by an expansion of the field amplitude to fifth order [7]. The simulated results in previous works [5,7], are focused on the generation solitonic shape of the output pulse only, so the initial pulse envelope has been chosen by using the hyperbolic secant solution with no chirp. 172 BUI XUAN KIEN, TRAN HAI HUNG, AND TRINH DINH CHIEN In this paper, we present numerical studies of a ring fiber laser passively modelocked by MQW saturable absorber with chirped fiber Bragg grating. The simulation is used for the initial Gaussian pulse with frequency- chirp. II. MODELING AND SIMULATION Our model of passively mode-locked ring fiber laser is illustrated in Fig. 1. Fig. 1. Schematic of the passively mode-locked ring fiber laser. The theory of mode-locking is based on the complex Ginzburg-Landau equation derived under the condition that round-trip changes to the intra-cavity pulse are small. The saturable absorption was modeled by two-level system rate equation, which accounts for the free carrier that generated refractive index changes in the semiconductor absorber. The cavity dispersion is determined from the chirped fiber grating and the cavity losses are estimated basing on the properties of the MQW sample. The average equation belonging to the class of generated complex Ginzburg-Landau equations has a form     ∂A ∂2 ∂2 2 2 4 = i D 2 + δ3 |A| A + Dg 2 + g − l + γ3 |A| − γ5 |A| A (1) ∂z ∂t ∂t Here A is the optical field complex amplitude, scaled so that its square has units of optical power; t is the time variable and z is the distance divided by the cavity round-trip length Lc . The first two terms on the right hand side are imaginary contributions; D is the cavity dispersion, δ3 is related to the effective Kerr nonlinearity of the cavity. The last four terms have real coefficients with the following physical meaning: Dg is the gain and intracavity INFLUENCE OF THE FREQUENCY-CHIRP ON PULSE IN PASSIVELY MODE-LOCKING ... 173 filter bandwidth limits; l − g is the net loss, where l and g are linear losses and gain per round-trip, respectively. Other physical parameters are defined as β2 Lc + Dgr g 1 D=− , Dg = 2 + 2 , δ3 = γLc + αγ3 (2) 2 Ωg Ωf where β2 is the group velocity dispersion (GVD) for the fiber component, Lc is the optical cavity round-trip length, Dgr is the dispersion from the chirped grating. Further parameters are: Ωg(f ) , gain (filter) bandwidth, the nonlinear absorption parameters are given by γ3 = q0 /Ps and γ5 = q0 /Ps2 , where q0 is the measured linear loss and Ps is the saturation power of the MQW element. The nonlinear dispersion has two contributions, one from the fiber γLc , where γ = 2.6/W km is the fiber’s Kerr coefficient [5,7], and the other from the saturable absorber. The enhancement parameter, α, is adjusted to include the nonlinear index change of the saturable absorber. The simulations applied the beam propagating algorithm. The initial pulse envelope will be chosen by the frequency -chirp Gaussian profile solution. The initial pulse is   (1 + iC) t2 (3) A = A0 exp − 2T02 Using (3) to (1), after some arrangements we have ! t2 − CD D ∂A 2 g =i A + δ3 |A0 |2 e T0 + ∂z T02 T02 2 + 2 2t t Dg 2Dg (1 − C 2 )t2 DC 4 − T02 2 − T02 − γ |A | e − 2 + 2 + + g − l + γ |A | e 5 0 3 0 T0 T0 T04 (4) ! A The physical parameters for our cavity are: Lc = 15 m, Dgr = −13 ps2 , β2 = −15 ps2 /km, Dg = 0.015 ps2 , l = 0.2/m, g = 0.5/m. The parameters q0 = 0.55 and Ps = 5 mW, give γ3 = 0.11/mW, and γ5 = 0.5/mW2 . The choice α = 5, yields δ3 = 0.5/mW. There are three open parameters in our model, the gain g, the enhancement parameter α, and the chirp parameter C. The two first parameters have been discussed and presented in previous work [5]. In this work we focus only on the influence of the chirp parameter on the output pulse. We consider the initial pulse appeared in the fiber is a frequency-chirp Gaussian with the peak amplitude of A0 = 1 mW and the half of duration of T0 = 10ps. The input pulse and pulses after some round-trip are simulated using equation (4) and illustrated in Fig. 2 for case without frequency-chirp, C = 0. Then it can see that the peak power of the pulses are amplified and its duration smoothly increased after each round-trip. These simulated results are in good agreement with that given in work of Spaulding [8], in which the nonlinear dynamics of the mode-locking optical fiber ring laser are theoretically investigated. For the pulse with frequency-chirp of C = −5 and C = −10 , the output pulses are illustrated in Fig. 3. After each round-trip the peak power of the pulse is reduced, however, the reduction magnitude is about 1% if C=-10. Meanwhile its duration is appreciably increased about two times (∼ 20ps) after 20th round-trip, that means the pulse is broadened by the 174 BUI XUAN KIEN, TRAN HAI HUNG, AND TRINH DINH CHIEN Fig. 2. Simulated without-chirp Gaussian pulses after round-trips in laser cavity unnormalous-dispersion and down-chirp [9]. Moreover, the peak reduction and duration broadening depend on the chirp parameter, which is shown in Fig. 3, where reduction magnitude is about 1% and 0.4% , and duration is about 20 ps and 13 ps after 20th roundtrip, for C=-10 and C=-5, respectively. Conversely, for the case of C = 10 (see Fig. 4) the peak power of the output pulse is appreciately amplified about two times after 20th round-trip. But the its duration is quite no changed. In other ways it can say that the pulse with up-chirp should be compressed in the laser cavity. (a) (b) Fig. 3. Simulated down-chirp Gaussian pulses after round-trips in laser cavity. a) C = −5, b) C = −10. INFLUENCE OF THE FREQUENCY-CHIRP ON PULSE IN PASSIVELY MODE-LOCKING ... (a) 175 (b) Fig. 4. Simulated up-chirp Gaussian pulses after round-trips in laser cavity. a) C = 5, b) C = 10. The character points in figures 3 and 4 are discussed for two cases of the dispersion with two distinguish values of the frequency chirp, only. To confirm this character, the output pulses after 20th round-trip with some values of the frequency-chirp parameter for the case of anomalous dispersion are simulated and illustrated in Fig. 5. The results shows that if the amplifier fiber is the anomalous dispersive one, the output pulses will be more and more compressed when the frequency chirp parameter is positive and increases, but they will be broadened when the frequency chirp parameter is negative and decreases. The results presented in Fig. 5 are similar to that in the normalous dispersion laser cavity with opposite sign of the frequency chirp parameter. From above discussions, it is clear that the frequency chirp parameter plays an impotant role in pulsing process in the laser cavity with dispersion. The amplification process in laser cavity will be continuous round-trip till the stationary condition, for which the pulse will not changed, will be satisfied. It means that the propagating length of the intracavity pulse must be chosen so that ∂A/∂z = 0. From (4), we can find out the condition for the initial pulse 2 t CDg D 2 − T02 + δ |A | e + =0 3 0 T02 T02 D 2D (1−C 2 )t2 − t2 2 (5) 2 − 2t2 and −g + l + DC − T 2g − g T 4 − γ3 |A0 |2 e T0 + γ5 |A0 |4 e T0 = 0 T02 0 0 We consider the stationary condition satisfies at every time,t, in the duration of the pulse i.e. satisfies at t = 0. From given initial pulse (3) and condition (5), the condition for parameters of the initial pulse is given by T02 |A0 |2 < |D + CDg | δ3 (6) 176 BUI XUAN KIEN, TRAN HAI HUNG, AND TRINH DINH CHIEN Fig. 5. Simulated up-chirp Gaussian pulses after 20th round-trip in the anomalous dispersion laser cavity with some frequency chirp parameters. and condition for the energy balance of a stationary solution is given by Dg DC g − l = 2 − 2 − γ3 |A0 |2 + γ5 |A0 |4 T0 T0 (7) With the physical parameters given above and C=5 the initial pulse with half duration of T02 = 10ps, consequently, with peak power of P0 = A20 = 0.134mW . This initial pulse evolves in cavity using the Ginzburg-Landau equation until a new steady-state is achieved. If initial and final solutions are quite close, which demonstrates that the pulse is well approximated by soliton parameters. This problem will be presented in the future, in which the condition to crease the optical soliton should be interested. III. CONCLUSION The passively mode-locked ring Erbium-doped fiber laser is designed using the MQW saturable absorber and chirped fiber Brag grating. The evolution of the frequency-chirp Gaussian pulse in the cavity is given by the Ginzburg-Landau equation. The influence of the chirp parameter is simulated for the cases of the Gaussian pulse without-chirp, down-chirp and up-chirp. The shaping of the pulse, broadening, narrowing, amplifying and reduction, in the cavity depends on the the chirp parameter, C. 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