Simplified method for numerical modeling of fiber lasers O.V. Shtyrina,1,2 I.A. Yarutkina,1,2,∗ and M.P. Fedoruk1,2 1

Novosibirsk State University, 2 Pirogova Street, Novosibirsk, 630090, Russia of Computational Technologies, Siberian Branch of the Russian Academy of Sciences, 6 Ac. Lavrentjev Avenue, Novosibirsk, 630090, Russia

2 Institute



[email protected]

Abstract: A simplified numerical approach to modeling of dissipative dispersion-managed fiber lasers is examined. We present a new numerical iteration algorithm for finding the periodic solutions of the system of nonlinear ordinary differential equations describing the intra-cavity dynamics of the dissipative soliton characteristics in dispersion-managed fiber lasers. We demonstrate that results obtained using simplified model are in good agreement with full numerical modeling based on the corresponding partial differential equations. © 2014 Optical Society of America OCIS codes: (140.3510) Fiber laser; (140.4050) Mode-locked lasers.

References and links 1. H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11(9), 736–746 (1975). 2. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992). 3. S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14(8), 2099–2111 (1997). 4. J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48(4), 629–678 (2006). 5. N. Akhmediev and A. Ankiewicz, Eds. Dissipative Solitons: Lecture Notes in Physics (Springer, 2005), V. 661. 6. S.K. Turitsyn, B. Bale, and M.P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521(4), 135–203 (2012). 7. T. Schreiber, B. Ortac, J. Limpert, and A. Tunnermann, “On the study of pulse evolution in ultra-short pulse mode-locked fiber lasers by numerical simulations,” Opt. Express 15(13), 8252–8262 (2007). 8. O. Shtyrina, M. Fedoruk, S. Turitsyn, R. Herda, and O. Okhotnikov, “Evolution and stability of pulse regimes in SESAM-mode-locked femtosecond fiber lasers,” J. Opt. Soc. Am. B 26(2), 346–352 (2009). 9. B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010). 10. X. Tian, M. Tang, X. Cheng, P. P. Shum, Y. Gong, and C. Lin, “High-energy wave-breaking-free pulse from all-fiber mode-locked laser system,” Opt. Express 17(9), 7222–7227 (2009). 11. B.G Bale, O.G. Okhotnikov, and S.K. Turitsyn, “Modeling and Technologies of Ultrafast Fiber Lasers” in Fiber Lasers, O. G. Okhotnikov, ed. (Wiley-VCH Verlag GmbH Co., 2012). 12. I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21(5), 327–329 (1996). 13. S. K. Turitsyn, “Breathing self-similar dynamics and oscillatory tails of the chirped dispersion-managed soliton,” Phys. Rev. E. 58(2), R1256–R1259 (1998). 14. S. K. Turitsyn and V. K. Mezentsev, “Dynamics of self-similar dispersion-managed soliton presented in the basis of chirped Gauss-Hermite functions,” JETP Lett. 67(9), 640–646 (1998). 15. E. G. Shapiro and S. K. Turitsyn, “Theory of guiding-center breathing soliton propagation in optical communication systems with strong dispersion management,” Opt. Lett. 22(20), 1544–1546 (1997). 16. S. K. Turitsyn, T. Sch¨afer, and V. K. Mezentsev, “Self-similar core and oscillatory tails of a path-averaged chirped dispersion-managed optical pulse,” Opt. Lett. 23(17), 1351–1353 (1998).

#222689 - $15.00 USD Received 11 Sep 2014; revised 13 Nov 2014; accepted 18 Nov 2014; published 16 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031814 | OPTICS EXPRESS 31814

17. S. Turitsyn and E. Shapiro, “Enhanced power breathing soliton in communication systems with dispersion management,” Phys. Rev. E. 56(5), R4951–R4955, (1997). 18. J. Holt, “Numerical solution of nonlinear two-point boundary problems by finite difference methods,” Commun. ACM. 7(6), 366–373 (1964). 19. H. Keller, Numerical Methods for Two-point Boundary Value Problem (Blaisdell Publishing Co., 1968). 20. S. Roberts and S. J. Shipman, Two-point Boundary Value Problems: Shooting Methods (Elsevier, 1972). 21. J. Nijhof, W. Forisiak, and N. Doran, “The averaging method for finding exactly periodic dispersion-managed solitons,” IEEE J. Sel. Top. Quant. 6(2), 330–336 (2000). 22. I. A. Yarutkina and O. V. Shtyrina, “Mathematical modelling of dispersion-managed thulium/holmium fibre lasers,” Quantum Electron. 43(11), 1019–1023 (2013). 23. F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (2008).

1.

Introduction

The most common approach to the mathematical modeling of pulse generation and propagation in modern fiber laser systems is to apply a specific numerical model for each element of a cavity (for e.g., [1–11] for details). In general, it is necessary to carry out independent calculations for each set of system parameters and follow the radiation evolution from one round trip to another until the beam stabilizes. This approach provides high accuracy and good agreement between theory and experiment; however, the process involves large amounts of computational of time and resources to carry out multiparametric optimization for a number of cavity parameters. In this light, a simplified time-saving approach has been proposed in [11]. In general, it is of urgent practical interest to develop new, efficient, and fast numerical methods that can describe and model the generation of pulses in complex fiber lasers systems. The soliton waveform in dispersion-managed fiber laser systems consists of two parts: the self-similar central core and the pulse tails [6, 12–17]. The central pulse core contains most of the pulse energy, and this is why the study of evolution of the characteristics of the pulse central core is of key interest. Such a split of the full dissipative soliton waveform into the self-similarly propagating part and the pulse tails provides an opportunity to simplify numerical consideration. In this paper, we present an iteration algorithm for determining the periodic solutions of the system of nonlinear ordinary differential equations (ODEs) describing the “fast” dynamics of the main dissipative soliton characteristics (pulse width, peak power, and chirp parameter) in dispersionmanaged fiber lasers for a random set of cavity parameters. This method may be very useful for multiparamteric optimization of complex laser systems. 2.

System of nonlinear ordinary differential equations and iteration algorithm.

Pulse propagation in an active fiber can be described by the generalized nonlinear Schr¨odinger equation (NLSE) as below [1–11]: 1 G iUZ − β2UT T + γ |U|2U = i (G − Γ)U + i 2 UT T , 2 Ωg

(1)

where U(z,t) denotes the electromagnetic field envelope, Z [m] the propagation distance along the fiber, T [ps] the time, β2 [ps2 /m] the group velocity dispersion, γ [W−1 m−1 ] the nonlinear coefficient, Γ [dB/m] the linear loss coefficient, Ωg [THz] is related to the parabolic spectral width, and G [dB/m] the gain coefficient. In general, G = G(z) =

G0 , 1 + E(Z)/(PsatG · TR ) 

∞ where G0 [dB/m] represents the small-signal gain, E(Z) = −∞ |U(Z, T )|2 dT the pulse energy, TR [ps] the round-trip time, and PsatG [W] the saturation power of the gain medium.

#222689 - $15.00 USD Received 11 Sep 2014; revised 13 Nov 2014; accepted 18 Nov 2014; published 16 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031814 | OPTICS EXPRESS 31815

Below we consider the steady-state single pulse generation. Under this condition, the previous round-trip pulses have already saturated the gain, and the balance between gain and losses have already been achieved. Thus we derive the unsaturated gain relation with the pulse energy by the definition of the laser cavity parameters Γ, PsatG , and TR . Let us introduce z = Z/L, where L denotes the dispersion map length; using these notations we obtain the scaled equation. Consequently, we have: iuz + d (z) uT T + ε |u|2 u = i (g − l0 ) u + iν guT T ,

(2)

where d (z) = −β2 L/2, ε = γ L, ν = G/Ω2g , l0 = Γ·L, and g = G·L. After the scaling, d (z + 1) = d (z). One can transform the Eq. 2 to the system of equations for the root-mean-square (RMS) characteristics of the pulse as it was described in details in [6, 11]. This system can be transformed to the closed form under the assumption that the phase of the optical pulse is parabolic and that the pulse is Gaussian and can be described as below [6, 11]    t2 (1 − iC (z)) + iϕ (z) . u (z,t) = P (z) exp − 2 2τ (z) Thus, the problem of describing the evolution of the central part of the dissipative soliton dynamics using the key pulse characteristics (width τ , peak power P, and chirp C) is reduced to the boundary problem of determining the periodic solutions of the ordinary nonlinear differential equations [6, 11], which is discussed below:

τz = 2d (z)

 C 1 − ν g C2 − 1 , τ τ

Cz = (2d (z) − 2ν gC) Pz = −2d (z) where

1 +C2 ε − √ P, 2 τ 2

CP P + 2 (g − l0 ) P − 2ν g 2 , τ2 τ

g = g(τ , P) =

(3) (4) (5)

g0 √ , 1 + Pτ π /ε0

with the periodic boundary condition

τ (0) = τ (1),

C(0) = C(1),

P(0) = P(1).

(6)

Let us rewrite the problem defined by Eqs. (3)-(5) with the boundary condition (6) in vector notation: → − d V (z) → − → − = F (z, V (z)), z ∈ (0, 1); (7) dz → − → − V (0) = V (1), (8) → − → − → − where V (z) = (τ (z),C(z), P(z))T , F (z, V (z)) represents the nonlinear column vector. We consider dispersion map consisting of a fiber segment with normal dispersion and a fiber segment with anomalous dispersion. Thus the accumulated dispersion can be defined as < D >= d1 + d2 ,

(9)

#222689 - $15.00 USD Received 11 Sep 2014; revised 13 Nov 2014; accepted 18 Nov 2014; published 16 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031814 | OPTICS EXPRESS 31816

where d1 and d2 denote the anomalous and normal dispersions of the fiber segments and their values can be changed by changing the fiber length. The dispersion depth D = d1 − d2

(10)

also varies with the fiber length. Consequently, we obtain the expression for the dispersion of the two fiber segments as d1 = D+ < D >,

d2 = −D+ < D > .

Let us consider the following typical dispersion map: ⎧ ⎨ d1 , 0 ≤ z < 0.25, d2 , 0.25 ≤ z < 0.75, d (z) = ⎩ d1 , 0.75 ≤ z < 1.

(11)

Fig. 1. The dispersion map scheme defined by Eq. (11). The green and red lines correspond to the fiber segments with anomalous and normal dispersion, respectively. Grey areas denote the corresponding dispersion depth.

Figure 1 depicts the scheme of the dispersion map defined by Eq. (11) that corresponds to the regime with anomalous accumulated dispersion. There are several numerical methods to solve the system of ordinary nonlinear differential equations that can describe solutions to the problem defined by Eqs. (7)-(8); these include finite difference methods, function approximation methods, and iteration methods (see, for example, [18–20]). The averaging method to determine an exact periodic dispersion-managed solution of the conservative nonlinear Schr¨odinger equation has been presented for the first time in [21]. In the current paper, we present a new iteration algorithm for determining the periodic solutions of the system of ordinary nonlinear differential equations describing the “fast” dynamics of dissipative solitons in dispersion-managed fiber lasers. The presented algorithm is a modification of that developed in [21] adapted for the ODEs model in case of saturated gain and linear losses. This method has a number of advantages in comparison with approaches described above, which include fast calculation time for this specific application and the negligible contribution of the initial approximation to finding the periodic solution of the system. Using this iteration algorithm and adjusting the approximate initial data of the Cauchy problem during the number of cavity round-trips, we can obtain the periodic solutions of the boundary problem that exactly correspond with parameters of Eqs. (3)-(6). Let us examine this algorithm in detail. 1. We transform the initial problem to the Cauchy problem and consider its solution behavior for random initial data: → − d V (z) → − → − = F (z, V (z)), z > 0 (12) dz #222689 - $15.00 USD Received 11 Sep 2014; revised 13 Nov 2014; accepted 18 Nov 2014; published 16 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031814 | OPTICS EXPRESS 31817

→ − → − V (0) = W ,

(13)

→ − → − where V (z) = (τ (z),C(z), P(z))T . We set the vector W = (1, 0, 1)T . We solve the Cauchy problem using the Runge−Kutta method with the fourth order of accuracy. Equations (12)-(13) describe the dynamics of the main characteristics of a dissipative dispersionmanaged soliton in the frame of the model of the optical gain medium with saturation. When the gain is more than losses incurred, gain saturation leads to energy stabilization during pulse propagation through the√active fiber. For the Gaussian pulse approximation the pulse energy equals E(z) = π P(z)τ (z). Next, we define the first initial ap→ −0 0 0 T proximation√of the iteration process as a random vector V = (τ 0 ,C √ √ √ , P ) such that 0 0 π P τ = π P(zk )τ (zk ), where |E(zk ) − E(zm )| = | π P(zk )τ (zk ) − π P(zm )τ (zm )| < ε˜ E(zk ), ∀zm > zk with k, m ∈ {0, N} and a small preassigned value of ε˜ > 0. Figure 2(a) depicts the dynamics of the energy stabilization for the following set of parameters: g0 = 3, l0 = 0.5, ν = 0.1, ε0 = 10, ε = 1, D = 2, < D >= −0.002, and ε˜ = 10−8 . The solid line show the energy dynamics along z. Dots illustrate the energy values at zk ∈ {0, N}. Thus, for the presented case the energy stabilizes after z = 30, that corresponds to 30 dispersion map round-trips. For the described case the stabilized en→ − ergy value is 6.32. It means that we can specify the vector V 0 = (1.88858, 0, 1.88858)T which allows to set the desirable value of the energy.

Fig. 2. a) Dynamics of the energy stabilization for the following set of parameters: g0 = 1, l0 = 0.5, ν = 0.1, ε0 = 10, ε = 1, and ε˜ = 10−8 ; b) Dynamics of characteristics of the pulse along z for the first step of the iteration process.

→ − 2. The iteration algorithm determines the next approximation V j+1 on the basis of the dy→ −j → −j namics analysis of the vector V (z). The vector V (z) is the solution to Eqs. (12)-(13). The initial conditions of this problem are defined by a vector of the previous approxima→ − tion V j : → −j d V (z) → − → −j = F (z, V (z)), z > 0 (14) dz → −j → −j (15) V (0) = V , → −j where V (z) = (τ j (z),C j (z), P j (z))T . #222689 - $15.00 USD Received 11 Sep 2014; revised 13 Nov 2014; accepted 18 Nov 2014; published 16 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031814 | OPTICS EXPRESS 31818

When solving the Cauchy problem defined by Eqs. (14)-(15), we pay particular attention to the power dynamics component P j (z) in zk ∈ {0, N}. We introduce the following new notations: zmax = zk , max{P j (zk )}, k ∈ N denotes the first local maximum point k

and zmin = zk , min{P j (zk )}, k ∈ N denotes the first local minimum point. Figure 2(b) k

demonstrates the dynamics of characteristics of the pulse along z for the first step of the iteration process for the case described above in the Fig. 2(a). The pulse characteristics demonstrate the similar dynamics during the next iteration steps. Here, we remark that the system of ODEs given by (3)-(5) has been written under the assumption of a Gaussian pulse with a parabolic phase profile. Consequently, the field distribution can be written as:     t2 → − 1 − iC j (z) + iϕ (z) , (16) u( V j (z),t) = P j (z) exp − j,2 2τ (z) wherein the linear phase progression ϕ (z) is random.

→ − Let us define the next iteration approximation of vector V j+1 implicity: → − → − αφ (u( V j (zmax ),t)) + (1 − α )φ (u( V j (zmin ),t)) → − j+1 → − u( V ,t) = |u( V 0 ,t)|2 dt,  → −j → −j |αφ (u( V (zmax ),t)) + (1 − α )φ (u( V (zmin ),t))|2 dt (17) where 0 < α < 1 denotes an arbitrary weight, and   → −  u( V j (˜z), 0) → t2  → − −j j j (˜ 1 − iC φ (u( V j (˜z),t)) = → (˜ z ),t) = P z ) exp − (˜ z ) . u( V − 2τ j2 (˜z) |u( V j (˜z), 0)| The field distribution given by Eq. (17) is a linear combination of two fields that are in phase at the peak power point t = 0. This condition prevents extinction of the pulsed solution. The presented normalization is used for pulse energy conservation, which corresponds to the gain and loss balance of the system defined by (3)-(5). → − The obtained field distribution u( V j+1 ,t) in general is not Gaussian. However, the approximation of the real shape by the Gaussian ansatz in many practical situations is not of significant importance. We can determine τ j+1 , C j+1 , P j+1 from the following rule:  −  2 arg u(→ j+1 ,t) d V   → − j+1 → − 2 V = (τ i+1 ,C j+1 , P j+1 )T = (τ j+1 , τ j+1 , |u( V j+1 , 0)|2 )T , dt 2   → − wherein τ j+1 satisfies the equation ln |u( V j+1 , τ j+1 )|2 /P j+1 + 1 = 0. As it was mentioned above, the sum of two Gaussian pulses is not Gaussian pulse, that is why the abovementioned approximation allows to estimate the next iteration approximation of the vector in the frame of the system of ordinary differential equations. Once the method converges, our assumption does not add any additional fluctuations in the solution. All assumptions are verified by comparison with direct modeling. 3. We use the iteration process with the following convergence criterion in the third step as below. |P j+1 (zmin ) − P j+1 (zmax )| < ε˜ P j+1 (zmax ), where ε˜ > 0 is a small preassigned value. In the example described in the Fig. 2 iteration process stops after 50 iterations for ε˜ = 10−8 . It should be noted that we used the same value of ε˜ in all of our calculations; and iteration process stopped after ≥ 50 iterations. #222689 - $15.00 USD Received 11 Sep 2014; revised 13 Nov 2014; accepted 18 Nov 2014; published 16 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031814 | OPTICS EXPRESS 31819

The efficiency of the abovementioned algorithm is defined by of closeness of the iteration process to the real dissipative chirped dispersion-managed pulse evolution. A dissipative soliton experiences periodic stretching and compression in a dispersion-managed laser cavity. This results in peak-power oscillations. This fact defines the efficiency of the abovementioned algorithm. Another advantage of this algorithm is that it offers the possibility to fix the pulse energy to a previously determined level that exactly corresponds with the gain and loss balance in system (7). For instance, the shooting method does not allow setting of the soliton solution energy to a previously obtained level. This makes it impossible to obtain the required solution. In general, the iteration process may not converge to the periodic solution of the ordinary nonlinear differential equation system. However, this non-convergence only indicates the absence of pulse oscillations in the dispersion-managed laser with a specified set of parameters. Therefore, the abovementioned numerical algorithm leads to reduction in the calculation time and allows us to study the areas of stable pulse generation in the frame of the ordinary nonlinear differential equation system describing the fast dynamics of the central part of the pulse. The proposed method based on the joint use of ODEs-based model and iteration algorithm described above allows us to determine the solution to the problem (3)-(5) by two orders of magnitude faster than the time required using the traditional NLSE-based model. This acceleration is possible due to use of the fast algorithm for the linear system (3)-(5). On average, it took more than 2000 iterations to approach the steady state when use the NLSE model, while the proposed method based on the ODEs model reduces the necessary number of iterations by half. 3.

Comparison of ODEs and NLSE-based models

In this section, we present the results of mathematical modeling obtained using the described algorithm. To analyze qualitatively the impact of the changing dispersion map strength D and the average cavity dispersion < D > let us consider the simplified model neglecting cavity losses. This is justified by the observation that in the stable asymptotic regime losses are exactly compensated by gain and in some cases are separated from dispersion-nonlinearity processes. This assumption has to be verified by direct modeling of the full equations. During the mathematical modeling, we compared the results obtained via ODEs with results obtained using the full NLSE-based model using the expression below. ˆ ) − l0 )u. iuZ − d(Z)uT T + ε |u|2 u = i(g(T

(18)

The effect of the gain spectral dependence on the operator gˆ is typically introduced in the frequency domain using the Lorentzian line shape g( ˆ ω) =

g × 1 + E/ε0

1+



1 ω − ω0 Ωg

2 .

(19)

Here ω0 = 2π c/λ0 denotes the central frequency of the gain, c the speed of light, λ0 the laser wavelength, and E the pulse energy. The details of this approach to the numerical modeling of fiber lasers are described in several studies (see for e.g., [6]). We achieve a close consistency of these results between our method and the NLSE method for regimes with anomalous dispersion. Figure 3 shows the contour plot comparison of the pulse energy E, pulse width τ , and peak power P obtained via NLSE (Fig. 3(a)) and ODEs (Fig. 3(b)) in the (< D >, D) plane for anomalous regimes. The white area corresponds to the end of the range of generation. A stable solution has not been determined in this region. #222689 - $15.00 USD Received 11 Sep 2014; revised 13 Nov 2014; accepted 18 Nov 2014; published 16 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031814 | OPTICS EXPRESS 31820

Fig. 3. Contour plots of the pulse energy E, pulse width τ and peak power P obtained via a) NLSE and b) ODEs in the (< D >, D) plane for anomalous regimes.

Next, we examine the intracavity dynamics and the pulse shape and its spectrum in the anomalous regime. Figure 4 shows the intracavity dynamics obtained via NLSE and ODEs. Figure 4(a) depicts the 3D pulse shape dynamics, while Fig. 4(c) illustrates the 3D spectrum dynamics. The green sections correspond to the fiber sections with anomalous dispersion, and the red section corresponds to the fiber region with normal dispersion. Figure 4(b) shows the intracavity dynamics of the peak power, while Fig. 4(d) provides a comparison of the spectral shapes at the output. It is obvious from these figures that our results are consistent with those obtained using the NLSE. The spectral shapes coincide up to the −20 dB level. 4.

Analysis of average power dependence on cavity length and accumulated dispersion

We next discuss the mathematical modeling of the fiber laser with real-world fiber parameters for anomalous regimes. The parameters of the corresponding cavity are listed in table 1. In our model, a fiber laser system with a ring cavity comprises a 1-m-long Er-doped active fiber (LIEKKI 110–4) and two passive fibers, with the first one being a single-mode fiber with anomalous dispersion (SMF-28) and the second one being a dispersion compensation fiber (Thorlabs DCF-38). The lengths of the passive fibers (LPF and LDCF ) are set to be variable. Thus, the total cavity length equals L = LAF + LPF + LDCF , and the length of the normal segment equals LAF + LDCF . The sign of the cavity-accumulated dispersion can be changed by varying the fiber lengths. We next examine the relations between the coefficients of the distributed model given by Eqs. (2) and (18) and the cavity parameters listed in table 1: l0 = α L − 10 log(1 − Rout )/ log(10), ν = (λ02 /(2π cΩg ))2 , ε = γ L, ε0 = PsatG TR , g = G0 LAF , < D >= −0.5(β2,AF LAF + β2,PF LPF + β2,DCF LDCF ), and D = −0.5(−β2,AF LAF − β2,DCF LDCF + β2,PF LPF ). Figure 5 shows the dependence of the pulse energy (a) and peak power (b) on the cavityaccumulated dispersion, that is varied by varying LDCF and LPF . Each curve corresponds to a given constant total cavity length. The termination point of each curve corresponds to the end

#222689 - $15.00 USD Received 11 Sep 2014; revised 13 Nov 2014; accepted 18 Nov 2014; published 16 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031814 | OPTICS EXPRESS 31821

Fig. 4. Comparison of results obtained via NLSE and ODE models for anomalous regime. a) 3D pulse shape dynamics; b) Intracavity peak-power dynamics; c) 3D pulse spectrum dynamics; d) Comparison of spectral shapes at output. The green sections correspond to the part of the cavity with anomalous dispersion, and the red section corresponds to that with normal dispersion.

Table 1. Values of fiber laser parameters

Element Active Er-doped fiber

Passive fiber SMF-28 Dispersion compensation fiber System

Parameter Length LAF Second-order dispersion β2,AF Nonlinear parameter γ Gain bandwidth Ωg Small-signal gain G0 Saturation power PsatG Second-order dispersion β2,PF Nonlinear parameter γ Second-order dispersion β2,DCF Nonlinear parameter γ Laser wavelength λ0 Out-coupling parameter Rout Fiber losses α

Value 1m 12.6 ps2 /km 3 1/W/km 20 nm 3 dB/m 1.25 mW -22.8 ps2 /km 3 1/W/km 48.4 ps2 /km 3 1/W/km 1550 nm 70% 0.2 dB/km

of the range of generation. From the figures, we can observe a close agreement of the results obtained using the NLSE and ODEs models for anomalous regimes. We observe a decrease in the pulse energy and peak power in the zero accumulated dispersion region [22]. Note that coefficients ε0 , ε , and l0 of the distributed model were variable due to the passive fiber lengths variation. Figure 6 depicts the average pulse power (the pulse energy divided by the round-trip time) dependence on the total cavity length. In this figure, the red curves correspond to the modeling results obtained via ODEs while the black curves indicate results obtained via NLSE. As regards Fig. 6(a), the increase in the total cavity length was achieved by increasing LPF while

#222689 - $15.00 USD Received 11 Sep 2014; revised 13 Nov 2014; accepted 18 Nov 2014; published 16 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031814 | OPTICS EXPRESS 31822

Fig. 5. Pulse energy (a) and peak power (b) dependence on the cavity-accumulated dispersion as a function of passive fiber length. The curves from top to bottom correspond to the total cavity lengths of 27, 25, 22, 20, 17, 15, and 12 m. Red curves correspond to the modeling results obtained via ODEs while the black curves correspond to those obtained via NLSE.

LDCF was kept constant. We observe a decrease in the average power with increase in total cavity length, which corresponds to < D > approaching zero. As regards Fig. 6(b), the total cavity length was varied by increasing of the dispersion compensation fiber length LDCF while the length of the passive fiber SMF-28 (LPF ) was maintained constant. In this figure, we can observe that only for the SMF-28 fiber length of 1 m, the average power dynamics decreases with increase in the total cavity length, whereas in all other cases, the average power increases with increasing cavity length. The increase in accumulated dispersion leads to increase in the average power when the chirp-free point shifts from the output point to the boundary of the anomalous and normal fibers. For the case in which we did not observe such a chirp-free point movement, the pulse energy changes very little with the cavity length, which leads to decrease in the average power. In particular, for regimes that do not contain the intracavity chirp-free point, the average pulse power decreases over the whole region. This case corresponds to the topmost curve in Fig. 6(b).

Fig. 6. Average pulse-power dependence on the total cavity length. Red curves correspond to the modeling results obtained via ODEs while the black ones correspond to those obtained via NLSE. a) The curves from left to right correspond dispersion compensation fiber lengths of 8, 10, 12, 15, and 17 m; b) The curves from top to bottom correspond to SMF-28 fiber lengths of 1, 2, 3, 4, 6, and 7 m.

#222689 - $15.00 USD Received 11 Sep 2014; revised 13 Nov 2014; accepted 18 Nov 2014; published 16 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031814 | OPTICS EXPRESS 31823

5.

Applicability of ODE-based model the energy analysis of regimes with normal cavityaccumulated dispersion

From our discussion in the previous sections, we note that our results exhibit a close consistency with those obtained via NLSE in terms of the pulse shape and characteristics for anomalous regimes. However, it is to be noted that the pulse dynamics significantly differs for normal regimes. However, the pulse spectrum shape of the normal regimes significantly differs from the Gaussian shape, thereby invalidating the assumption of the simplified model. The spectrum obtained via the Schr¨odinger equation has a typical shape with sharp drops at the edges [23]; on the other hand, the spectrum obtained via ODEs exhibits a typical Gaussian shape (Fig. 7(e)). However, from Figs. 7(a)-(c) we note that our results agree closely with the NLSE results as regards the pulse energy levels. Figure 7(a) and Fig. 7(b) depict the energy contour plots obtained via NLSE and ODEs, respectively, in the (< D >, D) plane. Figure 7(c) depicts the pulse energy dependence on the cavity-accumulated dispersion < D >. The red curves correspond to the results obtained via ODEs and the black curves correspond to those obtained via NLSE. Each curve corresponds to a given constant total cavity length. The cavity-accumulated dispersion varies with variation in the passive fiber lengths LDCF and LPF .

Fig. 7. a) and b) show the energy contour plots obtained via NLSE and ODEs, respectively, in the (< D >, D) plane for normal regimes. c) Pulse energy dependence on the cavityaccumulated dispersion < D >. Here, red curves correspond to the results obtained via ODEs while the black curves correspond to those obtained via NLSE. The curves from the top to bottom correspond to constant total cavity lengths of 85 m, 75 m, 65 m, and 55 m. The cavity-accumulated dispersion varies with variation in the passive fiber length. The termination point of each curve corresponds to the end of the range of generation. d) Pulse shape comparison for normal regimes. e) Pulse spectrum comparison for normal regimes.

From our study results, we note that the simplified modeling of a fiber laser via ODEs can be successfully used to predict of the pulse characteristics (including pulse width and peak power) for regimes with anomalous dispersion. In the case of normal regimes the agreement between the results of our method and the NLSE method is not satisfactory; however, the results exhibited a sufficient agreement in terms of the energy levels even in this case. 6.

Conclusion

In this study, we presented a new iteration algorithm to solve the system of ordinary nonlinear differential equations describing the dynamics of the main pulse characteristics in a dissipative #222689 - $15.00 USD Received 11 Sep 2014; revised 13 Nov 2014; accepted 18 Nov 2014; published 16 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031814 | OPTICS EXPRESS 31824

dispersion-managed soliton fiber laser. We compared our results with those obtained using a full model based on the nonlinear Schr¨odinger equation, and we observed good agreement between the two sets of results. The proposed ODE-based method does not replace modeling using NLSE, but it can be time-saving by ensuring that full modeling is applied only to the relevant regions in the space defined by the variation in the free parameters. Acknowledgment The authors acknowledge Prof. S.K. Turitsyn and E.V. Podivilov for helpful discussions. This research has been supported by the grant N 14-21-00110 of the Russian Science Foundation, the grant N 14-01-31160 of the Russian Foundation for Basic Research. The work of I.A. Yarutkina was supported by the Program No.43 of the Fundamental Researches of the RAS Presidium on the Strategical Areas of Science Development “Fundamental Problems of Mathematical Modeling”.

#222689 - $15.00 USD Received 11 Sep 2014; revised 13 Nov 2014; accepted 18 Nov 2014; published 16 Dec 2014 (C) 2014 OSA 29 Dec 2014 | Vol. 22, No. 26 | DOI:10.1364/OE.22.031814 | OPTICS EXPRESS 31825

Simplified method for numerical modeling of fiber lasers.

A simplified numerical approach to modeling of dissipative dispersion-managed fiber lasers is examined. We present a new numerical iteration algorithm...
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