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Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers
Optical Fiber Technology xxx (2014) xxx–xxx
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Optical Fiber Technology www.elsevier.com/locate/yofte
Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers Xudong Hu a,b, Tigang Ning a,b, Li Pei a,b,⇑, Qingyan Chen c, Jing Li a,b a
Key Lab of All Optical Network &Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing 100044, China Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China c Wuhan Vocational College of Software and Engineering, Wuhan 430205, China b
a r t i c l e
i n f o
Article history: Received 5 January 2014 Revised 3 April 2014 Available online xxxx Keywords: Yb3+-doped fiber laser Simple shooting method Excellent initial guess functions
a b s t r a c t Excellent initial guess functions, providing for setting suitable initial estimates for simple shooting method, are developed to solve high power Yb3+-doped fiber lasers model with boundary conditions. When the guess value of slope efficiency is greater than or equal to the critical guess value of slope efficiency, the initial guess values of the forward signal power and backward pump power, generating from the excellent initial guess functions, are good initial guess values for the simple shooting method. Then, comparing the simulation results using our simple shooting method and the number sequence transition method based on MATLAB BVP solvers (NSTM-BVPs), the difference of the simulation results are less than given absolute error tolerance for different fiber length. Finally, we can conclude that the critical guess value of slope efficiency is less than 0.3 for all the fiber length, Yb3+-doped concentration, signal reflectivity and pump power. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In recent years, Yb3+-doped fiber lasers have been attracting increasing attention [1–5]. Relative to conventional solid state lasers, fiber lasers have unique merits such as high output power, compactness, high efficiency, high spatial beam quality, good heat dissipation, and less pronounced thermal effects [6]. A 100 W-class linearly-polarized single-mode Yb3+-doped fiber laser at 1120 nm with an optical efficiency of 50% has been reported [1]. A highly-efficient cladding-pumped Yb3+-doped fiber laser, generating 1.36 kW of continuous-wave output power at 1.1 lm with 83% slope efficiency and near diffraction-limited beam quality, has been demonstrated [2]. A highly efficient cladding-pumped Yb3+-doped fiber laser, generating >2.1 kW of continuous-wave output power at 1.1 lm with 74% slope efficiency with respect to launched pump power, has been demonstrated [3]. High-power fiber lasers with continuous-wave, pulsed regimes and various specific applications have been reviewed and discussed [4]. The Yb3+-doped fiber laser with a narrow line-width of 0.2 nm, operating in the range of 1147–1200 nm with the slope efficiencies ranging from 11% to 60%, has been realized and investigated [5].
⇑ Corresponding author at: Key Lab of All Optical Network &Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing 100044, China. E-mail address: [email protected] (L. Pei).
Many algorithms such as the shooting methods [7–9] and relaxation method [9,10] have been applied for the solutions of high power Yb3+-doped fiber lasers model [11]. A fast and stable shooting algorithm, using the Newton–Raphson method to solve the two-point boundary value problem of linear-cavity Yb3+-doped DCFL, has been demonstrated [7]. However, the pump power, expressed as analytic formula, may induce the error of the exact solutions; and the initial estimate, given only several simple data, may bring the shooting method to fail or unable to converge to the exact solutions. An improved shooting algorithm [8] has been proposed for high power double-clad fiber lasers with the initial guessed power, which is lower than the truth value. Clearly, the initial guessed power is not sure for the unknown truth value. In addition, there is some doubt about the efficiency of the improved shooting algorithm for only considering the forward pump structure. A combined algorithm [9] with shooting method and relaxation method [10] has been studied for solving the model of an Yb3+-doped double clad fiber laser. Since both shooting method and relaxation method need suitable predicted variable initial value, the combined algorithm is very doubtful to succeed to converge. And the entire calculation process takes even tens of seconds on a commercial desktop computer for given boundary conditions relative error tolerance 1E2. In this paper, the high power Yb3+-doped fiber laser model is briefly introduced in Section 2. In Section 3, we develop excellent initial guess functions, providing for setting good initial trial values
http://dx.doi.org/10.1016/j.yofte.2014.04.003 1068-5200/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: X. Hu et al., Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.003
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X. Hu et al. / Optical Fiber Technology xxx (2014) xxx–xxx
in the simple shooting method, to solve the above mentioned problems. Then, the simulation results of our simple shooting method are compared with the simulation results using number sequence transition method based on MATLAB BVP solvers (NSTM-BVPs) [12]. Finally, we discuss how the fiber length, Yb3+-doped concentration, signal reflectivity and pump power to take effect on the excellent initial guess functions for the simple shooting method.
where R1 ðks Þ and R2 ðks Þ are the input and output mirror reflectivity at the signal wavelength ks , respectively. R1 ðkp Þ and R2 ðkp Þ denote the input and output mirror reflectivity at the pump wavelength kp , respectively. Pp0 and PpL represent the launched pump power at z = 0 and z = L, respectively.
2. High power Yb3+-doped fiber laser model
The advantage of the Newton methods and quasi-Newton methods for solving steady-state systems of nonlinear rate equations is their speed of convergence once a sufficiently accurate approximation is known. A weakness of these methods is that an accurate initial estimate to the solution is needed to ensure convergence [13]. Most shooting methods treated two-point boundary value problems (BVPs) with initial value problems (IVPs) by setting suitable predicted variable initial value. According to boundary conditions (4)–(7), we take the forward laser power P þ s ð0Þ and pump power P p ð0Þ as independent variables of the initial values. We use the boundary conditions (5) and (7) as nonlinear equations in the Newton method. That is,
A typical high power linear cavity rare earth doped fiber laser with end-pumping is illustrated in Fig. 1. Our analysis is based on a set of steady-state nonlinear coupled ordinary differential rate equations model, and the typical fiber laser rate equations have been reported [11]. In the fiber laser’s numerical model, signal stimulated emission and absorption, stimulated emission at the pump wavelength and scattering losses both for the signal and the pump are considered, but spontaneous emission and excited state absorption (ESA) are negligible for strong pumping conditions. Here, considering high power Yb3+-doped fiber laser, the steady-state rate equations are described by the following set of ordinary differential equations (1)–(3): Cp rap ðPþ p ðzÞþP p ðzÞÞkp
Cs ras ðP þ s ðzÞþP s ðzÞÞks
þ N2 ðzÞ hcA hcA ¼ C ðr þr ÞðPþ ðzÞþP þ p ap ep p ðzÞÞkp p N þ 1 þ Cs ðras þres ÞðPs ðzÞþPs ðzÞÞks hcA
s
ð1Þ
hcA
dPs ðzÞ
dz
f1 ðPþs ð0Þ; Pp ð0ÞÞ ¼ Ps ðLÞ R2 ðks ÞPþs ðLÞ ¼ 0
ð8Þ
f2 ðPþs ð0Þ; Pp ð0ÞÞ ¼ Pp ðLÞ R2 ðkp ÞPþp ðLÞ PpL ¼ 0
ð9Þ
Using the Newton method, the iteration formula can be denoted
dPp ðzÞ ¼ Cp frap N ðrap þ rep ÞN2 ðzÞgP p ðzÞ ap Pp ðzÞ dz
3. Excellent initial Guess functions for simple shooting method
ð2Þ
by
½P þs ð0Þ; Pp ð0Þ ¼ ½Cs f½res þ ras N2 ðzÞ ras Ng as Ps ðzÞ
Pþs ð0Þ ¼ R1 ðks ÞPs ð0Þ
ð4Þ
Ps ðLÞ ¼ R2 ðks ÞPþs ðLÞ
ð5Þ
ðiÞ
¼ ½Pþs ð0Þ; Pp ð0Þ ðiÞ
ð6Þ
Pp ðLÞ ¼ R2 ðkp ÞPþp ðLÞ þ PpL
ð7Þ
ð10Þ
where the Jacobin matrix J can be described by
2 J¼4
ðiÞ
3
ðiÞ
5
ðiÞ
ðiÞ
@f1 ðP þs ð0Þ; Pp ð0ÞÞ=@ðPp ð0ÞÞðiÞ
ðiÞ
ðiÞ
@f2 ðP þs ð0Þ; Pp ð0ÞÞ=@ðPp ð0ÞÞðiÞ
@f1 ðPþs ð0Þ; P p ð0ÞÞ=@ðP þs ð0ÞÞ @f2 ðPþs ð0Þ; P p ð0ÞÞ=@ðP þs ð0ÞÞ
ð11Þ where i is the iteration times, for each i P 1. The Jacobin matrix J associated with nonlinear equations (8) and (9) written [f1; f2] that the 22 partial derivatives be determined and evaluated. In most situations, the exact evaluation of the partial derivatives is not practical, although the problem has been made more tractable with the widespread use of symbolic computation systems. Thus, we can use finite difference approximations to the partial derivatives. That is, the partial derivatives can be expressed by ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
@f1 ðP þs ð0Þ;P p ð0ÞÞ=@ðP þs ð0ÞÞ ½f1 ðP þs ð0Þ þ hstep;P p ð0ÞÞ f1 ðP þs ð0Þ;P p ð0ÞÞ=hstep ðiÞ
@f1 ðP þs ð0Þ;P p ð0ÞÞ=@ðP p ð0ÞÞðiÞ ½f1 ðP þs ð0Þ;P p ð0Þ þ hstepÞ f1 ðP þs ð0Þ;P p ð0ÞÞ=hstep ðiÞ ðiÞ @f2 ðP þs ð0Þ;P p ð0ÞÞ=@ðP þs ð0ÞÞ
Pþp ð0Þ ¼ R1 ðkp ÞPp ð0Þ þ Pp0
ðiÞ
J 1 ½f1 ðPþs ð0Þ; P p ð0ÞÞ; f2 ðPþs ð0Þ; Pp ð0ÞÞ
ð3Þ
In Eqs. (1)–(3), N is the rare earth ion dopant concentration. N2(z) is the upper lasing level population density. P p ðzÞ and P s ðzÞ are the pump power and the laser signal power along the fiber, respectively. The plus and minus superscripts represent propagation along the positive or negative z-direction, respectively. Cp and Cs represent respectively the pump and laser signal filling factor in the core. rap and rep are the pump absorption and the emission cross-section, respectively. ras and res are the laser signal absorption and the emission cross-section, respectively. kp and ks are the pump and laser signal wavelengths, respectively. The scattering losses for the pump and laser signal powers are given by ap and as, respectively. A, h, c and s is the effective core area, the Planck’s constant, the light velocity and spontaneous lifetime, respectively. The two-point boundary conditions can be expressed as follows:
ðiþ1Þ
ðiÞ ðiÞ ½f2 ðP þs ð0Þ þ hstep;P p ð0ÞÞ f2 ðP þs ð0Þ;P p ð0ÞÞ=hstep
ðiÞ
ðiÞ
ðiÞ
@f2 ðP þs ð0Þ;P p ð0ÞÞ=@ðP p ð0ÞÞðiÞ ½f2 ðP þs ð0Þ;P p ð0Þ þ hstepÞ f2 ðP þs ð0Þ;P p ð0ÞÞ=hstep
ð12Þ 4
where hstep is small in absolute value, such as 10 . To solve the fault of traditional shooting method, we develop and provide excellent initial guess functions for simple shooting method. The specified flow chart of the simple shooting method in high power Yb3+-doped fiber laser are schematically described in Fig. 2. A straightforward integration of Eq. (3) is reduced to
Pþs ðzÞPs ðzÞ ¼ Pþs ð0ÞPs ð0Þ ¼ Pþs ðLÞPs ðLÞ
ð13Þ
Pþ s Ps
Fig. 1. Schematic illustration of high power fiber laser with end-pump.
where the product is a constant, independent of z. Using the boundary conditions (4), (5) and Eq. (13), we find
Please cite this article in press as: X. Hu et al., Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.003
X. Hu et al. / Optical Fiber Technology xxx (2014) xxx–xxx
3
dPp ðzÞ ¼ ðCp rap N þ ap ÞPp ðzÞ dz
ð17Þ
A straightforward integration of Eq. (17) is also approximately reduced to
Pþp ðzÞ P p0 eðCp rap Nþap Þz
ð18Þ
Pp ðzÞ ¼ P p ðLÞeðCp rap Nþap ÞðLzÞ
ð19Þ
From Eqs. (18) and (19), we get approximately the pump power at z = 0 and z = L, respectively
Pþp ðLÞ Pp0 eðCp rap Nþap ÞL
ð20Þ
Pp ð0Þ ¼ Pp ðLÞeðCp rap Nþap ÞL
ð21Þ
Employing the boundary condition (7), the initial value of P p ð0Þ is evaluated by
Pp ð0Þ ðR2 ðkp ÞPp0 eðCp rap Nþap ÞL þ PpL ÞeðCp rap Nþap ÞL
ð22Þ
Due to the known total launched pump power Pp0 + PpL, signal reflectivity R1 ðks Þ and R2 ðks Þ, pump reflectivity R2 ðkp Þ, pump filling factor Cp, pump absorption rap, scattering losses ap, as well as Yb3+-doped concentration N and fiber length L, the initial value of the backward pump power P p ð0Þ in Eq. (22) can be approximately calculated and the initial value of the forward signal power Pþ s ð0Þ in Eq. (16) varies approximate linearly with the slope efficiency g. That is, the independent variables of the forward laser power Pþ s ð0Þ and backward pump power P p ð0Þ can be transferred by the only independent variable of the forward laser power Pþ s ð0Þ. Thus, adjusting the slope efficiency g means changing the initial values of the forward laser power Pþ s ð0Þ. We only provided a good initial approximation guess for the forward laser power Pþ s ð0Þ by setting suitable slope efficiency guess g, the desired right solutions can be expected rapidly and directly using simple shooting method. 4. Application for Yb3+-doped fiber laser
Fig. 2. The flow chart of calculating the numerical solutions with simple shooting method for Yb3+-doped fiber lasers.
Pþs ð0Þ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ R1 ðks ÞR2 ðks ÞPs ðLÞ
In addition, the output power P out ¼
ð14Þ Pþ s ðLÞ½1
R2 ðks Þ and Pout =
g(Pp0 + PpL Pth), where Pth denotes the threshold pump power. Pþs ð0Þ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pp0 þ PpL Pth g R1 ðks ÞR2 ðks Þ 1 R2 ðks Þ
ð15Þ
where g and Pp0 + PpL are the slope efficiency and the total launched pump power, respectively. Generally, Pp0 + PpL Pth in strongly pumped fiber lasers, the initial values of Pþ s ð0Þ is now approximately described by
Pþs ð0Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pp0 þ PpL g R1 ðks ÞR2 ðks Þ 1 R2 ðks Þ
ð16Þ
Similarly, we approximately assume N2(z) N, i.e. (rap + rep)N2(z) rapN. Eq. (2) can be simplified to
In all examples, unless stated otherwise, the simulation parameters [14] are given in the following: h = 6.63 1034 J s, c = 3 108 ms1, kp ¼ 975 nm, ks ¼ 1090 nm, R1 ðks Þ ¼ 0:98, R2 ðks Þ ¼ 0:04, R1 ðkp Þ ¼ 0:04, R2 ðkp Þ ¼ 0:04, ap = 6 103 m1, as = 5 103 m1, Cp = 0.0046, Cs = 0.9, A = 8 1010 m2, 24 2 24 2 rep = 2.5 10 m [15], rap = 2.5 10 m [15], ras = 1.4 1027 m2 [15], res = 2 1025 m2 [15], s = 0.84 103 s [15], N = 4 1025 m3, Pp0 = PpL = 100 W, TOL = 104, AbsTol = 104 W and RelTol = 104. For simplicity, the slope efficiency guess g is generated from 0.01 to 1 with interval 0.01 and dual-end pump configuration fiber lasers are considered in our examples. The iteration process of the powers ðPþ s ð0Þ; P p ð0ÞÞ with the guess value of the slope efficiency for high power Yb3+-doped fiber lasers are schematically described in Table 1. In Table 1, when the slope efficiency guess g is less than 0.15, the forward signal power Pþ s ð0Þ or backward pump power P p ð0Þ is lower than 0 after only one Newton iteration steps, which lead the simple shooting method to fail. While the slope efficiency guess g is from 0.15 to 0.2, as shown in Table 1, the simple shooting method succeed to make the powers ðP þ s ð0Þ; P p ð0ÞÞ converge to (34.0244, 0.1257) with given accuracy after several Newton iteration steps. Note that the ‘‘s s s s s s’’ in Table 1 is that program fails to continue in corresponding Newton iteration step for the wrong forward signal power P þ s ð0Þ or backward pump power P p ð0Þ. And the ‘‘- - - - - -’’ in Table 1 denotes that the numerical solutions of forward signal power Pþ s ð0Þ and backward pump power P p ð0Þ can be rapidly
Please cite this article in press as: X. Hu et al., Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.003
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X. Hu et al. / Optical Fiber Technology xxx (2014) xxx–xxx
Table 1 3+ The iteration process of ðPþ s ð0Þ; P p ð0ÞÞ with the guess value of slope efficiency g for Yb -doped fiber lasers (W).
g
The value ðP þ s ð0Þ; P p ð0ÞÞ in different iteration steps
1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
0.4125, 0.8250, 1.2374, 1.6499, 2.0624, 2.4749, 2.8874, 3.2998, 3.7123, 4.1248, 4.5373, 4.9497, 5.3622, 5.7747, 6.1872, 6.5997, 7.0121, 7.4246, 7.8371, 8.2496,
0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921
Note
2
3
4
5
6
7
77.8036, 2.4125 210.3532, 3.1658 700.4097, 6.7643 1443.9282, 9.9597 430.1153, 2.2316 269.4026, 1.0791 202.7976, 0.6343 165.0133, 0.4021 140.6847, 0.2640 123.1654, 0.1728 110.5144, 0.1109 100.2917, 0.0651 92.1734, 0.0310 85.4985, 0.0047 79.8907, 0.0161 75.1453, 0.0325 71.069, 0.0459 67.5951, 0.0568 64.4380, 0.0660 61.6845, 0.0738
ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss 56.2864, 0.1104 51.9876, 0.1129 48.6934, 0.1149 46.1516, 0.1165 44.0763, 0.1178 42.4233, 0.1190
ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss 38.0486, 0.1225 36.9049, 0.1233 36.1001, 0.1239 35.5356, 0.1243 35.1204, 0.1247 34.8268, 0.1249
ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss 34.2303, 0.1255 34.1343, 0.1256 34.0833, 0.1256 34.0563, 0.1257 34.0415, 0.1257 34.0337, 0.1257
ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss 34.0250, 0.1257 34.0244, 0.1257 34.0244, 0.1257 34.0244, 0.1257 34.0244, 0.1257 34.0244, 0.1257
ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss 34.0244, 0.1257 -------------------------------
obtained getting through several iteration steps, program succeeds to converge the right result. Updating the forward signal power P þ s ð0Þ and backward pump power P p ð0Þ at different slope efficiency guess g is shown in Fig. 3(a); and the output power Pout as a function of the slope efficiency guess is detailed in Fig. 3(b). When the slope efficiency guess g is from 0.15 to 1, it is easy to see that the powers ðP þ s ð0Þ; P p ð0ÞÞ succeed to converge (34.0244 W, 0.1257 W) with given accuracy after less than seven iteration steps in Fig. 3(a); meanwhile the output laser powers Pout vary from 164.98527 W to 164.98532 W in Fig. 3(b). Obviously, the guess value of slope efficiency g P 0.15 have no influence on the output power with given accuracy in high power Yb3+-doped fiber lasers. Thus, we name the corresponding guess value of slope efficiency g = 0.15 as the critical guess value of slope efficiency gc. The initial guess values of the powers ðPþ s ð0Þ; P p ð0ÞÞ, generating from the excellent initial guess functions (16) and (22), are suitable initial guess values for the simple shooting method when the guess value of slope efficiency g is greater than or equal to the critical guess value of slope efficiency gc, detailed in Eq. (23). Hence excellent initial guess functions (22) and (23) by searching the suitable critical guess value of slope efficiency gc can provide the suitable initial
Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Succeed Succeed Succeed Succeed Succeed Succeed
guess values for the simple shooting method in high power Yb3+doped fiber lasers.
Pþs ð0Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pp0 þ PpL g R1 ðks ÞR2 ðks Þ 1 R2 ðks Þ
gc 6 g 6 1
ð23Þ
Clearly, ðPþ s ðzÞ; P p ðzÞÞ ¼ ð0; 0Þ is a solution of Eqs. (1)–(3), but the solution (0, 0) does not satisfy the physical situations of YDFLs when Pp0 + PpL > Pth and fiber length L > Lmin, where Lmin is the minimum fiber length. If the guess value of slope efficiency g is setting too small, the solution of ðP þ s ð0Þ; P p ð0ÞÞ is approaching to (0, 0) after one Newton iteration step in the simple shooting method. For example in Table 1, the guess value of slope efficiency g < 0.15, the solution of ðP þ s ð0Þ; P p ð0ÞÞ is close to (0, 0) which lead to þ Ps ð0Þ < 0 or P p ð0Þ < 0, are unable to meet the specific physical value ranges. Once the guess value of slope efficiency g is sufficiently enough large, the solution of ðP þ s ð0Þ; P p ð0ÞÞ will be away (0, 0) and approach to the exact solution. As detailed in Fig. 3(a), the guessed values of slope efficiency g P 0.15 cause ðP þ s ð0Þ; P p ð0ÞÞ rapidly convergent to the exact solutions and avoid the case Pþ s ð0Þ < 0 or P p ð0Þ < 0. The critical guess value of slope efficiency gc is the boundary value of the guess value of slope efficiency g. In Table 1 and Fig. 3, the critical guess value of
Fig. 3. (a) Updating powers ðP þ s ð0Þ; P p ð0ÞÞ as a function of iteration step with dual-end pump at g 2 [0.15, 1] and (b) the output power as a function of the slope efficiency guess g.
Please cite this article in press as: X. Hu et al., Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.003
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X. Hu et al. / Optical Fiber Technology xxx (2014) xxx–xxx
slope efficiency gc = 0.15. From Table 1 and Fig. 3, one can find that the exact output power is 164.9853 W and the corresponding slope efficiency is 82.49265% are obtained rapidly using the novel shooting method with excellent initial guess function (22) and (23). That is, when the guess value of slope efficiency g is larger than the critical guess value of slope efficiency gc such as gc = 0.15 in Table 1 and Fig. 3, the novel shooting method is smoothly used to correct the initial guesses until all the boundary conditions (4)–(7) of YDFLs are satisfied. Now, we explore how to search the critical guess value of slope efficiency gc for different the fiber length. The critical guess value of slope efficiency gc and output powers as a function of the fiber length for high power Yb3+-doped fiber lasers is detailed in Fig. 4. When the fiber length varies from 1 m to 3.8 m, the output power rises swift from 38.4663 W to 135.5537 W and the corresponding critical guess values of slope efficiency gc rapidly from 0.07 to 0.15, the exact value of the corresponding slope efficiency grows 19.2331–67.7769%. And the fiber length varies from 3.8 m to 16.8 m for the critical guess values of slope efficiency gc = 0.15, the corresponding output power and optimum fiber length are respectively 166.1718 W and 11.1 m. While the fiber length varies from 16.9 m to 20 m, the output power drops from 164.0457 W to 162.4105 W with gc = 0.14. Further, to test the accuracy of the initial guess functions (22) and (23) for the given guess value of slope efficiency g in high power Yb3+-doped fiber lasers, we take g = 0.5 P gc as example.
Fig. 4. The critical guess value of slope efficiency gc and output powers as a function of the fiber length.
Fig. 5. The output power Pout and critical guess value of slope efficiency gc as a function of the dopant concentration N.
The boundary conditions value error and output powers are correspondingly detailed in Table 2. Comparing the output power of our simple shooting method and the simulation results using NSTMBVPs [12], it is easy to find that the differences of output powers are less than 3 105 W for given accuracy 104 W when the fiber length is from 5 m to 20 m. The result shows that the simple shooting method with the excellent initial guess functions (22) and (23) achieve rapidly the exact numerical results, which meet the desired requirement high accuracy. Then, we investigate how the Yb3+-doped concentration N and signal reflectivity R2 ðks Þ to effect the critical guess value of slope efficiency gc in the excellent initial guess function (23) for the simple shooting method. The critical guess value of slope efficiency gc and output power as a function of the Yb3+-doped concentration N and signal reflectivity R2 ðks Þ is respectively shown in Figs. 5 and 6. When the Yb3+-doped concentration N varies from 1 1025 m3 to 10 1025 m3, as given by Fig. 5, the critical guess value of slope efficiency gc slightly fluctuates among 0.14, 0.15 and 0.16. Thus, the Yb3+-doped concentration N has little influence on the critical guess value of slope efficiency gc in Eq. (23) for the simple shooting method in high power Yb3+-doped fiber lasers. As detailed in Fig. 6, the critical guess value of slope efficiency gc decreases slightly from 0.15 to 0.1 when the signal reflectivity R2 ðks Þ varies from 0.05 to 0.6, while the critical guess value of slope efficiency gc drops rapidly as increasing the signal reflectivity R2 ðks Þ from 0.6 to 0.8. Thus, the Yb3+-doped concentration N has little influence on the critical guess value of slope efficiency gc in Eq. (23) for
Table 2 The boundary conditions value error and output power as a function of fiber length using different methods. L (m)
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Simple shooting method with excellent initial guess functions
NSTM-BVPs [12] Pout (W)
Pout (W)
þ P s ðLÞ R2 ðks ÞP s ðLÞ (W)
þ P p ðLÞ R2 ðkp ÞP p ðLÞ P pL (W)
150.29667 157.37782 161.60682 164.04960 165.37221 165.99206 166.17119 166.07429 165.80492 165.42826 164.98531 164.50187 163.99420 163.47257 162.94332 162.41053
1.947727747975137e06 6.298835962148530e06 1.123917024248300e05 1.488381649128456e05 1.673733252527399e05 1.700532429982360e05 1.623326744049081e05 1.493029574906046e05 1.323666366292997e05 1.157667139750629e05 9.903708251712828e06 8.304923064805791e06 6.737008870771888e06 5.172503029449160e06 3.533407852529535e06 1.769308191512664e06
2.740426111813576e06 8.455579092014887e06 1.464980140042371e05 1.903977194217532e05 2.114758743232414e05 2.129952972040883e05 2.019014316090306e05 1.842606224045085e05 1.618880580167570e05 1.397369258882009e05 1.169570623460459e05 9.431287153915946e06 7.085763229497388e06 4.574282968405896e06 1.728849412074851e06 1.618554549054352e06
150.29668 157.37782 161.60683 164.04961 165.37223 165.99209 166.17122 166.07429 165.80491 165.42824 164.98529 164.50186 163.99422 163.47256 162.94332 162.41053
Please cite this article in press as: X. Hu et al., Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.003
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X. Hu et al. / Optical Fiber Technology xxx (2014) xxx–xxx
Fig. 6. The output power Pout and critical guess value of slope efficiency gc as a function of output mirror reflectivity R2 ðks Þ at signal wavelength ks .
the simple shooting method in high power Yb3+-doped fiber lasers. On the other hand, the result that the critical guess value of slope efficiency gc is less than 0.3 for simple shooting method in high power Yb3+-doped fiber lasers. Similarly, the critical guess value of slope efficiency gc and output power as a function of pump powers with different end-pumping configuration is described schematically in Fig. 7(a)–(c). In Fig. 7, the critical guess value of slope efficiency gc goes dramatically up to a certain stable value when the pump power with different end-pumping configuration is up to a certain pump power, such as gc = 0.15 for dual-end pumping in Fig. 7(a),
Fig. 8. Iteration times as a function of the guess value of slope efficiency g with different end-pumping configuration.
gc = 0.23 for forward pumping in Fig. 7(b) and gc = 0.13 for backward pumping in Fig. 7(c). While the pump power exceed the above certain pump power, the critical guess value of slope efficiency gc is a constant such as Pp0 + PpL > 8 W in Fig. 7(a), Pp0 + PpL > 30 W in Fig. 7(b) and Pp0 + PpL > 15 W in Fig. 7(c). Therefore, the high pump power Pp0 + PpL > 30 W has little influence on the critical guess value of slope efficiency gc in Eq. (23) for the simple shooting method in high power Yb3+-doped fiber lasers. Finally, we take the total pump power 100 W as an example to explain how about the convergence of the novel shooting method.
Fig. 7. The output power Pout and critical guess value of slope efficiency gc as a function of pump power with different end-pumping configuration: (a) dual-end pumping, (b) forward pumping, and (c) backward pumping.
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As the above mentioned, the results show that the critical guess value of slope efficiency gc are less than 0.3 for the simple shooting method in YDFLs. Iteration times as a function of the guess value of slope efficiency g P 0:3 with different end-pumping configuration are further plotted in Fig. 8. Whether forward pumping, backward pumping or dual-end pumping in Fig. 8, iteration times is less than or equal to 4, i.e. iteration steps 6 5, once the guess value of slope efficiency is larger than or equal to the critical guess value of slope efficiency gc. When the guess value of slope efficiency are 0.71– 0.89, 0.55–0.66 and 0.62–0.77 for forward pumping, backward pumping and dual-end pumping, all the iteration times are less than 2. That is, the novel shooting method with (22) and (23) are highly efficient of YDFLs. 5. Conclusions We introduce briefly the high power Yb3+-doped fiber laser model. Then, we develop and discuss in detail the simple shooting method with excellent initial guess functions to solve the difficulty of setting initial values in traditional shooting method. Generating from the excellent initial guess functions, the initial guess values of the forward signal powers and backward pump powers are suitable initial guess values for the simple shooting method when the guess value of slope efficiency is greater than or equal to the critical guess value of slope efficiency in high power Yb3+-doped fiber lasers. Comparing the simulation results by our simple shooting method and the simulation results using NSTM-BVPs, we find clearly that both the difference of the simulation results are less than 3 105 W under given absolute error tolerance 104 W. That is, the simple shooting method with the excellent initial guess functions (22) and (23) is highly efficient and accurate for the high power Yb3+-doped fiber laser. Furthermore, we further found the fiber length and Yb3+-doped concentration have little influence on the critical guess value of slope efficiency. In addition, the small signal reflectivity (<0.45) and high pump power (>30 W) have slightly effect on the critical guess value of slope efficiency. Finally, we can conclude that the critical guess value of slope efficiency is less than 0.3 for all the fiber length, Yb3+-doped concentration, signal reflectivity and pump power.
7
Acknowledgments This work is jointly supported by the National Natural Science Foundation of China (61177069, 61275092, 60837002), and National Basic Research Program of China (2010CB328206). References [1] J. Wang, J. Hu, L. Zhang, X. Gu, J. Chen, Y. Feng, A 100 W all-fiber linearlypolarized Yb-doped single-mode fiber laser at 1120 nm, Opt. Express 20 (27) (2012) 28373–28382. [2] Y.E. Jeong, J. Sahu, D. Payne, J. Nilsson, Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power, Opt. Express 12 (2004) 6088– 6092. [3] Y. Jeong, A.J. Boyland, J.K. Sahu, S.-H. Chung, J. Nilsson, D.N. Payne, Multikilowatt single-mode ytterbium-doped large-core fiber laser, J. Opt. Soc. Korea 13 (2009) 416–422. [4] Y. Jeong, L.A. Vazquez-Zuniga, S.J. Lee, G. Choi, Y. Kwon, H. Kim, High-power fiber lasers, in: 17th, IEEE, 2012, Opto-Electronics and Communications Conference (OECC), 2012, pp. 580–581. [5] V. Dvoyrin, O. Medvedkov, I. Sorokina, YDFL Operating in the 1150–1200 nm Spectral Domain, 2013. [6] B. Rossi, Commercial fiber lasers take on industrial markets, Laser Focus World 33 (1997) 143–149. [7] Z. Luo, C. Ye, G. Sun, Z. Cai, M. Si, Q. Li, Simplified analytic solutions and a novel fast algorithm for Yb3+-doped double-clad fiber lasers, Opt. Commun. 277 (2007) 118–124. [8] J. Liu, C. Zhao, S. Wen, D. Fan, C. Shuai, An improved shooting algorithm and its application to high-power fiber lasers, Opt. Commun. 283 (2010) 3764– 3767. [9] L. Shang, L. Qi, Y. Liao, S. Zhang, A combined algorithm for simulating fiber lasers based on the shooting and relaxation methods, Opt. Fiber Technol. (2012). [10] W.H. Press, Numerical recipes, third ed., The Art of Scientific Computing, Cambridge University Press, 2007. [11] I. Kelson, A.A. Hardy, Strongly pumped fiber lasers, IEEE J. Quant. Electron. 34 (1998) 1570–1577. [12] X. Hu, T. Ning, L. Pei, Q. Chen, W. Jian, Number sequence transition method based on MATLAB BVP solvers for high power Yb3+-doped fiber lasers, Opt. Laser Technol. 58 (2014) 76–83. [13] J.D. Faires, R.L. Burden (Eds.), Numerical Analysis, Brooks/Cole, 2011. [14] D. Xue, Q. Lou, J. Zhou, Comparison of Yb-doped fiber laser with one-end and double-end pumping configuration, Opt. Laser Technol. 39 (2007) 871– 874. [15] R. Paschotta, J. Nilsson, A.C. Tropper, D.C. Hanna, Ytterbium-doped fiber amplifiers, IEEE J. Quant. Electron. 33 (1997) 1049–1056.
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Contents lists available at ScienceDirect
Optical Fiber Technology www.elsevier.com/locate/yofte
Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers Xudong Hu a,b, Tigang Ning a,b, Li Pei a,b,⇑, Qingyan Chen c, Jing Li a,b a
Key Lab of All Optical Network &Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing 100044, China Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China c Wuhan Vocational College of Software and Engineering, Wuhan 430205, China b
a r t i c l e
i n f o
Article history: Received 5 January 2014 Revised 3 April 2014 Available online xxxx Keywords: Yb3+-doped fiber laser Simple shooting method Excellent initial guess functions
a b s t r a c t Excellent initial guess functions, providing for setting suitable initial estimates for simple shooting method, are developed to solve high power Yb3+-doped fiber lasers model with boundary conditions. When the guess value of slope efficiency is greater than or equal to the critical guess value of slope efficiency, the initial guess values of the forward signal power and backward pump power, generating from the excellent initial guess functions, are good initial guess values for the simple shooting method. Then, comparing the simulation results using our simple shooting method and the number sequence transition method based on MATLAB BVP solvers (NSTM-BVPs), the difference of the simulation results are less than given absolute error tolerance for different fiber length. Finally, we can conclude that the critical guess value of slope efficiency is less than 0.3 for all the fiber length, Yb3+-doped concentration, signal reflectivity and pump power. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In recent years, Yb3+-doped fiber lasers have been attracting increasing attention [1–5]. Relative to conventional solid state lasers, fiber lasers have unique merits such as high output power, compactness, high efficiency, high spatial beam quality, good heat dissipation, and less pronounced thermal effects [6]. A 100 W-class linearly-polarized single-mode Yb3+-doped fiber laser at 1120 nm with an optical efficiency of 50% has been reported [1]. A highly-efficient cladding-pumped Yb3+-doped fiber laser, generating 1.36 kW of continuous-wave output power at 1.1 lm with 83% slope efficiency and near diffraction-limited beam quality, has been demonstrated [2]. A highly efficient cladding-pumped Yb3+-doped fiber laser, generating >2.1 kW of continuous-wave output power at 1.1 lm with 74% slope efficiency with respect to launched pump power, has been demonstrated [3]. High-power fiber lasers with continuous-wave, pulsed regimes and various specific applications have been reviewed and discussed [4]. The Yb3+-doped fiber laser with a narrow line-width of 0.2 nm, operating in the range of 1147–1200 nm with the slope efficiencies ranging from 11% to 60%, has been realized and investigated [5].
⇑ Corresponding author at: Key Lab of All Optical Network &Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing 100044, China. E-mail address: [email protected] (L. Pei).
Many algorithms such as the shooting methods [7–9] and relaxation method [9,10] have been applied for the solutions of high power Yb3+-doped fiber lasers model [11]. A fast and stable shooting algorithm, using the Newton–Raphson method to solve the two-point boundary value problem of linear-cavity Yb3+-doped DCFL, has been demonstrated [7]. However, the pump power, expressed as analytic formula, may induce the error of the exact solutions; and the initial estimate, given only several simple data, may bring the shooting method to fail or unable to converge to the exact solutions. An improved shooting algorithm [8] has been proposed for high power double-clad fiber lasers with the initial guessed power, which is lower than the truth value. Clearly, the initial guessed power is not sure for the unknown truth value. In addition, there is some doubt about the efficiency of the improved shooting algorithm for only considering the forward pump structure. A combined algorithm [9] with shooting method and relaxation method [10] has been studied for solving the model of an Yb3+-doped double clad fiber laser. Since both shooting method and relaxation method need suitable predicted variable initial value, the combined algorithm is very doubtful to succeed to converge. And the entire calculation process takes even tens of seconds on a commercial desktop computer for given boundary conditions relative error tolerance 1E2. In this paper, the high power Yb3+-doped fiber laser model is briefly introduced in Section 2. In Section 3, we develop excellent initial guess functions, providing for setting good initial trial values
http://dx.doi.org/10.1016/j.yofte.2014.04.003 1068-5200/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: X. Hu et al., Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.003
2
X. Hu et al. / Optical Fiber Technology xxx (2014) xxx–xxx
in the simple shooting method, to solve the above mentioned problems. Then, the simulation results of our simple shooting method are compared with the simulation results using number sequence transition method based on MATLAB BVP solvers (NSTM-BVPs) [12]. Finally, we discuss how the fiber length, Yb3+-doped concentration, signal reflectivity and pump power to take effect on the excellent initial guess functions for the simple shooting method.
where R1 ðks Þ and R2 ðks Þ are the input and output mirror reflectivity at the signal wavelength ks , respectively. R1 ðkp Þ and R2 ðkp Þ denote the input and output mirror reflectivity at the pump wavelength kp , respectively. Pp0 and PpL represent the launched pump power at z = 0 and z = L, respectively.
2. High power Yb3+-doped fiber laser model
The advantage of the Newton methods and quasi-Newton methods for solving steady-state systems of nonlinear rate equations is their speed of convergence once a sufficiently accurate approximation is known. A weakness of these methods is that an accurate initial estimate to the solution is needed to ensure convergence [13]. Most shooting methods treated two-point boundary value problems (BVPs) with initial value problems (IVPs) by setting suitable predicted variable initial value. According to boundary conditions (4)–(7), we take the forward laser power P þ s ð0Þ and pump power P p ð0Þ as independent variables of the initial values. We use the boundary conditions (5) and (7) as nonlinear equations in the Newton method. That is,
A typical high power linear cavity rare earth doped fiber laser with end-pumping is illustrated in Fig. 1. Our analysis is based on a set of steady-state nonlinear coupled ordinary differential rate equations model, and the typical fiber laser rate equations have been reported [11]. In the fiber laser’s numerical model, signal stimulated emission and absorption, stimulated emission at the pump wavelength and scattering losses both for the signal and the pump are considered, but spontaneous emission and excited state absorption (ESA) are negligible for strong pumping conditions. Here, considering high power Yb3+-doped fiber laser, the steady-state rate equations are described by the following set of ordinary differential equations (1)–(3): Cp rap ðPþ p ðzÞþP p ðzÞÞkp
Cs ras ðP þ s ðzÞþP s ðzÞÞks
þ N2 ðzÞ hcA hcA ¼ C ðr þr ÞðPþ ðzÞþP þ p ap ep p ðzÞÞkp p N þ 1 þ Cs ðras þres ÞðPs ðzÞþPs ðzÞÞks hcA
s
ð1Þ
hcA
dPs ðzÞ
dz
f1 ðPþs ð0Þ; Pp ð0ÞÞ ¼ Ps ðLÞ R2 ðks ÞPþs ðLÞ ¼ 0
ð8Þ
f2 ðPþs ð0Þ; Pp ð0ÞÞ ¼ Pp ðLÞ R2 ðkp ÞPþp ðLÞ PpL ¼ 0
ð9Þ
Using the Newton method, the iteration formula can be denoted
dPp ðzÞ ¼ Cp frap N ðrap þ rep ÞN2 ðzÞgP p ðzÞ ap Pp ðzÞ dz
3. Excellent initial Guess functions for simple shooting method
ð2Þ
by
½P þs ð0Þ; Pp ð0Þ ¼ ½Cs f½res þ ras N2 ðzÞ ras Ng as Ps ðzÞ
Pþs ð0Þ ¼ R1 ðks ÞPs ð0Þ
ð4Þ
Ps ðLÞ ¼ R2 ðks ÞPþs ðLÞ
ð5Þ
ðiÞ
¼ ½Pþs ð0Þ; Pp ð0Þ ðiÞ
ð6Þ
Pp ðLÞ ¼ R2 ðkp ÞPþp ðLÞ þ PpL
ð7Þ
ð10Þ
where the Jacobin matrix J can be described by
2 J¼4
ðiÞ
3
ðiÞ
5
ðiÞ
ðiÞ
@f1 ðP þs ð0Þ; Pp ð0ÞÞ=@ðPp ð0ÞÞðiÞ
ðiÞ
ðiÞ
@f2 ðP þs ð0Þ; Pp ð0ÞÞ=@ðPp ð0ÞÞðiÞ
@f1 ðPþs ð0Þ; P p ð0ÞÞ=@ðP þs ð0ÞÞ @f2 ðPþs ð0Þ; P p ð0ÞÞ=@ðP þs ð0ÞÞ
ð11Þ where i is the iteration times, for each i P 1. The Jacobin matrix J associated with nonlinear equations (8) and (9) written [f1; f2] that the 22 partial derivatives be determined and evaluated. In most situations, the exact evaluation of the partial derivatives is not practical, although the problem has been made more tractable with the widespread use of symbolic computation systems. Thus, we can use finite difference approximations to the partial derivatives. That is, the partial derivatives can be expressed by ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
ðiÞ
@f1 ðP þs ð0Þ;P p ð0ÞÞ=@ðP þs ð0ÞÞ ½f1 ðP þs ð0Þ þ hstep;P p ð0ÞÞ f1 ðP þs ð0Þ;P p ð0ÞÞ=hstep ðiÞ
@f1 ðP þs ð0Þ;P p ð0ÞÞ=@ðP p ð0ÞÞðiÞ ½f1 ðP þs ð0Þ;P p ð0Þ þ hstepÞ f1 ðP þs ð0Þ;P p ð0ÞÞ=hstep ðiÞ ðiÞ @f2 ðP þs ð0Þ;P p ð0ÞÞ=@ðP þs ð0ÞÞ
Pþp ð0Þ ¼ R1 ðkp ÞPp ð0Þ þ Pp0
ðiÞ
J 1 ½f1 ðPþs ð0Þ; P p ð0ÞÞ; f2 ðPþs ð0Þ; Pp ð0ÞÞ
ð3Þ
In Eqs. (1)–(3), N is the rare earth ion dopant concentration. N2(z) is the upper lasing level population density. P p ðzÞ and P s ðzÞ are the pump power and the laser signal power along the fiber, respectively. The plus and minus superscripts represent propagation along the positive or negative z-direction, respectively. Cp and Cs represent respectively the pump and laser signal filling factor in the core. rap and rep are the pump absorption and the emission cross-section, respectively. ras and res are the laser signal absorption and the emission cross-section, respectively. kp and ks are the pump and laser signal wavelengths, respectively. The scattering losses for the pump and laser signal powers are given by ap and as, respectively. A, h, c and s is the effective core area, the Planck’s constant, the light velocity and spontaneous lifetime, respectively. The two-point boundary conditions can be expressed as follows:
ðiþ1Þ
ðiÞ ðiÞ ½f2 ðP þs ð0Þ þ hstep;P p ð0ÞÞ f2 ðP þs ð0Þ;P p ð0ÞÞ=hstep
ðiÞ
ðiÞ
ðiÞ
@f2 ðP þs ð0Þ;P p ð0ÞÞ=@ðP p ð0ÞÞðiÞ ½f2 ðP þs ð0Þ;P p ð0Þ þ hstepÞ f2 ðP þs ð0Þ;P p ð0ÞÞ=hstep
ð12Þ 4
where hstep is small in absolute value, such as 10 . To solve the fault of traditional shooting method, we develop and provide excellent initial guess functions for simple shooting method. The specified flow chart of the simple shooting method in high power Yb3+-doped fiber laser are schematically described in Fig. 2. A straightforward integration of Eq. (3) is reduced to
Pþs ðzÞPs ðzÞ ¼ Pþs ð0ÞPs ð0Þ ¼ Pþs ðLÞPs ðLÞ
ð13Þ
Pþ s Ps
Fig. 1. Schematic illustration of high power fiber laser with end-pump.
where the product is a constant, independent of z. Using the boundary conditions (4), (5) and Eq. (13), we find
Please cite this article in press as: X. Hu et al., Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.003
X. Hu et al. / Optical Fiber Technology xxx (2014) xxx–xxx
3
dPp ðzÞ ¼ ðCp rap N þ ap ÞPp ðzÞ dz
ð17Þ
A straightforward integration of Eq. (17) is also approximately reduced to
Pþp ðzÞ P p0 eðCp rap Nþap Þz
ð18Þ
Pp ðzÞ ¼ P p ðLÞeðCp rap Nþap ÞðLzÞ
ð19Þ
From Eqs. (18) and (19), we get approximately the pump power at z = 0 and z = L, respectively
Pþp ðLÞ Pp0 eðCp rap Nþap ÞL
ð20Þ
Pp ð0Þ ¼ Pp ðLÞeðCp rap Nþap ÞL
ð21Þ
Employing the boundary condition (7), the initial value of P p ð0Þ is evaluated by
Pp ð0Þ ðR2 ðkp ÞPp0 eðCp rap Nþap ÞL þ PpL ÞeðCp rap Nþap ÞL
ð22Þ
Due to the known total launched pump power Pp0 + PpL, signal reflectivity R1 ðks Þ and R2 ðks Þ, pump reflectivity R2 ðkp Þ, pump filling factor Cp, pump absorption rap, scattering losses ap, as well as Yb3+-doped concentration N and fiber length L, the initial value of the backward pump power P p ð0Þ in Eq. (22) can be approximately calculated and the initial value of the forward signal power Pþ s ð0Þ in Eq. (16) varies approximate linearly with the slope efficiency g. That is, the independent variables of the forward laser power Pþ s ð0Þ and backward pump power P p ð0Þ can be transferred by the only independent variable of the forward laser power Pþ s ð0Þ. Thus, adjusting the slope efficiency g means changing the initial values of the forward laser power Pþ s ð0Þ. We only provided a good initial approximation guess for the forward laser power Pþ s ð0Þ by setting suitable slope efficiency guess g, the desired right solutions can be expected rapidly and directly using simple shooting method. 4. Application for Yb3+-doped fiber laser
Fig. 2. The flow chart of calculating the numerical solutions with simple shooting method for Yb3+-doped fiber lasers.
Pþs ð0Þ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ R1 ðks ÞR2 ðks ÞPs ðLÞ
In addition, the output power P out ¼
ð14Þ Pþ s ðLÞ½1
R2 ðks Þ and Pout =
g(Pp0 + PpL Pth), where Pth denotes the threshold pump power. Pþs ð0Þ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pp0 þ PpL Pth g R1 ðks ÞR2 ðks Þ 1 R2 ðks Þ
ð15Þ
where g and Pp0 + PpL are the slope efficiency and the total launched pump power, respectively. Generally, Pp0 + PpL Pth in strongly pumped fiber lasers, the initial values of Pþ s ð0Þ is now approximately described by
Pþs ð0Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pp0 þ PpL g R1 ðks ÞR2 ðks Þ 1 R2 ðks Þ
ð16Þ
Similarly, we approximately assume N2(z) N, i.e. (rap + rep)N2(z) rapN. Eq. (2) can be simplified to
In all examples, unless stated otherwise, the simulation parameters [14] are given in the following: h = 6.63 1034 J s, c = 3 108 ms1, kp ¼ 975 nm, ks ¼ 1090 nm, R1 ðks Þ ¼ 0:98, R2 ðks Þ ¼ 0:04, R1 ðkp Þ ¼ 0:04, R2 ðkp Þ ¼ 0:04, ap = 6 103 m1, as = 5 103 m1, Cp = 0.0046, Cs = 0.9, A = 8 1010 m2, 24 2 24 2 rep = 2.5 10 m [15], rap = 2.5 10 m [15], ras = 1.4 1027 m2 [15], res = 2 1025 m2 [15], s = 0.84 103 s [15], N = 4 1025 m3, Pp0 = PpL = 100 W, TOL = 104, AbsTol = 104 W and RelTol = 104. For simplicity, the slope efficiency guess g is generated from 0.01 to 1 with interval 0.01 and dual-end pump configuration fiber lasers are considered in our examples. The iteration process of the powers ðPþ s ð0Þ; P p ð0ÞÞ with the guess value of the slope efficiency for high power Yb3+-doped fiber lasers are schematically described in Table 1. In Table 1, when the slope efficiency guess g is less than 0.15, the forward signal power Pþ s ð0Þ or backward pump power P p ð0Þ is lower than 0 after only one Newton iteration steps, which lead the simple shooting method to fail. While the slope efficiency guess g is from 0.15 to 0.2, as shown in Table 1, the simple shooting method succeed to make the powers ðP þ s ð0Þ; P p ð0ÞÞ converge to (34.0244, 0.1257) with given accuracy after several Newton iteration steps. Note that the ‘‘s s s s s s’’ in Table 1 is that program fails to continue in corresponding Newton iteration step for the wrong forward signal power P þ s ð0Þ or backward pump power P p ð0Þ. And the ‘‘- - - - - -’’ in Table 1 denotes that the numerical solutions of forward signal power Pþ s ð0Þ and backward pump power P p ð0Þ can be rapidly
Please cite this article in press as: X. Hu et al., Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.003
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Table 1 3+ The iteration process of ðPþ s ð0Þ; P p ð0ÞÞ with the guess value of slope efficiency g for Yb -doped fiber lasers (W).
g
The value ðP þ s ð0Þ; P p ð0ÞÞ in different iteration steps
1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
0.4125, 0.8250, 1.2374, 1.6499, 2.0624, 2.4749, 2.8874, 3.2998, 3.7123, 4.1248, 4.5373, 4.9497, 5.3622, 5.7747, 6.1872, 6.5997, 7.0121, 7.4246, 7.8371, 8.2496,
0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921 0.0921
Note
2
3
4
5
6
7
77.8036, 2.4125 210.3532, 3.1658 700.4097, 6.7643 1443.9282, 9.9597 430.1153, 2.2316 269.4026, 1.0791 202.7976, 0.6343 165.0133, 0.4021 140.6847, 0.2640 123.1654, 0.1728 110.5144, 0.1109 100.2917, 0.0651 92.1734, 0.0310 85.4985, 0.0047 79.8907, 0.0161 75.1453, 0.0325 71.069, 0.0459 67.5951, 0.0568 64.4380, 0.0660 61.6845, 0.0738
ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss 56.2864, 0.1104 51.9876, 0.1129 48.6934, 0.1149 46.1516, 0.1165 44.0763, 0.1178 42.4233, 0.1190
ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss 38.0486, 0.1225 36.9049, 0.1233 36.1001, 0.1239 35.5356, 0.1243 35.1204, 0.1247 34.8268, 0.1249
ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss 34.2303, 0.1255 34.1343, 0.1256 34.0833, 0.1256 34.0563, 0.1257 34.0415, 0.1257 34.0337, 0.1257
ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss 34.0250, 0.1257 34.0244, 0.1257 34.0244, 0.1257 34.0244, 0.1257 34.0244, 0.1257 34.0244, 0.1257
ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss ssssss 34.0244, 0.1257 -------------------------------
obtained getting through several iteration steps, program succeeds to converge the right result. Updating the forward signal power P þ s ð0Þ and backward pump power P p ð0Þ at different slope efficiency guess g is shown in Fig. 3(a); and the output power Pout as a function of the slope efficiency guess is detailed in Fig. 3(b). When the slope efficiency guess g is from 0.15 to 1, it is easy to see that the powers ðP þ s ð0Þ; P p ð0ÞÞ succeed to converge (34.0244 W, 0.1257 W) with given accuracy after less than seven iteration steps in Fig. 3(a); meanwhile the output laser powers Pout vary from 164.98527 W to 164.98532 W in Fig. 3(b). Obviously, the guess value of slope efficiency g P 0.15 have no influence on the output power with given accuracy in high power Yb3+-doped fiber lasers. Thus, we name the corresponding guess value of slope efficiency g = 0.15 as the critical guess value of slope efficiency gc. The initial guess values of the powers ðPþ s ð0Þ; P p ð0ÞÞ, generating from the excellent initial guess functions (16) and (22), are suitable initial guess values for the simple shooting method when the guess value of slope efficiency g is greater than or equal to the critical guess value of slope efficiency gc, detailed in Eq. (23). Hence excellent initial guess functions (22) and (23) by searching the suitable critical guess value of slope efficiency gc can provide the suitable initial
Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Succeed Succeed Succeed Succeed Succeed Succeed
guess values for the simple shooting method in high power Yb3+doped fiber lasers.
Pþs ð0Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pp0 þ PpL g R1 ðks ÞR2 ðks Þ 1 R2 ðks Þ
gc 6 g 6 1
ð23Þ
Clearly, ðPþ s ðzÞ; P p ðzÞÞ ¼ ð0; 0Þ is a solution of Eqs. (1)–(3), but the solution (0, 0) does not satisfy the physical situations of YDFLs when Pp0 + PpL > Pth and fiber length L > Lmin, where Lmin is the minimum fiber length. If the guess value of slope efficiency g is setting too small, the solution of ðP þ s ð0Þ; P p ð0ÞÞ is approaching to (0, 0) after one Newton iteration step in the simple shooting method. For example in Table 1, the guess value of slope efficiency g < 0.15, the solution of ðP þ s ð0Þ; P p ð0ÞÞ is close to (0, 0) which lead to þ Ps ð0Þ < 0 or P p ð0Þ < 0, are unable to meet the specific physical value ranges. Once the guess value of slope efficiency g is sufficiently enough large, the solution of ðP þ s ð0Þ; P p ð0ÞÞ will be away (0, 0) and approach to the exact solution. As detailed in Fig. 3(a), the guessed values of slope efficiency g P 0.15 cause ðP þ s ð0Þ; P p ð0ÞÞ rapidly convergent to the exact solutions and avoid the case Pþ s ð0Þ < 0 or P p ð0Þ < 0. The critical guess value of slope efficiency gc is the boundary value of the guess value of slope efficiency g. In Table 1 and Fig. 3, the critical guess value of
Fig. 3. (a) Updating powers ðP þ s ð0Þ; P p ð0ÞÞ as a function of iteration step with dual-end pump at g 2 [0.15, 1] and (b) the output power as a function of the slope efficiency guess g.
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X. Hu et al. / Optical Fiber Technology xxx (2014) xxx–xxx
slope efficiency gc = 0.15. From Table 1 and Fig. 3, one can find that the exact output power is 164.9853 W and the corresponding slope efficiency is 82.49265% are obtained rapidly using the novel shooting method with excellent initial guess function (22) and (23). That is, when the guess value of slope efficiency g is larger than the critical guess value of slope efficiency gc such as gc = 0.15 in Table 1 and Fig. 3, the novel shooting method is smoothly used to correct the initial guesses until all the boundary conditions (4)–(7) of YDFLs are satisfied. Now, we explore how to search the critical guess value of slope efficiency gc for different the fiber length. The critical guess value of slope efficiency gc and output powers as a function of the fiber length for high power Yb3+-doped fiber lasers is detailed in Fig. 4. When the fiber length varies from 1 m to 3.8 m, the output power rises swift from 38.4663 W to 135.5537 W and the corresponding critical guess values of slope efficiency gc rapidly from 0.07 to 0.15, the exact value of the corresponding slope efficiency grows 19.2331–67.7769%. And the fiber length varies from 3.8 m to 16.8 m for the critical guess values of slope efficiency gc = 0.15, the corresponding output power and optimum fiber length are respectively 166.1718 W and 11.1 m. While the fiber length varies from 16.9 m to 20 m, the output power drops from 164.0457 W to 162.4105 W with gc = 0.14. Further, to test the accuracy of the initial guess functions (22) and (23) for the given guess value of slope efficiency g in high power Yb3+-doped fiber lasers, we take g = 0.5 P gc as example.
Fig. 4. The critical guess value of slope efficiency gc and output powers as a function of the fiber length.
Fig. 5. The output power Pout and critical guess value of slope efficiency gc as a function of the dopant concentration N.
The boundary conditions value error and output powers are correspondingly detailed in Table 2. Comparing the output power of our simple shooting method and the simulation results using NSTMBVPs [12], it is easy to find that the differences of output powers are less than 3 105 W for given accuracy 104 W when the fiber length is from 5 m to 20 m. The result shows that the simple shooting method with the excellent initial guess functions (22) and (23) achieve rapidly the exact numerical results, which meet the desired requirement high accuracy. Then, we investigate how the Yb3+-doped concentration N and signal reflectivity R2 ðks Þ to effect the critical guess value of slope efficiency gc in the excellent initial guess function (23) for the simple shooting method. The critical guess value of slope efficiency gc and output power as a function of the Yb3+-doped concentration N and signal reflectivity R2 ðks Þ is respectively shown in Figs. 5 and 6. When the Yb3+-doped concentration N varies from 1 1025 m3 to 10 1025 m3, as given by Fig. 5, the critical guess value of slope efficiency gc slightly fluctuates among 0.14, 0.15 and 0.16. Thus, the Yb3+-doped concentration N has little influence on the critical guess value of slope efficiency gc in Eq. (23) for the simple shooting method in high power Yb3+-doped fiber lasers. As detailed in Fig. 6, the critical guess value of slope efficiency gc decreases slightly from 0.15 to 0.1 when the signal reflectivity R2 ðks Þ varies from 0.05 to 0.6, while the critical guess value of slope efficiency gc drops rapidly as increasing the signal reflectivity R2 ðks Þ from 0.6 to 0.8. Thus, the Yb3+-doped concentration N has little influence on the critical guess value of slope efficiency gc in Eq. (23) for
Table 2 The boundary conditions value error and output power as a function of fiber length using different methods. L (m)
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Simple shooting method with excellent initial guess functions
NSTM-BVPs [12] Pout (W)
Pout (W)
þ P s ðLÞ R2 ðks ÞP s ðLÞ (W)
þ P p ðLÞ R2 ðkp ÞP p ðLÞ P pL (W)
150.29667 157.37782 161.60682 164.04960 165.37221 165.99206 166.17119 166.07429 165.80492 165.42826 164.98531 164.50187 163.99420 163.47257 162.94332 162.41053
1.947727747975137e06 6.298835962148530e06 1.123917024248300e05 1.488381649128456e05 1.673733252527399e05 1.700532429982360e05 1.623326744049081e05 1.493029574906046e05 1.323666366292997e05 1.157667139750629e05 9.903708251712828e06 8.304923064805791e06 6.737008870771888e06 5.172503029449160e06 3.533407852529535e06 1.769308191512664e06
2.740426111813576e06 8.455579092014887e06 1.464980140042371e05 1.903977194217532e05 2.114758743232414e05 2.129952972040883e05 2.019014316090306e05 1.842606224045085e05 1.618880580167570e05 1.397369258882009e05 1.169570623460459e05 9.431287153915946e06 7.085763229497388e06 4.574282968405896e06 1.728849412074851e06 1.618554549054352e06
150.29668 157.37782 161.60683 164.04961 165.37223 165.99209 166.17122 166.07429 165.80491 165.42824 164.98529 164.50186 163.99422 163.47256 162.94332 162.41053
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X. Hu et al. / Optical Fiber Technology xxx (2014) xxx–xxx
Fig. 6. The output power Pout and critical guess value of slope efficiency gc as a function of output mirror reflectivity R2 ðks Þ at signal wavelength ks .
the simple shooting method in high power Yb3+-doped fiber lasers. On the other hand, the result that the critical guess value of slope efficiency gc is less than 0.3 for simple shooting method in high power Yb3+-doped fiber lasers. Similarly, the critical guess value of slope efficiency gc and output power as a function of pump powers with different end-pumping configuration is described schematically in Fig. 7(a)–(c). In Fig. 7, the critical guess value of slope efficiency gc goes dramatically up to a certain stable value when the pump power with different end-pumping configuration is up to a certain pump power, such as gc = 0.15 for dual-end pumping in Fig. 7(a),
Fig. 8. Iteration times as a function of the guess value of slope efficiency g with different end-pumping configuration.
gc = 0.23 for forward pumping in Fig. 7(b) and gc = 0.13 for backward pumping in Fig. 7(c). While the pump power exceed the above certain pump power, the critical guess value of slope efficiency gc is a constant such as Pp0 + PpL > 8 W in Fig. 7(a), Pp0 + PpL > 30 W in Fig. 7(b) and Pp0 + PpL > 15 W in Fig. 7(c). Therefore, the high pump power Pp0 + PpL > 30 W has little influence on the critical guess value of slope efficiency gc in Eq. (23) for the simple shooting method in high power Yb3+-doped fiber lasers. Finally, we take the total pump power 100 W as an example to explain how about the convergence of the novel shooting method.
Fig. 7. The output power Pout and critical guess value of slope efficiency gc as a function of pump power with different end-pumping configuration: (a) dual-end pumping, (b) forward pumping, and (c) backward pumping.
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As the above mentioned, the results show that the critical guess value of slope efficiency gc are less than 0.3 for the simple shooting method in YDFLs. Iteration times as a function of the guess value of slope efficiency g P 0:3 with different end-pumping configuration are further plotted in Fig. 8. Whether forward pumping, backward pumping or dual-end pumping in Fig. 8, iteration times is less than or equal to 4, i.e. iteration steps 6 5, once the guess value of slope efficiency is larger than or equal to the critical guess value of slope efficiency gc. When the guess value of slope efficiency are 0.71– 0.89, 0.55–0.66 and 0.62–0.77 for forward pumping, backward pumping and dual-end pumping, all the iteration times are less than 2. That is, the novel shooting method with (22) and (23) are highly efficient of YDFLs. 5. Conclusions We introduce briefly the high power Yb3+-doped fiber laser model. Then, we develop and discuss in detail the simple shooting method with excellent initial guess functions to solve the difficulty of setting initial values in traditional shooting method. Generating from the excellent initial guess functions, the initial guess values of the forward signal powers and backward pump powers are suitable initial guess values for the simple shooting method when the guess value of slope efficiency is greater than or equal to the critical guess value of slope efficiency in high power Yb3+-doped fiber lasers. Comparing the simulation results by our simple shooting method and the simulation results using NSTM-BVPs, we find clearly that both the difference of the simulation results are less than 3 105 W under given absolute error tolerance 104 W. That is, the simple shooting method with the excellent initial guess functions (22) and (23) is highly efficient and accurate for the high power Yb3+-doped fiber laser. Furthermore, we further found the fiber length and Yb3+-doped concentration have little influence on the critical guess value of slope efficiency. In addition, the small signal reflectivity (<0.45) and high pump power (>30 W) have slightly effect on the critical guess value of slope efficiency. Finally, we can conclude that the critical guess value of slope efficiency is less than 0.3 for all the fiber length, Yb3+-doped concentration, signal reflectivity and pump power.
7
Acknowledgments This work is jointly supported by the National Natural Science Foundation of China (61177069, 61275092, 60837002), and National Basic Research Program of China (2010CB328206). References [1] J. Wang, J. Hu, L. Zhang, X. Gu, J. Chen, Y. Feng, A 100 W all-fiber linearlypolarized Yb-doped single-mode fiber laser at 1120 nm, Opt. Express 20 (27) (2012) 28373–28382. [2] Y.E. Jeong, J. Sahu, D. Payne, J. Nilsson, Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power, Opt. Express 12 (2004) 6088– 6092. [3] Y. Jeong, A.J. Boyland, J.K. Sahu, S.-H. Chung, J. Nilsson, D.N. Payne, Multikilowatt single-mode ytterbium-doped large-core fiber laser, J. Opt. Soc. Korea 13 (2009) 416–422. [4] Y. Jeong, L.A. Vazquez-Zuniga, S.J. Lee, G. Choi, Y. Kwon, H. Kim, High-power fiber lasers, in: 17th, IEEE, 2012, Opto-Electronics and Communications Conference (OECC), 2012, pp. 580–581. [5] V. Dvoyrin, O. Medvedkov, I. Sorokina, YDFL Operating in the 1150–1200 nm Spectral Domain, 2013. [6] B. Rossi, Commercial fiber lasers take on industrial markets, Laser Focus World 33 (1997) 143–149. [7] Z. Luo, C. Ye, G. Sun, Z. Cai, M. Si, Q. Li, Simplified analytic solutions and a novel fast algorithm for Yb3+-doped double-clad fiber lasers, Opt. Commun. 277 (2007) 118–124. [8] J. Liu, C. Zhao, S. Wen, D. Fan, C. Shuai, An improved shooting algorithm and its application to high-power fiber lasers, Opt. Commun. 283 (2010) 3764– 3767. [9] L. Shang, L. Qi, Y. Liao, S. Zhang, A combined algorithm for simulating fiber lasers based on the shooting and relaxation methods, Opt. Fiber Technol. (2012). [10] W.H. Press, Numerical recipes, third ed., The Art of Scientific Computing, Cambridge University Press, 2007. [11] I. Kelson, A.A. Hardy, Strongly pumped fiber lasers, IEEE J. Quant. Electron. 34 (1998) 1570–1577. [12] X. Hu, T. Ning, L. Pei, Q. Chen, W. Jian, Number sequence transition method based on MATLAB BVP solvers for high power Yb3+-doped fiber lasers, Opt. Laser Technol. 58 (2014) 76–83. [13] J.D. Faires, R.L. Burden (Eds.), Numerical Analysis, Brooks/Cole, 2011. [14] D. Xue, Q. Lou, J. Zhou, Comparison of Yb-doped fiber laser with one-end and double-end pumping configuration, Opt. Laser Technol. 39 (2007) 871– 874. [15] R. Paschotta, J. Nilsson, A.C. Tropper, D.C. Hanna, Ytterbium-doped fiber amplifiers, IEEE J. Quant. Electron. 33 (1997) 1049–1056.
Please cite this article in press as: X. Hu et al., Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers, Opt. Fiber Technol. (2014), http://dx.doi.org/10.1016/j.yofte.2014.04.003