Shooting method with excellent initial guess functions for multipoint pumping Yb3+-doped fiber lasers

Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Shooting method with excellent initial guess functions for multipoint pumping Yb3 þ -doped fiber lasers Xudong Hu a,b,n, Tigang Ning a,b, Li Pei a,b, Qingyan Chen c, Jing Li a,b, Jingjing Zheng a,b, Chuanbiao Zhang a,b a

Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China Key Lab of All Optical Network & Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing 100044, China c Wuhan Vocational College of Software and Engineering, Wuhan 430205, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 15 July 2014 Received in revised form 23 September 2014 Accepted 30 September 2014

Excellent initial guess functions for shooting method, providing for setting suitable initial estimates, are developed to solve multipoint pumping high power Yb3 þ -doped fiber lasers (YDFLs) model with boundary conditions. Simulation results show that shooting method with excellent initial guess functions can get rapidly the exact values of YDFLs after less than five iteration times. In excellent initial guess functions for signal power and pump power of 3-point pumping or 4-point pumping YDFLs, the guess value of conversion efficiency is greater than or equal to the critical guess value of conversion efficiency. The critical guess value of conversion efficiency is less than 0.21 for all the fiber length in three-point pumping or four-point pumping YDFLs. Comparing to number sequence transition method based on MATLAB BVP solvers of bvp4c(NSTM-bvp4c) and shooting method with answer ranges definition, the simple shooting method with excellent initial guess functions is more reliable and faster for multipoint pumping YDFLs. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Shooting method Yb3 þ -doped fiber Fiber lasers Multipoint pumping Initial guess function

1. Introduction In recent years, high power Yb3 þ -doped fiber lasers (YDFLs) and related pump technologies have been attracting increasing attention. A highly efficient cladding-pumped Yb3 þ -doped fiber laser generating 4 2.1 kW of continuous-wave output power at 1.1 μm has been demonstrated [1]. 3 kW single mode Yb3 þ -doped LMA fiber laser at 1080 nm has been presented by 976 nm diode directly pumping [2]. In end-pumping configuration, the inhomogeneous distribution inversion levels along the fiber lead to selfpulses [3], high transition loss of the polymer optical fiber reduce the laser power [4], and when using high pump powers, the high local intensity of the end fiber can even lead to the destruction of the fiber by overheating. In addition, the heat induced temperature gradient of the fiber core that creates thermal lens [5], and the thermal lens change the propagation direction of laser power which escape from the fiber core. That is, inhomogeneous gain distribution, the restricted fiber length [6], thermal effects occurring at the end of fiber and the limitation of pump point in endpumping scheme, can be solved by the multipoint side pump n Corresponding author at: Beijing Jiaotong University, Institute of Lightwave Technology, Key Lab of All Optical Network & Advanced Telecommunication Network of EMC, Beijing 100044, China. E-mail address: [email protected] (X. Hu).

technique. Thus, compared with end-pumping scheme, the sidepumping scheme in YDFLs has more pump point, homogeneous distribution inversion levels along the fiber, longer fiber and better managed thermal. Many pump technologies such as imbedded V-grooves [7], binary gold diffraction grating side-pumping [8], sub-wavelength grating [9], GT-Wave side-pumping [10], embedded-mirror side pumping [11], monolithic integrated all-glass combiner [12,13] and side-pump coupler with refractive index valley configuration [14] have been developed for multipoint side pumping YDFLs.” In addition, more than 1 kW output power of multipoint side -pumping geometry fiber laser [15] was realized by Akira Shirakawa. Including high optical-to-optical slope efficiency, large alignment tolerances, no obstruction of fiber ends, no loss for light propagating in the fiber core, compact, rugged packaging and low cost, a double-clad fiber lasers with embedded-mirror side pumping [11,16] are employed in practical commercial fiber lasers. The output power 102.5 W and slope efficiency 77.1% ytterbium-doped monolithic fiber laser with fused angle-polished side-pumping flexible configuration has been demonstrated [4]. Many algorithms such as the shooting methods have been applied for the solution of two-point end-pumping YDFLs model [17–19]. Shooting methods are sensitive to initial guess values and unsuitable initial guess values may lead shooting methods to fail. That is, when the guess is too poor for shooting methods, the

http://dx.doi.org/10.1016/j.optcom.2014.09.074 0030-4018/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: X. Hu, et al., Optics Communications (2014), http://dx.doi.org/10.1016/j.optcom.2014.09.074i

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backward pump power or backward signal power is lower than 0 which is unable to meet the practical physical problems [17]. A fast and stable shooting algorithm, using the Newton–Raphson method to solve the two-point boundary value problem of linearcavity YDFLs, has been demonstrated [18]. However, the initial estimate, given only several data, may bring the shooting method to fail or unable to converge to the exact solutions. A combined algorithm [19] with shooting method and relaxation method has been studied for solving the model of YDFLs. Since both shooting method and relaxation method need suitable predicted variable initial value, the combined algorithm is very doubtful to succeed to converge. Comparing to boundary conditions of two-point endpumping YDFLs model, the boundary conditions of multipoint pumping YDFLs model are more complicated which need more computation time. Excellent initial guess functions, providing for setting suitable initial estimates for simple shooting method, are proposed to solve two-point end-pumped YDFLs [20]. It is necessary to develop excellent initial guess functions for multipoint pumping YDFLs. In this paper, multipoint pumping YDFLs model is briefly introduced. Then a shooting method with excellent initial guess functions, providing for setting excellent initial trial values, are developed to solve the solution of 3-point and 4-point pumping YDFLs. Further, we discuss how fiber length to take effect on excellent initial guess functions for the shooting method in 3-point and 4-point pumping YDFLs. Finally, compared with the number sequence transition method based on MATLAB BVP solvers bvp4c (NSTM-bvp4c) [21], we verify the efficiency and speed of the simple shooting method.

2. Multipoint pumping Yb3 þ -doped fiber lasers model A typical multipoint pumping high power linear cavity Yb3 þ doped fiber laser with pump reflectivity, including n-point forward and backward pump, is described schematically in Fig. 1. In the Yb3 þ -doped fiber laser 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 multipoint pumping conditions [22]. Moreover, the lengths of side-coupling components are neglected [23]. The rate equations in high power Yb3 þ -doped fiber laser are described by the following set of nonlinear coupled ordinary differential equations (1)–(3) [21].

N2 (z) = N

((Γ σ

p ap

±

(Pp+ (z) + Pp− (z) )λ p )/hcA) + ( (Γs σas (Ps+ (z) + Ps− (z) ) λ s )/hcA) ((Γp (σap + σep )(Pp+ (z) + Pp− (z) )λ p )/hcA) + (1/τ) + (Γs (σas + σes ) (Ps+ (z) + Ps− (z) )λ s /hcA)

dPp± (z) dz

= g (Pp± (z)) = − Γp {σap N − (σap + σep ) N2 (z) } Pp± (z) − α p Pp± (z)

±

(1)

dPs± (z) = f (Ps± (z)) = [Γs {[σes + σas ] N2 (z) − σas N} − α s ] Ps± (z) dz

(2)

(3)

where N is the rare earth ion dopant concentration. N2 (z) is the upper lasing level population density. Ps± (z) and Pp± (z) are the laser signal power and the pump power along the fiber, respectively. The plus and minus superscripts represent propagation along the positive or negative z-direction, respectively. λ p and λ s are the pump and laser signal wavelengths, respectively. Γs and Γp represent respectively the laser signal and pump filling factor in the core. The scattering losses for the pump and laser signal powers are given by α p and α s , respectively. σap and σep are the pump absorption and the emission cross-section, respectively. σas and σes are the laser signal absorption and the emission cross-section, respectively. A, h, c and τ is the effective core area, the Planck's constant, the light velocity and spontaneous lifetime, respectively. The end-point boundary conditions can be expressed as follows:

Ps+ (0) = R1 (λ s ) Ps− (0)

(4-a)

Ps− (L) = R2 (λ s ) Ps+ (L)

(4-b)

Pp+ (0) = R1 (λ p ) Pp− (0) + η p0 P0f

(4-c)

Pp− (L) = R2 (λ p ) Pp+ (L) + η pL P Lb

(4-d)

where P0f and PLb represent the side pump power at z = 0 and z = L1, respectively. ηp0 and ηpL are the pump conversion efficiency at the pump positions z = 0 and z = L , respectively. R1 (λ s ) and R2 (λ s ) are the input and output mirror reflectivity at signal wavelength at the fiber length z = 0 and z = L , respectively. R1 (λ p ) and R2 (λ p ) are the input and output mirror reflectivity at pump wavelength at the fiber length z = 0 and z = L , respectively. In addition, the conditions [23,24] at side pump positions z = L i (1 ≤ i ≤ n − 1) are

P s+i+ 1 (L i ) = (1 − l si ) P s+i (L i )

(5-a)

P s−i (L i ) = (1 − l si ) P s−i+ 1 (L i )

(5-b)

P p+ i+ 1 (L i ) = (1 − l pi ) P p+i (L i ) + η pi P Lfi

(5-c)

P p−i (L i ) = (1 − l pi ) P p− i+ 1 (L i ) + η pi P Lbi

(5-d)

where PLfi and PLbi are the forward and backward pump powers at

Fig. 1. Schematic illustration of high power Yb3 þ -doped fiber laser with multipoint pumping.

side-pump position z = L i (1 ≤ i ≤ n − 1), respectively. l pi and ηpi are the pump leakage rate and coupling efficiency at pump position z = L i , correspondingly. The signal leakage rate of propagating in the fiber core on side-pumping position z = L i is l si , respectively. For simplicity, each the pump coupling efficiency,

Please cite this article as: X. Hu, et al., Optics Communications (2014), http://dx.doi.org/10.1016/j.optcom.2014.09.074i

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pump and signal transmission losses coefficient on the pump positions are the same, i.e. l pi = l p , ηpi = ηp , ηp0 = ηpL = ηp and l si = l s .

R1 (λ s ) P s−, 1 (0) =

3

R2 (λ s ) Ps+, 2 (1)

(10-b)

Moreover, the output power Pout = Ps+, 2 (1)[1 − R2 (λ s )] and

Pout = (P0f + PLf1 + PLb1 + PLb ) ηp η , thus the initial guess of the backward signal power Ps,− 1 (0) [17,20]

3. Shooting method with initial guess functions We take 3-point side pumping YDFLs as an example. The 3-point side pumping YDFLs can be transformed into two-point boundary value problems by coordinate scale transform and continuum boundary transform of pumping powers and laser powers, and then shooting methods can be applied to multipoint pumping YDFLs [25,26]. The two intervals [0, L1] and [L1, L] in 3-point pump YDFLs and the variables are formulated the ± ± signal and pump powers Ps,± 1 (z1), Ps,± 2 (z2 ), Pp, 1 (z1) , P p, 2 (z2 ) in interval [0,1], the difference of signal and pump powers ±dPs± (z)/dz = f (Ps± (z)), ±dPp± (z)/dz = g (Pp± (z)), the differential equations (2) and (3) are induced

±

dP p±, i (zi ) dzi

=

dz g (P p±, i (zi )) = (L i − L i − 1) g (P p±, i (zi )) dzi

P s−, 1 (0) =

ηc ≤ η ≤ 1

(11)

where η and P0f + PLf1 + PLb1 + PLb are conversion efficiency and total pump power, respectively. Similar to two-point pump YDFLs [20], conversion efficiency should be greater than some certain value ηc . Here ηc is named as the critical guess value of conversion efficiency. Similarly, according to the boundary conditions (10-a), the initial guess of the backward signal power Ps,− 2 (0)

Ps−, 2 (0) = (i = 1, 2)

b b f f R2 (λ s ) P0 + P L1 + P L1 + P L ηp η R1 (λ s ) 1 − R 2 (λ s )

P s−, 1 (0) + Ps, 2 (0)/P s+, 1 (0)

=

P s−, 1 (0) 1 + (1/ R1 (λ s ) R2 (λ s ) − 1) L1/L

(12)

(6) Here, the forward laser signal is assumed linearly with the fiber

±

dP s±, i (zi ) dzi

dz = f (P s±, i (zi )) = (L i − L i − 1) f (P s±, i (zi )) dzi

(i = 1, 2)

length in [0, L], then Ps+, 2 (0)/Ps+, 1 (0) = 1 + (1/ R1 (λ s ) R2 (λ s ) − 1) L1/L .

(7)

The point side-pumping boundary conditions (4) and (5) become

P s+, 1 (0) = R1 (λ s ) P s−, 1 (0)

(8-a)

Ps−, 2 (1) = R2 (λ s ) Ps+, 2 (1)

(8-b)

P p+, 1 (0) = R1 (λ p ) P p−, 1 (0) + ηp P0f

(8-c)

Pp−, 2 (1) = R2 (λ p ) Pp+, 2 (1) + ηp P Lb

(8-d)

Ps+, 2 (0) = (1 − l s ) P s+, 1 (1)

(8-e)

P s−, 1 (1) = (1 − l s ) Ps−, 2 (0)

(8-f)

efficiency ηp , total launched pump power P0f + PLf1 + PLb1 + PLb and fiber length L1, L , initial value of the forward signal power Ps,− 1 (0) of Eq. (11) and Ps,− 2 (0) of Eq. (12) varies approximate linearly with the conversion efficiency η . The population inversion N2 (z) can be assumed to be independent of z and equals to its averaged value N¯ 2 (z) [27]. And the pump reflectivity is small in practice. From Eqs. (2), (8-f) and (8-b), the − − initial guesses backward pump powers Pp, 1 (0) and P p, 2 (0) can be expressed as the following

P p−, 1 (0) = ηp P Lb1 exp (((Γp (σap + σep ) N¯ 2 − σap N) − α p ) L1) + ηp P Lb (1 − l p ) exp (((Γp (σap + σep ) N¯ 2 − σap N) − α p ) L) (13) Pp−, 2 (0) = ηp P Lb exp (((Γp (σap + σep ) N¯ 2 − σap N) − α p )(L − L1))

Pp+, 2 (0) = (1 − l p ) P p+, 1 (1) + ηp P Lf1

(8-g)

P p−, 1 (1) = (1 − l p ) Pp−, 2 (0) + ηp P Lb1

(8-h)

A straightforward integrations of (6) d (Ps±, i (zi ) Ps∓, i (zi ))/dzi = 0 (i = 1, 2) and (7) d (P p±, i (zi ) P p∓, i (zi ))/dzi = 0

(i = 1, 2) is reduced to

P s+, 1 (0) P s−, 1 (0) = P s+, 1 (1) P s−, 1 (1)

(9-a)

Ps+, 2 (0) Ps−, 2 (0) = Ps+, 2 (1) Ps−, 2 (1)

(9-b)

P p+, 1 (0) P p−, 1 (0) = P p+, 1 (1) P p−, 1 (1)

(9-c)

Pp+, 2 (0) Pp−, 2 (0) = Pp+, 2 (1) Pp−, 2 (1)

(9-d)

+ ηp P Lf1 R2 (λ p ) exp (2((Γp (σap + σep ) N¯ 2 − σap N) − α p )(L − L1))

(14)

where N¯ 2 = ( − ln (R1 (λ s ) R2 (λ s ))/L + Γs σas N + α s )/(Γs (σas + σes )) [27]. Due to the known pump reflectivity R2 (λ p ), pump filling factor Γp , pump absorption σap and emission cross-section σep , scattering losses α p , the forward power PLf1 and backward pump

Ps+Ps−

where is a constant, independent of z. Using the boundary conditions, we find

P s+, 1 (0) P s−, 1 (0) = P s+, 1 (1) P s−, 1 (1) = Ps+, 2 (0) Ps−, 2 (0) = Ps+, 2 (1) Ps−, 2 (1)

For the known signal reflectivity R1 (λ s ), R2 (λ s ), the pump conversion

(10-a)

power PLb1 at side-pump position z = L1, backward pump power PLb at pump position z = L , as well as Yb3 þ -doped concentration N and fiber lengthL1, L , the initial value of the backward pump power − − Pp, 1 (0) of Eq. (13) and P p, 2 (0) of Eq. (14) can be approximately calculated. The dependent variables of the initial guess values of the forward signal and pump powers Ps+, 1 (0), Ps+, 2 (0), Pp+, 1 (0), Pp+, 2 (0) are

P s+, 1 (0) = R1 (λ s ) P s−, 1 (0)

(15)

Ps+, 2 (0) = R1 (λ s ) P s−, 1 (0)2 /Ps−, 2 (0)

(16)

Please cite this article as: X. Hu, et al., Optics Communications (2014), http://dx.doi.org/10.1016/j.optcom.2014.09.074i

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P p+, 1 (0) = R1 (λ p ) P p−, 1 (0) + ηp P0f

(17)

Pp+, 2 (0) =

P p+, 1 (0) P p−, 1 (0) + ηp P Lf1 Pp−, 2 (0) + ηp2 P Lf1 P Lb1/(1 − l p ) Pp−, 2 (0)

+

ηp P Lb1/(1

− lp)

(18)

That is, the independent variables of the backward laser power Ps−, 1 (0), Ps−, 2 (0) and backward pump power Pp−, 1 (0), Pp−, 2 (0) can be transferred by the only independent variable of the backward laser power Ps−, 1 (0), Ps−, 2 (0). Thus, adjusting the conversion efficiency η means changing the initial values of the backward laser power Ps−, 1 (0), Ps−, 2 (0). We only provided a good initial approximation guess for the backward laser power Ps−, 1 (0), Ps−, 2 (0) by setting suitable conversion efficiency guess η , the desired right solutions can be expected rapidly and directly using simple shooting method. We use the boundary conditions (8-c), (8-f), (8-g) and (8-h) as nonlinear equations in the Newton method.

F1 (P s−, 1 (0), P p−, 1 (0), Ps−, 2 (0), Pp−, 2 (0)) = (1 − l s ) P s+, 1 (1) − Ps+, 2 (0) (19) F2 (P s−, 1 (0), P p−, 1 (0), Ps−, 2 (0), Pp−, 2 (0)) = P p−, 1 (1) − (1 − l p ) Pp−, 2 (0) − ηp P Lb1

(20)

Table 1 Excellent initial guess functions of variables in shooting method for 4-point side pump YDFLs. Variables

Excellent initial guess functions for 4-point pump YDFLs

Independent variables

Ps−,1(0) =

Ps−,2(0) =

R2(λs) R1(λs)

(1 − R2(λs))Ptolηpη

ηc ≤ η ≤ 1

P − (0) s,1 1 + (1 / R1(λs)R2(λs) − 1)L1 / L P − (0) s,1

Ps−,3(0)

=

Pp−,1(0)

= ηpPLb1exp(((Γp(σap + σep)N¯ 2 − σapN) − αp)L1)

1 + (1 / R1(λs)R2(λs) − 1)L2 / L

+ ηpPLb2(1 − lp)exp(((Γp(σap + σep)N¯ 2 − σapN) − αp)L2) + ηpPLb(1 − lp)2exp(((Γp(σap + σep)N¯ 2 − σapN) − αp)L) Pp−,2(0)

= ηpPLb2exp(((Γp(σap + σep)N¯ 2 − σapN) − αp)(L2 − L1)) + ηpPLb(1 − lp)exp(((Γp(σap + σep)N¯ 2 − σapN) − αp)(L − L1))

Pp−,3(0)

= ηpPLbexp(((Γp(σap + σep)N¯ 2 − σapN) − αp)(L − L2)) + ηpPLf R2(λ p)exp(2((Γp(σap + σep)N¯ 2 − σapN) − αp)(L − L2)) 2

Dependent variables

Ps+,1(0) = R1(λs)Ps−,1(0) Ps+,2(0) = R1(λs)(Ps−,1(0))2/Ps−,2(0) Ps+,3(0) = R1(λs)(Ps−,1(0))2/Ps−,3(0) Pp+,1(0) = ηpP0f + R1(λ p)Pp−,1(0) Pp+,2(0) =

Pp+,3(0) = Note

P − (0) + ηpP b / (1 − lp) L2 p,3

Ptol = P0f + ∑2i = 1 (PLf + PLb ) + PLb N¯ 2 =

Fig. 2. The flow chart of the shooting method for 3-point pump YDFLs.

P + (0)P − (0) + ηpP f P − (0) + ηp2P f P b / (1 − lp) L1 L1 L1 p,2 p,1 p,1 P − (0) + ηpP b / (1 − lp) L1 p,2 P + (0)P − (0) + ηpP f P − (0) + ηp2P f P b / (1 − lp) L2 L2 L2 p,3 p,2 p,2

−

i i 1 ln(R1(λs)R2(λs)) + ΓsσasN + αs L Γs(σas + σes)

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F3 (P s−, 1 (0),

P p−, 1 (0),

Ps−, 2 (0),

Pp−, 2 (0))

=

Ps−, 2 (1)

−

R2 (λ s ) Ps+, 2 (1)

(21)

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 are

(22)

∂F1(i) /∂ (P s−,1 (0))(i)

F4 (P s−, 1 (0), P p−, 1 (0), Ps−, 2 (0), Pp−, 2 (0)) = Pp−, 2 (1) − R2 (λ p ) Pp+, 2 (1) − ηp P Lb

5

Using the Newton method, the iteration formula can be denoted

≈ [F1(i) (P s−, 1 (0) + hstep, P p−, 1 (0), Ps−, 2 (0), Pp−, 2 (0)) − F1(i) (P s−, 1 (0), P p−, 1 (0), Ps−, 2 (0), Pp−, 2 (0))]/hstep

[P s−,1 (0); P p−,1 (0); Ps−,2 (0); Pp−,2 (0)](i + 1)

∂F2(i) /∂ (P s−,1 (0))(i)

= [P s−,1 (0); P p−,1 (0); Ps−,2 (0); Pp−,2 (0)](i) − J −1 [F s(,i)1; F p(i,)1; Fs(,i)2; F p(i,)2]

(23)

≈ [F2(i) (P s−, 1 (0) + hstep, P p−, 1 (0), Ps−, 2 (0), Pp−, 2 (0)) − F2(i) (P s−, 1 (0), P p−, 1 (0), Ps−, 2 (0), Pp−, 2 (0))]/hstep

where the Jacobian matrix J can be described

∂F3(i) /∂ (P s−,1 (0))(i) ≈ [F3(i) (P s−, 1 (0) + hstep, P p−, 1 (0), Ps−, 2 (0), Pp−, 2 (0))

J= ⎡ (i ) − (i ) ⎢ ∂F1 /(P s,1 (0)) ⎢ (i ) − ( i) ⎢ ∂F2 /(P s,1 (0)) ⎢ (i ) − (i ) ⎢ ∂F3 /(P s,1 (0)) ⎢ (i ) − ( ⎢⎣ ∂F4 /(P s,1 (0)) i)

− F3(i) (P s−, 1 (0), P p−, 1 (0), Ps−, 2 (0), Pp−, 2 (0))]/hstep ∂F1(i) /∂ (P p−,1 (0)) (i) ∂F1(i) /∂ (P s−,2 (0)) (i) ∂F2(i) /∂ (P p−,1 (0)) (i)

∂F2(i) /∂ (P s−,2 (0)) (i)

∂F3(i) /∂ (P p−,1 (0)) (i) ∂F3(i) /∂ (P s−,2 (0)) (i) ∂F4(i) /∂ (P p−,1 (0)) (i) ∂F4(i) /∂ (P s−,2 (0)) (i)

⎤ ∂F1(i) /∂ (P p−,2 (0)) (i) ⎥ ⎥ − (i ) i ( ) ∂F2 /∂ (P p,2 (0)) ⎥ ⎥ ∂F3(i) /∂ (P p−,2 (0)) (i) ⎥ ⎥ ∂F4(i) /∂ (P p−,2 (0)) (i) ⎥⎦ (24)

where i is iteration steps, for each iZ 1. The Jacobian matrix J associated with nonlinear Eqs. (19)–(22) that the 42 partial derivatives be determined and evaluated. The

∂F4(i) /∂ (P s−,1 (0))(i) ≈ [F4(i) (P s−, 1 (0) + hstep, P p−, 1 (0), Ps−, 2 (0), Pp−, 2 (0)) − F4(i) (P s−, 1 (0), P p−, 1 (0), Ps−, 2 (0), Pp−, 2 (0))]/hstep

(25) -8 W

where hstep is small in absolute value, such as 10 . The specified flow chart of the shooting method for 3-point pump YDFLs is schematically described in Fig. 2. The specific steps of the novel shooting method for 3-point side pump YDFLs are detailed in the following:

Table 2 The iteration process of the backward signal and pump powers with different guess value of the conversion efficiency η .

η

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

(Ps−, 1 (0), P p−, 1 (0), Ps−, 2 (0), P p−, 2 (0)) in first and second iteration steps for 3-point pump YDFLs (W) Iteration step 1

Iteration step 2

(0.5539, 0.1228, 0.1742, 0.1226) (1.1078, 0.1228, 0.3484, 0.1226) (1.6617, 0.1228, 0.5227, 0.1226) (2.2156, 0.1228, 0.6969, 0.1226) (2.7696, 0.1228, 0.8711, 0.1226) (3.3235, 0.1228, 1.0453, 0.1226) (3.8774, 0.1228, 1.2196, 0.1226) (4.4313, 0.1228, 1.3938, 0.1226) (4.9852, 0.1228, 1.5680, 0.1226) (5.5391, 0.1228, 1.7422, 0.1226) (6.0930, 0.1228, 1.9165, 0.1226) (6.6469, 0.1228, 2.0907, 0.1226) (7.2008, 0.1228, 2.2649, 0.1226) (7.7548, 0.1228, 2.4391, 0.1226) (8.3087, 0.1228, 2.6133, 0.1226) (8.8626, 0.1228, 2.7876, 0.1226) (9.4165, 0.1228, 2.9618, 0.1226) (9.9704, 0.1228, 3.1360, 0.1226) (10.5243, 0.1228, 3.3102, 0.1226) (11.0782, 0.1228, 3.4845, 0.1226)

(  100.3422, 5.4041,  51.3046, 2.1334) (  601.8657, 15.5622,  319.7476, 4.8958) (672.8142,  11.1504, 358.2051,  3.2263) (293.2228,  3.4911, 154.8905,  0.9694) (206.5028,  1.8737, 107.8000,  0.5052) (166.3875,  1.1913, 85.6939,  0.3110) (142.6274,  0.8235, 72.4242,  0.2056) (126.5663,  0.5970, 63.3432,  0.1399) (114.8650,  0.4460, 56.6579,  0.0954) (105.9847,  0.3398, 51.5398,  0.0634) (98.9289,  0.2617, 47.4417,  0.0392) (93.1540,  0.2022, 44.0641,  0.0204) (88.3590,  0.1558, 41.2425,  0.0054) (84.3053,  0.1190, 38.8443, 0.0068) (80.8221,  0.0890, 36.7743, 0.0170) (77.8370,  0.0646, 34.9905, 0.0255) (75.2174,  0.0442, 33.4200, 0.0328) (72.9062,  0.0270, 32.0290, 0.0390) (70.8565,  0.0124, 30.7908, 0.0444) (69.0511,  4.96e  6, 29.6973, 0.0491)

Note

Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail Fail

Table 3 The detailed iteration process of the backward signal and pump powers with η ¼ 0.21, 0.22 and 0.23. Iteration step

1 2 3 4 5 6

(Ps−, 1 (0), P p−, 1 (0), Ps−, 2 (0), P p−, 2 (0)) for 3-point pump YDFLs

η ¼ 0.21

η ¼ 0.22

η ¼ 0.23

(11.6321, 0.1228, 3.6587, 0.1226) (67.4038.0.0108,28.6957,0.0533) (49.5323, 0.0943.18.1448,0.0940) (47.5759.0.1065,16.3075,0.0957) (47.4882, 0.1068, 16.3120.0.0957) (47.4882, 0.1068.16.3120,0.0957)

(12.1860.0.1228, 3.8329, 0.1226) (65.9259.0.0202.27.7943.0.0569) (49.4222.0.0963.17.8790.0.0943) (47.5496.0.1066.16.3134.0.0957) (47.4882.0.1068.16.3120.0.0957) (47.4882.0.1068.16.3120.0.0957)

(12.7400, 0.1228, 4.0071, 0.1226) (64.5899, 0.0284.26.9773,0.0602) (49.2921, 0.0979.17.6560,0.0945) (47.5316, 0.1067, 16.3159.0.0957) (47.4882, 0.1068.16.3120,0.0957) (47.4882, 0.1068, 16.3120.0.0957)

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Step 1: the side-pumping position z = L1 divide the interval of integration into 2 regions [0, L1], [L1, L]. Step 2: let L 0 = 0, L 2 = L and z1 = (z − L 0 )/(L1 − L 0 ), z2 = (z − L1)/(L 2 − L1) and all the variable of 2 intervals are [0,1], reformulate the nonlinear coupled ordinary differential

equations (6)–(7) with boundary conditions (8-a)–(8-h) of the multipoint pump Yb3 þ -doped fiber laser in each region [0,1]. Step 3: set the absolute tolerance (AbsTol) and boundary conditions error tolerances (Tol).

Fig. 3. Updating the independent variables of (Ps−, 1 (0), P p−, 1 (0), Ps−, 2 (0), P p−, 2 (0)) as a function of iteration steps at different conversion efficiency guess value η ≥ 0.21 for 3-point pumping YDFLs.

Fig. 4. (a) Output powers and iteration times and (b) boundary value error as a function of the guess value of conversion efficiency η for 3-point pumping YDFLs.

Please cite this article as: X. Hu, et al., Optics Communications (2014), http://dx.doi.org/10.1016/j.optcom.2014.09.074i

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Step 4: the independent variables of backward signal and pump power (Ps−, 1 (0), Ps−, 2 (0), Pp−, 1 (0), Pp−, 2 (0)) are from guess functions (11)–(14), while the dependent variables of forward signal and pump power (Ps+, 1 (0), Ps+, 2 (0), Pp+, 1 (0), Pp+, 2 (0)) are from (15)–(18). Step 5: use the boundary conditions (19)–(22) as nonlinear equations in the shooting method. Step 6: generate the iteration formula (23) using the Newton method and evaluate (24) of the Jacobian matrix J. Step 7: output Ps±, 1 (z1), Pp±, 1 (z1), Ps±, 2 (z2 ), Pp±, 2 (z2 ) in interval [0,1]. Step 8: retransform the signal powers Ps± (z) and pump powers Pp± (z). Step 9: output Ps± (z), Pp± (z) in interval [0,L].

7

Similarly, the shooting method can be applied to the 4-point side pumping YDFLs, and the initial guess functions are schematically described in Table 1.

4. Simulation and discussion In all examples, unless stated otherwise, the simulation parameters are [28]: h = 6.63 × 10−34Js , c = 3 × 108ms−1, λ p = 974nm, λ s = 1100nm, R1 (λ s ) = 0.995, R2 (λ s ) = 0.035, R1 (λ p ) = 0.035 [23],

R2 (λ p ) = 0.035

[23],

σas = 1 × 10−27m2, σap = 26 × τ = 0.8 ×

ηp = 0.95,

10−25m2,

10−3s ,

σes = 1.6 × 10−25m2, Γp = 0.0059,

N = 2.96 ×

l p = 0.05,

α s = 4 × 10−4m−1,

α p = 2 × 10−3m−1,

σep = 26 × 10−25m2,

Γs = 0.82,

1025m−3,

l s = 0.01,

A = 7.07 × 10−10m2,

ηp0 = 0.95,

Ptol = 300W

ηpL = 0.95, Tol = 10−5W ,

AbsTol = 10−4W and RelTol = 10−4 . For simplicity, the i-th point side pump position with n point pumping is L i=L × i/(n − 1) 0 ≤ i ≤ n − 1

Fig. 5. Critical guess value of conversion efficiency and output powers as a function of fiber length for 300 W total pump powers.

, side pump power P0f = PLb = PLfi = PLbi = Ptol/(2n − 2) 1 ≤ i ≤ n − 2 and the conversion efficiency guessη is generated from 0.01 to 1 with interval 0.01. Note that all calculations are carried on an Intel CPU T2300 notebook-PC using MATLAB program. The first and second iteration process of the backward signal and pump powers (Ps−, 1 (0), Pp−, 1 (0), Ps−, 2 (0), Pp−, 2 (0)) with the guess value of the conversion efficiency η for 3-point pump YDFLs are schematically described in Table 2, respectively. When the conversion efficiency guess η is less than 0.20 in Table 2, Ps,− 1 (0) or − Pp, 1 (0) is lower than 0 W after only one iteration step, which leads the shooting method to fail. The detailed iteration process of (Ps−, 1 (0), Pp−, 1 (0), Ps−, 2 (0), Pp−, 2 (0)) at η ¼0.21, 0.22 and 0.23 for 3-point pump YDFLs are described in Table 3. The corresponding powers (Ps−, 1 (0), Pp−, 1 (0), Ps−, 2 (0), Pp−, 2 (0)) are greater than 0 W and succeed rapidly to converge to the same value (47.4882, 0.1068, 16.3120, 0.0957) W at the sixth iteration step in Table 3.

Table 4 Simulation and experiment with different pump power. Pump power (W)

Output power (W)

1.2 1.7 2 2.4 2.7 3

Experiment data [29]

Shooting method with initial guess functions

0 0.17 0.34 0.51 0.7 0.898

0 0.0372 0.2346 0.4971 0.6941 0.8910

Table 5 Output power and computation time as a function of fiber length using different methods. Fiber length (m) Shooting method with initial guess functions

5 10 15 20 25 30 35 40 45 50

NSTM-bvp4c [21]

Shooting method with answer ranges definition [30]

Output power (W)

Computation time (s)

Output power (W) Computation time (s) Output power (W)

328.31803 434.49487 467.46734 479.83841 484.74297 486.54098 486.95807 486.74190 486.23126 485.58374

1.303908 1.295142 1.229014 1.206181 1.313717 1.173159 1.114805 1.149700 1.122555 1.090587

328.31799 434.49487 467.46737 479.83840 484.74290 486.54083 486.95704 486.74032 486.22971 485.58226

1.457678 1.745663 1.978693 1.878122 2.571070 1.678128 2.537965 2.572989 2.613976 3.125511

1.9021E  121 1.7314E  137 Fail to converge Fail to converge Fail to converge Fail to converge Fail to converge Fail to converge Fail to converge 485.58374

Computation time (s) 1.406278 1.287751 – – – – – – – 3.360107

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Updating the independent variables of backward signal and pump power (Ps−, 1 (0), Pp−, 1 (0), Ps−, 2 (0), Pp−, 2 (0)) at different conversion efficiency guess η ≥ 0.21 for 3-point pump is shown in Fig. 3. From Fig. 3, all the powers (Ps−, 1 (0), Pp−, 1 (0), Ps−, 2 (0), Pp−, 2 (0)) are succeed to converge to (47.4882, 0.1068, 16.3120, 0.0957)W with given accuracy AbsTol = 10−4W after less than five iteration steps. The output powers, iteration times and boundary value errors as a function of the guess value of conversion efficiency η for 3-point pump YDFLs are respectively described in Fig. 4. Obviously, the output power is between 244.34301 W and 244.34303 W at η ≥ 0.21 in Fig. 4(a), and all of the pump and laser's boundary condition errors are less than givenTol = 10−5W in Fig. 4(b). The exact output power is 244.3430 W at given AbsTol = 10−4W . Hence, the guess values of conversion efficiency have no influence on the output power with given accuracy when the guess values of conversion efficiency η ≥ 0.21 in YDFLs. Furthermore, the speed of shooting method with initial guess functions is rapid for iteration times r5 in Fig. 4(a). The corresponding guess value of conversion efficiency η = 0.21 for 3-point pump is named as the critical guess value of conversion efficiency ηc . Obviously, the critical guess value of conversion efficiency ηc is the boundary value of the guess value of conversion efficiency η . When the guess value of conversion efficiency η is larger than the critical guess value of conversion efficiency ηc such as ηc = 0.21 in Table 3 and Fig. 4, the initial guess values of (Ps−, i (0), P p−, i (0)) (i = 1, 2), generating from initial guess functions (15)–(18), are suitable initial guess values for the shooting method. That is, in excellent initial guess functions for signal power and pump power of 3-point pumping YDFLs, the guess value of conversion efficiency is greater than or equal to the critical guess value of conversion efficiency. Like the 3-point pumping YDFLs, the guess value of conversion efficiency η for excellent initial guess functions of 4-point pumping YDFLs, as shown in Table 1, must be also greater than or equal to the critical guess value of conversion efficiency ηc . Now, we explore how fiber length to effect the critical guess value of conversion efficiency ηc in the excellent initial guess functions. The critical guess value of conversion efficiency ηc and output powers as a function of fiber length for 3-point and 4-point pump YDFLs is detailed in Fig. 5. The critical guess value of conversion efficiency ηc rises rapidly from 0.02 at fiber length 1 m to 0.21 at 8 m and the corresponding output power rises swift from 8.4758 W to 216.4637 W in 3-point pumping YDFLs; while the critical guess value of conversion efficiency ηc increases quickly from 0.02 at fiber length 1 m to 0.18 at 6 m and the corresponding output power goes swift up from 6.7710 W to 179.8863 W for 4-point pumping YDFLs. Clearly, the critical guess value of conversion efficiency ηc and corresponding output power of 4-point pumping YDFLs are less than those of 3-point pumping YDFLs. According to the experiment data from [29, Fig. 5], we take λ p = 980nm, λ s = 1100nm, R1 (λ s ) = 1, R2 (λ s ) = 0.035,

R1 (λ p ) = 0.0, R2 (λ p ) = 0.0, α p = 0m−1, α s = 0m−1, σas = 1 × 10−27m2, σes = 1.6 × 10−25m2, σep = 1.5 × 10−24m2, σap = 1.5 × 10−24m2, Γp = 0.00097, Γs = 0.82, ηp = 0.786 , l p = 0.127, l s = 0.0, L1 = 0.3m,

L 2 = 0.4m, L 3 = L = 38m, A = 1.131 × 10−10m2, τ = 0.8 × 10−3s , N = 1.435 × 1026m−3, Tol = 10−5W , AbsTol = 10−4W , RelTol = 10−4 , PLf1 = PLb2 = Ptol/2, and PLf0 = PLb1 = PLb2 = PLb3 = 0 as parameters to simulate the Yb3 þ -doped fiber laser. Simulation and experiment with different total pump power are correspondingly detailed in Table 4. Comparing the output power of the experiment data and the simulation results using simple shooting method, it is easy to find that the simulation results are very excellent in agreement with the experiment data when pump power is larger than 2.4 W. The results show that the simple shooting method with initial guess functions is easy to get the exact numerical results, which

meet the desired requirement high accuracy for strongly pumped fiber laser. Finally, to further verify the efficiency and speed of the simple shooting method, we investigate and compare simultaneously the output power and computation time for 4-point pump YDFLs with different fiber lengths and side pump power

P0f = PLb = PLf1 = PLb1 = PLf2 = PLb2 = 100W using the simple shooting method with initial guess functions, NSTM-bvp4c [21] and shooting method with answer ranges definition [30].The other parameters is the same as Fig. 5. In Table 5, compared the shooting method with initial guess functions with the NSTM-bvp4c, all the differences of output power are less than 0.0016 W. Moreover, the computation time of the shooting method with initial guess functions is shorter than that of the NSTM-bvp4c at all different fiber lengths. Simulation result using shooting method with answer ranges definition can succeed to converge to exact value only when fiber length is 50 m, while most of the simulation results fail to convergence or converge to values near to zero such as 1.9021E  121 and 1.7314E  137.These simulation results show that the unsuitable initial trial values cause the simulation results to fail to converge or converge to values near to zero in the shooting method with answer range. Among these three methods, the shooting method with initial guess functions is most reliable and fastest for multipoint pumping YDFLs.

5. Conclusions We develop and discuss in detail the shooting method with excellent initial guess functions to solve the difficulty of setting initial values in YDFLs. Simulation results show that shooting method with excellent initial guess functions can get rapidly the exact values of YDFLs after less than five iteration times. That is, generating from the excellent initial guess functions, the initial guess values of the signal powers and pump powers are suitable initial guess values for the shooting method. In excellent initial guess functions for signal power and pump power in 3-point pumping or 4-point pumping YDFLs, the guess value of conversion efficiency is greater than or equal to the critical guess value of conversion efficiency. The critical guess value of conversion efficiency is less than 0.21 in 3-point pumping or 0.18 in 4-point pumping YDFLs for all the fiber length. The critical conversion efficiency guess value of 4-point pumping YDFLs is lower than that of 3-point pumping YDFLs. Compared the simple shooting method with the NSTM-bvp4c, all the output power differences are less than 0.0016 W. In addition, the computation time of the simple shooting method is shorter than that of the NSTM-bvp4c at different fiber lengths. That is, the simple shooting method with the excellent initial guess functions is reliable and fast for multipoint pumping YDFLs. We are confident that the shooting method with excellent initial guess functions can be extended into more than four-point pumping YDFLs.

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).

Please cite this article as: X. Hu, et al., Optics Communications (2014), http://dx.doi.org/10.1016/j.optcom.2014.09.074i

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Please cite this article as: X. Hu, et al., Optics Communications (2014), http://dx.doi.org/10.1016/j.optcom.2014.09.074i

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