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Adaptive shooting method for 4-point side-pumping high power Yb3+-doped double-clad fiber lasers
Optical Fiber Technology 22 (2015) 13–22
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Adaptive shooting method for 4-point side-pumping high power Yb3+-doped double-clad fiber lasers Xudong Hu a,b, 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
a r t i c l e
i n f o
Article history: Received 27 July 2014 Revised 8 December 2014 Available online 13 January 2015 Keywords: Shooting method Simple identification process Side-pumping Yb3+-doped fiber Fiber lasers
a b s t r a c t An adaptive shooting method for solving 4-point side-pumping high power Yb3+-doped double-clad fiber lasers (YDCFLs) is developed. The adaptive shooting method combines the simple identification process and shooting method procedure. Simulation results show that the adaptive shooting method can identify automatically and easily eight different cases in the 4-point side-pumping YDCFLs with different pump schemes. The initial estimate values of pump powers as independent variables are given approximate expressions and the signal powers are set random functions to speed the adaptive shooting method. The adaptive shooting method can succeed rapidly to get the exact results after average less than eight iteration steps updating initial guess values. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In recent years, high power Yb3+-doped double-clad fiber lasers (YDCFLs) and related pump technologies have been attracting increasing attention. A highly efficient cladding-pumped Yb3+-doped fiber laser generating >2.1 kW of continuous-wave output power at 1.1 lm has been demonstrated [1]. Compared with end-pumping scheme, multipoint pump scheme can overcome the limitation of pump source and thermal effect in YDCFLs [2,3]. Many pump technologies such as embedded-mirror side pumping [4], monolithic integrated all-glass combiner [5,6] and side-pump coupler with refractive index valley configuration [7] have been developed for multipoint single or bidirectional side pumping Yb3+-doped fiber lasers. Many shooting methods have been applied for the solution of YDCFLs [8–10]. 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 backward pump power or backward signal power is lower than 0 which is unable to meet the practical physical problems [8]. A fast and stable shooting algorithm, using the Newton–Raphson method to solve the two-point boundary value problem of linear-cavity YDCFLs, has been demonstrated [9]. However, the initial estimate, given only several simple data, may bring the shooting method to ⇑ Corresponding author at: Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China. E-mail addresses: [email protected] (X. Hu), [email protected] (L. Pei). http://dx.doi.org/10.1016/j.yofte.2014.12.001 1068-5200/Ó 2014 Elsevier Inc. All rights reserved.
fail or unable to converge to the exact solutions. A combined algorithm [10] with shooting method and relaxation method has been studied for solving the model of YDCFLs. Since both shooting method and relaxation method need suitable predicted variable initial value, the combined algorithm is very doubtful to succeed to converge. Shooting method with simple control strategy successfully deal with poor initial guess values of the backward signal and pump powers by random functions in end pumping YDCFLs [8]. However, it is very difficult and slow for solving n-point pumping YDCFLs only using random functions adjusting initial guess values of 2(n 1) independent variables. In addition, the backward pump powers cannot be taken as independent variables when the backward pump powers are nearly to zero, which lead to fail or convergent to error results in multipoint pump YDCFLs. An adaptive shooting method is developed for solving the above problems in multipoint pump YDCFLs. For simplicity, the 4-point side-pumping YDCFLs is only discussed in this paper. 2. 4-Point side-pumping Yb3+-doped double-clad fiber lasers model A typical 4-point side-pumping high power linear cavity Yb3+-doped fiber laser is described schematically in Fig. 1. For the 4-point side-pumping YDCFLs, pump positions z = L1 and z = L2 divide Yb3+-doped fiber into 3 intervals [0, L1], [L1, L2], [L2, L]. 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
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X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
Fig. 1. Schematic illustration of 4-point side-pumping high power Yb3+-doped double-cladding fiber lasers (YDCFLs).
state absorption (ESA) are negligible for strong pumping conditions in the 4-point side-pumping YDCFLs [11]. The steady-state rate equations in the 4-point side-pumping YDCFLs are described by the following set of nonlinear coupled ordinary differential Eqs. (1)–(3) [12]. Cp rap ðP þ p ðzÞþP p ðzÞÞkp
Cs ras ðPþ s ðzÞþP s ðzÞÞks
þ N2 ðzÞ hcA hcA ¼ Cp ðrap þrep ÞðPþ Cs ðras þres ÞðPþ N p ðzÞþP p ðzÞÞkp s ðzÞþP s ðzÞÞks 1 þ þ hcA s hcA
ð1Þ
dP p ðzÞ ¼ gðP p ðzÞÞ ¼ Cp frap N ðrap þ rep ÞN2 ðzÞgP p ðzÞ ap P p ðzÞ ð2Þ dz dP ðzÞ ¼ f ðPs ðzÞÞ ¼ ½Cs f½res þ ras N2 ðzÞ ras Ng as P s ðzÞ ð3Þ s dz
where N is the rare earth ion dopant concentration. N 2 ðzÞ is the upper lasing level population density. P p ðzÞ and P s ðzÞ are the pump power and laser signal power along the fiber, respectively. The plus and minus superscripts represent propagation along 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 emission cross-section, respectively. ras and res are the laser signal absorption and the emission crosssection, 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, Planck’s constant, light velocity and spontaneous lifetime, respectively. The boundary conditions [13,14] at side pump positions z ¼ 0; z ¼ L, z = Li (i = 1, 2) are
Pþs ð0Þ ¼ R1 ðks ÞPs ð0Þ
ð4-aÞ
Ps ðLÞ
R2 ðks ÞPþs ðLÞ
ð4-bÞ
Pþp ð0Þ ¼ R1 ðkp ÞPp ð0Þ þ gp0 P f0
ð4-cÞ
Pp ðLÞ ¼ R2 ðkp ÞP þp ðLÞ þ pL PbL Pþsiþ1 ðLi Þ ¼ ð1 lsi ÞPþsi ðLi Þ ði ¼ Psi ðLi Þ ¼ ð1 lsi ÞPsiþ1 ðLi Þ ði ¼
ð4-dÞ
¼
g
Pþpiþ1 ðLi Þ
lpi ÞPþpi ðLi Þ
þg
1; 2Þ
ð4-eÞ
1; 2Þ
ð4-fÞ
f pi P Li
ði ¼ 1; 2Þ
ð4-gÞ
Ppi ðLi Þ ¼ ð1 lpi ÞP piþ1 ðLi Þ þ gpi PbLi ði ¼ 1; 2Þ
ð4-hÞ
¼ ð1
Fig. 2. The flow chart of simple identification process in the adaptive shooting method for 4-point side-pumping YDCFLs.
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X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
where P f0 is the side pump power at the end position z ¼ 0 and PbL denotes the side pump power at z ¼ L, respectively. PfLi and P bLi are the forward and backward pump powers at pump position z ¼ Li ði ¼ 1; 2Þ, respectively. R1 ðks Þ and R1 ðkp Þ are the input mirror reflectivity of signal and pump wavelength at z ¼ 0, respectively. R2 ðks Þ and R2 ðkp Þ are the output mirror reflectivity of signal and pump wavelength at z ¼ L, respectively. gp0 and gpL are the pump coupling efficiency at z ¼ 0 and z ¼ L, respectively. gpi are the pump coupling efficiency at pump position z ¼ Li . lpi and lsi are the pump and signal leakage rate at z ¼ Li , correspondingly. For simplicity, the pump coupling efficiency, pump and signal leakage rate on each pump position are the same, i.e., lpi ¼ lp , gpi ¼ gp , gp0 ¼ gpL ¼ gp and lsi ¼ ls . If L0 ¼ 0; L3 ¼ L, and z1 ¼ ðz L0 Þ=ðL1 L0 Þ, z2 ¼ ðz L1 Þ= ðL2 L1 Þ, z3 ¼ ðz L2 Þ=ðL3 L2 Þ in each segment fiber [15,16], then all the new variables of 3 segment interval are changed [0,1]. For each segment fiber, the new variables are formulated the pump and signal powers P p;i ðzi Þ ¼ P p ðzÞ, P s;i ðzi Þ ¼ P s ðzÞ ði ¼ 1; 2; 3Þ. The nonlinear coupled ordinary differential Eqs. (2) and (3) of the 4-point side-pumping YDCFLs in [0,1] are
dP p;i ðzi Þ dz ¼ g Pp;i ðzi Þ ¼ ðLi Li1 Þg Pp;i ðzi Þ ði ¼ 1; 2; 3Þ ð5Þ dzi dzi dP s;i ðzi Þ dz ¼ f Ps;i ðzi Þ ¼ ðLi Li1 Þf Ps;i ðzi Þ ði ¼ 1; 2; 3Þ ð6Þ dzi dzi
In addition, the 4-point boundary conditions in each interval [0,1] are
Pþs;1 ð0Þ ¼ R1 ðks ÞPs;1 ð0Þ Pþp;1 ð0Þ Ps;3 ð1Þ Pp;3 ð1Þ Pþs;i ð0Þ
¼
R1 ðkp ÞPp;1 ð0Þ
¼
R2 ðks ÞPþs;3 ð1Þ
¼
R2 ðkp ÞPþp;3 ð1Þ
¼ ð1
ð7-aÞ f p P0
þg
ð7-cÞ b p PL
ð7-dÞ
ði ¼ 2; 3Þ
ð7-eÞ
þg
ls ÞPþs;i1 ð1Þ
Pþp;i ð0Þ ¼ ð1 lp ÞPþp;i1 ð1Þ þ gp PfLi Ps;i1 ð1Þ Pp;i1 ð1Þ
ð7-bÞ
ði ¼ 2; 3Þ
¼ ð1
ls ÞPs;i ð0Þ
ði ¼ 2; 3Þ
¼ ð1
lp ÞPp;i ð0Þ
b p P Li1
þg
ð7-fÞ ð7-gÞ
ði ¼ 2; 3Þ
Fig. 3. The flow chart of shooting method procedure in the adaptive shooting method to calculate case I for 4-point side-pumping YDCFLs.
ð7-hÞ
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X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
3. Adaptive shooting method for 4-point side-pumping YDCFLs In the 4-point side-pumping YDCFLs, as mentioned Fig. 1, the backward signal powers P s;1 ðz1 Þ; P s;2 ðz2 Þ; P s;3 ðz3 Þ and backward pump powers P p;1 ðz1 Þ; P p;2 ðz2 Þ; P p;3 ðz3 Þ are generally viewed as independent variables for shooting method. According to all the different pump configurations in Fig. 1, there are eight cases: (I) P p;1 ðz1 Þ > 0; P p;2 ðz2 Þ > 0 and P p;3 ðz3 Þ > 0; (II) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ > 0 and P ðz Þ > 0; (III) P ðz Þ > 0; P ðz Þ ¼ 0 and P ðz Þ > 0; 3 1 2 3 p;3 p;1 p;2 p;3 (IV) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ > 0; (V) P p;1 ðz1 Þ > 0; P p;2 ðz2 Þ > 0 and P p;3 ðz3 Þ ¼ 0; (VI) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ > 0 and Pp;3 ðz3 Þ ¼ 0; (VII) P p;1 ðz1 Þ > 0; P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ ¼ 0; (VIII) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ ¼ 0.
Clearly, the backward pump powers P p;1 ð1Þ; P p;2 ð1Þ and P p;3 ð1Þ in 3+ the 4-point side-pumping Yb -doped double-clad fiber lasers are respectively the maximum value of P p;1 ðz1 Þ, P p;2 ðz2 Þ and P p;3 ðz3 Þ; i.e., Pp;1 ðz1 Þ 6 P p;1 ð1Þ; P p;2 ðz2 Þ 6 P p;2 ð1Þ and P p;3 ðz3 Þ 6 Pp;3 ð1Þ. And the absolute error tolerance (AbsTol) of the shooting method (S-method) is small in absolute value such as 104 W. The exact analytical solutions of Eqs. (1)–(3) have not been acquired up to now [17], while exact numerical solutions with given error tolerance of Eqs. (1)–(3) have been obtained by several methods [8,12,18]. The backward pump powers P p;1 ðz1 Þ, P p;2 ðz2 Þ or P p;3 ðz3 Þ are zero for numerical solution when the backward pump powers P p;1 ð1Þ or P p;2 ð1Þ or P p;3 ð1Þ is less than or equal to the absolute error tolerance (i.e., P p;1 ð1Þ 6 AbsTol, P p;2 ð1Þ 6 AbsTol or P p;3 ð1Þ 6 AbsTol).
Table 1 The variables and nonlinear equations with S-method procedures in the adaptive shooting method. Case
Old variables in interval [0,L]
(I) P p;1 ðz1 Þ > 0
Independent variables
New variables in interval [0,1]
Equations (see Appendix A)
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A2)–(A4)
P p1 ð0Þ; P p2 ðL1 Þ; P p3 ðL2 Þ
P p;1 ð0Þ; P p;2 ð0Þ; P p;3 ð0Þ
(A5), (A6), (A7)
þ þ Pþ s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8), (A9), (A10)
þ þ Pþ p1 ð0Þ; P p2 ðL1 Þ; P p3 ðL2 Þ
þ þ Pþ p;1 ð0Þ; P p;2 ð0Þ; P p;3 ð0Þ
(A11)–(A13)
P p;2 ðz2 Þ > 0
Dependent variables
P p;3 ðz3 Þ > 0
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14)–(A19) (see Appendix A)
(II) P p;1 ðz1 Þ ¼ 0
Independent variables
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
Pþ p1 ðL1 Þ; P p2 ðL1 Þ; P p3 ðL2 Þ
Pþ p;2 ð0Þ; P p;2 ð0Þ; P p;3 ð0Þ
(A23), (A6), (A7)
þ þ Pþ s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
þ Pþ p1 ð0Þ; P p3 ðL2 Þ
þ Pþ p;1 ð0Þ; P p;3 ð0Þ
(A24), (A13)
P p;2 ðz2 Þ > 0
Dependent variables
P p;3 ðz3 Þ > 0
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ?(A14), (A25), (A16)–(A19) (see Appendix A)
(III) P p;1 ðz1 Þ > 0
Independent variables
(A2)–(A4)
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
þ P p1 ð0Þ; P p2 ðL2 Þ; P p3 ðL2 Þ
þ P p;1 ð0Þ; P p;3 ð0Þ; P p;3 ð0Þ
(A5), (A26), (A7)
þ þ Pþ s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
þ Pþ p1 ð0Þ; P p2 ðL1 Þ
þ Pþ p;1 ð0Þ; P p;2 ð0Þ
(A11), (A27)
P p;2 ðz2 Þ ¼ 0
Dependent variables
P p;3 ðz3 Þ > 0
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14)–(A16), (A28), (A18), (A19) (see Appendix A)
(IV) P p;1 ðz1 Þ ¼ 0
Independent variables
(A2)–(A4)
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
þ Pþ p2 ðL1 Þ; P p3 ðL2 Þ; P p3 ðL2 Þ
þ Pþ p;2 ð0Þ; P p;3 ð0Þ; P p;3 ð0Þ
(A23), (A26), (A7)
þ þ Pþ s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
Pþ p1 ð0Þ
Pþ p;1 ð0Þ
(A24)
(A2)–(A4)
P p;2 ðz2 Þ ¼ 0
Dependent variables
P p;3 ðz3 Þ > 0
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14), (A25), (A16), (A28), (A18), (A19) (see Appendix A)
(V) P p;1 ðz1 Þ > 0
Independent variables
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ þ P p;1 ð0Þ; P p;2 ð0Þ; P p;3 ð0Þ
(A5), (A6), (A26)
P p;2 ðz2 Þ > 0
Dependent variables
þ P p1 ð0Þ; P p2 ðL1 Þ; P p3 ðL2 Þ þ þ ð0Þ; P ðL Þ; P Pþ 1 s1 s2 s3 ðL2 Þ þ Pþ p1 ð0Þ; P p2 ðL1 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
þ Pþ p;1 ð0Þ; P p;2 ð0Þ
(A11), (A12)
P p;3 ðz3 Þ ¼ 0
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14)–(A16), (A29), (A18), (A30) (see Appendix A)
(VI) P p;1 ðz1 Þ ¼ 0
Independent variables
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
þ Pþ p2 ðL1 Þ; P p2 ðL1 Þ; P p3 ðL2 Þ
þ Pþ p;2 ð0Þ; P p;2 ð0Þ; P p;3 ð0Þ
(A23), (A6), (A26)
þ þ Pþ s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
Pþ p1 ð0Þ
Pþ p;1 ð0Þ
(A24)
P p;2 ðz2 Þ > 0
Dependent variables
P p;3 ðz3 Þ ¼ 0
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14), (A25), (A16), (A29), (A18), (A30) (see Appendix A)
(VII) P p;1 ðz1 Þ > 0
Independent variables
P p;2 ðz2 Þ ¼ 0 P p;3 ðz3 Þ
¼0
(VIII) P p;1 ðz1 Þ ¼ 0 P p;2 ðz2 Þ ¼ 0 P p;3 ðz3 Þ ¼ 0
dependent variables
(A2)–(A4)
(A2)–(A4)
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
þ þ P p1 ð0Þ; P p2 ðL1 Þ; P p3 ðL2 Þ
þ þ P p;1 ð0Þ; P p;2 ð0Þ; P p;3 ð0Þ
(A5), (A23), (A26)
þ þ Pþ s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
Pþ p1 ð0Þ
Pþ p;1 ð0Þ
(A11)
(A2)–(A4)
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14), (A31), (A16), (A32), (A18), (A30) (see Appendix A) Independent variables
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A2)–(A4)
þ Pþ p;2 ð0Þ; P p;3 ð0Þ
(A23), (A26)
Dependent variables
þ Pþ p2 ðL1 Þ; P p3 ðL2 Þ þ þ ð0Þ; P Pþ s1 s2 ðL1 Þ; P s3 ðL2 Þ þ P p1 ð0Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
Pþ p;1 ð0Þ
(A24)
Nonlinear equations: Ds1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14), (A16), (A32), (A18), (A30) (see Appendix A)
X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
17
Fig. 4. Output powers and cases as a function of fiber length using adaptive shooting method in 4-point side-pumping YDCFLs with different pump configurations: (a) P fL1 ¼ 100 W;P fL2 ¼ 100 W;P bL1 ¼ 100 W;P bL2 ¼ 100 W;P f0 ¼ 100 W;P bL ¼ 100 W; (b) P bL1 ¼ 0 W;P fL1 ¼ 120 W;P fL2 ¼ 120 W;P bL2 ¼ 120 W;P f0 ¼ 120 W;P bL ¼ 120 W; (c) P bL2 ¼ 0 W;P fL1 ¼ 120 W;P fL2 ¼ 120 W;P bL1 ¼ 120 W;P f0 ¼ 120 W;P bL ¼ 120 W; (d) P bL1 ¼ 0 W;P fL1 ¼ 150 W;P fL2 ¼ 150 W;P f0 ¼ 150 W;P bL ¼ 150 W; (e) P f0 ¼ P fL1 ¼ P bL1 ¼ P fL2 ¼ P bL2 ¼ 120 W;P bL ¼ 0 W; (f) P f0 ¼ P fL1 ¼ P fL2 ¼ P bL2 ¼ 150 W;P bL1 ¼ P bL ¼ 0 W; (g) P f0 ¼ P fL1 ¼ P bL1 ¼ P fL2 ¼ 150 W;P bL2 ¼ P bL ¼ 0 W; (h) P f0 ¼ P fL1 ¼ P fL2 ¼ 200 W;P bL1 ¼ P bL2 ¼ P bL ¼ 0 W.
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X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
According to ordinary differential Eq. (5), boundary conditions (7) and N 2 N for the high power 4-point side-pumping YDCFLs [11], the analytical approximations of P p;1 ð1Þ, P p;2 ð1Þ and P p;3 ð1Þ are
Pp;1 ð1Þ gp PbL1 þ gp PbL2 ð1 lp Þ exp½ðCp rap N þ ap ÞðL2 L1 Þ 2
þ gp PbL ð1 lp Þ exp½ðCp rap N þ ap ÞðL L1 Þ Pp;2 ð1Þ Pp;3 ð1Þ
b p P L2
g
b p PL
g
b p P L ð1
þg þ
lp Þ exp½ðCp rap N þ ap ÞðL L2 Þ
R2 ðkp ÞPf0 ð1
þ
R2 ðkp ÞPfL1 ð1
þ
R2 ðkp ÞPfL2
ð8Þ ð9Þ
2
lp Þ exp½ðCp rap N þ ap ÞL
lp Þ exp½ðCp rap N þ ap ÞðL L1 Þ
exp½ðCp rap N þ ap ÞðL L2 Þ
ð10Þ
Hence, all the above mentioned cases (I), (II), (III), (IV), (V), (VI), (VII), and (VIII) can be identified easily by P p;1 ð1Þ, P p;2 ð1Þ and Pp;3 ð1Þ, as schematically shown in Fig. 2. And the specified flow chart of shooting method (S-method) procedure to calculate case I is schematically described in Fig. 3. All the variables and nonlinear equation of the eight cases with S-method procedures are detailed in Table 1. The detailed expressions of variables and nonlinear equations are given in Appendix A. Obviously, these eight cases, including different independent variables and nonlinear equations, are identified and classified after the simple identification process. In addition, each case is solved using S-method procedure. Therefore, this new method is named as adaptive shooting method, which combines the simple identification process and S-method procedure. The adaptive shooting method can adaptively handle different cases with different independent variables and nonlinear equations. Case (I) in the above 4-point side-pumping YDCFLs is taken as an example to illustrate how S-method procedure in the adaptive shooting method to deal with poor initial guesses. The backward signal powers P and pump powers s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ Pp;1 ð0Þ; Pp;2 ð0Þ; P p;3 ð0Þ are taken as independent variables. If the initial estimate of the backward signal power and pump power is too poor for solving the above 4-point side-pumping YDCFLs employing contraditional shooting method, P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ or P ð0Þ; P ð0Þ; P ð0Þ may be lower than 0, which lead the conp;1 p;2 p;3 traditional shooting method to fail to converge. The adaptive shooting method update P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ by uniform distribution random numbers of the slope efficiency g0 in (A2) (see Appendix A), laser signal gain GL1 in (A3) and GL2 L1 in (A4) in case of poor initial guess. As shown in Fig. 3, this adaptive shooting method can succeed to converge to the exact numerical solution with given tolerance adaptively.
4. Simulation and discussion For demonstrating the effects of the adaptive shooting method in 4-point side-pumping YDCFLs, simulation parameters are [2]: h = 6.63 1034 Js, c = 3 108 ms1, kp = 974 nm, ks = 1100 nm, R1 (ks) = 0.995, R2 (ks) = 0.035, R1 (kp) = 0.035, R2 (kp) = 0.035 [13], Cp = 0.0059, Cs = 0.82, ap = 2 103 m1, as = 4 104 m1, ras = 1 1027 m2, res = 1.6 1025 m2, rep = 26 1025 m2, rap = 26 1025 m2, gp = 0.95, lp = 0.05, ls = 0.01, A = 7.07 1010 m2, s = 0.8 103 s, N = 2.96 1025 m2. For simplicity, the two side pump position z = L1 and z = L2 are L/3, 2L/3, respectively. To explore the simple identification process in adaptive shooting method how to distinguish and identify automatically the eight cases of Section 3 for 4-point side-pumping YDCFLs, output powers and cases as a function of fiber length using adaptive shooting method in 4-point side-pumping YDCFLs with different pump scheme are detailed in Fig. 4. The total pump powers 600 W are coupled into YDCFLs at 4 pump positions z ¼ 0, z ¼ L=3, z ¼ 2L=3 and z ¼ L. Obviously, when the pump power PbL P 100 W > 0 W in Fig. 4(a–d) the pump power P p;3 ð1Þ from Eq. (10) is greater than the absolute error tolerance (i.e., P p;3 ð1Þ > AbsTol).Then there are only four cases such as case (I)–(IV) in Fig. 4(a–d). In Fig. 4(a), from Eqs. (8) and (9), the pump powers P p;1 ð1Þ and P p;2 ð1Þ are greater than the absolute error tolerance (i.e., P and p;1 ð1Þ > AbsTol P p;2 ð1Þ > AbsTol), thus the pump powers P p;1 ðz1 Þ and P p;2 ðz2 Þ are higher than 0, i.e., case (I) P p;1 ðz1 Þ > 0; P p;2 ðz2 Þ > 0. While PbL1 ¼ 0 W in Fig. 4(b) and PbL2 ¼ 0 W in Fig. 4(c) lead to P p;1 ð1Þ 6 AbsTol and P p;2 ð1Þ 6 AbsTol when the fiber length is longer than a certain value. That is, the case (II) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ > 0 in Fig. 4(b) and case (III) P p;1 ðz1 Þ > 0; Pp;2 ðz2 Þ ¼ 0 in Fig. 4(c) will occur when the fiber length is longer than 91 m. On the other hand, the case (I) P p;1 ðz1 Þ > 0; P p;2 ðz2 Þ > 0 in Fig. 4(b) and (c) will occur unless the fiber length is shorter than or equal to 91 m. In Fig. 4(d), the backward pump powers PbL1 ¼ 0 W;PbL2 ¼ 0 W, when the fiber length is shorter than 47 m, the pump powers P p;1 ð1Þ > AbsTol, P p;2 ð1Þ > AbsTol lead to the case (I) P p;1 ðz1 Þ > 0; P p;2 ðz2 Þ > 0; while the fiber length is longer than 92 m, P p;1 ð1Þ 6 AbsTol and P p;2 ð1Þ 6 AbsTol cause the case (IV) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ ¼ 0; In addition, the fiber length is between 47 m and 92 m, P p;1 ð1Þ 6 AbsTol and P p;2 ð1Þ > AbsTol cause the case (IV) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ > 0. Similarly, when the pump power at z ¼ L is zero detailed in Fig. 4(e–h), the simple identification process can also identify different cases. For example, there are case (IV) and case (VIII) distinguished with different fiber lengths in Fig. 4(h). Hence, the simple identification process in the adaptive shooting method can
Fig. 5. (a) Boundary condition error and (b) Iteration steps as a function of fiber length with adaptive shooting method.
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X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
Fig. 6. Signal and pump powers as a function of fiber position in YDCFLs.
distinguish automatically above eight cases at different pump struc tures through P p;1 ð1Þ, P p;2 ð1Þ and P p;3 ð1Þ. We take pump structure PfL1 ¼ P fL2 ¼ P bL1 ¼ PbL2 ¼ P f0 ¼ P bL ¼ 100 W in Fig. 4(a) as example to show the accuracy and speed of adaptive shooting method. In the adaptive shooting method, we use the boundary conditions (7-c), (7-d), (7-e) and (7-h) as nonlinear equations. The boundary condition errors of signal and pump powers jDs1 j, jDs2 j, jDs3 j, jDp1 j, jDp2 j, jDp3 j are obtained for given Tol ¼104 W, as shown in Fig. 5(a). Clearly, all of the laser and pump’s boundary condition error are less than Tol. The iteration as a function of fiber length is described in Fig. 5(b). The maximum iteration is 17 at fiber length 36 m, while the minimum
iteration is 4 at 12 m, 23 m, 34 m and 64 m. The average iteration is 7.36, that is, the adaptive shooting method can get the exact numerical solution of YDCFLs after average iteration steps less than 8. Now we take fiber length 30 m in Fig. 6 as example to show how the adaptive shooting method to deal with poor initial guess values of independent variables. The iteration steps updating the back ward signal and pump power P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ; P p;1 ð0Þ; Pp;2 ð0Þ; P p;3 ð0Þ are shown in Table 2. When Ps;1 ð0Þ < 0 or P s;2 ð0Þ < 0 or P s;3 ð0Þ < 0 or P p;1 ð0Þ < 0 or P p;2 ð0Þ < 0 or P p;3 ð0Þ < 0, the shooting method may fail to convergence, while the adaptive shooting method can solve the difficult problem using a new start by (A2)-(A7). The backward pump power ðP p;1 ð0Þ; P p;2 ð0Þ; P p;3 ð0ÞÞ, generating from (A5)-(A7) as start, are the same. While the back ward signal power ðP s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0ÞÞ, generating from (A2)– (A4), are random functions. Clearly, if the initial guess values of backward signal and pump powers are poor, P s;1 ð0Þ < 0 or P s;2 ð0Þ < 0 or P s;3 ð0Þ < 0 or P p;1 ð0Þ < 0 or P p;2 ð0Þ < 0 or P p;3 ð0Þ < 0 will occur after only one iteration step by iteration formula (A20). For example, the backward signal and pump powers P p;1 ð0Þ ¼ 7:2013 W and P p;2 ð0Þ ¼ 4:1958 W at iteration 2, Ps;3 ð0Þ ¼ 8:6611 W at iteration 4 and P p;1 ð0Þ ¼ 2:5651 W and P p;2 ð0Þ ¼ 0:1343 W at iteration 6 indicate that the initial guess values of the backward signal and pump powers from (A2)-(A7) at corresponding iteration 1, 3, and 5 are poor. The adaptive shooting method converges successfully and rapidly to the exact result after only four iteration steps by the suitable initial estimate values of backward signal and pump powers at iteration step 7. Note that the ‘‘- - - - - -’’ in Table 2 is that program continues in corresponding iteration step without resetting the initial value of the backward
Table 2 The iteration process of the signal and pump powers using the adaptive shooting method. Iteration steps
1 2 3 4 5 6 7 8 9 10 11
Signal powers (W)
Pump powers (W)
Note
P s;1 ð0Þ
P s;2 ð0Þ
P s;3 ð0Þ
P p;1 ð0Þ
P p;2 ð0Þ
P p;3 ð0Þ
11.8152 482.9115 85.8463 132.2985 9.3540 196.6576 88.6334 99.3021 95.1308 94.5675 94.5614
11.4798 645.3478 70.8712 58.8353 3.7773 122.6379 38.6239 46.8033 43.8906 43.4747 43.4706
0.0537 4.3936 62.1873 8.6611 0.9915 38.3967 35.5280 22.9894 25.2816 25.5517 25.5558
1.0032 7.2013 1.0032 1.2106 1.0032 2.5651 1.0032 1.2689 1.2768 1.2782 1.2783
1.0031 4.1958 1.0031 1.1511 1.0031 0.1343 1.0031 1.2029 1.2061 1.2065 1.2065
0.9933 1.0798 0.9933 1.0546 0.9933 0.9832 0.9933 1.1391 1.1446 1.1469 1.1469
Start Fail New Start Fail New Start Fail New Start ---------------Succeed
Fig. 7. Powers as a function of fiber length using the adaptive shooting method in 4-point side-pumping YDCFLs with given AbsTol ¼ 1 104 W for backward pump powers (a) P p;1 ð1Þ; (b) P p;2 ð1Þ; where the circle ‘‘s’’ is analytical solution and the dot ‘‘’’ is numerical solution.
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X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
signal powers P and pump powers s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ Pp;1 ð0Þ; Pp;2 ð0Þ; P p;3 ð0Þ. The signal and pump powers as a function of fiber length are shown in Fig. 6. The output power is 486.541 W and corresponding slope efficiency is about 81.09%. Now we take the pump structure P bL1 ¼ 0 W; PbL2 ¼ 0 W; P fL1 ¼ 150 W; P fL2 ¼ 150 W; P f0 ¼ 150 W; PbL ¼ 150 W in Fig. 4(d) as example to investigate the accuracy of distinguishing different cases using approximate solution of Eqs. (8) and (9). Powers as a function of fiber length using adaptive shooting method in 4-point side-pumping YDCFLs with given AbsTol ¼ 1 104 W for backward pump powers P p;1 ð1Þ and P p;2 ð1Þ are detailed in Fig. 7. From Fig. 7, the differences of the analytical solutions and numer ical solutions of backward pump powers P p;1 ð1Þ and P p;2 ð1Þ are decreasing with fiber length increasing when the numerical solutions is greater than 0. The difference of analytical solutions and exact solutions of backward pump powers P p;1 ð1Þ in Fig. 7(a) or 4 P W at P p;2 ð1Þ in Fig. 7(b) are less than 1 10 p;1 ð1Þ 6 AbsTol or P p;2 ð1Þ 6 AbsTol. Therefore, the analytical approximation solutions from Eqs. (8) and (9) instead of exact solutions in the judgment condition P p;1 ð1Þ 6 AbsTol and P p;2 ð1Þ 6 AbsTol of Fig. 2 are accurate and reliable.
5. Conclusions An adaptive shooting method is proposed for solving the solution of 4-point side-pumping YDCFLs. Simulation results demonstrate that the adaptive shooting method can identify automatically and easily eight different cases by only computing backward pump power at position L1, L2, and L in the 4-point side-pumping YDCFLs with different pump schemes. The initial estimate values of pump powers as independent variables are given approximate expressions and the signal powers are set random functions to speed the adaptive shooting method. By adjusting initial guess values from poor to suitable, the adaptive shooting method can succeed rapidly to get the exact results after average less than eight iteration steps. We are confident that the adaptive shooting method can be valid for more than 4-points side-pumping YDCFLs. Acknowledgments
Similarly, according to the boundary condition (7-g), the initial guess of the independent variable of the backward signal power P s;2 ð0Þ
Ps;2 ð0Þ ¼
P s;1 ð1Þ Ps;1 ð0ÞPþs;1 ð0Þ P s;1 ð0Þ 1 ¼ ¼ 1 ls 1 l s G L1 ð1 ls ÞPþs;1 ð1Þ
where GL1 is laser signal gain coefficient in the fiber interval ½0; L1 , GL1 > 1. Due to the unknown GL1 > 1, the guess of the value 1=GL1 is generated entirely arbitrary from interval [0,1], such as uniform distribution random number in [0,1]. Furthermore, according to the boundary condition (7-g), the initial guess of the independent variable of the backward signal power P s;3 ð0Þ
Ps;3 ð0Þ ¼
P s;2 ð1Þ Ps;2 ð0ÞPþs;2 ð0Þ P s;2 ð0Þ 1 ¼ ¼ 1 ls 1 ls GL2 L1 ð1 ls ÞPþs;2 ð1Þ
Pp;1 ð0Þ gp PbL1 exp½ðCp rap N þ ap ÞL1 þ gp PbL2 ð1 lp Þ exp½ðCp rap N þ ap ÞL2 2
þ gp PbL ð1 lp Þ exp½ðCp rap N þ ap ÞL
A.1. Approximate expression of case I-VIII for YDCFLs in the adaptive shooting method Case I: P p;1 ðz1 Þ > 0, P p;2 ðz2 Þ > 0 and P p;3 ðz3 Þ > 0 A straight forward integrations of (6) dðP s;i ðzi ÞP s;i ðzi ÞÞ= dz ¼ 0ði ¼ 1; 2; 3Þ and boundary conditions (7-a), (7-c), (7-e), þ (7-g), we find that P þ s;i ð0ÞP s;i ð0Þ ¼ P s;i ð1ÞP s;i ð1Þ ði ¼ 1; 2; 3Þ, and
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi R1 ðks ÞP s;1 ð0Þ ¼ R2 ðks ÞPþs;3 ð1Þ
ðA1Þ Pþ s;3 ð1Þ½1
In addition, the output power Pout ¼ R2 ðks Þ and P out ¼ P g0 gp ðPfo þ 2i¼1 ðPfLi þ PbLi Þ þ PbL Þ, thus the initial guess of the indepen-
dent variable of the backward signal powers P s;1 ð0Þ
Ps;1 ð0Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffi f f f b b b R2 ðks Þ P0 þ PL1 þ PL1 þ P L2 þ P L2 þ PL ¼ gp g0 R1 ðks Þ 1 R2 ðks Þ
ðA2Þ
where the guess of the slope efficiency g0 is generated entirely arbitrary from interval [0,1], such as uniform distribution random number in [0,1].
ðA5Þ
Pp;2 ð0Þ gp PbL2 exp½ðCp rap N þ ap ÞðL2 L1 Þ þ gp PbL ð1 lp Þ exp½ðCp rap N þ ap ÞðL L1 Þ
ðA6Þ
2
Pp;3 ð0Þ R2 ðkp ÞPf0 ð1 lp Þ exp½ðCp rap N þ ap Þð2L L2 Þ þ R2 ðkp ÞPfL1 ð1 lp Þ exp½ðCp rap N þ ap Þð2L L1 L2 Þ þ R2 ðkp ÞPfL2 exp½ðCp rap N þ ap Þð2L 2L2 Þ PbL ¼ 0 ðA7-aÞ
PbL > 0
ðA7-bÞ
According to the boundary condition (7-a), the dependent variables of signal powers Pþ s;1 ð0Þ
Pþs;1 ð0Þ ¼ R1 ðks ÞPs;1 ð0Þ Appendix A
ðA4Þ
where GL2 L1 is laser signal gain coefficient in interval ½L1 ; L2 ; GL2 L1 > 1. Obviously, the guess of the value 1=GL2 L1 can also be generated entirely arbitrary from interval [0,1], such as uniform distribution random number in [0,1]. Clearly N 2 N for the high power 4-point side-pumping YDCFLs, the initial guess of the independent variable of the backward pump powers P p;1 ð0Þ, P p;2 ð0Þ, P p;3 ð0Þ from ordinary differential Eq. (5)
Pp;3 ð0Þ gp PbL expððCp rap N þ ap ÞðL L2 ÞÞ
This work is jointly supported by the National Natural Science Foundation of China (61177069, 61471033, 61405007), and the National Natural Science Foundation of Beijing (No. 4154081).
ðA3Þ
ðA8Þ
þ þ þ According to Pþ s;1 ð0ÞP s;1 ð0Þ ¼ P s;1 ð1ÞP s;1 ð1Þ, P s;2 ð0ÞP s;2 ð0Þ ¼ P s;2 ð1Þ P ð1Þ and the boundary conditions (7-e), (7-g), the dependent varis;2 ables of signal powers P þ s;2 ð0Þ
Pþs;2 ð0Þ ¼
P þs;1 ð0ÞPs;1 ð0Þ Ps;2 ð0Þ
ðA9Þ
þ þ According to P þ Pþ s;1 ð0ÞP s;1 ð0Þ ¼ P s;1 ð1ÞP s;1 ð1Þ, s;3 ð0ÞP s;3 ð0Þ ¼ P s;3 ð1Þ Ps;3 ð1Þ and the boundary conditions (7-e), (7-g), the dependent variable of signal power Pþ s;3 ð0Þ
Pþs;3 ð0Þ ¼
P þs;1 ð0ÞPs;1 ð0Þ Ps;3 ð0Þ
ðA10Þ
According to the boundary condition (7-b), the dependent variable of pump power P þ p;1 ð0Þ
Pþp;1 ð0Þ ¼ R1 ðkp ÞPp;1 ð0Þ þ gp Pf0
ðA11Þ dðP p;i ðzi ÞP p;i ðzi ÞÞ=
A straight forward integrations of (5) dz ¼ 0ði ¼ 1; 2; 3Þ and according to the boundary conditions (7-f), þ (7-h), we find P þ p;i ð0ÞP p;i ð0Þ ¼ P p;i ð1ÞP p;i ð1Þ ði ¼ 1; 2; 3Þ, and the þ dependent variable of pump power Pp;2 ð0Þ
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X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
Pþp;2 ð0Þ ¼ Pþp;1 ð0ÞPp;1 ð0Þ þ gp PfL1 Pp;2 ð0Þ þ
g2p PbL1 PfL1
!, Pp;2 ð0Þ þ
1 lp
gp PbL1
! ðA12Þ
1 lp
According to the boundary conditions (7-f), (7-h), and þ Pþ p;2 ð0ÞP p;2 ð0Þ ¼ P p;2 ð1ÞP p;2 ð1Þ, the dependent variable of pump þ power P p;3 ð0Þ P þp;3 ð0Þ ¼
P þp;2 ð0ÞPp;2 ð0Þ þ
g
f p P L2 P p;3 ð0Þ þ
g2p PbL2 PfL2 1 lp
!, Pp;3 ð0Þ þ
gp PbL2
!
expressed as following:where hstep is small in absolute value, such as 108 W. Case II: P p;1 ðz1 Þ ¼ 0, P p;2 ðz2 Þ > 0 and P p;3 ðz3 Þ > 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers P ð0Þ; P ð0Þ can be acquired by (A2)–(A4), p;2 p;3 (A6), (A7). And P þ p;2 ð0Þ can be given
Pþp;2 ð0Þ ¼ Pþp2 ðL1 Þ
1 lp
gp PfL1 þ ð1 lp Þgp Pf0 exp½ðCp rap N þ ap ÞL1
ðA13Þ
The boundary conditions (7-e), (7-h), (7-c), (7-d) as nonlinear equations in the Newton method
Ds1 ¼ ð1 ls ÞP þs;1 ð1Þ P þs;2 ð0Þ ¼ 0
ðA14Þ
Dp1 ¼ Pp;1 ð1Þ gp PbL1 ð1 lp ÞPp;2 ð0Þ ¼ 0
ðA15Þ
Ds2 ¼ ð1 ls ÞP þs;2 ð1Þ P þs;3 ð0Þ ¼ 0
ðA16Þ
Dp2 ¼ Pp;2 ð1Þ gp PbL2 ð1 lp ÞPp;3 ð0Þ ¼ 0
ðA17Þ
Ds3 ¼ Ps;3 ð1Þ R2 ðks ÞPþs;3 ð1Þ ¼ 0
ðA18Þ
Dp3 ¼ Pp;3 ð1Þ gp PbL R2 ðkp ÞPþp;3 ð1Þ ¼ 0
ðA19Þ
ðA23Þ
þ þ The dependent variables of signal powers Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and þ pump powers Pp;3 ð0Þ can be acquired by (A8)–(A10), (A13). Pþ p;1 ð0Þ
and nonlinear equation Dp1 can be given
Pþp;1 ð0Þ ¼ Pþp ð0Þ ¼ R1 ðkp ÞPp ð0Þ þ gp Pf0 ¼ gp Pf0
Dp1 ¼
Pþp;2 ð0Þ
g
f p P L1
ð1
lp ÞPþp;1 ð1Þ
ðA24Þ
¼0
ðA25Þ
Ds1 Ds2 ; Dp2 ; Ds3 ; Dp3 can be expressed the same as Case I. Case III: P p;1 ðz1 Þ > 0, P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ > 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers Pp;1 ð0Þ; P p;3 ð0Þ can be acquired by (A2)–(A5), (A7). And P þ p;3 ð0Þ can be given
The iteration formula in the Newton method can be denoted by
2
ðiÞ
@ Ds1 =@ðPs;1 ð0ÞÞðiÞ
6 6 @ DðiÞ =@ðP ð0ÞÞðiÞ 6 p1 s;1 6 6 ðiÞ ðiÞ 6 @ Ds2 =@ðPs;1 ð0ÞÞ J¼6 6 ðiÞ ðiÞ 6 @ Dp2 =@ðPs;1 ð0ÞÞ 6 6 ðiÞ 6 @ Ds3 =@ðPs;1 ð0ÞÞðiÞ 4 ðiÞ @ Dp3 =@ðPs;1 ð0ÞÞðiÞ
h
ðiÞ
@ Ds1 =@ðPs;2 ð0ÞÞðiÞ
ðiÞ
@ Dp1 =@ðPs;2 ð0ÞÞðiÞ
ðiÞ
@ Ds2 =@ðPs;2 ð0ÞÞðiÞ
ðiÞ
@ Dp2 =@ðPs;2 ð0ÞÞðiÞ
@ Ds3 =@ðPp;1 ð0ÞÞðiÞ
ðiÞ
ðiÞ @ Dp3 =@ðPp;1 ð0ÞÞðiÞ
@ Ds1 =@ðPp;1 ð0ÞÞðiÞ @ Dp1 =@ðPp;1 ð0ÞÞðiÞ @ Ds2 =@ðPp;1 ð0ÞÞðiÞ @ Dp2 =@ðPp;1 ð0ÞÞðiÞ
ðiÞ
@ Ds1 =@ðPp;2 ð0ÞÞðiÞ
ðiÞ
@ Dp1 =@ðPp;2 ð0ÞÞðiÞ
ðiÞ
@ Ds2 =@ðPp;2 ð0ÞÞðiÞ
ðiÞ
@ Dp2 =@ðPp;2 ð0ÞÞðiÞ
@ Ds3 =@ðPs;2 ð0ÞÞðiÞ
ðiÞ
ðiÞ @ Dp3 =@ðPs;2 ð0ÞÞðiÞ
iðiþ1Þ Ps;1 ð0Þ; Pp;1 ð0Þ; Ps;2 ð0Þ; Pp;2 ð0Þ; P s;3 ð0Þ; Pp;3 ð0Þ h iðiÞ ¼ Ps;1 ð0Þ; P p;1 ð0Þ; Ps;2 ð0Þ; Pp;2 ð0Þ; Ps;3 ð0Þ; Pp;3 ð0Þ h i ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ J 1 Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3
ðiÞ
@ Ds1 =@ðPs;3 ð0ÞÞðiÞ
ðiÞ
ðiÞ
@ Dp1 =@ðPs;3 ð0ÞÞðiÞ
ðiÞ
@ Ds2 =@ðPs;3 ð0ÞÞðiÞ
ðiÞ
@ Dp2 =@ðPs;3 ð0ÞÞðiÞ
@ Ds3 =@ðPp;2 ð0ÞÞðiÞ
ðiÞ
@ Ds3 =@ðPs;3 ð0ÞÞðiÞ
ðiÞ @ Dp3 =@ðPp;2 ð0ÞÞðiÞ
@ Dp3 =@ðPs;3 ð0ÞÞðiÞ
ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ
ðiÞ
@ Ds1 =@ðPp;3 ð0ÞÞðiÞ
3
7 ðiÞ @ Dp1 =@ðPp;3 ð0ÞÞðiÞ 7 7 7 ðiÞ ðiÞ 7 @ Ds2 =@ðPp;3 ð0ÞÞ 7 7 7 ðiÞ @ Dp2 =@ðPp;3 ð0ÞÞðiÞ 7 7 7 ðiÞ @ Ds3 =@ðPp;3 ð0ÞÞðiÞ 7 5 ðiÞ @ Dp3 =@ðPp;;3 ð0ÞÞðiÞ
ðA21Þ
Pþp;3 ð0Þ ¼ Pþp3 ðL2 Þ 2
gp PfL2 þ gp Pf0 ð1 lp Þ exp½ðCp rap N þ ap ÞL2 þ gp PfL1 ð1 lp Þ exp½ðCp rap N þ ap ÞðL2 L1 Þ ðA20Þ
where the Jacobian matrix J can be described bywhere i is iteration steps, for each i P 1. The Jacobian matrix J associated with nonlinear Eqs. (A14)– (A19) that the 36 partial derivatives be determined and evaluated. The exact evaluation of the partial derivatives is not practical, thus, we can use finite difference approximations to the partial derivatives. For example, the partial derivatives ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ of @ Ds1 =@ P ; @ Dp1 =@ P ; @ Ds2 =@ P , @ Dp2 = s;1 ð0Þ s;1 ð0Þ s;1 ð0Þ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ @ P , @ Ds3 =@ P and @ Dp3 =@ P can be s;1 ð0Þ s;1 ð0Þ s;1 ð0Þ
ðA26Þ
þ þ The dependent variables of signal powers Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and þ pump powers Pþ ð0Þ can be acquired by (A8)–(A11). P p;1 p;2 ð0Þ and nonlinear equation Dp2 can be given
Pþp;2 ð0Þ
¼
Pþp;1 ð0ÞPp;1 ð0Þ
þ
g2p PbL1 PfL1 1 lp
!,
gp PbL1 1 lp
Dp2 ¼ Pþp;3 ð0Þ gp PfL2 ð1 lp ÞPþp;2 ð1Þ ¼ 0
ðA27Þ ðA28Þ
Ds1 ; Dp1 ; Ds2 ; Ds3 ; Dp3 can be expressed the same as Case I . Case IV: P p;1 ðz1 Þ ¼ 0, P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ > 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers P þ ð0Þ, P ð0Þ can be acquired by (A2)–(A4), p;3 p;3 (A26), (A7). And P þ p;2 ð0Þ can be given (A23).
h i ðiÞ ðiÞ ðiÞ @ Ds1 =@ðP s;1 ð0ÞÞðiÞ Ds1 ðP s;1 ð0ÞðiÞ þ hstep;P p;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ; Pp;2 ð0ÞðiÞ ;P s;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ Ds1 ðPs;1 ð0ÞðiÞ ;P p;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ; Pp;2 ð0ÞðiÞ ;P s;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ =hstep h i ðiÞ ðiÞ ðiÞ @ Dp1 =@ðP s;1 ð0ÞÞðiÞ Dp1 ðPs;1 ð0ÞðiÞ þ hstep;P p;1 ð0ÞðiÞ ; Ps;2 ð0ÞðiÞ ;P p;2 ð0ÞðiÞ ; Ps;3 ð0ÞðiÞ ; Pp;3 ð0ÞðiÞ Þ Dp1 ðP s;1 ð0ÞðiÞ ; Pp;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ;P p;2 ð0ÞðiÞ ; Ps;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ =hstep h i ðiÞ ðiÞ ðiÞ @ Ds2 =@ðP s;1 ð0ÞÞðiÞ Ds2 ðP s;1 ð0ÞðiÞ þ hstep;P p;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ; Pp;2 ð0ÞðiÞ ;P s;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ Ds2 ðPs;1 ð0ÞðiÞ ;P p;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ; Pp;2 ð0ÞðiÞ ;P s;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ =hstep h i ðiÞ ðiÞ ðiÞ @ Dp2 =@ðP s;1 ð0ÞÞðiÞ Dp2 ðPs;1 ð0ÞðiÞ þ hstep;P p;1 ð0ÞðiÞ ; Ps;2 ð0ÞðiÞ ;P p;2 ð0ÞðiÞ ; Ps;3 ð0ÞðiÞ ; Pp;3 ð0ÞðiÞ Þ Dp2 ðP s;1 ð0ÞðiÞ ; Pp;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ;P p;2 ð0ÞðiÞ ; Ps;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ =hstep h i ðiÞ ðiÞ ðiÞ @ Ds3 =@ðP s;1 ð0ÞÞðiÞ Ds3 ðP s;1 ð0ÞðiÞ þ hstep;P p;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ; Pp;2 ð0ÞðiÞ ;P s;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ Ds3 ðPs;1 ð0ÞðiÞ ;P p;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ; Pp;2 ð0ÞðiÞ ;P s;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ =hstep h i ðiÞ ðiÞ ðiÞ @ Dp3 =@ðP s;1 ð0ÞÞðiÞ Dp3 ðPs;1 ð0ÞðiÞ þ hstep;P p;1 ð0ÞðiÞ ; Ps;2 ð0ÞðiÞ ;P p;2 ð0ÞðiÞ ; Ps;3 ð0ÞðiÞ ; Pp;3 ð0ÞðiÞ Þ Dp3 ðP s;1 ð0ÞðiÞ ; Pp;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ;P p;2 ð0ÞðiÞ ; Ps;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ =hstep
ðA22Þ
22
X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
þ þ The dependent variables of signal powers P þ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ þ and pump powers P p;1 ð0Þ can be acquired by (A8)–(A10), (A24). And nonlinear equation Dp1 ; Dp2 can be given (A25), (A28); Ds1 , Ds2 ; Ds3 ; Dp3 can be expressed the same as Case I. Case V: P p;1 ðz1 Þ > 0, P p;2 ðz2 Þ > 0 and P p;3 ðz3 Þ ¼ 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers Pp;1 ð0Þ; P p;2 ð0Þ can be acquired by (A2)–(A6). And Pþ p;3 ð0Þ can be given be acquired by (A26). þ þ The dependent variables of signal powers P þ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ þ and pump powers P þ ð0Þ; P ð0Þ can be acquired by (A8)–(A12). p;1 p;2 The nonlinear equation Dp2 ; Dp3 can be given
Dp2 ¼ Pp;2 ð1Þ gp PbL2 ð1 lp ÞPp;3 ð0Þ ¼ Pp;2 ð1Þ gp P bL2 ¼ 0 Dp3 ¼
Pþp;3 ð0Þ
ð1
lp ÞPþp;2 ð1Þ
f p P L2
g
¼0
ðA29Þ ðA30Þ
Ds1 ; Ds2 ; Ds3 ; Dp1 can be expressed the same as Case I . Case VI: P p;1 ðz1 Þ ¼ 0, P p;2 ðz2 Þ > 0 and P p;3 ðz3 Þ ¼ 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers Pp;2 ð0Þ can be acquired by (A2)–(A4), (A6). And þ Pþ p;2 ð0Þ; P p;3 ð0Þ can be given by (A23), (A26). þ þ The dependent variables of signal powers P þ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers P þ ð0Þ can be acquired by (A8)–(A10), (A24). p;1 The nonlinear equation Dp1 ; Dp1 ; Dp3 can be given by (A25), (A29), (A30); Ds1 , Ds2 ; Ds3 can be expressed the same as Case I. Case VII: P p;1 ðz1 Þ > 0, P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ ¼ 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers Pp;1 ð0Þ can be acquired by (A2)–(A5). And þ Pþ p;2 ð0Þ; P p;3 ð0Þ can be given by (A23), (A26). þ þ The dependent variables of signal powers P þ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers P þ ð0Þ can be obtained by (A8)–(A11). Nonlinear p;1 equation Dp3 can be given by (A30). The nonlinear equation Dp1 ; Dp2 can be given
Dp1 ¼ Pp;1 ð1Þ gp PbL1 ð1 lp ÞPp;2 ð0Þ ¼ Pp;1 ð1Þ gp P bL1 ¼ 0
ðA31Þ
Dp2 ¼ Pþp;2 ð0Þ gp PfL1 ð1 lp ÞP þp;1 ð1Þ ¼ 0
ðA32Þ
Ds1 ; Ds2 ; Ds3 are the same as Case I. Case VIII: P p;1 ðz1 Þ ¼ 0, P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ ¼ 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ þ can be acquired by (A2)–(A4). And pump powers Pþ ð0Þ; P ð0Þ can p;2 p;3 be given be acquired by (A23), (A26). þ þ The dependent variables of signal powers P þ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ can be acquired by (A8)–(A10).
The dependent variables of pump powers Pþ p;1 ð0Þ can be given by (A24). The nonlinear equation Dp2 ; Dp3 can be given (A32), (A30); Ds1 ; Ds2 ; Ds3 can be expressed the same as Case I. References [1] Y.-C. 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. [2] P. Yan, M. Gong, C. Li, P. Ou, A. Xu, Distributed pumping multifiber series fiber laser, Opt. Express 13 (2005) 2699–2706. [3] Y. Wang, C.-Q. Xu, H. Po, Thermal effects in kilowatt fiber lasers, IEEE Photon. Technol. Lett. 16 (2004) 63–65. [4] J.P. Koplow, S.W. Moore, D.A. Kliner, A new method for side pumping of double-clad fiber sources, IEEE J. Quantum Electron. 39 (2003) 529–540. [5] J.K. Kim, C. Hagemann, T. Schreiber, T. Peschel, S. Böhme, R. Eberhardt, A. Tünnermann, Monolithic all-glass pump combiner scheme for high-power fiber laser systems, Opt. Express 18 (2010) 13194–13203. [6] T. Theeg, H. Sayinc, J. Neumann, L. Overmeyer, D. Kracht, Pump and signal combiner for bi-directional pumping of all-fiber lasers and amplifiers, Opt. Express 20 (2012) 28125–28141. [7] Q. Xiao, P. Yan, H. Ren, X. Chen, M. Gong, A side-pump coupler with refractive index valley configuration for fiber lasers and amplifiers, J. Lightwave Technol. 31 (2013) 2715–2722. [8] X. Hu, T. Ning, L. Pei, W. Jian, Novel shooting method with simple control strategy for fiber lasers, Optik – Int. J. Light Electron Opt. 125 (2014) 1975– 1979. [9] 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. [10] 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. 18 (2012) 502–508. [11] L. Xiao, P. Yan, M. Gong, W. Wei, P. Ou, An approximate analytic solution of strongly pumped Yb-doped double-clad fiber lasers without neglecting the scattering loss, Opt. Commun. 230 (2004) 401–410. [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] Y. Wang, C.-Q. Xu, H. Po, Analysis of Raman and thermal effects in kilowatt fiber lasers, Opt. Commun. 242 (2004) 487–502. [14] J. Wu, E. Zhou, K. Yan, W. Wei, B. Peng, Comparison of end-pumped and multipoint pumped Yb3+-doped gain guided and index antiguided fiber laser, Opt. Laser Technol. 44 (2012) 2371–2376. [15] U. Ascher, R.D. Russell, Reformulation of boundary value problems into ‘‘standard’’ form, SIAM Rev. 23 (1981) 238–254. [16] O. Pan, Z. Chun-xi, Y. Ping, G. Mali, Numerical analysis for multi-positionpumped double-clad fiber lasers, Laser Technol. 31 (2007) 150–152 (in Chinese). [17] Z. Mohammed, H. Saghafifar, Optimization of strongly pumped Yb-doped double-clad fiber lasers using a wide-scope approximate analytical model, Opt. Laser Technol. 55 (2014) 50–57. [18] X. Hu, T. Ning, L. Pei, Q. Chen, J. Li, Excellent initial guess functions for simple shooting method in Yb3+-doped fiber lasers, Opt. Fiber Technol. 20 (2014) 358– 364.
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Optical Fiber Technology www.elsevier.com/locate/yofte
Regular Articles
Adaptive shooting method for 4-point side-pumping high power Yb3+-doped double-clad fiber lasers Xudong Hu a,b, 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
a r t i c l e
i n f o
Article history: Received 27 July 2014 Revised 8 December 2014 Available online 13 January 2015 Keywords: Shooting method Simple identification process Side-pumping Yb3+-doped fiber Fiber lasers
a b s t r a c t An adaptive shooting method for solving 4-point side-pumping high power Yb3+-doped double-clad fiber lasers (YDCFLs) is developed. The adaptive shooting method combines the simple identification process and shooting method procedure. Simulation results show that the adaptive shooting method can identify automatically and easily eight different cases in the 4-point side-pumping YDCFLs with different pump schemes. The initial estimate values of pump powers as independent variables are given approximate expressions and the signal powers are set random functions to speed the adaptive shooting method. The adaptive shooting method can succeed rapidly to get the exact results after average less than eight iteration steps updating initial guess values. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In recent years, high power Yb3+-doped double-clad fiber lasers (YDCFLs) and related pump technologies have been attracting increasing attention. A highly efficient cladding-pumped Yb3+-doped fiber laser generating >2.1 kW of continuous-wave output power at 1.1 lm has been demonstrated [1]. Compared with end-pumping scheme, multipoint pump scheme can overcome the limitation of pump source and thermal effect in YDCFLs [2,3]. Many pump technologies such as embedded-mirror side pumping [4], monolithic integrated all-glass combiner [5,6] and side-pump coupler with refractive index valley configuration [7] have been developed for multipoint single or bidirectional side pumping Yb3+-doped fiber lasers. Many shooting methods have been applied for the solution of YDCFLs [8–10]. 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 backward pump power or backward signal power is lower than 0 which is unable to meet the practical physical problems [8]. A fast and stable shooting algorithm, using the Newton–Raphson method to solve the two-point boundary value problem of linear-cavity YDCFLs, has been demonstrated [9]. However, the initial estimate, given only several simple data, may bring the shooting method to ⇑ Corresponding author at: Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China. E-mail addresses: [email protected] (X. Hu), [email protected] (L. Pei). http://dx.doi.org/10.1016/j.yofte.2014.12.001 1068-5200/Ó 2014 Elsevier Inc. All rights reserved.
fail or unable to converge to the exact solutions. A combined algorithm [10] with shooting method and relaxation method has been studied for solving the model of YDCFLs. Since both shooting method and relaxation method need suitable predicted variable initial value, the combined algorithm is very doubtful to succeed to converge. Shooting method with simple control strategy successfully deal with poor initial guess values of the backward signal and pump powers by random functions in end pumping YDCFLs [8]. However, it is very difficult and slow for solving n-point pumping YDCFLs only using random functions adjusting initial guess values of 2(n 1) independent variables. In addition, the backward pump powers cannot be taken as independent variables when the backward pump powers are nearly to zero, which lead to fail or convergent to error results in multipoint pump YDCFLs. An adaptive shooting method is developed for solving the above problems in multipoint pump YDCFLs. For simplicity, the 4-point side-pumping YDCFLs is only discussed in this paper. 2. 4-Point side-pumping Yb3+-doped double-clad fiber lasers model A typical 4-point side-pumping high power linear cavity Yb3+-doped fiber laser is described schematically in Fig. 1. For the 4-point side-pumping YDCFLs, pump positions z = L1 and z = L2 divide Yb3+-doped fiber into 3 intervals [0, L1], [L1, L2], [L2, L]. 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
14
X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
Fig. 1. Schematic illustration of 4-point side-pumping high power Yb3+-doped double-cladding fiber lasers (YDCFLs).
state absorption (ESA) are negligible for strong pumping conditions in the 4-point side-pumping YDCFLs [11]. The steady-state rate equations in the 4-point side-pumping YDCFLs are described by the following set of nonlinear coupled ordinary differential Eqs. (1)–(3) [12]. Cp rap ðP þ p ðzÞþP p ðzÞÞkp
Cs ras ðPþ s ðzÞþP s ðzÞÞks
þ N2 ðzÞ hcA hcA ¼ Cp ðrap þrep ÞðPþ Cs ðras þres ÞðPþ N p ðzÞþP p ðzÞÞkp s ðzÞþP s ðzÞÞks 1 þ þ hcA s hcA
ð1Þ
dP p ðzÞ ¼ gðP p ðzÞÞ ¼ Cp frap N ðrap þ rep ÞN2 ðzÞgP p ðzÞ ap P p ðzÞ ð2Þ dz dP ðzÞ ¼ f ðPs ðzÞÞ ¼ ½Cs f½res þ ras N2 ðzÞ ras Ng as P s ðzÞ ð3Þ s dz
where N is the rare earth ion dopant concentration. N 2 ðzÞ is the upper lasing level population density. P p ðzÞ and P s ðzÞ are the pump power and laser signal power along the fiber, respectively. The plus and minus superscripts represent propagation along 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 emission cross-section, respectively. ras and res are the laser signal absorption and the emission crosssection, 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, Planck’s constant, light velocity and spontaneous lifetime, respectively. The boundary conditions [13,14] at side pump positions z ¼ 0; z ¼ L, z = Li (i = 1, 2) are
Pþs ð0Þ ¼ R1 ðks ÞPs ð0Þ
ð4-aÞ
Ps ðLÞ
R2 ðks ÞPþs ðLÞ
ð4-bÞ
Pþp ð0Þ ¼ R1 ðkp ÞPp ð0Þ þ gp0 P f0
ð4-cÞ
Pp ðLÞ ¼ R2 ðkp ÞP þp ðLÞ þ pL PbL Pþsiþ1 ðLi Þ ¼ ð1 lsi ÞPþsi ðLi Þ ði ¼ Psi ðLi Þ ¼ ð1 lsi ÞPsiþ1 ðLi Þ ði ¼
ð4-dÞ
¼
g
Pþpiþ1 ðLi Þ
lpi ÞPþpi ðLi Þ
þg
1; 2Þ
ð4-eÞ
1; 2Þ
ð4-fÞ
f pi P Li
ði ¼ 1; 2Þ
ð4-gÞ
Ppi ðLi Þ ¼ ð1 lpi ÞP piþ1 ðLi Þ þ gpi PbLi ði ¼ 1; 2Þ
ð4-hÞ
¼ ð1
Fig. 2. The flow chart of simple identification process in the adaptive shooting method for 4-point side-pumping YDCFLs.
15
X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
where P f0 is the side pump power at the end position z ¼ 0 and PbL denotes the side pump power at z ¼ L, respectively. PfLi and P bLi are the forward and backward pump powers at pump position z ¼ Li ði ¼ 1; 2Þ, respectively. R1 ðks Þ and R1 ðkp Þ are the input mirror reflectivity of signal and pump wavelength at z ¼ 0, respectively. R2 ðks Þ and R2 ðkp Þ are the output mirror reflectivity of signal and pump wavelength at z ¼ L, respectively. gp0 and gpL are the pump coupling efficiency at z ¼ 0 and z ¼ L, respectively. gpi are the pump coupling efficiency at pump position z ¼ Li . lpi and lsi are the pump and signal leakage rate at z ¼ Li , correspondingly. For simplicity, the pump coupling efficiency, pump and signal leakage rate on each pump position are the same, i.e., lpi ¼ lp , gpi ¼ gp , gp0 ¼ gpL ¼ gp and lsi ¼ ls . If L0 ¼ 0; L3 ¼ L, and z1 ¼ ðz L0 Þ=ðL1 L0 Þ, z2 ¼ ðz L1 Þ= ðL2 L1 Þ, z3 ¼ ðz L2 Þ=ðL3 L2 Þ in each segment fiber [15,16], then all the new variables of 3 segment interval are changed [0,1]. For each segment fiber, the new variables are formulated the pump and signal powers P p;i ðzi Þ ¼ P p ðzÞ, P s;i ðzi Þ ¼ P s ðzÞ ði ¼ 1; 2; 3Þ. The nonlinear coupled ordinary differential Eqs. (2) and (3) of the 4-point side-pumping YDCFLs in [0,1] are
dP p;i ðzi Þ dz ¼ g Pp;i ðzi Þ ¼ ðLi Li1 Þg Pp;i ðzi Þ ði ¼ 1; 2; 3Þ ð5Þ dzi dzi dP s;i ðzi Þ dz ¼ f Ps;i ðzi Þ ¼ ðLi Li1 Þf Ps;i ðzi Þ ði ¼ 1; 2; 3Þ ð6Þ dzi dzi
In addition, the 4-point boundary conditions in each interval [0,1] are
Pþs;1 ð0Þ ¼ R1 ðks ÞPs;1 ð0Þ Pþp;1 ð0Þ Ps;3 ð1Þ Pp;3 ð1Þ Pþs;i ð0Þ
¼
R1 ðkp ÞPp;1 ð0Þ
¼
R2 ðks ÞPþs;3 ð1Þ
¼
R2 ðkp ÞPþp;3 ð1Þ
¼ ð1
ð7-aÞ f p P0
þg
ð7-cÞ b p PL
ð7-dÞ
ði ¼ 2; 3Þ
ð7-eÞ
þg
ls ÞPþs;i1 ð1Þ
Pþp;i ð0Þ ¼ ð1 lp ÞPþp;i1 ð1Þ þ gp PfLi Ps;i1 ð1Þ Pp;i1 ð1Þ
ð7-bÞ
ði ¼ 2; 3Þ
¼ ð1
ls ÞPs;i ð0Þ
ði ¼ 2; 3Þ
¼ ð1
lp ÞPp;i ð0Þ
b p P Li1
þg
ð7-fÞ ð7-gÞ
ði ¼ 2; 3Þ
Fig. 3. The flow chart of shooting method procedure in the adaptive shooting method to calculate case I for 4-point side-pumping YDCFLs.
ð7-hÞ
16
X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
3. Adaptive shooting method for 4-point side-pumping YDCFLs In the 4-point side-pumping YDCFLs, as mentioned Fig. 1, the backward signal powers P s;1 ðz1 Þ; P s;2 ðz2 Þ; P s;3 ðz3 Þ and backward pump powers P p;1 ðz1 Þ; P p;2 ðz2 Þ; P p;3 ðz3 Þ are generally viewed as independent variables for shooting method. According to all the different pump configurations in Fig. 1, there are eight cases: (I) P p;1 ðz1 Þ > 0; P p;2 ðz2 Þ > 0 and P p;3 ðz3 Þ > 0; (II) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ > 0 and P ðz Þ > 0; (III) P ðz Þ > 0; P ðz Þ ¼ 0 and P ðz Þ > 0; 3 1 2 3 p;3 p;1 p;2 p;3 (IV) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ > 0; (V) P p;1 ðz1 Þ > 0; P p;2 ðz2 Þ > 0 and P p;3 ðz3 Þ ¼ 0; (VI) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ > 0 and Pp;3 ðz3 Þ ¼ 0; (VII) P p;1 ðz1 Þ > 0; P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ ¼ 0; (VIII) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ ¼ 0.
Clearly, the backward pump powers P p;1 ð1Þ; P p;2 ð1Þ and P p;3 ð1Þ in 3+ the 4-point side-pumping Yb -doped double-clad fiber lasers are respectively the maximum value of P p;1 ðz1 Þ, P p;2 ðz2 Þ and P p;3 ðz3 Þ; i.e., Pp;1 ðz1 Þ 6 P p;1 ð1Þ; P p;2 ðz2 Þ 6 P p;2 ð1Þ and P p;3 ðz3 Þ 6 Pp;3 ð1Þ. And the absolute error tolerance (AbsTol) of the shooting method (S-method) is small in absolute value such as 104 W. The exact analytical solutions of Eqs. (1)–(3) have not been acquired up to now [17], while exact numerical solutions with given error tolerance of Eqs. (1)–(3) have been obtained by several methods [8,12,18]. The backward pump powers P p;1 ðz1 Þ, P p;2 ðz2 Þ or P p;3 ðz3 Þ are zero for numerical solution when the backward pump powers P p;1 ð1Þ or P p;2 ð1Þ or P p;3 ð1Þ is less than or equal to the absolute error tolerance (i.e., P p;1 ð1Þ 6 AbsTol, P p;2 ð1Þ 6 AbsTol or P p;3 ð1Þ 6 AbsTol).
Table 1 The variables and nonlinear equations with S-method procedures in the adaptive shooting method. Case
Old variables in interval [0,L]
(I) P p;1 ðz1 Þ > 0
Independent variables
New variables in interval [0,1]
Equations (see Appendix A)
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A2)–(A4)
P p1 ð0Þ; P p2 ðL1 Þ; P p3 ðL2 Þ
P p;1 ð0Þ; P p;2 ð0Þ; P p;3 ð0Þ
(A5), (A6), (A7)
þ þ Pþ s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8), (A9), (A10)
þ þ Pþ p1 ð0Þ; P p2 ðL1 Þ; P p3 ðL2 Þ
þ þ Pþ p;1 ð0Þ; P p;2 ð0Þ; P p;3 ð0Þ
(A11)–(A13)
P p;2 ðz2 Þ > 0
Dependent variables
P p;3 ðz3 Þ > 0
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14)–(A19) (see Appendix A)
(II) P p;1 ðz1 Þ ¼ 0
Independent variables
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
Pþ p1 ðL1 Þ; P p2 ðL1 Þ; P p3 ðL2 Þ
Pþ p;2 ð0Þ; P p;2 ð0Þ; P p;3 ð0Þ
(A23), (A6), (A7)
þ þ Pþ s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
þ Pþ p1 ð0Þ; P p3 ðL2 Þ
þ Pþ p;1 ð0Þ; P p;3 ð0Þ
(A24), (A13)
P p;2 ðz2 Þ > 0
Dependent variables
P p;3 ðz3 Þ > 0
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ?(A14), (A25), (A16)–(A19) (see Appendix A)
(III) P p;1 ðz1 Þ > 0
Independent variables
(A2)–(A4)
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
þ P p1 ð0Þ; P p2 ðL2 Þ; P p3 ðL2 Þ
þ P p;1 ð0Þ; P p;3 ð0Þ; P p;3 ð0Þ
(A5), (A26), (A7)
þ þ Pþ s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
þ Pþ p1 ð0Þ; P p2 ðL1 Þ
þ Pþ p;1 ð0Þ; P p;2 ð0Þ
(A11), (A27)
P p;2 ðz2 Þ ¼ 0
Dependent variables
P p;3 ðz3 Þ > 0
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14)–(A16), (A28), (A18), (A19) (see Appendix A)
(IV) P p;1 ðz1 Þ ¼ 0
Independent variables
(A2)–(A4)
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
þ Pþ p2 ðL1 Þ; P p3 ðL2 Þ; P p3 ðL2 Þ
þ Pþ p;2 ð0Þ; P p;3 ð0Þ; P p;3 ð0Þ
(A23), (A26), (A7)
þ þ Pþ s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
Pþ p1 ð0Þ
Pþ p;1 ð0Þ
(A24)
(A2)–(A4)
P p;2 ðz2 Þ ¼ 0
Dependent variables
P p;3 ðz3 Þ > 0
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14), (A25), (A16), (A28), (A18), (A19) (see Appendix A)
(V) P p;1 ðz1 Þ > 0
Independent variables
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ þ P p;1 ð0Þ; P p;2 ð0Þ; P p;3 ð0Þ
(A5), (A6), (A26)
P p;2 ðz2 Þ > 0
Dependent variables
þ P p1 ð0Þ; P p2 ðL1 Þ; P p3 ðL2 Þ þ þ ð0Þ; P ðL Þ; P Pþ 1 s1 s2 s3 ðL2 Þ þ Pþ p1 ð0Þ; P p2 ðL1 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
þ Pþ p;1 ð0Þ; P p;2 ð0Þ
(A11), (A12)
P p;3 ðz3 Þ ¼ 0
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14)–(A16), (A29), (A18), (A30) (see Appendix A)
(VI) P p;1 ðz1 Þ ¼ 0
Independent variables
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
þ Pþ p2 ðL1 Þ; P p2 ðL1 Þ; P p3 ðL2 Þ
þ Pþ p;2 ð0Þ; P p;2 ð0Þ; P p;3 ð0Þ
(A23), (A6), (A26)
þ þ Pþ s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
Pþ p1 ð0Þ
Pþ p;1 ð0Þ
(A24)
P p;2 ðz2 Þ > 0
Dependent variables
P p;3 ðz3 Þ ¼ 0
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14), (A25), (A16), (A29), (A18), (A30) (see Appendix A)
(VII) P p;1 ðz1 Þ > 0
Independent variables
P p;2 ðz2 Þ ¼ 0 P p;3 ðz3 Þ
¼0
(VIII) P p;1 ðz1 Þ ¼ 0 P p;2 ðz2 Þ ¼ 0 P p;3 ðz3 Þ ¼ 0
dependent variables
(A2)–(A4)
(A2)–(A4)
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
þ þ P p1 ð0Þ; P p2 ðL1 Þ; P p3 ðL2 Þ
þ þ P p;1 ð0Þ; P p;2 ð0Þ; P p;3 ð0Þ
(A5), (A23), (A26)
þ þ Pþ s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
Pþ p1 ð0Þ
Pþ p;1 ð0Þ
(A11)
(A2)–(A4)
Nonlinear equations: Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14), (A31), (A16), (A32), (A18), (A30) (see Appendix A) Independent variables
P s1 ð0Þ; P s2 ðL1 Þ; P s3 ðL2 Þ
P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A2)–(A4)
þ Pþ p;2 ð0Þ; P p;3 ð0Þ
(A23), (A26)
Dependent variables
þ Pþ p2 ðL1 Þ; P p3 ðL2 Þ þ þ ð0Þ; P Pþ s1 s2 ðL1 Þ; P s3 ðL2 Þ þ P p1 ð0Þ
þ þ Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ
(A8)–(A10)
Pþ p;1 ð0Þ
(A24)
Nonlinear equations: Ds1 ; Ds2 ; Dp2 ; Ds3 ; Dp3 ? (A14), (A16), (A32), (A18), (A30) (see Appendix A)
X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
17
Fig. 4. Output powers and cases as a function of fiber length using adaptive shooting method in 4-point side-pumping YDCFLs with different pump configurations: (a) P fL1 ¼ 100 W;P fL2 ¼ 100 W;P bL1 ¼ 100 W;P bL2 ¼ 100 W;P f0 ¼ 100 W;P bL ¼ 100 W; (b) P bL1 ¼ 0 W;P fL1 ¼ 120 W;P fL2 ¼ 120 W;P bL2 ¼ 120 W;P f0 ¼ 120 W;P bL ¼ 120 W; (c) P bL2 ¼ 0 W;P fL1 ¼ 120 W;P fL2 ¼ 120 W;P bL1 ¼ 120 W;P f0 ¼ 120 W;P bL ¼ 120 W; (d) P bL1 ¼ 0 W;P fL1 ¼ 150 W;P fL2 ¼ 150 W;P f0 ¼ 150 W;P bL ¼ 150 W; (e) P f0 ¼ P fL1 ¼ P bL1 ¼ P fL2 ¼ P bL2 ¼ 120 W;P bL ¼ 0 W; (f) P f0 ¼ P fL1 ¼ P fL2 ¼ P bL2 ¼ 150 W;P bL1 ¼ P bL ¼ 0 W; (g) P f0 ¼ P fL1 ¼ P bL1 ¼ P fL2 ¼ 150 W;P bL2 ¼ P bL ¼ 0 W; (h) P f0 ¼ P fL1 ¼ P fL2 ¼ 200 W;P bL1 ¼ P bL2 ¼ P bL ¼ 0 W.
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X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
According to ordinary differential Eq. (5), boundary conditions (7) and N 2 N for the high power 4-point side-pumping YDCFLs [11], the analytical approximations of P p;1 ð1Þ, P p;2 ð1Þ and P p;3 ð1Þ are
Pp;1 ð1Þ gp PbL1 þ gp PbL2 ð1 lp Þ exp½ðCp rap N þ ap ÞðL2 L1 Þ 2
þ gp PbL ð1 lp Þ exp½ðCp rap N þ ap ÞðL L1 Þ Pp;2 ð1Þ Pp;3 ð1Þ
b p P L2
g
b p PL
g
b p P L ð1
þg þ
lp Þ exp½ðCp rap N þ ap ÞðL L2 Þ
R2 ðkp ÞPf0 ð1
þ
R2 ðkp ÞPfL1 ð1
þ
R2 ðkp ÞPfL2
ð8Þ ð9Þ
2
lp Þ exp½ðCp rap N þ ap ÞL
lp Þ exp½ðCp rap N þ ap ÞðL L1 Þ
exp½ðCp rap N þ ap ÞðL L2 Þ
ð10Þ
Hence, all the above mentioned cases (I), (II), (III), (IV), (V), (VI), (VII), and (VIII) can be identified easily by P p;1 ð1Þ, P p;2 ð1Þ and Pp;3 ð1Þ, as schematically shown in Fig. 2. And the specified flow chart of shooting method (S-method) procedure to calculate case I is schematically described in Fig. 3. All the variables and nonlinear equation of the eight cases with S-method procedures are detailed in Table 1. The detailed expressions of variables and nonlinear equations are given in Appendix A. Obviously, these eight cases, including different independent variables and nonlinear equations, are identified and classified after the simple identification process. In addition, each case is solved using S-method procedure. Therefore, this new method is named as adaptive shooting method, which combines the simple identification process and S-method procedure. The adaptive shooting method can adaptively handle different cases with different independent variables and nonlinear equations. Case (I) in the above 4-point side-pumping YDCFLs is taken as an example to illustrate how S-method procedure in the adaptive shooting method to deal with poor initial guesses. The backward signal powers P and pump powers s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ Pp;1 ð0Þ; Pp;2 ð0Þ; P p;3 ð0Þ are taken as independent variables. If the initial estimate of the backward signal power and pump power is too poor for solving the above 4-point side-pumping YDCFLs employing contraditional shooting method, P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ or P ð0Þ; P ð0Þ; P ð0Þ may be lower than 0, which lead the conp;1 p;2 p;3 traditional shooting method to fail to converge. The adaptive shooting method update P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ by uniform distribution random numbers of the slope efficiency g0 in (A2) (see Appendix A), laser signal gain GL1 in (A3) and GL2 L1 in (A4) in case of poor initial guess. As shown in Fig. 3, this adaptive shooting method can succeed to converge to the exact numerical solution with given tolerance adaptively.
4. Simulation and discussion For demonstrating the effects of the adaptive shooting method in 4-point side-pumping YDCFLs, simulation parameters are [2]: h = 6.63 1034 Js, c = 3 108 ms1, kp = 974 nm, ks = 1100 nm, R1 (ks) = 0.995, R2 (ks) = 0.035, R1 (kp) = 0.035, R2 (kp) = 0.035 [13], Cp = 0.0059, Cs = 0.82, ap = 2 103 m1, as = 4 104 m1, ras = 1 1027 m2, res = 1.6 1025 m2, rep = 26 1025 m2, rap = 26 1025 m2, gp = 0.95, lp = 0.05, ls = 0.01, A = 7.07 1010 m2, s = 0.8 103 s, N = 2.96 1025 m2. For simplicity, the two side pump position z = L1 and z = L2 are L/3, 2L/3, respectively. To explore the simple identification process in adaptive shooting method how to distinguish and identify automatically the eight cases of Section 3 for 4-point side-pumping YDCFLs, output powers and cases as a function of fiber length using adaptive shooting method in 4-point side-pumping YDCFLs with different pump scheme are detailed in Fig. 4. The total pump powers 600 W are coupled into YDCFLs at 4 pump positions z ¼ 0, z ¼ L=3, z ¼ 2L=3 and z ¼ L. Obviously, when the pump power PbL P 100 W > 0 W in Fig. 4(a–d) the pump power P p;3 ð1Þ from Eq. (10) is greater than the absolute error tolerance (i.e., P p;3 ð1Þ > AbsTol).Then there are only four cases such as case (I)–(IV) in Fig. 4(a–d). In Fig. 4(a), from Eqs. (8) and (9), the pump powers P p;1 ð1Þ and P p;2 ð1Þ are greater than the absolute error tolerance (i.e., P and p;1 ð1Þ > AbsTol P p;2 ð1Þ > AbsTol), thus the pump powers P p;1 ðz1 Þ and P p;2 ðz2 Þ are higher than 0, i.e., case (I) P p;1 ðz1 Þ > 0; P p;2 ðz2 Þ > 0. While PbL1 ¼ 0 W in Fig. 4(b) and PbL2 ¼ 0 W in Fig. 4(c) lead to P p;1 ð1Þ 6 AbsTol and P p;2 ð1Þ 6 AbsTol when the fiber length is longer than a certain value. That is, the case (II) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ > 0 in Fig. 4(b) and case (III) P p;1 ðz1 Þ > 0; Pp;2 ðz2 Þ ¼ 0 in Fig. 4(c) will occur when the fiber length is longer than 91 m. On the other hand, the case (I) P p;1 ðz1 Þ > 0; P p;2 ðz2 Þ > 0 in Fig. 4(b) and (c) will occur unless the fiber length is shorter than or equal to 91 m. In Fig. 4(d), the backward pump powers PbL1 ¼ 0 W;PbL2 ¼ 0 W, when the fiber length is shorter than 47 m, the pump powers P p;1 ð1Þ > AbsTol, P p;2 ð1Þ > AbsTol lead to the case (I) P p;1 ðz1 Þ > 0; P p;2 ðz2 Þ > 0; while the fiber length is longer than 92 m, P p;1 ð1Þ 6 AbsTol and P p;2 ð1Þ 6 AbsTol cause the case (IV) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ ¼ 0; In addition, the fiber length is between 47 m and 92 m, P p;1 ð1Þ 6 AbsTol and P p;2 ð1Þ > AbsTol cause the case (IV) P p;1 ðz1 Þ ¼ 0; P p;2 ðz2 Þ > 0. Similarly, when the pump power at z ¼ L is zero detailed in Fig. 4(e–h), the simple identification process can also identify different cases. For example, there are case (IV) and case (VIII) distinguished with different fiber lengths in Fig. 4(h). Hence, the simple identification process in the adaptive shooting method can
Fig. 5. (a) Boundary condition error and (b) Iteration steps as a function of fiber length with adaptive shooting method.
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X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
Fig. 6. Signal and pump powers as a function of fiber position in YDCFLs.
distinguish automatically above eight cases at different pump struc tures through P p;1 ð1Þ, P p;2 ð1Þ and P p;3 ð1Þ. We take pump structure PfL1 ¼ P fL2 ¼ P bL1 ¼ PbL2 ¼ P f0 ¼ P bL ¼ 100 W in Fig. 4(a) as example to show the accuracy and speed of adaptive shooting method. In the adaptive shooting method, we use the boundary conditions (7-c), (7-d), (7-e) and (7-h) as nonlinear equations. The boundary condition errors of signal and pump powers jDs1 j, jDs2 j, jDs3 j, jDp1 j, jDp2 j, jDp3 j are obtained for given Tol ¼104 W, as shown in Fig. 5(a). Clearly, all of the laser and pump’s boundary condition error are less than Tol. The iteration as a function of fiber length is described in Fig. 5(b). The maximum iteration is 17 at fiber length 36 m, while the minimum
iteration is 4 at 12 m, 23 m, 34 m and 64 m. The average iteration is 7.36, that is, the adaptive shooting method can get the exact numerical solution of YDCFLs after average iteration steps less than 8. Now we take fiber length 30 m in Fig. 6 as example to show how the adaptive shooting method to deal with poor initial guess values of independent variables. The iteration steps updating the back ward signal and pump power P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ; P p;1 ð0Þ; Pp;2 ð0Þ; P p;3 ð0Þ are shown in Table 2. When Ps;1 ð0Þ < 0 or P s;2 ð0Þ < 0 or P s;3 ð0Þ < 0 or P p;1 ð0Þ < 0 or P p;2 ð0Þ < 0 or P p;3 ð0Þ < 0, the shooting method may fail to convergence, while the adaptive shooting method can solve the difficult problem using a new start by (A2)-(A7). The backward pump power ðP p;1 ð0Þ; P p;2 ð0Þ; P p;3 ð0ÞÞ, generating from (A5)-(A7) as start, are the same. While the back ward signal power ðP s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0ÞÞ, generating from (A2)– (A4), are random functions. Clearly, if the initial guess values of backward signal and pump powers are poor, P s;1 ð0Þ < 0 or P s;2 ð0Þ < 0 or P s;3 ð0Þ < 0 or P p;1 ð0Þ < 0 or P p;2 ð0Þ < 0 or P p;3 ð0Þ < 0 will occur after only one iteration step by iteration formula (A20). For example, the backward signal and pump powers P p;1 ð0Þ ¼ 7:2013 W and P p;2 ð0Þ ¼ 4:1958 W at iteration 2, Ps;3 ð0Þ ¼ 8:6611 W at iteration 4 and P p;1 ð0Þ ¼ 2:5651 W and P p;2 ð0Þ ¼ 0:1343 W at iteration 6 indicate that the initial guess values of the backward signal and pump powers from (A2)-(A7) at corresponding iteration 1, 3, and 5 are poor. The adaptive shooting method converges successfully and rapidly to the exact result after only four iteration steps by the suitable initial estimate values of backward signal and pump powers at iteration step 7. Note that the ‘‘- - - - - -’’ in Table 2 is that program continues in corresponding iteration step without resetting the initial value of the backward
Table 2 The iteration process of the signal and pump powers using the adaptive shooting method. Iteration steps
1 2 3 4 5 6 7 8 9 10 11
Signal powers (W)
Pump powers (W)
Note
P s;1 ð0Þ
P s;2 ð0Þ
P s;3 ð0Þ
P p;1 ð0Þ
P p;2 ð0Þ
P p;3 ð0Þ
11.8152 482.9115 85.8463 132.2985 9.3540 196.6576 88.6334 99.3021 95.1308 94.5675 94.5614
11.4798 645.3478 70.8712 58.8353 3.7773 122.6379 38.6239 46.8033 43.8906 43.4747 43.4706
0.0537 4.3936 62.1873 8.6611 0.9915 38.3967 35.5280 22.9894 25.2816 25.5517 25.5558
1.0032 7.2013 1.0032 1.2106 1.0032 2.5651 1.0032 1.2689 1.2768 1.2782 1.2783
1.0031 4.1958 1.0031 1.1511 1.0031 0.1343 1.0031 1.2029 1.2061 1.2065 1.2065
0.9933 1.0798 0.9933 1.0546 0.9933 0.9832 0.9933 1.1391 1.1446 1.1469 1.1469
Start Fail New Start Fail New Start Fail New Start ---------------Succeed
Fig. 7. Powers as a function of fiber length using the adaptive shooting method in 4-point side-pumping YDCFLs with given AbsTol ¼ 1 104 W for backward pump powers (a) P p;1 ð1Þ; (b) P p;2 ð1Þ; where the circle ‘‘s’’ is analytical solution and the dot ‘‘’’ is numerical solution.
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X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
signal powers P and pump powers s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ Pp;1 ð0Þ; Pp;2 ð0Þ; P p;3 ð0Þ. The signal and pump powers as a function of fiber length are shown in Fig. 6. The output power is 486.541 W and corresponding slope efficiency is about 81.09%. Now we take the pump structure P bL1 ¼ 0 W; PbL2 ¼ 0 W; P fL1 ¼ 150 W; P fL2 ¼ 150 W; P f0 ¼ 150 W; PbL ¼ 150 W in Fig. 4(d) as example to investigate the accuracy of distinguishing different cases using approximate solution of Eqs. (8) and (9). Powers as a function of fiber length using adaptive shooting method in 4-point side-pumping YDCFLs with given AbsTol ¼ 1 104 W for backward pump powers P p;1 ð1Þ and P p;2 ð1Þ are detailed in Fig. 7. From Fig. 7, the differences of the analytical solutions and numer ical solutions of backward pump powers P p;1 ð1Þ and P p;2 ð1Þ are decreasing with fiber length increasing when the numerical solutions is greater than 0. The difference of analytical solutions and exact solutions of backward pump powers P p;1 ð1Þ in Fig. 7(a) or 4 P W at P p;2 ð1Þ in Fig. 7(b) are less than 1 10 p;1 ð1Þ 6 AbsTol or P p;2 ð1Þ 6 AbsTol. Therefore, the analytical approximation solutions from Eqs. (8) and (9) instead of exact solutions in the judgment condition P p;1 ð1Þ 6 AbsTol and P p;2 ð1Þ 6 AbsTol of Fig. 2 are accurate and reliable.
5. Conclusions An adaptive shooting method is proposed for solving the solution of 4-point side-pumping YDCFLs. Simulation results demonstrate that the adaptive shooting method can identify automatically and easily eight different cases by only computing backward pump power at position L1, L2, and L in the 4-point side-pumping YDCFLs with different pump schemes. The initial estimate values of pump powers as independent variables are given approximate expressions and the signal powers are set random functions to speed the adaptive shooting method. By adjusting initial guess values from poor to suitable, the adaptive shooting method can succeed rapidly to get the exact results after average less than eight iteration steps. We are confident that the adaptive shooting method can be valid for more than 4-points side-pumping YDCFLs. Acknowledgments
Similarly, according to the boundary condition (7-g), the initial guess of the independent variable of the backward signal power P s;2 ð0Þ
Ps;2 ð0Þ ¼
P s;1 ð1Þ Ps;1 ð0ÞPþs;1 ð0Þ P s;1 ð0Þ 1 ¼ ¼ 1 ls 1 l s G L1 ð1 ls ÞPþs;1 ð1Þ
where GL1 is laser signal gain coefficient in the fiber interval ½0; L1 , GL1 > 1. Due to the unknown GL1 > 1, the guess of the value 1=GL1 is generated entirely arbitrary from interval [0,1], such as uniform distribution random number in [0,1]. Furthermore, according to the boundary condition (7-g), the initial guess of the independent variable of the backward signal power P s;3 ð0Þ
Ps;3 ð0Þ ¼
P s;2 ð1Þ Ps;2 ð0ÞPþs;2 ð0Þ P s;2 ð0Þ 1 ¼ ¼ 1 ls 1 ls GL2 L1 ð1 ls ÞPþs;2 ð1Þ
Pp;1 ð0Þ gp PbL1 exp½ðCp rap N þ ap ÞL1 þ gp PbL2 ð1 lp Þ exp½ðCp rap N þ ap ÞL2 2
þ gp PbL ð1 lp Þ exp½ðCp rap N þ ap ÞL
A.1. Approximate expression of case I-VIII for YDCFLs in the adaptive shooting method Case I: P p;1 ðz1 Þ > 0, P p;2 ðz2 Þ > 0 and P p;3 ðz3 Þ > 0 A straight forward integrations of (6) dðP s;i ðzi ÞP s;i ðzi ÞÞ= dz ¼ 0ði ¼ 1; 2; 3Þ and boundary conditions (7-a), (7-c), (7-e), þ (7-g), we find that P þ s;i ð0ÞP s;i ð0Þ ¼ P s;i ð1ÞP s;i ð1Þ ði ¼ 1; 2; 3Þ, and
pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi R1 ðks ÞP s;1 ð0Þ ¼ R2 ðks ÞPþs;3 ð1Þ
ðA1Þ Pþ s;3 ð1Þ½1
In addition, the output power Pout ¼ R2 ðks Þ and P out ¼ P g0 gp ðPfo þ 2i¼1 ðPfLi þ PbLi Þ þ PbL Þ, thus the initial guess of the indepen-
dent variable of the backward signal powers P s;1 ð0Þ
Ps;1 ð0Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffi f f f b b b R2 ðks Þ P0 þ PL1 þ PL1 þ P L2 þ P L2 þ PL ¼ gp g0 R1 ðks Þ 1 R2 ðks Þ
ðA2Þ
where the guess of the slope efficiency g0 is generated entirely arbitrary from interval [0,1], such as uniform distribution random number in [0,1].
ðA5Þ
Pp;2 ð0Þ gp PbL2 exp½ðCp rap N þ ap ÞðL2 L1 Þ þ gp PbL ð1 lp Þ exp½ðCp rap N þ ap ÞðL L1 Þ
ðA6Þ
2
Pp;3 ð0Þ R2 ðkp ÞPf0 ð1 lp Þ exp½ðCp rap N þ ap Þð2L L2 Þ þ R2 ðkp ÞPfL1 ð1 lp Þ exp½ðCp rap N þ ap Þð2L L1 L2 Þ þ R2 ðkp ÞPfL2 exp½ðCp rap N þ ap Þð2L 2L2 Þ PbL ¼ 0 ðA7-aÞ
PbL > 0
ðA7-bÞ
According to the boundary condition (7-a), the dependent variables of signal powers Pþ s;1 ð0Þ
Pþs;1 ð0Þ ¼ R1 ðks ÞPs;1 ð0Þ Appendix A
ðA4Þ
where GL2 L1 is laser signal gain coefficient in interval ½L1 ; L2 ; GL2 L1 > 1. Obviously, the guess of the value 1=GL2 L1 can also be generated entirely arbitrary from interval [0,1], such as uniform distribution random number in [0,1]. Clearly N 2 N for the high power 4-point side-pumping YDCFLs, the initial guess of the independent variable of the backward pump powers P p;1 ð0Þ, P p;2 ð0Þ, P p;3 ð0Þ from ordinary differential Eq. (5)
Pp;3 ð0Þ gp PbL expððCp rap N þ ap ÞðL L2 ÞÞ
This work is jointly supported by the National Natural Science Foundation of China (61177069, 61471033, 61405007), and the National Natural Science Foundation of Beijing (No. 4154081).
ðA3Þ
ðA8Þ
þ þ þ According to Pþ s;1 ð0ÞP s;1 ð0Þ ¼ P s;1 ð1ÞP s;1 ð1Þ, P s;2 ð0ÞP s;2 ð0Þ ¼ P s;2 ð1Þ P ð1Þ and the boundary conditions (7-e), (7-g), the dependent varis;2 ables of signal powers P þ s;2 ð0Þ
Pþs;2 ð0Þ ¼
P þs;1 ð0ÞPs;1 ð0Þ Ps;2 ð0Þ
ðA9Þ
þ þ According to P þ Pþ s;1 ð0ÞP s;1 ð0Þ ¼ P s;1 ð1ÞP s;1 ð1Þ, s;3 ð0ÞP s;3 ð0Þ ¼ P s;3 ð1Þ Ps;3 ð1Þ and the boundary conditions (7-e), (7-g), the dependent variable of signal power Pþ s;3 ð0Þ
Pþs;3 ð0Þ ¼
P þs;1 ð0ÞPs;1 ð0Þ Ps;3 ð0Þ
ðA10Þ
According to the boundary condition (7-b), the dependent variable of pump power P þ p;1 ð0Þ
Pþp;1 ð0Þ ¼ R1 ðkp ÞPp;1 ð0Þ þ gp Pf0
ðA11Þ dðP p;i ðzi ÞP p;i ðzi ÞÞ=
A straight forward integrations of (5) dz ¼ 0ði ¼ 1; 2; 3Þ and according to the boundary conditions (7-f), þ (7-h), we find P þ p;i ð0ÞP p;i ð0Þ ¼ P p;i ð1ÞP p;i ð1Þ ði ¼ 1; 2; 3Þ, and the þ dependent variable of pump power Pp;2 ð0Þ
21
X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
Pþp;2 ð0Þ ¼ Pþp;1 ð0ÞPp;1 ð0Þ þ gp PfL1 Pp;2 ð0Þ þ
g2p PbL1 PfL1
!, Pp;2 ð0Þ þ
1 lp
gp PbL1
! ðA12Þ
1 lp
According to the boundary conditions (7-f), (7-h), and þ Pþ p;2 ð0ÞP p;2 ð0Þ ¼ P p;2 ð1ÞP p;2 ð1Þ, the dependent variable of pump þ power P p;3 ð0Þ P þp;3 ð0Þ ¼
P þp;2 ð0ÞPp;2 ð0Þ þ
g
f p P L2 P p;3 ð0Þ þ
g2p PbL2 PfL2 1 lp
!, Pp;3 ð0Þ þ
gp PbL2
!
expressed as following:where hstep is small in absolute value, such as 108 W. Case II: P p;1 ðz1 Þ ¼ 0, P p;2 ðz2 Þ > 0 and P p;3 ðz3 Þ > 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers P ð0Þ; P ð0Þ can be acquired by (A2)–(A4), p;2 p;3 (A6), (A7). And P þ p;2 ð0Þ can be given
Pþp;2 ð0Þ ¼ Pþp2 ðL1 Þ
1 lp
gp PfL1 þ ð1 lp Þgp Pf0 exp½ðCp rap N þ ap ÞL1
ðA13Þ
The boundary conditions (7-e), (7-h), (7-c), (7-d) as nonlinear equations in the Newton method
Ds1 ¼ ð1 ls ÞP þs;1 ð1Þ P þs;2 ð0Þ ¼ 0
ðA14Þ
Dp1 ¼ Pp;1 ð1Þ gp PbL1 ð1 lp ÞPp;2 ð0Þ ¼ 0
ðA15Þ
Ds2 ¼ ð1 ls ÞP þs;2 ð1Þ P þs;3 ð0Þ ¼ 0
ðA16Þ
Dp2 ¼ Pp;2 ð1Þ gp PbL2 ð1 lp ÞPp;3 ð0Þ ¼ 0
ðA17Þ
Ds3 ¼ Ps;3 ð1Þ R2 ðks ÞPþs;3 ð1Þ ¼ 0
ðA18Þ
Dp3 ¼ Pp;3 ð1Þ gp PbL R2 ðkp ÞPþp;3 ð1Þ ¼ 0
ðA19Þ
ðA23Þ
þ þ The dependent variables of signal powers Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and þ pump powers Pp;3 ð0Þ can be acquired by (A8)–(A10), (A13). Pþ p;1 ð0Þ
and nonlinear equation Dp1 can be given
Pþp;1 ð0Þ ¼ Pþp ð0Þ ¼ R1 ðkp ÞPp ð0Þ þ gp Pf0 ¼ gp Pf0
Dp1 ¼
Pþp;2 ð0Þ
g
f p P L1
ð1
lp ÞPþp;1 ð1Þ
ðA24Þ
¼0
ðA25Þ
Ds1 Ds2 ; Dp2 ; Ds3 ; Dp3 can be expressed the same as Case I. Case III: P p;1 ðz1 Þ > 0, P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ > 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers Pp;1 ð0Þ; P p;3 ð0Þ can be acquired by (A2)–(A5), (A7). And P þ p;3 ð0Þ can be given
The iteration formula in the Newton method can be denoted by
2
ðiÞ
@ Ds1 =@ðPs;1 ð0ÞÞðiÞ
6 6 @ DðiÞ =@ðP ð0ÞÞðiÞ 6 p1 s;1 6 6 ðiÞ ðiÞ 6 @ Ds2 =@ðPs;1 ð0ÞÞ J¼6 6 ðiÞ ðiÞ 6 @ Dp2 =@ðPs;1 ð0ÞÞ 6 6 ðiÞ 6 @ Ds3 =@ðPs;1 ð0ÞÞðiÞ 4 ðiÞ @ Dp3 =@ðPs;1 ð0ÞÞðiÞ
h
ðiÞ
@ Ds1 =@ðPs;2 ð0ÞÞðiÞ
ðiÞ
@ Dp1 =@ðPs;2 ð0ÞÞðiÞ
ðiÞ
@ Ds2 =@ðPs;2 ð0ÞÞðiÞ
ðiÞ
@ Dp2 =@ðPs;2 ð0ÞÞðiÞ
@ Ds3 =@ðPp;1 ð0ÞÞðiÞ
ðiÞ
ðiÞ @ Dp3 =@ðPp;1 ð0ÞÞðiÞ
@ Ds1 =@ðPp;1 ð0ÞÞðiÞ @ Dp1 =@ðPp;1 ð0ÞÞðiÞ @ Ds2 =@ðPp;1 ð0ÞÞðiÞ @ Dp2 =@ðPp;1 ð0ÞÞðiÞ
ðiÞ
@ Ds1 =@ðPp;2 ð0ÞÞðiÞ
ðiÞ
@ Dp1 =@ðPp;2 ð0ÞÞðiÞ
ðiÞ
@ Ds2 =@ðPp;2 ð0ÞÞðiÞ
ðiÞ
@ Dp2 =@ðPp;2 ð0ÞÞðiÞ
@ Ds3 =@ðPs;2 ð0ÞÞðiÞ
ðiÞ
ðiÞ @ Dp3 =@ðPs;2 ð0ÞÞðiÞ
iðiþ1Þ Ps;1 ð0Þ; Pp;1 ð0Þ; Ps;2 ð0Þ; Pp;2 ð0Þ; P s;3 ð0Þ; Pp;3 ð0Þ h iðiÞ ¼ Ps;1 ð0Þ; P p;1 ð0Þ; Ps;2 ð0Þ; Pp;2 ð0Þ; Ps;3 ð0Þ; Pp;3 ð0Þ h i ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ J 1 Ds1 ; Dp1 ; Ds2 ; Dp2 ; Ds3 ; Dp3
ðiÞ
@ Ds1 =@ðPs;3 ð0ÞÞðiÞ
ðiÞ
ðiÞ
@ Dp1 =@ðPs;3 ð0ÞÞðiÞ
ðiÞ
@ Ds2 =@ðPs;3 ð0ÞÞðiÞ
ðiÞ
@ Dp2 =@ðPs;3 ð0ÞÞðiÞ
@ Ds3 =@ðPp;2 ð0ÞÞðiÞ
ðiÞ
@ Ds3 =@ðPs;3 ð0ÞÞðiÞ
ðiÞ @ Dp3 =@ðPp;2 ð0ÞÞðiÞ
@ Dp3 =@ðPs;3 ð0ÞÞðiÞ
ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ
ðiÞ
@ Ds1 =@ðPp;3 ð0ÞÞðiÞ
3
7 ðiÞ @ Dp1 =@ðPp;3 ð0ÞÞðiÞ 7 7 7 ðiÞ ðiÞ 7 @ Ds2 =@ðPp;3 ð0ÞÞ 7 7 7 ðiÞ @ Dp2 =@ðPp;3 ð0ÞÞðiÞ 7 7 7 ðiÞ @ Ds3 =@ðPp;3 ð0ÞÞðiÞ 7 5 ðiÞ @ Dp3 =@ðPp;;3 ð0ÞÞðiÞ
ðA21Þ
Pþp;3 ð0Þ ¼ Pþp3 ðL2 Þ 2
gp PfL2 þ gp Pf0 ð1 lp Þ exp½ðCp rap N þ ap ÞL2 þ gp PfL1 ð1 lp Þ exp½ðCp rap N þ ap ÞðL2 L1 Þ ðA20Þ
where the Jacobian matrix J can be described bywhere i is iteration steps, for each i P 1. The Jacobian matrix J associated with nonlinear Eqs. (A14)– (A19) that the 36 partial derivatives be determined and evaluated. The exact evaluation of the partial derivatives is not practical, thus, we can use finite difference approximations to the partial derivatives. For example, the partial derivatives ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ of @ Ds1 =@ P ; @ Dp1 =@ P ; @ Ds2 =@ P , @ Dp2 = s;1 ð0Þ s;1 ð0Þ s;1 ð0Þ ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ @ P , @ Ds3 =@ P and @ Dp3 =@ P can be s;1 ð0Þ s;1 ð0Þ s;1 ð0Þ
ðA26Þ
þ þ The dependent variables of signal powers Pþ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and þ pump powers Pþ ð0Þ can be acquired by (A8)–(A11). P p;1 p;2 ð0Þ and nonlinear equation Dp2 can be given
Pþp;2 ð0Þ
¼
Pþp;1 ð0ÞPp;1 ð0Þ
þ
g2p PbL1 PfL1 1 lp
!,
gp PbL1 1 lp
Dp2 ¼ Pþp;3 ð0Þ gp PfL2 ð1 lp ÞPþp;2 ð1Þ ¼ 0
ðA27Þ ðA28Þ
Ds1 ; Dp1 ; Ds2 ; Ds3 ; Dp3 can be expressed the same as Case I . Case IV: P p;1 ðz1 Þ ¼ 0, P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ > 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers P þ ð0Þ, P ð0Þ can be acquired by (A2)–(A4), p;3 p;3 (A26), (A7). And P þ p;2 ð0Þ can be given (A23).
h i ðiÞ ðiÞ ðiÞ @ Ds1 =@ðP s;1 ð0ÞÞðiÞ Ds1 ðP s;1 ð0ÞðiÞ þ hstep;P p;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ; Pp;2 ð0ÞðiÞ ;P s;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ Ds1 ðPs;1 ð0ÞðiÞ ;P p;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ; Pp;2 ð0ÞðiÞ ;P s;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ =hstep h i ðiÞ ðiÞ ðiÞ @ Dp1 =@ðP s;1 ð0ÞÞðiÞ Dp1 ðPs;1 ð0ÞðiÞ þ hstep;P p;1 ð0ÞðiÞ ; Ps;2 ð0ÞðiÞ ;P p;2 ð0ÞðiÞ ; Ps;3 ð0ÞðiÞ ; Pp;3 ð0ÞðiÞ Þ Dp1 ðP s;1 ð0ÞðiÞ ; Pp;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ;P p;2 ð0ÞðiÞ ; Ps;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ =hstep h i ðiÞ ðiÞ ðiÞ @ Ds2 =@ðP s;1 ð0ÞÞðiÞ Ds2 ðP s;1 ð0ÞðiÞ þ hstep;P p;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ; Pp;2 ð0ÞðiÞ ;P s;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ Ds2 ðPs;1 ð0ÞðiÞ ;P p;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ; Pp;2 ð0ÞðiÞ ;P s;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ =hstep h i ðiÞ ðiÞ ðiÞ @ Dp2 =@ðP s;1 ð0ÞÞðiÞ Dp2 ðPs;1 ð0ÞðiÞ þ hstep;P p;1 ð0ÞðiÞ ; Ps;2 ð0ÞðiÞ ;P p;2 ð0ÞðiÞ ; Ps;3 ð0ÞðiÞ ; Pp;3 ð0ÞðiÞ Þ Dp2 ðP s;1 ð0ÞðiÞ ; Pp;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ;P p;2 ð0ÞðiÞ ; Ps;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ =hstep h i ðiÞ ðiÞ ðiÞ @ Ds3 =@ðP s;1 ð0ÞÞðiÞ Ds3 ðP s;1 ð0ÞðiÞ þ hstep;P p;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ; Pp;2 ð0ÞðiÞ ;P s;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ Ds3 ðPs;1 ð0ÞðiÞ ;P p;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ; Pp;2 ð0ÞðiÞ ;P s;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ =hstep h i ðiÞ ðiÞ ðiÞ @ Dp3 =@ðP s;1 ð0ÞÞðiÞ Dp3 ðPs;1 ð0ÞðiÞ þ hstep;P p;1 ð0ÞðiÞ ; Ps;2 ð0ÞðiÞ ;P p;2 ð0ÞðiÞ ; Ps;3 ð0ÞðiÞ ; Pp;3 ð0ÞðiÞ Þ Dp3 ðP s;1 ð0ÞðiÞ ; Pp;1 ð0ÞðiÞ ;P s;2 ð0ÞðiÞ ;P p;2 ð0ÞðiÞ ; Ps;3 ð0ÞðiÞ ;P p;3 ð0ÞðiÞ Þ =hstep
ðA22Þ
22
X. Hu et al. / Optical Fiber Technology 22 (2015) 13–22
þ þ The dependent variables of signal powers P þ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ þ and pump powers P p;1 ð0Þ can be acquired by (A8)–(A10), (A24). And nonlinear equation Dp1 ; Dp2 can be given (A25), (A28); Ds1 , Ds2 ; Ds3 ; Dp3 can be expressed the same as Case I. Case V: P p;1 ðz1 Þ > 0, P p;2 ðz2 Þ > 0 and P p;3 ðz3 Þ ¼ 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers Pp;1 ð0Þ; P p;2 ð0Þ can be acquired by (A2)–(A6). And Pþ p;3 ð0Þ can be given be acquired by (A26). þ þ The dependent variables of signal powers P þ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ þ and pump powers P þ ð0Þ; P ð0Þ can be acquired by (A8)–(A12). p;1 p;2 The nonlinear equation Dp2 ; Dp3 can be given
Dp2 ¼ Pp;2 ð1Þ gp PbL2 ð1 lp ÞPp;3 ð0Þ ¼ Pp;2 ð1Þ gp P bL2 ¼ 0 Dp3 ¼
Pþp;3 ð0Þ
ð1
lp ÞPþp;2 ð1Þ
f p P L2
g
¼0
ðA29Þ ðA30Þ
Ds1 ; Ds2 ; Ds3 ; Dp1 can be expressed the same as Case I . Case VI: P p;1 ðz1 Þ ¼ 0, P p;2 ðz2 Þ > 0 and P p;3 ðz3 Þ ¼ 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers Pp;2 ð0Þ can be acquired by (A2)–(A4), (A6). And þ Pþ p;2 ð0Þ; P p;3 ð0Þ can be given by (A23), (A26). þ þ The dependent variables of signal powers P þ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers P þ ð0Þ can be acquired by (A8)–(A10), (A24). p;1 The nonlinear equation Dp1 ; Dp1 ; Dp3 can be given by (A25), (A29), (A30); Ds1 , Ds2 ; Ds3 can be expressed the same as Case I. Case VII: P p;1 ðz1 Þ > 0, P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ ¼ 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers Pp;1 ð0Þ can be acquired by (A2)–(A5). And þ Pþ p;2 ð0Þ; P p;3 ð0Þ can be given by (A23), (A26). þ þ The dependent variables of signal powers P þ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ and pump powers P þ ð0Þ can be obtained by (A8)–(A11). Nonlinear p;1 equation Dp3 can be given by (A30). The nonlinear equation Dp1 ; Dp2 can be given
Dp1 ¼ Pp;1 ð1Þ gp PbL1 ð1 lp ÞPp;2 ð0Þ ¼ Pp;1 ð1Þ gp P bL1 ¼ 0
ðA31Þ
Dp2 ¼ Pþp;2 ð0Þ gp PfL1 ð1 lp ÞP þp;1 ð1Þ ¼ 0
ðA32Þ
Ds1 ; Ds2 ; Ds3 are the same as Case I. Case VIII: P p;1 ðz1 Þ ¼ 0, P p;2 ðz2 Þ ¼ 0 and P p;3 ðz3 Þ ¼ 0 The independent variables of signal powers P s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ þ can be acquired by (A2)–(A4). And pump powers Pþ ð0Þ; P ð0Þ can p;2 p;3 be given be acquired by (A23), (A26). þ þ The dependent variables of signal powers P þ s;1 ð0Þ; P s;2 ð0Þ; P s;3 ð0Þ can be acquired by (A8)–(A10).
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