- Email: [email protected]
Numerical analysis of multi-wavelength cascaded Raman fiber lasers based on genetic algorithm
Optics Communications 282 (2009) 1626–1630
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Numerical analysis of multi-wavelength cascaded Raman fiber lasers based on genetic algorithm Qin Zujun a,*, Zhou Xiaojun a, Wu Haocheng b, Zou Zili b a
Key Laboratory of Broadband Optical Fiber Transmission and Communication Networks, Ministry of Education, School of Opto-electronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China b Guilin Institute of Optical Communications, Guilin 541004, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 5 November 2008 Received in revised form 29 December 2008 Accepted 4 January 2009
A novel numerical method is presented for the calculations of the coupled propagation equations in multi-wavelength Raman fiber lasers. By taking the advantages of genetic algorithm and shooting method, only a few of the best individuals at each generation are chosen to implement several shootings in order to accelerate theirs converging. The output characteristics of an all-fiber three-wavelength Raman fiber laser have been analyzed based on the proposed algorithm. Results show that the total output power linearly depends on the pump power with a slope efficiency of 51%. For the three output Stokes, the slope efficiencies of the longer wavelengths are larger than that of the shorter ones because the optical energy at the Stokes with shorter wavelengths allows for a transfer of optical power to the longer Stokes via stimulated Raman scattering. We also find that the total output power degrades by less than 10% by adjusting the output-coupler reflectivity and is insensitive to the variation of the Raman fiber length over a large range. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Fiber lasers Stimulated Raman scattering Multi-wavelength lasers Genetic algorithm
1. Introduction Raman fiber amplifiers (RFAs) have been attracting increasing interests in long-haul and large-capacity fiber-optic transmission systems [1]. RFAs with broad gain bandwidth and small gain ripples have been demonstrated by using the well-know multi-wavelength pumping schemes [1–3]. Multi-wavelength cascaded Raman fiber lasers (MRFLs) with proper choice of the output wavelengths and powers are one of the promising pump sources for realizing such RFAs. They have simple all-fiber configurations, which are composed of double-clad fiber lasers acting as pumps, low-loss and high Raman-gain silica-based fibers doped with either GeO2 or P2O5 as gain mediums, and fiber Bragg gratings (FBGs) as feedback elements. What is more important is that MRFLs can emit several tens-of-nanometers-spaced wavelengths simultaneously because of the broad Raman gain spectrum of the gain fibers. In the past several years, MRFLs have received a lot of attention as pump sources for RFAs with desirable gain bandwidth and flatness [4–9]. The characteristics of MRFLs are governed by a first-order system of nonlinear two-point boundary-value ordinary differential equations (BVODEs) with boundary conditions constructed by the reflections of FBGs at both ends of the Raman fiber and by the in* Corresponding author. E-mail addresses: [email protected] (Z. Xiaojun).
(Q.
Zujun),
[email protected]
0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.01.003
jected pump power. For the BVODEs of single-wavelength Raman fiber lasers (SRFLs), some numerical and analytic methods have been well established in the last few years [10–13]. Unlike SRFLs, MRFLs operate at multiple wavelengths and it is often hard to obtain the analytical solutions. The results for the BVODEs of MRFLs are usually found numerically by shooting method. Solving by the shooting method with guessed arbitrary values as initial inputs, whereas, may lead to the calculating divergent or unstable. Numerical models for investigating MRFLs based on variable substitution [14] and differential evolution algorithm [15] have been reported. In this paper, taking the excellent searching ability of genetic algorithm and the fast converging speed of shooting method we propose a novel and efficient numerical algorithm for solving the BVODEs of MRFLs. A three-wavelength all-fiber Raman fiber laser has been analyzed by the method. Simulation results show that the proposed method has the advantages of good convergence and fast converging speed.
2. Theoretical model As shown in Fig. 1, a MRFL comprises of a pump source, a high Raman gain single-mode fiber with a few hundred meters, an input highly reflective (HR) FBGs set, and a matching output FBGs set including HR gratings and partially reflective ones with adjustable reflectivity acting as output couplers (OCs). All the HR FBGs are
Q. Zujun et al. / Optics Communications 282 (2009) 1626–1630
Fig. 1. Schematic of a MRFL (HR-FBG: highly reflective fiber Bragg gratings; A-FBG: Adjustable fiber Bragg gratings; OCs-Output couplers; A and B represent splicing points).
broadband to avoid the power leakage because of the powerdependent spectrum broadening of the intra-cavity Stokes. In such a configuration, the pump wavelength can be converted to the Stokes lines at desirable wavelengths after multiple stages of stimulated Raman scattering (SRS) in the cascaded cavities. Coupled-power differential equations are used to describe the stead-state SRS process in the cascaded resonators. Given that the spontaneous Raman scattering and backward Rayleigh scattering have negligible effects on MRFLs under strong pump, these noise terms are not included. Thus, the power evolutions of the pump and the Stokes along the Raman fiber in both directions of forward (+) and backward () are expressed as:
n X 1 dP 0 m0 g 0i ðPþi þ Pi Þ=Aeff ðm0 ; mi Þ ¼ a0 mi P0 dz i¼1
j1 X 1 dP j g ij ðPþi þ Pi Þ=Aeff ðmi ; mj Þ ¼ aj Pj dz i¼0
n X
i¼jþ1
g ji
mj
vi
ðPþi þ Pi Þ=Aeff ðmj ; mi Þ
n1 X 1 dP j g in ðPþi þ Pi Þ=Aeff ðmi ; mn Þ ¼ an Pn dz i¼0
ð1Þ
where P0, Pj refer to the pump and the jth order Stokes power (j = 1 n) and a denotes the fiber attenuation. As shown in the equations, depending on the Raman gain spectrum of the Raman fiber a Stokes line at mj will gain from the higher frequency radiations at mi(i = 0 j 1) at the same time it experiences losses caused by affording energy to the lower frequency Stokes at mi(i = j + 1 n). The parameter gji is the Raman gain coefficient at mi gained from mj, which scales inversely with pump wavelength and can be written by gji = g(mj, mi) = gR(Dm)mj/mR, where gR(Dm) is the Raman gain spectrum measured at a reference pump frequency mR, and Dm is the frequency shift. Finally, the quantity Aeff(mi, mj) represents the effective core area associated with the waves at mi and mj, and can be given by Aeff(mi, mj) = [Aeff(mi) + Aeff(mj)]/2 if the radial intensity distribution of the optical fields are approximated as Gaussian profiles. The two-point boundary conditions for the resonated waves are given by
Pþ0 ð0Þ ¼ Pin ; Pþj ð0Þ ¼ Pj ð0Þ Rj
ðj ¼ 1 nÞ;
Pj ðLÞ
ðj ¼ 0 nÞ;
¼
Pþj ðLÞ
Rþj
ð2Þ
Pjout ¼ ð1 Rþj ÞPþj ðLÞ ¼
ð1 Rþj ÞRj qffiffiffiffiffiffiffiffiffiffiffi Pj ð0Þ: Rþj Rj
Thus, finding the solutions of Eq. (1) is transformed to obtain the initial power P j ð0Þ. In the theoretical model of MRFLs described by Eq. (1) we treat the power at the pump and each Stokes line as being concentrated at the center wavelength of the corresponding FBG. These assumptions are reasonable for the pump and the Stokes in the highly reflective and broadband cavities [10–15]. For the output Stokes, however, it should consider the influences of spectrum broadening because the OCs are usually narrowband. The spectral broadening effect is caused mainly by the four-wave mixing interaction between the laser modes, which will modify the OCs’ feedback. Thus, in addition to the above treatments it is necessary to introduce the effective OCs reflectivity, which can be measured experimentally, for the output Stokes [14,16].
3. Numerical algorithm Shooting method is generally regarded as an effective approach for solving the classical two-point BVODEs such as Eq. (1), whereas, the initial values for shooting must be assigned properly. In the case of cascaded number n 6 3, satisfactory shooting results may be obtained via manual adjusting the initial inputs. For the higher order cascaded number case (n > 3), however, it is very difficult to determine the appropriate initial input for each shooting variable. To overcome the difficulty, genetic algorithms (GA), which has been turned out to be a powerful tool in the field of global optimization, is employed to search for the initial power values. One of the disadvantages of GA, however, is that it converges slowly or oscillates near the optimal space. Fortunately, this can be well solved by the shooting method. By taking the advantages of both a novel algorithm is proposed. At each generation of GA, a few of the fittest individuals after operations of selection, crossover, and mutation will be chosen to carry out several shootings in order to accelerate theirs converging for the individuals near the global optimum or diverging for the individuals situated in the local optimums. Before performing the method, we define an objective funcT tion, W, with the initial power P ð0Þ ¼ ½P 0 ð0Þ; P 1 ð0Þ; ; P n ð0Þ as variable. P (0) obtained by our algorithm will be substituted into Eq. (3) to calculate the output power.
W½P ð0Þ ¼ Rþ P ðLÞ=Pþ ðLÞ;
ð3Þ
ð4Þ
where P(0) is composed by powers of the backward propagating T waves at the starting point (z = 0), Rþ ¼ ½R0 þ; R1 þ; ; Rþ n denotes the reflectivity of the output FBGs set, and Pþ= ðLÞ ¼ ½P 0þ= ðLÞ; P1þ= ðLÞ; ; Pnþ= ðLÞT represent the calculated power of the forward- and backward-propagating waves at the output side (z = L) obtained by integrating Eq. (1) over the fiber length with P (0) as initial value. Therefore, RCal ¼ P ðLÞ=Pþ ðLÞ refers to the calculated reflectivity of the output FBGs set. The end of the proposed method is to find out a right P(0) to make the calculated reflectivity of the output FBGs set match the real one with any given precision, i.e.,
y ¼ kW½P ð0Þk < e;
where Pin refers to the injected pump power, L corresponds to the Raman fiber length, and R j represent the FBG reflectivity centered at frequency mj at the input () and output (+) side of the cavity, respectively. Based on Eq. (2), the output power at each output Stokes is defined by
1627
ð5Þ
where e is a prescribed error value. The value of y reveals the deviation of RCal from R+. The following is the procedure of our proposed algorithm: (1) Generate a population with N individuals of fP ð0Þgind¼N ind¼1 . All the individuals are guaranteed to be not divergent over the whole integrating fiber length by the rule of R L 0 < 0 dP ðzÞ < 1.
1628
Q. Zujun et al. / Optics Communications 282 (2009) 1626–1630
(2) Selection, crossover, and mutation. The elitist strategy is applied in GA. (3) Calculate the objective values and the deviation values of y based on Eqs. (4) and (5) and sort individuals according to the ascending order of the deviation values. Find out the first M (M < N) elite individuals with different y values. (4) Implement k shootings to the chosen M elite individuals. Let ½P ð0Þg be one of the elite individuals at generation g: (i) Calculate the objective value, Wf½P ð0Þg g, by Eq. (4). (ii) Calculate the Jacobian matrix J.
J¼
@W @W @W ; ; : @P 0 ð0Þ @P1 ð0Þ @P n ð0Þ
ð6Þ
Eq. (6) can not be calculated analytically due to that the objective function,W, is not capable of being expressed in a closed form, but can be approximated numerically as
Jmþ1
Wf½P ð0Þg ; Pm ð0Þ þ DPm ð0Þg Wf½P ð0Þg ; Pm ð0Þg ; DPm ð0Þ m ¼ 0; 1; ; n:
ð7Þ
ð0Þ g
1
(iii) Update [P (0)]g by ½P ¼ ½P ð0Þg J :Wf½P ð0Þg g, then calculate the objective function with the new n o ½P ð0Þ g . If the condition of kW ½P ð0Þ g k < e is satisfied then stop and output the optimal results, else ½P ð0Þg ¼ ½P ð0Þ g and go to (i) for the next shooting until k. (5) If any individual that satisfies Eq. (5) is present after the shooting step of (4) then stop and output the optimal results, else go to step (2) for the next generation.The whole process of the proposed numerical method is also generalized in Fig. 2.
Fig. 2. Flowchart of the proposed numerical algorithm. g is the generation number of GA. st is the shooting time.
4. Numerical results and discussion We have applied the proposed method to analyze a three-wavelength Raman fiber laser (RFL) based on phosphorus-doped silica fiber. We consider a pump at 1117 nm (k0) emitted from a double-clad ytterbium fiber laser will be converted to the desirable longer wavelengths at 1427 (k3), 1455 (k4) and 1480 nm (k5) after a cascade of two orders of Stokes (1312 (k1) and 1375 nm (k2)) [5]. The three output wavelengths are suitable to act as a pump source for a wideband Raman amplifier operating in both C- and L-bands. To attain these necessary wavelengths, P2O5-related (40 THz) and SiO2-related (3.4–26 THz) Stokes shifts are both employed in the frequency-shift schemes. All the FBGs used to construct resonant cavities are broadband and highly reflective except the three OCs centered at k3, k4, and k5, are partially reflective in order to couple the Stokes power out of the cavities. Fig. 3 shows the normalized Raman gain spectrum g R ðDmÞ=Aeff of the phosphosilicate fiber used in the numerical calculations [9]. The maximum Raman gain of the phosphorus-doped silica fiber measured by a reference pump at 1060 nm is assumed to be 1.5/(km W) at the frequency shift of 40 THz. The reflectivities of the three OCs are 0.4(k3), 0.3(k4) and 0.3(k5), respectively, the highly reflective FBGs are all set to be 0.99. The fiber length is 500 m, splicing losses for both splicing places in Fig. 1 are 0.05 dB, and all FBGs are assumed to have the averaged gray losses of 0.07 dB which include absorption, scattering, and transition losses generated in FBGs’ fabrication process and will affect all the transmitted Stokes [17]. Additionally, the parameters M, N, k, and the error tolerance e are 50, 5, 2, 106, respectively. Calculations have been performed by using our method and the traditional GA to explore the output characteristics of the threewavelength RFL. The comparison results between both methods are shown in Fig. 4. It can be seen that the best individual that fulfills Eq. (5) is found after iterations of only twelve generations by our method while the traditional GA can not converge to the given error value e even after one hundred generations because of its intrinsic disadvantages. The result indicates that the shootings exerted on the minority elite individuals accelerate the converging greatly. Furthermore, in view of the initial population of GA being generated randomly, one can see that our algorithm overcomes the necessity to allocate the proper initial values for the shooting variables which are required by the ‘pure’ shooting method. Fig. 5 shows the output characteristics of the three-wavelength RFL. An approximately linear relationship between the total output
Fig. 3. Raman gain spectrum of the phosphosilicate fiber. Inset shows the attenuation spectrum of the fiber in the window of 1000–1600 nm.
Q. Zujun et al. / Optics Communications 282 (2009) 1626–1630
Fig. 4. Comparison between our method and ‘pure’ GA. Pin = 5 W.
1629
is shown in Fig. 6. As the OC reflectivity increases, the power at k3 decreases, and the power reduction at k5 is observed by reason of lesser energy being achievable from k3. Simultaneously, there is a power growth at k4 due to its depletion by the Stokes k5 falls. In the practical applications, it is often important to determine the adjusting range of the OC reflectivity within which the output power at every output wavelength is grater than zero. The adjustable range for the OC reflectivity is 27–67% over which the total output power shows a fluctuation of <9%. The effects of the variations of the OCs reflectivities at k4 and k5 on the RFL are also examined. The adjustable intervals for these two OCs are 18–40% and 24–55%, respectively, over which their corresponding total output power ripples are 9.8% and 1.9%. We know from the above discussion that the total output power just displays a maximum degradation of <10% when one of the OCs reflectivity is changed. Therefore, it is necessary to know the effect of the Raman fiber length on the total output power. The dependence of the total output power on the fiber length is illustrated in Fig. 7. The maximum output power, Ptotal = 1.8029 W, is obtained with a fiber length of 420 m. It should be noticed that the total output power is insensitive to the variation of the Raman fiber length, in which the total output power reduces by less than 3% over a large range of 300m 6 L 6 600m. If the OCs reflectivities are altered within their adjustable range, the insensitive region is still valid.
Fig. 5. Output characteristics of the three-wavelength Raman fiber laser.
and the pump with a slope efficiency of 51% is observed, which is in agreement with the experimental result of 49.7% [5]. Different laser parameters used in our calculations and the experiment lead to somewhat different output power for the respective output wavelength. The output powers at the three out Stokes are interdependent, which depends primarily on the gain achieved from the waves with shorter wavelengths and losses caused by fiber attenuation and red-shift to the longer Stokes. In addition, the fiber length, OCs reflectivity, FBG gray losses, and splicing losses also have significant effects on the output power. With the increasing of the pump power, k4 reaches its threshold firstly and resonates because of its winning from mode competition among the three output wavelengths. The threshold of k4 is 1.7 W. A further increase of the pump power does not boost but deplete the output power of k4. The reason is that the Stokes lines k5 oscillates at 2.7 W and part of the energy in k4 tends to be transferred to k5 via SRS because the latter Stokes lies within the Raman gain spectrum of k4. Thus, it can be seen that the slope efficiency of k4 has a remarkable reduction when the pump power is larger than 2.7 W. Also observed in Fig. 5, the slope efficiency of k5 is the greatest and that of k3 is the minimum. This is believed due to the longer Stokes gain additional power at the expense of the shorter ones. The dependence of the output power on the OCs reflectivity and fiber length is also investigated, respectively. The output power versus the OC reflectivity at k3 with the other two OCs unvaried
Fig. 6. Output power as a function of the output-coupler reflectivity at 1427 nm.
Fig. 7. Total output power and residual pump power as a function of fiber length. Pin = 5 W.
1630
Q. Zujun et al. / Optics Communications 282 (2009) 1626–1630
With the fiber length chosen in this interval, the total output power of the three-wavelength RFL will appear in the vicinity of the maximal value. Thus the RFL can obtain the adjustable power spaces as large as possible for each output Stokes by varying the OCs reflectivity. Also shown in Fig. 5 and Fig. 7 is the residual pump power, P 0, which is the first element in each individual (see Eq. (4)), as a function of the launched pump power and evolution along the Raman fiber, respectively. The small searching space by GA for the respective element in individuals can speed up the computing greatly. From both figures, it can be seen that the residual pump power in unit of dBm decreases approximately linearly with the pump power and the fiber length. This can be considered as a result of the intensive interactions among the resonated waves which promote the conversion efficiency of pump-to-Stokes. For a Raman fiber of 500 m and a pump power of 5 W, only 10.2 mW of the residual pump power is un-depleted and returns to the input. Therefore, the searching range for the residual pump power in GA can be reduced considerably in the case of a large injected pump or a long fiber length. 5. Conclusions We have proposed an efficient numerical algorithm for the steady-state coupled power equations describing stimulated Raman scattering in Multi-wavelength Raman fiber lasers. Taking the excellent searching ability of genetic algorithm in the global space and the fast converging speed of shooting method in its convergence region, we perform several shootings to a few elite individuals after GA operations of selection, crossover, and mutation at each generation to accelerate theirs converging. An example of an all-fiber three-wavelength Raman fiber laser has been used to demonstrate the method. Results show that our method exhibits
a good convergence and the shootings exerted on the minority elite individuals accelerate the converging speed greatly. The simulated results show that the total output power is insensitive to the variation of the Raman fiber length, in which the total output power reduces by less than 3% over a large range of 300m 6 L 6 600m. We also find that the searching range set for the residual pump power in GA can be reduced considerably in the case of a large injected pump power or a long fiber length, which will also speed up the calculating. References [1] M.N. Islam, IEEE J. Sel. Top. Quantum. Electron. 8 (2002) 548. [2] H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, E. Rabarijaona, IEEE Photon. Technol. Lett. 11 (1999) 530. [3] V.E. Perlin, H.G. Winful, J. Lightwave Technol. 20 (2002) 250. [4] M.D. Mermelstein, C. Headley, J.-C. Bouteiller, P. Steinvurzel, C. Horn, K. Feder, B.J. Eggleton, IEEE Photon. Technol. Lett. 13 (2001) 1286. [5] X. Normandin, F. Leplingard, E. Bourova, C. Leclère, T. Lopez, J.-J. Guérin, D. Bayart, In: Proceedings of Optical Fiber Communication Conference, 2002, TuB2. [6] F. Leplingard, S. Borne, L. Lorcy, T. Lopez, J.-J. Guérin, C. Moreau, C. Martinelli, D. Bayart, Electron. Lett. 38 (2002) 886. [7] A.A. Demidov, A.N. Starodumov, X. Li, A. Martinez-Rios, H. Po, Opt. Lett. 28 (2003) 1540. [8] Z. Xiong, T. Chen, Opt. Fiber Technol. 13 (2007) 81. [9] M.N. Islam, Raman Amplifiers for Telecommunications. 2: Sub-Systems and Systems, Springer-Verlag, New York, 2004. [10] M. Rini, I. Cristiani, V. Degiorgio, IEEE J. Quantum Electron. 36 (2000) 1117. [11] S.D. Jackson, P.H. Muir, J. Opt. Soc. Am. B 18 (2001) 1297. [12] H. Kaidi, Z. Xiaojun, Q. Zujun, W. Haocheng, Z. Zili, Opt. Commun. 271 (2007) 257. [13] Q. Zujun, Z. Xiaojun, L. Qing, W. Haocheng, Z. Zili, J. Lightwave Technol. 25 (2007) 1555. [14] F. Leplingard, C. Martinelli, S. Borne, L. Lorcy, D. Bayart, F. Castella, P. Chartier, E. Faou, IEEE Photon. Technol. Lett. 16 (2004) 2601. [15] S. Cierullies, H. Renner, E. Brinkmeyer, Opt. Commun. 217 (2003) 233. [16] J.-C. Bouteiller, IEEE Photon. Technol. Lett. 15 (2003) 1698. [17] Y. Wang, H. Po, Meas. Sci. Technol. 14 (2003) 883.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Numerical analysis of multi-wavelength cascaded Raman fiber lasers based on genetic algorithm Qin Zujun a,*, Zhou Xiaojun a, Wu Haocheng b, Zou Zili b a
Key Laboratory of Broadband Optical Fiber Transmission and Communication Networks, Ministry of Education, School of Opto-electronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China b Guilin Institute of Optical Communications, Guilin 541004, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 5 November 2008 Received in revised form 29 December 2008 Accepted 4 January 2009
A novel numerical method is presented for the calculations of the coupled propagation equations in multi-wavelength Raman fiber lasers. By taking the advantages of genetic algorithm and shooting method, only a few of the best individuals at each generation are chosen to implement several shootings in order to accelerate theirs converging. The output characteristics of an all-fiber three-wavelength Raman fiber laser have been analyzed based on the proposed algorithm. Results show that the total output power linearly depends on the pump power with a slope efficiency of 51%. For the three output Stokes, the slope efficiencies of the longer wavelengths are larger than that of the shorter ones because the optical energy at the Stokes with shorter wavelengths allows for a transfer of optical power to the longer Stokes via stimulated Raman scattering. We also find that the total output power degrades by less than 10% by adjusting the output-coupler reflectivity and is insensitive to the variation of the Raman fiber length over a large range. Ó 2009 Elsevier B.V. All rights reserved.
Keywords: Fiber lasers Stimulated Raman scattering Multi-wavelength lasers Genetic algorithm
1. Introduction Raman fiber amplifiers (RFAs) have been attracting increasing interests in long-haul and large-capacity fiber-optic transmission systems [1]. RFAs with broad gain bandwidth and small gain ripples have been demonstrated by using the well-know multi-wavelength pumping schemes [1–3]. Multi-wavelength cascaded Raman fiber lasers (MRFLs) with proper choice of the output wavelengths and powers are one of the promising pump sources for realizing such RFAs. They have simple all-fiber configurations, which are composed of double-clad fiber lasers acting as pumps, low-loss and high Raman-gain silica-based fibers doped with either GeO2 or P2O5 as gain mediums, and fiber Bragg gratings (FBGs) as feedback elements. What is more important is that MRFLs can emit several tens-of-nanometers-spaced wavelengths simultaneously because of the broad Raman gain spectrum of the gain fibers. In the past several years, MRFLs have received a lot of attention as pump sources for RFAs with desirable gain bandwidth and flatness [4–9]. The characteristics of MRFLs are governed by a first-order system of nonlinear two-point boundary-value ordinary differential equations (BVODEs) with boundary conditions constructed by the reflections of FBGs at both ends of the Raman fiber and by the in* Corresponding author. E-mail addresses: [email protected] (Z. Xiaojun).
(Q.
Zujun),
[email protected]
0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.01.003
jected pump power. For the BVODEs of single-wavelength Raman fiber lasers (SRFLs), some numerical and analytic methods have been well established in the last few years [10–13]. Unlike SRFLs, MRFLs operate at multiple wavelengths and it is often hard to obtain the analytical solutions. The results for the BVODEs of MRFLs are usually found numerically by shooting method. Solving by the shooting method with guessed arbitrary values as initial inputs, whereas, may lead to the calculating divergent or unstable. Numerical models for investigating MRFLs based on variable substitution [14] and differential evolution algorithm [15] have been reported. In this paper, taking the excellent searching ability of genetic algorithm and the fast converging speed of shooting method we propose a novel and efficient numerical algorithm for solving the BVODEs of MRFLs. A three-wavelength all-fiber Raman fiber laser has been analyzed by the method. Simulation results show that the proposed method has the advantages of good convergence and fast converging speed.
2. Theoretical model As shown in Fig. 1, a MRFL comprises of a pump source, a high Raman gain single-mode fiber with a few hundred meters, an input highly reflective (HR) FBGs set, and a matching output FBGs set including HR gratings and partially reflective ones with adjustable reflectivity acting as output couplers (OCs). All the HR FBGs are
Q. Zujun et al. / Optics Communications 282 (2009) 1626–1630
Fig. 1. Schematic of a MRFL (HR-FBG: highly reflective fiber Bragg gratings; A-FBG: Adjustable fiber Bragg gratings; OCs-Output couplers; A and B represent splicing points).
broadband to avoid the power leakage because of the powerdependent spectrum broadening of the intra-cavity Stokes. In such a configuration, the pump wavelength can be converted to the Stokes lines at desirable wavelengths after multiple stages of stimulated Raman scattering (SRS) in the cascaded cavities. Coupled-power differential equations are used to describe the stead-state SRS process in the cascaded resonators. Given that the spontaneous Raman scattering and backward Rayleigh scattering have negligible effects on MRFLs under strong pump, these noise terms are not included. Thus, the power evolutions of the pump and the Stokes along the Raman fiber in both directions of forward (+) and backward () are expressed as:
n X 1 dP 0 m0 g 0i ðPþi þ Pi Þ=Aeff ðm0 ; mi Þ ¼ a0 mi P0 dz i¼1
j1 X 1 dP j g ij ðPþi þ Pi Þ=Aeff ðmi ; mj Þ ¼ aj Pj dz i¼0
n X
i¼jþ1
g ji
mj
vi
ðPþi þ Pi Þ=Aeff ðmj ; mi Þ
n1 X 1 dP j g in ðPþi þ Pi Þ=Aeff ðmi ; mn Þ ¼ an Pn dz i¼0
ð1Þ
where P0, Pj refer to the pump and the jth order Stokes power (j = 1 n) and a denotes the fiber attenuation. As shown in the equations, depending on the Raman gain spectrum of the Raman fiber a Stokes line at mj will gain from the higher frequency radiations at mi(i = 0 j 1) at the same time it experiences losses caused by affording energy to the lower frequency Stokes at mi(i = j + 1 n). The parameter gji is the Raman gain coefficient at mi gained from mj, which scales inversely with pump wavelength and can be written by gji = g(mj, mi) = gR(Dm)mj/mR, where gR(Dm) is the Raman gain spectrum measured at a reference pump frequency mR, and Dm is the frequency shift. Finally, the quantity Aeff(mi, mj) represents the effective core area associated with the waves at mi and mj, and can be given by Aeff(mi, mj) = [Aeff(mi) + Aeff(mj)]/2 if the radial intensity distribution of the optical fields are approximated as Gaussian profiles. The two-point boundary conditions for the resonated waves are given by
Pþ0 ð0Þ ¼ Pin ; Pþj ð0Þ ¼ Pj ð0Þ Rj
ðj ¼ 1 nÞ;
Pj ðLÞ
ðj ¼ 0 nÞ;
¼
Pþj ðLÞ
Rþj
ð2Þ
Pjout ¼ ð1 Rþj ÞPþj ðLÞ ¼
ð1 Rþj ÞRj qffiffiffiffiffiffiffiffiffiffiffi Pj ð0Þ: Rþj Rj
Thus, finding the solutions of Eq. (1) is transformed to obtain the initial power P j ð0Þ. In the theoretical model of MRFLs described by Eq. (1) we treat the power at the pump and each Stokes line as being concentrated at the center wavelength of the corresponding FBG. These assumptions are reasonable for the pump and the Stokes in the highly reflective and broadband cavities [10–15]. For the output Stokes, however, it should consider the influences of spectrum broadening because the OCs are usually narrowband. The spectral broadening effect is caused mainly by the four-wave mixing interaction between the laser modes, which will modify the OCs’ feedback. Thus, in addition to the above treatments it is necessary to introduce the effective OCs reflectivity, which can be measured experimentally, for the output Stokes [14,16].
3. Numerical algorithm Shooting method is generally regarded as an effective approach for solving the classical two-point BVODEs such as Eq. (1), whereas, the initial values for shooting must be assigned properly. In the case of cascaded number n 6 3, satisfactory shooting results may be obtained via manual adjusting the initial inputs. For the higher order cascaded number case (n > 3), however, it is very difficult to determine the appropriate initial input for each shooting variable. To overcome the difficulty, genetic algorithms (GA), which has been turned out to be a powerful tool in the field of global optimization, is employed to search for the initial power values. One of the disadvantages of GA, however, is that it converges slowly or oscillates near the optimal space. Fortunately, this can be well solved by the shooting method. By taking the advantages of both a novel algorithm is proposed. At each generation of GA, a few of the fittest individuals after operations of selection, crossover, and mutation will be chosen to carry out several shootings in order to accelerate theirs converging for the individuals near the global optimum or diverging for the individuals situated in the local optimums. Before performing the method, we define an objective funcT tion, W, with the initial power P ð0Þ ¼ ½P 0 ð0Þ; P 1 ð0Þ; ; P n ð0Þ as variable. P (0) obtained by our algorithm will be substituted into Eq. (3) to calculate the output power.
W½P ð0Þ ¼ Rþ P ðLÞ=Pþ ðLÞ;
ð3Þ
ð4Þ
where P(0) is composed by powers of the backward propagating T waves at the starting point (z = 0), Rþ ¼ ½R0 þ; R1 þ; ; Rþ n denotes the reflectivity of the output FBGs set, and Pþ= ðLÞ ¼ ½P 0þ= ðLÞ; P1þ= ðLÞ; ; Pnþ= ðLÞT represent the calculated power of the forward- and backward-propagating waves at the output side (z = L) obtained by integrating Eq. (1) over the fiber length with P (0) as initial value. Therefore, RCal ¼ P ðLÞ=Pþ ðLÞ refers to the calculated reflectivity of the output FBGs set. The end of the proposed method is to find out a right P(0) to make the calculated reflectivity of the output FBGs set match the real one with any given precision, i.e.,
y ¼ kW½P ð0Þk < e;
where Pin refers to the injected pump power, L corresponds to the Raman fiber length, and R j represent the FBG reflectivity centered at frequency mj at the input () and output (+) side of the cavity, respectively. Based on Eq. (2), the output power at each output Stokes is defined by
1627
ð5Þ
where e is a prescribed error value. The value of y reveals the deviation of RCal from R+. The following is the procedure of our proposed algorithm: (1) Generate a population with N individuals of fP ð0Þgind¼N ind¼1 . All the individuals are guaranteed to be not divergent over the whole integrating fiber length by the rule of R L 0 < 0 dP ðzÞ < 1.
1628
Q. Zujun et al. / Optics Communications 282 (2009) 1626–1630
(2) Selection, crossover, and mutation. The elitist strategy is applied in GA. (3) Calculate the objective values and the deviation values of y based on Eqs. (4) and (5) and sort individuals according to the ascending order of the deviation values. Find out the first M (M < N) elite individuals with different y values. (4) Implement k shootings to the chosen M elite individuals. Let ½P ð0Þg be one of the elite individuals at generation g: (i) Calculate the objective value, Wf½P ð0Þg g, by Eq. (4). (ii) Calculate the Jacobian matrix J.
J¼
@W @W @W ; ; : @P 0 ð0Þ @P1 ð0Þ @P n ð0Þ
ð6Þ
Eq. (6) can not be calculated analytically due to that the objective function,W, is not capable of being expressed in a closed form, but can be approximated numerically as
Jmþ1
Wf½P ð0Þg ; Pm ð0Þ þ DPm ð0Þg Wf½P ð0Þg ; Pm ð0Þg ; DPm ð0Þ m ¼ 0; 1; ; n:
ð7Þ
ð0Þ g
1
(iii) Update [P (0)]g by ½P ¼ ½P ð0Þg J :Wf½P ð0Þg g, then calculate the objective function with the new n o ½P ð0Þ g . If the condition of kW ½P ð0Þ g k < e is satisfied then stop and output the optimal results, else ½P ð0Þg ¼ ½P ð0Þ g and go to (i) for the next shooting until k. (5) If any individual that satisfies Eq. (5) is present after the shooting step of (4) then stop and output the optimal results, else go to step (2) for the next generation.The whole process of the proposed numerical method is also generalized in Fig. 2.
Fig. 2. Flowchart of the proposed numerical algorithm. g is the generation number of GA. st is the shooting time.
4. Numerical results and discussion We have applied the proposed method to analyze a three-wavelength Raman fiber laser (RFL) based on phosphorus-doped silica fiber. We consider a pump at 1117 nm (k0) emitted from a double-clad ytterbium fiber laser will be converted to the desirable longer wavelengths at 1427 (k3), 1455 (k4) and 1480 nm (k5) after a cascade of two orders of Stokes (1312 (k1) and 1375 nm (k2)) [5]. The three output wavelengths are suitable to act as a pump source for a wideband Raman amplifier operating in both C- and L-bands. To attain these necessary wavelengths, P2O5-related (40 THz) and SiO2-related (3.4–26 THz) Stokes shifts are both employed in the frequency-shift schemes. All the FBGs used to construct resonant cavities are broadband and highly reflective except the three OCs centered at k3, k4, and k5, are partially reflective in order to couple the Stokes power out of the cavities. Fig. 3 shows the normalized Raman gain spectrum g R ðDmÞ=Aeff of the phosphosilicate fiber used in the numerical calculations [9]. The maximum Raman gain of the phosphorus-doped silica fiber measured by a reference pump at 1060 nm is assumed to be 1.5/(km W) at the frequency shift of 40 THz. The reflectivities of the three OCs are 0.4(k3), 0.3(k4) and 0.3(k5), respectively, the highly reflective FBGs are all set to be 0.99. The fiber length is 500 m, splicing losses for both splicing places in Fig. 1 are 0.05 dB, and all FBGs are assumed to have the averaged gray losses of 0.07 dB which include absorption, scattering, and transition losses generated in FBGs’ fabrication process and will affect all the transmitted Stokes [17]. Additionally, the parameters M, N, k, and the error tolerance e are 50, 5, 2, 106, respectively. Calculations have been performed by using our method and the traditional GA to explore the output characteristics of the threewavelength RFL. The comparison results between both methods are shown in Fig. 4. It can be seen that the best individual that fulfills Eq. (5) is found after iterations of only twelve generations by our method while the traditional GA can not converge to the given error value e even after one hundred generations because of its intrinsic disadvantages. The result indicates that the shootings exerted on the minority elite individuals accelerate the converging greatly. Furthermore, in view of the initial population of GA being generated randomly, one can see that our algorithm overcomes the necessity to allocate the proper initial values for the shooting variables which are required by the ‘pure’ shooting method. Fig. 5 shows the output characteristics of the three-wavelength RFL. An approximately linear relationship between the total output
Fig. 3. Raman gain spectrum of the phosphosilicate fiber. Inset shows the attenuation spectrum of the fiber in the window of 1000–1600 nm.
Q. Zujun et al. / Optics Communications 282 (2009) 1626–1630
Fig. 4. Comparison between our method and ‘pure’ GA. Pin = 5 W.
1629
is shown in Fig. 6. As the OC reflectivity increases, the power at k3 decreases, and the power reduction at k5 is observed by reason of lesser energy being achievable from k3. Simultaneously, there is a power growth at k4 due to its depletion by the Stokes k5 falls. In the practical applications, it is often important to determine the adjusting range of the OC reflectivity within which the output power at every output wavelength is grater than zero. The adjustable range for the OC reflectivity is 27–67% over which the total output power shows a fluctuation of <9%. The effects of the variations of the OCs reflectivities at k4 and k5 on the RFL are also examined. The adjustable intervals for these two OCs are 18–40% and 24–55%, respectively, over which their corresponding total output power ripples are 9.8% and 1.9%. We know from the above discussion that the total output power just displays a maximum degradation of <10% when one of the OCs reflectivity is changed. Therefore, it is necessary to know the effect of the Raman fiber length on the total output power. The dependence of the total output power on the fiber length is illustrated in Fig. 7. The maximum output power, Ptotal = 1.8029 W, is obtained with a fiber length of 420 m. It should be noticed that the total output power is insensitive to the variation of the Raman fiber length, in which the total output power reduces by less than 3% over a large range of 300m 6 L 6 600m. If the OCs reflectivities are altered within their adjustable range, the insensitive region is still valid.
Fig. 5. Output characteristics of the three-wavelength Raman fiber laser.
and the pump with a slope efficiency of 51% is observed, which is in agreement with the experimental result of 49.7% [5]. Different laser parameters used in our calculations and the experiment lead to somewhat different output power for the respective output wavelength. The output powers at the three out Stokes are interdependent, which depends primarily on the gain achieved from the waves with shorter wavelengths and losses caused by fiber attenuation and red-shift to the longer Stokes. In addition, the fiber length, OCs reflectivity, FBG gray losses, and splicing losses also have significant effects on the output power. With the increasing of the pump power, k4 reaches its threshold firstly and resonates because of its winning from mode competition among the three output wavelengths. The threshold of k4 is 1.7 W. A further increase of the pump power does not boost but deplete the output power of k4. The reason is that the Stokes lines k5 oscillates at 2.7 W and part of the energy in k4 tends to be transferred to k5 via SRS because the latter Stokes lies within the Raman gain spectrum of k4. Thus, it can be seen that the slope efficiency of k4 has a remarkable reduction when the pump power is larger than 2.7 W. Also observed in Fig. 5, the slope efficiency of k5 is the greatest and that of k3 is the minimum. This is believed due to the longer Stokes gain additional power at the expense of the shorter ones. The dependence of the output power on the OCs reflectivity and fiber length is also investigated, respectively. The output power versus the OC reflectivity at k3 with the other two OCs unvaried
Fig. 6. Output power as a function of the output-coupler reflectivity at 1427 nm.
Fig. 7. Total output power and residual pump power as a function of fiber length. Pin = 5 W.
1630
Q. Zujun et al. / Optics Communications 282 (2009) 1626–1630
With the fiber length chosen in this interval, the total output power of the three-wavelength RFL will appear in the vicinity of the maximal value. Thus the RFL can obtain the adjustable power spaces as large as possible for each output Stokes by varying the OCs reflectivity. Also shown in Fig. 5 and Fig. 7 is the residual pump power, P 0, which is the first element in each individual (see Eq. (4)), as a function of the launched pump power and evolution along the Raman fiber, respectively. The small searching space by GA for the respective element in individuals can speed up the computing greatly. From both figures, it can be seen that the residual pump power in unit of dBm decreases approximately linearly with the pump power and the fiber length. This can be considered as a result of the intensive interactions among the resonated waves which promote the conversion efficiency of pump-to-Stokes. For a Raman fiber of 500 m and a pump power of 5 W, only 10.2 mW of the residual pump power is un-depleted and returns to the input. Therefore, the searching range for the residual pump power in GA can be reduced considerably in the case of a large injected pump or a long fiber length. 5. Conclusions We have proposed an efficient numerical algorithm for the steady-state coupled power equations describing stimulated Raman scattering in Multi-wavelength Raman fiber lasers. Taking the excellent searching ability of genetic algorithm in the global space and the fast converging speed of shooting method in its convergence region, we perform several shootings to a few elite individuals after GA operations of selection, crossover, and mutation at each generation to accelerate theirs converging. An example of an all-fiber three-wavelength Raman fiber laser has been used to demonstrate the method. Results show that our method exhibits
a good convergence and the shootings exerted on the minority elite individuals accelerate the converging speed greatly. The simulated results show that the total output power is insensitive to the variation of the Raman fiber length, in which the total output power reduces by less than 3% over a large range of 300m 6 L 6 600m. We also find that the searching range set for the residual pump power in GA can be reduced considerably in the case of a large injected pump power or a long fiber length, which will also speed up the calculating. References [1] M.N. Islam, IEEE J. Sel. Top. Quantum. Electron. 8 (2002) 548. [2] H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, E. Rabarijaona, IEEE Photon. Technol. Lett. 11 (1999) 530. [3] V.E. Perlin, H.G. Winful, J. Lightwave Technol. 20 (2002) 250. [4] M.D. Mermelstein, C. Headley, J.-C. Bouteiller, P. Steinvurzel, C. Horn, K. Feder, B.J. Eggleton, IEEE Photon. Technol. Lett. 13 (2001) 1286. [5] X. Normandin, F. Leplingard, E. Bourova, C. Leclère, T. Lopez, J.-J. Guérin, D. Bayart, In: Proceedings of Optical Fiber Communication Conference, 2002, TuB2. [6] F. Leplingard, S. Borne, L. Lorcy, T. Lopez, J.-J. Guérin, C. Moreau, C. Martinelli, D. Bayart, Electron. Lett. 38 (2002) 886. [7] A.A. Demidov, A.N. Starodumov, X. Li, A. Martinez-Rios, H. Po, Opt. Lett. 28 (2003) 1540. [8] Z. Xiong, T. Chen, Opt. Fiber Technol. 13 (2007) 81. [9] M.N. Islam, Raman Amplifiers for Telecommunications. 2: Sub-Systems and Systems, Springer-Verlag, New York, 2004. [10] M. Rini, I. Cristiani, V. Degiorgio, IEEE J. Quantum Electron. 36 (2000) 1117. [11] S.D. Jackson, P.H. Muir, J. Opt. Soc. Am. B 18 (2001) 1297. [12] H. Kaidi, Z. Xiaojun, Q. Zujun, W. Haocheng, Z. Zili, Opt. Commun. 271 (2007) 257. [13] Q. Zujun, Z. Xiaojun, L. Qing, W. Haocheng, Z. Zili, J. Lightwave Technol. 25 (2007) 1555. [14] F. Leplingard, C. Martinelli, S. Borne, L. Lorcy, D. Bayart, F. Castella, P. Chartier, E. Faou, IEEE Photon. Technol. Lett. 16 (2004) 2601. [15] S. Cierullies, H. Renner, E. Brinkmeyer, Opt. Commun. 217 (2003) 233. [16] J.-C. Bouteiller, IEEE Photon. Technol. Lett. 15 (2003) 1698. [17] Y. Wang, H. Po, Meas. Sci. Technol. 14 (2003) 883.