Gain optimization of fiber optical parametric amplifier based on genetic algorithm with pump depletion Yulin Liu, Zeyong Wang,* Hongna Zhu, and Xiaorong Gao School of Physical Science and Technology, Southwest Jiaotong University, Chengdu, Sichuan 610031, China *Corresponding author: [email protected] Received 24 July 2013; revised 1 October 2013; accepted 1 October 2013; posted 2 October 2013 (Doc. ID 194425); published 21 October 2013
A method to optimize the gain of a fiber optical parametric amplifier (FOPA) is presented by using a genetic algorithm (GA), which can determine the parameters of FOPA and avoid the trouble of trial and error to achieve it. The effect of pump depletion on the gain characteristic of the FOPA is emphasized, and the effects of the fiber length, the wavelength, and the power of two pumps on bandwidth, flatness, and magnitude of the gain spectrum are also studied. According to the presentation, fiber length and the wavelength of the two pumps are selected to be the variable parameters in the GA. When the parameters of the fiber are determined, with the numerical simulation, the optimum combination scheme between those chosen variables could be obtained by the algorithms with the result of the gain optimization of FOPA. © 2013 Optical Society of America OCIS codes: (060.2320) Fiber optics amplifiers and oscillators; (190.4380) Nonlinear optics, four-wave mixing; (000.4430) Numerical approximation and analysis. http://dx.doi.org/10.1364/AO.52.007445
1. Introduction
With its bandwidth and gain advantages, the fiber optical parametric amplifier (FOPA) is attracting more and more attention. Especially in the wavelength division multiplexing communication system, the optical amplifier requires a larger bandwidth, and these are precisely the advantages of FOPA [1–5]. Lots of research has been done to optimize the gain of the FOPA up to now [6–16], such as combining standard and high-SBS threshold highly nonlinear fibers [6], introducing a wavelength-dependent loss in the idler band [7], using the cascaded structure with different zero-dispersion-wavelength high nonlinear fibers (HNLFs) [8], inserting dispersion compensating fibers (DCFs) to compensate the HNLF dispersion [9], and so on. In addition, Gao et al. used genetic algorithm (GA) to optimize the design of FOPA without pump depletion, with the result of better gain characteristics, where the optimization variables are fiber length and second-order dispersion 1559-128X/13/317445-04$15.00/0 © 2013 Optical Society of America
coefficient in FOPA with multisection HNLF configuration [10–12]. The cascade structure has its own advantages [6], but insertion loss is inevitable [13]. Furthermore, in order to get the largest signal gain the pump depletion should be considered [14,15]. Gain characteristics of FOPA are affected by many parameters. In addition to the internal parameters of optical fiber, the influence of the optical fiber length and some parameters of pump light on FOPA is also crucial [16–18]. It is inconvenient to determine those parameters and their combination in most cases. When the parameters of the fiber have been determined, changing the pump light and fiber lengths are the only way to optimize the gain [13]. What we want to do is, through the GA, to find the optimal combination between the fiber length and the pump wavelength so that the parametric amplification system has a wide and flat gain. Simulation results show that this idea is feasible. 2. Theory and Model of FOPA
The waves are assumed to propagate in the z direction along the fiber with continuous-wave (CW) pumping. The four waves of pump1, pump2, signal, 1 November 2013 / Vol. 52, No. 31 / APPLIED OPTICS
7445
and idler propagate simultaneously with the angular frequencies ωp1 , ωp2 , ωs , and ωi , respectively, where the frequencies satisfy ωp1 ωp2 ωs ωi. When the fiber losses are neglected and the pumps are depleted, and under the assumption that the four waves are sufficiently separated in frequency, the nonlinear Schrödinger equation can be reduced to a group of coupled differential equations. The basic equations for describing the process of FOPA in terms of optical power and phase are as follows [1,19,20]: dPp1 −4γPp1 Pp2 Ps Pi 1∕2 sin θ; dz
(1)
dPp2 −4γPp1 Pp2 Ps Pi 1∕2 sin θ; dz
(2)
(1) Coding and initialization. Every individual in the algorithm is made up of a genetic unit; each genetic unit represents a variable and is composed of the binary-coded M-bit. That is to say, an individual consists of three variables, so the individuals will be composed by 3 M-bit binary code. In particular, we used crossed code instead of simple concatenated coding parameters to improve the convergence problem of this algorithm. (2) Fitness function. Fitness function as an individual fitness evaluation index is crucial to the algorithm. Facing the multiobjective optimization problem, we adopt the weight coefficient method to transform the multiobjective problem into a single one. Fitness values are expressed in the following formula: F
dPs 4γPp1 Pp2 Ps Pi 1∕2 sin θ; dz
(3)
dPi 4γPp1 Pp2 Ps Pi 1∕2 sin θ; dz
(4)
dθ Δβ γPp1 Pp2 − Ps − Pi dz 2γPp1 Pp2 Pi ∕Ps 1∕2 Pp1 Pp2 Ps ∕Pi 1∕2 − Pp2 Ps Pi ∕Pp1 1∕2 − Pp1 Ps Pi ∕Pp2 1∕2 cos θ; (5) where Pp1 , Pp2 , Ps , Pi are the power of pumps, the signal, and the idler waves, respectively; z is the length of fiber, γ is the nonlinearity coefficient of fiber, θz is the relative phase difference between the involved waves, and Δβ βs βi − βp1 − βp2 is the linear phase mismatch factor. Numerical solutions of the coupled wave equations were carried out with a four-order Runge–Kutta technique, and then we can get the data of the parametric process. 3. GA
GA is a kind of multivariate stochastic optimization algorithm which is based on natural selection and genetic evolution. It has proved to be an efficient advanced optimization algorithm [21]. When the appropriate population and number of iterations are selected, we can get good optimization results. Gain optimization is a multiobjective and multivariate problem. We consider the fiber length (L) and the wavelengths of two pump (λp1 , λp2 ) as the operating variables. The peak gain, bandwidth, and gain flatness are the optimization goal. Therefore we have combined the general GA with the gain optimization problem, and then we implement the algorithm. The process of GA is given in Algorithm 1. 7446
Algorithm 1. Genetic Algorithm
APPLIED OPTICS / Vol. 52, No. 31 / 1 November 2013
3 X
Ai × f i ;
(6)
i1
where Ai , f i are the weight coefficient and fitness value of the optimization goals, respectively. Along the way, we introduce the variance to measure the gain ripple as the gain flatness evaluation index. (3) Selection function. Here we use the roulette wheel selection. The higher fitness values lead to the higher probability of being selected into the next generation. In conclusion, the probability of being selected can be written as follows: Pi F i
X n
Fj;
(7)
j1
where the number of individuals in a population is n and F i is the fitness of the individuals. (4) Crossover and mutation. Crossover and mutation are important operators in GA, where crossover can achieve the same information exchange between different individuals in a population and mutation can restore the community diversity by random changes in individual genes, so GA could be solved with the global optimal solution. Finally, we control the number of iterations by setting genetic algebras. As a result of the numerical simulation, we obtain a flat gain with bandwidth of nearly 180 nm by using about 30 generations basic loop iterations. 4. Result and Discussion
For the two-pump FOPA, the parameters of the optical fiber are determined according to the determination of optical fiber. We find the optimal combination by adjusting the wavelength of the two pumps and the length of the optical fiber. Searching with GA, we got a level of about 180 nm (1471–1651 nm) gain bandwidth with a series of normal fiber parameters. The range of the gain bandwidth is 0.5 dB, the variance is 0.55, and the average gain is 63.5 dB.
Fig. 1. Gain curves of two-pump FOPA with pump depletion. (a) Best result after GA iteration, where the length of HNLF is L 106 m and the wavelengths of the two pumps are λp1 1591.5 nm, λp2 1521.0 nm. (b), (c), (d) Some of the results in the process of GA iteration.
Those HNLF parameters that are set as constants include λ0 1556 nm, γ 20 W−1 km−1 , β3 0.49 × 10−40 s3 m−1 , β4 −5.8 × 10−56 s4 m−1 , and Pp1 Pp2 2.5 W. Partial result of GA iteration is shown in Fig. 1. The pump depletion can often be ignored when we analyze the gain of FOPA, which can be convenient. But the pump depletion does exist, and in order to get a larger signal peak gain the pump depletion should be considered. From Fig. 2 we can find the pump depletion and its effect on the peak gain clearly. As depicted in Fig. 2, pump depletion will always be there. In the beginning, the pump loss occurs at a low level. With the increase of fiber length and deepening of the four-wave mixing effect, pump depletion becomes serious; where the lowest number is 0.1 W, and then energy transferred from the signal light to the pumps, the pump power increased. In particular, the maximization gain of signal light is 63.8 dB. But the depletion of pump power is 0.45 dB when the signal light energy gain is 54 dB, and the gain flatness is terrible, which will be discussed in the following section. We can say that the pump
Fig. 2. (a) Energy transformation between the signal and pump light in the FOPA, where λp1 1591.5 nm, λp2 1521.0 nm, λs 1550.0 nm. (b) Gain of the signal power.
Fig. 3. Signal gain curves of two-pump FOPA with different fiber lengths where the λp1 1591.5 nm, λp2 1521.0 nm.
depletion cannot be ignored when we want the greatest signal gain. Fiber length affects the four-wave mixing process so as to impact the gain amplitude and flatness of parametric amplification. The gain spectra of FOPA with different fiber lengths of 10, 30, 70, 106, and 135 m are shown in Fig. 3. Pump power will affect the gain bandwidth, amplitude, and flatness. We can find the influence trend from Fig. 4. With the increasing of pump power the gain characteristic can be improved significantly and we can get an ideal gain characteristic in a shorter fiber length. But we do not have the pump power as an optimization variable. The influence of pump power on the FOPA is approximately linear. That is to say, the greater the pump power is, the better the gain characteristic of the FOPA. What is more, an excessive pumping power will bring unexpected nonlinear optical effects. From what has been discussed above, we have set the pump power as a modest value, that is Pp1 Pp2 2.5 W. Moreover, we set the normal fiber parameters when the GA was carrying out its iterative result. If we chose some special fiber parameters, a better gain spectrum GA can be obtained through the GA. For example, when the γ 40 W−1 km−1 and β4 −1.0 × 10−56 s4 m−1 , and other parameters are same as those used in Fig. 1, the gain spectrum
Fig. 4. Signal gain curves of two-pump FOPA with different pump power and modest fiber length where the λp1 1591.5 nm, λp2 1521.0 nm. 1 November 2013 / Vol. 52, No. 31 / APPLIED OPTICS
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Fig. 5. Gain curves of two-pump FOPA with different fiber parameters from Fig. 1 where γ 40 W−1 km−1 and β4 −1.0 × 10−56 s4 m−1 .
which is shown in Fig. 5 can be obtained; the optimum pump wavelengths calculated by GA are 1532.7 and 1579.5 nm, respectively, and the fiber length is 57 m. In this gain spectrum, the range of the gain bandwidth is 0.5 dB and the gain bandwidth is about 300 nm (1420–1720 nm) with an average gain of 63.5 dB. 5. Conclusion
We present a method to obtain the optimal combination of parameters in FOPA with GA. The effects of pump depletion, fiber length, pump wavelength, and power on the gain characteristic of FOPA are discussed. We chose the fiber length and the wavelength of the two pumps for FOPA as the variable parameter and the gain amplitude, the bandwidth, and its flatness as the optimized objectives. The numerical simulation showed that the GA is an appropriate method for this multiobjective and multivariate problem. We can obtain a flat gain optimization combination scheme easily with the given optical fiber parameters. This work was supported by the Fundamental Research Funds for the Central Universities (SWJTU11ZT24). References 1. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002). 2. J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a fiber-optic parametric amplifier in phase-sensitive and phase-insensitive operation,” Opt. Express 18, 4130–4137 (2010). 3. A. Bogris and D. Syvridis, “40 Gb/s all-optical regeneration based on the pump depletion effect in fiber parametric amplification,” Opt. Fiber Technol. 14, 63–71 (2008).
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4. N. E. Dahdah, D. S. Govan, M. Jamshidifar, N. J. Doran, and M. E. Marhic, “Fiber optical parametric amplifier performance in a 1-Tb/s DWDM communication system,” IEEE J. Quantum Electron. 18, 950–957 (2012). 5. M. C. Ho, K. Uesaka, M. Marhic, Y. Akasaka, and L. G. Kazovsky, “200-nm-bandwidth fiber optical amplifier combining parametric and Raman gain,” J. Lightwave Technol. 19, 977–981 (2001). 6. F. Da Ros, K. Rottwitt, and C. Peucheret, “Gain optimization in fiber optical parametric amplifiers by combining standard and high-SBS threshold highly nonlinear fibers,” in Conference on Lasers and Electro-Optics 2012, OSA Technical Digest (Optical Society of America, 2012), paper CM4N.5. 7. K. Xu, H. Y. Liu, Y. T. Dai, J. Wu, and J. T. Lin, “Synthesis of broadband and flat parametric gain by idler loss in optical fiber,” Opt. Commun. 285, 790–794 (2012). 8. L. Provino, A. Mussot, E. Lantz, T. Sylvestre, and H. Maillotte, “Broadband and flat parametric amplifiers with a multisection dispersion-tailored nonlinear fiber arrangement,” J. Opt. Soc. Am. B 20, 1532–1537 (2003). 9. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Quantum Electron. 8, 506–520 (2002). 10. M. Y. Gao, C. Jiang, W. S. Hu, and J. Wang, “Optimized design of two-pump fiber optical parametric amplifier with twosection nonlinear fibers using genetic algorithm,” Opt. Express 12, 5603–5613 (2004). 11. W. Zhang, C. G. Wang, J. W. Shu, C. Jiang, and W. S. Hu, “Design of fiber-optical parametric amplifiers by genetic algorithm,” IEEE Photon. Technol. Lett. 16, 1652–1654 (2004). 12. M. Gao, J. Wang, C. Jiang, W. Hu, and H. Ren, “Two-pump fiber optical parametric amplifiers using optimized photonic crystal fiber by genetic algorithm.” Appl. Phys. B 84, 433–438 (2006). 13. Y. Tian, X. Xiao, S. Gao, S. Lu, and C. Yang, “Ultra-flat and broadband two-pump optical parametric amplifiers using a single-section highly nonlinear fiber,” Opt. Commun. 263, 116–119 (2006). 14. H. Steffensen, J. R. Ott, K. Rottwitt, and C. J. McKinstrie, “Full and semi-analytic analyses of two-pump parametric amplification with pump depletion,” Opt. Express 19, 6648–6656 (2011). 15. J. Hansryd and P. A. Andrekson, “Broad-band continuouswave-pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE J. Photon. Technol. Lett. 13, 194–196 (2001). 16. M. Jamshidifar, A. Vedadi, and M. E. Marhic, “Continuouswave one-pump fiber optical parametric amplifier with 270 nm gain bandwidth,” IEEE. Opt. Commun. 35, 1–2 (2009). 17. Y. Q. Xu and S. G. Murdoch, “Gain spectrum of an optical parametric amplifier with a temporally incoherent pump,” Opt. Lett. 35, 169–171 (2010). 18. M. E. Marhic, K. Y. Wong, and L. G. Kazovsky, “Wide-band tuning of the gain spectra of one-pump fiber optical parametric amplifiers,” IEEE J. Quantum Electron 10, 1133–1141 (2004). 19. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007). 20. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Academic, 2008). 21. D. E. Goldberg, “Genetic algorithms in search, optimization, and machine learning,” Mach. Learn. 3, 95–99 (1988).
A method to optimize the gain of a fiber optical parametric amplifier (FOPA) is presented by using a genetic algorithm (GA), which can determine the parameters of FOPA and avoid the trouble of trial and error to achieve it. The effect of pump depletion on the gain characteristic of the FOPA is emphasized, and the effects of the fiber length, the wavelength, and the power of two pumps on bandwidth, flatness, and magnitude of the gain spectrum are also studied. According to the presentation, fiber length and the wavelength of the two pumps are selected to be the variable parameters in the GA. When the parameters of the fiber are determined, with the numerical simulation, the optimum combination scheme between those chosen variables could be obtained by the algorithms with the result of the gain optimization of FOPA. © 2013 Optical Society of America OCIS codes: (060.2320) Fiber optics amplifiers and oscillators; (190.4380) Nonlinear optics, four-wave mixing; (000.4430) Numerical approximation and analysis. http://dx.doi.org/10.1364/AO.52.007445
1. Introduction
With its bandwidth and gain advantages, the fiber optical parametric amplifier (FOPA) is attracting more and more attention. Especially in the wavelength division multiplexing communication system, the optical amplifier requires a larger bandwidth, and these are precisely the advantages of FOPA [1–5]. Lots of research has been done to optimize the gain of the FOPA up to now [6–16], such as combining standard and high-SBS threshold highly nonlinear fibers [6], introducing a wavelength-dependent loss in the idler band [7], using the cascaded structure with different zero-dispersion-wavelength high nonlinear fibers (HNLFs) [8], inserting dispersion compensating fibers (DCFs) to compensate the HNLF dispersion [9], and so on. In addition, Gao et al. used genetic algorithm (GA) to optimize the design of FOPA without pump depletion, with the result of better gain characteristics, where the optimization variables are fiber length and second-order dispersion 1559-128X/13/317445-04$15.00/0 © 2013 Optical Society of America
coefficient in FOPA with multisection HNLF configuration [10–12]. The cascade structure has its own advantages [6], but insertion loss is inevitable [13]. Furthermore, in order to get the largest signal gain the pump depletion should be considered [14,15]. Gain characteristics of FOPA are affected by many parameters. In addition to the internal parameters of optical fiber, the influence of the optical fiber length and some parameters of pump light on FOPA is also crucial [16–18]. It is inconvenient to determine those parameters and their combination in most cases. When the parameters of the fiber have been determined, changing the pump light and fiber lengths are the only way to optimize the gain [13]. What we want to do is, through the GA, to find the optimal combination between the fiber length and the pump wavelength so that the parametric amplification system has a wide and flat gain. Simulation results show that this idea is feasible. 2. Theory and Model of FOPA
The waves are assumed to propagate in the z direction along the fiber with continuous-wave (CW) pumping. The four waves of pump1, pump2, signal, 1 November 2013 / Vol. 52, No. 31 / APPLIED OPTICS
7445
and idler propagate simultaneously with the angular frequencies ωp1 , ωp2 , ωs , and ωi , respectively, where the frequencies satisfy ωp1 ωp2 ωs ωi. When the fiber losses are neglected and the pumps are depleted, and under the assumption that the four waves are sufficiently separated in frequency, the nonlinear Schrödinger equation can be reduced to a group of coupled differential equations. The basic equations for describing the process of FOPA in terms of optical power and phase are as follows [1,19,20]: dPp1 −4γPp1 Pp2 Ps Pi 1∕2 sin θ; dz
(1)
dPp2 −4γPp1 Pp2 Ps Pi 1∕2 sin θ; dz
(2)
(1) Coding and initialization. Every individual in the algorithm is made up of a genetic unit; each genetic unit represents a variable and is composed of the binary-coded M-bit. That is to say, an individual consists of three variables, so the individuals will be composed by 3 M-bit binary code. In particular, we used crossed code instead of simple concatenated coding parameters to improve the convergence problem of this algorithm. (2) Fitness function. Fitness function as an individual fitness evaluation index is crucial to the algorithm. Facing the multiobjective optimization problem, we adopt the weight coefficient method to transform the multiobjective problem into a single one. Fitness values are expressed in the following formula: F
dPs 4γPp1 Pp2 Ps Pi 1∕2 sin θ; dz
(3)
dPi 4γPp1 Pp2 Ps Pi 1∕2 sin θ; dz
(4)
dθ Δβ γPp1 Pp2 − Ps − Pi dz 2γPp1 Pp2 Pi ∕Ps 1∕2 Pp1 Pp2 Ps ∕Pi 1∕2 − Pp2 Ps Pi ∕Pp1 1∕2 − Pp1 Ps Pi ∕Pp2 1∕2 cos θ; (5) where Pp1 , Pp2 , Ps , Pi are the power of pumps, the signal, and the idler waves, respectively; z is the length of fiber, γ is the nonlinearity coefficient of fiber, θz is the relative phase difference between the involved waves, and Δβ βs βi − βp1 − βp2 is the linear phase mismatch factor. Numerical solutions of the coupled wave equations were carried out with a four-order Runge–Kutta technique, and then we can get the data of the parametric process. 3. GA
GA is a kind of multivariate stochastic optimization algorithm which is based on natural selection and genetic evolution. It has proved to be an efficient advanced optimization algorithm [21]. When the appropriate population and number of iterations are selected, we can get good optimization results. Gain optimization is a multiobjective and multivariate problem. We consider the fiber length (L) and the wavelengths of two pump (λp1 , λp2 ) as the operating variables. The peak gain, bandwidth, and gain flatness are the optimization goal. Therefore we have combined the general GA with the gain optimization problem, and then we implement the algorithm. The process of GA is given in Algorithm 1. 7446
Algorithm 1. Genetic Algorithm
APPLIED OPTICS / Vol. 52, No. 31 / 1 November 2013
3 X
Ai × f i ;
(6)
i1
where Ai , f i are the weight coefficient and fitness value of the optimization goals, respectively. Along the way, we introduce the variance to measure the gain ripple as the gain flatness evaluation index. (3) Selection function. Here we use the roulette wheel selection. The higher fitness values lead to the higher probability of being selected into the next generation. In conclusion, the probability of being selected can be written as follows: Pi F i
X n
Fj;
(7)
j1
where the number of individuals in a population is n and F i is the fitness of the individuals. (4) Crossover and mutation. Crossover and mutation are important operators in GA, where crossover can achieve the same information exchange between different individuals in a population and mutation can restore the community diversity by random changes in individual genes, so GA could be solved with the global optimal solution. Finally, we control the number of iterations by setting genetic algebras. As a result of the numerical simulation, we obtain a flat gain with bandwidth of nearly 180 nm by using about 30 generations basic loop iterations. 4. Result and Discussion
For the two-pump FOPA, the parameters of the optical fiber are determined according to the determination of optical fiber. We find the optimal combination by adjusting the wavelength of the two pumps and the length of the optical fiber. Searching with GA, we got a level of about 180 nm (1471–1651 nm) gain bandwidth with a series of normal fiber parameters. The range of the gain bandwidth is 0.5 dB, the variance is 0.55, and the average gain is 63.5 dB.
Fig. 1. Gain curves of two-pump FOPA with pump depletion. (a) Best result after GA iteration, where the length of HNLF is L 106 m and the wavelengths of the two pumps are λp1 1591.5 nm, λp2 1521.0 nm. (b), (c), (d) Some of the results in the process of GA iteration.
Those HNLF parameters that are set as constants include λ0 1556 nm, γ 20 W−1 km−1 , β3 0.49 × 10−40 s3 m−1 , β4 −5.8 × 10−56 s4 m−1 , and Pp1 Pp2 2.5 W. Partial result of GA iteration is shown in Fig. 1. The pump depletion can often be ignored when we analyze the gain of FOPA, which can be convenient. But the pump depletion does exist, and in order to get a larger signal peak gain the pump depletion should be considered. From Fig. 2 we can find the pump depletion and its effect on the peak gain clearly. As depicted in Fig. 2, pump depletion will always be there. In the beginning, the pump loss occurs at a low level. With the increase of fiber length and deepening of the four-wave mixing effect, pump depletion becomes serious; where the lowest number is 0.1 W, and then energy transferred from the signal light to the pumps, the pump power increased. In particular, the maximization gain of signal light is 63.8 dB. But the depletion of pump power is 0.45 dB when the signal light energy gain is 54 dB, and the gain flatness is terrible, which will be discussed in the following section. We can say that the pump
Fig. 2. (a) Energy transformation between the signal and pump light in the FOPA, where λp1 1591.5 nm, λp2 1521.0 nm, λs 1550.0 nm. (b) Gain of the signal power.
Fig. 3. Signal gain curves of two-pump FOPA with different fiber lengths where the λp1 1591.5 nm, λp2 1521.0 nm.
depletion cannot be ignored when we want the greatest signal gain. Fiber length affects the four-wave mixing process so as to impact the gain amplitude and flatness of parametric amplification. The gain spectra of FOPA with different fiber lengths of 10, 30, 70, 106, and 135 m are shown in Fig. 3. Pump power will affect the gain bandwidth, amplitude, and flatness. We can find the influence trend from Fig. 4. With the increasing of pump power the gain characteristic can be improved significantly and we can get an ideal gain characteristic in a shorter fiber length. But we do not have the pump power as an optimization variable. The influence of pump power on the FOPA is approximately linear. That is to say, the greater the pump power is, the better the gain characteristic of the FOPA. What is more, an excessive pumping power will bring unexpected nonlinear optical effects. From what has been discussed above, we have set the pump power as a modest value, that is Pp1 Pp2 2.5 W. Moreover, we set the normal fiber parameters when the GA was carrying out its iterative result. If we chose some special fiber parameters, a better gain spectrum GA can be obtained through the GA. For example, when the γ 40 W−1 km−1 and β4 −1.0 × 10−56 s4 m−1 , and other parameters are same as those used in Fig. 1, the gain spectrum
Fig. 4. Signal gain curves of two-pump FOPA with different pump power and modest fiber length where the λp1 1591.5 nm, λp2 1521.0 nm. 1 November 2013 / Vol. 52, No. 31 / APPLIED OPTICS
7447
Fig. 5. Gain curves of two-pump FOPA with different fiber parameters from Fig. 1 where γ 40 W−1 km−1 and β4 −1.0 × 10−56 s4 m−1 .
which is shown in Fig. 5 can be obtained; the optimum pump wavelengths calculated by GA are 1532.7 and 1579.5 nm, respectively, and the fiber length is 57 m. In this gain spectrum, the range of the gain bandwidth is 0.5 dB and the gain bandwidth is about 300 nm (1420–1720 nm) with an average gain of 63.5 dB. 5. Conclusion
We present a method to obtain the optimal combination of parameters in FOPA with GA. The effects of pump depletion, fiber length, pump wavelength, and power on the gain characteristic of FOPA are discussed. We chose the fiber length and the wavelength of the two pumps for FOPA as the variable parameter and the gain amplitude, the bandwidth, and its flatness as the optimized objectives. The numerical simulation showed that the GA is an appropriate method for this multiobjective and multivariate problem. We can obtain a flat gain optimization combination scheme easily with the given optical fiber parameters. This work was supported by the Fundamental Research Funds for the Central Universities (SWJTU11ZT24). References 1. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002). 2. J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a fiber-optic parametric amplifier in phase-sensitive and phase-insensitive operation,” Opt. Express 18, 4130–4137 (2010). 3. A. Bogris and D. Syvridis, “40 Gb/s all-optical regeneration based on the pump depletion effect in fiber parametric amplification,” Opt. Fiber Technol. 14, 63–71 (2008).
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