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Gain ripple minimization in fiber Raman amplifiers based on variational method
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Optics Communications 281 (2008) 1545–1557 www.elsevier.com/locate/optcom
Gain ripple minimization in fiber Raman amplifiers based on variational method A.R. Bahrampour a,b,*, A. Ghasempour a, L. Rahimi b a b
Faculty of Physics, Sharif University of Technology, Tehran, Iran Department of Physics, Shahid Bahonar University of Kerman, Iran
Received 14 July 2007; received in revised form 17 October 2007; accepted 12 November 2007
Abstract In this paper, the variational method is employed for minimizing the gain ripple of multi-wavelength fiber Raman amplifiers. The variance of gain spectrum of the fiber Raman amplifier is regarded as the cost function, restriction on total pump power and average gain is given as the constraints of the minimization problem. It is shown that the minimization problem with any necessary constraints on the pump powers, average gain and signal to noise ratio, is reduced to a two-point boundary value problem. The method gives the entire possible local and global solutions. The method is applied to different examples of fiber Raman amplifiers with different lengths from 25 km to 100 km and different numbers of pumps from 4 to 20 to determine the pump powers and wavelengths for minimum gain ripple. It was obtained for a 100 km fiber Raman amplifier the gain ripple can be about 0.1 dB with on–off gain more than 20 dB. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Raman amplifier; Raman gain flatness; Variational method
1. Introduction Raman amplifier is based on the stimulated Raman scattering effect in silica-based optical fibers. Raman gain arises from the transfer of power from one optical beam to another that is down shifted by the energy of an optical phonon (a vibrational mode of the medium) [1]. There are two kinds of phonon, one is acoustic phonon and the other is optical phonon. Raman scattering is the scattering between photon and optical phonon while the one between photon and acoustic phonon is called Brillouin scattering. Because optical phonon has almost uniform dispersion relation versus wave number, the phase matching is easily obtained for arbitrary relative direction between the pump and signal waves. Therefore, Raman amplifiers can take both co-pump and counter-pump schemes [2]. To increase the bandwidth of the amplifier for amplification in wavelength division multiplexing (WDM) systems, authors have proposed multi-wavelength pumping technique [1,3]. The idea of multi-wavelength pumping (MWP) is to prepare a set of pumps operating at different wavelengths combined through WDM couplers into a single fiber to create a composite Raman gain. The composite Raman gain, created by different wavelengths of pumps, can be so designed as to be arbitrarily broad and flat. Principally, the MWP can obtain the arbitrary gain spectra by the proper choice of pump wavelengths and powers. However, the strong Raman interaction of pump to pump, signal to signal and pump to signal makes the problem of *
Corresponding author. Address: Faculty of Physics, Sharif University of Technology, Tehran, Iran. Tel.: +98 21 6616 4527. E-mail address: [email protected] (A.R. Bahrampour).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.11.057
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the required gain spectral somewhat more difficult for optimal design of MWP. Therefore, the MWP design presents a grand challenge to numerical optimization. It involves multiple powers and wavelengths. Recently, some soft computational methods have been proposed to select the appropriate pump wavelengths and powers in fiber Raman amplifiers. Perlin and Zhou et al. employ traditional genetic algorithm (GA) to optimize the gain flatness and gain bandwidth performance [4–6]. The optimal results are exciting but they only obtain a single optimal solution in each domain, and even their methods may be trapped in local optimal solution of the search space due to intrinsic weakness of the traditional GA [7]. To overcome these difficulties a hybrid genetic algorithm (HGA) based on techniques such as clustering, sharing, crowding and adaptive probability is introduced and is used for optimal design of multi-wavelength pumping Raman fiber amplifiers [7–9]. In [10], employing the neural network method optimizes the Raman gain spectrum. However, it obtains the optimization only and partly in pump powers excluding wavelengths [7]. The Raman governing equations are transformed to a system of integral equations and the Picard method is applied for the solution of integral equations. On the basis of integral equations, a method for designing optimal Raman amplifier is introduced [11–14]. Some other compound methods are introduced by previous authors [15]. Variational method is a high power tool with strong mathematical basis for optimizing a cost function in the presence of constraints in the form of differential equations and/or algebraic equations [16]. In employing this high power tool, the main goal parameter such as gain ripple or signal to noise ratio can be regarded as the cost function and others are given as the inequality and equality constraints in the optimization problem. In this paper, the variance of gain spectrum is regarded as the cost function, the upper limit of the total pump power and lower bound of average gain are given as the constraints of the minimization problem. The variational method is employed to minimize the gain ripple of multiwavelength pumping fiber Raman amplifiers. The results can be extremely helpful in the design of the multi-wavelength pumping fiber Raman amplifiers. This paper is organized as follows: In Section 2 the governing equation and constraints are presented. The mathematical model for the optimal design is given in Section 3. In Section 4 the numerical solution for the optimal design is obtained. Finally, the paper ends with some conclusions. 2. Governing equations and constraints Wave propagation in a multi-pump Raman amplifier is characterized by a large number of effects. The most important of these effects, for the purposes of the present consideration, are pump to pump, pump to signal and signal to signal Raman interaction and wavelength dependent linear attenuation, experienced by both pump and signal waves. In copumped multi-pump fiber Raman amplifiers where signal and pump lights travel along the same direction in the fiber, fluctuation in pump power can be coupled as noise onto signal power and serious crosstalk that will be produced under the condition of saturated amplification. On the other hand, backward pumped multi-pump fiber Raman amplifiers have no such shortages [17]. Hence, only backward pumped multi-pump fiber Raman amplifiers are considered in this paper. Because the powers of back scattering pumps and signals are lower by 30 dB and 20 dB than the original power, the power of forward and backward noises is less than that of input signals by 30 dB [18], so noise effects such spontaneous Raman scattering, Rayleigh backscattering and thermal factor are neglected in calculating the amplifier gain profile [4,15]. The governing equations of signal powers (Si(z); i = 1, 2, . . . , Ns) and pump powers (Pi(z); i = 1, 2, . . . , Np)in the steady state are a system of coupled differential equations given in the following [9]: 0 1 Np Ns X X dS i ðzÞ ¼ S i ðzÞ@aðmi Þ þ gðmi ; mj ÞS j ðzÞ þ gðmi ; m^j ÞP ^j ðzÞA; i ¼ 1; 2; . . . ; N s ð1Þ dz ^j¼1 j¼1 ! Np Ns X X dP i ðzÞ ¼ P i ðzÞ aðmi Þ þ gðmi ; mj ÞS j ðzÞ þ gðmi ; mj ÞP j ðzÞ ; i ¼ 1; 2; . . . ; N p ð2Þ dz j¼1 j¼1 where Np is the number of pumps and Ns is the number of signals. a(mi) is the intrinsic fiber attenuation at frequency mi. The following relation gives the Raman gain coefficient g(mi,mj) – [19]: 8 m2i gref ðmi mj Þ > > < mj mref CAeff ; mi > mj ð3Þ gðmi ; mj Þ ¼ 0; mi ¼ mj > > : mj gref ðmj mi Þ CAeff ; mi < mj mreff where mi and mj are the frequencies of the interacting waves in the Raman effect. The factor C accounts for polarization randomization effect, Aeff is the effective fiber core area and gref(n g) is the Raman gain coefficient measured at reference pump frequency mreff which describes the power transferred by the stimulated Raman scattering from wave with frequency n to the one with frequency g. The Raman gain coefficient spectrum gref(Dm) is represented in Fig. 1 [20].
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–14
x 10
Raman gain cofficient,m / W
6 5 4 3 2 1 0
0
5
10
15
20
25
30
35
Pump–signal frequency difference,THz
Fig. 1. Typical Raman gain coefficient [20].
The minus sign on the left side of Eq. (2) denotes backward propagating pump waves. In other words, for a fiber span of length L, the boundary values are given at z = 0 for signal waves and at z = L for pump waves. Also, we have: P i ðzÞ P 0
ð4Þ
where Pi(z) is the power of the ith pump at point z. For applying the constraint (4), it is assumed that P i ðzÞ ¼ Q2i ðzÞ, where Qi(z) is a real valued function. Therefore the governing equations for pumps are rearranged with respect to Qi(z). ! Np Ns X X dQi ðzÞ Qi ðzÞ 2 ¼ aðmi Þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ ; i ¼ 1; 2; . . . ; N p ð5Þ dz 2 j¼1 j¼1 To avoid thermal effect in the fiber, it is assumed as a restriction that the total of pump powers is upper limited. Its upper limit value is denoted by Pmax i.e. Np X
Q2i ðLÞ 6 P max
ð6Þ
i¼1
The inequality (6) can be written in the form of an equality: P max
Np X
Q2i ðLÞ ¼ D2
ð7Þ
i¼1
In the above equation D is a real number. Obviously maximum flatness with no gain and even with some loss is possible. In order to avoid these solutions, it is assumed that the average gain of signals G is lower limited by a defined value G0 i.e. G P G0 where the gain average G is defined in the following: Ns Ns 1 X 1 X S k ðLÞ G¼ Gk ¼ N s k¼1 N s k¼1 S k ð0Þ
ð8Þ
ð9Þ
In the above relation Sk(0) and (Sk(L)) is the kth input (output) signal power and Gk is the gain of the kth signal. Eq. (8) is written in the following form: G G0 ¼ E 2
ð10Þ
where E is a real number. The distance between the gain spectrum and target predefined gain spectrum is given as the cost function [11] which can be considered by this method. Now our problem is to minimize the gain ripple in a Raman amplifier subject to the constraint (1), (5), (7) and (10). In this analysis the noise effects are neglected [4,15], otherwise the lower limit on the signal to noise ratio and governing equations on the noise evolution along the fiber length should be regarded as other constraints of the minimization problem [11].
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3. Theoretical model for the gain ripple minimization The following relation defines the cost function F as a characteristic of flatness in the fiber Raman amplifier: F ðS 1 ðLÞ; S 2 ðLÞ; . . . ; S N s ðLÞÞ ¼
Ns X 2 ðGk GÞ
ð11Þ
k¼1
Now the problem is to find a Np tuple P ¼ ðP 1 ðLÞ; . . . ; P N p ðLÞ; kP 1 ; kP 2 ; . . . ; kP N p Þ among all possible Np tuple P ¼ ðP 1 ðLÞ; P 2 ðLÞ; . . . ; P N p ðLÞ; kP 1 ; kP 2 ; . . . ; kP N p Þ, which minimizes the cost function F subject to the constraints (1),(5),(7) and (10). The generalized cost function J is defined by formula (A8) in Appendix A. Therefore, the optimization problem is reduced to the following classical variational problem: dJ ¼ 0
ð12Þ
After some mathematical manipulations (see Appendix A), it is easy to show that the solution of the above variational problem is equivalent to the solution of a complex two-point boundary value problem. The governing differential equations on the components of the state variable vector X ¼ ðQ1 ðzÞ; . . . ; QN p ; S 1 ðzÞ; . . . ; S N s ðzÞ; b1 ðzÞ; . . . ; bN p ðzÞ; c1 ðzÞ; . . . ; cN s ðzÞÞ are given by ! Np Ns X X dQi ðzÞ Qi ðzÞ 2 ¼ aðmi Þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ ; i ¼ 1; 2; . . . ; N p ð13Þ dz 2 j¼1 j¼1 ! Np Ns X X dbi ðzÞ bi ðzÞ 2 þ aðmi Þ þ gðmi ; mj ÞQj ðzÞ þ gðmi ; mj ÞS j ðzÞ dz 2 j¼1 j¼1 ! Np Ns X X þ Qi ðzÞ gðmj ; mi Þbj ðzÞQj ðzÞ 2 gðmj ; mi ÞS j ðzÞcj ðzÞ ¼ 0 i ¼ 1; 2; . . . ; N p ð14Þ j¼1
dS i ðzÞ ¼S i ðzÞ aðmi Þ þ dz
j¼1 Ns X
gðmi ; mj ÞS j ðzÞ þ
j¼1
Np X
! gðmi ; mj ÞQ2j ðzÞ
;
i ¼ 1; 2; . . . ; N s
ð15Þ
j¼1
Ns p Ns X X X dci ðzÞ þ ci ðzÞðaðmi Þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQ2j ðzÞÞ gðmj ; mi ÞS j ðzÞcj ðzÞ dz j¼1 j¼1 j¼1
þ
Np 1X gðmj ; mi ÞQj ðzÞbj ðzÞ ¼ 0; 2 j¼1
N
i ¼ 1; 2; . . . ; N s
ð16Þ
where bi(z) (i = 1, 2, . . . , Np) and ci(z) (i = 1, 2, . . . , Ns) are conjugate variables corresponding to the pump and signal powers, respectively. For differential Eqs. (13)–(16), the boundary conditions are as follows (Appendix A): bi ð0Þ ¼ 0;
i ¼ 1; 2; . . . ; N p
bi ðLÞ ¼ 2xQi ðLÞ; i ¼ 1; 2; . . . ; N p 2 u ðGi GÞ ; ci ðLÞ ¼ S i ð0Þ N s S i ð0Þ
ð17Þ ð18Þ i ¼ 1; 2; . . . ; N s
ð19Þ
There is no restriction on ci(0) (i = 1, 2, . . . , Ns) and they are evaluated from the solution of differential equations. It is assumed that the input power of signals at z = 0 is specified (Si(0) = Si0 (i = 1, 2, . . . , Ns)) so from this assumption and Eqs. (17)–(19), the number of boundary conditions, 2(Np + Ns), is just equal to the number of governing differential equations. In addition, the Lagrange multipliers x,u must satisfy the following switching equations: Dx ¼ 0
ð20Þ
Eu ¼ 0
ð21Þ
where x is the Lagrange multiplier corresponding to the restriction due to the upper-bound of the sum of pump powers and u is the Lagrange multiplier corresponding to the lower bound of the gain average restriction. The system of algebraic equations corresponding to the vanishing variational derivatives with respect to the pump wavelengths (kk; k = 1, 2, . . . , Np) is as following:
A.R. Bahrampour et al. / Optics Communications 281 (2008) 1545–1557
X Np Ns X ogðmj ; mk Þ ogðmk ; mj Þ ogðmk ; mj Þ oaðmk Þ Ajk þ Bkj C kj þ Dk þ ¼0 omk omk omk omk j¼1 j¼1
k ¼ 1; 2; . . . ; N p
1549
ð22Þ
where Aik, Bkj, Ckj, Dk are defined in Appendix A. That is, the optimization is converted to a two-point boundary value problem. The number of unknown functions (bi,Qi,ck,Sk; i = 1, 2, . . . , Np,k = 1, 2, . . . , Ns) is 2(Np + Ns), which is the same as the number of differential Eqs. (13)–(16). The number of unknown parameters (D, u, x, E) is four, which is the same as the number of algebraic Eqs. (7), (10), (20) and (21). The number of unknown pump wavelengths corresponding to the minimum gain ripple is equal to the number of algebraic Eq. (22). To obtain the pump power and wavelength corresponding to the optimal condition the system of governing Eqs. (7), (10), (13)–(21) and (22) are solved numerically. 4. Numerical solution and results The system of governing equations are organized into four classes, the system of nonlinear differential Eqs. (13)–(16), the set of equations for the initial and final values of governing ordinary differential Eqs. (17)–(19), algebraic Eqs. (7), (10) and (22) and finally the system of switching Eqs. (20) and (21). To formulize the switching equations a two component binary vector b is defined such that, when the total power of the pumps clamps to maximum value Pmax the first component of b is zero, otherwise all bi(L)s (i = 1, 2, . . . , Np) are zero. The zero values of the second component are corresponding to the clamping of G to its lower value G0 (i.e. G ¼ G0 ) and if it equals one, then the final value of the conjugate functions ci(L)s (i = 1, 2, . . . , Ns) is determined from the following equation: ci ðLÞ ¼
2 u ðGi GÞ ; S i ð0Þ N s S i ð0Þ
i ¼ 1; 2; . . . ; N s
ð23Þ
The dimension of the binary vector b is independent of the number of pumps and the number of signals. In our problem binary vector b makes four possible states; the boundary conditions corresponding to each binary vector bi (i = 1,2,3,4) are summarized and represented in Appendix B. To introduce the algorithm of numerical computation, we start the solution of the system of differential equations, and the system of algebraic equations among the initial and final values of pump powers, signal powers and their conjugate variables. The algorithm of computation is introduced as follows: (1) (2) (3) (4) (5)
Specify the number of pumps and signals and choose the uniform distribution for pump wavelengths. Put i = 1. Specify the binary vector bi, and its related boundary conditions according to Appendix B. Try to solve two-point boundary value problems. Is there any solution to the two-point boundary value problem? If so, determine the corresponding pump power and wavelengths. If not, put Fi = 500 (500 is greater than the actual values of the cost function) then go to step (7). (6) Calculate the cost function F related to the solution of the third step and store in Fi. (7) If i is less than four, put i = i + 1 and return to step 3. (8) Put minimum value of Fi in M (i.e. M is the minimum value of {F1,F2,F3,F4}). The two-point boundary value problems are solved by the finite difference method. The state corresponding to M is the solution of the optimization problem. As an example this method is applied to a fiber Raman amplifier with 20 pumps and the following parameter set is fixed: there are 50 signal channels spaced with equal distances in 1528–1578 nm (the corresponding bandwidth is about 50 nm), the signal power of each channel is 0.1 mW (Si(0) = 0.1 mW; i = 1, 2, . . . , Ns), the fiber length is 25 km, mreff = 200 THz and C = 2. Calculations are done for SMF-28 fiber where its mode effective area (Aeff(k)) and attenuation coefficient (a(k)) versus wavelength are presented in Fig. 2 [21]. For a fiber Raman amplifier with 20 pumps, the pump wavelengths with equal distances are chosen in the bandwidth (1408–1490 nm), maximum total power is 512 mW(Pmax = 512 mW) and the value of 0.47 dB is selected for the lower bound of average net gain, (G0 = 0.47 dB). These values are selected so that the output power of each signal is not less than its input power, that is the Raman gain can compensate the loss of each signal. Obviously, the selected value of Pmax and G0 are dependent on some parameters such as the number and characteristics of pumps. The gain spectrum and pump powers corresponding to the binary vectors b2, b3 and b4 are presented in Fig. 3a–c, respectively. Calculation PN p shows there is no solution to the binary vector b1; this means that the average gain ðGÞ and the total pump power i¼1 P i ðLÞ can not clamp to the G0 and Pmax simultaneously. The results corresponding to different binary vectors are compared in Table 1 and it is shown that the binary vector b3 gives the global minimum gain ripple
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attenuation cofficient
effective area
80
0.05
Effective area, μm2
Attenuation cofficint,1/km
0.1
70
0 1400
1450
1500 Wavelength,nm
1550
1600
b
1
gain,dB 1450
1500 wavelength,nm
input pump power,mW
c
0
1550
100
3
50
2
0 1400
1450
1500 wavelength,nm
150
3
100
2
50
1
1550
1
gain ,dB
0 1400
input pump power,mW
50
input pump power,mW
a
gain,dB
Fig. 2. Spectral dependence of a and Aeff for a SMF-28 fiber [21].
0 1400
1450
1500 wavelength.nm
1550
0
Fig. 3. Optimal Raman gain profile and pump spectrum corresponding to a 25 km SMF-28 optical fiber for different binary vector (bi). (a) Corresponding to binary vector b2, (b) corresponding to binary vector b3, (c) corresponding to binary vector b4.
(0.031 dB). It is interesting to know that for binary vector b4, the power of 10 pumps from 20 pumps is zero and the corresponding gain ripple is not far from the global minimum gain ripple. For both b3 and b4, the average net gain of the fiber Raman amplifier is more than 2 dB. The evolution of signal powers along the fiber length for different binary vectors is shown in Fig. 4.
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Table 1 Comparison of gain ripple, gain average, total pump power and signal bandwidth for different binary vector of optimization in a 20 pumps fiber Raman amplifier Binary vector
b1
b2
b3
b4
Gain ripple (dB) Gain average (dB) Total pump power(mW) Signal band width(nm) Number of pump
No solution No solution No solution 1528–1578 20
0.06 0.47 415 1528–1578 20
0.031 2.0 512 1528–1578 20
0.042 2.1 487 1528–1578 20
a
b 0.16
0.11
signal power,mW
signal power,mW
0.12
0.1 0.09
0.14 0.12 0.1
0.08 0.08 1580 1580
30
1560 1560
1540 signal wavelength,nm
0
20 15 10 fiber length,km
5
25
20
1540 signal wavelength,nm
10 1520
0
fiber length,km
c
signal power,mW
0.18 0.16 0.14 0.12 0.1 0.08 1580 30
1560
20
1540 signal wavelength,nm
10 1520
0
fiber length,km
Fig. 4. Signal power evolution along a 25 km SMF-28 optical fiber for different binary vector (bi). (a) Corresponding to binary vector b2, (b) corresponding to binary vector b3, (c) corresponding to binary vector b4.
The results are in agreement with those obtained by a full numerical method and have been published by the previous investigators [21]. As a second example, calculations are also done for 31 pumps and signals are distributed on the interval 1528–1590 nm, the results corresponding to different binary vectors are presented and compared in Table 2. Table 2 Comparison of gain ripple, gain average, total pump power and signal bandwidth for different binary vector of optimization in a 31 pumps fiber Raman amplifier Binary vector
b1
b2
b3
b4
Gain ripple (dB) Gain average (dB) Total pump power(mW) Signal band width(nm) Number of pump
No solution No solution No solution 1528–1590 31
0.07 0.5 526 1528–1590 31
0.17 2.2 583.7 1528–1590 31
0.04 1 482.6 1528–1590 31
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300
5
200
4
100
3
gain,dB
input pump power,mW
The method is employed to optimize gain ripple a 100 km fiber Raman amplifier with 20 pumps uniformly distributed in the pump bandwidth (1408–1490 nm) and average net gain more than 4 dB, the average on–off gain more than 20 dB, the minimum global gain ripple 0.096 and results are shown in Figs. 5 and 6.
0 2 1400 1420 1440 1460 1480 1500 1520 1540 1560 1580 wavelength,nm
Fig. 5. Optimal Raman gain profile and pump spectrum for a 100 km SMF-28 optical fiber corresponding to global minimum gain ripple.
signal power,mW
0.4 0.3 0.2 0.1
0 1580 100
1560 50
1540 signal wavelength,nm
1520
0
fiber length,km
300
1
250
0
200
–1
150
–2
gain,dB
input pump power,mW
Fig. 6. Signal power evolution along a 100 km SMF-28 optical fiber corresponding to global minimum gain ripple.
100 –3 1420 1440 1460 1480 1500 1520 1540 1560 1580 wavelength,nm
Fig. 7. The gain and pump power spectrum for a gain ripple minimized, 100 km SMF-28, 4 pumps fiber Raman amplifier. The average net gain is 0.5 dB and the minimum gain ripple is 0.72 dB. The effective core area of fiber is constant and its value is 50 lm2.
A.R. Bahrampour et al. / Optics Communications 281 (2008) 1545–1557
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signal power,mW
0.2 0.15 0.1 0.05 0 1580 100
1560 50
1540 wavelength,nm
1520
0
fiber length,km
Fig. 8. Signal power evolution along the fiber length of a 100 km SMF-28 with constant effective core area (Aeff = 50 lm2), 4 pumps fiber Raman amplifier with minimum gain ripple, minimized by the variational method.
As a final example, the minimization of gain ripple with respect to the pump powers and wavelengths in a 100 km SMF28, 4 pumps fiber Raman amplifier is done and results corresponding to the global minimum are presented in Figs. 7 and 8. The average on–off gain is greater than 20 dB. The gain spectrum and corresponding input pump power are presented in Fig. 7. The signal evolution along the fiber length for this system of fiber Raman amplifier is shown in Fig. 8. The global minimum of gain ripple is less than 0.72 dB and decreases rapidly as the number of pumps increases. The signal to noise ratio for the optimal condition can be obtained by the standard methods presented in references [11,12]. The minimization of gain ripple in fiber Raman amplifier with wide-band continuous pump spectrum can be obtained by simulating the pump spectrum with a large number of monochromatic pumps [21]. 5. Conclusion The calculus of variation is a high power tool for optimizing the gain flatness in fiber Raman amplifiers. This method is applied to optimize the gain ripple of multi-pump fiber Raman amplifiers versus the variation of pump powers and pump wavelengths. It is assumed that the total power of pumps is upper limited and average gain is lower limited. As the variational method has a strong mathematical basis than the other methods, results show that the calculus of variation is precise but the speed of calculation is slower than some other methods. This method is suitable for the design of fiber Raman amplifier but it is slow for the online adjustment of pump powers and wavelengths during the line reconfiguration. Increasing the speed of the numerical calculations is under investigation and the results will be published in near future. Appendix A The optimization problem is reduced to the minimization of cost function F = var(G) subject to the following constraints: ! Np Ns X X dQi ðzÞ Qi ðzÞ 2 ¼ aðmi Þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ ; i ¼ 1; 2; . . . ; N p ðA1 Þ dz 2 j¼1 j¼1 ! Np Ns X X dS i ðzÞ 2 ¼ S i ðzÞ aðmi Þ þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ ; i ¼ 1; 2; . . . ; N s ðA2 Þ dz j¼1 j¼1 P i ðzÞ ¼ Q2i ðzÞ
ðA3 Þ
where Pi(z) and Sj(z) are the powers of the ith pump and the jth signal at the point z respectively, g(mi, mj) gives the Raman gain coefficient. Quantities with hated index are related to the pumps and that without hats correspond to signals. Other constraints are as follows: P max
Np X
Q2i ðLÞ ¼ D2
ðA4 Þ
i¼1
G G0 ¼ E 2
ðA5 Þ
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L is the length of the fiber, Pmax is the upper limit of the total pump powers, G is the average of gain and G0 is its lower limit. The average gain is defined as: Ns Ns 1 X 1 X S k ðLÞ G¼ Gk ¼ ðA6 Þ N s k¼1 N s k¼1 S k ð0Þ The cost function F is defined by: F ðS 1 ðLÞ; S 2 ðLÞ; . . . ; S N s ðLÞÞ ¼
Ns X 2 ðGk GÞ
ðA7 Þ
k¼1
The generalized cost function is written: ! Np X 2 2 Qi ðLÞ D þ u G G0 E2 J ¼ F þ x P max Z
"
i¼1
!# Np Ns X X dS i ðzÞ 2 þ S i ðzÞ aðmi Þ ci ðzÞ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ dz þ dz 0 i¼1 j¼1 j¼1 " !# Np Z L Np X X dQi ðzÞ aðmi ÞQi ðzÞ 1 Ns þ Qi ðzÞ m gðmi ; mj ÞS j ðzÞ þ þ bi ðzÞ gðmi ; mj ÞQ2j ðzÞ dz j¼1 dz 2 2 0 i¼1 j¼1 Ns X
L
ðA8 Þ
where bi(z) (i = 1, 2, . . . , Np), ci(z) (i = 1, 2, . . . , Ns), x and u are the Lagrange multipliers corresponding to the Eqs. (A1), (A2), (A4) and (A5). The gain ripple minimization problem is reduced to the following variational problem: ðA9 Þ
dJ ¼ 0 The total variation of J can be written as: ! Np X 2 2 dJ ¼ dx P max Qi ðLÞ D 2xDdD þ duðG G0 E2 Þ 2uEdE i¼1
!# Np Ns X X dS i ðzÞ þ S i ðzÞ aðmi Þ þ dzdci ðzÞ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQ2j ðzÞ dz 0 i¼1 j¼1 j¼1 " !# Np Z L Np Ns X X X dQi ðzÞ Qi ðzÞ 2 þ aðmi Þ þ dzdbi ðzÞ gðmi ; mj ÞS j ðzÞ þ gðmi ; mj ÞQj ðzÞ dz 2 0 i¼1 j¼1 j¼1 " ! Np Z L Np Ns X dbi ðzÞ aðmi Þ 1 X 1X 2 þ bi ðzÞ þ þ dzdQi ðzÞ gðmi ; mj ÞS j ðzÞ þ gðmi ; mj ÞQj ðzÞ dz 2 2 j¼1 2 j¼1 0 i¼1 !# Np Ns X X þ Qi gðmj ; mi ÞQj ðzÞbj ðzÞ 2 gðmj ; mi ÞS j ðzÞcj ðzÞ Ns Z X
L
j¼1
"
þ
Np X i¼1
"
j¼1
! Np Ns X X dci ðzÞ 2 þ ci ðzÞ aðmi Þ þ dzdS i ðzÞ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ dz 0 i¼1 j¼1 j¼1 # Np Ns Ns X X 1X 2 u ðGi GÞ þ þ ci ðLÞ dS i ðLÞ gðmj ; mi ÞS j ðzÞcj ðzÞ þ gðmj ; mi ÞQj ðzÞbj ðzÞ þ 2 j¼1 S i ð0Þ N s S i ð0Þ j¼1 i¼1 Ns Z X
L
½ðbi ðLÞ 2xQi ðLÞÞdQi ðLÞ bi ð0ÞdQi ð0Þ "
Z Z Z Ns Ns X ogðmi ; mk Þ L 1 oaðmk Þ L 1X ogðmk ; mj Þ L ci ðzÞQ2k ðzÞdz bk ðzÞQk ðzÞ þ bk ðzÞQk ðzÞS j ðzÞdz omk 2 omk 2 j¼1 omk 0 0 0 k¼1 i¼1 # Z Z L Np Ns 1X ogðmk ; mj Þ 1X ogðmi ; mk Þ L 2 2 þ bk ðzÞQk ðzÞQj ðzÞdz þ bi ðzÞQi ðzÞQk ðzÞdz ðA10 Þ 2 j¼1 omk 2 i¼1 omk 0 0
þ
Np X
dmk
A.R. Bahrampour et al. / Optics Communications 281 (2008) 1545–1557
In the above equation we have used the following by parts integration: Z L Z L ddQi ðzÞ db ðzÞ ¼ bi ðLÞdQi ðLÞ bi ð0ÞdQi ð0Þ dzbi ðzÞ dzdQi ðzÞ i ; i ¼ 1; 2; . . . ; N p dz dz 0 0 Z L Z L ddS i ðzÞ dci ðzÞ ¼ ci ðLÞdS i ðLÞ ci ð0ÞdS i ð0Þ ; i ¼ 1; 2; . . . ; N s dzci ðzÞ dzdS i ðzÞ dz dz 0 0
1555
ðA11 Þ ðA12 Þ
The input power of each signal is known, so we have dSi(0) = 0. Since Pi(z)(i = 1, 2, . . . , Np), Si(z)(i = 1, 2, . . . , Ns), E, D and Lagrange multipliers x and u are independent of each other, dJ can be zero if and only if the coefficients of variations (dy) are zero, where y can be each of the above variables. Therefore, in addition to the constraint Eqs. (A1), (A2), (A4) and (A5) the following equation are obtained as the necessary conditions for the optimal state (1) The set of differential equations:
! Np Ns X X dQi ðzÞ Qi ðzÞ 2 ¼ aðmi Þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ ; dz 2 j¼1 j¼1
i ¼ 1; 2; . . . ; N p
! Np Ns X X dbi ðzÞ bi ðzÞ 2 þ aðmi Þ þ gðmi ; mj ÞQj ðzÞ þ gðmi ; mj ÞS j ðzÞ dz 2 j¼1 j¼1 ! Np Ns X X þ Qi ðzÞ gðmj ; mi Þbj ðzÞQj ðzÞ 2 gðmj ; mi ÞS j ðzÞcj ðzÞ ¼ 0; i ¼ 1; 2; . . . ; N p j¼1
ðA13 Þ
ðA14 Þ
j¼1
! Np Ns X X dS i ðzÞ 2 ¼S i ðzÞ aðmi Þ þ gðmi ; mj ÞS j ðzÞ þ gðmi ; mj ÞQj ðzÞ ; dz j¼1 j¼1
i ¼ 1; 2; . . . ; N s
ðA15 Þ
! Np Ns Ns X X X dci ðzÞ þ ci ðzÞ aðmi Þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQ2j ðzÞ gðmj ; mi ÞS j ðzÞcj ðzÞ dz j¼1 j¼1 j¼1 þ
Np 1X gðmj ; mi ÞQj ðzÞbj ðzÞ ¼ 0; 2 j¼1
i ¼ 1; 2; . . . ; N s
ðA16 Þ
(2) The boundary condition corresponding to the conjugate variables: bi ð0Þ ¼ 0; i ¼ 1; 2; . . . ; N p bi ðLÞ ¼ 2xQi ðLÞ; i ¼ 1; 2; . . . ; N p 2 u ðGi GÞ ; ci ðLÞ ¼ S i ð0Þ N s S i ð0Þ
ðA17 Þ ðA18 Þ i ¼ 1; 2; . . . ; N s
ðA19 Þ
And there is no restriction on the initial values of ci. (3) Switching equations: Dx ¼ 0 Eu ¼ 0
ðA20 Þ ðA21 Þ
(4) The set of algebraic equations: Ns X j¼1
Ajk
X Np ogðmj ; mk Þ ogðmk ; mj Þ ogðmk ; mj Þ oaðmk Þ þ Bkj C kj þ Dk þ ¼0 omk omk omk omk j¼1
ðA22 Þ
k ¼ 1; 2; . . . ; N p where Aik, Bkj, Ckj, Dk are defined by the following integrals: Z L Aik ¼ ci ðzÞQ2k ðzÞdz 0 Z 1 L b ðzÞQk ðzÞS j ðzÞdz Bkj ¼ 2 0 k
ðA23 Þ ðA24 Þ
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1 2
C kj ¼
Z
1 Dk ¼ 2
L
bk ðzÞQk ðzÞQ2j ðzÞdz
ðA25 Þ
0
Z
L
bk ðzÞQk ðzÞ
ðA26 Þ
0
Appendix B Eqs. (A20) and (A21)make four possible states for any optimization problem. Each state is specified by a two-dimensional vector with components zero or one which are called binary vectors and are denoted by bi (i = 1,2,3,4). The zero and one values of the first component of bi are correspond to D = 0 and x = 0 respectively. The E = 0 and u = 0 correspond to the 0 and 1 values for the second component of bi, respectively. Each state satisfies Eqs. (A4), (A5), (A13)–(A21). This leads to a system of algebraic equations. By using above definition for the binary vectors, we have: (1) b1 ¼
0 0
)
8 Np > < P Q2 ðLÞ ¼ P ^
^i
max
i¼1 > : G ¼ G0
ðB1 Þ
(2) b2 ¼
1
) G ¼ G0
ðB2 Þ
Np X 0 b3 ¼ ) Q^2i ðLÞ ¼ P max 1 ^i¼1
ðB3 Þ
1 ) no special boundary condition b4 ¼ 1
ðB4 Þ
0
(3)
(4)
References [1] M.N. Islam, Raman Amplifier for Telecommunications, vol. 1, Springer-Verlag, New York, 2004 (Chapter 1). [2] S. Namiki, Y. Emori, Ultra-broad band Raman amplifiers pumped and gain equalized by wavelength division multiplexed high power laser diodes, IEEE J. Sel. Top. Quant. Electron. 7 (January/February) (2001) 3. [3] Y. Emory, S. Kado, S. Namiki, Broadband flat gain and low-noise Raman amplifier pumped by wavelength multiplexed high power laser diodes, J. Opt. Fiber Technol. 8 (February) (2002) 107. [4] V.E. Perlin, H.G. Winful, Optimal design of flat-gain wide-band fiber Raman amplifiers, IEEE J. Lightwave Technol. 20 (February) (2002) 250. [5] V.E. Perlin, H.G. Winful, Efficient design method for multi pump flat gain for fiber Raman amplifiers, in: OFC 2002, p. 57. [6] X. Zhou, C. Lu, P. Shum, T.H. Cheng, A simplified model and optimal design of a multi wavelength backward-pumped of fiber Raman amplifier, IEEE Photon. Technol. Lett. 13 (September) (2001) 945. [7] X. Liu, Y. Li, Optimizing the bandwidth and noise performance of distributed multi pump Raman amplifiers, J. Opt. Commun. 230 (February) (2004) 425. [8] X. Liu, Y. Li, Optimal bandwidth for distributed multi-pump Raman amplifier based on Hybrid genetic algorithm, Chinese Phys. Lett. 21 (1) (2004) 84. [9] X. Lu, Optimization and comparison for different Raman amplifier configurations, Chinese Phys. Lett. 21 (9) (2004) 1756. [10] P. Xiao, Q. Zeng, J. Huang, J. Liu, A new optimal algorithm for multi-pump sources of distributed of fiber Raman amplifier, IEEE Photon. Technol. Lett. 15 (February) (2003) 206. [11] N. Park, P. Kim, J. Park, L.K. Choi, Integral form expansion of fiber Raman amplifier problem, J. Opt. Fiber Technol. 11 (2005) 111. [12] J. Park, P. Kim, J. Park, H. Lee, N. Park, Closed integral form expansion of Raman equation for efficient gain optimization process, J. IEEE Photon. Technol. Lett. 16 (7) (2004) 1649. [13] L.K. Choi, J. Park, P. Kim, J. Park, N. Park, Closed integral form expansion of Raman equation and its iteration method for highly efficient Raman amplifier analysis, in: Proceeding of Optoelectronics and Communications Conference, Yokahama, 2004, paper 13P-85. [14] P. Kim, J. Park, H. Yoon, J. Park, N. Park, In situ design method for multi-channel gain of distributed Raman amplifier with the multi-wave OTDR, IEEE Photon. Technol. Lett. 14 (2002) 1683. [15] He Wen, Xin Yang, Wanyi Gu, Novel methods for fast fiber Raman amplifier pump configuration, J. Lightwave Technol. 23 (4) (2005).
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[16] Donald E. Kirk, Optimal Control Theory: An Introduction, Prentice Hall, 1970. [17] S. Wang, C. Fan, Distributed fiber Raman amplifiers: analytical expression of noise characteristics under complex conditions, J. Opt. Commun. 198 (October) (2001) 65. [18] X. Liu, B. Lee, A fast and stable method for Raman amplifier propagation equations, J. Opt. Express 11 (18) (2003) 2163. [19] G. Agrawal, Nonlinear Fiber Optics, Elsevier, 2001 (Chapter 8). [20] C. Headely, G. Agrawal (Eds.), Raman Amplification in Fiber Optical Communication Systems, Elsevier Academic Press, 2005, ISBN 0-12-044506-9 (Chapter 2). [21] S.M. Kobtsev, A.A. Pustovkikh, Fiber Raman amplifier pumped by continuous–spectrum radiation, J. Quant. Electron. 34 (6) (2004) 575.
Optics Communications 281 (2008) 1545–1557 www.elsevier.com/locate/optcom
Gain ripple minimization in fiber Raman amplifiers based on variational method A.R. Bahrampour a,b,*, A. Ghasempour a, L. Rahimi b a b
Faculty of Physics, Sharif University of Technology, Tehran, Iran Department of Physics, Shahid Bahonar University of Kerman, Iran
Received 14 July 2007; received in revised form 17 October 2007; accepted 12 November 2007
Abstract In this paper, the variational method is employed for minimizing the gain ripple of multi-wavelength fiber Raman amplifiers. The variance of gain spectrum of the fiber Raman amplifier is regarded as the cost function, restriction on total pump power and average gain is given as the constraints of the minimization problem. It is shown that the minimization problem with any necessary constraints on the pump powers, average gain and signal to noise ratio, is reduced to a two-point boundary value problem. The method gives the entire possible local and global solutions. The method is applied to different examples of fiber Raman amplifiers with different lengths from 25 km to 100 km and different numbers of pumps from 4 to 20 to determine the pump powers and wavelengths for minimum gain ripple. It was obtained for a 100 km fiber Raman amplifier the gain ripple can be about 0.1 dB with on–off gain more than 20 dB. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Raman amplifier; Raman gain flatness; Variational method
1. Introduction Raman amplifier is based on the stimulated Raman scattering effect in silica-based optical fibers. Raman gain arises from the transfer of power from one optical beam to another that is down shifted by the energy of an optical phonon (a vibrational mode of the medium) [1]. There are two kinds of phonon, one is acoustic phonon and the other is optical phonon. Raman scattering is the scattering between photon and optical phonon while the one between photon and acoustic phonon is called Brillouin scattering. Because optical phonon has almost uniform dispersion relation versus wave number, the phase matching is easily obtained for arbitrary relative direction between the pump and signal waves. Therefore, Raman amplifiers can take both co-pump and counter-pump schemes [2]. To increase the bandwidth of the amplifier for amplification in wavelength division multiplexing (WDM) systems, authors have proposed multi-wavelength pumping technique [1,3]. The idea of multi-wavelength pumping (MWP) is to prepare a set of pumps operating at different wavelengths combined through WDM couplers into a single fiber to create a composite Raman gain. The composite Raman gain, created by different wavelengths of pumps, can be so designed as to be arbitrarily broad and flat. Principally, the MWP can obtain the arbitrary gain spectra by the proper choice of pump wavelengths and powers. However, the strong Raman interaction of pump to pump, signal to signal and pump to signal makes the problem of *
Corresponding author. Address: Faculty of Physics, Sharif University of Technology, Tehran, Iran. Tel.: +98 21 6616 4527. E-mail address: [email protected] (A.R. Bahrampour).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.11.057
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A.R. Bahrampour et al. / Optics Communications 281 (2008) 1545–1557
the required gain spectral somewhat more difficult for optimal design of MWP. Therefore, the MWP design presents a grand challenge to numerical optimization. It involves multiple powers and wavelengths. Recently, some soft computational methods have been proposed to select the appropriate pump wavelengths and powers in fiber Raman amplifiers. Perlin and Zhou et al. employ traditional genetic algorithm (GA) to optimize the gain flatness and gain bandwidth performance [4–6]. The optimal results are exciting but they only obtain a single optimal solution in each domain, and even their methods may be trapped in local optimal solution of the search space due to intrinsic weakness of the traditional GA [7]. To overcome these difficulties a hybrid genetic algorithm (HGA) based on techniques such as clustering, sharing, crowding and adaptive probability is introduced and is used for optimal design of multi-wavelength pumping Raman fiber amplifiers [7–9]. In [10], employing the neural network method optimizes the Raman gain spectrum. However, it obtains the optimization only and partly in pump powers excluding wavelengths [7]. The Raman governing equations are transformed to a system of integral equations and the Picard method is applied for the solution of integral equations. On the basis of integral equations, a method for designing optimal Raman amplifier is introduced [11–14]. Some other compound methods are introduced by previous authors [15]. Variational method is a high power tool with strong mathematical basis for optimizing a cost function in the presence of constraints in the form of differential equations and/or algebraic equations [16]. In employing this high power tool, the main goal parameter such as gain ripple or signal to noise ratio can be regarded as the cost function and others are given as the inequality and equality constraints in the optimization problem. In this paper, the variance of gain spectrum is regarded as the cost function, the upper limit of the total pump power and lower bound of average gain are given as the constraints of the minimization problem. The variational method is employed to minimize the gain ripple of multiwavelength pumping fiber Raman amplifiers. The results can be extremely helpful in the design of the multi-wavelength pumping fiber Raman amplifiers. This paper is organized as follows: In Section 2 the governing equation and constraints are presented. The mathematical model for the optimal design is given in Section 3. In Section 4 the numerical solution for the optimal design is obtained. Finally, the paper ends with some conclusions. 2. Governing equations and constraints Wave propagation in a multi-pump Raman amplifier is characterized by a large number of effects. The most important of these effects, for the purposes of the present consideration, are pump to pump, pump to signal and signal to signal Raman interaction and wavelength dependent linear attenuation, experienced by both pump and signal waves. In copumped multi-pump fiber Raman amplifiers where signal and pump lights travel along the same direction in the fiber, fluctuation in pump power can be coupled as noise onto signal power and serious crosstalk that will be produced under the condition of saturated amplification. On the other hand, backward pumped multi-pump fiber Raman amplifiers have no such shortages [17]. Hence, only backward pumped multi-pump fiber Raman amplifiers are considered in this paper. Because the powers of back scattering pumps and signals are lower by 30 dB and 20 dB than the original power, the power of forward and backward noises is less than that of input signals by 30 dB [18], so noise effects such spontaneous Raman scattering, Rayleigh backscattering and thermal factor are neglected in calculating the amplifier gain profile [4,15]. The governing equations of signal powers (Si(z); i = 1, 2, . . . , Ns) and pump powers (Pi(z); i = 1, 2, . . . , Np)in the steady state are a system of coupled differential equations given in the following [9]: 0 1 Np Ns X X dS i ðzÞ ¼ S i ðzÞ@aðmi Þ þ gðmi ; mj ÞS j ðzÞ þ gðmi ; m^j ÞP ^j ðzÞA; i ¼ 1; 2; . . . ; N s ð1Þ dz ^j¼1 j¼1 ! Np Ns X X dP i ðzÞ ¼ P i ðzÞ aðmi Þ þ gðmi ; mj ÞS j ðzÞ þ gðmi ; mj ÞP j ðzÞ ; i ¼ 1; 2; . . . ; N p ð2Þ dz j¼1 j¼1 where Np is the number of pumps and Ns is the number of signals. a(mi) is the intrinsic fiber attenuation at frequency mi. The following relation gives the Raman gain coefficient g(mi,mj) – [19]: 8 m2i gref ðmi mj Þ > > < mj mref CAeff ; mi > mj ð3Þ gðmi ; mj Þ ¼ 0; mi ¼ mj > > : mj gref ðmj mi Þ CAeff ; mi < mj mreff where mi and mj are the frequencies of the interacting waves in the Raman effect. The factor C accounts for polarization randomization effect, Aeff is the effective fiber core area and gref(n g) is the Raman gain coefficient measured at reference pump frequency mreff which describes the power transferred by the stimulated Raman scattering from wave with frequency n to the one with frequency g. The Raman gain coefficient spectrum gref(Dm) is represented in Fig. 1 [20].
A.R. Bahrampour et al. / Optics Communications 281 (2008) 1545–1557
1547
–14
x 10
Raman gain cofficient,m / W
6 5 4 3 2 1 0
0
5
10
15
20
25
30
35
Pump–signal frequency difference,THz
Fig. 1. Typical Raman gain coefficient [20].
The minus sign on the left side of Eq. (2) denotes backward propagating pump waves. In other words, for a fiber span of length L, the boundary values are given at z = 0 for signal waves and at z = L for pump waves. Also, we have: P i ðzÞ P 0
ð4Þ
where Pi(z) is the power of the ith pump at point z. For applying the constraint (4), it is assumed that P i ðzÞ ¼ Q2i ðzÞ, where Qi(z) is a real valued function. Therefore the governing equations for pumps are rearranged with respect to Qi(z). ! Np Ns X X dQi ðzÞ Qi ðzÞ 2 ¼ aðmi Þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ ; i ¼ 1; 2; . . . ; N p ð5Þ dz 2 j¼1 j¼1 To avoid thermal effect in the fiber, it is assumed as a restriction that the total of pump powers is upper limited. Its upper limit value is denoted by Pmax i.e. Np X
Q2i ðLÞ 6 P max
ð6Þ
i¼1
The inequality (6) can be written in the form of an equality: P max
Np X
Q2i ðLÞ ¼ D2
ð7Þ
i¼1
In the above equation D is a real number. Obviously maximum flatness with no gain and even with some loss is possible. In order to avoid these solutions, it is assumed that the average gain of signals G is lower limited by a defined value G0 i.e. G P G0 where the gain average G is defined in the following: Ns Ns 1 X 1 X S k ðLÞ G¼ Gk ¼ N s k¼1 N s k¼1 S k ð0Þ
ð8Þ
ð9Þ
In the above relation Sk(0) and (Sk(L)) is the kth input (output) signal power and Gk is the gain of the kth signal. Eq. (8) is written in the following form: G G0 ¼ E 2
ð10Þ
where E is a real number. The distance between the gain spectrum and target predefined gain spectrum is given as the cost function [11] which can be considered by this method. Now our problem is to minimize the gain ripple in a Raman amplifier subject to the constraint (1), (5), (7) and (10). In this analysis the noise effects are neglected [4,15], otherwise the lower limit on the signal to noise ratio and governing equations on the noise evolution along the fiber length should be regarded as other constraints of the minimization problem [11].
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A.R. Bahrampour et al. / Optics Communications 281 (2008) 1545–1557
3. Theoretical model for the gain ripple minimization The following relation defines the cost function F as a characteristic of flatness in the fiber Raman amplifier: F ðS 1 ðLÞ; S 2 ðLÞ; . . . ; S N s ðLÞÞ ¼
Ns X 2 ðGk GÞ
ð11Þ
k¼1
Now the problem is to find a Np tuple P ¼ ðP 1 ðLÞ; . . . ; P N p ðLÞ; kP 1 ; kP 2 ; . . . ; kP N p Þ among all possible Np tuple P ¼ ðP 1 ðLÞ; P 2 ðLÞ; . . . ; P N p ðLÞ; kP 1 ; kP 2 ; . . . ; kP N p Þ, which minimizes the cost function F subject to the constraints (1),(5),(7) and (10). The generalized cost function J is defined by formula (A8) in Appendix A. Therefore, the optimization problem is reduced to the following classical variational problem: dJ ¼ 0
ð12Þ
After some mathematical manipulations (see Appendix A), it is easy to show that the solution of the above variational problem is equivalent to the solution of a complex two-point boundary value problem. The governing differential equations on the components of the state variable vector X ¼ ðQ1 ðzÞ; . . . ; QN p ; S 1 ðzÞ; . . . ; S N s ðzÞ; b1 ðzÞ; . . . ; bN p ðzÞ; c1 ðzÞ; . . . ; cN s ðzÞÞ are given by ! Np Ns X X dQi ðzÞ Qi ðzÞ 2 ¼ aðmi Þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ ; i ¼ 1; 2; . . . ; N p ð13Þ dz 2 j¼1 j¼1 ! Np Ns X X dbi ðzÞ bi ðzÞ 2 þ aðmi Þ þ gðmi ; mj ÞQj ðzÞ þ gðmi ; mj ÞS j ðzÞ dz 2 j¼1 j¼1 ! Np Ns X X þ Qi ðzÞ gðmj ; mi Þbj ðzÞQj ðzÞ 2 gðmj ; mi ÞS j ðzÞcj ðzÞ ¼ 0 i ¼ 1; 2; . . . ; N p ð14Þ j¼1
dS i ðzÞ ¼S i ðzÞ aðmi Þ þ dz
j¼1 Ns X
gðmi ; mj ÞS j ðzÞ þ
j¼1
Np X
! gðmi ; mj ÞQ2j ðzÞ
;
i ¼ 1; 2; . . . ; N s
ð15Þ
j¼1
Ns p Ns X X X dci ðzÞ þ ci ðzÞðaðmi Þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQ2j ðzÞÞ gðmj ; mi ÞS j ðzÞcj ðzÞ dz j¼1 j¼1 j¼1
þ
Np 1X gðmj ; mi ÞQj ðzÞbj ðzÞ ¼ 0; 2 j¼1
N
i ¼ 1; 2; . . . ; N s
ð16Þ
where bi(z) (i = 1, 2, . . . , Np) and ci(z) (i = 1, 2, . . . , Ns) are conjugate variables corresponding to the pump and signal powers, respectively. For differential Eqs. (13)–(16), the boundary conditions are as follows (Appendix A): bi ð0Þ ¼ 0;
i ¼ 1; 2; . . . ; N p
bi ðLÞ ¼ 2xQi ðLÞ; i ¼ 1; 2; . . . ; N p 2 u ðGi GÞ ; ci ðLÞ ¼ S i ð0Þ N s S i ð0Þ
ð17Þ ð18Þ i ¼ 1; 2; . . . ; N s
ð19Þ
There is no restriction on ci(0) (i = 1, 2, . . . , Ns) and they are evaluated from the solution of differential equations. It is assumed that the input power of signals at z = 0 is specified (Si(0) = Si0 (i = 1, 2, . . . , Ns)) so from this assumption and Eqs. (17)–(19), the number of boundary conditions, 2(Np + Ns), is just equal to the number of governing differential equations. In addition, the Lagrange multipliers x,u must satisfy the following switching equations: Dx ¼ 0
ð20Þ
Eu ¼ 0
ð21Þ
where x is the Lagrange multiplier corresponding to the restriction due to the upper-bound of the sum of pump powers and u is the Lagrange multiplier corresponding to the lower bound of the gain average restriction. The system of algebraic equations corresponding to the vanishing variational derivatives with respect to the pump wavelengths (kk; k = 1, 2, . . . , Np) is as following:
A.R. Bahrampour et al. / Optics Communications 281 (2008) 1545–1557
X Np Ns X ogðmj ; mk Þ ogðmk ; mj Þ ogðmk ; mj Þ oaðmk Þ Ajk þ Bkj C kj þ Dk þ ¼0 omk omk omk omk j¼1 j¼1
k ¼ 1; 2; . . . ; N p
1549
ð22Þ
where Aik, Bkj, Ckj, Dk are defined in Appendix A. That is, the optimization is converted to a two-point boundary value problem. The number of unknown functions (bi,Qi,ck,Sk; i = 1, 2, . . . , Np,k = 1, 2, . . . , Ns) is 2(Np + Ns), which is the same as the number of differential Eqs. (13)–(16). The number of unknown parameters (D, u, x, E) is four, which is the same as the number of algebraic Eqs. (7), (10), (20) and (21). The number of unknown pump wavelengths corresponding to the minimum gain ripple is equal to the number of algebraic Eq. (22). To obtain the pump power and wavelength corresponding to the optimal condition the system of governing Eqs. (7), (10), (13)–(21) and (22) are solved numerically. 4. Numerical solution and results The system of governing equations are organized into four classes, the system of nonlinear differential Eqs. (13)–(16), the set of equations for the initial and final values of governing ordinary differential Eqs. (17)–(19), algebraic Eqs. (7), (10) and (22) and finally the system of switching Eqs. (20) and (21). To formulize the switching equations a two component binary vector b is defined such that, when the total power of the pumps clamps to maximum value Pmax the first component of b is zero, otherwise all bi(L)s (i = 1, 2, . . . , Np) are zero. The zero values of the second component are corresponding to the clamping of G to its lower value G0 (i.e. G ¼ G0 ) and if it equals one, then the final value of the conjugate functions ci(L)s (i = 1, 2, . . . , Ns) is determined from the following equation: ci ðLÞ ¼
2 u ðGi GÞ ; S i ð0Þ N s S i ð0Þ
i ¼ 1; 2; . . . ; N s
ð23Þ
The dimension of the binary vector b is independent of the number of pumps and the number of signals. In our problem binary vector b makes four possible states; the boundary conditions corresponding to each binary vector bi (i = 1,2,3,4) are summarized and represented in Appendix B. To introduce the algorithm of numerical computation, we start the solution of the system of differential equations, and the system of algebraic equations among the initial and final values of pump powers, signal powers and their conjugate variables. The algorithm of computation is introduced as follows: (1) (2) (3) (4) (5)
Specify the number of pumps and signals and choose the uniform distribution for pump wavelengths. Put i = 1. Specify the binary vector bi, and its related boundary conditions according to Appendix B. Try to solve two-point boundary value problems. Is there any solution to the two-point boundary value problem? If so, determine the corresponding pump power and wavelengths. If not, put Fi = 500 (500 is greater than the actual values of the cost function) then go to step (7). (6) Calculate the cost function F related to the solution of the third step and store in Fi. (7) If i is less than four, put i = i + 1 and return to step 3. (8) Put minimum value of Fi in M (i.e. M is the minimum value of {F1,F2,F3,F4}). The two-point boundary value problems are solved by the finite difference method. The state corresponding to M is the solution of the optimization problem. As an example this method is applied to a fiber Raman amplifier with 20 pumps and the following parameter set is fixed: there are 50 signal channels spaced with equal distances in 1528–1578 nm (the corresponding bandwidth is about 50 nm), the signal power of each channel is 0.1 mW (Si(0) = 0.1 mW; i = 1, 2, . . . , Ns), the fiber length is 25 km, mreff = 200 THz and C = 2. Calculations are done for SMF-28 fiber where its mode effective area (Aeff(k)) and attenuation coefficient (a(k)) versus wavelength are presented in Fig. 2 [21]. For a fiber Raman amplifier with 20 pumps, the pump wavelengths with equal distances are chosen in the bandwidth (1408–1490 nm), maximum total power is 512 mW(Pmax = 512 mW) and the value of 0.47 dB is selected for the lower bound of average net gain, (G0 = 0.47 dB). These values are selected so that the output power of each signal is not less than its input power, that is the Raman gain can compensate the loss of each signal. Obviously, the selected value of Pmax and G0 are dependent on some parameters such as the number and characteristics of pumps. The gain spectrum and pump powers corresponding to the binary vectors b2, b3 and b4 are presented in Fig. 3a–c, respectively. Calculation PN p shows there is no solution to the binary vector b1; this means that the average gain ðGÞ and the total pump power i¼1 P i ðLÞ can not clamp to the G0 and Pmax simultaneously. The results corresponding to different binary vectors are compared in Table 1 and it is shown that the binary vector b3 gives the global minimum gain ripple
1550
A.R. Bahrampour et al. / Optics Communications 281 (2008) 1545–1557 90
attenuation cofficient
effective area
80
0.05
Effective area, μm2
Attenuation cofficint,1/km
0.1
70
0 1400
1450
1500 Wavelength,nm
1550
1600
b
1
gain,dB 1450
1500 wavelength,nm
input pump power,mW
c
0
1550
100
3
50
2
0 1400
1450
1500 wavelength,nm
150
3
100
2
50
1
1550
1
gain ,dB
0 1400
input pump power,mW
50
input pump power,mW
a
gain,dB
Fig. 2. Spectral dependence of a and Aeff for a SMF-28 fiber [21].
0 1400
1450
1500 wavelength.nm
1550
0
Fig. 3. Optimal Raman gain profile and pump spectrum corresponding to a 25 km SMF-28 optical fiber for different binary vector (bi). (a) Corresponding to binary vector b2, (b) corresponding to binary vector b3, (c) corresponding to binary vector b4.
(0.031 dB). It is interesting to know that for binary vector b4, the power of 10 pumps from 20 pumps is zero and the corresponding gain ripple is not far from the global minimum gain ripple. For both b3 and b4, the average net gain of the fiber Raman amplifier is more than 2 dB. The evolution of signal powers along the fiber length for different binary vectors is shown in Fig. 4.
A.R. Bahrampour et al. / Optics Communications 281 (2008) 1545–1557
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Table 1 Comparison of gain ripple, gain average, total pump power and signal bandwidth for different binary vector of optimization in a 20 pumps fiber Raman amplifier Binary vector
b1
b2
b3
b4
Gain ripple (dB) Gain average (dB) Total pump power(mW) Signal band width(nm) Number of pump
No solution No solution No solution 1528–1578 20
0.06 0.47 415 1528–1578 20
0.031 2.0 512 1528–1578 20
0.042 2.1 487 1528–1578 20
a
b 0.16
0.11
signal power,mW
signal power,mW
0.12
0.1 0.09
0.14 0.12 0.1
0.08 0.08 1580 1580
30
1560 1560
1540 signal wavelength,nm
0
20 15 10 fiber length,km
5
25
20
1540 signal wavelength,nm
10 1520
0
fiber length,km
c
signal power,mW
0.18 0.16 0.14 0.12 0.1 0.08 1580 30
1560
20
1540 signal wavelength,nm
10 1520
0
fiber length,km
Fig. 4. Signal power evolution along a 25 km SMF-28 optical fiber for different binary vector (bi). (a) Corresponding to binary vector b2, (b) corresponding to binary vector b3, (c) corresponding to binary vector b4.
The results are in agreement with those obtained by a full numerical method and have been published by the previous investigators [21]. As a second example, calculations are also done for 31 pumps and signals are distributed on the interval 1528–1590 nm, the results corresponding to different binary vectors are presented and compared in Table 2. Table 2 Comparison of gain ripple, gain average, total pump power and signal bandwidth for different binary vector of optimization in a 31 pumps fiber Raman amplifier Binary vector
b1
b2
b3
b4
Gain ripple (dB) Gain average (dB) Total pump power(mW) Signal band width(nm) Number of pump
No solution No solution No solution 1528–1590 31
0.07 0.5 526 1528–1590 31
0.17 2.2 583.7 1528–1590 31
0.04 1 482.6 1528–1590 31
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300
5
200
4
100
3
gain,dB
input pump power,mW
The method is employed to optimize gain ripple a 100 km fiber Raman amplifier with 20 pumps uniformly distributed in the pump bandwidth (1408–1490 nm) and average net gain more than 4 dB, the average on–off gain more than 20 dB, the minimum global gain ripple 0.096 and results are shown in Figs. 5 and 6.
0 2 1400 1420 1440 1460 1480 1500 1520 1540 1560 1580 wavelength,nm
Fig. 5. Optimal Raman gain profile and pump spectrum for a 100 km SMF-28 optical fiber corresponding to global minimum gain ripple.
signal power,mW
0.4 0.3 0.2 0.1
0 1580 100
1560 50
1540 signal wavelength,nm
1520
0
fiber length,km
300
1
250
0
200
–1
150
–2
gain,dB
input pump power,mW
Fig. 6. Signal power evolution along a 100 km SMF-28 optical fiber corresponding to global minimum gain ripple.
100 –3 1420 1440 1460 1480 1500 1520 1540 1560 1580 wavelength,nm
Fig. 7. The gain and pump power spectrum for a gain ripple minimized, 100 km SMF-28, 4 pumps fiber Raman amplifier. The average net gain is 0.5 dB and the minimum gain ripple is 0.72 dB. The effective core area of fiber is constant and its value is 50 lm2.
A.R. Bahrampour et al. / Optics Communications 281 (2008) 1545–1557
1553
signal power,mW
0.2 0.15 0.1 0.05 0 1580 100
1560 50
1540 wavelength,nm
1520
0
fiber length,km
Fig. 8. Signal power evolution along the fiber length of a 100 km SMF-28 with constant effective core area (Aeff = 50 lm2), 4 pumps fiber Raman amplifier with minimum gain ripple, minimized by the variational method.
As a final example, the minimization of gain ripple with respect to the pump powers and wavelengths in a 100 km SMF28, 4 pumps fiber Raman amplifier is done and results corresponding to the global minimum are presented in Figs. 7 and 8. The average on–off gain is greater than 20 dB. The gain spectrum and corresponding input pump power are presented in Fig. 7. The signal evolution along the fiber length for this system of fiber Raman amplifier is shown in Fig. 8. The global minimum of gain ripple is less than 0.72 dB and decreases rapidly as the number of pumps increases. The signal to noise ratio for the optimal condition can be obtained by the standard methods presented in references [11,12]. The minimization of gain ripple in fiber Raman amplifier with wide-band continuous pump spectrum can be obtained by simulating the pump spectrum with a large number of monochromatic pumps [21]. 5. Conclusion The calculus of variation is a high power tool for optimizing the gain flatness in fiber Raman amplifiers. This method is applied to optimize the gain ripple of multi-pump fiber Raman amplifiers versus the variation of pump powers and pump wavelengths. It is assumed that the total power of pumps is upper limited and average gain is lower limited. As the variational method has a strong mathematical basis than the other methods, results show that the calculus of variation is precise but the speed of calculation is slower than some other methods. This method is suitable for the design of fiber Raman amplifier but it is slow for the online adjustment of pump powers and wavelengths during the line reconfiguration. Increasing the speed of the numerical calculations is under investigation and the results will be published in near future. Appendix A The optimization problem is reduced to the minimization of cost function F = var(G) subject to the following constraints: ! Np Ns X X dQi ðzÞ Qi ðzÞ 2 ¼ aðmi Þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ ; i ¼ 1; 2; . . . ; N p ðA1 Þ dz 2 j¼1 j¼1 ! Np Ns X X dS i ðzÞ 2 ¼ S i ðzÞ aðmi Þ þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ ; i ¼ 1; 2; . . . ; N s ðA2 Þ dz j¼1 j¼1 P i ðzÞ ¼ Q2i ðzÞ
ðA3 Þ
where Pi(z) and Sj(z) are the powers of the ith pump and the jth signal at the point z respectively, g(mi, mj) gives the Raman gain coefficient. Quantities with hated index are related to the pumps and that without hats correspond to signals. Other constraints are as follows: P max
Np X
Q2i ðLÞ ¼ D2
ðA4 Þ
i¼1
G G0 ¼ E 2
ðA5 Þ
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A.R. Bahrampour et al. / Optics Communications 281 (2008) 1545–1557
L is the length of the fiber, Pmax is the upper limit of the total pump powers, G is the average of gain and G0 is its lower limit. The average gain is defined as: Ns Ns 1 X 1 X S k ðLÞ G¼ Gk ¼ ðA6 Þ N s k¼1 N s k¼1 S k ð0Þ The cost function F is defined by: F ðS 1 ðLÞ; S 2 ðLÞ; . . . ; S N s ðLÞÞ ¼
Ns X 2 ðGk GÞ
ðA7 Þ
k¼1
The generalized cost function is written: ! Np X 2 2 Qi ðLÞ D þ u G G0 E2 J ¼ F þ x P max Z
"
i¼1
!# Np Ns X X dS i ðzÞ 2 þ S i ðzÞ aðmi Þ ci ðzÞ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ dz þ dz 0 i¼1 j¼1 j¼1 " !# Np Z L Np X X dQi ðzÞ aðmi ÞQi ðzÞ 1 Ns þ Qi ðzÞ m gðmi ; mj ÞS j ðzÞ þ þ bi ðzÞ gðmi ; mj ÞQ2j ðzÞ dz j¼1 dz 2 2 0 i¼1 j¼1 Ns X
L
ðA8 Þ
where bi(z) (i = 1, 2, . . . , Np), ci(z) (i = 1, 2, . . . , Ns), x and u are the Lagrange multipliers corresponding to the Eqs. (A1), (A2), (A4) and (A5). The gain ripple minimization problem is reduced to the following variational problem: ðA9 Þ
dJ ¼ 0 The total variation of J can be written as: ! Np X 2 2 dJ ¼ dx P max Qi ðLÞ D 2xDdD þ duðG G0 E2 Þ 2uEdE i¼1
!# Np Ns X X dS i ðzÞ þ S i ðzÞ aðmi Þ þ dzdci ðzÞ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQ2j ðzÞ dz 0 i¼1 j¼1 j¼1 " !# Np Z L Np Ns X X X dQi ðzÞ Qi ðzÞ 2 þ aðmi Þ þ dzdbi ðzÞ gðmi ; mj ÞS j ðzÞ þ gðmi ; mj ÞQj ðzÞ dz 2 0 i¼1 j¼1 j¼1 " ! Np Z L Np Ns X dbi ðzÞ aðmi Þ 1 X 1X 2 þ bi ðzÞ þ þ dzdQi ðzÞ gðmi ; mj ÞS j ðzÞ þ gðmi ; mj ÞQj ðzÞ dz 2 2 j¼1 2 j¼1 0 i¼1 !# Np Ns X X þ Qi gðmj ; mi ÞQj ðzÞbj ðzÞ 2 gðmj ; mi ÞS j ðzÞcj ðzÞ Ns Z X
L
j¼1
"
þ
Np X i¼1
"
j¼1
! Np Ns X X dci ðzÞ 2 þ ci ðzÞ aðmi Þ þ dzdS i ðzÞ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ dz 0 i¼1 j¼1 j¼1 # Np Ns Ns X X 1X 2 u ðGi GÞ þ þ ci ðLÞ dS i ðLÞ gðmj ; mi ÞS j ðzÞcj ðzÞ þ gðmj ; mi ÞQj ðzÞbj ðzÞ þ 2 j¼1 S i ð0Þ N s S i ð0Þ j¼1 i¼1 Ns Z X
L
½ðbi ðLÞ 2xQi ðLÞÞdQi ðLÞ bi ð0ÞdQi ð0Þ "
Z Z Z Ns Ns X ogðmi ; mk Þ L 1 oaðmk Þ L 1X ogðmk ; mj Þ L ci ðzÞQ2k ðzÞdz bk ðzÞQk ðzÞ þ bk ðzÞQk ðzÞS j ðzÞdz omk 2 omk 2 j¼1 omk 0 0 0 k¼1 i¼1 # Z Z L Np Ns 1X ogðmk ; mj Þ 1X ogðmi ; mk Þ L 2 2 þ bk ðzÞQk ðzÞQj ðzÞdz þ bi ðzÞQi ðzÞQk ðzÞdz ðA10 Þ 2 j¼1 omk 2 i¼1 omk 0 0
þ
Np X
dmk
A.R. Bahrampour et al. / Optics Communications 281 (2008) 1545–1557
In the above equation we have used the following by parts integration: Z L Z L ddQi ðzÞ db ðzÞ ¼ bi ðLÞdQi ðLÞ bi ð0ÞdQi ð0Þ dzbi ðzÞ dzdQi ðzÞ i ; i ¼ 1; 2; . . . ; N p dz dz 0 0 Z L Z L ddS i ðzÞ dci ðzÞ ¼ ci ðLÞdS i ðLÞ ci ð0ÞdS i ð0Þ ; i ¼ 1; 2; . . . ; N s dzci ðzÞ dzdS i ðzÞ dz dz 0 0
1555
ðA11 Þ ðA12 Þ
The input power of each signal is known, so we have dSi(0) = 0. Since Pi(z)(i = 1, 2, . . . , Np), Si(z)(i = 1, 2, . . . , Ns), E, D and Lagrange multipliers x and u are independent of each other, dJ can be zero if and only if the coefficients of variations (dy) are zero, where y can be each of the above variables. Therefore, in addition to the constraint Eqs. (A1), (A2), (A4) and (A5) the following equation are obtained as the necessary conditions for the optimal state (1) The set of differential equations:
! Np Ns X X dQi ðzÞ Qi ðzÞ 2 ¼ aðmi Þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQj ðzÞ ; dz 2 j¼1 j¼1
i ¼ 1; 2; . . . ; N p
! Np Ns X X dbi ðzÞ bi ðzÞ 2 þ aðmi Þ þ gðmi ; mj ÞQj ðzÞ þ gðmi ; mj ÞS j ðzÞ dz 2 j¼1 j¼1 ! Np Ns X X þ Qi ðzÞ gðmj ; mi Þbj ðzÞQj ðzÞ 2 gðmj ; mi ÞS j ðzÞcj ðzÞ ¼ 0; i ¼ 1; 2; . . . ; N p j¼1
ðA13 Þ
ðA14 Þ
j¼1
! Np Ns X X dS i ðzÞ 2 ¼S i ðzÞ aðmi Þ þ gðmi ; mj ÞS j ðzÞ þ gðmi ; mj ÞQj ðzÞ ; dz j¼1 j¼1
i ¼ 1; 2; . . . ; N s
ðA15 Þ
! Np Ns Ns X X X dci ðzÞ þ ci ðzÞ aðmi Þ gðmi ; mj ÞS j ðzÞ gðmi ; mj ÞQ2j ðzÞ gðmj ; mi ÞS j ðzÞcj ðzÞ dz j¼1 j¼1 j¼1 þ
Np 1X gðmj ; mi ÞQj ðzÞbj ðzÞ ¼ 0; 2 j¼1
i ¼ 1; 2; . . . ; N s
ðA16 Þ
(2) The boundary condition corresponding to the conjugate variables: bi ð0Þ ¼ 0; i ¼ 1; 2; . . . ; N p bi ðLÞ ¼ 2xQi ðLÞ; i ¼ 1; 2; . . . ; N p 2 u ðGi GÞ ; ci ðLÞ ¼ S i ð0Þ N s S i ð0Þ
ðA17 Þ ðA18 Þ i ¼ 1; 2; . . . ; N s
ðA19 Þ
And there is no restriction on the initial values of ci. (3) Switching equations: Dx ¼ 0 Eu ¼ 0
ðA20 Þ ðA21 Þ
(4) The set of algebraic equations: Ns X j¼1
Ajk
X Np ogðmj ; mk Þ ogðmk ; mj Þ ogðmk ; mj Þ oaðmk Þ þ Bkj C kj þ Dk þ ¼0 omk omk omk omk j¼1
ðA22 Þ
k ¼ 1; 2; . . . ; N p where Aik, Bkj, Ckj, Dk are defined by the following integrals: Z L Aik ¼ ci ðzÞQ2k ðzÞdz 0 Z 1 L b ðzÞQk ðzÞS j ðzÞdz Bkj ¼ 2 0 k
ðA23 Þ ðA24 Þ
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1 2
C kj ¼
Z
1 Dk ¼ 2
L
bk ðzÞQk ðzÞQ2j ðzÞdz
ðA25 Þ
0
Z
L
bk ðzÞQk ðzÞ
ðA26 Þ
0
Appendix B Eqs. (A20) and (A21)make four possible states for any optimization problem. Each state is specified by a two-dimensional vector with components zero or one which are called binary vectors and are denoted by bi (i = 1,2,3,4). The zero and one values of the first component of bi are correspond to D = 0 and x = 0 respectively. The E = 0 and u = 0 correspond to the 0 and 1 values for the second component of bi, respectively. Each state satisfies Eqs. (A4), (A5), (A13)–(A21). This leads to a system of algebraic equations. By using above definition for the binary vectors, we have: (1) b1 ¼
0 0
)
8 Np > < P Q2 ðLÞ ¼ P ^
^i
max
i¼1 > : G ¼ G0
ðB1 Þ
(2) b2 ¼
1
) G ¼ G0
ðB2 Þ
Np X 0 b3 ¼ ) Q^2i ðLÞ ¼ P max 1 ^i¼1
ðB3 Þ
1 ) no special boundary condition b4 ¼ 1
ðB4 Þ
0
(3)
(4)
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[16] Donald E. Kirk, Optimal Control Theory: An Introduction, Prentice Hall, 1970. [17] S. Wang, C. Fan, Distributed fiber Raman amplifiers: analytical expression of noise characteristics under complex conditions, J. Opt. Commun. 198 (October) (2001) 65. [18] X. Liu, B. Lee, A fast and stable method for Raman amplifier propagation equations, J. Opt. Express 11 (18) (2003) 2163. [19] G. Agrawal, Nonlinear Fiber Optics, Elsevier, 2001 (Chapter 8). [20] C. Headely, G. Agrawal (Eds.), Raman Amplification in Fiber Optical Communication Systems, Elsevier Academic Press, 2005, ISBN 0-12-044506-9 (Chapter 2). [21] S.M. Kobtsev, A.A. Pustovkikh, Fiber Raman amplifier pumped by continuous–spectrum radiation, J. Quant. Electron. 34 (6) (2004) 575.