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The moment method for fiber Raman amplifier gain ripple minimization
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Optics Communications 281 (2008) 3673–3680 www.elsevier.com/locate/optcom
The moment method for fiber Raman amplifier gain ripple minimization A.R. Bahrampour a,b,*, F. Farman a, A. Ghasempour a a
Department of Physics, Sharif University of Technology, Room 328, P.O. Box 11365-9161, Azadi Avenue, Tehran, Iran b Valiasr University, Rafsanjan, Iran Received 20 November 2007; received in revised form 10 March 2008; accepted 10 March 2008
Abstract We aim to propose a novel fiber Raman amplifier modeling based on the moment method, which is previously introduced for modeling the inhomogeneous Erbium doped fiber amplifiers and recently employed to analyze the fiber Raman amplifier with continuous pump spectrum. In this model, the number of governing equations is independent of the number of signals and according to the degree of accuracy it is proportional to the number of pumps. This method is employed to analyze the Raman fiber amplifiers with an arbitrary input signal line shape and to minimize the gain ripple of the fiber Raman amplifier with respect to the pump powers and pump frequencies. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Multi-wavelength pump fiber Raman amplifiers (FRA) can increase the transmission bandwidth. While multi-wavelength pumps can be used to equalize gain over a large bandwidth, flattening of the gain in wideband Raman amplifiers is one of the most important tasks for the designers of Raman amplifiers [1–5]. By a proper choice of the pump powers and wavelengths it is possible to reduce the gain-ripple of the corresponding Raman gain in a broad wavelength range. The statistics reported in [6–8] shows that more and more FRAs are adapted in experimental systems to enhance the system performance as the pumping technique matures [9,10]. The amplifier performance, such as the available bandwidth [11], gain flatness [12], optical signal to noise ratio (OSNR) [13–15] and nonlinear penalty, is greatly affected by the pump configuration, including pump powers, pump wavelengths, pumping scheme and so on. * Corresponding author. Address: Department of Physics, Sharif University of Technology, Room 328, P.O. Box 11365-9161, Azadi Avenue, Tehran, Iran. Tel.: +98 21 6616 4527; fax: +98 21 66022711. E-mail addresses: [email protected] (A.R. Bahrampour), [email protected] (F. Farman), [email protected] (A. Ghasempour).
0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.03.040
For the optimal design of the Raman amplifier, various approaches have been proposed for the efficient and accurate modeling of the Raman amplifier [16,17]. Several search algorithms, such as simulated annealing (SA) [18], genetic algorithm (GA) [19–22], neural network [23] and pump power integral method [24], were used to search the pump powers and wavelength pairs. During the search process, the pump powers and wavelength pairs were directly substituted into the governing fiber Raman equations to calculate the gain profile. The governing equations are a system of coupled nonlinear ordinary differential equations with two-point boundary values for the pump and signal powers at the ends of the fiber. The method of two-point boundary value problem solving was involved in a large amount of numerical computations, which was inevitably time consuming and rarely got the real optimal solution in real time. When the line shape of input channels is taken into consideration the system of governing equations is a system of uncountable nonlinear differential equations [3]. The moment method [24–26] is used to get a discrete and reduced number of governing equations and also to decouple the pump and signal equations. In reference [26] authors employed the moment method to obtain the gain ripple in Raman fiber amplifiers with continuous pump spectrum. In this paper, we employ the
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moment method for continuous signal spectrum, in addition to considering the effect of line shape, on the output parameters. The number of governing equations of twopoint boundary value problem is reduced to a number proportional to the number of pumps which is less than the number of signals. While in the continuous pump spectrum in reference [26] the number of governing equations is proportional to the number of signals which is greater than the number of pumps. In this paper, in order to reduce the time of calculation, the integral form for expansion of the pump equations [16] is employed. Also in this paper, the moment method is employed to minimize the gain ripple in a WDM pump Raman fiber amplifier. This paper is organized into seven sections as follows. Section 2 introduces governing equations, and is followed by the moment method and the corresponding equations, in Section 3. The gain spectrum of the amplifier is presented in Section 4, and in Section 5 the gain ripple minimization is described in detail. The numerical results and discussion will be given in Section 6 with some calculations presented in Section 7. Finally two numerical methods for calculating the average value of signal and Raman tilt along the fiber length is illustrated in Appendices A and B, respectively.
where S(m,z) is the signal power spectrum and P(mi,z) is the pump power of frequency mi, the frequencies are numerated in decreasing order. The function a(m) is the fiber loss rate at frequency m. The gain coefficient g(m,m0 is equal to gR(m,m0 )/CAeff, (m0 /m)gR(m,m0 )/C Aeff and 0, for mhm0 , mi m0 and m = m0 , respectively. gR(m, m0 ) is the Raman gain coefficient from the wave of frequency m to the frequency m0 . The factor C accounts for the polarization randomization effects, Aeff is the effective area of the optical fiber. ml and mh are the lower and upper band of the input signal spectrum, respectively and N is the number of pumps. The number of equations corresponding to Eq. (1) is the same as the number of real numbers. The system of governing equations is a system of uncountable coupled nonlinear differential equations. 3. Moments and their evolution equations The moment method [24–26] is employed to reduce the system of governing equations into a countable system of equations. The linear operators F mk1 ...kl : L2 ðRÞ ! L2 ðRÞ are defined as follows: Z 1 ðmÞ F k1 ;...kl ½q ¼ am ðm0 Þgðm; m0 Þgðm0 ; mk1 Þ . . . gðm0 ; mkl ÞS 0
ðm0 ; zÞqðm0 ; zÞdm0
2. Governing equations In calculating the gain profile of the distributed fiber Raman amplifier (DFRA), noise effects such as spontaneous Raman scattering, Rayleigh back scattering and thermal factor etc. can be neglected. The interactions of pump to pump, signal to signal, pump to signal and the attenuation of pump and signal as the major effects are taken into consideration. Even we assume that the laser line shape are negligible and can be approximated by a sum of Dirac delta functions, the spectrum of the modulated signal is the convolution of the signal spectrum and the laser spectrum. Hence, the bandwidth of the modulated signal at least is equal to the signal bandwidth. As an example for each STM 256 channel this bandwidth is more than 40 GHz and the signal spectrum can be approximated by a sum of Lorentzian line shape. Hence, in this work it is assumed that the spectrum of signal power is continuous. In the steady state, the coupled nonlinear equations for stimulated Raman gain spectrum can be described as [3] " X dSðm; zÞ ¼ Sðm; zÞ aðmÞ þ gðm; mi ÞP ðmi ; zÞ dz i Z mh þ Sðm0 ; zÞgðm; mÞdm0 m 2 ½ml ; mh ð1Þ ml " X dP ðmi ; zÞ ¼ P ðmi ; zÞ aðmi Þ gðmi ; mj ÞP ðmj ; zÞ dz j Z mh þ Sðm0 ; zÞgðmi ; m0 Þdm0 i ¼ 1; . . . ; N ð2Þ ml
ð3Þ
where q(m,z) is an arbitrary function of S(m,z). The integral part on the right hand side of Eq. (1) is given by qð0;0Þ ðm; zÞ ¼ F ð0Þ ½1
ð4Þ
(0,0)
The moment q (m,z) is denoted by Q(m,z) and moment ðm;nÞ functions qk1 ;...:kl ðm; zÞ are defined by the following relation ðm;nÞ
ðmÞ
qk1 ;...:kl ðm; zÞ ¼ F k1 ...kl ½Qn
ð5Þ
The governing equations for moment functions are easily obtained:
ðm;0Þ qk1 ;...:kl ðm; zÞ
ðm;0Þ
N X dqk1 ;...kl ðm; zÞ ðmþ1;0Þ ðm;0Þ ¼ qk1 ;...kl ðm; zÞ þ P i ðzÞqk1 ;...kl ;ki ðm; zÞ dz i¼1 ðm;1Þ
þ qk1 ;...kl ðm; zÞ
ð6Þ
It is shown that the right hand side of Eq. (6) decreases and rapidly approaches zero by increasing each of the indices (m,n,l) of the moment functions. More generally, the governing equations of moment functions qm;n k 1 ;...:k l ðm; zÞ are as follows ðm;nÞ
N X dqk1 ;...kl ðm;zÞ ðmþ1;nÞ ðm;nÞ ¼ qk1 ;...kl ðm; zÞ þ P i ðzÞqk1 ;...kl ;ki ðm;zÞ dz i¼1 ( ðm;nþ1Þ
ðmÞ
þ qk1 ;...kl ðm; zÞ þ n F k1... kl ½Qn1 qð1;0Þ þ
X
) ðmÞ ðmÞ pi ðzÞF k1... kl ½Qn1 qð0;0Þ þ F k1... kl ½Qn1 qð0;1Þ
i
ð7Þ
A.R. Bahrampour et al. / Optics Communications 281 (2008) 3673–3680 ðmÞ
Since F k1... kl is a linear operator, the governing equations of ðmÞ F k1... kl ½qðm; zÞ can be written easily. The effects of signal to signal interactions are presented in Q(m, z) which is defined by Eq. (4) and appeared in the moment functions. The moment functions are symmetric with respect to all the lower indices. The governing equations are rewritten with respect to the moment functions: " # X dSðm; zÞ ¼ Sðm; zÞ aðmÞ þ gðm; mi ÞP ðmi ; zÞþQðm; zÞ dz i m 2 ½ml ; mh "
X dP ðmi ; zÞ ¼ P ðmi ; zÞ aðmi Þ gðmi ; mj ÞP ðmj ; zÞQðmi ; zÞ dz j i ¼ 1; . . . ; N
#
ð8Þ
ð9Þ
The boundary conditions for the governing equations are Sðm; 0Þ ¼ S 0 ðmÞ P ðmi ; LÞ ¼ P iL
ð10Þ ð11Þ
where PiL is the ith pump input power and S0(m) is the line shape of input signals and depends on the assumption. For example S0(m) can be a sum of Lorentzian line shapes corresponding to each channel, a flat spectral density for simplicity of calculations and a sum of delta functions for discrete input signal spectrum. Initial conditions corresponding to the moment funcðm;nÞ tions qk1 ;...kl ðm; 0Þ are easily obtained by the following relation: Z mh ðm;nÞ am ðm0 ÞQn ðm0 ; 0ÞSðm0 ; 0Þgðm; m0 Þ qk1 ...kl ðm; 0Þ ¼ ml
gðm0 ; mk1 Þ . . . gðm0 ; mkl Þdm0
ð12Þ
The initial value of the other moment functions can be obtained by the same method. 0 Since the magnitudes of, g(m,m ), Q(m) and a(m) are smaller than one, the moment functions are decreasing functions with respect to the indices (m,n,l) and rapidly converge to zero as the indices (m,n,l) increase. Because 0 the order of the magnitude of g(m,m ) and Q(m) are less than that of a(m), the rate of convergence with respect to the number of g(m,m0 ) and powers of Q(m) are faster than that of the powers of a(m). The system of Eq. (7) at the pump frequencies (m = mi, i = 1, . . . ,N) and (9) are explicitly independent of the signal Eq. (6) and are solved by truncating the system of moment equations by a finite number of indices (m,n,l). Variation of the pump powers P(mi,z) (i = 1, . . . ,N) are simply obtained by the solution of a system of two-point boundary value problem. Higher order truncation gives precise solution for the pumps variation along the fiber length. First order truncation of moment function equations are reduced to a system of 2N differential equations. Numerical calculations show that the first order truncation is accurate for the Raman fiber amplifier design.
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4. Gain spectrum of the amplifier Variations of pump powers along the fiber length are obtained by solving the moment Eq. (7) at pump frequencies (m = mi,i = 1, . . .,N) and pump Eq. (9). The system of uncountable governing equations is converted to a system of countable differential equations by using the moment functions. The system of countable equations of moment functions is truncated with respect to the indices (m,n,l) to obtain a system of a finite number of differential equations. As it is expected, calculations show that the speed of convergence is very fast. The precise pump evolution along the fiber can be obtained by the solution of a system of 2N nonlinear differential equations. Of course, higher order moments give more precise solutions for the pumps evolution along the fiber length. In order to have a gain spectrum of the FRA by the moment method, the pumps distribution along the fiber length which is obtained in the previous step will be applied to the truncated moment equations for any m 2 R and the system of the initial value problem will be solved (Appendix B). The numerical calculations show that the solution of the moment equations is in agreement with that obtained by the numerical method introduced in Appendix A. By solving the truncated system of equations, the Q(m,z) will be found. In order to have the gain spectrum, results are substituted in Eq. (8): GsðmÞ ¼ ð10loge10 ÞLðaðmÞ þ
N X
gðm; mi ÞP i þ QðmÞÞ
ð13Þ
i¼1
where P i and QðmÞ are the average value of Pi(z) and Q(m,z) along the fiber length, respectively. 5. Gain ripple minimization Since Q(m,z) is a complicated function of input pump powers Pi (L) and pump frequencies mi, the optimization of gain ripple is a complicated variational problem. In the case of negligible QðmÞ, some optimization methods for discrete spectrum signals are presented [3]. The numerical calculations show that the signal to signal interaction in Eq. (13) has perturbation effects on the gain spectrum of the FRA. Of course, effects of signal to signal interaction on the pump and signal behaviors along the fiber length are also considered in the moment equations. The signal exponential gain spectrum per unit length is written as follows: N X 1 Sðm; LÞ GeðmÞ ¼ Ln ¼ aðmÞ þ gðm; mi ÞC i þ QðmÞ L Sðm; 0Þ i¼1
m 2 ½ml; mh
ð14Þ
where Ci is the average value of the ith pump power along the fiber length. The variance of the gain spectrum is
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chosen as a scaling factor (cost function) for the gain ripple specification Z mh 2 ½GeðmÞ G0 ðmÞ dm ð15Þ V ¼ ml
where G0(m) is the target gain spectrum. The values of Ci are always positive. Now the problem is to pick up the values of Ci and mi which produce a gain that minimizes the RHS of (15). By neglecting the dependence of QðmÞ on the Ci and mi(i = 1, . . . ,N), picking the best Ci s which give a set of average powers, is fairly straightforward because (15) is just a quadratic function of Cis. Picking the best set of frequencies is a more complicated nonlinear problem. The Lagerange multiplier method can be employed to the minimization problem if there is any constraint on the pump powers or pump frequencies. Otherwise, the necessary condition for the minimization of the gain ripple (V) is obtained by taking the derivative with respect to mj and Cj(j = 1, . . . N) and putting the results equal to zero. Z mh N X gðm0 ; mj Þ½aðm0 Þ þ gðm0 ; mi ÞC i þ Qðm0 Þ ml
Z ml
i¼1
mh
G0 ðm0 Þdm0 ¼ 0j ¼ 1; . . . N N X ogðm0 ; mj Þ ½aðm0 Þ þ gðm0 ; mi ÞC i þ Qðm0 Þ omj i¼1 G0 ðmprime Þdmprime ¼ 0j ¼ 1; . . . N
ð16Þ
ð17Þ
The frequencies and average powers corresponding to the optimal pumps and optimum gain spectrum are obtained by solving the system of algebraic Eqs. (16) and (17). To obtain the pump powers, the pump Eq. (9) is integrated over the fiber length and rewritten as follows: " # gðmX i ;mk ÞC k P i ðLÞ ¼ exp ai L Qðmi Þ ð18Þ P i ð0Þ k The QðmÞ and Qðmi Þ are unknown and have perturbative effects in Eqs. (16)–(18). There are several methods for the initial guess of QðmÞ. In this analysis, for the initial guess we have assumed that in the first half length of the fiber, the main process is the fiber intrinsic loss and in the last half length the Raman amplification is the main process. In this step of optimization, it is assumed that the Raman exponential gain per unit length is constant and its value is determined by the target net gain of the FRA. Now, the RHS of Eq. (18) is completely specified, hence the ratio of input pump powers to the output pump powers is completely specified. The system of pump equations is converted to a two-point boundary value problem which can be solved by a standard nonlinear shooting method or the integral form expansion method [16]. Of course, the QðmÞ can also be determined by the numerical method presented in Appendix A. The new QðmÞis employed in Eqs. (16)–(18) to obtain the new optimal solution with a higher degree of precision. This process is continued up to obtain the desired degree of accuracy.
6. Numerical results The procedure to practically implement the above derived analytic formulation into numerical domain depending on their applications can be summarized as follows: 6.1. Gain prediction Step 1. Moment equations step The pump power distributions are obtained by solving the moment Eqs. (7) and (9). Depending on the order of the magnitude of a(m), Q(m) and g(m,m0 ), integers (m,n,l) are chosen and the system of Eq. (7) is truncated. By choosing the input signal spectrum S(m,0) and using Eq. (12) the initial values of moment functions can be determined. In the first step of truncation, the system of Eqs. (B.1) and (B.2) is a system of two-point boundary value problem which can be solved by the standard nonlinear shooting method or the iteration methods presented in Appendix A or Appendix B. In this paper, by employing an initial guess forQ(mi,z), the two-point boundary value problem is broken in two separate initial and final value problems which are solved by the fourth order Runge Kuta method. Higher precision solutions are obtained by a higher order of truncation of moment equations. Results are employed in Eqs. (B.1) and (B.2), using the iterative procedure the precise average values of pumps and moments along the fiber length can be determined. In this method the number of equations is proportional to the number of pumps, while in direct method the number of equations is proportional to the number of signals. Our numerical calculations show that the precision of the first order truncation is enough for the FRA calculations. Step 2. Gain spectrum calculation The gain spectrum is determined by Eq. (14). The average values of the pump powers and moment QðmÞ are determined by solving the pump and moment equations by the iterative method. The initial guess for signal to signal interaction term (Raman tilt) is determined by the method presented in Section 5.To understand the speed of convergence in the moment method, the gain spectrum at different moment order is calculated and compared with each other. In Fig. 1, the results corresponding to the three first orders of approximation for a system of flat input spectral intensity are compared with that of a system of discrete input spectrum. It is assumed that 50 signals are at the wavelength of 1528–1578 nm with 1 nm spacing at the signal power of 10 dB m/ch. For the counter directional pumping scheme, we used 20 pumps (1408–1490 nm). We employed 25 km
A.R. Bahrampour et al. / Optics Communications 281 (2008) 3673–3680
150
2.1
100
2
Input pump power,mW
Gain,dB
2.2
90
a effective area
attenuation cofficient
0.1
80
0.05
Effective area, μm2
200 pump power
first iteration second iteration discrete zeroth iteration
Attenuation cofficint,1/km
2.3
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50
70
1.9 1400
1450
1500
1550
0 1400
0 1600
1450
Wavelength,nm
of single mode fiber (SMF-28) as the test gain medium. Corresponding pump powers and their wavelengths are also presented in Fig. 1. The fiber loss spectrum, effective core area and Raman gain spectrum of (SMF-28) fiber are presented in Fig. 2. As it is obvious in Fig. 1, the suggested method shows that there is an excellent rate of convergence and accuracy in the first order of truncation. The accuracy improvement from the next order of truncation becomes less than (0.008 dB).Meanwhile, the required computation time to get the exact Raman gain and signal/pump powers distribution was less than a second for the counter directionally pumping configurations, with a conventional PC (2. GHz CPU clock). Fig. 3 shows the pump and signal spectrum evolution along the fiber. The excellent agreement between the results of the simulation and the conventional method shows the validity of our suggested algorithm. 6.2. Raman gain ripple minimization Step1. Average pump powers and pump frequencies Eq. (14) shows that the exponential gains are linear functions of average pump powers. It is noted that pumps are also appeared implicitly in the average moment QðmÞ, hence the solution of optimal problem is complicated and can be obtained by the variational method. Fortunately, as it is shown in previous section, the average signal spectrum has perturbation effects on the exponential gain spectrum. In this analysis, the approximated relation described in Section 5 is employed for Raman tilt calculation. We start the gain ripple minimization by defining the target gain spectrum G0 under a given set of constraints. The gain spectrum, average pump powers and pump frequencies corresponding to the optimal conditions, are obtained
1550
1600
−14
x 10
b 6
Raman gain cofficient,m / W
Fig. 1. Comparison of the gain spectrum corresponding to the first three order of approximation for a flat input spectral density in a 25 km SMF-28 Raman fiber amplifier shows the rapid speed of convergence of the model.
1500
Wavelength,nm
5
4
3
2
1
0 0
5
10
15
20
25
30
35
Pumpsignal frequency difference,THz
Fig. 2. Characteristics of single mode fiber (SMF-28). (a) The loss spectrum and effective core area [17]. (b) Raman gain spectrum [27].
by solving the system of algebraic Eqs. (16) and (17). The results that are obtained by the iterative method are presented in Appendices A and B of this paper. Step 2. Pump calculations In the first order of approximation, the moment function q(0,0)(m ,z) = Q(m,z) is determined by the initial guess. The average values of the moment ð0;0Þ functions qi are obtained with respect to the initial value of the signal spectrum. The ratios of the initial value to the final value of pumps are ð0;0Þ obtained by substituting qi and the optimal Ci into Eq. (18). In order to get the input power of pumps, the system of two-point boundary value problem of pumps (9) is easily solved. The results of a minimization problem with respect to the pump powers are presented in Fig. 3. We set net gain G0 to 0.8 dB over the 70 nm (1520–1590 nm) spectral range. Considering practical restrictions in pump WDM wavelength allocation, 15 equally spaced, counter-directional pumps at 1400–1490 nm. Calculations are done for a
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a
A.R. Bahrampour et al. / Optics Communications 281 (2008) 3673–3680
a
0.9 0.85
0.5 0
0.8 −0.5 −1
0.7
Gain,dB
Gain ,dB
0.75
0.65 first iteration second iteration zeroth iteration discrete
0.6 0.55
−2 −2.5
0.5
−3
0.45 0.4 1510
first iteration second iteration zeroth hteration discrete
−1.5
1520
1530
1540
1550
1560
1570
1580
1590
−3.5 1520
1600
1530
Wanelength,nm
1540
1550
1560
1570
1580
1590
Signal wavelength,nm
b 140
b
100 90
120
Input pump power,mW
Input pump power,mW
80
100
80
60
40
70 60 50 40 30 20
20
10
0 1400
1420
1440
1460
1480
1500
Wavelength,nm
Fig. 3. Gain optimization of a 50 km SMF-28 Raman fiber amplifier with respect to the pump powers and uniformly distributed 15 pumps over the 1408–1490 nm pump bandwidth. (a) The gain spectrum of the Raman fiber amplifier. (b) The input pump powers.
50 km SMF-28 Raman fiber amplifier. After the first step, the pump evolution sufficiently converges to that of a real value and the gain curve extremely overlaps that of an ideal target gain. The pump powers spectrum corresponding to the optimal condition is presented in Fig. 3b. To consider the effect of iteration steps on the minimization process, the gain profiles corresponding to the different steps of iteration are presented in Fig. 3a. The convergence error from the iteration procedure becomes almost negligible after the first iteration step. The observed absolute error between target gain profile and iteratively obtained gain profile after the first iteration step was less than 0.06 dB over the 70 nm gain spectral range. A fiber Raman amplifier with eight pumps is optimized with respect to the pump frequencies and pump powers. Results are presented in Fig. 4. Obviously, by increasing the number of pumps, the optimum gain approached the gain profile corresponding to the uniform distributed pumps. Naturally, the level of gain accuracy could be further improved with additional level of moment equations and
0 1380
1400
1420
1440
1460
1480
1500
Wavelength,nm
Fig. 4. Gain optimization of a 50 km SMF-28 Raman fiber amplifier with respect to the wave length of pumps and pump powers. (a) The optimal gain spectrum of the Raman fiber amplifier. (b) The input pump spectrum.
iteration steps. The convergence rate of gain spectrum profile is measured in the sense of L2 space metric; by considering most of practical applications under experimental error, the first step of approximation was generally sufficient. The required computation time to get the result was less than 8 s with a conventional PC (2.0 GHz CPU clock). 7. Conclusion The moment method which has been previously employed to analyze the inhomogeneous Erbium doped fiber amplifier (EDFA) and continuous pump spectrum Raman fiber amplifier, is applied to the RFA with continuous input spectrum. The number of governing equations for a RFA with continuous spectrum input signals is uncountable. By using the moment method the system of uncountable differential equations reduces to a countable system of ordinary differential equations. By truncating the system of moment equations, the number of governing
A.R. Bahrampour et al. / Optics Communications 281 (2008) 3673–3680
equations is reduced to a number of equations proportional to the number of pump equations. The speed of convergence of the signal continuous spectrum moment method is faster than that of the pump continuous spectrum method which has been presented in our previous publication [26]. Calculations show that the presented method is a precise and fast method.
ðnÞ
Rij ¼ where
j X
3679
ðn1Þ
ðA:6Þ
Ail Blj
l¼0 ðn1Þ Bil
is defined by the following matrix 1 ðnÞ Gik Rkl X A ¼ exp @ 0
ðnÞ
Bil
ðA:7Þ
k
Appendix A. An efficient numerical method for calculation of SðmÞ
Using these formulations, the signal average power for each frequency can be obtained with matrix multiplication procedure
The gain spectrum of the amplifier is obtained by the solution of Eq. (8). After dividing Eq. (8) by S(m,z) and integrating over z we get " Z z N X Sðm; zÞ ¼ Sðm; 0Þ exp aðmÞz þ gðm; mi Þ P ðmi ; z0 Þdz0
1 Sðmi Þ ¼ RiN ðA:8Þ L where Sðmi Þ is the average value of signal along the fiber length at frequency mi. The exponential gain spectrum is given by
þ
Z
1 0
gðm; m Þdm
0
i
Z
0
z 0
0
Sðm ; z Þdz
0
0
ðA:1Þ
0
With an additional integration process; we get the following integral form of Raman wave equations: " Z Z z N X Sðm; z0 Þdz0 ¼ Sðm; 0Þ dz0 aðmÞz þ gðm; mi Þ 0
Z
Z
z0
0 z0
P ðmi ; z00 Þdz00 þ # 0
00
i¼1
Z
1
Sðm ; z Þdz
ðA:2Þ
Rz By the definition of Rðm; zÞ ¼ 0 Sðm; z0 Þdz0 , we can rewrite Eq. (A.2) into a closed integral form for the R(m,z) Z 1 Z z Rðm; zÞ ¼ Aðm; z0 Þ exp gðm; m0 ÞRðm0 ; z0 Þdm0 dz0 ðA:3Þ 0
Where A(m,z) is given by the following relation " # Z z N X gðmi ; mÞ P ðmi ; z0 Þdz0 Aðm; zÞ ¼ Sðm; 0Þ exp aðmÞz þ 0
ðA:4Þ To solve Eq. (A.3), we employ the Picard iteration method, taking R(m,z) as the interim target solution. At nth iteration step, Eq. (A.3) becomes Z mh Z z RðnÞ ðm; zÞ ¼ Aðm; z0 Þ exp gðm; m0 ÞRðn1Þ ðm0 ; z0 Þdm0 dz0 0
ðA:9Þ
j
where the P j is the average pump power along the fiber length at the frequency mj. Of course the Q(m,z) can be obtained directly by solving Eq. (B.2) at the arbitrary frequency mm 2 [ml,mh]; results are in agreement with those obtained by this method. Appendix B. A semi analytical solution for QðmÞ
00
i¼1
j
gðm; m0 Þdm0 0
0
0
1 Sðm; LÞ GeðmÞ ¼ Ln L Sðm; 0Þ X X ¼ aðmÞ þ gðm; mj ÞP j þ gðm; mj ÞRjN
ml
ðA:5Þ In order to implement the above equations into the numerical domain, we now construct matrices Rij R(mi,zj) and Aij A(mi,zj) to assign the value of R and A at the ith frequency at the position zj (the whole fiber link is covering by discrete position elements with step size of Dz). The matrix Gij = g(mi,mj) is defined as the value of Raman gain coefficient at the frequencies mi and mj.
To calculate the Q(m,z) and its average value along the fiber length ðQðmÞÞ an iterative method can be employed. The governing equations corresponding to the pumps and moment Q(m, z) = q(0,0)(m,z) evolution along the fiber are as follows N X dP ðnÞ ðmi ; zÞ ¼ P ðnÞ ðmi ; zÞ½aðmi Þ gðmi ; mj ÞP ðnÞ ðmj ; zÞ dz j¼1 Qðnþ1Þ ðmi ; zÞ
ðB:1Þ
N X dQðnÞ ðm; zÞ ðn1Þ ð0;0Þ ¼ qð1;0Þ ðm; zÞ þ P j ðzÞqj ðm; zÞ dz j¼1
þ qð0;1Þ ðm; zÞ (n)
ðB:2Þ (n)
where the Q (m,z) and P (mi,z) are Q(m, z) and P(mi,z) at the nth iteration step. Since the derivative of the moments, ð0;0Þ q(1,0)(m,z), qj ðm; zÞ and q(0,1)(m,z) are functions of higher order moments, the variations of these moments along the fiber length are very slow and it is assumed that they are independent of z which is confirmed by the numerical calculations. Q(m,z) is obtained by integrating both side of the Eq. (B.2): N Z z X ðn1Þ ð0;0Þ ðnÞ ð0;0Þ Q ðm; zÞ ¼ q ðm; 0Þz þ P j ðz0 Þqj ðm; 0Þdz0 j¼1
þq
ð0;1Þ
ðm; 0Þz
0
ðB:3Þ
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The average value of Q(n)(m,z) along the fiber length is Z z N X L ð0;0Þ ðn1Þ QðnÞ ðmÞ ¼ qð0;0Þ ðm; 0Þ þ qj ðm; 0Þ P j ðz0 Þdz0 2 0 j¼1 L þ qð0;1Þ ðm; 0Þ 2
ðB:4Þ
References [1] Victor E. Perlin, Herbert G. Winful, Journal of Light Wave Technology 20 (3) (2002). [2] S.M. Kobtsev, A.A. Pustouskikth, Laser Physics 14 (12) (2004) 1488. [3] Andrew R. Grant, IEEE Journal of Quantum Electronics 38 (11) (2002). [4] He Wen, Xin Yang, Wanyi Gu, Journal of Light Wave Technology 23 (4) (2005). [5] S. Martin-Lopez, M. Gonzalez-Herraez, P. Corredera, M.L. Hernanz, A. Carrasco, Optic Communications 242 (2004) 463. [6] J. Bromage, Journal of Light Wave Technology 22 (1) (2004) 79. [7] Evgeny M. Dianov, Journal of Light Wave Technology 20 (8) (2002) 1457. [8] mohammad N. Islam, IEEE Journal of Selected Topics in Quantum Electronics 8 (3) (2002). [9] R.P. Epsindola, K.L. Bacher, K. Kojima, N. Chand, S. Srinivasan, G.C. Cho, F. Jin, C. Fuchs, V. Millner. W.C. Dautremont-Smith, High power, low RIN spectraly broadened 14xx DFB pump for application in co-pumped Raman amplification, Presented at the EOR.Conference, Optical Communications, 2001, Paper P.D.F.H.7. [10] M.D. Mermelstein, C. Horn, Z. Huang, M. Luvalle, J.C. Bouteiller, C. Headley, B.J. Eggleton, Configurability of a three-wavelength Raman fiber laser for gain ripple minimization and power partitioning, Presented at the Optical Fiber Communication Conference, 2002, Paper TUJ2.
[11] P.C. Reeves- Hall, D.A. Chestnut, C.J.S. De Matos, J.R. Taylor, Electronics Letters 37 (14) (2001) 883. [12] Y. Emori, S. Namiki, 100 nm bandwidth flat gain Raman amplifiers pumped and gain-equalized by 12 wave length-channel WDM high power laser diodes, Presented at the Optical Fiber Communication Conference, 1999, Paper PD19. [13] S. Kado, Y. Emori, S. Namiki, Gain and noise tilt control in multiwavelength bi-directionally pumped Raman amplifier, Presented at the Optical Fiber Communication Conference, 2002. Paper TUJ4. [14] J.D. Ania-Castanon, S.K. Turitsyn, Optic Communications 224 (2003) 107. [15] Sihai Wang, Chongcheng Fan, Optic Communications 198 (2001) 65. [16] Namkyoo park, Pilhan Kim, Jaehyoung Park, Lark Kwon Choi, Optical Fiber Technology 11 (2005) 111. [17] S.M. Kobtsev, A.A. Pustovskikh, Quantum Electronics 34 (6) (2004) 575. [18] M. Yan, J. Chen, W. Jiang, J. Li, J. Chen, X. Li, IEEE Photonics Technology Letters 13 (9) (2001) 948. [19] X. Zhou, C. Lu, P. Shum, T.H. Cheng, IEEE Photonics Technology Letters 13 (9) (2001) 945. [20] V.E. Perlin, H.G. Winful, Journal of Light Wave Technology 20 (2) (2002) 250. [21] X. Liu, B. Lee, Journal of Light Wave Technology 21 (12) (2003) 3446. [22] Xueming Liu, Yanhe Li, Optics Express 12 (4) (2004) 564. [23] P.C. Xiao, Q.J. Zeng, J. Huang, J.M. Liu, IEEE Photonics Technology Letters 15 (2) (2003) 206. [24] A.R. Bahrampour, M. Mahjoei, Journal of Lightwave Technology 19 (8) (2001) 1130. [25] A.R. Bahrampour, Sh. Keivaninia, M. Karvar, Optical Fiber Technology, in press. [26] A.R. Bahrampour, M.H. Teimourpour, L. Rahimi, Optic Communications 281 (2008) 587. [27] C. Headly, G. Agrawal, Elsevier Academic Press, ISBN 0-12-0445069, 2005 (Chapter 2).
Optics Communications 281 (2008) 3673–3680 www.elsevier.com/locate/optcom
The moment method for fiber Raman amplifier gain ripple minimization A.R. Bahrampour a,b,*, F. Farman a, A. Ghasempour a a
Department of Physics, Sharif University of Technology, Room 328, P.O. Box 11365-9161, Azadi Avenue, Tehran, Iran b Valiasr University, Rafsanjan, Iran Received 20 November 2007; received in revised form 10 March 2008; accepted 10 March 2008
Abstract We aim to propose a novel fiber Raman amplifier modeling based on the moment method, which is previously introduced for modeling the inhomogeneous Erbium doped fiber amplifiers and recently employed to analyze the fiber Raman amplifier with continuous pump spectrum. In this model, the number of governing equations is independent of the number of signals and according to the degree of accuracy it is proportional to the number of pumps. This method is employed to analyze the Raman fiber amplifiers with an arbitrary input signal line shape and to minimize the gain ripple of the fiber Raman amplifier with respect to the pump powers and pump frequencies. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Multi-wavelength pump fiber Raman amplifiers (FRA) can increase the transmission bandwidth. While multi-wavelength pumps can be used to equalize gain over a large bandwidth, flattening of the gain in wideband Raman amplifiers is one of the most important tasks for the designers of Raman amplifiers [1–5]. By a proper choice of the pump powers and wavelengths it is possible to reduce the gain-ripple of the corresponding Raman gain in a broad wavelength range. The statistics reported in [6–8] shows that more and more FRAs are adapted in experimental systems to enhance the system performance as the pumping technique matures [9,10]. The amplifier performance, such as the available bandwidth [11], gain flatness [12], optical signal to noise ratio (OSNR) [13–15] and nonlinear penalty, is greatly affected by the pump configuration, including pump powers, pump wavelengths, pumping scheme and so on. * Corresponding author. Address: Department of Physics, Sharif University of Technology, Room 328, P.O. Box 11365-9161, Azadi Avenue, Tehran, Iran. Tel.: +98 21 6616 4527; fax: +98 21 66022711. E-mail addresses: [email protected] (A.R. Bahrampour), [email protected] (F. Farman), [email protected] (A. Ghasempour).
0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.03.040
For the optimal design of the Raman amplifier, various approaches have been proposed for the efficient and accurate modeling of the Raman amplifier [16,17]. Several search algorithms, such as simulated annealing (SA) [18], genetic algorithm (GA) [19–22], neural network [23] and pump power integral method [24], were used to search the pump powers and wavelength pairs. During the search process, the pump powers and wavelength pairs were directly substituted into the governing fiber Raman equations to calculate the gain profile. The governing equations are a system of coupled nonlinear ordinary differential equations with two-point boundary values for the pump and signal powers at the ends of the fiber. The method of two-point boundary value problem solving was involved in a large amount of numerical computations, which was inevitably time consuming and rarely got the real optimal solution in real time. When the line shape of input channels is taken into consideration the system of governing equations is a system of uncountable nonlinear differential equations [3]. The moment method [24–26] is used to get a discrete and reduced number of governing equations and also to decouple the pump and signal equations. In reference [26] authors employed the moment method to obtain the gain ripple in Raman fiber amplifiers with continuous pump spectrum. In this paper, we employ the
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moment method for continuous signal spectrum, in addition to considering the effect of line shape, on the output parameters. The number of governing equations of twopoint boundary value problem is reduced to a number proportional to the number of pumps which is less than the number of signals. While in the continuous pump spectrum in reference [26] the number of governing equations is proportional to the number of signals which is greater than the number of pumps. In this paper, in order to reduce the time of calculation, the integral form for expansion of the pump equations [16] is employed. Also in this paper, the moment method is employed to minimize the gain ripple in a WDM pump Raman fiber amplifier. This paper is organized into seven sections as follows. Section 2 introduces governing equations, and is followed by the moment method and the corresponding equations, in Section 3. The gain spectrum of the amplifier is presented in Section 4, and in Section 5 the gain ripple minimization is described in detail. The numerical results and discussion will be given in Section 6 with some calculations presented in Section 7. Finally two numerical methods for calculating the average value of signal and Raman tilt along the fiber length is illustrated in Appendices A and B, respectively.
where S(m,z) is the signal power spectrum and P(mi,z) is the pump power of frequency mi, the frequencies are numerated in decreasing order. The function a(m) is the fiber loss rate at frequency m. The gain coefficient g(m,m0 is equal to gR(m,m0 )/CAeff, (m0 /m)gR(m,m0 )/C Aeff and 0, for mhm0 , mi m0 and m = m0 , respectively. gR(m, m0 ) is the Raman gain coefficient from the wave of frequency m to the frequency m0 . The factor C accounts for the polarization randomization effects, Aeff is the effective area of the optical fiber. ml and mh are the lower and upper band of the input signal spectrum, respectively and N is the number of pumps. The number of equations corresponding to Eq. (1) is the same as the number of real numbers. The system of governing equations is a system of uncountable coupled nonlinear differential equations. 3. Moments and their evolution equations The moment method [24–26] is employed to reduce the system of governing equations into a countable system of equations. The linear operators F mk1 ...kl : L2 ðRÞ ! L2 ðRÞ are defined as follows: Z 1 ðmÞ F k1 ;...kl ½q ¼ am ðm0 Þgðm; m0 Þgðm0 ; mk1 Þ . . . gðm0 ; mkl ÞS 0
ðm0 ; zÞqðm0 ; zÞdm0
2. Governing equations In calculating the gain profile of the distributed fiber Raman amplifier (DFRA), noise effects such as spontaneous Raman scattering, Rayleigh back scattering and thermal factor etc. can be neglected. The interactions of pump to pump, signal to signal, pump to signal and the attenuation of pump and signal as the major effects are taken into consideration. Even we assume that the laser line shape are negligible and can be approximated by a sum of Dirac delta functions, the spectrum of the modulated signal is the convolution of the signal spectrum and the laser spectrum. Hence, the bandwidth of the modulated signal at least is equal to the signal bandwidth. As an example for each STM 256 channel this bandwidth is more than 40 GHz and the signal spectrum can be approximated by a sum of Lorentzian line shape. Hence, in this work it is assumed that the spectrum of signal power is continuous. In the steady state, the coupled nonlinear equations for stimulated Raman gain spectrum can be described as [3] " X dSðm; zÞ ¼ Sðm; zÞ aðmÞ þ gðm; mi ÞP ðmi ; zÞ dz i Z mh þ Sðm0 ; zÞgðm; mÞdm0 m 2 ½ml ; mh ð1Þ ml " X dP ðmi ; zÞ ¼ P ðmi ; zÞ aðmi Þ gðmi ; mj ÞP ðmj ; zÞ dz j Z mh þ Sðm0 ; zÞgðmi ; m0 Þdm0 i ¼ 1; . . . ; N ð2Þ ml
ð3Þ
where q(m,z) is an arbitrary function of S(m,z). The integral part on the right hand side of Eq. (1) is given by qð0;0Þ ðm; zÞ ¼ F ð0Þ ½1
ð4Þ
(0,0)
The moment q (m,z) is denoted by Q(m,z) and moment ðm;nÞ functions qk1 ;...:kl ðm; zÞ are defined by the following relation ðm;nÞ
ðmÞ
qk1 ;...:kl ðm; zÞ ¼ F k1 ...kl ½Qn
ð5Þ
The governing equations for moment functions are easily obtained:
ðm;0Þ qk1 ;...:kl ðm; zÞ
ðm;0Þ
N X dqk1 ;...kl ðm; zÞ ðmþ1;0Þ ðm;0Þ ¼ qk1 ;...kl ðm; zÞ þ P i ðzÞqk1 ;...kl ;ki ðm; zÞ dz i¼1 ðm;1Þ
þ qk1 ;...kl ðm; zÞ
ð6Þ
It is shown that the right hand side of Eq. (6) decreases and rapidly approaches zero by increasing each of the indices (m,n,l) of the moment functions. More generally, the governing equations of moment functions qm;n k 1 ;...:k l ðm; zÞ are as follows ðm;nÞ
N X dqk1 ;...kl ðm;zÞ ðmþ1;nÞ ðm;nÞ ¼ qk1 ;...kl ðm; zÞ þ P i ðzÞqk1 ;...kl ;ki ðm;zÞ dz i¼1 ( ðm;nþ1Þ
ðmÞ
þ qk1 ;...kl ðm; zÞ þ n F k1... kl ½Qn1 qð1;0Þ þ
X
) ðmÞ ðmÞ pi ðzÞF k1... kl ½Qn1 qð0;0Þ þ F k1... kl ½Qn1 qð0;1Þ
i
ð7Þ
A.R. Bahrampour et al. / Optics Communications 281 (2008) 3673–3680 ðmÞ
Since F k1... kl is a linear operator, the governing equations of ðmÞ F k1... kl ½qðm; zÞ can be written easily. The effects of signal to signal interactions are presented in Q(m, z) which is defined by Eq. (4) and appeared in the moment functions. The moment functions are symmetric with respect to all the lower indices. The governing equations are rewritten with respect to the moment functions: " # X dSðm; zÞ ¼ Sðm; zÞ aðmÞ þ gðm; mi ÞP ðmi ; zÞþQðm; zÞ dz i m 2 ½ml ; mh "
X dP ðmi ; zÞ ¼ P ðmi ; zÞ aðmi Þ gðmi ; mj ÞP ðmj ; zÞQðmi ; zÞ dz j i ¼ 1; . . . ; N
#
ð8Þ
ð9Þ
The boundary conditions for the governing equations are Sðm; 0Þ ¼ S 0 ðmÞ P ðmi ; LÞ ¼ P iL
ð10Þ ð11Þ
where PiL is the ith pump input power and S0(m) is the line shape of input signals and depends on the assumption. For example S0(m) can be a sum of Lorentzian line shapes corresponding to each channel, a flat spectral density for simplicity of calculations and a sum of delta functions for discrete input signal spectrum. Initial conditions corresponding to the moment funcðm;nÞ tions qk1 ;...kl ðm; 0Þ are easily obtained by the following relation: Z mh ðm;nÞ am ðm0 ÞQn ðm0 ; 0ÞSðm0 ; 0Þgðm; m0 Þ qk1 ...kl ðm; 0Þ ¼ ml
gðm0 ; mk1 Þ . . . gðm0 ; mkl Þdm0
ð12Þ
The initial value of the other moment functions can be obtained by the same method. 0 Since the magnitudes of, g(m,m ), Q(m) and a(m) are smaller than one, the moment functions are decreasing functions with respect to the indices (m,n,l) and rapidly converge to zero as the indices (m,n,l) increase. Because 0 the order of the magnitude of g(m,m ) and Q(m) are less than that of a(m), the rate of convergence with respect to the number of g(m,m0 ) and powers of Q(m) are faster than that of the powers of a(m). The system of Eq. (7) at the pump frequencies (m = mi, i = 1, . . . ,N) and (9) are explicitly independent of the signal Eq. (6) and are solved by truncating the system of moment equations by a finite number of indices (m,n,l). Variation of the pump powers P(mi,z) (i = 1, . . . ,N) are simply obtained by the solution of a system of two-point boundary value problem. Higher order truncation gives precise solution for the pumps variation along the fiber length. First order truncation of moment function equations are reduced to a system of 2N differential equations. Numerical calculations show that the first order truncation is accurate for the Raman fiber amplifier design.
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4. Gain spectrum of the amplifier Variations of pump powers along the fiber length are obtained by solving the moment Eq. (7) at pump frequencies (m = mi,i = 1, . . .,N) and pump Eq. (9). The system of uncountable governing equations is converted to a system of countable differential equations by using the moment functions. The system of countable equations of moment functions is truncated with respect to the indices (m,n,l) to obtain a system of a finite number of differential equations. As it is expected, calculations show that the speed of convergence is very fast. The precise pump evolution along the fiber can be obtained by the solution of a system of 2N nonlinear differential equations. Of course, higher order moments give more precise solutions for the pumps evolution along the fiber length. In order to have a gain spectrum of the FRA by the moment method, the pumps distribution along the fiber length which is obtained in the previous step will be applied to the truncated moment equations for any m 2 R and the system of the initial value problem will be solved (Appendix B). The numerical calculations show that the solution of the moment equations is in agreement with that obtained by the numerical method introduced in Appendix A. By solving the truncated system of equations, the Q(m,z) will be found. In order to have the gain spectrum, results are substituted in Eq. (8): GsðmÞ ¼ ð10loge10 ÞLðaðmÞ þ
N X
gðm; mi ÞP i þ QðmÞÞ
ð13Þ
i¼1
where P i and QðmÞ are the average value of Pi(z) and Q(m,z) along the fiber length, respectively. 5. Gain ripple minimization Since Q(m,z) is a complicated function of input pump powers Pi (L) and pump frequencies mi, the optimization of gain ripple is a complicated variational problem. In the case of negligible QðmÞ, some optimization methods for discrete spectrum signals are presented [3]. The numerical calculations show that the signal to signal interaction in Eq. (13) has perturbation effects on the gain spectrum of the FRA. Of course, effects of signal to signal interaction on the pump and signal behaviors along the fiber length are also considered in the moment equations. The signal exponential gain spectrum per unit length is written as follows: N X 1 Sðm; LÞ GeðmÞ ¼ Ln ¼ aðmÞ þ gðm; mi ÞC i þ QðmÞ L Sðm; 0Þ i¼1
m 2 ½ml; mh
ð14Þ
where Ci is the average value of the ith pump power along the fiber length. The variance of the gain spectrum is
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chosen as a scaling factor (cost function) for the gain ripple specification Z mh 2 ½GeðmÞ G0 ðmÞ dm ð15Þ V ¼ ml
where G0(m) is the target gain spectrum. The values of Ci are always positive. Now the problem is to pick up the values of Ci and mi which produce a gain that minimizes the RHS of (15). By neglecting the dependence of QðmÞ on the Ci and mi(i = 1, . . . ,N), picking the best Ci s which give a set of average powers, is fairly straightforward because (15) is just a quadratic function of Cis. Picking the best set of frequencies is a more complicated nonlinear problem. The Lagerange multiplier method can be employed to the minimization problem if there is any constraint on the pump powers or pump frequencies. Otherwise, the necessary condition for the minimization of the gain ripple (V) is obtained by taking the derivative with respect to mj and Cj(j = 1, . . . N) and putting the results equal to zero. Z mh N X gðm0 ; mj Þ½aðm0 Þ þ gðm0 ; mi ÞC i þ Qðm0 Þ ml
Z ml
i¼1
mh
G0 ðm0 Þdm0 ¼ 0j ¼ 1; . . . N N X ogðm0 ; mj Þ ½aðm0 Þ þ gðm0 ; mi ÞC i þ Qðm0 Þ omj i¼1 G0 ðmprime Þdmprime ¼ 0j ¼ 1; . . . N
ð16Þ
ð17Þ
The frequencies and average powers corresponding to the optimal pumps and optimum gain spectrum are obtained by solving the system of algebraic Eqs. (16) and (17). To obtain the pump powers, the pump Eq. (9) is integrated over the fiber length and rewritten as follows: " # gðmX i ;mk ÞC k P i ðLÞ ¼ exp ai L Qðmi Þ ð18Þ P i ð0Þ k The QðmÞ and Qðmi Þ are unknown and have perturbative effects in Eqs. (16)–(18). There are several methods for the initial guess of QðmÞ. In this analysis, for the initial guess we have assumed that in the first half length of the fiber, the main process is the fiber intrinsic loss and in the last half length the Raman amplification is the main process. In this step of optimization, it is assumed that the Raman exponential gain per unit length is constant and its value is determined by the target net gain of the FRA. Now, the RHS of Eq. (18) is completely specified, hence the ratio of input pump powers to the output pump powers is completely specified. The system of pump equations is converted to a two-point boundary value problem which can be solved by a standard nonlinear shooting method or the integral form expansion method [16]. Of course, the QðmÞ can also be determined by the numerical method presented in Appendix A. The new QðmÞis employed in Eqs. (16)–(18) to obtain the new optimal solution with a higher degree of precision. This process is continued up to obtain the desired degree of accuracy.
6. Numerical results The procedure to practically implement the above derived analytic formulation into numerical domain depending on their applications can be summarized as follows: 6.1. Gain prediction Step 1. Moment equations step The pump power distributions are obtained by solving the moment Eqs. (7) and (9). Depending on the order of the magnitude of a(m), Q(m) and g(m,m0 ), integers (m,n,l) are chosen and the system of Eq. (7) is truncated. By choosing the input signal spectrum S(m,0) and using Eq. (12) the initial values of moment functions can be determined. In the first step of truncation, the system of Eqs. (B.1) and (B.2) is a system of two-point boundary value problem which can be solved by the standard nonlinear shooting method or the iteration methods presented in Appendix A or Appendix B. In this paper, by employing an initial guess forQ(mi,z), the two-point boundary value problem is broken in two separate initial and final value problems which are solved by the fourth order Runge Kuta method. Higher precision solutions are obtained by a higher order of truncation of moment equations. Results are employed in Eqs. (B.1) and (B.2), using the iterative procedure the precise average values of pumps and moments along the fiber length can be determined. In this method the number of equations is proportional to the number of pumps, while in direct method the number of equations is proportional to the number of signals. Our numerical calculations show that the precision of the first order truncation is enough for the FRA calculations. Step 2. Gain spectrum calculation The gain spectrum is determined by Eq. (14). The average values of the pump powers and moment QðmÞ are determined by solving the pump and moment equations by the iterative method. The initial guess for signal to signal interaction term (Raman tilt) is determined by the method presented in Section 5.To understand the speed of convergence in the moment method, the gain spectrum at different moment order is calculated and compared with each other. In Fig. 1, the results corresponding to the three first orders of approximation for a system of flat input spectral intensity are compared with that of a system of discrete input spectrum. It is assumed that 50 signals are at the wavelength of 1528–1578 nm with 1 nm spacing at the signal power of 10 dB m/ch. For the counter directional pumping scheme, we used 20 pumps (1408–1490 nm). We employed 25 km
A.R. Bahrampour et al. / Optics Communications 281 (2008) 3673–3680
150
2.1
100
2
Input pump power,mW
Gain,dB
2.2
90
a effective area
attenuation cofficient
0.1
80
0.05
Effective area, μm2
200 pump power
first iteration second iteration discrete zeroth iteration
Attenuation cofficint,1/km
2.3
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70
1.9 1400
1450
1500
1550
0 1400
0 1600
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Wavelength,nm
of single mode fiber (SMF-28) as the test gain medium. Corresponding pump powers and their wavelengths are also presented in Fig. 1. The fiber loss spectrum, effective core area and Raman gain spectrum of (SMF-28) fiber are presented in Fig. 2. As it is obvious in Fig. 1, the suggested method shows that there is an excellent rate of convergence and accuracy in the first order of truncation. The accuracy improvement from the next order of truncation becomes less than (0.008 dB).Meanwhile, the required computation time to get the exact Raman gain and signal/pump powers distribution was less than a second for the counter directionally pumping configurations, with a conventional PC (2. GHz CPU clock). Fig. 3 shows the pump and signal spectrum evolution along the fiber. The excellent agreement between the results of the simulation and the conventional method shows the validity of our suggested algorithm. 6.2. Raman gain ripple minimization Step1. Average pump powers and pump frequencies Eq. (14) shows that the exponential gains are linear functions of average pump powers. It is noted that pumps are also appeared implicitly in the average moment QðmÞ, hence the solution of optimal problem is complicated and can be obtained by the variational method. Fortunately, as it is shown in previous section, the average signal spectrum has perturbation effects on the exponential gain spectrum. In this analysis, the approximated relation described in Section 5 is employed for Raman tilt calculation. We start the gain ripple minimization by defining the target gain spectrum G0 under a given set of constraints. The gain spectrum, average pump powers and pump frequencies corresponding to the optimal conditions, are obtained
1550
1600
−14
x 10
b 6
Raman gain cofficient,m / W
Fig. 1. Comparison of the gain spectrum corresponding to the first three order of approximation for a flat input spectral density in a 25 km SMF-28 Raman fiber amplifier shows the rapid speed of convergence of the model.
1500
Wavelength,nm
5
4
3
2
1
0 0
5
10
15
20
25
30
35
Pumpsignal frequency difference,THz
Fig. 2. Characteristics of single mode fiber (SMF-28). (a) The loss spectrum and effective core area [17]. (b) Raman gain spectrum [27].
by solving the system of algebraic Eqs. (16) and (17). The results that are obtained by the iterative method are presented in Appendices A and B of this paper. Step 2. Pump calculations In the first order of approximation, the moment function q(0,0)(m ,z) = Q(m,z) is determined by the initial guess. The average values of the moment ð0;0Þ functions qi are obtained with respect to the initial value of the signal spectrum. The ratios of the initial value to the final value of pumps are ð0;0Þ obtained by substituting qi and the optimal Ci into Eq. (18). In order to get the input power of pumps, the system of two-point boundary value problem of pumps (9) is easily solved. The results of a minimization problem with respect to the pump powers are presented in Fig. 3. We set net gain G0 to 0.8 dB over the 70 nm (1520–1590 nm) spectral range. Considering practical restrictions in pump WDM wavelength allocation, 15 equally spaced, counter-directional pumps at 1400–1490 nm. Calculations are done for a
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a
0.9 0.85
0.5 0
0.8 −0.5 −1
0.7
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Gain ,dB
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−2 −2.5
0.5
−3
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first iteration second iteration zeroth hteration discrete
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1600
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b 140
b
100 90
120
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Input pump power,mW
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60
40
70 60 50 40 30 20
20
10
0 1400
1420
1440
1460
1480
1500
Wavelength,nm
Fig. 3. Gain optimization of a 50 km SMF-28 Raman fiber amplifier with respect to the pump powers and uniformly distributed 15 pumps over the 1408–1490 nm pump bandwidth. (a) The gain spectrum of the Raman fiber amplifier. (b) The input pump powers.
50 km SMF-28 Raman fiber amplifier. After the first step, the pump evolution sufficiently converges to that of a real value and the gain curve extremely overlaps that of an ideal target gain. The pump powers spectrum corresponding to the optimal condition is presented in Fig. 3b. To consider the effect of iteration steps on the minimization process, the gain profiles corresponding to the different steps of iteration are presented in Fig. 3a. The convergence error from the iteration procedure becomes almost negligible after the first iteration step. The observed absolute error between target gain profile and iteratively obtained gain profile after the first iteration step was less than 0.06 dB over the 70 nm gain spectral range. A fiber Raman amplifier with eight pumps is optimized with respect to the pump frequencies and pump powers. Results are presented in Fig. 4. Obviously, by increasing the number of pumps, the optimum gain approached the gain profile corresponding to the uniform distributed pumps. Naturally, the level of gain accuracy could be further improved with additional level of moment equations and
0 1380
1400
1420
1440
1460
1480
1500
Wavelength,nm
Fig. 4. Gain optimization of a 50 km SMF-28 Raman fiber amplifier with respect to the wave length of pumps and pump powers. (a) The optimal gain spectrum of the Raman fiber amplifier. (b) The input pump spectrum.
iteration steps. The convergence rate of gain spectrum profile is measured in the sense of L2 space metric; by considering most of practical applications under experimental error, the first step of approximation was generally sufficient. The required computation time to get the result was less than 8 s with a conventional PC (2.0 GHz CPU clock). 7. Conclusion The moment method which has been previously employed to analyze the inhomogeneous Erbium doped fiber amplifier (EDFA) and continuous pump spectrum Raman fiber amplifier, is applied to the RFA with continuous input spectrum. The number of governing equations for a RFA with continuous spectrum input signals is uncountable. By using the moment method the system of uncountable differential equations reduces to a countable system of ordinary differential equations. By truncating the system of moment equations, the number of governing
A.R. Bahrampour et al. / Optics Communications 281 (2008) 3673–3680
equations is reduced to a number of equations proportional to the number of pump equations. The speed of convergence of the signal continuous spectrum moment method is faster than that of the pump continuous spectrum method which has been presented in our previous publication [26]. Calculations show that the presented method is a precise and fast method.
ðnÞ
Rij ¼ where
j X
3679
ðn1Þ
ðA:6Þ
Ail Blj
l¼0 ðn1Þ Bil
is defined by the following matrix 1 ðnÞ Gik Rkl X A ¼ exp @ 0
ðnÞ
Bil
ðA:7Þ
k
Appendix A. An efficient numerical method for calculation of SðmÞ
Using these formulations, the signal average power for each frequency can be obtained with matrix multiplication procedure
The gain spectrum of the amplifier is obtained by the solution of Eq. (8). After dividing Eq. (8) by S(m,z) and integrating over z we get " Z z N X Sðm; zÞ ¼ Sðm; 0Þ exp aðmÞz þ gðm; mi Þ P ðmi ; z0 Þdz0
1 Sðmi Þ ¼ RiN ðA:8Þ L where Sðmi Þ is the average value of signal along the fiber length at frequency mi. The exponential gain spectrum is given by
þ
Z
1 0
gðm; m Þdm
0
i
Z
0
z 0
0
Sðm ; z Þdz
0
0
ðA:1Þ
0
With an additional integration process; we get the following integral form of Raman wave equations: " Z Z z N X Sðm; z0 Þdz0 ¼ Sðm; 0Þ dz0 aðmÞz þ gðm; mi Þ 0
Z
Z
z0
0 z0
P ðmi ; z00 Þdz00 þ # 0
00
i¼1
Z
1
Sðm ; z Þdz
ðA:2Þ
Rz By the definition of Rðm; zÞ ¼ 0 Sðm; z0 Þdz0 , we can rewrite Eq. (A.2) into a closed integral form for the R(m,z) Z 1 Z z Rðm; zÞ ¼ Aðm; z0 Þ exp gðm; m0 ÞRðm0 ; z0 Þdm0 dz0 ðA:3Þ 0
Where A(m,z) is given by the following relation " # Z z N X gðmi ; mÞ P ðmi ; z0 Þdz0 Aðm; zÞ ¼ Sðm; 0Þ exp aðmÞz þ 0
ðA:4Þ To solve Eq. (A.3), we employ the Picard iteration method, taking R(m,z) as the interim target solution. At nth iteration step, Eq. (A.3) becomes Z mh Z z RðnÞ ðm; zÞ ¼ Aðm; z0 Þ exp gðm; m0 ÞRðn1Þ ðm0 ; z0 Þdm0 dz0 0
ðA:9Þ
j
where the P j is the average pump power along the fiber length at the frequency mj. Of course the Q(m,z) can be obtained directly by solving Eq. (B.2) at the arbitrary frequency mm 2 [ml,mh]; results are in agreement with those obtained by this method. Appendix B. A semi analytical solution for QðmÞ
00
i¼1
j
gðm; m0 Þdm0 0
0
0
1 Sðm; LÞ GeðmÞ ¼ Ln L Sðm; 0Þ X X ¼ aðmÞ þ gðm; mj ÞP j þ gðm; mj ÞRjN
ml
ðA:5Þ In order to implement the above equations into the numerical domain, we now construct matrices Rij R(mi,zj) and Aij A(mi,zj) to assign the value of R and A at the ith frequency at the position zj (the whole fiber link is covering by discrete position elements with step size of Dz). The matrix Gij = g(mi,mj) is defined as the value of Raman gain coefficient at the frequencies mi and mj.
To calculate the Q(m,z) and its average value along the fiber length ðQðmÞÞ an iterative method can be employed. The governing equations corresponding to the pumps and moment Q(m, z) = q(0,0)(m,z) evolution along the fiber are as follows N X dP ðnÞ ðmi ; zÞ ¼ P ðnÞ ðmi ; zÞ½aðmi Þ gðmi ; mj ÞP ðnÞ ðmj ; zÞ dz j¼1 Qðnþ1Þ ðmi ; zÞ
ðB:1Þ
N X dQðnÞ ðm; zÞ ðn1Þ ð0;0Þ ¼ qð1;0Þ ðm; zÞ þ P j ðzÞqj ðm; zÞ dz j¼1
þ qð0;1Þ ðm; zÞ (n)
ðB:2Þ (n)
where the Q (m,z) and P (mi,z) are Q(m, z) and P(mi,z) at the nth iteration step. Since the derivative of the moments, ð0;0Þ q(1,0)(m,z), qj ðm; zÞ and q(0,1)(m,z) are functions of higher order moments, the variations of these moments along the fiber length are very slow and it is assumed that they are independent of z which is confirmed by the numerical calculations. Q(m,z) is obtained by integrating both side of the Eq. (B.2): N Z z X ðn1Þ ð0;0Þ ðnÞ ð0;0Þ Q ðm; zÞ ¼ q ðm; 0Þz þ P j ðz0 Þqj ðm; 0Þdz0 j¼1
þq
ð0;1Þ
ðm; 0Þz
0
ðB:3Þ
3680
A.R. Bahrampour et al. / Optics Communications 281 (2008) 3673–3680
The average value of Q(n)(m,z) along the fiber length is Z z N X L ð0;0Þ ðn1Þ QðnÞ ðmÞ ¼ qð0;0Þ ðm; 0Þ þ qj ðm; 0Þ P j ðz0 Þdz0 2 0 j¼1 L þ qð0;1Þ ðm; 0Þ 2
ðB:4Þ
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