An analytical approach for gain optimization in multimode fiber Raman amplifiers Junhe Zhou* Dept. of electronics science and engineering, Tongji University, Shanghai 200092, China * [email protected]
Abstract: In this paper, an analytical approach is proposed to minimize the mode dependent gain as well as the wavelength dependent gain for the multimode fiber Raman amplifiers (MFRAs). It is shown that the optimal power integrals at the corresponding modes and wavelengths can be obtained by the non-negative least square method (NNLSM). The corresponding input pump powers can be calculated afterwards using the shooting method. It is demonstrated that if the power overlap integrals are not wavelength dependent, the optimization can be further simplified by decomposing the optimization problem into two sub optimization problems, i.e. the optimization of the gain ripple with respect to the modes, and with respect to the wavelengths. The optimization results closely match the ones in recent publications. ©2014 Optical Society of America OCIS codes: (060.0060) Fiber optics and optical communications; (060.2320) Fiber optics amplifiers and oscillators.
References and links 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
H. R. Stuart, “Dispersive Multiplexing in Multimode Optical Fiber,” Science 289(5477), 281–283 (2000). N. Bai, E. Ip, T. Wang, and G. Li, “Multimode fiber amplifier with tunable modal gain using a reconfigurable multimode pump,” Opt. Express 19(17), 16601–16611 (2011). S. Namiki and Y. Emori, “Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelengthdivision-multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. 7(1), 3–16 (2001). M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 548–559 (2002). B. Inan, B. Spinnler, F. Ferreira, D. van den Borne, A. Lobato, S. Adhikari, V. A. Sleiffer, M. Kuschnerov, N. Hanik, and S. L. Jansen, “DSP complexity of mode-division multiplexed receivers,” Opt. Express 20(10), 10859–10869 (2012). R. Ryf, A. Sierra, R. Essiambre, S. Randel, A. Gnauck, C. A. Bolle, M. Esmaeelpour, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, D. Peckham, A. McCurdy, and R. Lingle, “Mode-Equalized Distributed Raman Amplification in 137-km Few-Mode Fiber,” in 37th European Conference and Exposition on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Th.13.K.5. R. Ryf, R. Essiambre, J. Hoyningen-Huene, and P. Winzer, “Analysis of Mode-Dependent Gain in Raman Amplified Few-Mode Fiber,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2012), paper OW1D.2. V. E. Perlin and H. G. Winful, “On Distributed Raman Amplification for Ultrabroad-Band Long-Haul WDM Systems,” J. Lightwave Technol. 20(3), 409–416 (2002). W. Zhang, X. Feng, J. Peng, and X. Liu, “A Simple Algorithm for Gain Spectrum Adjustment of BackwardPumped Distributed Fiber Raman Amplifiers,” IEEE Photon. Technol. Lett. 16(1), 69–71 (2004). V. E. Perlin, “Novel configurations in nonlinear fiber optics Raman scattering and Bragg gratings,” Ph.D. dissertation, Dept. of electrical engineering, University of Michigan, 2003. A. R. Grant, “Calculating the Raman Pump Distribution to Achieve Minimum Gain Ripple,” IEEE J. Quantum Electron. 38(11), 1503–1509 (2002). R. Bro and S. De Jong, “A fast non-negativity-constrained least squares algorithm,” J. Chemometr. 11(5), 393– 401 (1997). C. Antonelli, A. Mecozzi, and M. Shtaif, “Raman amplification in multimode fibers with random mode coupling,” Opt. Lett. 38(8), 1188–1190 (2013). F. Poletti and P. Horak, “Description of ultra-short pulse propagation in multi-mode optical fibers,” J. Opt. Soc. Am. B 25(10), 1645–1654 (2008). C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Random coupling between groups of degenerate fiber modes in mode multiplexed transmission,” Opt. Express 21(8), 9484–9490 (2013). X. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express 11(12), 1452–1461 (2003).
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21393
17. C. Fludger, A. Maroney, N. Jolley, and R. Mears, “An analysis of the improvements in OSNR from distributed Raman amplifiers using modern transmission fibres,” in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2000), paper FF2.
1. Introduction Recently, mode division multiplexing (MDM) has been proposed as a promising technique for the future fiber communication systems. By modulating signals on different modes of the multimode fibers and implementing optical multiple-input and multiple-output (MIMO) technique, one is able to greatly increase the transmission capacity [1]. Optical amplifiers for multimode fibers will be the key components to realize MDM systems [2]. Fiber Raman amplifiers (FRAs) [3] are very promising optical amplifiers in traditional optical communication systems. Because FRAs can provide distributed amplification, the equivalent noise figure (NF) is relatively low in comparison with the discrete amplifiers, such as the erbium doped fiber amplifiers (EDFAs) [4]. Furthermore, the distributed nature will enable a lower input signal power, resulting in a lower nonlinear noise during the transmission. These merits become even more prominent for a MDM system, because it is highly reliant on the digital signal processing (DSP) technique and therefore has a more stringent requirement on the noise performance [5]. To implement FRAs in MDM systems, their performance should be carefully optimized [6]. The key requirement for the multimode fiber Raman amplifiers (MFRAs) is the gain flatness with respect to the modes and the wavelengths. The unequal gain on the different modes/wavelengths will cause uneven optical signal to noise ratio and therefore induce penalty at the receiver and impact the overall system capacity [7]. Therefore, the gain among the modes as well as the wavelengths should be flattened. Currently, most of the studies on the optimization of the FRAs focus on the optimization of the gain spectrum of the FRAs with respect to the wavelengths in single mode fibers. Numerous techniques have been proposed on this topic, such as the genetic algorithm [8] and the semi-analytical expression based algorithms [9–11]. For multimode FRAs, there have been experimental and theoretical demonstrations of a MFRA pumped at a single wavelength by Ryf et al [6,7], who adjusted the pump power distributions among different modes to ensure an equalized gain for the signals at different modes. Although there have been these pioneering studies [6–11], there has been lack of a systematic optimization approach to achieve equalized gain for different modes in multimode FRAs. Furthermore, for a MFRA pumped by multiple pumps at different modes and wavelengths, a method which could achieve simultaneous optimization of the mode dependent gain and the wavelength dependent gain is highly required, but has not been discussed in the published literatures. In this paper, MFRAs pumped at different modes and wavelengths are studied. We propose an analytical approach to optimize gain among different modes and different wavelengths simultaneously. The method is based on the analytical solution of the optimal pump power integrals using the least square method (LSM) [10,11], but modified to incorporate the non-negativity-constraint on the pump power integrals [12]. After obtaining the optimal pump power integrals, full scale simulation is performed. The input pump powers at the different modes and wavelengths can be decided by the shooting method [8]. The method is performed on a single wavelength pumped MFRA which is the same as the one in [7], and the optimization results match very closely. A multi-wavelength-pumped MFRA is studied afterwards, and less than 1dB total gain variation is achieved among 4 different propagation modes in the wavelength range of 40nm. 2. Mathematical modeling 2.1 General mathematical formulation Assuming that there are total N signal wavelengths and M pump wavelengths in the MFRA and the number of the propagation modes is K, the evolution of the powers of the signals and the pumps obeys the following coupled equations [6, 7, 10, 11, 13, 14]:
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21394
= −α i Si , m + g (ν i ,ν j ) f i , m , j , n Pj , n Si , m + g (ν i ,ν i ' ) fi , m ,i ', m ' Si ', m ' Si , m dz i ′≠ i , m ' j ,n (1) dPj , n − = −α j Pj , n + g (ν j ,ν j ' ) f j , n, j ', n ' Pj ', n ' Pj , n + g (ν j ,ν i ) f j , n,i , m Si , m Pj , n dz i,m j '≠ j , n ' where S and P stand for the powers of the signals and the pumps respectively, fi,m,j,n stands for the power overlap integral between the modes m and n at the wavelengths i and j, dSi , m
+∞ +∞
I ( x, y ) I ( x, y ) dxdy i,m
fi ,m, j ,n =
+∞ +∞
−∞ −∞
j ,n
+∞ +∞
(2)
I ( x, y ) dxdy I ( x, y ) dxdy i,m
j ,n
−∞ −∞
−∞ −∞
the Raman gain coefficient is described by γ R (ν j −ν i ) (ν i < ν j ) g (ν i ,ν j ) = ν i (3) − γ R (ν i −ν j ) (ν i > ν j ) νj where γR indicates the Raman gain between the frequency νi and νj. Equation (1) is derived based on the theories and principles described in [13,14], and is analogous to the equations in [6,7]. First of all, the coupling between the modes with large propagation constant difference is ignored [15]. The modes with identical or similar propagation constants are considered as one mode in Eq. (1), because the strong coupling among these modes equalizes the gain within the mode group [7, 13] and they are equivalent to one mode during the mathematical treatment. Secondly, the multimode transmission fiber usually has the length of tens of kilometers. The long fiber introduces large mode dispersion, and therefore the random coupling processes for the pumps and the signals are independent. So the random variations of the intensity distribution of the pump and signal waves are decorrelated and will not contribute to the amplification process. It should be noted that other propagation effects, such as the mode coupling and four wave mixing, are outside the scope of this paper. Usually they can be ignored during the gain design for the multimode Raman amplifiers as explained by [13]. The first equation describes the signal gain and can be rewritten as L L S ( L) = −α i L + g (ν i ,ν j ) f i , m , j , n Pj , n dz + g (ν i ,ν i ' ) f i , m ,i ', m ' Si ', m ' dz (4) ln i , m S (0) 0 0 j ,n i ′≠ i , m ' i,m The first term on the RHS of the equation is the fiber loss, the second term is the Raman on-off gain, and the third term is the signal-signal Raman interactions. Usually, the signalsignal interactions are neglected due to their small values (or to be considered in the later optimizations as demonstrated in section 4), and the Raman on-off gain is the main parameters to be optimized. The on-off gain at the ith signal wavelength and the mth signal mode can be described by the following equation
L
Gainon − off = 10 log e g (ν i ,ν j ) f i , m , j , n Pj , n dz j ,n
(5)
0
Equation (5) can be rewritten into the matrix form as
Gain on − off = 10 log eCp (6) where the (iK + m) element of the vector Gainon-off is described by Eq. (5) and the element of the matrix C is calculated by th
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21395
g (ν i ,ν j ) f i , m , j , n
th
(7)
the (iK + m) element of the vector p is the pump power integral L
p ( jK + n ) = Pj , n dz
(8)
0
Assuming that the targeted gain for each signal wavelength and mode is described by a vector t, the optimization problem will be minimized by the following formula Gain on − off − t
2
(9)
According to LSM [10,11], the optimal solution for the pump power integral vector is −1 1 CT C ) CT t (10) ( 10 log e It should be noted that Eq. (10) gives the optimal power integrals without constraints. However, the power integrals should always be positive while Eq. (10) could lead to negative results. This might not be a problem in single mode FRA optimization, because the number of variables to be optimized is much smaller than the number of the elements in the target vector. However, in MFRA optimization, especially in the mode-dependent gain optimization, the number of variables to be optimized could be the same as the number of variables in the target vector, and an optimal solution with negative values is very likely to occur. Therefore, we propose to optimize the function, i.e. Eq. (9) via non-negative least square method (NNLSM) [12]. The method is briefly described below. If the optimal power integrals are positive, we define the corresponding non-negativityconstraints to be passive (not active). If some of the non-negativity-constraints do function during the search for the optimal power integrals, the corresponding pump power integrals will be 0 and we define the constraints to be active. If the active set and the passive set for the constraints are known, the optimal power integral can be obtained as follows. For the power integrals whose constraints are passive, their optimal values can be calculated via formulas similar to Eq. (10), while for the power integrals whose constraints are active, they are set to be 0. To determine whether the constraints are active or passive, an iterative procedure is designed [12]. For the details of the algorithm, the readers are encouraged to refer to [12]. For readers' convenience, it is summarized as follows [12].
p=
1, The initial passive set P is assumed to have no elements and the active set A is assumed to contain all of the non-negative constraints. The initial optimal pump power integrals are assumed to be 0. 2, Calculate the derivative of Eq. (9) at the optimal pump power integral values, i.e. 2 (10 log e ) CT ( CPopi − t ) 2
(11)
3, If all of elements of the above vector are non-negative, the optimal pump power integrals are obtained. If any of the elements are negative, find the minimum element and add the corresponding non-negative constraint to the passive set P and remove it from the active set A. 4, Find the optimal pump power integrals corresponding to the passive set P according to a formula similar to Eq. (10), while setting the pump power integrals corresponding to the active set A to be 0. Denote the optimal pump power integral vector as Popt1. 5, If all of the elements of Popt1 are non-negative, update Popt as Popt1. Go to step 2. If any elements of Popt1 are negative, calculate the constant β and update Popt as
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21396
β = − min Popt ( n ) / ( Popt ( n ) − Popt1 ( n ) ) n∈P
Popt
update
(12)
= Popt + β ( Popt1 − Popt )
and update set A and set P. Go to step 4. It should be noted that the optimization seldom enters step 5 [12]. With the above procedures, all the optimal power integrals are calculated and the input pump power can be obtained by the shooting method [8, 16]. 2.2 The case of non-wavelength-dependent power overlap integrals In the previous sub-section, there are totally NK optimal pump power integrals to be determined. Therefore, it is quite computational expensive. In this sub-section, a simpler model is developed to resolve this problem. In multimode fibers, the mode profile varies with respect to the wavelengths, but the variation is relatively small within the wavelength range of 1400nm-1600nm. Therefore, Ryf et. al. [6,7] proposed to simplify the analysis by assuming that the overlap integrals of the modal powers are not wavelength dependent. In such cases, the gain expression can be simplified as L
Gainon − off = 10 log e g (ν i ,ν j ) f m , n Pj , n dz j
n
(13)
0
The corresponding matrix formula is Gain on − off = 10 log e ( g ⊗ f ) p (14) where ⊗ denotes the Kronecker product of the matrix. The elements of matrices g, f and the vector p are
g ( i, j ) = g (ν i ,ν j )
f ( m, n ) = f m , n
(15)
L
p ( jK + n ) = Pj , n dz 0
It is reasonable to assume the expected gain target for each mode at the same wavelength to be the same (flat). In such cases, t = t g ⊗ t f , where tg stands for the targeted gain related to the wavelength and tf stands for the targeted gain related to the mode. Hence, the optimal pump power integrals without constraints are
(
) (
)
−1 −1 1 gT g ) gT t g ⊗ ( f T f ) f T t f (16) ( 10 log e From Eq. (16), it can be concluded that if the power overlap integrals are not wavelength dependent, the problem of optimizing MFRA gain can be decomposed into two sub-problems, which are the optimization of the gain with respect to the wavelengths and the optimization of the gain with respect to the modes. All of the pumps at the different wavelengths should have the same power fractions at different modes. To implement the non-negative constraint, similar procedures to determine the active or passive constraints will be performed on the two sub-optimization problems. However, during the optimization of the wavelength dependent gain, this is usually not necessary. The decomposition greatly simplifies the optimization efforts in comparison with the ones in the previous sub-section. However, it should be noted that when the power overlap integrals are wavelength dependent, the method in the previous sub-section will predict a better optimal solution for the pump power integrals.
p=
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21397
3. Verification of the proposed formulas and the optimization algorithm To verify the proposed formulas and the optimization method, we apply them to the published results in [6,7]. In [7], a multimode Raman fiber amplifier in a step index (SI) few mode fiber (FMF) is studied. The cladding refractive index and the core refractive index of the fiber are 1.46 and 1.4673 respectively [7]. The normalized frequency V = 5 and therefore, it will support four mode groups. The power overlap integrals are listed below and assumed to be wavelength independent [7]. The pump wavelength is 1455nm, while the signal wavelength is 1560nm [7]. Since there are no pump-pump interactions in a single-wavelength pumped MFRA, the optimal pump power integrals are directly related to optimal input pump powers at different modes, and there is no requirement to use the shooting method. Table 1. The power overlap integrals from [7] (in 109/m2) LP01
LP11
LP21
LP02
LP01
6.24
4.12
2.85
4.62
LP11
4.12
4.36
3.81
2.33
LP21
2.85
3.81
3.88
2.12
LP02
4.62
2.33
2.12
6.15
Based on NNLSM, the optimal percentage for the four pump power integrals/input powers at the four pump modes to achieve the flat modal gain are listed as follows. Compared with the optimized simulation values in [7], there is an exact agreement. Table 2. The optimal pump power percentage on each mode obtained by NNLSM. pump integral in Percentage
Modes
NNLSM
Reference [7]
LP01
0.00%
0.00%
LP11
0.00%
0.00%
LP21
69.67%
69.60%
LP02
30.33%
30.40%
One thing worth mentioning is that the degenerated modes, which have identical propagation constants, are regarded as one mode group in the analysis. For example, LP11a and LP11b are combined as LP11, because the strong coupling will equalize the gain within the mode group [7, 13]. The intensities of the mode groups result from the combination of the original degenerated modes [7, 14]. (The details of the intensities of the mode groups can be found in [7].)Therefore, the gain equalization for the degenerated modes is not necessary [7, 13]. 4. An optimization example for a multimode Raman amplifier with multi-wavelength pumps To further demonstrate the capability of the optimization algorithm, a MFRA with multiple pump wavelengths is used as an example. The Raman amplifier is based on the same multimode fiber discussed in section 3. The length of the multimode fiber is assumed to be 50km. The Raman gain coefficient is from the published literature [17]. Four pumps are used in the simulation, which are located at the wavelengths of 1420nm, 1430nm, 1450nm, and 1465nm. Totally 50 signal wavelengths are used in the simulation, which are within the Cband, starting from 1525.2nm to 1565nm with 100GHz channel spacing. The total signal input power is −3dBm, which corresponds to −20dBm input power per wavelength. Counter pumping scheme is assumed in the simulations. #210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21398
It is assumed that the power overlap integrals are not wavelength dependent [7]. Therefore, the optimization can be accomplished via two steps, i.e. the steps to optimize the mode dependent gain and the wavelength dependent gain. By employing NNLSM, the optimal pump power integrals on each mode and wavelength are obtained. The input pump powers are calculated afterwards using the shooting method. First of all, the targeted on-off gain for the Raman amplifier is set to be 10dB. The optimized on-off gain profile for each mode is plotted in Fig. 1. It can be inspected from the figure that the residual wavelength dependent gain for each mode is only 0.5dB and the maximum mode dependent gain is about 0.15dB. The overall gain variation is about 0.7dB. The calculated input pump powers are listed in Table 3. It can be observed that the optimal input pump power ratio on each mode matches the optimal pump integral ratio illustrated in Table 2. 10.4 10.3 10.2
on-off gain(dB)
10.1 10 9.9 9.8 9.7 9.6 9.5 9.4 1525
LP01 gain LP11 gain LP21 gain LP02 gain 1530
1535
1540 1545 1550 wavelength(nm)
1555
1560
1565
Fig. 1. The gain profile of an optimized C-band MFRA with the targeted gain of on-off 10dB. Table 3. The optimal pump power at each wavelength and mode when the target on-off gain is 10dB.
wavelength(nm)
1420
1430
1450
1465
mode
power(mW)
percentage
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
190.81
69.67%
LP02
83.05
30.33%
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
171.31
69.67%
LP02
74.57
30.33%
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
258.48
69.67%
LP02
112.51
30.33%
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
150.57
69.67%
LP02
65.54
30.33%
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21399
To illustrate the robustness of the algorithm, the targeted on-off gain is changed to 15dB. The gain profile for each mode is plotted in Fig. 2. It can be observed that the wavelengthdependent gain for each mode is less than 0.8dB and the total gain variation is less than 1dB. The effectiveness of the method has been demonstrated. The calculated input pump power is listed in Table 4. Although there are small deviations, the optimal input pump power ratio on each mode still matches the ratio in Table 2. 15.6
15.4
LP01 gain LP11 gain LP21 gain LP02 gain
on-off gain(dB)
15.2
15
14.8
14.6
14.4
14.2 1525
1530
1535
1540 1545 1550 wavelength(nm)
1555
1560
1565
Fig. 2. The on-off gain profile of an optimized C-band MFRA with the targeted on-off gain of 15dB. Table 4. The optimal pump power at each wavelength and mode when the target on-off gain is 15dB.
wavelength(nm)
1420
1430
1450
1465
mode
power(mW)
percentage
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
326.48
69.73%
LP02
141.72
30.27%
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
280.14
69.69%
LP02
121.84
30.31%
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
367.10
69.67%
LP02
159.78
30.33%
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
186.61
69.64%
LP02
81.35
30.36%
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21400
It should be noted that the algorithm cannot only optimize the flattened on-off gain spectrums, but also be used to optimize the on-off gain spectrums with a non-zero slope. For instance, the example illustrated in Fig. 2 shows a flat on-off gain spectrum. However, due to the loss slope and the signal-signal Raman interactions, there is a higher net gain at the longer wavelength as illustrated in Fig. 3. Since the passive loss of the 50km fiber is about 10dB, Fig. 3 (net gain) differs from Fig. 2 (on-off gain) by about 10dB in average gain. To optimize the net gain profile of the multimode Raman amplifier, one needs to set the targeted on-off gain profile with the inverse gain slope with respect to Fig. 3. Using the new targeted on-off gain profile, the Raman gain is re-optimized and the final net gain is flat as illustrated in Fig. 4, which is a flattened net gain profile among the modes and wavelengths. 6.4 6.2 6
LP01 gain LP11 gain LP21 gain LP02 gain
Netgain gain(dB)
5.8 5.6 5.4 5.2 5 4.8 4.6 4.4 1525
1530
1535
1540 1545 1550 wavelength(nm)
1555
1560
1565
Fig. 3. The net gain profile of the optimized C-band MFRA with the targeted on-off gain of 15dB.
6 5.8 5.6
net gain(dB)
5.4 5.2 5 4.8 4.6 4.4 1525
LP01 gain LP11 gain LP21 gain LP02 gain 1530
1535
1540 1545 1550 wavelength(nm)
1555
1560
1565
Fig. 4. The flattened net gain profile of the optimized C-band MFRA with the targeted on-off gain of 15dB.
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21401
5. Conclusion In this paper, an approach to optimize the gain profiles of the MFRAs is proposed. Analytical formulas are derived and they are combined with NNLSM to optimize the pump powers. It is revealed that under the non-wavelength-dependent power overlap integral assumption, the gain optimization problem with respect to the wavelengths and the modes can be decomposed into two sub-problems. The formulas and optimization algorithm are verified by applying them to a published MFRA and very good agreement is achieved by comparing the published results with those from the algorithm. Finally, the method is used to optimize a four-mode MFRA with 50 signal wavelengths and 4 pump wavelengths. Since full scale simulation is conducted within the algorithm, the method is both valid in the unsaturated and saturated regimes. Acknowledgment The author would like to thank three anonymous reviewers for their evaluable comments during the review process. This work is partially supported by the National science foundation of China (Grant No. 61201068).
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21402
Abstract: In this paper, an analytical approach is proposed to minimize the mode dependent gain as well as the wavelength dependent gain for the multimode fiber Raman amplifiers (MFRAs). It is shown that the optimal power integrals at the corresponding modes and wavelengths can be obtained by the non-negative least square method (NNLSM). The corresponding input pump powers can be calculated afterwards using the shooting method. It is demonstrated that if the power overlap integrals are not wavelength dependent, the optimization can be further simplified by decomposing the optimization problem into two sub optimization problems, i.e. the optimization of the gain ripple with respect to the modes, and with respect to the wavelengths. The optimization results closely match the ones in recent publications. ©2014 Optical Society of America OCIS codes: (060.0060) Fiber optics and optical communications; (060.2320) Fiber optics amplifiers and oscillators.
References and links 1. 2. 3. 4. 5. 6.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
H. R. Stuart, “Dispersive Multiplexing in Multimode Optical Fiber,” Science 289(5477), 281–283 (2000). N. Bai, E. Ip, T. Wang, and G. Li, “Multimode fiber amplifier with tunable modal gain using a reconfigurable multimode pump,” Opt. Express 19(17), 16601–16611 (2011). S. Namiki and Y. Emori, “Ultrabroad-band Raman amplifiers pumped and gain-equalized by wavelengthdivision-multiplexed high-power laser diodes,” IEEE J. Sel. Top. Quantum Electron. 7(1), 3–16 (2001). M. N. Islam, “Raman amplifiers for telecommunications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 548–559 (2002). B. Inan, B. Spinnler, F. Ferreira, D. van den Borne, A. Lobato, S. Adhikari, V. A. Sleiffer, M. Kuschnerov, N. Hanik, and S. L. Jansen, “DSP complexity of mode-division multiplexed receivers,” Opt. Express 20(10), 10859–10869 (2012). R. Ryf, A. Sierra, R. Essiambre, S. Randel, A. Gnauck, C. A. Bolle, M. Esmaeelpour, P. J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, D. Peckham, A. McCurdy, and R. Lingle, “Mode-Equalized Distributed Raman Amplification in 137-km Few-Mode Fiber,” in 37th European Conference and Exposition on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Th.13.K.5. R. Ryf, R. Essiambre, J. Hoyningen-Huene, and P. Winzer, “Analysis of Mode-Dependent Gain in Raman Amplified Few-Mode Fiber,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2012), paper OW1D.2. V. E. Perlin and H. G. Winful, “On Distributed Raman Amplification for Ultrabroad-Band Long-Haul WDM Systems,” J. Lightwave Technol. 20(3), 409–416 (2002). W. Zhang, X. Feng, J. Peng, and X. Liu, “A Simple Algorithm for Gain Spectrum Adjustment of BackwardPumped Distributed Fiber Raman Amplifiers,” IEEE Photon. Technol. Lett. 16(1), 69–71 (2004). V. E. Perlin, “Novel configurations in nonlinear fiber optics Raman scattering and Bragg gratings,” Ph.D. dissertation, Dept. of electrical engineering, University of Michigan, 2003. A. R. Grant, “Calculating the Raman Pump Distribution to Achieve Minimum Gain Ripple,” IEEE J. Quantum Electron. 38(11), 1503–1509 (2002). R. Bro and S. De Jong, “A fast non-negativity-constrained least squares algorithm,” J. Chemometr. 11(5), 393– 401 (1997). C. Antonelli, A. Mecozzi, and M. Shtaif, “Raman amplification in multimode fibers with random mode coupling,” Opt. Lett. 38(8), 1188–1190 (2013). F. Poletti and P. Horak, “Description of ultra-short pulse propagation in multi-mode optical fibers,” J. Opt. Soc. Am. B 25(10), 1645–1654 (2008). C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Random coupling between groups of degenerate fiber modes in mode multiplexed transmission,” Opt. Express 21(8), 9484–9490 (2013). X. Liu and B. Lee, “Effective shooting algorithm and its application to fiber amplifiers,” Opt. Express 11(12), 1452–1461 (2003).
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21393
17. C. Fludger, A. Maroney, N. Jolley, and R. Mears, “An analysis of the improvements in OSNR from distributed Raman amplifiers using modern transmission fibres,” in Optical Fiber Communication Conference, OSA Technical Digest Series (Optical Society of America, 2000), paper FF2.
1. Introduction Recently, mode division multiplexing (MDM) has been proposed as a promising technique for the future fiber communication systems. By modulating signals on different modes of the multimode fibers and implementing optical multiple-input and multiple-output (MIMO) technique, one is able to greatly increase the transmission capacity [1]. Optical amplifiers for multimode fibers will be the key components to realize MDM systems [2]. Fiber Raman amplifiers (FRAs) [3] are very promising optical amplifiers in traditional optical communication systems. Because FRAs can provide distributed amplification, the equivalent noise figure (NF) is relatively low in comparison with the discrete amplifiers, such as the erbium doped fiber amplifiers (EDFAs) [4]. Furthermore, the distributed nature will enable a lower input signal power, resulting in a lower nonlinear noise during the transmission. These merits become even more prominent for a MDM system, because it is highly reliant on the digital signal processing (DSP) technique and therefore has a more stringent requirement on the noise performance [5]. To implement FRAs in MDM systems, their performance should be carefully optimized [6]. The key requirement for the multimode fiber Raman amplifiers (MFRAs) is the gain flatness with respect to the modes and the wavelengths. The unequal gain on the different modes/wavelengths will cause uneven optical signal to noise ratio and therefore induce penalty at the receiver and impact the overall system capacity [7]. Therefore, the gain among the modes as well as the wavelengths should be flattened. Currently, most of the studies on the optimization of the FRAs focus on the optimization of the gain spectrum of the FRAs with respect to the wavelengths in single mode fibers. Numerous techniques have been proposed on this topic, such as the genetic algorithm [8] and the semi-analytical expression based algorithms [9–11]. For multimode FRAs, there have been experimental and theoretical demonstrations of a MFRA pumped at a single wavelength by Ryf et al [6,7], who adjusted the pump power distributions among different modes to ensure an equalized gain for the signals at different modes. Although there have been these pioneering studies [6–11], there has been lack of a systematic optimization approach to achieve equalized gain for different modes in multimode FRAs. Furthermore, for a MFRA pumped by multiple pumps at different modes and wavelengths, a method which could achieve simultaneous optimization of the mode dependent gain and the wavelength dependent gain is highly required, but has not been discussed in the published literatures. In this paper, MFRAs pumped at different modes and wavelengths are studied. We propose an analytical approach to optimize gain among different modes and different wavelengths simultaneously. The method is based on the analytical solution of the optimal pump power integrals using the least square method (LSM) [10,11], but modified to incorporate the non-negativity-constraint on the pump power integrals [12]. After obtaining the optimal pump power integrals, full scale simulation is performed. The input pump powers at the different modes and wavelengths can be decided by the shooting method [8]. The method is performed on a single wavelength pumped MFRA which is the same as the one in [7], and the optimization results match very closely. A multi-wavelength-pumped MFRA is studied afterwards, and less than 1dB total gain variation is achieved among 4 different propagation modes in the wavelength range of 40nm. 2. Mathematical modeling 2.1 General mathematical formulation Assuming that there are total N signal wavelengths and M pump wavelengths in the MFRA and the number of the propagation modes is K, the evolution of the powers of the signals and the pumps obeys the following coupled equations [6, 7, 10, 11, 13, 14]:
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21394
= −α i Si , m + g (ν i ,ν j ) f i , m , j , n Pj , n Si , m + g (ν i ,ν i ' ) fi , m ,i ', m ' Si ', m ' Si , m dz i ′≠ i , m ' j ,n (1) dPj , n − = −α j Pj , n + g (ν j ,ν j ' ) f j , n, j ', n ' Pj ', n ' Pj , n + g (ν j ,ν i ) f j , n,i , m Si , m Pj , n dz i,m j '≠ j , n ' where S and P stand for the powers of the signals and the pumps respectively, fi,m,j,n stands for the power overlap integral between the modes m and n at the wavelengths i and j, dSi , m
+∞ +∞
I ( x, y ) I ( x, y ) dxdy i,m
fi ,m, j ,n =
+∞ +∞
−∞ −∞
j ,n
+∞ +∞
(2)
I ( x, y ) dxdy I ( x, y ) dxdy i,m
j ,n
−∞ −∞
−∞ −∞
the Raman gain coefficient is described by γ R (ν j −ν i ) (ν i < ν j ) g (ν i ,ν j ) = ν i (3) − γ R (ν i −ν j ) (ν i > ν j ) νj where γR indicates the Raman gain between the frequency νi and νj. Equation (1) is derived based on the theories and principles described in [13,14], and is analogous to the equations in [6,7]. First of all, the coupling between the modes with large propagation constant difference is ignored [15]. The modes with identical or similar propagation constants are considered as one mode in Eq. (1), because the strong coupling among these modes equalizes the gain within the mode group [7, 13] and they are equivalent to one mode during the mathematical treatment. Secondly, the multimode transmission fiber usually has the length of tens of kilometers. The long fiber introduces large mode dispersion, and therefore the random coupling processes for the pumps and the signals are independent. So the random variations of the intensity distribution of the pump and signal waves are decorrelated and will not contribute to the amplification process. It should be noted that other propagation effects, such as the mode coupling and four wave mixing, are outside the scope of this paper. Usually they can be ignored during the gain design for the multimode Raman amplifiers as explained by [13]. The first equation describes the signal gain and can be rewritten as L L S ( L) = −α i L + g (ν i ,ν j ) f i , m , j , n Pj , n dz + g (ν i ,ν i ' ) f i , m ,i ', m ' Si ', m ' dz (4) ln i , m S (0) 0 0 j ,n i ′≠ i , m ' i,m The first term on the RHS of the equation is the fiber loss, the second term is the Raman on-off gain, and the third term is the signal-signal Raman interactions. Usually, the signalsignal interactions are neglected due to their small values (or to be considered in the later optimizations as demonstrated in section 4), and the Raman on-off gain is the main parameters to be optimized. The on-off gain at the ith signal wavelength and the mth signal mode can be described by the following equation
L
Gainon − off = 10 log e g (ν i ,ν j ) f i , m , j , n Pj , n dz j ,n
(5)
0
Equation (5) can be rewritten into the matrix form as
Gain on − off = 10 log eCp (6) where the (iK + m) element of the vector Gainon-off is described by Eq. (5) and the element of the matrix C is calculated by th
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21395
g (ν i ,ν j ) f i , m , j , n
th
(7)
the (iK + m) element of the vector p is the pump power integral L
p ( jK + n ) = Pj , n dz
(8)
0
Assuming that the targeted gain for each signal wavelength and mode is described by a vector t, the optimization problem will be minimized by the following formula Gain on − off − t
2
(9)
According to LSM [10,11], the optimal solution for the pump power integral vector is −1 1 CT C ) CT t (10) ( 10 log e It should be noted that Eq. (10) gives the optimal power integrals without constraints. However, the power integrals should always be positive while Eq. (10) could lead to negative results. This might not be a problem in single mode FRA optimization, because the number of variables to be optimized is much smaller than the number of the elements in the target vector. However, in MFRA optimization, especially in the mode-dependent gain optimization, the number of variables to be optimized could be the same as the number of variables in the target vector, and an optimal solution with negative values is very likely to occur. Therefore, we propose to optimize the function, i.e. Eq. (9) via non-negative least square method (NNLSM) [12]. The method is briefly described below. If the optimal power integrals are positive, we define the corresponding non-negativityconstraints to be passive (not active). If some of the non-negativity-constraints do function during the search for the optimal power integrals, the corresponding pump power integrals will be 0 and we define the constraints to be active. If the active set and the passive set for the constraints are known, the optimal power integral can be obtained as follows. For the power integrals whose constraints are passive, their optimal values can be calculated via formulas similar to Eq. (10), while for the power integrals whose constraints are active, they are set to be 0. To determine whether the constraints are active or passive, an iterative procedure is designed [12]. For the details of the algorithm, the readers are encouraged to refer to [12]. For readers' convenience, it is summarized as follows [12].
p=
1, The initial passive set P is assumed to have no elements and the active set A is assumed to contain all of the non-negative constraints. The initial optimal pump power integrals are assumed to be 0. 2, Calculate the derivative of Eq. (9) at the optimal pump power integral values, i.e. 2 (10 log e ) CT ( CPopi − t ) 2
(11)
3, If all of elements of the above vector are non-negative, the optimal pump power integrals are obtained. If any of the elements are negative, find the minimum element and add the corresponding non-negative constraint to the passive set P and remove it from the active set A. 4, Find the optimal pump power integrals corresponding to the passive set P according to a formula similar to Eq. (10), while setting the pump power integrals corresponding to the active set A to be 0. Denote the optimal pump power integral vector as Popt1. 5, If all of the elements of Popt1 are non-negative, update Popt as Popt1. Go to step 2. If any elements of Popt1 are negative, calculate the constant β and update Popt as
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21396
β = − min Popt ( n ) / ( Popt ( n ) − Popt1 ( n ) ) n∈P
Popt
update
(12)
= Popt + β ( Popt1 − Popt )
and update set A and set P. Go to step 4. It should be noted that the optimization seldom enters step 5 [12]. With the above procedures, all the optimal power integrals are calculated and the input pump power can be obtained by the shooting method [8, 16]. 2.2 The case of non-wavelength-dependent power overlap integrals In the previous sub-section, there are totally NK optimal pump power integrals to be determined. Therefore, it is quite computational expensive. In this sub-section, a simpler model is developed to resolve this problem. In multimode fibers, the mode profile varies with respect to the wavelengths, but the variation is relatively small within the wavelength range of 1400nm-1600nm. Therefore, Ryf et. al. [6,7] proposed to simplify the analysis by assuming that the overlap integrals of the modal powers are not wavelength dependent. In such cases, the gain expression can be simplified as L
Gainon − off = 10 log e g (ν i ,ν j ) f m , n Pj , n dz j
n
(13)
0
The corresponding matrix formula is Gain on − off = 10 log e ( g ⊗ f ) p (14) where ⊗ denotes the Kronecker product of the matrix. The elements of matrices g, f and the vector p are
g ( i, j ) = g (ν i ,ν j )
f ( m, n ) = f m , n
(15)
L
p ( jK + n ) = Pj , n dz 0
It is reasonable to assume the expected gain target for each mode at the same wavelength to be the same (flat). In such cases, t = t g ⊗ t f , where tg stands for the targeted gain related to the wavelength and tf stands for the targeted gain related to the mode. Hence, the optimal pump power integrals without constraints are
(
) (
)
−1 −1 1 gT g ) gT t g ⊗ ( f T f ) f T t f (16) ( 10 log e From Eq. (16), it can be concluded that if the power overlap integrals are not wavelength dependent, the problem of optimizing MFRA gain can be decomposed into two sub-problems, which are the optimization of the gain with respect to the wavelengths and the optimization of the gain with respect to the modes. All of the pumps at the different wavelengths should have the same power fractions at different modes. To implement the non-negative constraint, similar procedures to determine the active or passive constraints will be performed on the two sub-optimization problems. However, during the optimization of the wavelength dependent gain, this is usually not necessary. The decomposition greatly simplifies the optimization efforts in comparison with the ones in the previous sub-section. However, it should be noted that when the power overlap integrals are wavelength dependent, the method in the previous sub-section will predict a better optimal solution for the pump power integrals.
p=
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21397
3. Verification of the proposed formulas and the optimization algorithm To verify the proposed formulas and the optimization method, we apply them to the published results in [6,7]. In [7], a multimode Raman fiber amplifier in a step index (SI) few mode fiber (FMF) is studied. The cladding refractive index and the core refractive index of the fiber are 1.46 and 1.4673 respectively [7]. The normalized frequency V = 5 and therefore, it will support four mode groups. The power overlap integrals are listed below and assumed to be wavelength independent [7]. The pump wavelength is 1455nm, while the signal wavelength is 1560nm [7]. Since there are no pump-pump interactions in a single-wavelength pumped MFRA, the optimal pump power integrals are directly related to optimal input pump powers at different modes, and there is no requirement to use the shooting method. Table 1. The power overlap integrals from [7] (in 109/m2) LP01
LP11
LP21
LP02
LP01
6.24
4.12
2.85
4.62
LP11
4.12
4.36
3.81
2.33
LP21
2.85
3.81
3.88
2.12
LP02
4.62
2.33
2.12
6.15
Based on NNLSM, the optimal percentage for the four pump power integrals/input powers at the four pump modes to achieve the flat modal gain are listed as follows. Compared with the optimized simulation values in [7], there is an exact agreement. Table 2. The optimal pump power percentage on each mode obtained by NNLSM. pump integral in Percentage
Modes
NNLSM
Reference [7]
LP01
0.00%
0.00%
LP11
0.00%
0.00%
LP21
69.67%
69.60%
LP02
30.33%
30.40%
One thing worth mentioning is that the degenerated modes, which have identical propagation constants, are regarded as one mode group in the analysis. For example, LP11a and LP11b are combined as LP11, because the strong coupling will equalize the gain within the mode group [7, 13]. The intensities of the mode groups result from the combination of the original degenerated modes [7, 14]. (The details of the intensities of the mode groups can be found in [7].)Therefore, the gain equalization for the degenerated modes is not necessary [7, 13]. 4. An optimization example for a multimode Raman amplifier with multi-wavelength pumps To further demonstrate the capability of the optimization algorithm, a MFRA with multiple pump wavelengths is used as an example. The Raman amplifier is based on the same multimode fiber discussed in section 3. The length of the multimode fiber is assumed to be 50km. The Raman gain coefficient is from the published literature [17]. Four pumps are used in the simulation, which are located at the wavelengths of 1420nm, 1430nm, 1450nm, and 1465nm. Totally 50 signal wavelengths are used in the simulation, which are within the Cband, starting from 1525.2nm to 1565nm with 100GHz channel spacing. The total signal input power is −3dBm, which corresponds to −20dBm input power per wavelength. Counter pumping scheme is assumed in the simulations. #210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21398
It is assumed that the power overlap integrals are not wavelength dependent [7]. Therefore, the optimization can be accomplished via two steps, i.e. the steps to optimize the mode dependent gain and the wavelength dependent gain. By employing NNLSM, the optimal pump power integrals on each mode and wavelength are obtained. The input pump powers are calculated afterwards using the shooting method. First of all, the targeted on-off gain for the Raman amplifier is set to be 10dB. The optimized on-off gain profile for each mode is plotted in Fig. 1. It can be inspected from the figure that the residual wavelength dependent gain for each mode is only 0.5dB and the maximum mode dependent gain is about 0.15dB. The overall gain variation is about 0.7dB. The calculated input pump powers are listed in Table 3. It can be observed that the optimal input pump power ratio on each mode matches the optimal pump integral ratio illustrated in Table 2. 10.4 10.3 10.2
on-off gain(dB)
10.1 10 9.9 9.8 9.7 9.6 9.5 9.4 1525
LP01 gain LP11 gain LP21 gain LP02 gain 1530
1535
1540 1545 1550 wavelength(nm)
1555
1560
1565
Fig. 1. The gain profile of an optimized C-band MFRA with the targeted gain of on-off 10dB. Table 3. The optimal pump power at each wavelength and mode when the target on-off gain is 10dB.
wavelength(nm)
1420
1430
1450
1465
mode
power(mW)
percentage
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
190.81
69.67%
LP02
83.05
30.33%
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
171.31
69.67%
LP02
74.57
30.33%
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
258.48
69.67%
LP02
112.51
30.33%
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
150.57
69.67%
LP02
65.54
30.33%
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21399
To illustrate the robustness of the algorithm, the targeted on-off gain is changed to 15dB. The gain profile for each mode is plotted in Fig. 2. It can be observed that the wavelengthdependent gain for each mode is less than 0.8dB and the total gain variation is less than 1dB. The effectiveness of the method has been demonstrated. The calculated input pump power is listed in Table 4. Although there are small deviations, the optimal input pump power ratio on each mode still matches the ratio in Table 2. 15.6
15.4
LP01 gain LP11 gain LP21 gain LP02 gain
on-off gain(dB)
15.2
15
14.8
14.6
14.4
14.2 1525
1530
1535
1540 1545 1550 wavelength(nm)
1555
1560
1565
Fig. 2. The on-off gain profile of an optimized C-band MFRA with the targeted on-off gain of 15dB. Table 4. The optimal pump power at each wavelength and mode when the target on-off gain is 15dB.
wavelength(nm)
1420
1430
1450
1465
mode
power(mW)
percentage
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
326.48
69.73%
LP02
141.72
30.27%
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
280.14
69.69%
LP02
121.84
30.31%
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
367.10
69.67%
LP02
159.78
30.33%
LP01
0.00
0.00%
LP11
0.00
0.00%
LP21
186.61
69.64%
LP02
81.35
30.36%
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21400
It should be noted that the algorithm cannot only optimize the flattened on-off gain spectrums, but also be used to optimize the on-off gain spectrums with a non-zero slope. For instance, the example illustrated in Fig. 2 shows a flat on-off gain spectrum. However, due to the loss slope and the signal-signal Raman interactions, there is a higher net gain at the longer wavelength as illustrated in Fig. 3. Since the passive loss of the 50km fiber is about 10dB, Fig. 3 (net gain) differs from Fig. 2 (on-off gain) by about 10dB in average gain. To optimize the net gain profile of the multimode Raman amplifier, one needs to set the targeted on-off gain profile with the inverse gain slope with respect to Fig. 3. Using the new targeted on-off gain profile, the Raman gain is re-optimized and the final net gain is flat as illustrated in Fig. 4, which is a flattened net gain profile among the modes and wavelengths. 6.4 6.2 6
LP01 gain LP11 gain LP21 gain LP02 gain
Netgain gain(dB)
5.8 5.6 5.4 5.2 5 4.8 4.6 4.4 1525
1530
1535
1540 1545 1550 wavelength(nm)
1555
1560
1565
Fig. 3. The net gain profile of the optimized C-band MFRA with the targeted on-off gain of 15dB.
6 5.8 5.6
net gain(dB)
5.4 5.2 5 4.8 4.6 4.4 1525
LP01 gain LP11 gain LP21 gain LP02 gain 1530
1535
1540 1545 1550 wavelength(nm)
1555
1560
1565
Fig. 4. The flattened net gain profile of the optimized C-band MFRA with the targeted on-off gain of 15dB.
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21401
5. Conclusion In this paper, an approach to optimize the gain profiles of the MFRAs is proposed. Analytical formulas are derived and they are combined with NNLSM to optimize the pump powers. It is revealed that under the non-wavelength-dependent power overlap integral assumption, the gain optimization problem with respect to the wavelengths and the modes can be decomposed into two sub-problems. The formulas and optimization algorithm are verified by applying them to a published MFRA and very good agreement is achieved by comparing the published results with those from the algorithm. Finally, the method is used to optimize a four-mode MFRA with 50 signal wavelengths and 4 pump wavelengths. Since full scale simulation is conducted within the algorithm, the method is both valid in the unsaturated and saturated regimes. Acknowledgment The author would like to thank three anonymous reviewers for their evaluable comments during the review process. This work is partially supported by the National science foundation of China (Grant No. 61201068).
#210983 - $15.00 USD Received 28 Apr 2014; revised 29 Jun 2014; accepted 15 Aug 2014; published 27 Aug 2014 (C) 2014 OSA 8 September 2014 | Vol. 22, No. 18 | DOI:10.1364/OE.22.021393 | OPTICS EXPRESS 21402
