## Abstract

A mid-point shooting algorithm using the Newton–Raphson method is adopted for solving nonlinear coupled equations describing bidirectionally pumped broadband Raman amplifiers. A series of novel backward-differentiation methods are constructed for the first time to our knowledge. Their combination can form a powerful solution for fiber amplifiers. Numerical results show that the approach can solve Raman amplifier propagation equations on various conditions including co-, counter-, and bidirectionally pumped cases. The computation speed of the present methods is about four times that of the backward-differentiation methods previously adopted.

© 2004 Optical Society of America

## 1. Introduction

Wavelength-division multiplexed (WDM) transmission using distributed Raman amplification, counterpumped Raman/erbium-doped fiber amplification, and hybrid amplification of semiconductor optical amplifiers (SOAs) and Raman amplifiers has recently been reported [1–6]. Theoretical analysis and experimental results demonstrate that longer wavelength channels have larger output optical signal-to-noise ratio (OSNR) than shorter wavelength channels in backward-pumped Raman amplifiers [3,7]. However, a bidirectional pumping scheme can equalize OSNR tilt by addition of proper forward pump diodes [8, 9].

In WDM systems with fiber Raman amplifiers, the analysis of the bidirectional pump/signal propagation in fibers is essential. Therefore, the boundary value problems of model equations must be solved in fiber amplifier systems, and a shooting algorithm is adopted typically [10]. In the co- and counterpumped Raman amplifiers, the gain profile and noise performance can be obtained numerically [3, 8, 11, 12]. However, the traditional algorithms fail to accommodate bidirectional pumping schemes with multiple pump diodes and high power. In this paper, a powerful solution is proposed to overcome the difficulty of bidirectionally multipumped Raman amplifiers, for the first time to our knowledge.

## 2. Physical model and mathematical algorithm

#### 2.1. Physical model

Wave propagation and noise propagation in fiber Raman amplifiers are characterized by a variety of physical effects, and they can be described by a set of coupled equations in the steady state [11, 12], i.e.,

$$\pm h{v}_{i}\sum _{\mu >v}\frac{{g}_{\mu v}}{{A}_{\mathit{eff}}}[{P}_{\mu}^{+}+{P}_{\mu}^{-}][1+{({e}^{\frac{h(\mu -v)}{\mathit{KT}}}-1)}^{-1}]\Delta v$$

$$\mp {P}_{v}^{\pm}\sum _{\mu >v}\frac{v}{\mu}\frac{{g}_{\mu v}}{\Gamma {A}_{\mathit{eff}}}[{P}_{\mu}^{+}+{P}_{\mu}^{-}]\mp 2h{v}_{i}{P}_{v}^{\pm}\sum _{\mu >v}\frac{v}{\mu}\frac{{g}_{\mu v}}{{A}_{\mathit{eff}}}[1+{({e}^{\frac{h(v-\mu )}{\mathit{KT}}}-1)}^{-1}]\Delta v,$$

where ${P}_{v}^{+}$
and ${P}_{v}^{-}$
are optical power of forward- and backward-propagating waves within the infinitesimal bandwidth around *v*, respectively; *α*_{v}
, *η*_{v}
, *h*, *K*, and *T* are attenuation coefficient, Rayleigh-backscattering coefficient, Planck’s constant, Boltzmann constant, and temperature, respectively; *A*_{eff}
is effective area of optical fiber; *g*_{µv}
is Raman gain parameter at frequency *v* resulting from the pump at frequency *µ*; and the factor of Γ accounts for polarization randomization effects, whose value lies between 1 and 2.

Obviously, Eq. (1) is a two-point boundary value problem, and it is usually solved by a shooting algorithm [13, 14]. In our previous reports [10, 11], we proposed a “pure” shooting algorithm, where the integration proceeds from the end point (*z*=*L*) to the starting point (*z*=0), and we try to match boundary conditions at the input port of signals. Although this algorithm can solve the boundary value problems of distributed multipump Raman amplifiers on the general conditions, it is rather sensitive to starting conditions and even possibly fails for bidirectional multipump Raman amplifiers with higher power and longer span.

Conveniently, Eq.(1) is rewritten as the standard two-point boundary value problem [15, 16], i.e.,

At *z*
_{1} (i.e., *z*=0) and *z*
_{2} (i.e., *z*=*L*), the solutions are supposed to satisfy

Here, we specify a set of *N* coupled differential equations of Eq. (1), satisfying *n*
_{1} boundary conditions at the starting point *z*
_{1}, and a remaining set of *n*
_{2}=*N*-*n*
_{1} boundary conditions at the end point *z*
_{2}.

#### 2.2. Shooting algorithm

Now we propose a mid-point shooting algorithm, where the solution of Eq. (2) is implemented by integrating from both sides of the interval and trying to match continuity conditions at an intermediate point *z*_{f}
(e.g., *z*_{f}
=*L*/2). At the same time, the multidimensional Newton–Raphson method is employed in the implementation of the new algorithm. The detailed procedure is as follows:

Step 1 Given an *n*
_{2}-vector **V**
_{(1)} of starting parameters at *z*
_{1}, a particular *P*_{i}
(*z*
_{1}) is generated from Eq. (2.b), i.e., ${P}_{i}\left({z}_{1}\right)={P}_{i}({z}_{1};{V}_{\left(1\right)1,}{V}_{\left(1\right)2,\dots ,}{V}_{\left(1\right){n}_{2}})$. Then, Eq. (2.a) is integrated in the interval [*z*
_{1}, *z*_{f}
], and an *N*-vector **F**1 is constructed from the solution * P*=(

*P*

_{1},

*P*

_{2},…,

*P*

_{N})

^{T}of Eq. (2.a), i.e.,

**F**1=

**F**[

*(*

**P***z*

_{f},

**V**

_{(1)})].

Step 2 Given an *n*
_{1}-vector **V**
_{(2)} of final parameters at *z*
_{2}, a particular *P*_{i}
(*z*
_{2}) is produced from Eq. (2.c), i.e., ${P}_{i}\left({z}_{2}\right)={P}_{i}({z}_{2};{V}_{\left(2\right)1,}{V}_{\left(2\right)2,\dots ,}{V}_{\left(1\right){n}_{1}})$. Then, Eq. (2.a) is integrated in the interval [*z*
_{2}, *z*_{f}
], and an *N*-vector **F2** is formed from the solution * P*=(

*P*

_{1},

*P*

_{2},…,

*P*

_{N})

^{T}of Eq. (2.a), i.e.,

**F2**=

**F**[

*(*

**P***z*

_{f},

**V**

_{(2)})].

Step 3 Now, we want to find a vector of **V**=(**V**
_{(1)}, **V**
_{(2)}) that zeros the vector value of **W**
_{(V)}=**F1**-**F2**, i.e.,

**W**
_{(V)}=**F**[* P*(

*z*

_{f},

**V**

_{(1)})]-

**F**[

*(*

**P***z*

_{f},

**V**

_{(2)})]=0.

Based on the Newton–Raphson method, we can solve above nonlinear equations as δ**V**=-[**J**]^{-1}
**W**, where

$\delta \mathbf{V}={(\delta {V}_{\left(1\right)1,}\delta {V}_{\left(1\right)2,\dots ,}\delta {V}_{\left(1\right){n}_{2}},\delta {V}_{\left(2\right)1,}\delta {V}_{\left(2\right)2,\dots ,}{\delta V}_{\left(1\right){n}_{1}})}^{T}$

and Jacobian matrix

$\left[\mathbf{J}\right]=[\partial \mathbf{W}\u2044\partial \mathbf{V}]{|}_{z={z}_{f}}={[\partial \mathbf{W}\u2044\partial {\mathbf{V}}_{\left(1\right)},\partial \mathbf{W}\u2044\partial {\mathbf{V}}_{\left(2\right)}]|}_{z={z}_{f}}$

Step 4 Then, the revised value ${{\mathbf{V}}_{\left(1\right)}}^{\text{new}}$ and ${{\mathbf{V}}_{\left(2\right)}}^{\text{new}}$ are obtained, i.e.,

${{\mathbf{V}}_{\left(1\right)}}^{\text{new}}$=**V**
^{(1)}+δ**V**
_{(1)}

and

${{\mathbf{V}}_{\left(2\right)}}^{\text{new}}$=**V**
_{(2)}+δ**V**
_{(2)}.

Step 5 If ‖(**F1**-**F2**)/(**F1**+**F2**)‖>*ε*(*ε* is the specified relative error), go to Step 1. Otherwise, output results.

#### 2.3. Novel backward-differentiation methods

For co- and bipumped Raman amplifiers with high power and multiple diodes, Eq. (1) will be stiff differential equations. The traditional methods such as the Runge–Kutta and Adams methods may fail to solve Eq. (1). And backward-differentiation methods (BDFs) have to be applied to solve it. Although the traditional BDFs can solve stiff equations, their efficiency is poor because of Jacobian matrix calculation and multiple iterations at per step. To improve the computational efficiency, a series of novel BDFs are constructed.

Similar to the operational procedure in Refs. [10, 11, 13], Eq. (2.a) can be solved from the traditional BDF formulas, i.e.,

where * Pt* and

**P**_{t-j}are the value of

*at the*

**P***t*th step and (

*t*-

*j*)th step, and the step size Δ

*z*=

*z*

_{t+1}-

*z*

_{t}. Here we use bold scripts only to shorten notation for a set of equations, i.e.,

*=(*

**P***P*

_{1},

*P*

_{2},…,

*P*

_{N}) and

*=(*

**f***f*

_{1},

*f*

_{2},…,

*f*

_{N}). Hence, hereafter the division or multiplication of bold scripts does not have any meaning of vector calculus. After the manipulation, Eq. (3.a) is simplified as

The coefficients of Eq. (3) up to order *k*=6 are demonstrated in Table 1. For an implicit method such as BDFs, a nonlinear system of equations must be solved at each step [15], namely,

By using the Newton-Raphson method, we can obtain the iterative expression of Eq. (3)from Eq. (4), i.e.,

where the superscript *n* is the current iteration, and *n*=0, 1, 2, ….

## 3. Simulation results

From the proposed shooting algorithm and BDFs, a bidirectionally distributed multipump Raman amplifier is simulated. In the calculations, we assumed that Γ=2, *L*=100 km, *α*=0.2, and 0.35 dB/km for signals and pumps, respectively, and ignored ASE, Rayleigh scattering, and other noises. There are 19 signal channels spaced 200 GHz from 188.85 to 192.45 THz. The signal power of each channel is 1 mW. Four backward-propagating and three forward-propagating pumps are used; their powers are all 300 mW, and their wavelengths are 1425, 1445, 1465, 1485, 1435, 1455, and 1475 nm. The estimated values of each signal and each copropagating pump at* z*=*L* are 1 mW and 0.1 mW, and each counterpropagating pump at *z*=0 is estimated as 0.1 mW, which is shown in Fig. 1(a).

Figure 1 shows the power evolution of pumps and signals along the fiber position, where for (a) first iteration (i.e., estimated values), (b) second iteration, (c) third iteration, and (d) fourth iteration in the proposed shooting algorithm (see Subsection 2.2). One can see that, from Fig. 1, only four iterations can solve model equations of bidirectionally multipumped Raman amplifiers accurately, and numerical results also show that the relative error ε<10^{-6} at the fourth iteration. Simulated results demonstrate that various cases of co-, counter-, and bidirectionally pumped Raman amplifiers can be solved by the proposed methods under normal conditions. However, some methods in the previous reports [10, 11] and some commercial software (e.g., VPItransmissionMaker) may not be successful at solving the case of bidirectionally pumped Raman amplifiers. For example, when the power of each pump is enhanced to more than 150 mW in Fig.1, these methods mostly fail. The numerical results demonstrate that, when the span length and power of each pump are greater than 150 km and 1 W, respectively, the proposed algorithm also fails to solve Eq. (1). But, under general experimental conditions for distributed Raman amplifiers, each pump power is less than 1 W.

Moreover, the novel algorithm is less sensitive to the estimated starting parameters in the calculations so that it is greatly more robust than the traditional algorithms. The assumed value of intermediate point *z*_{f}
affects the efficiency and stability of the proposed shooting algorithm. Usually, it can be specified as *z*_{f}
=*L*/2. Although the traditional BDFs can also obtain Fig. 1 on the basis of our shooting algorithm, its CPU time is increased about four times in comparison with the novel BDFs under the same conditions.

## 4. Discussions

When fiber Raman amplifiers work in the saturation region or/and the bidirectional signal transmission, ASE and Rayleigh scattering will affect the gain characteristics. Specially, double Rayleigh scattering is suggested as the major limiting factor for fiber Raman transmission systems, and its impairment grows with the increase of distributed gain [17, 18]. Because of the intrinsic great stability, the proposed BDFs can solve the stiff cases of Eq. (1) if including ASE and Rayleigh scattering terms. However, because the power of backscattering pumps and backscattering signals is lower by ~30 dB and ~20 dB than their original power, and because the power of forward and backward noise is less than that of input signals by ~30 dB [11, 13, 14], we divide the simulation procedure into two steps. Firstly, ignoring all noise terms, the gain profile is obtained. From the obtained gain profile, second, ASE and Rayleigh scattering can be calculated from Eq. (1). In fact, some researchers have simulated the performance of Rayleigh scattering on the assumption that the pump is nondepleted (i.e., the small signal model) [19]. The numerical results show that, compared with the exact values, the error of our two-step method is less than 3%, whereas that of the small signal model is more than 20% in simulating the noise performance of Rayleigh scattering. Therefore, our method can have much less error in comparison with the method in Ref. [19].

By comparing with the “pure” shooting algorithm [10, 11], we see that the mid-point shooting algorithm can offer greater stability. For example, when the fiber length in Fig. 1 is reduced to be one third of original length and other parameters remain the same, the “pure” shooting algorithm still fails to solve it. Therefore, the effective span length solved by the mid-point shooting algorithm is over 3 times as long as that solved by the “pure” shooting algorithm. Simulations demonstrate that, on the basis of the parameter set in Ref. [8], the results obtained by the present methods are consistent with the reports in Ref. [8].

## 5. Conclusions

We have constructed a series of new backward-differentiation methods for the first time to our knowledge. An effective shooting algorithm for Raman amplifier propagation equations has been proposed. Their combination can offer a powerful solution for fiber amplifiers. Simulated results demonstrate that the proposed algorithms can be used effectively to solve model equations of Raman amplifiers under various conditions including co-, counter-, and bidirectionally pumped cases. New methods increase the computing speed by about four times in comparison with backward-differentiation methods in our simulations.

Xueming Liu is currently engaged in research at the School of Electrical Engineering, Seoul National University, Korea.

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