## Abstract

In this paper a Raman Fiber Lasers (RFLs) with several embedded cavities are studied. A novel algorithm is proposed to solve the coupled equations describing the optical power evolution in a RFL. By using some invariant constants as the boundary condition at the output end, the problem of solving ordinary differential equations (ODEs) with guessing boundary value is translated into a two-boundary-condition ODE problem. The algorithm is based on Newton-Raphson method and proved rather fast and stable. Quantitative analysis is performed based on the algorithm.

© 2006 Optical Society of America

## 1. Introduction

Raman fiber lasers (RFLs) have been widely studied as efficient all-fiber wavelength converters. These cost-effective lasers are depolarized and have larger output lasing power compared with semiconductor lasers. Therefore, they are suitable pump sources for multiwavelength pumped Raman fiber amplifiers (RFAs).[1] RFLs with multiple nested cavities have been paid special attention since they can transfer optical power from short wavelengths such as 1067 or 1318.8 nm to desired wavelengths in the C band. To better design and analyze RFLs, the famous coupled equations have to be solved numerically to investigate the power evolution along the fiber and good agreement with the experimental measurement has been achieved [2–6]. However, the lasing power is reflected at the ends of the cavities and the boundary condition of the ODE is not a simple constant value. Hence, the initial value has to be guessed while using the shooting method. Since the target value at the output end varies during every iteration, conventional mathematical tools such as Newton Raphson method can not be applied. Consequently, the algorithm is not only time consuming, but also unstable if the guessed initial value is not properly chosen. To accelerate the calculation speed and ensure the stability, several techniques have been proposed [7–10]. Ref. [7–9] try to accelerate the integration speed by using the modified Runge-Kouta method. However, this does not solve the problem of guessing suitable initial values. In Ref. [10] the coupled equations for RFL are impressively transformed into two-boundary-condition ODEs to ensure the stability of the solution. However, there are still some unknown constants that remain undetermined in the modified equations before integration. To guess these constants takes additional computational time. In this paper we propose a novel method to transform the problem of ODEs with guessed boundary values into a problem of ODEs with constant two-boundary conditions, where no additional unknown constants are involved. The mathematical translation makes it possible to apply an algorithm based on the Newton-Raphson method to analyze RFLs with several embedded cavities. Fast convergence and good accuracy are achieved.

## 2. Numerical model

The power evolution of the pump and Stokes waves in the optical fiber can be characterized by the following coupled equations [11]:

with the boundary condition of:

$${P}_{1}^{-}\left(L\right)={R}_{L1}{P}_{1}^{+}\left(L\right)\phantom{\rule{7.1em}{0ex}}$$

$$\left.\begin{array}{c}{\phantom{\rule{.5em}{0ex}}P}_{i}^{+}\left(0\right)={R}_{0i}{P}_{i}^{-}\left(0\right)\\ {\phantom{\rule{.5em}{0ex}}P}_{i}^{-}\left(L\right)={R}_{\mathit{Li}}{P}_{i}^{+}\left(L\right)\end{array}\right\}\left(i=2\cdots n\right)\phantom{\rule{3.2em}{0ex}}$$

where n is the number of waves including the pump wave and the Stokes wave, *P*_{i}
and *P*_{j}
are the powers of the ith and jth wave propagating along the fiber, +/- stands for the wave forward/backward propagation, *g*(*ν*_{i}
,*ν*_{j}
) is the Raman gain coefficient between frequency *ν*_{i}
and frequency *ν*_{j}
:

where *A*
_{eff} is the effective area of the fiber, factor 2 stands for the consideration of the polarization effect, *gr*(*ν*_{i}
- *ν*_{j}
) is the Raman gain spectrum measure at the frequency of *ν*_{i}
which has a peak at the frequency of 14T Hz, R is the reflectivity of corresponding mirrors, subscript 0 and L stand for the input and output ends.

We did not include the amplified spontaneous emission (ASE) and the Raleigh backward scattering in Eq. (1). They may slightly affect the output power near the threshold; however, their effect is negligible when the input pump power is far beyond the threshold power.

From Eq. (1), we may easily obtain the relation of ${P}_{i}^{+}$
${P}_{i}^{-}$ = *C*_{i}
, where *C*_{i}
is a constant along the z axis. By dealing with the boundary condition and resorting to ${P}_{i}^{+}$
${P}_{i}^{-}$ = *C*_{i}
, one can obtain

$$\frac{{P}_{i}^{+}\left(L\right)}{{P}_{i}^{+}\left(0\right)}=\sqrt{\frac{1}{{R}_{0i}{R}_{\mathit{Li}}}}i=2\cdots n$$

Since ${P}_{i}^{+}$(0) and ${P}_{i}^{-}$(0) are correlated, there are only n unknown inputs,${P}_{i}^{+}$(0)(*i* = 2⋯*n*) and ${P}_{i}^{-}$(0). We treat$\frac{{\left({P}_{1}^{-}\left(L\right)\right)}^{2}}{{R}_{L1}{P}_{1}^{-}\left(0\right)}$and $\frac{{P}_{i}^{+}\left(L\right)}{{P}_{i}^{+}\left(0\right)}\left(i=2\cdots n\right)$ as outputs. Now the problem is to find suitable input values that lead to corresponding outputs satisfying Eq. (3). To clarify the fact, we rewrite the mathematical relation mentioned above into the following equations:

where:

$$\mathbf{P}\left(0\right)=\left(\begin{array}{c}{P}_{i}^{-}\left(0\right)\\ {P}_{2}^{+}\left(0\right)\\ \vdots \\ {P}_{n}^{+}\left(0\right)\end{array}\right)$$

the target output is:

The New-Raphson method can be used to obtain the solution that gives the target output with assistance of the Jacobi matrix. The Jacobi matrix **J** is defined as $\mathbf{J}\left(i,j\right)=\frac{\partial {\mathit{output}}_{i}}{\partial {P}_{j}}$.

According to its definition, it can be obtained by the following means:

First, the output of the steady state should be stored. Then one should increase the ith element of the input *P*_{i}
(0) by a small disturbance Δ*P*_{i}
(0) and obtain the output. By subtracting the state output from the output and dividing the result by Δ*P*_{i}
(0), the ith column of the matrix is obtained.

The procedure of Newton-Raphson method is as follows:

- Initial guessing values are given at the input.
- The coupled equation is integrated by Runge-Kouta method, and the error between the output and the target output
**Δoutput(**is obtained.*L*) - The Jacobi matrix is calculated and the input is updated by adding
**ΔP(0) = J**^{-1}**Δoutput(L)** - Iteration stops if the error is below a small threshold, else, go to procedure A.

## 3. Simulation results and discussion

We have calculated a Raman laser with three embedded cavities. The gain media is a piece of dispersion shifted fiber (DSF) with the length of 500m. The structure is illustrated in Fig. 1. We choose the pump wavelength at 1318.8 nm, so that the high power Yag laser can provide enough pump power at this wavelength [12]. The three cavities can be formed by six fiber Brag gratings (FBGs) and the pump power is reflected back by another FBG at 1318.8nm to have it fully utilized.

The pump power and the lasing power evolution along the fiber are illustrated in Fig. 2. From the figure we can see that the boundary condition of Eq. (1) is satisfied automatically. The error is less then 1e-4. The calculation of the solution takes less than one second using a conventional personal computer, i.e., Intel Pentium 4, 2.0 GHz. The powers of the Stokes waves at 1405nm and 1485nm maintain almost constant during the propagation. This is caused by two reasons. Firstly, the net gains of the forward and backward propagated waves at the two wavelengths are close to 0 dB, hence, the powers at the input end and the output end are almost equal. Secondly, the fiber length is rather short so that the pump power is not fully depleted.

Figure 3 shows the relationship between the input pump and the output lasing power. It can be seen that there is almost linear relationship between the input pump power and the output lasing power. The threshold pump power is about 0.6W. It is worth noting that the algorithm proposed above is only suitable for calculation when the pump power is beyond the threshold power, i.e., when there exists lasing power. Else, the calculated power will be negative. One may simply adjust the lasing power to zero when the negative power value occurs.

In Fig. 4, the reflective coefficient at the lasing wavelength at the output end varies and so does the output lasing power. The pump power is fixed to be 1W. It can be seen that the laser reaches the maximum output when the reflectivity is around 75%.

## 4. Conclusion

We theoretically investigated the Raman fiber laser with several embedded cavities. Using some simple relations, we obtain some constants at the output end and therefore turn the ODEs with unknown values at two boundaries into ODEs with definite boundary conditions and no additional unknown constants. Based on the Newton-Raphson method, the coupled equations are solved. Our results show that the algorithm is stable and efficient.

## Acknowledgments

This work is partially supported by NSFC (ID: 60377013, 90204006, 60507013), Ministry of Education, China (ID:20030248035) and STCSM(ID: 036105009)

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