1. Introduction

Stimulated Raman scattering (SRS) is a nonlinear process where the optical electric field interacts with phonons in the gain medium. The all-fiber design provides a long gain medium with a good confinement and beam quality. Furthermore, since silica is an amorphous medium, there is a continuum of phonon energy which yields a broad gain spectrum. All-fiber Raman lasers have attracted much attention because of their high power and customizable operating wavelength [1,2]. They are mostly used to pump amplifiers in long distance optical fiber links. The equations describing the steady-state regime of the cascaded Raman fiber laser has been previously solved [3–5]. We are herein looking at the transient regime Raman lasers from the switch-on to steady-state.

2. Theoretical model

 

Figure 1. Scheme for a nth-order cascaded Raman fiber laser. λ _j represents the wavelength of the jth Stokes wave. Only the output coupler (OC) is not highly reflective. The vertical lines represent the Bragg gratings used as reflectors with reflectivity R±_j.

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The laser cavity is shown in Fig. 1. The classical treatment of the stimulated Raman scattering process in optical fibers yields the following system of 2n+2 first-order coupled partial differential equations [6,7], where n is the number of Stokes waves.

Here h is the Planck constant, f_j the frequency of the jth Stokes wave (j=0 being the pump) and

is the noise due to spontaneous Raman scattering. It follows a Bose-Einstein distribution with k_B , Boltzmann constant and T, the temperature [8]. All fields are assumed to be unpolarized. The positive and negative superscripts stand for forward and backward propagation respectively at the group velocity v_gj. P_j refers to the power of the jth Stokes wave. The stimulated Raman gain is given by g_j in (W·m)^-1 and scales inversely with wavelength and the mode confinement [3,8]. The intrinsic loss caused by Rayleigh scattering and OH^- impurities is represented by α_j in m^-1 at frequency f_j . The jth Stokes wave interacts with the (j-1)^th and (j+1)^th Stokes waves traveling in both directions. The boundary conditions are given by the injected pump power and by the Bragg gratings at each end of the fiber. It is assumed that the linewidth of the laser is smaller than the Bragg gratings spectral width but is large enough for the stimulated Brillouin scattering to neglected.

The positive and negative superscripts of the reflectivity refer to the output side and the injection side of the cavity respectively and P _in is the injected power. The reflectors thus couple the forward-propagating Stokes waves with the backward-propagating ones of the same order j^th . Typically only $R_n^{+}$ does not have a high reflectivity (since it is the output mirror). To analyze the transient regime from the first injection, we use the following initial conditions:

where P _in is the injected pump power.

3. Numerical method

To solve Eqs. (1) we use the so-called method of lines which consists of a discretization along one of the independent variables (z or t) in order to transform the system of partial differential equations into a system of ordinary differential equations (ODE). Each of the N discrete parts is considered a new variable, so that one has N times as many variables and thus equations to solve. For instance, if the z axis is separated in 100 parts (z ₁ to z ₁₀₀), each P_k (t)=P(t, z_k ) is treated as a new variable. In our case, it is easier to discretize along z because it yields 2N(n+1) time-dependent ODEs with initial conditions. If we were to discretize along t, we would have to solve 2N(n+1) z-dependent ODEs with boundary conditions which is a more difficult task. The spatial (∂/∂z) derivatives are transformed into their finite forward difference equivalent, thus coupling two adjacent P_k (t) (we do not use the centered scheme even though it is more precise because it is fundamentally unstable for transport equations such as Eqs. (1)). The boundary conditions are included in the new equations by substituting the finite difference elements by Eq. (3) at the fiber ends. To solve the ODE system, we use a standard adaptive stepsize Runge-Kutta integrator with the initial conditions given by Eq. (4) [9].

4. Results: Switch-on

The simulated laser is a sixth-order cavity (n=6), of length L=150 m pumped at λ ₀=1064 nm (with an output at 1480 nm), with $R_j^{\pm }$=96.7% except for the output coupler which is $R_6^{+}$=50%. The fiber is single mode at 980 nm with characteristics similar to PureMode 1060^TM from Corning; they are given in Table 1. The transient regime from first injection to the steady-state is shown in Fig. 2 for all the 6 Stokes waves in the center of the cavity. Figure 3 shows a close-up of the first 20 microseconds. Figure 4 shows the whole transient regime. The roundtrip time in the cavity is

The pump injected at z=0 in the cavity propagates until it is reflected at z=L by the Bragg grating ($R_0^{+}$=0.967). It then goes back and exits the cavity by the injection side (z=0). However, while propagating, a small fraction of the pump power is transfered to the first Stokes wave and amplifies the spontaneous noise. The power transfer is small because it is proportional to the product of the pump and the first Stokes wave power, the latter being only spontaneous emission at this point. At t=5 µs≈3τ, considerable power has been transfered to the first Stokes wave which raises the transfer rate. The process reinforces itself since the rate increases with the first Stokes power. Thus all the power is rapidly transfered to the first Stokes wave

Table 1. Fiber characteristics.

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until the pump is depleted. The pump remains depleted as long as there is a high power level in the first Stokes wave. The double peak feature of the first Stokes wave is due to the fact that the power distribution in the cavity is not uniform; there is a large pulse which is longer than the roundtrip time. Hence, the power in the middle of the cavity lowers and raises with the displacement of the pulse.

 

Figure 2. Evolution through time of Stokes power in the middle of the cavity at z=L/2 during switch-on. The laser is a 6 ^th -order cascade with a 150 m long cavity and 96.7% mirrors except for the 50% output coupler. The curves are offset by 10 W for comparison.

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Since the first Stokes wave is in a Fabry-Perot resonator, its power can be larger than the injected power inside the cavity. The transfer rate to the second Stokes wave is thus much faster (the first Stokes wave is thus rapidly depleted), resulting in a narrower pulse. The propagation of this pulse leads to the three peaks we see in $P_2^{+}$. Meanwhile, the pump power rises in the cavity because the power of the first Stokes wave fell back to a level too low to significantly deplete it. The second Stokes wave then empties itself into the third and so on while the process is beginning anew with the pump. The peak powers attained by the Stokes waves are as high as 60 W (for the fourth Stokes wave). The process continues this way until it reaches the last Stokes wave at t=15 µs≈10τ.

The last Stokes wave loses the majority of its power through the output mirror, so that less power can be accumulated inside the cavity and thus tends to flatten the power distribution and to eliminate high power spikes. As can be seen in Fig. 2, the last Stokes wave is the first one to stabilize (around t=30 µs) and the others follow in reverse order. There are however residual slowly relaxing oscillations. They are caused by the nonlinear nature of the transfer rate, which successively over- and undershoots the steady-state transfer rate.

 

Figure 3. Evolution through the first 20 µs of Stokes power in the middle of the cavity during switch-on. The Stokes waves’ power levels are shown superimposed so as to highlight the transfer from one wave to the other.

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Figure 4. Evolution of Stokes power in the cavity during switch-on. The colors white, red, green, blue, cyan, magenta and yellow stand for the pump and Stokes waves from n=1 to n=6. Note that the vertical scales change at t=20, 25, and 32 µs (2.6 MB).

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The high power spikes are caused for one part by the Fabry-Perot resonators that store the injected power. Due to the power-dependent tranfer rate, the spikes peak power increases with the Stokes order up to n=4. Higher orders have lower peak power because of the leakage through the output coupler for n=6. The steady-state is reached after 58 µs, corresponding to about 40 roundtrips in the cavity. The steady-stade is comparable to the one we calculated using a shooting method; the discrepancies come from the finite forward difference we had to use for stability, which is only precise to the first order.

5. Results: Switch-off

From the steady-state the laser is now switched-off. This transient relaxation regime is shown in Fig. 5. Since there is no more power accumulation in the cavity, no power peaks are present. There are however small oscillations in the power level of each Stokes due to the abrupt change of pump power. Each Stokes wave transfers its power to the following one until it is depleted. The transfer in this case gets slower because the power in each Stokes wave diminishes.

 

Figure 5. Evolution of Stokes power in the cavity during switch-off. The colors white, red, green, blue, cyan, magenta and yellow stand for the pump and Stokes waves from n=1 to n=6. Note that the vertical scales change at t=8, 23, and 36 µs (1.8 MB).

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Even though some energy is lost through the intermediate reflectors, it mostly comes out at the output coupler, which explains why the last Stokes waves is the last one to be depleted. The whole process takes 67 µs which is a little longer than the time required to reach the switch-on steady-state.

6. Conclusion

The transient regime of a n^th order cascaded Raman fiber laser has been numerically calculated. High power spikes are observed during switch-on. They are caused by the Fabry-Perot resonators and the nonlinear nature of the interaction. The duration of those spikes is comparable to the cavity roundtrip time. The steady-state is reached after about 40 roundtrips corresponding to 60 microseconds for a six cascade laser (n=6) with a 150 meter long resonator. These peaks are not observed during switch-off.