Bidirectionally pumped silicon Raman lasers

In this section we introduce the bidirectionally pumped silicon Raman laser, where pump power is injected from both ends of the waveguide instead of only one end, see Fig. 7.6.

This pumping scheme significantly increases the tolerance of silicon Raman lasers against FCA and leads to more efficient lasing [KRB05c].

Figure 7.6.: Schematic of a bidirectionally pumped silicon Raman laser.

7.2.1. Model

Our model for bidirectionally pumped silicon Raman lasers is essentially that which we have used in Sect. 7.1 to analyze single-side-pumped lasers. The only change is a straightforward modification of the boundary conditions: the reflection of the waves at the two end faces of the silicon waveguide (at z = 0 and z =L) with the reflectivities R{p,s},{l,r} and the input coupling of the left-hand and right-hand pump powers P_0,{l,r}

with efficienciesT_p,{l,r} are taken into account by the new boundary conditions (see also Fig. 7.6),

P_p⁺(0) =T_p,lP_0,l+R_p,lP_p⁻(0), P_s⁺(0) =R_s,lP_s⁻(0), (7.8) P_p⁻(L) =T_p,rP_0,r+R_p,rP_p⁺(L), P_s⁻(L) = R_s,rP_s⁺(L). (7.9) The output power of the laser is P_out = P_s⁺(L)(1 −R_s,r). We will consider only lasers which have been prepared such that they have a left-hand Stokes reflectivity of R_s,l = 100% (i. e., we want to have all the laser output power on the right-hand side), whereas the other reflectivities have the Si–air Fresnel-reflectivity value ofR_p,l =R_p,r = R_s,r = 30%.

In practice, bidirectional pumping of the Raman laser could be achieved by the use of two separate pump-laser diodes at the left-hand and right-hand ends of the silicon waveguide, the power of which could be controlled independently. For clarity, however, we choose to present our results for a situation where we have only one pump laser, the output powerP₀ of which is split between the left-hand and right-hand ends according to the splitting ratioρ, i. e., the left-hand and right-hand pump powers areP_0,l = (1−ρ)P₀ and P_0,r =ρP₀. Thus, the caseρ= 0 corresponds to conventional single-side pumping.

7.2.2. Lasing and shutdown thresholds

We start by demonstrating the effect of the introduction of bidirectional pumping on the threshold powers of the laser. Fig. 7.7a shows the lasing and shutdown thresholds for single-side pumping (ρ = 0) as a function of the laser length for various carrier lifetimesτ_eff. For a givenτ_eff, lasing is possible only inside the corresponding closed egg-shaped curve delimited by the two thresholds as discussed in Sect. 7.1.2. For example, for τ_eff = 4.0 ns, lasing is only possible for waveguide lengths between 2.5 and 8.1 cm (marked with dots in Fig. 7.7a). In particular, the laser with L= 7 cm (marked with a vertical arrow in Fig. 7.7a) starts lasing at a pump power of 1.8 W, and stops lasing again at the shutdown threshold of 4.7 W due to excessive FCA, delivering maximum output

Figure 7.7.: Threshold pump-laser pow-ers for lasing (solid) and shutdown (dot-ted) versus laser length L for several effective lifetimes τ_eff. (a) single-side-pumped lasers (ρ = 0), (b) lasers are bidirectionally pumped with a pump-power splitting ratio of ρ = 50%, i. e., the pump-laser power is split equally between the two waveguide ends.

shrinks, and it closes completely at the maximum tolerable lifetime of about 4.65 ns.

Fig. 7.7b shows the corresponding curves for the case where the pump power is split equally between the left-hand and right-hand ends, i. e., the pump-power splitting ratio is ρ = 50%. The maximum tolerable lifetime is now about 5.82 ns, which is higher by 25% as compared to the single-side-pumped case, an indication of higher tolerance against FCA. For shorter lifetimes, the laser length can now be chosen more freely.

The reason for the improved tolerance against FCA is easily illustrated. Consider the laser with τ_eff = 5 ns and L = 10 cm at a pump-laser power of P₀ = 1.8 W. When bidirectionally pumped with ρ = 50%, the laser is exactly at threshold (marked with a thick dot in Fig. 7.7b). The corresponding pump-power distribution is plotted as the thick solid and dashed lines in Fig. 7.8a, and the thick line in Fig. 7.8b shows the distribution of the resulting net Stokes gain (defined in Eq. (7.7)). Its integral along the waveguide amounts to 2.6 dB. In contrast, the thin curves in Figs. 7.8a and 7.8b correspond to the case of single-side-pumping. It can be seen that here the pump power drops more rapidly at the left-hand side of the waveguide due to higher FCA than in the bidirectionally pumped case. As a consequence, the net Stokes gain is lower: its integral is only −0.077 dB, meaning net loss. Thus, bidirectional pumping is more efficient in providing Stokes gain for the same pump-laser power.

Figure 7.8.: Comparison between single-side-pumped (thin) and bidirec-tionally pumped (thick) waveguides at the same total pump powerP0. (a) lon-gitudinal distribution of pump powers (solid: forward, dashed: backward),(b) local Stokes gain ¯γ.

0 1 2 3 4 5 6 7 8

0 5 10 15 20 25 30 35 40

Pump-laser power P

0 [W]

OutputpowerPout[mW]

r= 0%

r= 90%

r= 20%

r= 70%

r= 50%

Figure 7.9.: Input-output characteristics of the laser marked with a vertical arrow in Fig. 7.7a for several values of the pump-power splitting ratioρ.

Input-output characteristics

Now we look at the influence of bidirectional pumping on the laser characteristics. Con-sider the laser configuration marked with a vertical arrow in Fig. 7.7a, i. e., the laser with L = 7 cm, τ_eff = 4 ns and single-side pumping (ρ= 0). Its characteristic is shown as the solid curve in Fig. 7.9. The dashed curves in the same figure show the character-istics when the same laser is bidirectionally pumped with the same total pump power but several different splitting ratios. All of these lasers are more efficient than their single-side-pumped counterpart, the most dramatic efficiency increase by a factor of 2.7 occuring at a pump-power splitting ratio of ρ= 50%.

Figure 7.10.: (a): Maximum output powers of single-side-pumped (solid) and bidirectionally pumped (dashed) silicon Raman lasers as a function of the effective carrier lifetime τ_eff. Length L and pump-power splitting ratio ρ have been varied to find the maximum out-put power for each τeff. The optimiza-tion results for L and ρ are plotted in part(b).

Optimization for various lifetimes

In this section we investigate how the laser-efficiency increase provided by bidirectional pumping depends on the effective carrier lifetime τ_eff. For each value of τ_eff, we first optimize the lengthL of a single-side-pumped laser such that it produces the maximum possible output power. Simultaneously, we optimize the length L and the pump-power splitting ratioρ of a bidirectionally pumped laser. We assume that the available pump-laser power is limited to 8 W. The resulting maximum output powers for both pump-laser types are shown in Fig. 7.10a as a function ofτ_eff. As expected, bidirectionally pumped lasers are more efficient than single-side-pumped lasers, and the lasing efficiency is improved the more dramatically the larger τ_eff is. For example, when the lifetime is τ_eff = 3 ns, bidirectional pumping increases the maximum output power by 2.3 dB, whereas atτ_eff = 4.5 ns, the increase is already 10 dB. It can also be seen, in agreement with Fig. 7.7a, that single-side-pumped configurations do not lase as τ_eff exceeds 4.6 ns. Fig. 7.10a also shows that this maximum tolerable lifetime is larger for bidirectionally pumped lasers.

Fig. 7.10b shows the laser lengths and splitting ratiosρcorresponding to the optimized lasers from Fig. 7.10a. For large τ_eff, the optimized bidirectionally pumped lasers have a splitting ratio of 50%. The reason for this is as follows: because the Stokes powers are low, they nearly see the undepleted-pump gain which in turn is maximal when the left-hand and right-hand pump powers are kept as low as possible, i. e., when the pump power is equally split between both ends (ρ = 50%, see Fig. 7.8). The optimal device

Figure 7.11.: Schematic of a tapered silicon Raman laser.

lengths increase with τeff until the maximum tolerable lifetimes of 4.65 and 5.82 ns, respectively, are reached (see Sect. 7.2.2). The optimal device lengths at those values of τeff correspond to the points towards which the closed threshold curves in Figs. 7.7a and 7.7b contract for increasingτeff.