Conversion efficiency versus threshold

4 5 6 7

Longitudinal coordinate z [m]

Powers[W]

Stokes forward

Stokes backward

Pump forward

}

Figure 3.2.: Longitudinal pump- and Stokes-power distribution inside a Raman fiber laser.

is the single-pass gain experienced by both the forward- and backward-propagating Stokes waves. By then making use of the boundary conditions (3.3), one obtains the statement

P_s⁺(0)·(1−R_lR_rG²) = 0, (3.8) which must always be fulfilled by any solution of our model, no matter how large the pump power. If the laser is beyond threshold,P_s⁺(0) >0 and Eq. (3.8) shows that then the round-trip condition

R_lR_rG² = 1 (3.9)

must be fulfilled. At the lasing threshold, the depletion of the pump power by the Stokes waves can be neglected, and the longitudinal pump-power distribution is a simple exponential decay due to the linear losses, see Eq. (3.1). Thus, G can be evaluated explicitly. Inserting this into the round-trip condition (3.9) and rearranging for the pump power, one obtains the threshold pump power for arbitrary pump backreflectors Rp,

P_th= α_p

g · α_sL− ¹₂ln(R_lR_r)

(1−e^−αpL)(1 +R_pe^−αpL). (3.10) ForR_p = 0, this formula reduces to that derived in Ref. [AY79].

3.1.4. Conversion efficiency versus threshold

In this section, we show that the threshold power and the conversion efficiency at large pump powers can not be designed independently in SC-RFLs as shown in Fig. 3.1. Later in Sect. 3.2, the concept of DC-RFLs is introduced, which does not have such a restric-tion. For the sake of simplicity, we concentrate on RFLs with single-pass pumping only

0 20 40 60 80 100 Reflectivity R_r[%]

0 200 400 600 800

Fiber length L [m]

0 0.5 1 1.5 2 2.5 3 3.5

Output power P_r[W]

Figure 3.3.: The right-hand output powerP_rof a single-cavity Raman fiber laser as a function of the right-mirror reflectivityRrand the fiber lengthL. The left-mirror reflectivity isR_l= 99%, and the pump power isP₀= 4 W.

(R_p = 0). Qualitatively, however, RFLs behave the same for R_p >0 (see Sect. 3.2.3).

We start by calculating the output power of various SC-RFLs for a fixed input pump power of P₀ = 4 W, launched from the left-hand fiber end. As usually done in practi-cally realized RFLs [HBM⁺02], we choose a mirror for the Stokes wavelength with high reflectivity (HR), R_l = 99%, at the same end of the fiber, so that light is coupled out essentially only at the right-hand side of the RFL (Fig. 3.1). In order to find a con-figuration that is optimal in the sense that it emits maximum output power, we vary the two remaining free parameters, namely the right-hand reflectivity R_r and the fiber lengthL.

Fig. 3.3 shows the calculated right-hand output power P_r as a function of R_r and L.

At R_r = 51.2% and L = 280 m, P_r has its maximum value of 3.18 W. However, this maximum is not very pronounced and its exact location can not be made out clearly in the graph. In fact, there is a relatively large range of parametersR_r and Lthat yield an RFL with almost maximal conversion efficiency [KST⁺01]. However, we will show next that all these near-optimal lasers have a similar threshold pump power, i. e., it is not possible to find an RFL that has near-maximal conversion efficiency and at the same time a considerably lower threshold pump power.

To this aim, we pose the (arbitrary) requirement that the RFL to be designed have an output power not below 99% of the maximum obtainable output power 3.18 W. We thus restrict ourselves to a certain allowed range for the parameters R_r and L. This range is shown in Fig. 3.4 by the longish grey area in the R_r–L plane, where the black dot indicates the maximum-output-power configuration. Also included in Fig. 3.4 are the

0 200 400 600 800 1000

0 20 40 60 80 100

FiberlengthL[m]

Output mirror reflectivity R [%]r

P_th 0.2 W 0.4 W

0.6 W

Figure 3.4.: Lines of constant threshold power P_th for a single-cavity, single-pass-pumped Raman fiber laser separated by steps of 0.2 W, in the plane of the right-mirror reflectivityRr

and the fiber lengthL. AtP₀ = 4 W, the right-hand output power of the configurations inside the grey area is greater than 99% of the maximum output power of 3.18 W, which is obtained forRr= 51.2% andL= 280 m (black dot).

lines of constant threshold power, calculated from (3.10) with R_p = 0. As can be seen, the allowed parameter range is oriented just along the lines of constant threshold power, with the result that the threshold power of the maximum-output-power RFL (which is P_th = 1.17 W) can be lowered at most by 0.2 W by choosing another configuration from the right upper edge of the grey area. The threshold power can be lowered further only at the expense of considerably reduced conversion efficiency at P₀ = 4 W. This demonstrates the collision of the two optimization criteria “large output power” and

“low threshold power”.

In order to further illustrate the collision of the two requirements “large output power”

and “low threshold power” in SC-RFL designs, we now consider a variety of RFLs with a much lower, fixed threshold power ofP_th⁰ = 0.2 W (in contrast, the maximum-output-power RFL found at the beginning of this section had a threshold maximum-output-power ofP_th= 1.17 W) and look at their output powers when pumped with P₀ = 4 W. Suitable parameters R_r and L for the desired low-threshold RFLs can be found directly from Eq. (3.10) and correspond to the dashed line labeled “0.2 W” in Fig. 3.4. Note that each of the low-threshold RFLs can be uniquely identified by its fiber length L. In Fig. 3.5, the output power atP₀ = 4 W is plotted versus the length Lof the considered low-threshold RFLs.

The maximum output power achievable with the low-threshold RFLs at P₀ = 4 W is about 1.1 W, while that of the maximum-output-power RFL is 3.18 W, for which we posed no restrictions on the threshold power.

The results of this section show clearly that one has to find a trade-off between the threshold power and the conversion efficiency when designing an SC-RFL. The same situation arises in many conventional lasers. The reason is that a low threshold requires low cavity round-trip losses, i. e., cavity mirrors with a high reflectivity, see Eq. (3.10).

However, a high reflectivity implies low transmission, so that for mirror reflectivities approaching 100% (which would result in the lowest possible threshold), the Stokes power actually coupled out of the cavity is becoming ever lower and vanishes for the case where the threshold pump power is lowest.

0.6 0.7 0.8 0.9 1 1.1 1.2

0 500 1000 1500 2000 2500 3000

Output power Pr [W]

Fiber length L [m]

Figure 3.5.: Output power for a pump power P₀ = 4 W of single-cavity Raman fiber lasers with the same threshold power of 0.2 W versus the fiber length L.