We present a comprehensive study of power output characteristics of random distributed feedback Raman fiber lasers. The calculated optimal slope efficiency of the backward wave generation in the one-arm configuration is shown to be as high as ~90% for 1 W threshold. Nevertheless, in real applications a presence of a small reflection at fiber ends can appreciably deteriorate the power performance. The developed numerical model well describes the experimental data.
©2012 Optical Society of America
Lasers with random feedback owing to Rayleigh scattering in optical fibers  have attracted a great interest for a past couple of years because of specific properties compared to other types of random lasers (see [2,3] for a review). A number of papers demonstrating new laser schemes and applications have been published recently [4–13]. Basing on a balance equation set, numerical simulations  and analytical considerations [10,14] predicting the longitudinal power distribution, generation thresholds and output powers in symmetrical configuration with two pump lasers are made. In addition, noise transfer from the pumps to the signal could be also calculated using this approach 
The balance equation set provides also a possibility to optimize the performance of the laser. Indeed, rather good predictions for power performance of conventional Raman fiber lasers (RFLs) are made, see, for example [16–19]. However, to obtain a quantitative agreement between calculations and experimental data, one needs to take into account nonlinear spectral broadening in some phenomenological way, for example, by introducing effective transmission coefficient [19, 20]. In the case of random DFB fiber laser, the balance equation set should give even better result as the laser has no spectrally selective mirrors.
In the present paper we make a power optimization of the random DFB fiber laser, and show that a one-arm configuration with a single pump provides better performances than the initially demonstrated symmetrical configuration .
2. Numerical model
We study the single pump configuration where the pump light is coupled into a long fiber span from one side, see Fig. 1 . The laser generation owing to random distributed feedback is easily achieved both in 1.5 μm  and 1.1 μm  spectral region with a threshold power of only around 1 W. Such a laser has a simplest possible design as it does not comprise any elements except fiber and a pump light source.
To model the power performances of a random DFB fiber laser, we use the well-known power balance equation set:
Here P, S, 2S denote pump, first and second Stokes waves correspondingly; “+” and “-” denote forward and backward propagating waves relative to the pump wave propagation direction, i is a light frequency, is a Raman gain spectral width, αi, gi, and εI are losses, Raman gain coefficient and Rayleigh scattering coefficients correspondingly. εi is defined as εi = αi·Q, where Q = 0.002 is the geometrical confinement factor. Boundary conditions read PS,S2+(0) = 0, PS,S2– (L) = 0.
The balance equation set does not deal with any of spectral features of laser generation as well as cannot describe the time dynamics of the laser and statistical properties of the radiation. To do this an alternative approach based, for example, on nonlinear Schrödinger equation (NLSE) should be developed similar to the way how RFLs with conventional cavities [21–23] or Ytterbium doped fiber lasers  have been modelled recently. NLSE-based approach describes well statistical properties of Raman fiber laser generation including the emergence of rare extreme events [25,26]. However, in the case of a random DFB fiber laser one should face a challenge of modelling very long fiber (several tens of kilometres) while taking into account Rayleigh backscattering from the counter-propagating wave using a small integration step.
Here L is fiber length. The values of generation threshold extracted from integral balance condition (2) and numerical calculation using equation set (1) are in excellent agreement, Fig. 2(a) . In both cases we use TrueWave fiber parameters taken from both producer datasheets and our own measurements (can be found in ).
3. Comparison with experiment and role of parasitic reflection
We calculate the power performance of the random DFB lasers which were experimentally studied in . Firstly, we model the laser operating at 1.2 μm and based on 10.7 km of OFS TrueWave XL fiber. The equation set (1) provides us a good prediction for the first Stokes wave power for the both forward and backward fiber laser outputs as well as for the first Stokes wave generation threshold, Fig. 3(a) , dash lines. However, big discrepancy turned out in second Stokes wave generation threshold: 12.4 W in numeric, 6.6 W in experiment.
In general, the threshold could become lower if there is an additional feedback in the cavity. To check this, we modify the boundary conditions introducing parasitic point-like reflection due to surface scattering at fiber ends, RL and RF., i.e. , .The parasitic reflection values of only RL = RF = 4·10−5 give a good agreement for the second Stokes generation threshold keeping an accuracy for prediction of the first Stokes wave power. It is important that in the experimental conditions such a small value of the parasitic reflection is simply achievable because of dust and dirt on the fiber end surface, or physical damage of output connectors, as even low-reflection angle-polished connectors usually reflect no less than 10−6.
The shorter fiber, the more important role of the parasitic reflection on laser performances should be. In the random DFB laser based on 2 km of 1060-XP fiber (see  for fiber specification), the calculated power performances are far away from experimentally measured if one neglects parasitic reflection, Fig. 3(b). However, the parasitic reflection of RL = 1.5·10−3 and RF = 2·10−4 gives a very good agreement between numerical predictions and experimental data.
Important fact is that the parasitic scattering changes the balance between forward and backward waves: forward output power is higher than backward in experiment, while the situation is reverse in ideal situation of no additional feedback, Fig. 3(b).
The integral value of the random distributed feedback is as small as αS·Q·L ~ 10−3-10−4, so even a tiny parasitic feedback can sufficiently influence the generation properties. Despite that the parasitic scattering can make the generation threshold sufficiently lower, basically it deteriorates the laser performances. Indeed, one of the advantages of the random DFB fiber laser against conventional RFL is the possibility of high power first Stokes wave generation that is difficult to achieve in conventional RFL because of generation of higher order Stokes waves [27, 28]. In random DFB fiber lasers generation thresholds of higher order Stokes waves are very high, so the first Stokes wave power up to 100 W and more could be potentially achieved in short fibers under appropriate pumping. However, even very small parasitic scattering could sufficiently reduce the second Stokes wave generation threshold, Fig. 2(b), and thus limit the maximum achievable power in the first Stokes wave.
Finally, parasitic reflection affects the generation efficiency also. The differential efficiency in experiment with a parasitic scattering is lower than in numerical simulation with zero point-like scattering, Fig. 3(b). Thus, one needs to carefully manage parasitic reflection in experiments to achieve the desired laser performances.
4. Power optimization
The balance equation set can be used for power optimization of the random DFB fiber laser. In the following calculations we do not consider parasitic point reflections. It has been found that the forward and backward output powers depend completely differently on pump power. While the pump power increases, the forward output power starts to be saturated, while the backward output power increases linearly, Fig. 4(a) . It is important that the backward output power is always higher than the forward power.
Saturation of the forward output power is obviously caused by the shifting of the point z = LRS where the saturated Raman gain is equal to losses, gS·PP(LRS) = αS. While the pump power increases, the value of LRS is reduced due to the pump wave depletion. Thus the forward wave propagates more in a loss region at z>LRS, and the forward output power is saturated. From the practical point of view, such a saturation could be an important drawback of the laser. However, comparing the single pump one-arm scheme under study with classical symmetrical scheme with two pumps  reveals that the single arm scheme is more favourable as LRS decreases as only , Fig. 4(b), insert, contrary to the law of in symmetrical configuration . This means that saturation effects in forward output power are less pronounced in the one-arm single-pump configuration.
Decreasing LRS at higher pump power does not affect the backward wave output power, as the typical length of backward wave amplifying is always less than LRS, and backward wave is mainly amplified near the fiber end where the pump wave is undepleted, Fig. 4(b). As a result, the backward wave output power depends always linearly on pump power without any saturation. In addition, the backward output of the random DFB fiber laser does not include any residual pump, so it is preferably to use a single-pump one-arm configuration with backward output in practical applications.
Finally, we have calculated how the generation efficiency depends on fiber parameters. Surprisingly, the generation efficiency is almost constant over fiber length for the backward output power and reaches ~90% that is close to the quantum limit of 95%, Fig. 5(a) . For the system under study with the threshold of 1 W, total efficiency reaches the value of 70% at pump level of 7 W. It’s worth noting that such total efficiency is comparable with that of most efficient Raman fiber lasers with FBG based feedback [27, 29, 30]. For the forward output power the efficiency decreases with exponential law exp(-αS·L) [4, 10] similar to the symmetrical random laser configuration .
The backward output power efficiency varies only slightly over broad range of Raman gain and loss values, Fig. 5(b). For example, at typical losses near 1.2 μm of 0.4 dB/km, the efficiency differs only by few percent while Raman gain doubling.
We have presented the detailed analysis of power performances of the one-arm single-pump random DFB fiber laser. It is found that even tiny parasitic reflection at fiber ends play a key role determining the laser power performance. The laser radiates mainly in the direction backward to the pump power. Single-pump one-arm configuration is more preferable comparing to symmetrical configuration with 2 pumps from the fiber center , as saturation effect in forward output wave are less pronounced, and the backward wave efficiency almost does not depend on fiber length, losses and Raman gain coefficient.
The authors acknowledge support from the Russian Ministry of Science and Education, Russian Foundation for Basic Research, the Leverhulme Trust, the European Research Council, the Marie Curie FP7 Program IRSES, the Dynasty Foundation, Russian science support foundation.
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