1. Introduction

Lasers with random feedback owing to Rayleigh scattering in optical fibers [1] 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 [4] 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 [15]

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 [1].

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 [4] and 1.1 μm [7] 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.

 

Fig. 1 One-arm random distributed feedback fiber laser.

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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, $v$_i is a light frequency, $\Delta v$ is a Raman gain spectral width, α_i, g_i, and ε_I are losses, Raman gain coefficient and Rayleigh scattering coefficients correspondingly. ε_i is defined as ε_i = α_i·Q, where Q = 0.002 is the geometrical confinement factor. Boundary conditions read P_S,_S2⁺(0) = 0, P_S,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 [24] 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.

To test the numerical scheme we calculate the generation threshold using the equation set (1) and using the integral balance of gain and losses over the one-arm cavity shown on the Fig. 1 [7]:

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 [7]).

 

Fig. 2 (a) Numerically (symbols) and analytically (line) calculated generation threshold of first Stokes wave. (b) Numerically calculated generation threshold of second Stokes wave with (blue) and without (black) parasitic reflections of R_L = R_F = 4·10⁻⁵ at the fiber ends.

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3. Comparison with experiment and role of parasitic reflection

We calculate the power performance of the random DFB lasers which were experimentally studied in [7]. 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.

 

Fig. 3 First Stokes (forward – red, backward - black) and Second Stokes (backward - blue) output powers for the random DFB laser based on 10.7 km TrueWave fiber (a) and 2 km 1060-XP fiber (b): experimental data (boxes), numerical calculation without (dash) and with (solid) parasitic reflection.

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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, R_L and R_F., i.e. $P_{S,S2}^{+}(0)=R_L\cdot P_{S,S2}^{-}(0)$, $P_{S,S2}^{-}(L)=R_F\cdot P_{S,S2}^{+}(L)$_.The parasitic reflection values of only R_L = R_F = 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 [7] 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 R_L = 1.5·10⁻³ and R_F = 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 α_S·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.

 

Fig. 4 (a) The forward and backward output powers for different fiber lengths (red – forward waves, black – backward waves). (b) The typical longitudinal power distribution of the pump wave and generated forward and backward wave. Pump power is 4 W. Calculations are made for TrueWave fiber and lasing at 1.2 μm. Inset: L_RS dependence on pump power for different fiber lengths.

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Saturation of the forward output power is obviously caused by the shifting of the point z = L_RS where the saturated Raman gain is equal to losses, g_S·P_P(L_RS) = α_S. While the pump power increases, the value of L_RS is reduced due to the pump wave depletion. Thus the forward wave propagates more in a loss region at z>L_RS, 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 [1] reveals that the single arm scheme is more favourable as L_RS decreases as only $1/\sqrt{P_P}$, Fig. 4(b), insert, contrary to the law of $1/P$ in symmetrical configuration [14]. This means that saturation effects in forward output power are less pronounced in the one-arm single-pump configuration.

Decreasing L_RS at higher pump power does not affect the backward wave output power, as the typical length of backward wave amplifying is always less than L_RS, 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(-α_S·L) [4, 10] similar to the symmetrical random laser configuration [14].

 

Fig. 5 (a) Generation efficiency of the forward and backward waves over fiber length and (b) of backward wave over Raman gain and fiber losses. Calculations are for TrueWave fiber of 11 km.

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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.

5. Conclusion

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 [1], 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.