1. Introduction

Recently, Raman fiber lasers (RFLs) based on a multimode graded-index fiber (GIF) attract great deal of attention due to the opportunity of efficient Raman conversion of highly-multimode (M²=20-30) radiation of high-power laser diodes (LDs) into a high-quality (M²=2-3) Stokes beam in an all-fiber scheme with in-core fiber Bragg grating (FBG) cavity and fiber coupling of pump radiation, see [1] for a review. Using commercially available passive telecom GIFs and high-power 9xx-nm LDs, such RFLs may generate in the short-wavelength range of a single-mode Yb-doped fiber laser (YDFL) near 1,02 μm when pumped by 976-nm LDs [2,3], as well as in the broad range below 1 μm when pumped by 915-960 nm LDs, thus offering new wavelengths at which generation of high-power YDFLs is hardly possible [4].

Connecting several (up to 3) high-power LDs to a GIF with 100-μm core through a multimode 100-μm fiber pump combiner allows one to increase coupled pump power to ∼200W level in the all-fiber scheme and to generate Stokes radiation with beam quality M²=2-3 and output power 50-60 W at wavelengths 954 nm [5] and 976 nm [6] using 915 and 938 nm LDs, respectively. Note that the first attempt with one 938-nm LD coupled to a GIF through bulk optics resulted in only ∼3W output power at 976 nm [7]. Optimization of transverse profile of FBGs inscribed by femtosecond point-by-point technique in near-axis part of GIF core allows one to improve beam quality to M²≈2 without significant loss of conversion efficiency amounting to about 70% [6]. Though the Raman clean-up effect in graded-index fibers is well-known, including that one in a multimode fiber with FBGs pumped by a multimode laser [8], its explanation is rather qualitative [9] based on the complicated analysis of overlap integrals between different transverse modes of pump and Stokes beams showing higher small-signal gain (i.e. near the threshold of Raman laser) for low-order Stokes modes at random pump launching conditions in GIF.

Here we report on further improvements of the 976-nm Raman laser based on 100-μm GIF [6] resulting in the record pump-to-Stokes brightness enhancement and measurements of pump beam profiles both near the Raman threshold and well above the threshold (where the pump depletion becomes important) together with corresponding Stokes beam profiles generated at 976 nm. Analytical model for the analysis of Raman gain and beam profiles near the threshold has been developed, as well as its extension to pump-depleted regime. The role of different linear and nonlinear effects in the pump depletion and the output Stokes beam shaping and brightness enhancement is discussed.

2. Experiment and results

The experiment was performed on the base of the setup described in [6] with the modifications directed onto detailed beam shape measurements for the output Stokes and pump beams. Pump radiation from three fiber pigtailed high-power LDs operating at the wavelength of ∼938 nm is added together by means of a 3 × 1 multimode fiber pump combiner, see Fig. 1. The input ports of the fused pump combiner are made of multimode step-index fiber with 105-μm core and output port is made of a 100-μm core GIF with numerical aperture of 0.29. It is spliced to the same 1-km-long 100-μm core GIF Draka 100/140 in which the Raman gain is provided by LD pumping. The linear laser cavity for the 1st Stokes wave is formed in the GIF by a high-reflective (∼90%) UV-inscribed FBG and an output low-reflective (∼4%) fs-inscribed FBG providing cavity feedback at the Stokes wavelength of 976 nm. The output Stokes and pump beams were characterized by a Thorlabs M² meter by using an appropriate selective mirrors M_1,2 and interference filters IF to select corresponding wavelength. For measurement of a detailed output beam profile we adjusted the focus on the fiber end facet given that the defocusing from the optimal focus point blurred the spatial details of the measured beam profile, for more details see [10]. Corresponding power values were measured by power meter (PM), which replaces the M² meter for this measurement.

 

Fig. 1. Experimental setup: LD1,2,3 – multimode 938-nm laser diodes; UV FBG – UV-written high-reflection FBG; FS FBG – low-reflection FBG written by a point-by-point femtosecond pulse technique; L – collimating lens; M1, M2 – dichroic mirrors; IF – interference filter; M² – beam quality analyzer; PM – power meter.

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The measured output power evolution for the pump and Stokes waves is shown in Fig. 2(a). Due to optimization of the scheme (reducing losses at splices with FBG, etc.), the threshold pump power for the generation of Stokes radiation at 976 nm is slightly reduced from ∼108 W in [6] to ∼105 W so that the maximum Stokes power slightly increases (from 49 W to 52 W) at maximum input pump power of ∼186 W, whereas the slope efficiency remains to be nearly the same (70%) At the same time, the generated Stokes beam quality is improved from M²≈1.9 in [6] to M²≈1.7 here (both measured at output Stokes power around 10 W), see Fig. 2(b). The measured M² value monotonically grows with increasing power to M² = 1.95-2 at output power ∼50 W that is better than reported in [6]. As a result, the pump-to-Stokes brightness enhancement BE = [P/(M²λ)²]_Stokes / [P/(M²λ)²]_Pump reaches 73 tending to saturation at maximum powers. As far as we know, it is the best BE for GIF RFL, as well as for any Raman fiber laser with average power level >0.1 W, see [11] for a review. It should be noted that maximum power at multimode pumping exceeds 1 kW for Raman lasers based on special triple-clad fiber [11] and 2 kW for GIF-based Raman amplifier [12] at output beam quality M²=2.7-2.8 and BE=7-11.

 

Fig. 2. Output Stokes and pump power evolution with increasing input power and corresponding brightness enhancement BE value (a); beam quality (M²) measurements (b) and corresponding shapes (c) for the transmitted pump beam below Raman threshold at 17 W input power (top left) and above the threshold at 128 W input power (top right) and corresponding Stokes beam shape at output power of ∼10 W (bottom right).

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Since the Stokes power becomes comparable with the residual pump power [see Fig. 1(a)], the highly multimode (M²∼34) pump beam becomes depleted non-homogeneously: the output pump beam has got a visible hole [Fig. 2(c), right top], while the near-diffraction limited Stokes beam is rather narrow [Fig. 2(c), right bottom].To analyze the beam profiles in more detail, corresponding X-axis cross-section of the 2D beam profile has been extracted and drawn in Fig. 3. In Fig. 3(a) the residual pump beam profiles at different input pump powers are shown. One can see that in the absence of Stokes generation, the output pump beam has a shape similar to parabola and its amplitude grows with the input pump power. When the Stokes generation begins, the shape of the pump beam profile changes. Near the threshold it becomes closer to П-shaped one, then a dip in the center is formed and its amplitude and width increase with increasing Stokes power. As the dip corresponds to the converted pump power, we estimate the dip shape evolution more precisely by subtracting the depleted output pump profile from the non-depleted parabolic profile (at corresponding optical power), see Fig. 3(b). At the same time, the measured Stokes beam shape appears to be close to Gaussian and does not change significantly with increasing power, see Fig. 3(c). Stokes width is much narrower than the pump beam width and, moreover, than the dip width, whereas its amplitude grows much faster than the dip amplitude, in rough correspondence with the depleted pump.

 

Fig. 3. Residual pump (a) and Stokes (c) beam profiles at different powers. Difference between the experimental output pump profile and attenuated input pump beam approximated by parabola at corresponding power (b).

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3. Balance model and its comparison with experimental results

If we consider within the geometric optics the propagation of randomly directed pump beams (j=1,…, N) filling the entire core of a fiber with a parabolic profile of refractive index, we obtain an analytical expression for the intensity of the random pump field (corresponding to the equidistribution of transverse modes due to their random mixing) [10]:

At increasing pump power, higher-order modes also exceed the threshold, but insertion of predominant (by ∼10 dB) feedback for fundamental mode by the output FBG fs-inscribed in the central part of GIF core allows to keep predominant generation of fundamental mode (the detailed characteristics of the used FBGs can be found in [6]). Besides, at high Stokes powers the pump becomes to be depleted, in this case radially-dependent balance equations can be treated under the assumption of a large number of transverse modes [10]:

Here I_s – Stokes intensity, k_p and k_S are the pump and Stokes wavenumbers; α is the absorption coefficient of the fiber. These equations have analytical solutions:

Using boundary condition ${I_S}(r,0) = \sqrt {\rho (r)} {I_S}(r,L)$, where ρ(r) is the reflectivity of FBG (of Gaussian shape), and substituting (8) into (6) we can numerically calculate output Stokes I_s(r,L) and pump I_p(r,L) intensity profiles [Eq. (7)]. For our calculations, we used an experimentally measured pump beam profile before SRS threshold as input pump beam ${I_P}(r,0)$ (after normalization to an appropriate power level).

An analysis of the curves obtained from these equations shows that the generation of Stokes wave is followed by appearance of the dip in the depleted pump beam profile that qualitatively agrees with the experiment, in spite of that the model is obtained at the rather rough assumption of large number of transverse modes corresponding to local interaction of pump and Stokes beam in each radial point. As a result of this assumption, the depletion effect at low number of modes in the Stokes beam with predominant generation of fundamental mode (n=0) leads to the narrow dip in the pump profile with steep edges and the width nearly corresponding to the width of the Stokes beam defined by the FBG reflection profile [ Fig. 4(a)] that differs quantitatively from the experimental data.

 

Fig. 4. Output pump and Stokes profile I(r) at 155 W input pump power: balance model results for different Stokes beam radius defined by FBG (a) and experimental data for the input, attenuated and depleted pump beam and corresponding Stokes beam generated in GIF with FBG cavity (b).

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The experimental profiles normalized to the GIF core diameter (Fig. 4(b)) show that the pump dip is significantly broadened so that the pump depletion is almost homogeneous across the beam, whereas the amplitude (peak intensity) of the narrow Stokes beam becomes much higher than that for output and even input pump beam thus defining the BE effect. Such increase in the Stokes intensity nearly corresponds to the dip (depleted power) in the pump beam, which is sufficiently broader than the Stokes beam and additionally broadens with increasing power [see Figs. 3(b), 4(b)], whereas in the balance model the dip in the pump beam nearly correspond to the Stokes beam [Fig. 4(a)] and brightness enhancement is weak.

Though initially large width of the dip may be explained by the overlap of fundamental Stokes mode with higher order pump modes in GIF [9], its broadening with power occurs likely due to the random mode mixing, because the mode overlap does not change with power (and corresponding increase in the dip amplitude). This effect is similar to the formation of parabolic shape of the undepleted multimode pump beam via random mode mixing at its propagation in GIF. Note that the pump depletion occurs in a relatively short part of GIF fiber near the output FBG where the generated Stokes power reaches its maximum value, similar to single-mode RFL with low-reflection FBG [13]. Though such distance is sufficient for washing out the dip in the pump beam, one can see that the Stokes beam quality does not suffer from the mixing. This may mean the importance of nonlinear effects at propagation, in particular the Kerr self-cleaning effect that is very sensitive to initial mode composition of the beam and is usually seen at low impurity of high-order modes [14], whereas an opposite effect may be observed for a highly-multimode beam.

4. Conclusion

In conclusion, the performed experimental study of output beam shapes below and well above the threshold of Raman lasing at 976 nm in multimode GIF pumped by highly-multimode radiation of laser diodes at 938 nm demonstrates a record pump-to-Stokes brightness enhancement (BE=73) and reveals some interesting features. First, the low-power pump beam takes after propagation in GIF a parabolic shape corresponding to the equidistribution of transverse modes in GIF. Second, above the Raman threshold when the near-diffraction limited (M²∼1.7) Stokes beam is generated, the pump beam becomes to be non-homogeneously depleted with a dip formed in its central area. At that, the dip is significantly broader than the Stokes beam even near the threshold and further broadens with increasing power at constant Stokes beam width.

To analyze the observed effects, the balance model with radially dependent pump and Stokes intensities at the assumption of highly-multimode radiation, has been developed. It allows for the simple and clear explanation of the Raman beam cleanup effect near the threshold when the small-signal Raman gain takes its maximum value for the fundamental mode in correspondence with the parabolic pump beam shape. When the Stokes beam power increases, it is important that in-core FBGs provide predominant (by ∼10 dB) feedback for the fundamental mode thus keeping generation of the near-diffraction limited Stokes beam. Herewith, the pump beam depletion can be qualitatively described by the radially-dependent balance model, but the dip shape significantly differs from that obtained in the experiment where the dip appear to be significantly broadened. We suppose that the random mode mixing in the GIF output part (where the Stokes wave reaches its maximum value and the pump depletion prevails) can explain the dip broadening with power whereas near the threshold the dip shape is defined by the overlap of Stokes and pump modes. As the high-intensity Stokes beam does not suffer from the mixing, this may also indicate the importance of nonlinear effects at propagation which finally define the observed BE effect. For a detailed analysis of this regime one should develop a comprehensive model treating intra-cavity dynamics of individual modes in longitudinal and transverse directions and their interaction via linear (random mixing, FBG) and nonlinear (Raman, Kerr) effects.