1. Introduction

The development of technologies for coherent combining of laser beams has led to the appearance of fiber laser systems with record power unattainable when using single-core optical fibers that is especially important for high-brightness CW narrowband and ultra-short pulse fiber laser systems, see [1] for a review. These technologies are based on a multiple channel system of fiber amplifiers, a system for coherent combining and compression of laser pulses. In [2], with coherent combining of radiation from 12 fiber amplifiers based on 5 m long non-polarization-maintaining Yb-doped step-index fiber with a 20 µm core and a 400 µm cladding diameter, the authors managed to obtain a record 10.4 kW average power of laser radiation with a pulse duration of 254 fs. Although all fiber amplifiers share a common master oscillator, a set of quarter- and half-wave plates is employed for polarization control in all channels, except for one reference channel, where a piezo-actuated mirror is used for phase stabilization. In addition, to ensure coherent summation in free space after the amplifiers, a set of motorized components is used to control the spatiotemporal overlap of the output beams. Thus, to ensure coherent combining of radiation from different independent channels (fiber cores) of the amplifier, a complex system is required for phase and time synchronization of laser pulses. Similar problems are encountered when using multicore fiber (MCF) with weakly coupled cores as the active medium of a laser/amplifier, where spatial light modulators [3], or 2D piezo-driven array of mirrors [4], along with special algorithms for controlling the feedback system for phase-locked operation, are used to correct the phase of radiation from the cores.

An alternative solution providing inherent phase-locked operation is a laser system based on MCF with closely spaced (coupled) cores, in which, due to crosstalk through the evanescent wave coupling, radiation propagates as supermodes with coupled phases in the cores. There are so-called in-phase supermodes, in which the phase in neighboring cores is the same, and out-of-phase supermodes, in which the phase changes by π in neighboring cores. Previously, selective excitation and effective amplification of out-of-phase supermode up to 18 kW total peak power have been demonstrated in 7-core Yb-doped fiber using a spatial light modulator [5]. Application of Yb-doped MCF with coupled large-mode area cores and an external Talbot resonator for mode selection allowed to obtain 115 W of in-phase mode with the slope efficiency of 61% and beam quality factor M2 of 1.43 [6]. In [7], by using a special aperture that introduced a higher cavity loss for high-order modes, it was possible to select the fundamental in-phase supermode in a 6-core Yb-doped fiber laser with output power up to 33.9 W and slope efficiency of 52%.

However, in all above-mentioned schemes of lasers/amplifiers external optical elements are used to couple laser radiation with an MCF, as well as to form a laser cavity, which increases the complexity of laser scheme and leads to undesirable losses on its elements. An all-fiber laser based on 19-core Er/Yb-doped coupled-core fiber MCF was implemented in [8], where the cavity of a laser was formed by mirrors deposited on the end face of a coreless fiber of a certain length, which also selected the fundamental in-phase supermode by using the Talbot effect [9]. In [10] a highly reflective fiber Bragg grating (FBG) written in a multimode fiber and a fiber tip cleaved at a right angle formed cavity mirrors of a laser based on a 4-core Yb-doped fiber. The disadvantage of this scheme was the increase of the number of lasing cores with the growth of the pump power, which was explained by the slightly different reflection coefficient of the multimode Bragg grating for various core modes. Experiment with single-mode FBG coupled to one core of similar 4-core Yb-doped fiber [11] has shown that the generated beam is spread between all cores due to their coupling in the spooled fiber while their generation wavelength is the same and corresponds to the FBG reflection wavelength. Writing of Bragg gratings in the cores of a Yb-doped MCF for the use as a complex rear-reflector of a laser cavity was demonstrated in [12]. The laser performance of the MCF with the femtosecond-pulse inscribed FBGs at its end shows a similar performance to lasing with a free-space commercial volume Bragg grating. However, the absence of the need to align the mirrors in the first case makes the scheme more attractive from the point of practical use.

The main goal of the present paper is to investigate the effect of active fiber bending on the distribution of transverse and longitudinal modes at the output of a 4-core Yb-doped fiber laser, as well as their time dynamics. In particular, we study the lasing regimes with winding the active fiber on spools of different diameter (in the 6.5–21 cm range) that leads to various cross-coupling conditions between the cores. The laser cavity of the laser is formed by fiber Bragg gratings on one side, inscribed in each of the 4 cores using the femtosecond point-by-point technique, and by a fiber tip cleaved at a right angle on the other side. It is observed that when fiber is wound with a diameter of 6.5 cm, the laser operates in a regime in which lasing is concentrated predominantly in one core, and the power fraction in this core exceeds 92% of the total power. In addition, the mode composition of the laser radiation generated by individual cores is investigated by measuring and analyzing their optical and radio-frequency (RF) spectra. The regime of laser wavelength self-sweeping in time similar to that in single-core single-mode fiber lasers [13] is observed and characterized in the studied MCF laser, for the first time to our knowledge.

2. Experiments and results

2.1 Fiber laser schematic

The schematic representation of the fiber laser, characteristics of which are studied in this work, is shown in Fig. 1(a). The laser was pumped by a multimode (MM) laser diode (LD) with an output power of ∼5.5 W at ∼976 nm. The radiation from the MM LD passed through the MM bandpass filter and got to a 4.15-m-long section of a home-made 4-core Yb-doped fiber with a polymer waveguiding cladding and the same geometric parameters as in [11]: the core diameter is d_co = 6.5 ± 0.2 μm, the distance between the core centers is d_dist = 28 ± 0.7 μm, the core-cladding refractive index difference is Δn = 0.008, Yb ions concentration is estimated as ∼5×10¹⁹ cm⁻³; a microphotograph of the end tip of the fiber is shown in Fig. 1(b). The cores coupling length calculated based on coupled-mode theory [14] is estimated as ∼4 km at ∼1030 nm. The laser cavity was formed by highly reflective FBGs with a resonant wavelength of ∼1030 nm at the rear end of the Yb-doped MCF (Fig. 1(c)) and the fiber end face cleaved at a right angle thereby forming an output laser mirror with a Fresnel reflection coefficient of ∼4% on the other fiber side. A Thorlabs BC106N-VIS/M CCD camera beam profiler was used to measure the distribution of the pump and lasing field intensity, and a Coherent PM300F-50 thermopile sensor to measure the total output power. A collimator and an iris diaphragm installed after the output mirror of the laser made it possible to measure the lasing signal from the cores. Spectral dynamics of the laser radiation in a chosen core was studied using a HighFinesse/Angstrom WLS-6/200 wavelength meter at times of ∼1-1000 ms; fast dynamics (at times of ∼1-100 μs) was studied by measuring the RF spectra using a Thorlabs DET08CFC 5-GHz photodiode and a LeCroy WavePro725Zi-A oscilloscope.

 

Fig. 1. (a) Schematic of a 4-core Yb-doped fiber laser. (b) Microphotograph of the 4-core fiber end. (c) Spectra of FBGs inscribed in the corresponding core (marked as 1-4) of a 4-core fiber.

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Since the used active 4-core fiber is not photosensitive for UV radiation, the FBGs were inscribed in the fiber using the IR femtosecond (fs) point-by-point technique. Previously, it was shown that the advantage of the technique to inscribe an FBG in a strictly defined core opens additional performance capabilities for creating new laser schemes based on MCFs [15,16]. A Light Conversion Pharos 6W laser system (λ = 1026 nm, Δt = 232 fs, f = 1 kHz, and E_p ∼ 100 nJ) was used as a source of fs pulses and an Aerotech ABL1000 high-precision linear stage translated the fiber relative to the static focusing point of fs radiation during FBG inscription process. A Mitutoyo 50X Plan Apo NIR HR microobjective (NA = 0.65) made it possible to focus fs pulses into the selected core. Four FBGs with a length of 2.5 mm and a period of ∼0.71 µm (2-nd order of reflection) were inscribed in the same cross-section of the fiber, following which a right-angle cleaving was made near the section. Then, using a fiber splicer as a high-precision 3D positioner, each of the cores of the 4-core fiber was sequentially aligned with a single-mode single-core fiber from a circulator port. Using a Thorlabs SLD1050S superluminescent diode as a broadband radiation source and a Yokagawa AQ6370 optical spectrum analyzer, the FBG reflection spectra were measured (Fig. 1(c)). As can be seen, the FBGs had a central resonance wavelength in the range of 1030.7−1030.95 nm, and reflectance R > 80% for cores 1, 3, 4 and R ∼ 50% for core 2. We believe that the variation in the parameters is related to the heterogeneity and circular asymmetry of the cores, as well as to the positioning error of the focus point of the fs radiation relative to the core.

2.2 Output characteristics of the laser

The output characteristics of the laser were measured at different winding diameters (D₁ = 21 cm, D₂ = 9 cm, and D₃ = 6.5 cm) of the active 4-core fiber, which allowed us to change pump modes composition and the cross-coupling between the cores [11]. The beginning of the 4-core fiber (point of splicing with the pump fiber tail) was fixed in all experiments; the first and last 0.5 m were in the straightened state. The central part of the 4-core fiber (∼3 m) was loosely wound on cylindrical surfaces turn to turn. Although we did not control the fiber twist, several facts indicate that it was weak: due to an increased cladding diameter (∼200 µm) a higher torque force is required to twist the fiber as compared to a standard 125-µm diameter fiber, the section of the wound 4-core fiber was relatively short, we saw approximately the same cores orientation on the beam profiler when repeatedly rewinding the 4-core fiber.

The total output power including both the unabsorbed pump and lasing powers was measured for each of the diameters (Fig. 2(a)). An iris diaphragm installed at a distance from the collimator allowed us to select the central part of the beam, which contained the lasing signal (Fig. 2(b)). As far as the divergence of the pump beam is larger than for core beams, the pump power passed through the diaphragm decreases significantly with increase of distance from the diaphragm to collimator. Moreover, the pump signal is absorbed mainly in the central region of the fiber and its contribution to the transmitted power becomes insignificant. As can be seen, the lowest lasing threshold was reached for winding diameter D₂ = 9 cm at a pump power of 3.15 W, while for winding diameters D₃ = 6.5 cm and D₁ = 21 cm the threshold pump power amounts to ∼3.35 W and ∼3.55 W, respectively.

 

Fig. 2. Total output power (a) and lasing power (b) as a function of pump power for different winding diameters of a 4-core Yb-doped fiber.

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A beam profiler was used to separate lasing signals going out of individual cores. First, the intensity profile of the radiation at the output mirror (the end face of the optical fiber) was measured (Fig. 3), then the power fraction in each of the cores was calculated as a function of the winding diameter and the input pump power (Fig. 4). Since the intensity in each pixel of the profile is proportional to the power, summation of the intensities of all pixels nearby one selected core gives us a value proportional to the lasing power in this core. In our case the summation was performed in the radius of ∼10 µm from the center of the core by means of a script written in Python language. After the procedure was done for each of the cores, we normalized the obtained power values by the total power across all cores. In this way we obtained relative power values for each of the cores. It can be seen from the Fig. 3(a) that in the case of winding diameter D₁ = 21 cm the laser generation starts first in core 4, then in core 1, and then in cores 3 and 2. Note that the order in which the laser generation occurs in the cores correlates well with the reflectance spectra of highly reflective FBGs (Fig. 1(c)) — core 4, in which laser generation occurs first, corresponds to the FBG with the highest reflectivity and spectral width, and core 2, in which the generation occurs last, corresponds to the FBG with reflection coefficient and spectral width less than the others. This indicates a weak mutual influence of the cores on each other at the largest winding diameter. Figure 4(a) shows the lasing power in each individual core as well as their sum (total lasing power) as a function of input pump power. As can be seen, depending on the core number, the maximum output power in individual cores varies from 56 to 99 mW, which is 18% to 32% of the total lasing power, respectively. At ∼2 times smaller winding diameter D₂ = 9 cm, the behavior of the modes changes (Fig. 3(b)) — the lasing starts first in core 3, and then, over the entire pump power range, the lasing power in this core prevails and is more than 3 times higher than that in the neighboring cores. Thus, the maximum power in core 3 reaches 185 mW and accounts for 66% of the total power, whereas the power share for each of the other cores does not exceed 14% (Fig. 4(b)). At last, with the smallest winding diameter of the active fiber D₃ = 6.5 cm more than 92% of the total lasing power is concentrated in core 3 over the entire power range (Fig. 3(с) and Fig. 4(c)).

 

Fig. 3. Profiles of the output radiation intensity measured depending on the pump power for different winding diameter of the 4-core Yb-doped fiber: 21 cm (column a), 9 cm (column b), and 6.5 cm (column c).

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Fig. 4. Lasing power in each individual core depending on the pump power and the winding diameter of a 4-core Yb-doped fiber: (a) D₁ = 21 cm, (b) D₂ = 9 cm, (c) D₃ = 6.5 cm.

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Comparing the results obtained for different winding diameters of the 4-core fiber (Fig. 5), we can conclude that, from the point of view of the total efficiency, the best result is demonstrated by the laser with the largest winding diameter D₁ = 21 cm, however, from the point of lasing efficiency in an individual core, the best result is obtained for the smallest winding diameter D₃ = 6.5 cm.

 

Fig. 5. Lasing power at the brightest core (blue) and the total lasing power from all cores (red) as a function of the curvature (R_b^-1) of a 4-core Yb-doped fiber on the spool.

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It is known that when high-power laser radiation propagates along the MCF with coupled cores, the power concentration of radiation in one core (or a group of cores) occurs due to nonlinear effects [17,18]. However, in our case, in view of the low pump and lasing powers, it is appropriate to consider only linear effects. In particular, one of the mechanisms for the power concentration of radiation in an individual core can be the deformation of the pump modes propagating in the fiber cladding, as well as of the core modes, due to fiber bending. Indeed, when a fiber is bent the refractive index of the material changes in the radial direction due to the appearance of a non-uniform stress field. On the neutral plane (center of the cross section) the refractive index does not change, in the stretching region it increases, and in the compression region it decreases, which can be expressed as n_eq(r,θ) ≈ n(r,θ)[1 + r/R_b cos(θ)] [19], where n(r,θ) is the initial radial distribution of the refractive index, and R_b is the bending radius of the fiber. Thus, for multimode pump radiation, the deformation of the mode field occurs with a shift of the maximum outside the neutral plane [20]. Likewise, for MCFs with coupled cores, the radiation power is concentrated in a particular core undergoing maximum elongation.

As noted earlier, in our laser, we assume only a weak twist of the 4-core fiber section, regardless of the winding diameter. This is due to the fact that the beginning of the 4-core fiber was fixed, and the orientation of the output end face was repeated from winding to winding. In addition, the wound fiber section was relatively short, and the increased diameter of the fiber cladding required higher torque force compared to a standard 125 µm diameter fiber. Thus, when bending this 4-core active fiber, the core 3 was subjected to maximum expansion, which promoted a greater absorption of the pump power, as well as the concentration of the laser power in this core.

In the next section, we analyze the laser spectra in optical and RF domains, as well as their dynamics depending on the winding diameter of an active fiber.

2.3 Spectral dynamics

In the developed laser, we observed the temporal dynamics of longitudinal modes of the radiation in the form of a reverse wavelength self-sweeping. To begin with, let us consider the case of winding an active fiber on a spool with small diameter (D₃ = 6.5 cm), when the lasing occurs predominantly in a single core. The dynamics of longitudinal modes is given by pulse-to-pulse wavelength shift (Fig. 6(a)) at the microsecond self-sustained pulsations (Fig. 6(b)), similar to that in self-sweeping lasers based on single-mode fibers [13]. The repetition rate of self-pulsations as well as the absolute value of the sweep rate increases with the pump power, from 50 to 200 kHz and from 4 to 15 nm/s, respectively. Characteristic frequency hop between neighbor pulses can be estimated as a product of sweep rate and repetition rate and is equal to 22.6 MHz, which is close to the fundamental intermode beating frequency (IBF = 23.15 MHz) of the linear cavity. This behavior was previously observed in the so-called single-frequency self-sweeping laser [21], where each pulse consists of a single longitudinal mode of the cavity and the laser frequency changes between pulses by a fixed value defined by the cavity mode spacing.

 

Fig. 6. Characteristic dynamics of the spectrum (a) and intensity (b) of the laser at pump power of 4.85 W.

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The observed self-sweeping laser operation is explained by a dynamic grating formed by a standing wave of one initial longitudinal mode. A dynamic grating is narrowband selector (due to relatively long length) that determines the mode composition after the initial mode. The instant narrowband optical spectrum is described by the composition of longitudinal modes. The heterodyne technique with a single-frequency probe radiation [22] is commonly applied for correct measurements of fast mode dynamics. Due to the lack of the probe source at the required wavelength, the analysis of mode composition was made based on the RF spectra of signals. An RF spectrum is related to the longitudinal mode composition and shows beating between modes. Two different measurements of the RF spectrum in the range up to 200 MHz (Fig. 7), where the frequency axis was normalized to the fundamental IBF = 23.15 MHz, show that the laser signal consists of a small number of longitudinal modes (∼3-4 pcs). In particular, Fig. 7(a) corresponds to the case when the radiation consists of three longitudinal modes (one of which records the dynamic grating), beating of which give the difference frequency at the 2-nd and 3-rd fundamental frequencies. The peak near the 1st fundamental frequency corresponds to their difference frequency.

 

Fig. 7. RF spectra of the laser pulses with different longitudinal mode composition measured at two different instant times.

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Instant RF spectrum changes from measurement to measurement as Fig. 8(a) shows, where the series of 30 RF spectra taken at different times is presented. As can be seen, the RF spectrum consists of a four peak periodic structure (with 23.15 MHz period), which probably is associated with different fiber cores. It was also experimentally observed that positions of the peaks within the periodic structure changes when the fiber is exposed to external disturbance (for example, bending).

 

Fig. 8. Series of RF spectra of the laser pulses measured at different instant times for the different winding diameters of a 4-core Yb-doped fiber: (a) D₃= 6.5 cm and (b) D₁ = 21 cm.

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Based on the above, the following mode structure of the output radiation seems to correspond the observations: each core has its own mode composition, which period in each core is the same and equal to IBF (Fig. 9); otherwise, there was no periodicity in the RF-spectrum. At the same time, the mode compositions in each core are slightly shifted from each other in frequency domain, which is probably associated with a different phase incursion. The first generating longitudinal mode (with zero number) forms dynamic grating (due to longitudinal modulation of the gain and refractive index), the next generating modes are determined by the reflection spectrum of the existing dynamic grating. In particular, the grating formed by mode from one core can interact with modes from other cores. Figure 9 schematically shows the case of simultaneous lasing of the first mode (red ellipse) of the first core and the second and third modes of the third core (green ellipse). In this case crossmode beating frequency (XBF) should be observed in addition to the fundamental intermode beating. As a result, the final signal in one core consists of longitudinal modes of different cores.

 

Fig. 9. Representation of the mode structure in different cores.

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As mentioned early, the instant RF spectrum in Fig. 7(a) consists of 2nd and 3rd modes of the main core. However, the mode composition in Fig. 7(b) differs despite the presence of the same number of peaks in the RF spectrum. In this case, three peaks are observed: 1) at the first fundamental mode, 2) between the 6th and 7th fundamental modes, and 3) between the 7th and 8th fundamental modes. The frequencies of the second and third peaks are not equal to a multiple of the IBF and correspond to beating between modes located in different cores. As a result, the second peak is the beating of the initial mode in one core and 6th mode in the other core, the third peak is the beating of the initial mode in one core and 7th mode in the other core, and the first peak corresponds to the beating of 6th and 7th modes in the same core.

Similar analysis of mode dynamics was performed for the largest winding diameter of a 4-core fiber (D₁ = 21 cm). The main difference between the winding diameters lies in the mode composition observed in the RF spectra. As one can see from Fig. 8(b), a smaller number of peaks is observed in a spectrum for a larger diameter. This result can be explained by a weaker interaction and mixing of modes in the fiber cores. In the limiting case, when the cores are uncoupled, one can expect the mode composition to be different and independent in each individual core. On the contrary, a small winding diameter of a 4-core fiber leads to strong cross-coupling and mixing of longitudinal modes, and laser radiation generated in all cores can be observed in each core. As a result, each output laser pulse consists of a small number of modes despite the strong mixing between them. This fact can be associated with high selectivity of the dynamic grating formed by the standing wave due to its rather large length (∼ 4 meters).

3. Conclusion

Thus, we have shown that a winding of a 4-core Yb-doped fiber on a spool with a smaller diameter has a stronger effect on the core selection at lasing in a cavity formed by high-reflection FBGs directly inscribed in each core with femtosecond laser pulses and 4% Fresnel reflection from the output fiber end face. Although with such a winding the total lasing power decreases, the output power and efficiency for the selected individual core increases greatly so that the share of one core exceeds 92% for the winding diameter of 6.5 cm. It is shown that the power is concentrated in the most elongated core (with higher refractive index) at the fiber winding performed without twist. The performed analysis of spectra has shown that the laser generates narrowband radiation consisting of several longitudinal modes with spacing corresponding to fundamental frequency (inverse period) of the cavity with the periodic linear sweeping of central wavelength within the FBG spectrum. It has been also shown that the modes corresponding to different cores are shifted against each other so that crosstalk frequency shift is observed in the RF spectrum. Although for weakly bent fiber the generation in the cores occurs almost independently with a weak crosstalk, a strong fiber bending results in a strong cross-coupling of the cores and mixing of their longitudinal modes. As a result, the parts of frequency combs generated in four cores are observed in the output from one core where the laser power is concentrated. Thus, the regime of laser wavelength self-sweeping, previously observed in single-core single-mode fiber lasers, is observed and characterized in the studied MCF laser, for the first time to our knowledge. Moreover, this regime has got new and quite interesting features in the MCF requiring a more detailed investigation.