Power-dependent effective reflection of fiber Bragg grating as output coupler of Ytterbium-doped fiber laser

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YDFLs are also capable to operate in the long-wavelength (LW) range, viz. from 1120 nm to 1200 nm [16][17][18]. YDFLs of this class are of special practical interest because of combination of their excellent energetic properties and the operational spectral band in which diode lasers cannot compete due to the technological problems. Specifically, LW-YDFLs are applicable as pump sources for a line of FLs (such as Ho-doped and Raman (random) FLs [19][20][21][22][23]) and for frequencydoubling ("yellow light" [24,25]). Besides, laser sources operating in this range found other important applications, say, in the minimally invasive harmonic generation microscopy [26], medical diagnostics [27] etc.
However, a disadvantage inherent to LW-YDFLs is that laser signal in this case competes with a strong amplified spontaneous emission (ASE) arising within the "conventional" spectral band, ~1060 to 1100 nm [16,28], due to a very high gain in YDF in this spectral range as compared that inherent to LWs. Moreover, the higher pump power, the stronger ASE signal is; so with increasing pump power above some critical level, CW parasitic lasing (PL) arises at some wavelength within this spectral band [28,29] when a high YDF gain gets compensating a high-level loss of the parasitic cavity formed by weak reflections from the laser-cavity components and fiber splices. With pump power growth, the efficiency of lasing at a LW drops due to both the increase of ASE spectral density and PL power. Meanwhile, parasitic PL may, in turn, produce high-power pulsing ignited by stimulated Brillouin scattering (SBS) [29,30], the phenomenon that usually stands behind the damage of YDF used as an active fiber [29,31]. Also note that SBS pulsing in YDFLs operating in the 'conventional' wavelength range is inherent in laser cavities with a low Q-factor [30].
To clarify the physics behind PL in LW-YDFLs, we studied the processes relevant to the pump-dependent variation in 'effective' reflectivity of the laser output coupler (a weakly reflective fiber Bragg grating, FBG). This variation leads to a change of 'effective' Q-factor of the cavity and, hence, to a change in the active fiber gain, too. We show that the effective (factual) reflection of the FBG dramatically depends on the laser power, the effect related to strong broadening of CW YDFL spectrum [32,33] through the nonlinear effects in optical fiber, namely, by self-phase modulation (SPM) originated from Kerr effect and initiated by a fluctuating laser power [32,34,35]. Furthermore, we present a simple experimental technique permitting one to measure the effective FBG reflection as a function of YDFL laser power and its mathematical assessment including simple analytic formulae that describe variations in the FBG effective reflection. We believe that our present work is important for understanding the physics involved in CW FLs operating, including in the LW range, at moderate powers (watts to tens watts) and, also, for their modeling and optimization.
Note that the use of FBGs as selective reflectors permits one to assemble FL cavity in a robust all-fiber geometry; they also may modify the laser operation mode after a correspondent fiber treatment related, for instance, to the novel materials based on the nonlinear electromagnetically induced transparency [36,37]. Also note that the term "effective" length of a FBG placed inside a short Fabry-Pérot laser cavity, introduced in our earlier works [38,39], defines the factual length of the laser cavity and, consequently, the real optical frequencies of longitudinal modes established in it.

Laser setup and basic characteristics
Our YDFL was assembled in Fabry-Pérot geometry, common for high-power FLs. Fig. 1(a) presents the experimental setup. A standard double-clad YDF (Nufern, SM-YDF-5/130-VIII) was used as an active (gain) medium. YDF was relatively long: its length was 30 m that produces a high absorption (~20 dB) at the pump wavelength (λ p = 915 nm) and ensures ease of lasing in the LW Yb 3+ emission band. Note that YDF was pumped by two 30-W laser diodes (LDs) with multimode fiber outputs through a commercial (2 + 1) × 1 pump combiner with the waveguide parameters of its signal and pump fibers similar to those of both the YDF and LDs' output fibers.
Two home-made FBGs with the reflection coefficients of ~100% (high-reflection (HR)-FBG) and ~38% (low-reflection (LR)-FBG) where used as the rear and output cavity couplers, respectively; the peak reflection of the LR-FBG is herewith designated as R 0 . The FBGs' reflection spectra are demonstrated in Fig. 1(b). It is seen from this figure that reflection maxima of both FBGs were at λ s = 1134.7 nm (the laser wavelength that belong to the LW Yb3 + emission band), and that the FBGs' bandwidths measured at FWHM (full width as half of maximum) were correspondingly 480 pm (HR-FBG) and 220 pm (LR-FBG). Resolution of the optical spectrum analyzer (OSA, Yokogawa, model AQ6370D, wavelength range from 600 nm to 1700 nm) used in this experiment was 31 pm at the laser wavelength.
The reflection band of the HR-FBG has been managed to be as broad as possible (to maintain approximately 100%-reflection in the whole range of the laser power from 0 to 30 W) with an aim to minimize the decreasing of the laser power transmitted by any HR-FBG due to "spectral surrounding" of its reflection spectrum, the effect eventually resulted from spectral broadening of a laser signal at a high laser power [32][33][34]. The reflectivity of the LR-FBG was high enough to ensure maintaining of the tuning range of pump power as high as 60 W without arising PL within the "conventional" Yb 3+ band (see above). For the same reason, the YDF was heated up to 40 • C [16,28]. Furthermore, to diminish the thermal shift of the gratings' Bragg wavelengths, they were mounted on aluminum heat dissipaters; this allowed us to minimize the spectral shift of the laser peak wavelength lesser than 1 pm per 1 Watt of laser power.
A cladding mode stripper (CMS) was used to remove the residual pump at the input of the LR-FBG written in a single-mode fiber. To decrease the Fresnel reflections we used 7 • -angled fiber cuts on both sides of the laser.
The laser power was measured by a commercial power meter (Thorlabs, model PM100D) with a thermal head (Thorlabs, model S322C, maximum power is 200 W). From Fig. 2(a), it is seen that the output laser power depends linearly on the pump power; the laser threshold was 0.6 W, the power slope efficiency was 53% and the quantum slope efficiency was 66%. Note that such moderate laser efficiencies are common for LW-YDFLs [27][28][29] because of the large Stokes shift (~25% relatively to the pump wavelength) and strong ASE arising within the "conventional" Yb 3+ emission band. Fig. 2(b) shows the broadband YDFL spectrum measured at four pump powers. It is seen that ASE spectrum is always within the range ~1060 nm to ~1120 nm; its peak value pronounceably grows with increasing pump power, with the rate of 0.45 dB/W, while the laser spectral peak grows with the smaller rate, of 0.29 dB/W or less. Simultaneously, ASE spectral peak shifts to the anti-Stokes side, from 1090 nm to 1070 nm, where YDF gain is higher. Worth noticing, the CW PL at ~1060…1080 nm, potentially dangerous as igniting SBS pulsing (see above), arises in our experimental conditions at the highest pump power (around 60 W). Fig. 3(a) shows the YDFL spectra measured in the whole range of laser power. As it is seen, the laser spectrum dramatically broadens with increasing laser power, the effect related to the nonlinear phenomena in optical fibers (mostly due to SPM), arisen at such power values at which the Kerr-effect produces nonlinear phase exceeding ~1 rad. [35]; this condition is satisfied in the YDFL under study when CW laser power exceeds 10 W. Furthermore, since the narrow-band CW YDFLs demonstrate excessive photon noise with the photon statistics described by the Bose-Einstein distribution [8] when noise peaks reach power levels above the mean (CW) laser power by an order of value or morethe laser spectrum gets broadened yet at lower laser powers, of ~1 W. Note here that the laser spectra, measured at low laser powers (below 10-15 W) and plotted in the logarithmic (dB) scale, are characterized by symmetrical triangular shape, which corroborates with the noise behavior of the laser signal. That is, the derivative of the instantaneous laser power, responsible for the laser spectrum broadening, demonstrates similar (triangular) shaping of the probability density function likewise that of the laser spectrum, refer to Fig. 3.11 in ref. [40]. At further growth of the laser power, a notable dip arises at the spectrum center, which is due to overlapping of the transmission spectrum of the LR-FBG with the broadened laser spectrum: see Fig. 3(b). Note that such kind of the spectral hole burning effect was recently reported in [41].

Effective reflection of the output FBG: theory
Given that the YDFL spectrum strongly broadens with laser power growth, a part of the laser signal with spectral components outside the LR-FBG spectrum, dramatically increases. This part is not reflected by the grating at all or is reflected with smaller efficiency. This effect can be treated in terms of the power-dependent effective reflection of the grating, which decreases with the laser power increase. The latter results in diminishing the Q-factor of the laser cavity and enlarging the YDF gain that compensates the cavity loss growth.
We consider the effective reflection of the output (LR) FBG as a ratio of laser power reflected by the grating to the incidence power: where P refl is the optical power reflected by the FBG, P out is the laser output power, and P cav is the intracavity laser power at the FBG input. Thus, to calculate R eff one needs to know the output laser power, P out , spectrum of the output laser signal, S out (λ), and the FBG reflection spectrum, R FBG (λ). These data are easily obtained from the power and spectral measurements, see Fig. 2(a), 3 and 4. To simplify the simulation of the intracavity laser spectra, a Gaussian fit, obtained with high precision (adjusted R-square is 0.993), was used as the FBG reflection spectra (see Fig. 1(b)). The intracavity laser spectrum at the FBG input (further, the intracavity laser spectrum) is found as follows: where T FBG (λ) = 1 − R FBG (λ) is the FBG transmission spectrum. The intracavity laser power at the same point of the laser cavity is obtained as: where the proportionality coefficient, ξ, is found from the following formula: In these two equations, the interval from λ 1 to λ 2 is that within which the FBG and the laser spectra are not equal to zero.    4 shows two examples of simulating of S cav (λ) using S out (λ) along with the FBG reflection and transmission spectra. At very low laser powers (e.g. at 0.25 W, Fig. 4(a)), the laser output and the intracavity spectra are similar; their magnitudes differ by approximately 60% that equals to the peak FBG transmission. This result is explained by the fact that the laser spectrum is quite narrow as compared to that of the FBG (their widths measured at FWHM are 45 pm and 220 pm, respectively) whilst the variation in the FBG reflection near its peak is small. At high laser powers (e.g. at 30 W, Fig. 4(b)), the shapes of the spectra differ much stronger: the broadened laser output spectrum has two peaks that arise due to reflection from the spectrally narrow LR-FBG (compare with the spectra in Fig. 3(b) for laser power above 20 W). In the meantime, the intracavity laser spectrum demonstrates a single peak while its sections outside the FBG spectrum are identical to the output laser spectrum. Fig. 5 shows the intracavity laser spectra for the whole range of laser powers, calculated using the formulae presented above. As it is seen from Fig. 5(a), these spectra do not demonstrate a double-peak structure at high laser power: compare them with the spectra shown in Fig. 3. Then, as seen from Fig. 5(b) (where the same spectra but zoomed are shown), the maximum density of the laser spectral peak is observed in the range of laser powers between 8 and 13 W. This effect results from the strong spectral broadening that arises in the YDFL when the laser power exceeds 10 W and so the nonlinear phase increases up to the values well above 1 rad. [35]. We found, using the theory presented in [35], that the laser power at which the nonlinear phase is equal to 1 rad. is about 8 W; this value is close to that at which the spectral density at the laser peak wavelength takes a maximum. Note that the spectral broadening is so strong that the spectral density at the top decreases despite the integral spectral power increases. This detail is virtually undetectable when one examines the original laser spectra (Fig. 3).
Using the procedure described above, which is based on (i) the spectral analysis of the laser output signal (Fig. 3) along with the LR-FBG spectrum ( Fig. 1(b)) and (ii) the data on the laser output power (Fig. 2  (a)), we can obtain the intracavity laser power, P cav (see formulas (2)-(4)), and the effective reflection, R eff , of the grating (see formula (1)). Since all the data used in the calculus are experimental, we consider the resultant data for R eff (see Fig. 7 below) as experimental, too.

Effective reflection of the output FBG: simulation
First, we found, using Eqs. (2)-(4), the intracavity power P cav at the LR-FBG input for the whole range of laser (output) power; the result is shown in Fig. 6. It is seen that P cav is always above P out , the difference between them is the power reflected by the LR-FBG: ΔP = P cav − P out (see Fig. 6(a)). At low P out , this difference increases exponentially ( Fig. 6(b)) until it gets saturated at some ΔP 0 value (2.75 W in our case). Thus, the LR-FBG reflects approximately the same power (P refl = ΔP 0 = 2.75 W) for any laser power exceeding approximately 15 W. This effect results in that the blue line fitting data for the intra-cavity power above 15 W becomes parallel to the black one corresponding to the laser output power, see Fig. 6(a).
For comparison, we also show the dependence of the laser spectrum width, measured as FWHM, with the laser output power increase, see Fig. 6(b). It is seen that the spectrum width increases faster at high laser power, above 15 W, than it takes place at lower power; the difference in the slope of this dependence above and below 15 W is about two times. This effect is related to the rise, with increasing pump power, of the nonlinear phase shift up to the value above which the laser spectral line gets broaden faster, see the discussion in the previous section.
Let us consider that P refl increases with laser output power by the saturating exponential law: where P 0 = 4.39 W is the power at which P refl differs by (1 − e − 1 ) times from the maximum value. Despite this law is empirical, it fits the experimental points shown in Fig. 6(b) by curve 1 with a very high accuracy: R-square is higher than 0.999. Using Eq. (5) one can rewrite Eq.
(1) as follows: Using this equation, we fitted the set of R eff points found from Eqs. (1)-(4) with a high accuracy, too (R-square is higher than 0.996); see Fig. 7. At very low laser power, R eff = ΔP 0 /(P 0 + ΔP 0 ) ≈ R 0 , given that the laser spectrum is narrow due to the absence of nonlinear effects and that it matches the LR-FBG peak wavelength (refer to Fig. 4(a)). At higher laser power, R eff decreases with increasing P out as the hyperbolic function: R eff ≈ ΔP 0 /P out . Furthermore, one can see from Fig. 7 that R eff steadily decays by approximately-one order of magnitude, from ~40% to ~8%, with laser power growth. The experimental R eff points are fitted well by formula (6) when P 0 = 4.54 W. This value is very close to that found from the fitting of the power reflected by the LR-FBG (P 0 = 4.39 W). Note that both the parameters determining the grating's effective reflection P 0 and ΔP 0 , depend on its reflection spectrum width, too: the narrower the reflection peak, the smaller is the reflected power, and vice versa. Thus, a proper choice of the reflection band of a LR-FBG is advisable for estimating its effective (factual) reflection and, consequently, for realistic understanding of how the active fiber gain in a FL depends upon laser power.
A remark: in the above modeling, we ignored a small thermal shift to the Stokes side of the LR-FBG Bragg wavelength with increasing laser power since it is much less than the 3 dB-bandwidth of the HR-FBG used as the rear reflector of the cavity, see our discussion in Section "Laser setup and basic characteristics".

Conclusions
In this paper, we introduced the effective reflection (R eff ) of an output (low-reflecting, LR) FBG coupler of an YDFL as a ratio of the optical power returned by the FBG to the laser cavity to the optical power at the FBG incidence. A CW YDFL based on a standard Ytterbiumdoped double-clad fiber with 5-μm core operated at a long wavelength (~1.135 µm) was chosen for the current study. To find R eff we proposed a simple experimental technique permitting an analysis of output laser spectra, FBG spectrum, and laser power; we also derived simple formulae for simulating R eff .
We showed that the power reflected by the LR-FBG coupler back to the laser cavity increases by the exponential law at relatively low (<10 W) laser powers. Meanwhile, at laser power exceeding 10 W, the LR-FBG reflects the same power independently of laser power. The latter is resulted from the strong broadening of the laser line due to the nonlinear (Kerr) effect so that the spectral density at the laser spectrum peak (at the LR-FBG incidence) and in the narrow area in its proximity virtually does not change despite the integral spectral power increases linearly with increasing the laser output power.
It was found that the effective reflection of the output LR-FBG decreases with increasing laser power, which happens because of broadening of the laser spectrum and so results in passing by a considerable part of laser power off the LR-FBG reflection spectrum. At a high laser power, the effective decrease in LR-FBG's reflection is close to the hyperbolic law: R eff ~ (P out ) − 1 .
The revealed effect results in increasing the laser cavity loss and, therefore, enlarging the active fiber gain needed for its compensation to maintain the CW laser regime. The latter feature may result, in the case of an YDFL operating at a long wavelength, in the appearance of parasitic lasing within the "conventional" Yb 3+ emission band (1060…1100 nm), deteriorating the laser regime.
We believe that the clue element of the present study, viz. the role of the effective (power-dependent) reflection of a FBG output coupler, can be important for understanding the detailed physics of YDFLs operating at moderate laser powers (Watts to tenths of Watts) at longer operational wavelengths and their proper modeling.

Funding
This work was supported in part by the Centro de Investigaciones en   6. (a) The intracavity power (P cav , the orange circles and curve) vs laser output power; the laser output power (P out , the black squares and line) is shown for comparison; ΔP is the difference between these two graphs. The symbols stand for the experimental data and the lines are their fits. The blue line shows a linear fit of the intracavity power at P out > 12 W. (b) Curve 1: ΔP vs laser output power (the left scale); the points stand for the experimental data and the curve is their exponential fit. Curve 2: Spectrum width of the intracavity laser signal vs laser power (the right scale); the points are data obtained for the spectra shown in Fig. 5 and the lines are linear fits for low (below 15 W) and high (above 15 W) laser power. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Optica, A.C. (Mexico) through the internal project "2-µm fiber laser", the Ministerio de Ciencia e Innovación/Agencia Estatal de Investigación (MCIN/AEI/https://doi.org/10.13039/501100011033), co-funded by the European Union "ERDF A way of making Europe" under grant PDI2019-104276RB-I00 and the Generalitat Valenciana (PROMETEO/ 2019/048) (both from Spain).

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability
Data will be made available on request.