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We present a method for the estimation of the reflection spectra of transient gratings in rare-earth doped fiber lasers having a self-sweeping of laser wavelength. We show that high reflectivities of several tens of percent can be achieved. An example of this is demonstrated through the use of an experimental Yb-doped Fabry-Perot fiber laser. The gratings' spectra are highly asymmetric due to the apodization of the refractive index modulation. The importance of the self-sweeping regime for triggering self-Q-switched laser instabilities is discussed.
© 2014 Optical Society of America
Corrections
P. Peterka, P. Honzátko, P. Koška, F. Todorov, J. Aubrecht, O. Podrazký, and I. Kašík, "Reflectivity of transient Bragg reflection gratings in fiber laser with laser-wavelength self-sweeping: erratum," Opt. Express 24, 16222-16223 (2016)https://opg.optica.org/oe/abstract.cfm?uri=oe-24-14-16222
1. Introduction
The recent expansion of applications related to fiber lasers has brought significant attention to various types of fiber laser instabilities. In particular, a frequent and dangerous instability is represented by modes of self-Q-switched pulse regimes, which are characterized by intense sub-µs pulses having more or less a chaotic repetition rate [1]. The self-pulsation regimes have been attributed, e.g., to reabsorption in an unpumped part of the active fiber [2], Raman and Brillouin scattering processes [1], external mechanical perturbations or pump instabilities and ion pairs formation [3]. Formation of a standing wave in fiber lasers and associated dynamic fiber grating was often mentioned as a potential source of fiber laser instabilities [4–8]. Another self-pulsing regime is the so-called self-induced laser line sweeping (SLLS). This description reflects the fact that the self-pulsing in the form of self-sustained relaxation oscillations coexists with spectacular laser wavelength drift with time. The main characteristic of this effect is the self-scanning of the laser wavelength (usually drifting from shorter to longer wavelengths), spanning over several nanometers, and its instantaneous bouncing backward. In fiber lasers, such a laser wavelength drift with time has been observed in an Yb-doped-fiber ring laser [7, 8], in Yb-doped Fabry-Perot laser cavities [5, 6], in Er-doped fiber lasers at around 1560 nm [9], and in a Tm-doped fiber laser at 1915-1930 nm [10]. Laser wavelength sweeping in the reverse direction (i.e., from longer towards to shorter wavelengths) has been observed under specific circumstances in both Yb- and Er-doped fiber lasers [9].
SLLS can be explained by spatial-hole burning (SHB) in the active medium caused by a standing-wave in the laser cavity. Superposition of the narrow-linewidth, contra-propagating beams creates an interference pattern of nodes and antinodes in the active medium. While the gain is high at nodes, it is lower at antinodes. Therefore, the actual laser signal suppresses gain for itself and promotes lasing of longitudinal modes with better overlap with regions of higher gain. From the spectral point of view the self-written grating of nodes and antinodes exhibit itself as an inhomogeneous gain broadening. The gain/loss spectrum is then modulated by a function proportional to sinc(C(λ-λ0)) where the constant C depends primarily on the grating length and λ0 is the wavelength of high-power laser signal. The overall threshold inversion determines the gain spectrum and thus contributes to the selection of the sweeping direction. The overall inversion level is influenced by several factors including active fiber length and the pump wavelength. For example, with tuning the pump wavelength by changing the pump laser diode temperature we achieved switching of the sweeping direction [9]. Lobach et al. [5] observed that the sweep rate is inversely proportional to the length of the active fiber and that the sweep rate increases with square root of the laser output power. The discovery of the SLLS effect in fiber lasers pointed out their single- or few-longitudinal mode operation in a relatively wide range of pump powers above the laser threshold. Similarly to other fiber laser instabilities, the SLLS effect is an unwanted instability to be avoided in most cases. However, simple construction and relatively high power of the SLLS fiber laser may be interesting for applications of swept lasers, e.g, in interrogation of optical fiber sensor systems. Narrow linewidth nature of SLLS fiber lasers was used for characterization of spectrally narrow features in optical fiber components [11]. Another field of practical exploitation is all-fiber self-Q-switched fiber lasers because understanding of triggering mechanisms should lead to substantial improvement of all-fiber Q-switched laser sources.
The narrow linewidth of SLLS fiber lasers provides the opportunity to create dynamic refractive index gratings along the fiber laser cavity. Dynamic gratings in rare-earth doped fibers have already been investigated for variety of applications, including narrowband fiber optical filters [12], single-frequency fiber lasers [13], and adaptive interferometers [14].To the best our knowledge, we present here for the first time a quantitative analysis of the spectral reflectivity of a dynamic grating in a SLLS fiber laser. We report on a method for the estimation of the reflection spectra of transient gratings in rare-earth doped fiber lasers with self-sweeping. We present application of the method using an example of Yb-doped Fabry-Perot fiber laser, whose pulsing behavior was presented in a conference [15]. We also discuss origins of the self-Q-switched regime.
2. Experiment
The experimental setup of the SLLS fiber laser is shown in Fig. 1(a). The gain medium was a 10-m long double-clad fiber with Yb-doped phospho-silicate core, fabricated in-house, having a peak absorption of about 800 dB/m at 976 nm in the 5-µm diameter core, with a numerical aperture of NAmax = 0.18. The length of the experimental fiber was optimized for highest power. It should be noted that SLLS effect was observed also for the other fiber lengths in the cut-back measurements from 17 to 9 m. The fiber core supported a single-transversal mode of operation at the wavelength of the laser around 1.06 µm. The inner cladding has an outer diameter of 125 µm in a hexagonal shape; see Fig. 1(b), whereas the outer cladding was made from a low refractive index polymer. The average total Yb concentration of 4.6 × 1026 m−3 was determined by the electron microprobe analysis. The measured Yb absorption spectrum is shown in Fig. 2(a). The Fabry-Perot resonator was formed by perpendicularly cleaved fiber ends. Note that self-sweeping behavior was reported also for other resonator configurations, e.g., for ring-fiber laser with weak (<1%) parasitic reflection at the output coupler [7, 8], or for Fabry-Perot lasers with one high-reflectivity mirror, either in the form of broadband fiber loop mirror [5, 9] or narrowband fiber Bragg grating (FBG) [5]. We used a multimode-pump laser diode with central wavelength of about 975 nm at a maximum driving current of 7 A, where the case temperature was set to 25 °C. The pump and signal were combined in a tapered fiber bundle combiner. A 1% fiber tap was inserted into the cavity to monitor the laser output. The temporal dynamics were detected using an InGaAs photodiode with 1.2 GHz bandwidth and displayed on a digital oscilloscope (100 MHz bandwidth, 1 GSa/s). The spectra were monitored by a grating-monochromator-based optical spectrum analyzer ANDO AQ-1425. The overall laser cavity length was ~13 m. The output power of the fiber laser vs. the pump power launched into the double-clad fiber is shown in Fig. 2(b); because the pump wavelength varies with driving current, this dependence is not strictly linear. The laser operated in the SLLS regime up to ~0.5 W of the output power. Above the laser threshold, the laser wavelength drifted slowly from 1060 to 1064.5 nm, see Fig. 3(a). Taking into account constant scanning speed of the used optical spectrum analyzer (44 seconds per single scan), one can easily estimate the sweeping rate and the sweeping period from the measured spectrograms in Fig. 3 (a) and from the measurements of time evolution of the laser output at the wavelength 1062 nm shown in Fig. 3(b). The time evolution was obtained by using the optical spectrum analyzer in the power meter mode. The sweeping period of 5.3 s and correspondingly the sweeping rate of 0.81 nm/s were evaluated for the output power of 0.14 W, while for the 0.4 W output power the average sweeping period of 1.8 s and sweeping rate of 1.5 nm/s were estimated. The above mentioned method of basic characterization of the SLLS regime was verified by using CCD spectrometer for the case of Fabry-Perot and ring fiber lasers [8, 9]. In the contrast to the phospho-silicate-core fiber we report in this paper, the Yb-doped fiber in [7–9] was of alumino-silicate fiber-core composition. Despite the different host medium, we observed qualitatively similar self-sweeping behavior. Self-sweeping we observed also in core-pumped erbium-doped fiber laser [9] with much lower level concentration of rare-earth ions, the peak core absorption at around 980 nm was only 5 dB/m. Therefore, we assume that the high doping concentration, typical for fibers for cladding pumping, is not prerequisite for the SLLS effect.
Fig. 2 (a) Absorption spectrum of the experimental Yb-doped phospho-silicate fiber. (b) Laser output power vs. pump power.
Fig. 3 (a) Output spectra for three different laser output powers. The resolution (input slit width) of the optical spectrum analyzer was set to higher value for the low output power. (b) Optical spectrum analyzer output in the power meter mode at the wavelength of 1062 nm.
Since the laser resonator length gave a longitudinal mode separation as low as 8 MHz, the successive oscillation of tens of thousands longitudinal modes is possible within one sweep. Taken into account the sweeping wavelength interval, the sweeping rate, and the number of longitudinal modes, one can estimate that the oscillation of one longitudinal mode dominates within a time period of 20-40 µs. If the sweeping did not occur by successive hopping between the nearest longitudinal modes, but some mode(s) were skipped, the oscillation of one mode could dominate for even a longer period.
Further increases of the pump power led to a self-Q-switched regime. A laser spectrum of up to 0.6 nm FWHM was observed beyond the output power of 0.5 W, see the lowest spectrum in Fig. 3(a). The peak power of the self-Q-switched regime can be of the order of several kW as it was reported for the similar setup and pump-power levels [1]. Therefore, we protected the spectrum analyzer from eventual damage by inserting another 1% tap to further attenuate the laser pulse. The lowest spectrum in Fig. 3(a) was varying in time due to gain competition among the multitude of longitudinal modes. The background ASE is not seen in the Fig. 3 because of the high signal attenuation in front of the optical spectrum analyzer. Similar to other reports [5, 8], we have not observed continuous-wave regime in this Fabry-Perot, SLLS laser configuration within the available pump power levels. It should be noted that the transition from SLLS to self-Q-switched regime was not abrupt, but was rather a gradual process. The sweeping regime became less regular and giant pulses started to appear when the output power exceeded the level of ~0.4 W. Typical temporal characteristics of the output power for different pump power levels are shown in Fig. 4. A train of ~µs long narrow-linewidth pulses was generated in the SLLS regime. The laser in SLLS regime oscillates either in a single or only a few longitudinal modes [8] and [11]. Indeed, a sinusoidal beat signal was often detected under the pulse envelope, see details of selected pulses in Fig. 4(b). While increasing the pump power, a gradual transition of the sinusoidal signal to giant pulses shorter than 10 ns could be observed.
3. Theory
The formation of the dynamic refractive-index grating in an Yb-doped fiber is due to the refractive index modulation that is proportional to the population inversion of Yb ions. The index of refraction is connected with the Yb absorption through the Kramers-Krönig relations around the peaks of absorption. At the operational wavelength of the experimental fiber laser (around 1062 nm), the UV absorption band contribution to the refractive index change dominates over the contribution of the absorption of the pump band itself, and can be written as [16]
where e is the elementary electric charge, n0 is the refractive index of the host medium, m is the electron mass, ɛ0 is the vacuum permittivity, and c0 is the speed of light in vacuum. The factor S expresses the effect of transitions between the ground and excited states at the laser wavelength. Using the oscillator strengths and transition wavelengths of Yb given by Arkwright et al. in [16], the value of S at 1062 nm is S = 1.21 × 10−25 m·s. The longitudinal distribution of the metastable level population N2(z) at the respective position z along the fiber is given by the pump and the laser signal optical power. The evaluation of N2(z) consists of two steps, which we explain here using the fiber laser example described in the previous section.Firstly, we estimated the evolution of optical power along the fiber without interference effects, i.e., the optical power is assumed constant in a section of fiber Δz that obeys L»Δz»Λ, where L is the overall fiber length and Λ = 0.36 µm is the FBG pitch enforced by the wavelength of the single frequency laser. The optical power along the fiber was calculated using a comprehensive, spectrally and spatially resolved numerical model of a rare-earth doped fiber laser [17]. The results of the numerical calculations for a total laser output power of 400 mW are shown in Fig. 5(a). The laser signal that propagates along the direction of the pump is denoted as the forward signal (P+), and the laser signal that travels against the pump is labeled as the backward signal (P-). The metastable level population N2avg is also assumed constant (averaged) in a section of fiber with length Δz, see Fig. 5(b). Limitations of the model are noted here as (i) the aforementioned averaging along Δz, as well as (ii) an assumption that there exists monochromatic pump and homogenous absorption along the fiber. In fact, the pump spectrum is up to 4 nm wide and the spatial pump modes having a higher overlap of the core are absorbed faster than the rest of the modes. There is still room for improvement of this model; however, even under the assumed approximations it provides a basic quantitative estimation of the optical power distribution along the fiber.
Fig. 5 (a) Optical power of the pump, forward and backward laser signal and interference fringes visibility along the resonator according to the numerical model for the output power of 360 mW. (b) Modulation of the refractive index along the active fiber. The fast modulation is shown in the inset in the position of maximum modulation depth.
Secondly, we involve effects of interference. In the case of a single frequency fiber laser we can assume that the interference pattern would be created according to the interference equation, and accordingly, the laser signal power would oscillate between the values Pmax and Pmin as Pmax/min = P+ + P- ± 2 (P+ P-)1/2. All the variables are dependent on z. The interference fringes visibility distribution along the fiber is dependent on the relative power in the co- and counter propagation laser signal field, as shown in Fig. 5(a). The highest visibility coincides with the position of equal electric field of the interfering beam. The metastable population can be evaluated from the laser rate equation [17] as
where the fluorescence lifetime of the Yb metastable level is assumed to be τ = 1.5 ms. Wa and We are the absorption and emission rates, respectively, hν is the photon energy, A is the core area, Γ is the so called overlap factor describing the fraction of modal power overlapping with the doped core, and σa and σe are the Yb absorption and emission cross sections, respectively. The modulation depth of the refractive index variation along the fiber laser is then given by Eqs. (1) and (2). We can distinguish between a fast oscillation, having period Λ and modulation depth ΔnAC, and a slow variation of refractive index change ΔnDC, averaged spatially over a section of length Δz. Such a section of the fiber is assumed as a partial FBG with a uniform refractive index modulation. The refractive index modulation is shown in Fig. 5(b). For sake of clarity, only the maximum, average and minimum refractive index differences are plotted. The maximum and minimum refractive index difference are given as Δnmax/min = ΔnDC ± ΔnAC. Finally, with the knowledge of the refractive index distribution we can calculate the reflectivity of the complex fiber grating of the whole Yb doped fiber by the transfer matrix method, described in detail by Erdogan [18].4. Results and discussion
The reflection spectra shown in the upper curve in Fig. 6(a) were calculated for FBGs having the modulation and apodization of refractive index profiles in Fig. 5(b). A peak reflectivity of more than 50% can be achieved. Interestingly, the reflectivity spectrum exhibits an unexpected feature: the actual laser wavelength obtains almost no reflection. Conversely, it is assumed that the gain-induced grating created in the SLLS affect mostly the laser wavelength itself [5, 6] and [8]. The FBG reflection spectra are highly asymmetric, with high-spectral frequency sidelobes having decreasing reflectivity towards longer wavelengths. Yet at a distance of 38 pm of the Brillouin Stokes wave one can get a reflectivity of about −24.0 dB. Note that the SBS gain peak is shifted in the silica-based fibers by about 10 GHz, i.e., the Stokes wave is 38 pm away from the laser wavelength of 1062 nm. The values of the reflectivity at the Brillouin Stokes wavelength are small, ~0.4%, and of similar order of magnitude reflection as the Rayleigh scattering, which is also considered as a possible promoting mechanism of some self-Q-switched lasers [19].
Fig. 6 (a) Reflectivity of the dynamic FBG for three different levels of metastable level excitations. (b) Detail of the reflectivity in close proximity of the immediate laser wavelength.
The upper reflection in Fig. 6(a) was calculated under the assumption that the maximum modulation depth of the inversion population and correspondingly the maximum modulation depth of the refractive index were achieved. However, the saturation time of the inversion population built-up tsat = τ/(1 + Ppump/ Ppumpsat + Psignal/ Psignalsat) [20] is about 100 µs. This value was estimated using both the optical powers in Fig. 5(a) as well as the saturation powers of our fiber laser, where Ppumpsat ~600 mW and Psignalsat ~15 mW. Since one longitudinal mode dominates over the time of the order of tens of µs, the modulation depth of the inversion population would be smaller than that of the steady-state case. The reflection spectra for two different fractions of the maximum modulation depth are shown in Fig. 6. A reflectivity of as high as 5% can be achieved even for only 25% of the maximum modulation depth. The reflection spectra in Fig. 6 were calculated for the FBG of the whole 10-m long Yb-doped fiber. In fact, the laser signal undergoes reflections distributed along the FBGs. The reflectivities of shorter sections of the transient grating around the peak modulation depth are shown in Fig. 7. From the point of view of the distributed reflection, the FBG also influences the single-frequency laser signal itself. It should be noted that just recently Ivan Lobach et al. presented an alternative approach to describe transient phase-grating in SLLS fiber lasers. They linked the phase change with the spectral gain modulation induced by SHB, and furthermore, proposed a possible influence of the phase grating to the origin of self-sweeping [21]. A comparison of the approach presented here and in [21] is beyond the scope of this paper.
Fig. 7 Reflectivity of short sections around the peak refractive index modulation. The section length is 40 cm (a) and 10 cm (b). Maximum metastable level excitation was assumed.
To date, all reported SLLS fiber lasers in a Fabry-Perot arrangement exhibit a self-Q-switched mode if the lasers were studied for pump powers above the pump power range of the SLLS regime [5, 6] and [8]. Under certain conditions, even the SLLS fiber lasers in a unidirectional fiber-ring configuration with a perpendicularly cleaved output coupler fiber end (allowing single-fixed reflection to the Yb-doped fiber) exhibited a self-Q-switching regime for high pump powers, not only the continuous-wave regime. Such ring lasers and Fabry-Perot lasers can be described as fiber lasers in “similar experimental conditions” [6] in the sense that a regular regime of transient standing-wave can be created and that the laser output coupling is high. On the other hand, the unidirectional fiber ring lasers with careful suppression of any fixed reflections stay in continuous-wave-regime only [8, 9]. Indeed, for the free-running operation of a unidirectional fiber laser with a travelling wave (standing-wave is not allowed), any possible transient gratings are smoothed out along the rare-earth doped fiber, and correspondingly, no distributed reflections can occur. The laser output spectra are highly multimode: neither single- nor few-longitudinal mode oscillations are possible as it is in the case of the SLLS regime. Based on these comparisons, it can be assumed that the fiber laser SLLS regime is an important mechanism for triggering the self-Q-switched regime. The importance of the narrow-linewidth nature of the SLLS regime for the promotion of an SBS-based self-Q-switching operation has already been shortly noted [5]. In addition, the reflectivity transient FBG build-up of the SLLS fiber laser presented here may significantly change the quality Q-factor of the resonator, and furthermore, contribute to triggering the self-Q-switched regimes.
5. Conclusion
We have developed a method for the quantitative estimation of the grating spectra created within the regular process of SLLS in a Fabry-Perot Yb-doped fiber laser. We also evaluated the reflectivity of transient gratings for an experimental fiber laser. We have shown that the regular process of SLLS can build transient gratings of a high reflectivity of more than 50%. The spectral shape of the reflectivity may exhibit a high asymmetry due to the apodization of the refractive index modulation. We highlight the importance of the SLLS for triggering the self-Q-switched regime, notably because a fiber laser with the SLLS regime above a critical threshold under similar cavity conditions may pass on to the self-Q-switched regime, while a fiber laser without the SLLS regime may not.
Acknowledgments
The authors gratefully acknowledge fruitful discussion with Jiří Čtyroký about the FBG modelling. The research was supported by the Czech Science Foundation under the project P205/11/1840.
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