1. Introduction

Current investigations of dynamic gratings recorded via local saturation of optical absorption/gain in rare-earth doped optical fibers were started more than decade ago with the pioneer publication [1]. The interest is stimulated by possible applications of such gratings in tunable narrowband fiber optical filters [1–4], single-frequency standing-wave fiber lasers [5–8], adaptive interferometers for detection of mechanical vibrations [9,10], and also for suppression of the carrier frequency in modulated optical signals [11]. In the majority of the above-mentioned publications the Er-doped fibers (with or without optical pumping) were used with recording of the grating at the wavelength around λ ≈ 1530 nm corresponding to the wavelength of maximal absorption/gain of Er⁺³ ions in silica [12]. In some papers on single-frequency standing-wave fiber lasers, utilization of Yb-doped fibers with saturable absorption at the operation wavelength ≈ 1064 nm was, however, also reported [7,8].

Unshifted amplitude dynamic gratings are needed in the majority of the above-mentioned applications except for the adaptive interferomtery, where unshifted phase (i.e. refractive index) gratings, which ensure linear response in the transient two-wave mixing (TWM) configuration, are preferable. Special attention was given in [9,13,14] to evaluation of a relative contribution of the amplitude/phase components in the recorded population gratings. The experiments, performed in Er-doped fibers both with [13] and without [9,14] optical pumping demonstrated that in the center of the absorption/gain spectrum line of Er⁺³ (1520–1550 nm) the recorded gratings are essentially of an amplitude type, while, at the sides of this spectrum, the contribution of a phase component is also significant. This effect is especially pronounced at the short wavelength side (1480–1500 nm) of the absorption line, where the phase component in the grating recorded in the fiber without optical pumping [14] proved to be even stronger than the amplitude one.

Below we report results on separation of the phase/amplitude contributions in the population gratings recorded in Yb-doped fibers with saturable absorption (some preliminary results were reported in [15]). These data are important for different possible applications of population gratings in Yb-doped fibers. Indeed, the above-mentioned application in single-frequency lasers [7,8] needs the amplitude-type gratings, while the phase gratings are preferable in adaptive interferometry.

2. Experimental setup and results

The experiments were performed using transient TWM of two counter-propagating recording waves R and S [9,14] in configuration of a linear interferometer (Fig. 1) similar to those reported in [10,13] earlier. We utilized FEPL-100-1060T-DBR single-mode single-frequency semiconductor laser (“Frankfurt Laser Company”) with maximal output power about 50 mW at 1064 nm. As in [10], we used piezoelectric transducer with an attached mirror for phase modulation of the back-reflected wave S. We also used the rectangular profile of phase modulation with the amplitude controlled through the applied modulation voltage U_mod.

There was no special collimating optical system at the output end of the fiber in our experimental configuration. For this reason, the coupling coefficient for the back-reflected wave was, in general, quite low (typically < 10%) and the grating in the fiber was recorded by two counterpropagating waves of essentially different powers (P_S << P_R). To reduce unwanted reflection from the free fiber end we used the phase matching liquid to fill the space between it and the vibrating mirror surface.

 

Fig. 1. Experimental set up utilized in experiments on transient TWM in Yb-doped fibers. Inset shows profiles of the rectangular modulating signal (a), and typical transient TWM responses expected for unshifted amplitude (b) and phase (c) dynamic gratings.

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The optical power reflected from this interferometric configuration and detected at terminal 4 of the input 50/50 coupler (see Fig. 1) proves to be modulated because of fast periodic shift between the interference pattern and the inertial grating recorded in the doped fiber. As it was discussed in [14] earlier, the shape of the detected transient TWM signal depends on type of the recorded dynamic grating strongly. For purely unshifted amplitude grating the TWM response is essentially of an even type (with all transient peaks of the same, negative sign), while for purely phase unshifted grating the response is of an odd type (every second peak is of the opposite sign) - see inset to Fig. 1. The complex grating, which possesses both the amplitude and phase component, is characterized by more complex response which can be presented as a sum of two above-mentioned typical responses [14]. In both cases the maximal amplitudes of the TWM peaks relate to the amplitudes of the corresponding grating components, while the characteristic relaxation time of the trailing edge of the TWM peaks corresponds to the grating relaxation time τ_g.

In the presented experiments we utilized two different samples of the Yb-doped fibers. One of them was commercial “INO” single-clad single-mode 1-m long fiber Yb103 with the core diameter of 4.1 ± 1 μm. Relatively low ytterbium concentration in this fiber (with maximum absorption ≈ 30 dB/m@976 nm) ensured also quite low initial, i.e. not saturated, optical density at the recording wavelength 1064 nm. Using the data provided by supplier it could be evaluated as α₀L ≈ 0.05. This fiber is addressed as fiber #1 below. Another fiber sample was 1.9-m long double-clad single-mode Yb-doped fiber with the core diameter of ≈ 6 μm and with maximum absorption ≈ 700 dB/m@976 nm. Direct measurements showed that the initial optical density of this fiber (which we address below as fiber #2) at the recording wavelength was α₀L ≈ 0.8.

To evaluate spontaneous relaxation time τ₀ of the excited meta-stable level and the fiber saturation power P_sat we performed auxiliary measurements in fiber #1 with the fluorescence excited by the recording wave R only. To this end, we removed the piezoelectric modulator and immersed the output end of the fiber. The fluorescence spectra detected at terminal 4 of the input coupler are shown in Fig. 2(a) for different input laser power P_in.

Figure 2(b) presents the input power dependence of the growth rate, i.e. the inverse characteristic growth time τ_f, and of the dark decay rate (τ₀ ^-1) of the detected integral fluorescence signal observed in case of rectangular modulation of the laser output power. The latter parameter was practically the light power independent and allowed us to evaluate the dark fluorescence relaxation time τ₀ as ≈ 0.95 ms in fiber #1. Approximating the former set of the experimental points with linear dependence characteristic for the two-level saturable system [16,17] also allowed us to evaluate the saturation power P_sat of this fiber as ≈ 3.7 mW. Similar value of the saturation power (as it was done for Er-doped fiber in [16]) can also be estimated from the light power dependence of the fluorescence signal amplitude observed in its maximum at 1625 nm (see inset to Fig. 2(a)). Note that the obtained P_sat is approximately 10 times larger than typical values of the saturation power (typically 0.2 – 0.3 mW) observed in the maximum of the absorption line in Er-doped fibers [16,17].

 

Fig. 2. (a) Fluorescence spectra observed under excitation by the direct light wave R of different input power P_in, mW: 0.8, 6.8 and 15.0 (from the bottom to the top curve, λ = 1064 nm, fiber #1). Inset shows intensity dependence of the fluorescence signal detected at 1025 nm and corresponding theoretical fit. (b) Light power dependence of the fluorescence dark decay (circles) and growth (squares) rates. Solid and dashed lines represent theoretical fits.

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Figure 3 presents typical profiles of the transient TWM response which is observed in the investigated Yb-doped fibers. To record these signals we usually utilized averaging (typically over 512 oscilloscope traces), which was done to improve signal-to-noise ratio and to suppress slow fluctuations of the DC signal level and parasitic rectangular shape signals due to interference among the spurious reflections from different optical contacts of our fiber configuration [9]. Comparing experimentally observed TWM response with the expected profiles presented in inset to Fig. 1, one can see that the gratings recorded in Yb-doped fibers under above-mentioned conditions are predominantly of the phase type.

 

Fig. 3. Typical transient TWM signals observed in fiber #1 (a) and in fiber #2 (b) at U_mod ≈ U_π/2 (P_in = 5 and 13 mW for fiber #1 and fiber #2 respectively, with averaging over 512 traces).

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To make qualitative analysis of the TWM response we resolve it into the odd component (presented by curve c in inset to Fig. 1) with the relative amplitude of a total swing (U_max - U_min)/U_st and the even component (curve b in inset to Fig. 1) with the relative amplitude (U_max + U_min - 2U_st)/2U_st - see Fig. 2(b). As it was mentioned above, the former of these two contributions relates to the phase, i.e. to the refractive index, dynamic grating. The relative amplitude of this signal, presented in Fig. 4(a) as a function of the modulation voltage amplitude, follows the theoretically expected dependence ∝ sin(πU_mod/U_π) [14]. In turn, the latter TWM signal component, which can be attributed to the amplitude grating, also follows the corresponding theoretical dependence ∝[1-cos(πU_mod/U_π)]/2 = sin(πU_mod/2U_π)² [9,14] -see Fig. 4(a). Comparing maximal amplitudes of these two contributions one can conclude that the phase (i.e. refractive index) grating is approximately 3 times stronger than the amplitude one in fiber #1. The amplitude of the phase grating recorded in this fiber is also presented in Fig. 4(b) as a function of the incident recording power P_in.

Similar dependences on the modulation amplitude and on the recording light power were also observed for phase and for amplitude grating components in fiber #2 [15]. The main difference with the dependences presented in Fig. 4 for fiber #1 are only quantitative. Indeed, the TWM signal associated with the phase grating in fiber #2 was much stronger (see e.g. Fig. 2) with the maximal relative amplitude ≈ 0.4. In its turn, the even-to-odd component ratio (≈ 1/3 in fiber #1) proved to be significantly smaller and was about ≈ 1/10 in this fiber.

 

Fig. 4. (a) Experimental dependences of relative amplitudes of odd (circles) and even (squares) TWM signal components on U_mod (fiber #1, P_in = 5 mW). Solid lines present fitting by expected theoretical dependences (U_π ≈ 7.5 V_p-p). (b) Experimental dependence of relative amplitude of the odd TWM response component on input light power P_in (U_mod = 1 V_p-p).

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As it was mentioned above, the characteristic time of the TWM signal relaxation is related to the formation time τ_g of the corresponding dynamic grating. Experimentally observed decay of the TWM peaks can, indeed, be approximated by exponential functions quite well - see Fig. 5(a). The characteristic time of the TWM signal relaxation τ_g associated with more powerful phase grating was found to be very close to τ₀ ≈ 0.95 ms at low recording power and demonstrated continuous reduction with the growing P_in - see Fig. 5(b).

 

Fig. 5. (a) Decay of the TWM peaks observed at different input light power P_in (fiber #1, U_mod = 1.0 V_p-p). (b) Experimental dependence of the TWM peak relaxation rate as a function of P_in. Dashed line shows linear fit for power dependence of the fluorescence growth rate - Fig. 2(b).

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3. Discussion

The general analysis of the TWM efficiency performed in [15] predicts that in case of equal input powers of the recording waves (P_s,in = P_R,in) the relative amplitude of the even component of the TWM signal in fibers of low optical density (α₀L ≤ 1) is to be around ≈ α₀L/4. This maximum is observed for the total input light power around saturation power of the fiber P_sat. Detailed analysis shows that in case of essentially nonsymmetric conditions of recording in our configuration (when P_s << P_R) the relative TWM signal amplitude detected in weaker output wave S can be nearly two times higher (i.e. ≈ α₀L/2). This means that for the fibers under consideration with estimated optical densities ≈ 0.05 and ≈ 0.8 one can expect maximal amplitudes of the TWM even response around ≈ 0.025 and ≈ 0.4 respectively.

Both TWM components (i.e. even and odd) reached their maxima at the recording power close to the fiber saturation power (Fig. 4(b), see also [15]). However, the experimentally observed even TWM response amplitudes (0.008 and 0.02) were significantly inferior to the above-mentioned theoretical expectations for both fibers. Note that strong disagreements between the experimental values of the grating amplitude and their theoretical estimations based on measurements of the fiber initial optical density and saturation power are also typical for the amplitude gratings recorded in Er-doped fibers in the central part (1520 – 1550 nm) of the Er⁺³ absorption spectrum [9,14]. It is difficult to make similar theoretical evaluations for the amplitude of the odd TWM response components, since we do not know a priori the ratio between corresponding changes of the fiber optical absorption and the refractive index under saturation of the fundamental optical transition in Yb⁺³.

The main physical effects which were considered earlier in relation with similar reduction of the amplitude grating component in Er-doped fibers are the resonance migration of excitation among neighboring Er⁺³ ions [17], the polarization mismatch due to random birefringence of the fiber [18], presence of not saturable absorption, or significant contribution of the spontaneous fluorescence in the detected signal. Note also that in Er-doped fibers, the above mentioned mechanisms failed to explain significant reduction of the grating amplitude recorded in central/long wavelength side of Er⁺³ absorption spectrum. On the contrary, the grating recorded in the short-wavelength side of this spectrum was nearly as efficient as it is theoretically predicted from consideration of two-level model [16].

All these mechanisms can also contribute to similar reduction of the grating amplitude in Yb-doped fiber as well, but their detailed analysis needs additional experiments. After consideration of the presented results, one can conclude, however, that the first two mentioned-above reasons are probably not so important in case under consideration. Indeed, the close correspondence of the grating formation rates (Fig. 5(b)) and the fluorescence growth rates (Fig. 3(b)) implies that the diffusion of the excited state is not very efficient under conditions of the reported experiments. Relatively small length of the fibers also prevents from efficient influence of the random fiber birefringence. On the opposite, significant discrepancy between the modal diameters in the doped and conventional fibers, can increase the level of the background radiation in the fiber jacket, and, in this way, reduce estimated value of the TWM signal relative amplitude.

The main difference between the population dynamic gratings recorded in Er-doped fibers and those recorded in Yb-doped fibers at 1064 nm is that in the latter case the grating is essentially of the phase type. This can not be considered, however, as an unexpected result since the recording wavelength used is essentially out of the maximum of Yb⁺³ optical absorption located at ≈ 975 nm. This makes these fibers quite attractive for applications in adaptive interferometers for detection of mechanical vibrations, where the phase dynamic gratings directly ensure linear TWM response without any linearization techniques [10].

4. Conclusion

Summarizing, we have reported the first results on investigation of the dynamic population gratings recorded at λ = 1064 nm in Yb-doped fibers with saturable absorption. The gratings formation time proved to be approximately one order of magnitude shorter and the recording cw light power approximately one order of magnitude larger than it is typically observed for similar population gratings in Er-doped fibers in spectral range 1480 – 1570 nm. The population gratings are shown to be predominantly of the phase type with the amplitude component significantly weaker than it can be estimated from the fiber optical density.