1. Introduction

One of the main mechanisms of the refractive index change (RIC) in pumped rare-earth-doped optical fibers is associated with changes of population of the ion states with different polarizabilities [1,2]. This electronic effect, which is important both for laser crystals and glasses, is intensively investigated during several years [3–10]. The origin of the polarizability difference (PD) in rear-earth ions used for doping the laser fibers is also widely discussed. Some authors believe that the main contribution to the PD comes from located far from the resonance strong UV transitions [1], similar to those observed in laser crystals [4,8]. An alternative model suggests the predominant contribution from the near-resonance IR transitions [2,5–7]. Correct understanding of these electronic phenomena in Yb-doped fibers [3,5,7,11] is very important because the pump-induced index changes influence dynamics of the fiber lasers and amplifiers, fiber Bragg gratings, etc. significantly. Besides, the enhanced nonlinear phase shift can be used for optical switching [1] and coherent beam combining [5,12–14].

In this paper we report original results on observation of RIC in standard single-mode Yb-doped optical fibers under diode pumping at 980 nm in configuration of Mach-Zehnder interferometer operating in IR far from the absorption and emission Yb³⁺ ion resonances. The main objective is to explore the RIC dynamics and to determine the values of the PD in standard Yb-doped fibers in spectral range 1460–1620 nm.

2. Experimental setup

The Yb-doped fibers under investigation ware included in one arm of the all-fiber spliced Mach-Zehnder interferometer (Fig. 1(a)). The CW-radiation of diode laser “Tunics” with the coherence length ~10 m was used as a probe wave and was detected at the interferometer output by fast photodiode. The probe wavelength λ_T could be continuously tuned from 1460 to 1620 nm. The fibers under test were pumped by a standard pumping laser diode operating at λ_P≈980 nm with the power up to ~145 mW. The RIC was evaluated from response to the rectangular pump pulse of 10 µs - 10 ms duration (Fig. 1(b)).

 

Fig. 1. Experimental setup for testing of Yb-doped fibers (a); (b): the pump pulse profile (red), recorded oscilloscope trace (black), and reconstructed phase trace (blue).

Download Full Size | PDF

The induced phase shift δφ(t) is evaluated from the oscilloscope trace U(t) as δφ(t)=φ(t)-φ(0), where $\phi \left(t\right)={\left(-1\right)}^k\arcsin \left(\frac{2U\left(t\right)-U_{max}-U_{min}}{U_{max}-U_{min}}\right)+\pi k$ and k=0, 1, 2… ensures continuity of δφ(t). Several fiber samples of a different length, profile geometry and Yb³⁺-ion concentration have been examined. All the fibers used (Fibers 1–4) are the single-mode ones at the pump and the probe wavelengths. Fiber 1 (fabricated in Fiber Optics Research Center, Russia) and Fibers 2–4 (Yb-198, -118, -103, of CorActive, Canada) have rectangular and circular cladding profile, respectively, with the cladding side/diameter ~125 µm. The fibers peak absorption near at 980 nm (α_P), the modal field diameter at 1060 nm (w) and the cut-off wavelength (λ_c) are α_P=900, 1073, 245, 35dB m, w=4.5, 3.6, 4.5, 3.6 µm and λ_c=810, 870, 680, 816 nm for Fibers 1–4, respectively. All of them are aluminum silicate fibers with a Gaussian distribution of Yb³⁺-ion concentration in the core.

3. Experimental results and their interpretation

The phase shifts (up to ~4π) shown in Fig. 2 indicate strong RICs induced in Fiber 1 by different pump pulses. Being dependent on the total absorbed energy, the effect exhibits smooth saturation at some steady-state level that also depends on the pulse amplitude (Fig. 2(a)) and on the pulse duration (Fig. 2(b)) in a similar manner. Decaying parts of the phase traces describe the refractive index relaxation after the end of the excitation pumping pulse. They are perfectly fitted by the exponential function φ(t)~exp(-t/τ_sp) with the relaxation constant corresponding to the excited state life-time of Yb-ion τ_sp≈850 µs (nearly the same for all Fibers 1–4). All normalized curves shown in the inserts in Fig.2 seem to be similar, despite they correspond to different pumping conditions. A multi-exponential relaxation behavior that might be attributed to the temperature induced RIC has not been observed in our experiments. The above-presented relaxation behavior indicates to the electronic mechanism of RIC associated with population redistribution between the Yb-ion levels (²F_5/2 and ²F_7/2) possessing different polarizabilities.

 

Fig. 2. Phase shifts at 1550 nm induced in Fiber 1 of 2m length by pulses of different (0.02–2.5 ms) pulse duration (a) and of different (2–145 mW) pulse amplitude (b). Inserts show the same phase shifts normalized to their maximal levels (only decaying parts are presented).

Download Full Size | PDF

In simplified two-level approximation, the electronic RIC δn caused by the excited state population (²F_7/2) change δN ₂ can be expressed as [3,4]:

where n ₀ is the unperturbed refractive index; F_L=(n ₀ ²+2)/3 is the Lorentz factor; Δp=p ₂-p ₁, p ₁ and p ₂ are the polarizabilities of Yb³⁺ ions in the ground (²F_7/2) and excited state (²F_5/2), respectively.

The corresponding phase shift detected in the probe wave at λ_T in the fiber length L can be evaluated from Eq. (1) by integration with ρ_T(r) as a weight function:

where $\delta {\mathbf{N}}_2^{\sum }=2\pi \underset{0}{\overset{L}{\int }}\underset{0}{\overset{\infty }{\int }}{N}_2(z,r)\mathrm{rdrdz}$ is the pump-induced change in the number of the excited Yb³⁺ ions in the whole fiber volume, ρ _T(r) is the normalized power radial distribution of the probe light. Parameter $\overline{\eta }$ ρ_T(0) approximates the efficiency of the probe mode interaction with the population changes δN ₂(r) induced in the fiber core. Here we intentionally separated the factor ρ_T(0)/λ_T that includes the major part of the dependence on λ_T and the correction factor $\overline{\eta }$ accounting for a distributed character of the population changes δN ₂(r) within the doped area core. If all changes are considered to occur near the fiber axis η̄→1 and for uniform Yb-ion distribution on the fiber core η≈0.7. For a Gaussian ion distribution the factor $\overline{\eta }$ ρ_T(0)~ā ^-2 has been quantified from a step-index fiber approximation with the core radius ā=w[1.3+0.864(λ_S/λ_c)^3/2+0.0298(λ_S/λ_c)⁶]^-1 [15,16] evaluated from Fiber 1–4 specifications.

Equation (2) predicts that the phase shift is proportional to the pump-induced change in the whole number of the excited ions. Therefore, the dynamics of both of them is governed by the same rate equation:

where h is the Plank constant, ν_P, ν_S are the average frequencies of the pump and the amplified spontaneous emission (ASE) (or lasing), P_in, P_out are the input and output (residual) powers at ν_P, and P_ASE is the emitted power at ν_S.

In case of total pump absorption and low spontaneous emission, Eqs. (2,3) reduce to the following simple expression that is validated for highly doped Fibers 1–3 below:

Here K=2πF ² _LΔp$\overline{\eta }$ ρ_T(0)λ_P/hcλ_T and P ₀ is the pump pulse amplitude.

 

Fig. 3. Phase shifts at 1550 nm induced by 4ms pulses in 2 m long Fiber 2 as functions of time (a), normalized time τ=1-exp(-t/τ_sp) (b), pulse amplitude (c). (d) shows the slope δφ/δt|_t→0.

Download Full Size | PDF

In highly doped fibers the pump power is completely absorbed by Yb-ions. The experimental traces shown in Fig. 3 follow the key features predicted by Eq. (4) very well. The exponential character of the phase growth to its steady-state (Fig. 3(b)), and the linear dependence on the pump pulse amplitude (Fig. 3(c)) are observed. If the pass gain and ASE level remain low, the slopes δφ/δt|_t→0 can be measured for different pulse amplitudes (Fig. 3(d)), and their linear fit allows one to determine the factor K=0.056π rad ms^-1mW^-1 that proves to be the only fitting parameter in Eqs. (2–4).

To validate the model at higher excitation level, the output powers detected at both fiber ends during the pulse excitation were recorded together with the phase traces. The spectra of the optical emission at both fiber ends highlights ASE as the reason of saturation of the monotonic growth of the phase shift at high pump energies (Fig. 4(a,b)) (ripples are attributed to a specific coupling with the analyzer). Notice that due to absorption saturation and reabsorption processes, the ASE pulse observed in the forward direction is delayed with respect to the backward pulse providing a break in the leading edge of the total power (Fig. 4(c)). Using Eqs. (2,3) we can present the ASE power by the following equation:

which is in a good quantitative agreement with the recorded traces (Fig. 4(c,d)). Note that the reconstructed pulse edge also exhibits a break as the experimentally observed ASE pulse.

 

Fig. 4. ASE from Fiber 2 in backward (B) and forward (F) directions during the excitation pulse: optical spectra (a,b), and detected power (c). Curve (d) presents the reconstructed power.

Download Full Size | PDF

Factor K is the only parameter relating the induced phase shift and the pump power in the fiber samples and at the probe wavelengths used in our experiments. Phase dynamics observed in two different samples of Fiber 3 demonstrates the factor K to be independent on the fiber length (Fig. 5(a)). Reduction in the fiber length just causes reduction of the saturation energy, but does not affect the slope in phase changes. On other hand, the phase shift dependences observed in Fibers 1–4, provided that both the fiber length and other conditions are the same, reveal different slopes and other distinct features (Fig. 5(b)). In particular, Fiber 4 with lowest Yb-ion concentration exhibits lower saturation power than others fibers, but slope in this fiber is nearly the same as that in Fiber 3 and is larger than the slope in Fiber 2 and Fiber 1. Approximation by Eq. (5) gives K ₄≈K ₃≈1.21K ₂≈1.57K ₁≈0.067π rad ms^-1mW^-1 that highlights also the ratios between the parameters K for Fibers 4-1, respectively. These results are in good agreement with Eqs. (2–4) that predict that factor K is to be independent on Yb-ion concentration and to be inversely proportional to the square of the fiber core radius ā. They allow us estimation of the polarizability difference at 1550 nm which proves to be the same for all tested fibers. Evaluation of $\overline{\eta }$ ρ_T(0) for a step-index fiber and Gaussian Yb-ion distribution with diameter equal to ā($\overline{\eta }$≈0.85) gives Δp ₁₅₅₀≈7.5·10^-26 cm³. We expect ~20% error in this value due to uncertainty of the doped area size. Notice that above mentioned PD value is comparable with those reported for Yb-doped laser crystals earlier [4].

 

Fig.5. Phase shift induced by 145-mW pulses: in different lengths of Fiber 3 (a), in 2 m long Fibers 1–4 (b), and in Fiber 2 at different probe wavelengths (c). Figure (d) shows the relative polarizability difference (points) in comparison with the resonance and non-resonance PD contributions at ~1µm (blue), and ~0.4 µm (black), and the dependence ~ρ_T(0)/λ_T (red).

Download Full Size | PDF

To characterize the PD dispersion in spectral range of 1450–1620 nm, the Fiber 2 was also additionally tested at different probe wavelengths λ_T. As it was reported for the two-core fibers earlier [3], significantly different slopes δφ/δτ (Fig. 5(c)) are observed at different λ_T. In our case this difference is explained by the dependence ~ρ_T(0)/λ_T calculated for Fiber 2 (Fig. 5(d)) mainly. The experimentally observed dispersion Δp(λ_T) is to be compared with the Lorentz-line dispersion profiles associated with near-IR and UV transitions of Yb³⁺ ions at ~1µm and ~0.4 µm, respectively. The reconstructed PD profile matches the UV-line wing and has the same value in all investigated spectral range.

4. Conclusion

Summarizing, we have reported observation of a strong RIC at 1460–1620 nm in standard single-mode Yb-doped fibers pumped in the absorption line of Yb³⁺ ions. The RIC exhibits a typical excited population dynamics and is explained in two-level approximation. These results clearly indicate to the electronic mechanism of PD between the excited and unexcited Yb³⁺ ion states and allow us to estimate its value. The PD dispersion curve indicates the dominant nonresonant contribution of the UV transitions.