1. Introduction

During the past two decades, Ytterbium-doped fibers (YFs) have been shown to be an ideal choice for lasing and amplifying in the 1.0 – 1.2-μm spectral region [1–4]. Output powers available from these devices are attainable now from a few Watts to a few kW. Further development in this field is mainly aimed on searching for novel architectures of active YFs (e.g., of the microstructure fiber design), optimizing chemical composition of YF, and also overcoming the problems stemming from un-wanted effects like clustering of Ytterbium ions in heavily-doped YF. The effect of clustering is notable at high concentrations of Yb³⁺ in a fiber and limits implementation of single-frequency Ytterbium fiber lasers. On the other hand, the “primary” stage of clustering of Yb³⁺ ions in YF, the formation of Yb³⁺ - Yb³⁺ ion-pairs (IPs), leads to the cooperative processes, where two adjacent Yb³⁺ ions forming a pair can act–absorb or emit collectively.

The cooperative processes in a solid state doped with Ytterbium–emission and absorption are known from seventies; see, correspondingly, Refs. [5] and [6]. In the first case, two Yb³⁺ ions excited by IR (1-μm) radiation can emit a photon with double energy (in the 0.5-μm region); in the second case an Yb³⁺ - Yb³⁺ IP is allowable to absorb in the visible. Note that despite the cooperative luminescence and absorption have been investigated for a long time in a variety of crystalline, glass, and wave-guiding systems (see, e.g., Refs. [7–15]), some their features stay unclear.

Regarding the cooperative luminescence and absorption in YF, there are a very few works addressing these phenomena (see, respectively, Refs. [11, 12] and [14, 15]). That’s why further treatment of these effects is interesting itself as well as their relation to other phenomena caused by the presence of Yb³⁺ - Yb³⁺ IPs in YF. First of all, we mean here the quenching effect which limits efficiency of a fiber laser through the reduction of the lifetime of the excited ²F_5/2 state of Ytterbium and the presence of effectively “non-working” ions in the active fiber. Thus, considering an Yb³⁺ - Yb³⁺ IP as a simplest type of a cluster, one may draw the ways towards overcoming the clustering problem in heavily-doped YF.

In the present work, we report a study of the two appearances of the presence of Yb3⁺ - Yb³⁺ IPs in a set of YFs with quite different concentrations of Yb³⁺ ions:

2. Experimental

2.1. Cooperative luminescence and absorption in Ytterbium-doped fibers

Experiments were conducted with Ytterbium-doped silica fibers (YFs) fabricated using the solution-doping technique in the Fiber Optics Research Center of A. M. Prokhorov General Physics Institute (RAS, Moscow, Russia) [16]. We had in our disposal YFs ## 1–5 with different Yb³⁺ concentrations (see Table 1, where the main fiber parameters are given); all the fiber samples have been additionally co-doped with Al³⁺ for homogenizing the spatial distribution of the Yb³⁺ main dope. A set of photos of the fibers ## 1, 3, and 5, which represent the cases of the lowest, medium, and highest concentration of Yb³⁺ ions, is shown in Fig. 1. These images have been obtained at the white-light illumination. Note that the core and wave-guiding radii of the YFs with different Yb³⁺ contain were essentially different, but the outer fiber diameter was practically the same, 125 ± 2 μm The fiber geometry was chosen to provide single-mode generation / amplifying in the 1-μm spectral range.

 

Fig. 1. Images of YFs ## 1 (a), 3 (b), and 5 (c) obtained at white-light illumination.

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Table 1. Parameters of Ytterbium-doped fibers used in experiments.

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All the inspected YF samples demonstrated, under the 980-nm diode-laser pumping, together with a strong IR luminescence centered at λ ≈ 1 μm, also a pronounceable visible luminescence near λ ≈ 0.5 μm (Fig. 2).

 

Fig. 2. Visible cooperative luminescence from YF # 3 pumped at λ = 980 nm.

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The correspondent luminescence spectra of YFs ## 1, 3, and 5 in the spectral range 0.4–1.4 μm are shown in Fig. 3. Notice that the presented spectra have been obtained in the lateral geometry, where the emission signal is detected from the lateral surface of the end-pumped fibers, near its splice with a pig-tail of the diode laser (see Fig. 2). This geometry differs from the previously reported ones [11, 12, 15], where the frontal IR and visible emissions from YFs were registered and therefore a significant influence of the re-absorption effects was the case.

 

Fig. 3. Luminescence spectra of YFs ## 1, 3, and 5 under the diode laser pumping (λ = 980 nm, pump power P_in = 200 mW).

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The examples of the dependences of IR and visible emission powers versus 980-nm pump power measured in these two geometries for low-power excitation are demonstrated in Fig. 4. It is seen that for the lateral geometry [see Fig. 4(a)] the correspondent slopes of the curves are 2.24 for visible emission and 1.07 for IR emission, giving their ratio of about 2. Thus, one can conclude on the quadratic dependence of visible emission on the pump power that allows one to affirm this emission as the cooperative process [5]. Meanwhile, in the case of the frontal observation geometry [see Fig. 4(b)] the slopes of the curves for visible and IR emissions are nearby the same, of about 1. An effective decrease of the slope for cooperative emission can be explained by the saturation and re-absorption effects [17] which are pronounceable at propagation along the YF length.

Notice here that all the YFs under study were almost free from contaminating by other unwanted rare-earth dopes (i.e., Holmium, Erbium, Thulium, etc.), since the fibers have been fabricated from the specially purified materials. It is seen from Fig. 3, where no steps of these un-wanted dopes are seen. (In principle, there might be a unique alternative to affirm the visible line near 0.5 μm in Fig. 3 to the emission / absorption peaks of Tm³⁺ (the ¹G₄↔³H₆ transitions), but it is well-known (see e.g. Ref [18]) that in silica fiber this dope should also have the much more pronounceable emission / absorption lines near 680, 790, 1200, 1470, and 1650 nm; it is evident that none of these lines presents in the spectra of our YFs.).

Therefore, the green emission near 0.5 μm is to be solely attributed to the cooperative emission effect, which stems from the presence in YF of Yb³⁺ - Yb³⁺ IPs [11, 12].

 

Fig. 4. Dependences of IR (black curves) and visible cooperative (green curves) luminescence power versus IR (λ = 980 nm) pump power in two measurement geometries: lateral (a) and frontal (b). Slopes of the curves characterize the value of emission quanta per quanta of pumping radiation.

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The absorption spectra of YFs ## 1, 3, and 5 in the IR (a) and visible (b) spectral regions are shown in Fig. 5. Whilst the intensive IR absorption band centered at λ = 980 nm [Fig. 5(a)] in YFs is the appearance of the fiber doping with Yb³⁺ ions, the presence of the absorption peak at 0.47–0.48 μm [Fig. 5(b)] may be explained by the cooperative absorption effect, which has been recently observed in heavily-doped Ytterbium-Holmium [14] and purely-Ytterbium [15] silica fibers as well.

 

Fig. 5. Absorption spectra of YFs ## 1, 3, and 5 in IR (a) and visible (b) spectral ranges.

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In order to show that the visible emission / absorption in our fibers (see Figs. 3 and 5) is explained by the pairs’ effect, i.e. they have the cooperative character, we have performed a modeling of the visible emission / absorption spectra, applying the procedure of self-convolution of the correspondent IR spectra (see, e.g., Refs. [8–14]):

where E is the energy (in cm^-1) of the emission or absorption quanta and I_VIS , I_IR are the correspondent intensities of the spectral components within the IR (fundamental, single Yb³⁺ ions) and visible (cooperative, Yb³⁺ - Yb³⁺ IPs) spectral bands. We have modeled the visible spectra for all the YFs under scope, starting from the experimental IR spectra, and compared the calculation results of the self-convolution integral (1) with the corresponding experimental visible spectra. The example of these results for YF #3 is shown in Fig. 6.

 

Fig. 6. Absorption (a) and emission (b) spectra in the visible (lower scale) and IR (upper scale): Curve 1 (black) - experimental IR spectra; Curve 2 (red) - self-convoluted IR spectra; and Curve 3 (blue) - experimental visible spectra. All the data are for YF #3.

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It is seen from Fig. 6 that a good agreement is observed in the spectral positions of the main components in the experimentally registered visible emission/absorption spectra and the modeled self-convoluted IR ones. However, the intensities of the components within the bands differ. Meanwhile, as it is shown in the earlier reports (see e.g. Refs. [6, 9, 10]), the maxima of the absorption/emission spectra in the visible due to the pairs’ effect may not be exactly at the double energy of the correspondent IR peaks. The re-distribution of “powers” of the absorption/emission components within the visible band can be in particular connected with the lattice local stresses and deformations. In our case, where mostly heavily-doped YFs are treated, the latter is the case. On the other hand, the self-convolution integral (1) is a formal mathematical procedure that is the multiplication of the IR spectrum of single Yb³⁺ ions with itself, which doesn’t take into account the processes mentioned above.

Thus, the whole data set obtained (see Figs. 2–6) allows concluding on the presence of Yb³⁺ - Yb³⁺ IPs in our fibers as to be responsible for the strong cooperative absorption and emission processes.

Consider at the last an impact of Ytterbium concentration upon the cooperative processes in YFs. The IR absorption coefficients of single Yb³⁺ ions (in the 1-μm region) in our YFs are listed in Table 1. Plotting the values of the absorption coefficient in the visible (λ ≈ 0.5 μm) versus the one in the IR (λ ≈ 1 μm) for all available YFs, we obtain the dependence shown in Fig. 7(a). This dependence has a practically linear law that may be an indication of a growth of the cooperative absorption in YFs (or Yb³⁺ - Yb³⁺ IPs’ concentration) with an increase of Yb³⁺ single ions’ concentration. Similar features are observed for luminescence [see Fig. 7(b)]. Plotting the ratio of total powers of cooperative visible (λ ≈ 0.5 μm) and IR fundamental (λ ≈ 1 μm) emissions versus IR absorption coefficient, one can see an incremental character of this dependence. Note that both the emissions in Fig. 7(b) were counted at the 980-nm pump power of 100 mW, well above the saturation level, 5-10 mW.

 

Fig. 7. Dependences of cooperative absorption coefficient in visible (λ = 475 nm) (a) and ratio of cooperative visible to fundamental IR luminescence powers (b) on IR absorption coefficient (λ = 976 nm) in YFs ## 1 – 5.

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2.2. Impact of Ytterbium ion pairs’ presence in Ytterbium-doped fiber on its nonlinear transmission coefficient at 980-nm excitation

As it is seen from the above experiments, in YFs both the cooperative absorption and emission effects are quite notable. The presented results as well as the previous ones addressing these effects in YFs [11, 12, 14, 15] force one to take into account the role of Yb³⁺ -Yb³⁺ IPs at using of the fibers for lasing and amplifying. The presence of IPs in YFs inevitably results in an interaction between the sub-systems of single Yb³⁺ ions and Yb³⁺ -Yb³⁺ IPs. When an active YF is pumped by IR radiation, excitation in the sub-system of single Yb³⁺ ions will be removed not only via 1-μm (wanted) emission, but also via relaxation within Yb³⁺-Yb³⁺ IPs. This results in appearance of additional losses in YF and an effective lifetime decrease of excited single Yb³⁺ ions.

Evidently, the last features have to be reflected on the transmission characteristics of YF pumped by IR radiation, which falls into the fundamental (0.9–1.0-μm) spectral band of single Yb³⁺ ions. The correspondent experiment on measuring the YF nonlinear transmission coefficient was performed in the arrangement where a piece of YF is pumped by a fibered cw diode laser (wavelength, λ = 980 nm; maximal output power $P_{\mathit{\text{in}}}^{\text{max}}$ =200 mW). The tested piece of YF was spliced with the pumping laser output, and the transmission coefficient of the former was calculated as T = P_out /P_in , where P_in and P_out are the pump powers on the input and output of the fiber. A set of dependences of the transmission coefficient versus launched pump power T(P_in ) was obtained for different YF lengths L. We fulfilled these measurements for all available YFs ## 1–5 (see Table 1).

The experimentally measured dependences T(P_in ,L) are shown in left column of Fig. 8 (see respectively plots (a) (YF #1), (b) (YF #3), and (c) (YF #5)). It is seen that the dependences T(P_in ) in all the cases demonstrate the “bleaching” character that is the resonant absorption saturation effect characteristics. On the other hand, the dependences T(L) (compare the curves on each plot (a), (b), and (c)) have also a predictable behavior, being caused by “interplay” between the resonant absorption and gain and depleting of pump power propagating along the fiber length.

 

Fig. 8. Experimental (left) and theoretical (right) dependences of transmission coefficient T of YFs ## 1 (a), 3 (b), and 5 (c) on launched pump power P_in at different lengths L of the fiber samples.

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Another interesting feature seen from Fig. 8 is that the full-saturated transmission coefficient of YFs is considerably less than 100% that is an indication of the presence of strong nonlinear losses in the fibers.

Our attempts to fit the experimental dependences shown on the left-hand of Fig. 8 by the theoretical ones, accounting for the saturation effect solely in the sub-system of single Yb³⁺ ions, were failed. So, we have arrived at the conclusion that an account of the Yb³⁺ - Yb³⁺ IPs’ contribution in the fiber transmission nonlinearity is necessary for matching the modeling results with the experiment (as it has been done for heavily-doped Erbium fiber, Ref. [19]).

3. Theoretical model and discussion

3.1. Modeling of the transmission coefficient of Ytterbium-doped fiber

We have derived a system of equations that addresses the propagation of a pump wave in YF at the wavelength λ_p falling into the resonant (²F_7/2 → ²F_5/2) absorption band of Ytterbium; in the circumstances of the above experiments, λ_p = 980 nm.

We consider the scheme of energy levels of Ytterbium to have the simplified structure shown in Fig. 9, where the absorption (σ₁₂ ) and stimulated emission (σ₂₁ ) cross-sections relate to the transitions between the ground state “1” (²F_7/2) and excited state “2” (²F_5/2) of Yb³⁺. We also take that equally with the induced transitions (²F_7/2 → ²F_5/2) and (²F_5/2 → ²F_7/2) a single excited ion of Yb³⁺ can relax from the state “2” to the state “1” spontaneously with the decay time τ₀ , but its relaxation within the Stark components of the states “1” and “2” is instantaneous. Under the action of pump radiation, the transmission coefficient T of YF at the pump wavelength takes the nonlinear character due to the absorption saturation effect in the generalized two-level scheme of single Yb³⁺ ions. However, supposing existence of Yb³⁺ -Yb³⁺ IPs in YF, we need to consider as well the following processes:

 

Fig. 9. Scheme of energy levels and main processes involved at excitation of YF with λ ≈ 1-μm pump light.

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At these assumptions, the balance equations for the pump-wave power P (in s^-1) and dimensionless (normalized on the total concentration N₀ of single Yb³⁺ ions in the fiber core) population n ₂ (0 ≤ n ₂ ≤ 1) of the metastable state “2” are as follows:

where α₀ = N₀ σ₁₂ is the un-saturated (small-signal) absorption coefficient of YF at the pump wavelength; γ₀ is the linear loss coefficient; ξ= (σ₁₂ + σ₂₁ )/σ₁₂ is the coefficient which stands for the ratio between the absorption and stimulated emission cross-sections of single Yb³⁺ ions; τ_p is the effective pairs’ relaxation parameter (it addresses the cooperative decay of excitation of two excited adjacent Yb³⁺ ions composing a pair); and Γ = 1-exp(-S_r /S_w ) ≈ 1 -exp[-2(r/w ₀)²] is the overlap factor for the pump wave and fiber doped core, where S_w = ${\pi w}_0^2$/2 and S_r = πr² are, respectively, the geometrical cross-sections of the beam and core and w₀ and r are their radii.

As seen from Eqs. (2) and (3), the effect of Ytterbium IPs is accounted for in our model by the coefficient τ_p = β/N₀ . The coefficient τ_p [s] itself is dependent of concentration N₀ of single Yb³⁺ ions in YF, whilst the coefficient β[cm³s] is the “microscopic” parameter characterizing the Ytterbium IP binding strength. The dependence of the last term in Eq. (3) on $n_2^2$ stems from the accepted model of Yb³⁺ - Yb³⁺ IPs, where undertaken is the mechanism of down-conversion of excitation of two adjacent excited Yb³⁺ ions composing an Yb³⁺ - Yb³⁺ IP. When both ions in the pair are excited (being in the state “2”), they are allowable, according to the cooperative emission postulate, to emit cooperatively a photon with the double energy, or relax via phonons’ generation. From here, the quadratic dependence of the relaxation time of the pair composed of two neighbor excited Yb³⁺ ions stems.

Table 2. Parameters taken at modeling.

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The numerical calculations were performed for various pump (λ_p = 980 nm) powers P_in and various values of YF length L. All the parameters, apart from the pairs’ coefficient β, are given by the experimental arrangement or taken from the literature, see Refs. [1, 2, 20–22]. The wave-guiding radii w₀ of the fibers were measured to be from 3.2 (#1) to 7.1 (#5) μm. For all the YFs, almost the same normalized frequency value, V ≈ 1.18 has been calculated; this leads to the values of the fibers’ core radii, lying in the interval r = 1.1÷2.4 μm (see Table 1). Concerning the value of the parameter β (or τ_p at given N₀ ), characterizing the presence of Yb³⁺-Yb³⁺ IPs in YF, it was a single free parameter in numerical calculations.

3.2. Comparison of theory VS experiment and discussion

The modeling results for the transmission coefficient T(P_in ,L) of YFs ## 1, 3, and 5 at 980-nm pumping are shown in the right-handed column of Figs. 8(a), 8(b), and 8(c), which are the theoretical counter-parts for the experimental data (see the left-handed graphs (a), (b), and (c) in Fig. 8). The values of all the parameters used in the routine are listed in Tables 1 and 2; the values of YF doped core radii were taken, respectively, as r = 1.0 (YF #1), 1.7 (YF #3), and 2.2 (YF #5) μm, leading to virtually the same pump beam / fiber core overlap factor Γ= 0.2 ± 0.02.

A comparison of the experimental and theoretical data allows concluding on a good agreement between them. Notice that such a coincidence is obtained for all the inspected YF samples with considerably different Ytterbium concentrations N0, if one takes into account the effect of IPs in the fibers. Otherwise-if this effect is not incorporated in the modeling, i.e., the last term in Eq. (3) is omitted-significant deviations arise between the theory and experiment. In the last case, one cannot get in modeling neither the saturated value of the transmission coefficient T notably less than 100% (as it is observed experimentally at high pump powers), nor obtain the particular shapes of the curves T(P_in ,L) for ever fiber.

Another important result of the modeling is that the best fitting of the experimental data by the theory has allowed us to get the values of the pairs’ parameters τ_p and β for YFs with different Yb³⁺ concentrations. Whilst the decay time τ_p provided by the presence of Yb³⁺ -Yb³⁺ pairs in YF weakly depends on overall Ytterbium concentration in the fibers: τ_p ~ 0.1 -0.2 ms [see Fig. 10(b)], the value of the “microscopic” parameter β, which characterizes the binding strength of ions in a pair, depends practically linearly on the absorption coefficient α₀ (or concentration N₀ of single Yb³⁺ ions) [Fig. 10(a)]. The latter observation is not surprising, because, according to the above definition, the parameter β is the multiplication of the pairs’ relaxation time τ_p and Yb³⁺ concentration N₀ .

 

Fig. 10. Dependences of pairs’ coefficients β (a) and τ_p (b) on IR absorption coefficient (λ = 976 nm) in YFs ## 1 – 5.

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In order to check the values of the pairs’ parameters τ_p and β = τ_pN₀ obtained from the modeling, we have performed an independent experimental study of the fundamental λ ≈ 1-μm emission decay of Yb³⁺ in YFs under the 980-nm pumping. Experimentally, we registered the 1-μm lateral luminescence decay immediately after switching-off the pump light. An example of the measured dependences is shown in Fig. 11. Our attempts to fit the resultant signal (open circles) by a single exponent were failed (see red curve 1 in Fig. 11). On the other hand, this signal is perfectly fitted by two exponents (see green curve 2 in Fig. 11) with the characteristic decay times τ⁽¹⁾ ≈ 0.75 ms and τ⁽²⁾ ≈ 0.17 ms. The first time τ⁽¹⁾ is evidently the lifetime τ₀ of single Yb³⁺ ions in the excited ²F_5/2 state (this value coincides finely with the previously reported data for YFs of this type [16]), while the second time τ⁽²⁾ should be attributed to the process of excitation relaxation in Yb³⁺ - Yb³⁺ IPs. Notice that the last value, τ⁽²⁾ = 0.17 ms, whole agrees with the value of the pairs’ parameter τ_p = 0.15 ms obtained from the above modeling of the YFs’ nonlinear transmission coefficient T. It is also noticeable that the amplitudes of the two exponents, fitting the experimental decay curves, are comparable. So, the effect of pairs upon the Ytterbium population dynamics is significant and should be accounted for at implementing fiber lasers and amplifiers using heavily-doped YFs. In our opinion, the last experiment serves an additional confirmation for both the undertaken hypotheses of the role of the pairs’ effect in YFs and applicability of the developed theory.

 

Fig. 11. Fundamental 1-μm luminescence decay in YF # 3 after pump switching-off: Open circles – experimental data; curves 1 (red line) and 2 (green line) – respectively, its fits by one and two exponents.

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Notice at the last that our measurements of the decay time τ_coop of visible cooperative emission at the wavelength λ ≈ 0.5 μm in our YFs led to the following estimate of the visible signal decay, τ_coop ~ 0.2-0.3 ms. From the other side, the pairs’ parameter τ_p actually describes an overall relaxation time of two excited adjacent Yb³⁺ ions composing a pair, which involves the two collective processes–the cooperative visible emission of a photon with double energy (respectively to the fundamental energy quanta) and multi-phonon relaxation. So, the process of excitation relaxation of Yb³⁺-Yb³⁺ IPs via phonons (if this process exists indeed) should be much shorter than τ_coop .

4. Conclusions

We have reported the experimental and theoretical studies of the processes occurred in Ytterbium-doped alumino-silicate fibers with various concentrations of Yb³⁺ ions at 980-nm laser-diode pumping. The main attention has been paid to the features regarding the presence of Yb³⁺-Yb³⁺ ion-pairs in the fibers heavily-doped with Ytterbium-to the effects of visible cooperative luminescence and absorption. We have addressed in details the nonlinear transmission coefficient T of YFs at 980-nm pump, obtaining its dependences on Yb³⁺ concentration N₀ , pump power P_in , and fiber length L. An account of the cooperative luminescence and absorption effects as indicative for the presence of Yb³⁺-Yb³⁺ ion pairs in the fibers has allowed us to perform a modeling of the experimental data, which incorporates the pairs’ effect. This modeling has shown a significant role of Yb³⁺-Yb³⁺ IPs, especially in the fibers with high Yb³⁺ concentration, in effective shortening of Yb³⁺ excitation decay and, as a consequence, in notable change in the behavior of the nonlinear transmission coefficient T(P_in ,L). The modeling has also allowed us to find the coefficients τ_p and β, addressing the pairs’ effect in YFs, and thereafter to reach a good agreement between the theory and the experiment.