1. Introduction

Due to their outstanding performance in the field of telecommunication, the optical fiber technologies are being customized to be applied in medicine, nuclear facilities, space missions [1–3], etc. In these new areas, optical fibers are exposed to ionizing radiations, which impacts their optical parameters. In this context, we want to focus mainly on two demanding applications: space telecommunication and ionizing radiations dosimetry. To fulfill the requirements of these new applications, it is essential to know and to control the fiber sensitivity versus the amount of radiations. Accurate models are thus required. The purpose of this paper is to target the most widely spread Erbium-doped optical fibers : Er$\require{mhchem}^{3+}$ doped $\ce {SiO_2}$:$\ce {GeO_2}$/$\ce {Al2O_3}$ fibers. Indeed, this type of fiber is known to be the most sensitive to radiations [3].

At a microscopic level, it is well known that ionizing radiations induce a significant degradation in glasses, main constituent of optical fibers. This is due to chemical species changes through physical processes that in most cases involve a change in the number of charges in ions, atoms or molecules [4,5]. These modifications lead to a deep alteration of the absorption spectrum, be it in the UV-VIS wavelength range or in the near infrared, especially at the optical telecommunication wavelengths [6,7]. This alteration is called "RIA" for "radiation-induced absorption". The spectral distribution as well as the magnitude of this RIA are strongly influenced by the kind and concentration of the doping elements added in the glass matrix, whether they are used to define the waveguide structure or to obtain specific functionalities, like optical amplification.

2. Physical chemistry of glass under irradiation

Figure 1 illustrates the most common processes that happen when a basic $\ce {SiO_2}$ glass is irradiated. The additional processes that occur in the case of the more complex aluminum- and erbium-doped glass will be described in § 3.

 

Fig. 1. Different $\ce {SiO_2}$ network configurations: a) Ideal Quartz. b) Glass with defects. c) Changes induced by irradiations in the glass. d) Energy diagram for typical exchange. Whereas $\ce {Si}$ is tetravalent, only 3 bonds are displayed in this 2D representation.

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Compared to the canonical (crystalline) case of the quartz that contains only regular $\ce {#Si-O-Si#}$ bonds (Fig. 1(a)), a glass contains configurational disorder (Fig. 1(b)), i.e. dangling bonds (case of $\ce {#Si-O^{.}}$) or unexpected bonds such as $\ce {#Si-Si#}$. All are called "defects", and mainly come from mechanical and thermal stress during the manufacturing process. The most common unexpected bond is a lack of oxygen, like in $\ce {#Si-Si#}$. It is called Oxygen Deficient Center (ODC) [7], and in most cases finds its origin in the stoichiometry of the gazes used during the synthesis.

The starting material thus contains a variety of chemical species : regular ($\ce {#Si-O-Si#}$), unexpected ($\ce {#Si-Si#}$) and dangling ($\ce {#Si-O^{.}}$) bonds. During the irradiation process, new kinds of defects are created from each of the chemical species present in the initial material, and this will occur through three distinct processes:

This transformation of initial chemical species into others can be summarized by an energy diagram (as in Fig. 1(d)), in which the pairs generation and the trapping processes are identified by their energy levels and their populations. The phenomenological approach allowed by these energy diagrams is well known and widely used for the qualitative modeling of the glass degradation under irradiation, for example in the works by Chen et al. [8,9]. It leads to models that describe the time evolution of excited free carriers and chemical species populations that contribute to RIA through trapping phenomena, by means of balance equations.

The modeling of more elaborate or realistic glasses, containing $\ce {Cl}$, $\ce {Ge}$, $\ce {Al}$ and $\ce {Er}$ can be performed using the same tool, but is more complex because of the resulting diversity of defects after irradiation. At this time, the state of the art is the model developed by Mady et al. [10]. It describes correctly the mechanisms of degradation in Al-Er fibers, but has proven its efficiency only for heavily doped Erbium fibers. In fact, Mady model identifies Erbium-ion induced-traps fill up as the only mechanism able to relax HEES population, and then to balance electron generation. In these conditions, the accounting for the situations where Er traps are absent or in a negligible quantity (that is these of white or weakly Erbium doped fiber) leads to divergent simulations. In this work, we provide a more general and more accurate model, that cannot lead to diverging simulations and exhibits short computation times. For that purpose, we first describe the defects and physical phenomena that are involved in the model.

3. Involved defects in SiO₂:AlO₂/GeO₂ optical fibers

As illustrated in Fig. 1, from the Silica matrix itself, irradiations commonly induce two kinds of defects:

Finally, the $\ce {Er^{3+}}$ behaves like an electron trapping precursor, through the following reaction [19]:

4. Towards quantitative modeling

In addition to the selection of the most important traps, we identified that the setting of an accurate model for Er$^{3+}$ doped $\ce {SiO_2}$:$\ce {Al_2O_3}$/$\ce {GeO_2}$ fibers requires to include a charge competition mechanism between Erbium and Aluminum. This effect is suggested in the scientific literature [20–23] and confirmed by transmittance experiments we performed in the 400nm–1$\mu$m wavelength range on various Er-doped optical fibers with the same Aluminum amount, as illustrated in Fig. 2.

As a matter of fact, each spectrum obtained after irradiation can be separated in two regions:

 

Fig. 2. Absorption spectra before and after a 300 Gy irradiation for various Erbium concentrations. These fibers contain a similar mass concentration of Aluminum, estimated to be in the range 6–8$\%wt$ of Al.

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For these three curves, the Erbium concentration increase is identified as expected by the 980 nm absorption peak augmentation. However, the absorption plateau in the 400 nm–800 nm range decreases with the Erbium concentration. Since doping elements other than Erbium and Aluminum exhibit very low concentration, we assume that this experiment hilights the Er/Al competition mechanism, at a macroscopic level. The RIA evolution versus Er concentration (at constant Al concentration) in the 400 nm–800 nm range is interpreted as a screening of Erbium by Aluminum in the frame of the competition mechanism controlled by Al/Er relative concentrations.

From these data, the RIA is extracted at the conventional 980 nm pump wavelength: it increases versus Erbium concentration. The trivalent rare earth $\ce {Er^{3+}}$ ion accepts electrons and is reduced to $\ce {Er^{2+}}$ (Eq. (6)), which absorption is nearly uniform in the UV-Visible range [24,25]. This first analysis can be approached by considering that the Erbium doping element, through its change of chemical conformation due to radiations, may interfere with the spectral signature of the Aluminum, one of the main co-doping elements in most Erbium-doped fibers. This is the most fundamental added value of the model we present in this paper. This consequently leads, as demonstrated in the following, to high accuracy in the prediction of the RIA, whatever the absolute and relative concentration of Er and Al in a wide range (0–10$\%wt$ for Al, 0–1500$ppm$ for Er). This was not available before in the scientific literature, even with the most advanced models.

5. The theoretical model

5.1 Implementation

The defects described in §3 allows to express the trap population kinetics through the energy diagram represented in Fig. 3.

 

Fig. 3. Energy level scheme of the model describing the degradation of a Silica glass doped with Germanium, Aluminum and/or Erbium.

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In this model, all the pair generation processes are summed up by the rate $g$, that feeds both HEES and HEHS. Each trapping process described in §3 is identified by its corresponding energy level. For a defect associated with atom $X$, the probability of the reaction is identified by coefficients called $A_{X}$ for electron trapping processes and $B_{X}$ for hole trapping processes. The amount of precursors of the reaction is called $N_{X}$ for electron trapping (resp. $P_{X}$ for hole trapping), and the amount of defects already created is called $n_{X}$ (resp. $p_{X}$ for hole trapping). Obviously: $n_{X} < N_{X}$ and $p_{X} < P_{X}$. A specific case is the one of $\ce {Si}$: the presence of $\ce {Cl}$ is considered here to be part of the fabrication process of the Silica matrix and is thus associated to the Silica itself. That’s why the hole trap corresponding to $\ce {Cl}$ is identified by $B_{Si}$, $P_{Si}$ and $p_{Si}$.

The screening effect of Erbium and Aluminum traps evidenced in Fig. 2 can be considered as charge losses of the Al traps due to the presence of filled Er ones. We model then it, in a first approximation, through recombination processes which tend to empty the considered traps. For this purpose, we introduce in the equations of $n_{Er}$ and $p_{Al}$ of system (8) a transition of rate $C$. Note that we do not introduce such a transition between $n_{Ge}$ and $n_{Er}$ since we do not observe curve crossing around $900\ nm$ on similar absorption spectra resulting from experiments considering fibers with high concentrations of Ge, weak Al concentrations, and different Er concentrations.

Moreover, it is worth noting that, by introducing trap levels associated to pure Silica material, this modeling can also predict asymptotically the behavior under radiations of passive fibers, with or without co-doping elements, such as Ge or Al, which was not the case for the formerly proposed theoretical models. It is thus able to model the RIA for any proportion for any of the doping elements considered here, in a wide range (0–25$\%wt$ for Ge, 0–10$\%wt$ for Al and 0–1500$ppm$ for Er). For example, for stronger Aluminum concentrations, the model should be expanded to take into account that changes of Al coordination leading to the formation of $\ce {AlO_5}$ and $\ce {AlO_6}$ species [26] will strongly affects the variety and amount of Al related defects.

From the energy diagram (Fig. 3), we can now write the time evolution equations governing the different populations involved in these specialty fibers.

For the defect traps populations:

5.2 Coefficients

This model uses a microscopic phenomenologic approach and thus depends on many coefficients. However a robust prediction tool should not use too many fitting coefficients. Moreover, these coefficients should be accurately evaluated.

The generation rate $g$ is evaluated by considering the fiber as a bulk $\ce {SiO_2}$ glass (other atoms are only for doping), by considering the ratio between the energy brought by the radiation quantum to the glass and the ionization energy of the glass:

The transition rates $A_X$ and $B_X$ are difficult to estimate accurately and taken from [10,23] to be $\approx 10^{-14}m^{3}/s$.

For the precursors coming from doping elements, the associated densities, respectively $N_{X}$ and $P_{X}$ are estimated by applying a proportional coefficient respectively $\mathcal {A}^{-}$ or $\mathcal {A}^{+}$ to the concentration of doping elements itself:

The value of the $C$ parameter, which expresses the inter-band charge-competition process, is discussed in section 6.

The traps cross sections $\sigma _{X,\lambda }$ are estimated from the literature [4,6] for $\ce {Si}$ and $\ce {Ge}$, and measured from various optical fibers available in the frame of this work for $\ce {Er}$ and $\ce {Al}$.

Table 1 lists all the parameters with their description and value.

Table 1. Degradation model parameters.

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5.3 Optimisation of the computation time

In first intention, we solved the equation system (8)-(10) considering the data reported in Table 1 and short time steps. The obtained population behavior is reported in Fig. 4. We observe that high-energy states and traps have different characteristic times. On one hand, HEES and HEHS densities $n$ and $p$ reach a stationary state in very short times, which are of the order of the 100 $\mu$s, while, on the other hand, trapped carriers reach their stationary states in very long times, which are seconds, hours or years. As a consequence, both populations can not be correctly investigated within the same simulation scale and a two-step approach must be considered. For this reason, we consider that high-energy state densities reach their stationary values from the beginning of the simulation, so that they evolve independently of the trap populations. The quantities $n$ and $p$ are then determined analytically. As concerns trapped carrier densities, we analyse their evolution through a dedicated system numerically solved by using sufficiently long time steps.

 

Fig. 4. Simulated trap (indexed quantities) and high energy states populations ($n$ for electrons and $p$ for holes) as a function of time. In the time interval of this example, $n_{Ge}$ is close to zero.

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Considering stationary regime is reached in the high energy states allows to write $d/dt=0$ for both Eqs. (9) and (10). Moreover, since $n$ and $p$ reach their steady state value while traps are still empty, it is relevant to neglect the traps populations compared to the precursors ones ($n_{X} \ll N_{X}$ and $p_{X} \ll P_{X}$). Equations (9) and (10) are then rewritten as:

6. Results and validation

We performed RIA experiments and simulations on a large amount of germano-alumino-silicate optical fiber samples (see Table 2), with various doping and co-doping element concentrations.

Table 2. Composition of the fibers under test.

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Using our automated RIA measurement setup described in Ref. [17], we extracted the normalized RIA at 980 nm and at 1550 nm, for a zero up to 300 Gy dose deposit and a 0.4 Gy/h dose-rate; under these conditions, the RIA exhibits a linear evolution versus the radiation dose (inset Fig. 5). Both experimental and simulation results at $200$ to $300$ Gy are reported in Fig. 5.

 

Fig. 5. Normalized RIA for the fibers under test at the highest available total dose (200 to 300Gy) and a dose rate of 0.4 Gy/h at 980 nm (a) and 1550 nm (b). Data at 1550 nm are not available for some fibers due to experimental set-up limits. Inset in (a) : Simulated (line) and measured (symbols) evolutions of the RIA at 980 nm, as a function of the received dose, for an erbium doped germano-alumino-silicate fiber (#3).

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We first want to comment about the relation between the RIA and the Aluminum concentration on these experimental data. Previous studies performed on fibers without $\ce {Er^{3+}}$ showed that increasing the concentration of $\ce {Al2O3}$ slightly increases the RIA [32]: this is the argument that motivated, as an initial approach, the optimization of the Aluminum concentration versus the optimal length in previous works [16–18].

However, the literature also mentions that other chemical species can interact with $\ce {Al2O3}$ and influence significantly the RIA [33]. More specifically, this is the case with Erbium-doped optical fibers, in which the interaction of $\ce {Er^{3+}}$ with $\ce {Al2O3}$ leads to a reduction of the RIA if a correct stoichiometry is applied [34]. The experimental results presented here illustrate that last points: for example, measurements on fibers #10 to #14, at constant Aluminum concentration but with strongly changing Erbium doping indicate significant RIA variations. This specific behavior cannot be explained by the simple coupling induced by the HEHS level population dynamics: that what limits the previous models of the literature and that justifies, at a macroscopic level, the introduction of the $C$ parameter in our model.

To reach a quantitative approach, we used the model to perform simulations and compared results to our experiments. Among the 18 parameters of the model (Table 1), as we have the knowledge of the glass composition, the only unknown parameter is $C$, which expresses the inter-band charge-competition process. Whereas it may a priori depend on Aluminum and Erbium concentrations, we found that a single value of $C=1.7\cdot 10^{-20}$ cm$^{3}$/s can be used for all the concentrations tested in this work, that change a lot from a fiber sample to another.

We observe a relevant agreement on the whole range of these samples, with a median relative error below 4 % on the prediction of the RIA, which validates our model and the associated parameters for low doses (below 1kGy). We want to notice that this great agreement has been obtained by adjusting only a single value, the value of $C$. Indeed, it has been chosen constant across all the optical fibers under test (in spite of huge variations in the glass composition), as well as across various experimental conditions : spectral region (Fig. 5(a), Fig. 5(b) and 6(a)) and dose-rate variations (Fig. 5 is at 0.4 Gy/h and Fig. 6(a) at 1.2 Gy/h) ; all the other parameters were indeed estimated from experiments or from the literature. Setting the single value of the $C$ parameter is straightforward and thus leads to a robust predictive model.

 

Fig. 6. a) Experiments and simulations performed at 1.2 Gy/h, for 980 nm and 1550 nm. b) Study of the simulated $\ce {Er^{2+}}$ concentration versus the [$\ce {Al}$]/[$\ce {Er^{3+}}$] ratio at 0.4 Gy/h after 300 Gy irradiation.

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Finally, a good picture of the interplay between the main doping and co-doping atoms within the $C$ coefficient is obtained by studying the relative [$\ce {Er^{2+}}$] as a function of the [$\ce {Al}$]/[$\ce {Er^{3+}}$] ratio (Fig. 6(b)). We first want to notice that the $\ce {Er^{2+}}$ is always weak (below 5 % for all fibers at 300 Gy), but has a strong impact on the RIA due to its huge cross-section at both 980 nm and 1550 nm. The global tendencies observed in Fig. 6(b) can be understood as follows.

The Erbium recombination center forms $\ce {Er^{2+}}$ due to the electron population available on HEES. These can go back to $\ce {Er^{3+}}$ by recombination with holes, but in most practical cases this amount is small. This is due to an imbalance between the electrons and holes available at the recombination center. This imbalance can find two main causes :

7. Conclusion

We have developed a set of equations that models the degradation of Er-Al doped Silica fibers under radiations. The model takes into account the degradation kinetics of the Silica matrix under radiations as a function of the dose deposit, enhanced by the Erbium chemical conformation modification and including the main trap levels relevant to other elements ($\ce {SiO_2}$, $\ce {Ge}$ and $\ce {Al}$). Some parameters are provided by the literature and others come from various spectroscopic analyses. The $C$ parameter, which stands for the transition rate between Al and traps and is specific to this model, has been adjusted to a unique value that satisfies all the experiments performed in this work. The linear evolution of the RIA versus dose defines the validity area of this model, thus it is fully appropriate to the context of space missions.

In spite of widely changing doping and co-doping elements concentration ( in the range (0–25$\%wt$ for Ge, 0–10$\%wt$ for Al and 0–1500$ppm$ for Er), we observe a relevant agreement between the whole experimental measurements and the numerical simulations of the RIA. Our model allows to predict the fiber degradation versus a dose deposit up to typically 1 kGy, as a function of the fiber core composition. Thanks to its accuracy, this model enables the engineering of radiation hardened or radiation sensitive optical fibers. As a consequence, this model appears to be very promising to design specific radiation-hardened optical amplifiers.

Next enhancement of the model will take into consideration the trap relaxation through an optical or a thermal way. This will improve the model accuracy in order to fully assess the respective contributions of both degradation and recovery kinetics.