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Abstract
In this study, the response of regenerated fiber Bragg gratings (RFGBs) to axial forces was investigated in a temperature range from room temperature to 900 °C. For the first time, the transition from pure elastic to viscoelastic behavior around 700 °C of a standard SMF28 optical fiber was measured with an inscribed RFBG. An elastic model with linear temperature dependencies of Young’s modulus and Poisson’s ratio was established, and showed good agreement with the measurements up to temperatures of ∼500 °C. In the temperature range up to 900 °C, the RFBG response could be well described with a simple, single-material approach and a Burgers model that consists of a Kelvin and a Maxwell part. Based on the elastic parameter of the Maxwell part, the temperature-dependent force sensitivity of the RFBG was determined, and it showed a linear decrease in the range from room temperature to ∼500 °C, constant values in the range between ∼500 °C and ∼600 °C, and a strong increase at higher temperatures. While fulfilling the condition to operate in the elastic domain of the silica fiber, the investigations demonstrate that RFBGs can be used as force sensors up to temperatures of ∼600 °C - the range in which temperature-dependent force sensitivities have to be considered. The temperature-dependent parameters of the effective single-material model (elastic and viscoelastic part) are essential to describe the effective mechanical behavior of the optical fiber at high temperatures.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nowadays, fiber Bragg grating (FBG) sensors constitute a common type of fiber optic sensors. They have several distinctive advantages, like their ability to multiplex different FBGs in a single fiber [1]. This permits the building of a multi-point sensor within a single cable, which can save space and costs [1,2]. Another advantage is their relatively small dimension. The typical diameter of an optical fiber is about 125 µm and therefore, they have often been embedded into host materials like fiber-reinforced composites or metal [3,4]. They are also immune to electromagnetic fields, a property which enables them to be used in power generation [5].
Fiber Bragg gratings (FBGs) [6] are optical band-pass filters. The reflected light has a specific spectrum, which is centered around the Bragg wavelength $\lambda _B$. This Bragg wavelength, $\lambda _B=2 n_{eff} \Lambda$, depends on the refractive index modulation period $\Lambda$ and the effective refractive index $n_{eff}$ [7]. The Bragg wavelength is sensitive to temperature, strain, and force [8–12]. If the temperature of an FBG is increasing, the Bragg wavelength will increase due to the thermo-optic effect and the thermal expansion of the fiber [8]. Under strain, the Bragg wavelength will change due to the elasto-optic effect of the fiber and the elongation of the grating period [8]. The force sensitivity of an FBG depends on the strain response and Young’s modulus, due to Hook’s law.
Over the past few years, interest in high-temperature FBG-based sensing has grown [13–16]. Numerous applications of temperature measurements at high temperatures have been reported, such as the temperature gradient of hot sodium in a nuclear reactor [17], the temperature distribution in a gas turbine [1,18], the solidification process of an aluminum alloy during casting [19] or the dynamic temperatures of a Radiant Syngas Cooler [20]. In addition, the measurement of further physical parameters, like force and strain at high temperatures, are of great interest, too. For example, high strains have been measured during a casting process [21].
At high temperatures, silica shows a viscous behavior, which has been studied by many researchers [22]. For the temperature dependence of viscosity, different models have been established for different temperature regions above the glass transition temperature, such as the VTK model, the Adam and Gibbs model [23,24]. If glass cools down below the glass transition temperature, it will always be in a disordered, "frozen" amorphous state [25]. This is related to the cooling rate. The thermal history affects, amongst other things, the relaxation time, the density, fictive temperature, molar volume, enthalpy and viscosity of the glass [26–30].
FBGs written in a photosensitive silica fiber are called Type I FBGs and are most commonly used for sensing purposes [31]. At temperatures above 200 °C, their refractive index modulation decays [32]. For sensing at high temperatures, different kinds of gratings are needed. There are two types of high-temperature resistant FBGs: Type II FBGs, most often inscribed with femtosecond-laser [33] and regenerated fiber Bragg gratings (RFBGs) [34,35]. RFBGs have the advantages of good spectral reflection line shapes, and due to the high-temperature fabrication and annealing process, they show low wavelength drifts. RFBGs are fabricated in two steps. At first, a Type I FBG is written in a hydrogen-loaded fiber. Secondly, this FBG is exposed to a high-temperature treatment, where it transforms into an RFBG [36]. The strain sensitivity of RFBGs have been characterized for different doped optical fibers up to high temperatures [17,37,38]. Under constant mechanical load, RFBGs have been used to characterize the pure viscous behavior of a standard SMF28 optical fiber [39,40].
In order to employ RFBGs as a force sensors at high temperatures, it is essential to know whether the fiber shows a pure elastic or a viscous mechanical behavior at the given temperature. Therefore, in this study, the transition of an RFBG-based force sensor from pure elastic behavior at low temperatures to a viscoelastic behavior at high temperatures was investigated. We measured the wavelength response of a regenerated fiber Bragg grating under increasing and decreasing mechanical loads in a temperature range from room temperature to 900 °C. With these measurements, and in combination with the Burgers model, the temperature dependencies of the Young’s modulus and the viscosity could be derived. In addition, the temperature-dependence of the force sensitivity of RFBGs were obtained in the elastic temperature regime of the RFBGs.
2. Theory
The viscosity of glass depends greatly on temperature - with viscosity decreasing with increasing temperature. Thus, glass behaves in a fully elastic manner at lower temperatures, in a fully viscous manner at higher temperatures, and viscoelastically at intermediate temperatures. A standard optical fibre is composed of a pure silica cladding and a Ge-doped core. Because of the Ge doping, the material parameters of the core, as for example the glass transition temperature T$G$ and Young’s modulus are lower than those of pure silica [41,42]. This has the consequence that at high temperatures and under mechanical load, stresses in the core and cladding are not the same and will vary with temperature. In addition to that, the mechanism underlying the refractive index variation of regenerated Bragg gratings are not completely solved and some explanations for the RFBG include periodic variations in the stress at the core-cladding interface [35,43–45]. Therefore, in reality, complex, temperature-dependent stress fields are expected within the mode field of an optical fibre. For the data evaluation reported here, we apply a very simplified, single-material model, where the optical fibre with the integrated RFBG is described as a homogeneous material, neglecting the core-cladding material mix. Therefore, all model parameters are to be understood as effective values, which represent the combination of the contributions of the core and the cladding to the mode field. In this section, we will first introduce a temperature-dependent elastic model that applies to lower temperatures, and secondly, a viscoelastic model for an RFBG in a silica optical fiber that also applies to higher temperatures.
2.1 Elastic model
The Bragg wavelength of an FBG increases if temperature increases or strains are applied. With increasing temperature, the wavelength of an FBG will increase due to the thermo-optic effect and the thermal expansion of the fiber. If strain is applied to an FBG, the Bragg wavelength will change due to the elasto-optic effect of the fiber and the elongation of the grating period. The Bragg wavelength as a function of temperature and strain can be written as [46]
Here, $\lambda _B$ is the Bragg wavelength, $n_{eff}$ the effective refractive index and $\Lambda$ the grating period. At a constant temperature ($T=const.$) the wavelength shift of an RFBG caused by strain can be formulated with a linear model [46]: $\Delta \lambda _B$ represents the wavelength shift, $K_\varepsilon$ the strain sensitivity and $\varepsilon$ the strain of the fiber. The strain sensitivity $K_\varepsilon$ of an RFBG can be expressed analytically as [10]
Table 1. Temperature-dependent optomechanical parameters of the optical fiber with RFBG.
2.2 Viscoelastic model
In this section, a single-material viscoelastic model of an optical fiber is used to describe the wavelength shift of an RFBG, which is particularly important at high temperatures. A frequently employed viscoelastic model is the Burgers model [52]. Zhou et al. used the Burgers model for precise lens modeling and fabrication, and found good agreement between this model and the measurements [53]. Van den Brink showed that silicate glasses doped with different components deform under stress viscoelastically according to the Burgers model [54]. Koide et al. used the Burgers model to model the relaxation of SiO$2$PbO below the glass transition temperature. They showed that the Burgers model described the measurements very well [55].
In Fig. 1(a), a schematic of the Burgers model is shown [52]. It consists of four parts: the elastic part (spring representing Young’s modulus $R_1$), the viscous part (dashpot representing viscosity $\eta _1$), and the viscoelastic part (spring $R_2$ & dashpot $\eta _2$ connected in parallel). $F$ is the axial force acting on the fiber and $A$ is the cross section of the fiber. The spring and dashpot connected in serial are known as the Maxwell model and the spring and dashpot connected in parallel are called Kelvin model. In Fig. 1(b) the time-dependent response of the Burgers model to a stepwise increase of force of size $\hat {F}$ is shown. At first, there is an instantaneous increase of the strain (spring) followed by a viscoelastic (delayed elastic) behavior and a transition into a linear growth of the strain (viscous). If the force is removed, the strain follows the behavior in reverse, which is shown after time $t_1$ in Fig. 1(b). The complete strain as a function of time can be formulated as a sum of all parts of the model:
Fig. 1. (a) The Burgers model consists of a Maxwell and Kelvin model in series. (b) Time-dependent behavior of the Burgers model after a force step of amplitude $\hat {F}$ is applied at $T = 0^{\circ}\textrm{C}$ and released at $t=t_1$. Adapted from [52].
A strain response of the fiber to a step in axial force can be expressed as [52]
Fig. 2. Schematic representation of the Boltzmann principle. The resulting strain as a function of time - if stress $\sigma _i$ is increased or decreased at different points $\xi _i$ in time. Adapted from [52].
3. Experiments and results
The FBGs used in this study were fabricated in our laboratory using an excimer laser operating at a 248 nm wavelength and the phase mask method [56]. The seed FBGs were written in smf28 standard telecommunication fibers, which were loaded with H$2$ at 120 bar for 14 days. Two sensor fibers, each including one FBG at the wavelength of 1550 nm, were produced. A schematic drawing of the experimental setup is depicted in Fig. 3. All measurements employed a four-channel interrogator (sm125, Micron Optics, Atlanta, USA) for measuring the reflection spectra of the FBGs. The high-temperature tube furnace (ROS 20/250/12, Thermconcept GmbH, Bremen, Germany) was vertically aligned. The ceramic tube had an inner diameter of about 2 cm, a length of about 50 cm, and there was a 10 cm homogenous temperature region in the middle. The acrylate coatings of the fibers were removed mechanically over a length of 25 cm at each side of the FBG. Both fibers and a Type-K thermocouple were suspended inside the oven with the positions of the FBGs and the thermocouple in the middle of the furnace. As a first step, both seed FBGs were annealed at 80 °C for 40 h. During this treatment, the hydrogen diffused out of the fibers. After the heat treatment, back at room temperature, the FBG reflection lines had shifted by about 1 nm to shorter wavelengths as shown in Fig. 4(a) (red dashed line). The increase in reflected power when compared with the measurement after the inscription can be attributed to different insertion losses of the fiber connectors used. The fibers with the FBGs were heated up to 800 °C and were kept at this temperature for 66 h. The evolution of the reflected power during this process is shown in Fig. 4b. After 1 h at 800 °C, the reflectivity of the FBGs decayed to almost zero, which is depicted in the inset of Fig. 4(b) in more detail. Afterwards, new gratings (called RFBGs) emerged, and these gratings reached stable reflectivity values after $\approx$30 h. The RFBGs were annealed at 800 °C for an additional 36 h to reduce the wavelength drift. Then, the furnace was switched off and the fibers cooled down slowly to room temperature. The reflected spectrum of one RFBG at room temperature is shown in Fig. 4(a) (blue dotted line). The maximum reflected power was about 45 % compared to the maximum reflected power before the regeneration process, and a wavelength shift toward shorter wavelengths of about 1.4 nm occurred.
Fig. 3. Schematic drawing of the high-temperature measurement setup. It consists of a high-temperature oven with a ceramic tube, a thermocouple Type-K with a data acquisition (DAQ) system, an FBG interrogator (sm125), a PC for capturing the data, and a clamp for loading the fiber with weights.
Fig. 4. (a) The spectra of one FBG at room temperature after inscription (solid black line), after the hydrogen had diffused out of the fiber (red dash line) and after the regeneration process had taken place (blue dotted line). (b) Reflected power of the FBG during the regeneration and annealing processes.
For determining the influence of the regeneration process, the force sensitivity was measured before and after the regeneration and annealing process at room temperature. To apply different forces on the fibers, four weights were added one by one and removed one by one. All weights (25 g each) were determined with a high-precision scale (Scout STX 223, OHAUS Europe GmbH, Switzerland). At room temperature, the force sensitivity was measured while the furnace was kept off and 60 s were recorded before the next weight was applied or removed. This measurement at room temperature was performed before (after the H$2$ was diffused out of the fiber) and after the regeneration and annealing process. During the regeneration and annealing process, both fibers had no weight attached. The averaged wavelength shifts as a function of the applied force (weight force of the added weights) are shown in Fig. 5(a). All measurements show a linear behavior and linear functions were fitted to each measurement. It can be seen in Fig. 5(a) that the measurements taken after the regeneration have slightly smaller sensitivities than the measurements taken before regeneration. This is more apparent in Fig. 5(b), where the residues of the fits on the data before the regeneration process are depicted. The residues of the measurements after the regeneration show significantly smaller slopes than the measurement before the regeneration. Using the slopes of the fit functions, Young’s moduli of both fibers before and after the regeneration were calculated according to Eq. (5) and are shown in Fig. 5(c). After the regeneration process, Young’s moduli had increased. On the second axis of Fig. 5(c), the force sensitivities are shown with an inverted axis scaling.
Fig. 5. (a) The wavelength shift of both FBGs (before and after the regeneration) as a function of force. A linear fit was applied to each measurement. (b) Residues of all data to the linear fits to the data measured before regeneration. The residues of the measurements after the regeneration process show a significantly different slope than the measurement before regeneration. (c) Force sensitivity before and after the regeneration process of both FBGs. On the left axis, Young’s modulus is displayed. On the second axis (right-hand side), the corresponding force sensitivity (inverted) is shown.
For determining the force sensitivities in the temperature range from 100 °C to 900 °C, the furnace was heated in steps of 100 °C and the Bragg wavelengths were continuously captured at a measurement rate of 1 Hz, while various weights were applied to one fiber, with the other fiber served as a reference. At each temperature step we waited about 90 min until the oven had reached a stable temperature level before the measurements were performed. At temperatures below 400 °C, the oven temperature showed oscillations due to non-optimized temperature control. For example, at 100 °C, the furnace showed a temperature amplitude of about 4 °C and an oscillation period of 10 min (see Figs. 6(a), 6(b), and 9(a)). Above 400 °C, no significant temperature oscillations were present. After a weight was applied, the data acquisition was performed for 30 min before the next weight was added. All measured wavelengths as a function of time are displayed in Fig. 6(a). At 900 °C, the force-loaded RFBG showed a strong non-elastic behavior and a permanent wavelength shift during the measurement. The reference RFBG measured any wavelength changes which were not related to loads, such as temperature-induced wavelength drifts of the fiber. In Fig. 6(b), the wavelengths of the reference RFBG as a function of time at different temperatures are shown. Only at 900 °C can a small wavelength drift be observed with regard to the reference RFBG.
Fig. 6. (a) Bragg wavelengths as a function of time at different temperatures and different axial forces. (b) Bragg wavelengths of the reference RFBG at different temperatures as a function of time.
4. Discussion
4.1 The influence of the regeneration process on force sensitivity
At room temperature, the force sensitivities of both RFBGs were measured before and after the regeneration process. As shown in Fig. 5, the regeneration and annealing process changed the force sensitivity of the RFBGs. Before the regeneration process, the force sensitivity was found to be $K_F = 1.3615 \,\frac{\textrm{nm}}{\textrm{N}}$ (average value from both FBGs) and after regeneration, the force sensitivity was $K_F = 1.341 \,\frac{\textrm{nm}}{\textrm{N}}$, 0.0205 $\frac{\textrm{nm}}{\textrm{N}}$ lower, which corresponds to a change of 1.5 %. Roths et al. measured a force sensitivity of $K_F = 1.3471 \,\frac{\textrm{nm}}{\textrm{N}}$ with the same type of fiber but at a different wavelength of 1533.7 nm [48]. To compare both measurements, the relative force sensitivity
which is independent of the Bragg wavelength, should be used. The corresponding values are $k_F^{Roths} = 8.785 \times 10^{-4} \,\frac{1}{\textrm{N}}$ and this investigation $k_F = 8.79 \times 10^{-4} \,\frac{1}{\textrm{N}}$. The measurements display significant accordance. After the regeneration process, the force sensitivity decreased, which corresponds to an increase in the Young’s modulus from $E_0 = 73.8 \,\textrm{GPa}$ to $E_{0,reg} = 74.9 \,\textrm{GPa}$. This increase in the Young’s modulus is most likely caused by the compaction of the glass during the heat treatment of the regeneration and annealing process [57].4.2 Force- and temperature-induced permanent wavelength shifts
To quantify the wavelength shifts of the RFBGs at each temperature, the differences between the wavelengths before and after each loading cycle were calculated for both the reference RFBG and the RFBG that was loaded with force:
Here, $\Delta \lambda _{B,Drift}(T)$ is the wavelength shift, $\lambda _{B,Start}(T)$ is the wavelength at $t = 0 \,\textrm{h}$ before the weights were added, and $\lambda _{B,End}(T)$ is the wavelength at $t = 4.5 \,\textrm{h}$ after the weights had been applied and once again removed. In Fig. 7(a), these wavelength differences are depicted. Up to temperatures of 600 °C, no wavelength drift was observed in both RFBGs. At 700 °C, a minor wavelength drift of the loaded RFBG was observed. At 800 °C and 900 °C, both RFBGs showed a permanent wavelength change, although the shifts for the loaded RFBG were significantly larger. This indicates a viscous deformation of the fiber due to mechanical and thermal load.Fig. 7. (a) The wavelength drift as a function of temperature. The blue squares represent the RFBG under mechanical load and the red circles represent the reference RFBG without any load. (b) Mean wavelengths as a function of applied force at different temperatures. The dashed lines are linear functions fitted to the wavelength data taken at the same temperature. At 800 °C and 900 °C, a strong nonlinear behavior was observed.
4.3 Evaluation of the temperature-dependent force sensitivity based on an elastic model
For each load, the measured wavelengths and temperatures were averaged over $\sim 30 \,\textrm{min}$, which gives mean wavelength values for each force and temperature combination. In Fig. 7(b), the force-dependent mean wavelengths at different mean temperatures of the RFBG are shown. For each temperature, linear functions ($\lambda _B (F)=\lambda _{B,0}+K_F^{lin.Fit}(T)F$) were fitted to all wavelengths. As can be seen in Fig. 7(b), from room temperature up to 600 °C, the force response of the wavelength shift was linear but at higher temperatures, the viscous behavior of the fiber caused a nonlinear wavelength shift under force. This effect was most pronounced at 900 °C. All measured force sensitivities, $K_F^{lin.Fit}(T)$, are shown in Fig. 8 and Table 2. Up to 400 °C, the force sensitivities decreased linearly and between 400 °C and 600 °C, the force sensitivities showed almost constant values. Above 700 °C, the sensitivities showed a strong increase with temperature. However, as mentioned above, the elastic model is not appropriate in this high-temperature range due to the viscous behavior of the fiber. The transition from an elastic to a viscoelastic behavior will be addressed in Section 4.4. The linear decrease below 500 °C in the force sensitivity was described by Eq. (11) of the linear elastic model introduced in Section 2.1. The parameters used in this model are listed in Table 1, and the Young’s modulus $E_0$ at $T = 0^{\circ}\textrm{C}$ was approximated by the measured value $E_{0,reg}$ as described in Section 4.1. The force sensitivity based on the elastic model (Section 2.1) with the Young’s modulus is
Fig. 8. Force sensitivity as a function of temperature. The full triangles represent the force sensitivity of the RFBG using a linear fit, the grey circles are the viscoelastic model and the elastic model is shown as a grey dashed line. At 500 °C and above, the deviation of the measured force sensitivity to the elastic model increased.
Table 2. Measured force sensitivity, elastic model, and viscoelastic model.
4.4 Evaluation of the temperature-dependent force sensitivity based on a viscoelastic model
As can be seen in Fig. 6(a), at high temperatures, the fiber showed a viscoelastic behavior. Hence, in this temperature range, the force sensitivity cannot be described adequately using the elastic model. Therefore, the single-material viscoelastic model (Section 2.2) was used to analyze the force response of the fiber. Using this model (Eq. (16)) we assumed that the effective model parameters $R_1 (T)$, $R_2 (T)$, $\eta _1 (T)$, and $\eta _2 (T)$ depend on temperature, but the elasto-optic effect was constant with temperature. This single-material viscoelastic model was applied to all data measured at all temperatures, giving the temperature dependencies of the model parameters. As a result, the transition from an elastic to viscoelastic behavior can be described and the effective viscosity of the fiber below the glass transition temperature can be determined.
As examples, Figs. 9(a), 9(c), and 9(e) depict RFBG wavelength data at 100 °C, at 500 °C and at 900 °C, together with the wavelengths of the reference RFBG as a function of time. In Fig. 9(a), the wavelength shift at 100 °C of both RFBGs showed large oscillations with a period of about 10 min and an amplitude of about 40 pm, which were due to poor temperature control of the oven, as described above. At 500 °C and above, the oscillations caused by the electronic controller of the furnace were gone. At 500 °C (Fig. 9(c)), the fiber is still fully elastic and showed no wavelength drift. In Fig. 9(e), the viscosity of the fiber caused a permanent wavelength shift of the loaded RFBG in the order of 2.8 nm. An additional thermal wavelength drift in the order of 0.15 nm was present, as shown by the reference RFBG.
Fig. 9. Wavelength shift of the loaded RFBG and the reference RFBG as a function of time at (a) 100 °C (c) 500 °C (e) 900 °C. After every 30 min a weight was applied and after 2 hours the weights were removed after 30 min, one by one. The wavelength of the reference RFBG subtracted from the loaded RFBG at (b) 100 °C (d) 500 °C (f) 900 °C. The fit with the Burgers model corresponds very well with the data.
The wavelengths of the loaded RFBG were corrected by subtracting the wavelength shift of the reference RFBG. These corrected data are depicted in Figs. 9(b), 9(d), and 9(f), respectively. At 100 °C, the temperature oscillations of the oven are reduced in the corrected data (Fig. 9(b)). The wavelength shifts at 900 °C, showed a significant influence of a viscous behavior. The solid lines in Figs. 9(b), 9(d), and 9(f) are fits of the single-material Burgers model (Eq. (16)) to the corrected wavelength data. As can be seen, the single-material Burgers model described the response of the RFBG at all temperatures very well and there is no need to apply more sophisticated model approaches. The instantaneous shift of the wavelength if a weight was applied or removed is described by the elastic part of the Burgers model, $K_F^{vis.model}$ (Eq. (17)). It was followed by a slow increase or decrease, which is caused by the viscoelastic part, and it transitioned to a constant increase in the wavelength due to the viscous part. The force sensitivity $K_F^{vis.model}$ calculated from the elastic part of the Burgers model is displayed in Fig. 8 (grey circles). They showed a good agreement with the measured $K_F^{lin.Fit}$ values. In Fig. 10(a), the fitted Young’s moduli ($R_1(T)$) from the Maxwell part are depicted. In the temperature range up to 400 °C, the Young’s modulus showed a linear increase. With increasing temperature, the Young’s modulus reached a plateau, and above 700 °C, it decreased significantly. The strength of the Kelvin part ($1/R_2 (T)$) is shown in Fig. 10(b). For temperatures up to 600 °C, the strength is almost zero, thus the influence of the Kelvin part is negligible. Above 600 °C, it increases considerably, which shows that the fiber is starting to behave viscoelastically. Both values, $R_1$ (Fig. 10(a)) and $1/R_2$ (Fig. 10(b)), indicate that the transition from elastic to viscoelastic behavior occurs in the temperature range between 600 °C and 700 °C for an smf28 fiber.
Fig. 10. (a) Spring constant for the Maxwell part, which showed a nonlinear temperature dependence. The open square represents Young’s modulus determined at room temperature as described in Section 3.1 (see Fig. 5(c)) (b) Strength of the Kelvin part as a function of temperature. Up to 600 °C, the strength of the Kelvin part is negligible. (c) Viscosity of the Burgers’ model for the Maxwell part and d) Kelvin part. At 600 °C and above, the viscosities were decreasing exponentially.
In Figs. 10(c) and 10(d), both viscosities $\eta _1$ and $\eta _2$ of the Burgers model are shown as a function of temperature. The vertical axes have natural logarithm scales. For temperatures up to 600 °C, the measured viscosities showed almost no temperature dependence and the values were in the range of 1018 Pa s to 1019 Pa s for $\eta _1$ and for 1016 Pa s to 1019 Pa s for $\eta _2$. Thus, the viscosity for the Maxwell part ($\eta _1$) is larger than that of the Kelvin part ($\eta _2$). Zhou observed that the Maxwell part is lager by an order of magnitude of 1-2 for BaO doped glass and Bernard reported this for a TAS-fiber, too [53,59]. At temperatures above 600 °C, both viscosities increased almost linear with inverse temperature, as can be seen in the insets in Figs. 10(c) and 10(d), which have a reciprocal temperature on their horizontal axes. This also indicates the transition from elastic to viscoelastic behavior in this temperature range. Using an RFBG as a high-temperature force or strain sensor, a pure elastic behavior is required. Therefore, these results show that RFBG-based strain or force sensing is limited to a maximum temperature of $\sim$700 °C. The change in the viscosity of glass over a small temperature range can be described using an Arrhenius function [60]:
Here, $\eta _0$ is a material constant, $E_a$ the activation energy and $R$ the universal gas constant. This equation can be rewritten as For both viscosities $\eta _1$ and $\eta _2$, the data (insets in Figs. 10(c) and 10(d)) between 600 °C (873 K) and 900 °C (1173 K) were fitted with a linear function. The slopes of the functions contain the activation energies, according to Eq. (23). The activation energy for the Maxwell part was found to be $E_a = 270 \,\frac{\textrm{kJ}}{\textrm{mol}}$ and for the Kelvin part $E_a = 132 \,\frac{\textrm{kJ}} {\textrm{mol}}$. The activation energy of the Maxwell part was almost twice the activation energy of the Kelvin part. The activation energies (measured here) for glass were lower than values reported by several other researchers [40,61,62], which had been determined at higher temperatures in the region of $T_g$ of silica. Yet, the measured lower activation energy corresponds with the observation of Kiode et al., who measured the relaxation of glass fiber in a bending experiment below the glass transition temperature [63]. They measured an activation energy of $E_a = 145 \,\frac{\textrm{kJ}} {\textrm{mol}}$ for a Ge-doped SiO$2$ glass fiber. In this paper, a simple, single-material model was used that assumes homogeneous material parameters instead of a two-material model that considers different material parameters for the fiber core and for the cladding. Due to this simplification, only effective material parameters can be deduced from the single-material model fits to experimental data. This might explain a part of the discrepancies between our effective model parameters and the corresponding parameters from bulk materials reported in the literature5. Conclusion
The temperature dependence of the force sensitivity of a regenerated fiber Bragg grating (RFBG) in a standard telecommunication fiber (Type SMF28) was investigated in a temperature range from room temperature to 900 °C. The RFBG showed an elastic behavior at and below $\sim$600 °C and a viscoelastic behavior at $\sim$700 °C and above. A temperature-dependent model for the elastic behavior of the RFBG with linear increases of Young’s modulus and Poisson’s ratio with temperature was introduced. For temperatures up to 500 °C, the elastic model and the measurement showed good agreement. To describe the behavior of the fiber at higher temperatures, the Burgers model and a homogeneous fiber material, e.g. no distinction between core and cladding material parameters, was chosen. Using the elastic component of the Maxwell part of this model, the force sensitivity up to 900 °C could be obtained. After a linear decrease in the force sensitivity from room temperature to $\sim$500 °C, the force sensitivity showed almost constant values in the temperature range from $\sim$500 °C to $\sim$600 °C and it strongly increased at higher temperatures. The effective viscosity of the fiber was determined with the viscoelastic (Kelvin) part of the Burgers model as well and it was found that the effective viscosity could be neglected for temperatures below $\sim$700 °C. Measurements at room temperature of the effective Young’s modulus before and after the regeneration process showed an increase from $E_0 = 73.8 \,\textrm{GPa}$ to $E_{0,reg} = 74.9 \,\textrm{GPa}$ (+1.5 %), which was attributed to compaction of the silica fiber during the regeneration process. The results demonstrates that up to temperatures of $\sim$600 °C RFBGs in SMF28 optical fibers can be used as high-temperature force or strain sensors, because in this temperature range, the material still operates in its elastic domain. However, the temperature dependence of the force sensitivity has to be considered if employed as a force sensor. The effective temperature-dependent parameters of the elastic and viscoelastic part of the single-material model are essential for deploying optical fibers at high temperatures.
Funding
Deutsche Forschungsgemeinschaft (KO 2111/11-2, RO 4145/3-2, VO 1487/11-2).
Acknowledgments
This work was financially supported through the Open Access Publication fund of the Munich University of Applied Sciences (MUAS).
Disclosures
The authors declare no conflicts of interest.
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