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Abstract
Measurements of the thermo-optic coefficient (dn/dT) are made on a binary phosphosilicate core glass optical fiber. Using these results, dn/dT for the P2O5 constituent is found to be −13.3 × 10−6 ± 8.0% K−1, a value much lower in magnitude than reported in the literature for this system. Its accurate elucidation is especially useful in guiding the design of low- or negative-thermo-optic glasses and optical fibers. The phosphosilicate core also has a coefficient of thermal expansion that is higher than that of the pure silica cladding. As a result, the clad fiber geometry slightly lessens the effectiveness of the negative-valued contribution to dn/dT by P2O5 relative to bulk glass.
© 2017 Optical Society of America
1. Introduction
Phosphorus pentoxide (P2O5) is a popular dopant incorporated into silica glass optical fibers partly due to its reported large and negative thermo-optic coefficient (TOC, dn/dT), being useful for many applications ranging from sensors [1] and gratings [2] to lasers [3]. The value of −92.2 × 10−6 K−1 [1,4] is often quoted but, to the best of Authors’ ability, the origins of this number could not be found. Oftentimes, literature searches lead back to a single work [5] specifying this value, but offered without an apparent reference. Such a large, negative value would logically suggest that, at concentrations of as little as 10 mole% P2O5 in silica (dn/dT = + 10.4 × 10−6 K−1), an athermal (dn/dT = 0) glass can be realized. Other previous work on the determination of dn/dT in multicomponent phosphates and silicates can be found [6–8], however, the case of the additivity of P2O5 in silicates appears to be lacking (e.g. see Table 2 in [6]).
As such, using a binary phosphosilicate core fiber, dn/dT for the P2O5 component in phosphosilicate glass is found to be much smaller in magnitude than the value listed above, but indeed being negative: −13.3 × 10−6 ± 8.0% K−1. This value is much more consistent with those of other negative-valued TOC materials [9]. Measurements such as these, therefore, clearly are useful in guiding the design of low- or negative thermo-optic glass fibers. Finally, the effect of cladding the phosphosilicate glass in pure silica is analyzed. It is found that the cladding has the effect to slightly offset the reduction in dn/dT due to the negative TOC of P2O5 alone. As will be described, this arises from a mismatch in the coefficient of thermal expansion (CTE) between the core and cladding and a resultant net compression experienced by the core at elevated temperatures. This is believed to be the first systematic materials-based analysis of the thermo-optic effect in a P2O5-doped silica fiber.
2. Optical fiber
The optical fiber used in this investigation, fabricated by INO of Canada, has been described in detail elsewhere [10,11]. Nevertheless for convenience the refractive index profile (RIP) is provided in Fig. 1. The core is a P2O5-doped SiO2 glass, with 12.2 mole% P2O5 at the center. There is a slightly depressed inner cladding region at a radial position spanning roughly 3 to 8 μm. This depressed cladding layer contains both P2O5 and F, but both are present in concentrations of < 1 mole%. The principal cladding is pure silica with 125 μm outer diameter and has a standard polymeric coating. The RIP is a good representation of the P2O5 distribution within the core, and the central burnout of phosphorus is minimal.
3. dn/dT measurements
The apparatus utilized for the determination of dn/dT for the optical fiber is shown in Fig. 2 [11]. Most generally, the apparatus is a simple fiber ring laser. The pump is injected into the cavity via a wavelength division multiplexer (WDM) and into a segment of erbium doped fiber (EDF). Two isolators ensure unidirectional laser operation. No care was taken to optimize the spectral characteristics of the laser. As will be discussed, the presence of a large number of cavity modes enhances the measurement sensitivity. The output of the system is taken through the 10% arm of a 90/10 tap coupler. This output signal is sent directly into a detector whose signal is interrogated using an electrical spectrum analyzer (ESA). The laser operates at a nominal wavelength of 1554 nm. The fiber under test (FUT, typically 4 – 6 m in length) forms part of the laser cavity and this fiber is coiled and held in a heated water bath in order to accurately control its temperature.
Fig. 2 Block diagram of the ring laser apparatus used to measure the thermo-optic coefficient of the P2O5 doped fiber.
The free spectral range (FSR) of the ring laser can be written generally as where the subscript i simply refers to one of the M various different fibers, with length d and modal index n, associated with the different components (e.g. isolator, couplers, erbium doped fiber, etc.) encountered within the cavity. The FUT’s temperature can be controlled and thus for it via the linear CTE and via the TOC. Courtesy of the cavity modes beating on the detector, the FSR of the ring laser can be read directly from the ESA. Differentiating the FSR versus T gives the following expression,
which is utilized to determine the TOC of the fiber from measurements of FSR(T) (Eq. (1) represents the slope of FSR(T)). In Eq. (1), NL is the refractive index-length product summed for all fiber segments whose temperatures are invariant (fiber comprising the WDM, coupler, EDF, etc.). In addition, α is the CTE (K−1), and l0 is the room temperature length of the FUT. n and l are the mode index and length of the FUT, respectively, but also as linear functions of temperature. Since the core is held rigidly in a much more voluminous pure silica cladding, the linear CTE of the core, α, is taken to be that of silica (0.55 × 10−6 K−1 [12]). In order to enhance measurement sensitivity, the beat signal from cavity mode pairs that are separated by greater than one FSR can instead be used. The sensitivity of the measurement increases by a factor of the number of FSRs that separate them, and hence the advantage of having a larger number of laser cavity modes becomes apparent.Figure 3 provides an example set of collected data. The change of the FSR from the starting value is plotted as a function of FUT temperature. Equation (1) is then fit (after subtracting the room temperature FSR) to the measurement data (points) utilizing the TOC as the only fitting parameter. Measurements were repeated several times using different FUT lengths, coil diameters, and cavity mode pairs. In all cases, Eq. (1) could be fitted with R2 values > 0.999. The error in the TOC measurement for the fiber is taken conservatively to be the magnitude of the largest departure from the mean value. The result of these measurements gives TOC = 8.27 × 10−6 ± 1.5% K−1. This is roughly 20% lower than that of pure silica [13], apparently having been reduced by the addition of P2O5.
Fig. 3 Sample set of data in the measurements of the TOC of the FUT. The change in FSR is measured as a function of temperature (points). The data are normalized to the fundamental FSR. The blue line is a fit of Eq. (1) to the data.
4. Material dn/dT
The TOC value determined for the fiber is that of the optical mode. The mode field diameter (MFD) of the fiber is 7.55 μm (Petermann II method) and therefore the mode overlaps with regions of the fiber possessing differing compositions. In order to model this effect, the RIP of Fig. 1 is approximated by a 5-layer step profile. One of these layers is the outer cladding and one is the depressed-index inner region. The central core, therefore, is approximated by 3 layers. Each layer has a unique composition and therefore a unique refractive index and TOC. The central layer of the approximation has 12.2 mole% P2O5, 87.8 mole% SiO2, and has a refractive index difference of 10.2 × 10−3 relative to the pure silica cladding.
For binary glasses, an additivity model may be utilized to determine both refractive index and TOC as a function of composition (in this case, as a function of P2O5 concentration) [3]. More specifically, these can be found from
In Eq. (2), the g’s are the parameter to be added (index n, or TOC, dn/dT), and the subscripts refer to the pure bulk values of the constituent. The averaging parameter, m, is the fraction of the total volume occupied by constituent 1 (taken to be P2O5 in this case). This is given in terms of the mole percent of P2O5 ([P2O5]) and SiO2 ([SiO2]) aswhere MV is the molar volume (molar mass divided by density ρ). It is important to point out that Eq. (2) does not necessarily give any insight into the physical structure of the glass, but rather it forms a reliable empirical model that can be fitted to measured data. While the gi values derived for the ‘bulk’ additives to SiO2, such as P2O5, have been shown to hold with a high degree of accuracy over moderate compositional ranges [3,10], the sudden appearance of any new glass morphologies in wider compositional ranges may lead the fitting associated with Eq. (2) to break down. This would require the determination of new gi values in those regimes.Using Eqs. (2) and (3) the refractive index as a function of temperature can be determined for each layer. This can then be used to calculate the modal index as a function of temperature and therefore the TOC for the fiber can be found. However, an important consideration must be made with respect to the fiber geometry. Since the P2O5-doped silica will have a CTE that is larger than that of the surrounding pure silica [14], the core will experience a net compression as the temperature is raised [15]. Since the refractive index of silica is known to increase as a function of pressure [16], this can offset the reduction of the TOC caused by P2O5 doping. In other words, adding P2O5 lowers the material TOC, but it also increases the CTE of the core. As the temperature is elevated this imparts pressure on the core which, in turn, results in a slightly increased index. This, then, requires a modified expression for the refractive index with respect to its use in Eq. (2). In his well-known work, Prod’homme [17] showed that the TOC is an interplay between two competing processes: 1) a decrease in density associated with thermal expansion, resulting in a decreased index and 2) a concomitant increase in polarizability as the electrons become more loosely bound to their nuclei, thus increasing the refractive index. If the former process dominates, then the TOC is negative. Rather than modeling these processes separately for both SiO2 and P2O5, the cladded material is assumed compressed relative to a freely expanded thermal equilibrium state, and the photoelastic effect instead is invoked here.
The application of pressure on a material imparts a change in the refractive index via the photoelastic effect (and coefficients). It is assumed that the pressure is volumetric, and therefore 3 principle axes must be considered for a linearly polarized optical wave: two orthogonal to the polarization and one parallel to it. The change in refractive index as a function of strain ε in the two directions orthogonal to optical polarization can be written as [18]
while in the parallel direction can be written aswhere ν is the Poisson ratio for the constituent and the p’s are the Pockels coefficients. The CTE mismatch between the core and cladding imparts compression ω (negative strain) on the core. If the thermal expansion of the whole system is restricted uniformly by the cladding, then [15]where T0 is some starting temperature (e.g. room temperature) and α is once again the CTE. This leads to the following full expression for the temperature-dependent refractive index that can be used for g1,2 in Eq. (2)where the subscripts denoting constituent have been left off for convenience. For modeling, αcore is instead taken as the CTE in each layer of the step-wise approximation to the RIP.The physical properties of SiO2 and P2O5, as used in the modeling, are provided in Table 1. The P2O5 CTE was identified by fitting Eq. (2) with gi = CTEi to data found in [19]. The results of this fit are shown in Fig. 4. The model extrapolates to a CTE value for pure P2O5 as shown in Table 1. While the fit to the data is quite good, confidence should only be extended in the range of compositions available, which fortunately includes the present case. With this, the only unknown is the TOC of the P2O5 constituent. This can therefore be used as a fitting parameter in the modeling.
Table 1. Physical properties of the constituent materials used in the modeling.a
Fig. 4 CTE data for P2O5:SiO2 glass as a function of [P2O5] from [19] (points) and a fit using Eq. (2).
To summarize, all relevant values required for Eqs. (2) – (7) have been identified. For each layer of the model a refractive index as a function of temperature can be calculated from the composition. This can then be used to determine the modal index as a function of temperature, from which modal TOC can be found (slope of the n(T) curve). For each layer the only unknown is a single ‘bulk’ TOC value for P2O5. This number is then simply adjusted until the calculated modal TOC matches the measured TOC value as provided above. To determine the uncertainty in the TOC of P2O5, the model was fitted to the measured fiber TOC value with the largest departure from the mean value (see Section 3).
One approximation was made to the present system involving the slightly depressed cladding layer. Since this layer possesses less than 1 mole percent of both P2O5 and F, and the influence of F on these parameters is unknown, the CTE and TOC of this layer are taken to be that of pure silica, but with an index that is 0.00054 lower. This index difference was not neglected since it renders a slightly smaller mode with greater overlap with the high-P content central layers. In order to validate this approximation, a calculation assuming 1 mole percent P2O5 in that layer was performed, and that modified the modal TOC by only 1.8%. That the error associated with this assumption is small can be attributed to the mostly silica composition and low overlap with the optical mode.
A TOC for bulk P2O5 is obtained to be −13.3 × 10−6 ± 8.0% K−1. This value is much lower in magnitude than has been previously reported (−92.2 × 10−6 K−1 [1,5]). If this larger value of TOC were instead assumed in the model, a nearly athermal (but slightly negative) modal value of −0.06 × 10−6 K−1 is calculated for the fiber used in this study, again vastly different from the observations presented here. For completeness, the effect of core-cladding CTE mismatch is considered. If the material CTE of the various regions were assumed to be that of pure silica, the modal TOC is calculated to be 7.82 × 10−6 ± 1.5% K−1 or only about 6% lower than the observed value.
To conclude, a simple calculation is presented that may be utilized as a general guide in the design of phosphosilicate glass fibers. Using the data from Table 1, the P2O5 TOC measured here, and assuming a simple 2-layer core-cladding structure with the optical mode being tightly confined (mode index approximately equal to the material index), dn/dT is plotted as a function of [P2O5] in Fig. 5 (orange curve). An athermal (dn/dT = 0) composition can be found in the vicinity of 30 mole% P2O5 content. Also provided in Fig. 5 is a similar calculation for an un-clad bulk glass (blue curve), showing the influence of the CTE mismatch. Generally, it has the effect to raise the TOC in silicate glasses and increase the amount of phosphorus needed to achieve an athermal fiber.
Fig. 5 TOC for P2O5:SiO2 glass as a function of [P2O5] for a 2-layer (core-cladding) step index fiber with a tightly confined mode (orange) and bulk un-clad glass (blue).
5. Conclusions
Utilizing a phosphosilicate glass optical fiber, the thermo-optic coefficient (dn/dT) of the P2O5 constituent has been investigated. Measurements reveal a value of −13.3 × 10−6 ± 8.0% K−1, which is indeed negative, but with a magnitude much lower than the value typically quoted in the literature [1,5]. This suggests that a much larger P2O5 content is required for the reversal of the sign of dn/dT for silicate glasses, with an athermal (dn/dT = 0) composition that can be found near [P2O5] = 30 mole%. Cladding the phosphosilicate glass in pure silica has the effect to slightly offset the reduction in dn/dT due to the negative TOC of P2O5 alone.
Funding
Joint Technology Office (FA9451-15-D-0009/0002). The J. E. Sirrine Foundation.
Acknowledgments
The Authors gratefully acknowledge INO of Canada who fabricated the fiber.
The Authors gratefully acknowledge financial support from the JTO. The J. E. Sirrine Foundation also is gratefully acknowledged for supporting the efforts of Authors (MC and JB).
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