Optical fibers are ubiquitous tools for global communications and, despite all the effort that has gone into their development, the accurate modeling and deduction of selected properties such as the linear coefficient of thermal expansion ($\alpha$) lag behind those of other properties such as the linear refractive index. The thermal expansion coefficient is particularly important since, for example, its use in design has led to the realization of several practical types of polarization-maintaining fibers [1]. On the other hand, $\alpha$ can also lead to parasitic stress-optic effects [2], and it influences other fiber properties such as the thermo-optic coefficient, $dn/dT$ [3], which can strongly impact modal performance and lead to mode instabilities in large mode area (LMA) fiber lasers [4,5]. Therefore, understanding and better predicting the coefficient of thermal expansion of these optical systems is crucial in the design of next-generation fibers.

Several models have been proposed over the past century to determine the material additivity of $\alpha$ based on empirical formulae. For example, and among others, Winkelmann and Schott [6], English and Turner [7], and Hall [8] used empirical linear coefficients as a function of oxide percent of the glass, whereas Gilard and Dubrul [9] used a “parabolic equation,” where $\alpha$ additivity is described by a second order polynomial function $ax+b{x}^2$ where $x$ is the weight percent of oxide and $a$ and $b$ are fit parameters. More recently, Doweidar [10] developed an additive model where $\alpha$ is a function of the structural units per that component in the glass. In this Letter, we propose a new additive model of the coefficient of thermal expansion in silicate glasses based on its derivation in isotropic solids such as glasses, and including the notion of structural units from Ref. [10]. The model is developed for additives into ${\mathrm{SiO}}_2$ for the canonical dopants (i.e., ${\mathrm{GeO}}_2$, ${\mathrm{P}}_2{\mathrm{O}}_5$, ${\mathrm{Al}}_2{\mathrm{O}}_3$, and ${\mathrm{B}}_2{\mathrm{O}}_3$), along with an emerging network modifier dopant (i.e., BaO) which is gaining attention for the developing of intrinsically low Brillouin gain fibers [11]. The novelty of this Letter is to show that the evolution of $\alpha$ in different ubiquitous silicate systems scales with the density variation induced by the volume fraction of the structural units that form the glass network.

The additivity of density ($\rho$) in silicate glasses can be described as a function of the volume fraction of each individual component (e.g., ${\mathrm{SiO}}_2$ or ${\mathrm{GeO}}_2$) from which the glass is made [12–14]. First, the density can be expressed in its most trivial form as a function of mass (m) and volume (V):

Then, it is assumed that, in a multicomponent glass, each individual constituent “$i$” will additively contribute to the total molar mass and volume. Therefore, the aggregate value $\rho$ of a silicate glass system can be written as follows:

Here, [$A$] and [$B$] are the molar percent of components $A$ and $B$, and $M_A$, $M_B$, ${\rho }_A$, and ${\rho }_B$ are their respective molar masses and densities. Now, by differentiating $\rho$ with respect to the temperature $T$, we obtain

Next, we recall the expression of the relative derivative of the volume (here expressed in terms of density) for isotropic solids:

With Eq. (6), a simple expression for the linear coefficient of the thermal expansion in glasses based on the additivity argument of $\rho$ as enunciated earlier is derived.

As noted above, a series of ${\mathrm{GeO}}_2\text{-}{\mathrm{SiO}}_2$, ${\mathrm{P}}_2{\mathrm{O}}_5\text{-}{\mathrm{SiO}}_2$, ${\mathrm{B}}_2{\mathrm{O}}_3\text{-}{\mathrm{SiO}}_2$, ${\mathrm{Al}}_2{\mathrm{O}}_3\text{-}{\mathrm{SiO}}_2$, and $\mathrm{BaO}\text{-}{\mathrm{SiO}}_2$ binary glasses was chosen to validate this additivity approach. The necessary material properties to perform the calculation of the additivity are summarized in Table 1. The data are fit using Eq. (6), with $\alpha$ for fused ${\mathrm{SiO}}_2$ taken as $5.5\times {10}^{-7}{\mathrm{K}}^{-1}$ for all systems investigated [18]. Thus, the value of $\alpha$ for the non-silica compound is taken as a fit parameter. The data, along with graphical fits using Eq. (6), are displayed in Fig. 1. Since the molar volume of each individual constituent is employed in the Eq. (6) calculation, the $x$ axis of Fig. 1 is expressed in mole percent. It can clearly be seen that Eq. (6) works very well in the cases of the ${\mathrm{GeO}}_2\text{-}{\mathrm{SiO}}_2$, ${\mathrm{P}}_2{\mathrm{O}}_5\text{-}{\mathrm{SiO}}_2$, and $\mathrm{BaO}\text{-}{\mathrm{SiO}}_2$ systems. However, it does not fit the data in the ${\mathrm{Al}}_2{\mathrm{O}}_3\text{-}{\mathrm{SiO}}_2$ and ${\mathrm{B}}_2{\mathrm{O}}_3\text{-}{\mathrm{SiO}}_2$ systems.

Table 1. Molar Mass (M) and Density ($\rho$) Used for the Calculations

View Table | View all tables in this article

 

Fig. 1. Linear coefficient of thermal expansion ($\alpha$) for some typical binary silicate systems as a function of non-silica content. The fitting of these data is performed with Eq. (6). Data where found in [19,20] for ${\mathrm{GeO}}_2\text{-}{\mathrm{SiO}}_2$, in [21] for ${\mathrm{B}}_2{\mathrm{O}}_3\text{-}{\mathrm{SiO}}_2$ and $\mathrm{BaO}\text{-}{\mathrm{SiO}}_2$, citing [22] and [23], respectively, [24] for ${\mathrm{P}}_2{\mathrm{O}}_5\text{-}{\mathrm{SiO}}_2$, and [25] for ${\mathrm{Al}}_2{\mathrm{O}}_3\text{-}{\mathrm{SiO}}_2$. All the systems together are valid within a common range of 25°C–100°C.

Download Full Size | PDF

This suggests that Eq. (6) is not sufficient to entirely describe the behavior of the coefficient of thermal expansion, and other factors must account for the additivity. As suggested in [10,26], a glass can be considered as the sum of structural units that form the glass network and, therefore, the coefficient of thermal expansion of a glass must be related to the relative proportion of a structural unit that each mole of constituent occupies in the glass. For instance, in the ${\mathrm{GeO}}_2\text{-}{\mathrm{SiO}}_2$ system, a ${\mathrm{SiO}}_4$ tetrahedron is substituted by a ${\mathrm{GeO}}_4$ tetrahedron. In the case of the ${\mathrm{P}}_2{\mathrm{O}}_5\text{-}{\mathrm{SiO}}_2$ system, the addition of phosphorous leads to the formation of single and double phosphorous centers in the silica network [27]. In Fig. 10 of the same reference, it is shown that as ${\mathrm{P}}_2{\mathrm{O}}_5$ content increases, there is an increase of double phosphorous centers, and a decrease of single phosphorous centers.

For this reason, we consider here ${\mathrm{P}}_2{\mathrm{O}}_5$ to progressively enter the silica network as double phosphorous centers, locally forming a “pair” of $\mathrm{O}={\mathrm{PO}}_3$ tetrahedral units bonded with a common oxygen. In the case of ${\mathrm{B}}_2{\mathrm{O}}_3\text{-}{\mathrm{SiO}}_2$, the substitution of one mole of ${\mathrm{SiO}}_2$ for one mole of ${\mathrm{B}}_2{\mathrm{O}}_3$ will result in the substitution of one ${\mathrm{SiO}}_4$ unit for two ${\mathrm{BO}}_3$ units in the glass network. Additionally, we consider ${\mathrm{Al}}_2{\mathrm{O}}_3$ to enter the network by forming ${\mathrm{AlO}}_4$ tetrahedral units. Therefore, to appropriately take these structural effects into consideration, Eq. (6) is modified to

If we consider the ${\mathrm{GeO}}_2\text{-}{\mathrm{SiO}}_2$, ${\mathrm{P}}_2{\mathrm{O}}_5\text{-}{\mathrm{SiO}}_2$, and $\mathrm{BaO}\text{-}{\mathrm{SiO}}_2$ binary systems, $T({\mathrm{SiO}}_2)=T({\mathrm{GeO}}_2)=T({\mathrm{P}}_2{\mathrm{O}}_5)=T(\mathrm{BaO})=1$. Again, the substitution of ${\mathrm{SiO}}_2$ for ${\mathrm{P}}_2{\mathrm{O}}_5$ results in the formation of a “pair” of $\mathrm{O}={\mathrm{PO}}_3$ units sharing a common oxygen. Therefore, $T({\mathrm{P}}_2{\mathrm{O}}_5)=1$, not 2, as the structural unit is “$2\times \mathrm{O}={\mathrm{PO}}_3$.” However, for a ${\mathrm{B}}_2{\mathrm{O}}_3\text{-}{\mathrm{SiO}}_2$ system, $T({\mathrm{B}}_2{\mathrm{O}}_3)=2$ since one mole of ${\mathrm{B}}_2{\mathrm{O}}_3$ leads to two distinct (i.e., not connected) ${\mathrm{BO}}_3$ structural units. For the same reason as ${\mathrm{B}}_2{\mathrm{O}}_3$, $T({\mathrm{Al}}_2{\mathrm{O}}_3)=2$ as we consider ${\mathrm{Al}}_2{\mathrm{O}}_3$ incorporated into ${\mathrm{SiO}}_2$ by forming independent ${\mathrm{AlO}}_4$ units. Obviously, some limitations can arise from the difficulty to estimate the structural role that each dopant plays in the glassy network, as it is typically a strong function of glass composition. Nevertheless, using these assumptions, the data can be fitted using Eq. (7), instead of Eq. (6). The new fits are displayed in Fig. 2, and the correlation with the data is excellent.

 

Fig. 2. Modified additivity prediction of the coefficient of thermal expansion ($\alpha$) in binary silicate glasses using Eq. (7), instead of Eq. (6), and considering the additivity of $\alpha$ to be a function of the variation of density per structural glass unit.

Download Full Size | PDF

Table 2 reports the determined values of the coefficient of thermal expansion for the glassy non-silica compounds using Eqs. (6) and (7), and are compared with data found in literature where appropriate.

Table 2. Estimated Coefficient of Thermal Expansion of Amorphous Materials, Based on Additivity of Eqs. (6) and (7)

View Table | View all tables in this article

Next, the utility in developing such a model is demonstrated with a practical example. As mentioned above, a novel BaO-doped optical fiber was previously developed [11] for applications requiring a reduced Brillouin gain coefficient. $dn/dT$ for the BaO glass constituent was determined for one of the fabricated fibers to be $1.940\times {10}^{-5}{\mathrm{K}}^{-1}$ (Fiber C, Table II in [11]). This value is considerably larger than that of ${\mathrm{SiO}}_2$ and other materials, suggesting that silicate fibers possessing baria may be more susceptible to thermal mode instabilities [4,5]. However, as is seen in Fig. 2, a barium silicate core fiber will have an $\alpha$ that can be much larger than that of a surrounding pure silica cladding. This effect was not taken into consideration in [11], and the $dn/dT$ value listed above for BaO simply represents how the network itself is modified due to its presence in silica as a binary system, but also within a cladded geometry. Its value is essentially invalid for use in more complicated systems such as in ternary glasses [31], especially if another dopant brings about additional changes to $\alpha$.

In principle, as the temperature of a fiber is increased, both the core and cladding will thermally expand. However, when ${\alpha }_{\text{core}}>{\alpha }_{\text{cladd}\text{in}g}$, the expansion of the core should be greater than that of the cladding. However, due to the presence of a much more voluminous and massive cladding surrounding it, thermal expansion of the core is restricted, and it appears as compressed relative to an unbound, freely expanding core. This pressure exerted on the core then also modifies the refractive index via the photoelastic effect [32,33]. This pressure causes a relative increase to the index of ${\mathrm{SiO}}_2$ (a positive-$p_{12}$ photoelasticity material) [34], whereas it results in a relative decrease in the index of BaO, since it is a negative-$p_{12}$ material [11,35]. Using the procedure outlined in [36] and, by incorporating the model for $\alpha$ derived here, a global $dn/dT$ value for the bulk BaO constituent can be determined.

In short, Eqs. (2) and (3) can be used to calculate the refractive index of a binary glass if the replacement $\rho \to n$ is made. To account for the $\alpha$ mismatch, it is shown [36] that $n_i$ can be modified to

This Letter proposes and validates a new model for the coefficient of thermal expansion additivity in the binary silicate glasses employed most frequently in fiber optic systems. This simplistic model can be used as a tool to predict the additivity of $\alpha$ for a variety of silicate-based glass compositions. Additionally, it was shown that a good prediction of $\alpha$ in the case of the $\mathrm{BaO}\text{-}{\mathrm{SiO}}_2$ system can be successfully used to determine $dn/dT$ for amorphous BaO by removing the $\alpha$ mismatch due to fiber geometry.