1. Introduction

The growing demand for optical fibers with enhanced performance, especially as relates to high power laser systems, has led to a renaissance in fiber materials [1] and waveguide designs [2]. These two principal design considerations are not mutually exclusive and are, indeed, symbiotic as there are long established connections between the impact of the fiber’s core glass composition and the optical and acoustic properties that lead to an optimum design. The most canonical connection between fiber materials and waveguide design is whether a given dopant raises or lowers the effective refractive index of the resultant glass when added to the predominantly silica host; see, for example Fig. 4.2 in Ref. [2]. Secondary considerations, though no less important in high energy fiber applications, are how said dopants influence nonlinearities such as Brillouin and Raman scattering, and transverse mode instabilities [3,4].

There are a greater number of index-raising additives that can be doped into silica than index-lowering ones. This is further compounded by the processes used to fabricate silica optical fiber preforms, which further limits the available index modifying compounds and their doping concentrations into silica due to the very high processing temperatures [1]. Index-lowering compounds are important because, as noted, most additives raise the index. Thus, having species to counter this trend is critical to controlling the refractive index profile in the fiber, hence related fiber characteristics, such as the number of propagating modes and mode field diameter.

Commonly employed index-lowering species are fluorine (F) and boria, B₂O₃ [5–8]. Another index-lowering compound that has grown in importance is AlPO₄ which results during preform fabrication from the reaction of P₂O₅ and Al₂O₃ [9]. AlPO₄ is isostructural with SiO₂ in the glass matrix and, in addition to lowering the effective refractive index, also decreases the glasses’ thermo-optic coefficient and Brillouin gain coefficient [10].

Recently reported in Xiang, et al., is molecular modeling of silica glasses doped with AlPO₄ and BPO₄ [11]. Their theoretical results suggest that doping SiO₂ with BPO₄ results in a lower refractive index than when silica is doped with AlPO₄. Indeed, having another index-lowering dopant to silica would be noteworthy. Unfortunately, this computational result is not consistent with experimental literature dating back at least to 1950, nor does it agree with straight-forward additivity models. This Comment summarizes the literature reports on glass in the BPO₄-SiO₂ system and corroborates those historical experimental results with simple additivity relations.

2. Results and discussion

Xiang, et al., employ ab initio molecular dynamic simulations to construct SiO₂-AlPO₄ and SiO₂-BPO₄ glass structural models. Density functional theory was then used to estimate various properties, including refractive indices. Major findings therein include that “The refractive index of the SiO₂, SiO₂-AlPO₄, and SiO₂-BPO₄ are 1.4752, 1.4856, and 1.4766 at 760 nm” (page 942) and “It can be found that the refractive index of the SiO₂-BPO₄ is lower than that of the SiO₂-AlPO₄, as shown in Figures S8 and S9, which is consistent with the above calculation results (96 atoms)” (supplemental information). For the purposes of this Comment, the BPO₄ and AlPO₄ concentrations in Xiang, et al., are interpreted as being about 3.2 mole percent, though the value is never explicitly stated. This is inferred from the simulations starting with 96 atoms (32 SiO₂ units), from which two SiO₂ units are replaced by an XPO₄, yielding a presumed concentration of 100/(32-1) mole% XPO₄. Arguably, this is relatively light doping.

The refractive index along the BPO₄-SiO₂ join will be discussed in greater detail below. However, it is worth noting that the stated index trends cannot be right on their face because the addition of AlPO₄ into SiO₂ has been known for almost 40 years to result in a lower refractive index [9]. Xiang, et al., compute the refractive index of AlPO₄ doped SiO₂ to be higher than SiO₂ across the entirety of the visible spectral range (see Fig. 6 in Ref. [11]).

2.1 Established results in the literature

The refractive index of SiO₂ glass is well established and best fit by the Sellmeier relationship given in Malitson [12]. At a wavelength of 760 nm, the refractive index should be about 1.455, measurably lower than the 1.475 modeled in Xiang, et al. The refractive indices for AlPO₄ and BPO₄ are slightly more difficult to define since they are not classical glass forming compounds. As crystals, like quartz (crystalline SiO₂ at equilibrium ambient conditions), AlPO₄ (Berlinite) and BPO₄ are uniaxial, possessing two refractive indices. The ordinary and extraordinary refractive indices for BPO₄ were measured on high quality single crystals and provided as raw data and in Sellmeier form by Li, et al. [13]. Based on this experimentally determined data, the Sellmeier forms yield refractive indices of 1.594 (ordinary) and 1.588 (extraordinary) at a wavelength of 760 nm. Corroboration of the BPO₄ refractive index dates back to 1950 [14], which places the refractive indices between 1.595 and 1.601. Though the exact wavelength of measurement was not specified in those works, given the publication date, the values would almost certainly have been in the visible.

More importantly for this Comment, glasses along the BPO₄-SiO₂ join were fabricated over a nearly complete composition range and their refractive indices reported in 1955 [15]. Clearly shown there, and included in Fig. 1 here, is that the refractive index along the BPO₄-SiO₂ join increases markedly with the addition of BPO₄ into SiO₂, in opposition to the assertion in Xiang, et al., that “the refractive index of the SiO₂-BPO₄ is close to the pure silica, thus not altering the refractive index profile of the silica-based fiber core” [11]. Additionally, as noted, the refractive index of AlPO₄ doped SiO₂ decreases with increasing AlPO₄ concentration. Calculations found in Xiang, et al., suggest instead that the index increases.

 

Fig. 1. (a) Refractive index along the BPO₄-SiO₂ join including experimental data on glasses [15], calculated values based on additivity of the end-member indices, and theorized results [11]. Also included for reference are the deduced values for the AlPO₄-SiO₂ join based on modeled and experimental measurements [10]. (b) Deduced refractive index along the BPO₄-SiO₂ join overlaid by the experimental data [15]. Connecting lines are guides-to-the-eye.

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2.2 Material additivity results

The stated bulk indices of BPO₄ and SiO₂ would naturally suggest that the refractive index of a composition within the binary (BPO₄-SiO₂) system would be somewhere between the end-member values. Since there are no known or presumed reactions between the BPO₄ and SiO₂, unlike the case between Al₂O₃ and P₂O₅ yielding AlPO₄, material property additivity should apply and provide a reasonable estimation of refractive index, among other properties, including the density [16]. To illustrate this, linear additivity is applied to the Ref. [15] data shown in Fig. 1(a). The additivity of the refractive index is quite simple. The aggregate glass takes on a net refractive index that is the weighted sum of those of the constituents (BPO₄ and SiO₂). The weighting is against the total volume fraction occupied by each constituent. Converting to mole% from volume fraction requires knowledge of the molar volume, or alternatively the mass density. As it turns out, from a reasonably complete set of compositional data, the latter can be estimated. The procedure is as follows: The base SiO₂ refractive index was set to that of Ref. [15] (1.456, note that a wavelength was not provided) and its density was assumed to be 2200 kg/m³ as in [11]. Then, both the refractive index and density for glassy BPO₄ can become fit parameters in the model. The former value sets the endpoint of the fitting (where the BPO₄ concentration is 100%) and the latter sets the curvature of the graph. Due to some variation in the data, a range of densities 2640 ± 300 kg/m³ around n_BPO4 = 1.545 best fits the data (the extremes are shown plotted in Fig. 1(b)). In the modeling, a single BPO₄ unit has a molar mass of 105.78 g/mol and the reaction P₂O₅ + B₂O₃ → 2BPO₄ is assumed to take place with 100% efficiency. In comment, and to a large extent as expected, both the refractive index and density of glassy BPO₄ are somewhat lower than what is observed in the crystalline phase (density ∼2800 kg/m³ [17]). However, both are significantly lower in pure SiO₂. Using the silica values from [11] and 2640 kg/m³, n_BPO4 = 1.504 is deduced from [11]. Similarly, using a density of 2233 kg/m³ for AlPO₄ [10], n_AlPO4 = 1.640 is deduced from [11]. In short, the former index is lower and the latter much higher than reported literature values.

A similar set of refractive index data can be found in a more recent paper [18]. That data cuts across the compositional range (30-x)P₂O₅-xB₂O₃-70SiO₂ and index measurements were performed in the visible. That data exhibits a discontinuity at the BPO₄ join and the system is well modeled as a quaternary glass (SiO₂, P₂O₅, B₂O₃, BPO₄) with index and density values being (1.462, 1.549, 1.452, 1.540) and (2200, 2390, 1820, 2640) kg/m³, respectively. Again, it is assumed that the reaction P₂O₅ + B₂O₃ → 2BPO₄ is 100% efficient. Due to limited space, that fitting will not be shown here. However, it is worth noting that the two data sets, published 50 years apart, corroborate extremely well indicating that the impact of doping BPO₄ into SiO₂ has been known for some time, but so far perhaps overlooked by the broader optical fiber community. Being experimentalists, the authors of this Comment are not positioned to know the source of uncertainties that can be introduced in molecular dynamics modeling. That said, this is a cautionary note of the possible impact of purely theoretical and modeling efforts absent thorough literature analysis or simple additivity calculations.

3. Conclusions

This Comment aims to correct statements and conclusions made in the recent work by Xiang et al. [11]. Based both on direct experimental evidence and as illustrated with additivity relationships on selected experimental results, the refractive index along the BPO₄-SiO₂ join is higher than that for AlPO₄-SiO₂ and is expected to significantly raise, not lower, and the addition of AlPO₄ into SiO₂ reduces, not raises, the refractive index.