1. Introduction

With few exceptions, nonlinear optical fiber phenomena are a double-edged sword, metaphorically. On the one edge, they can be utilized for practical applications, such as in distributed Brillouin scattering-based sensing systems [1,2]. On the other, they represent leading limitations in certain types of fiber-based lightwave systems, such as Brillouin scattering in telecommunications [3] and fiber laser systems [4]. The control or management of these nonlinear phenomena is commonly accomplished through the implementation of appropriate waveguide designs (i.e., so-called specialty optical fibers). In the case of Brillouin scattering, classic specialty fiber methods to suppress it include varying core size [5] and/or dopant concentrations [6] along the fiber length. Active methods include applied temperature [7] and/or mechanical stress or strain gradients [8]. Each of these “longitudinal methods” serves to chirp the Brillouin spectrum along the fiber, thereby decreasing the peak gain experienced by the lightwave during the stimulated process (stimulated Brillouin scattering or SBS). More recently, “transverse methods” to suppress SBS were introduced [9–12] and, similar to the longitudinal methods, they give rise to a Brillouin gain spectrum (BGS) that is spread across a much wider range of frequencies. The enhancement of certain Brillouin characteristics for applications such as distributed sensing can also be achieved via appropriate fiber design [13,14].

More recently, methods to control Brillouin scattering through compositional engineering and the use of less conventional, though industrially practical, fiber fabrication methods [15] have been demonstrated [16–18]. With the appropriate combination of materials, optical fibers of much simpler geometries than the aforementioned specialty designs with Brillouin scattering frequencies that are immune to either temperature (athermal) or strain (atensic) can be realized. With this comes the potential to simplify distributed sensor systems utilizing fibers in environments where they are susceptible to both temperature and strain simultaneously. The reduction of the Brillouin gain coefficient (BGC or $g_B$) in compositionally tailored fibers is realized through the use of materials with large acoustic velocity, wide Brillouin spectral width, and low photoelastic constant. Furthermore it has been postulated that, with the appropriate combination of materials with oppositely signed photoelastic constants, the condition of zero Brillouin gain can be achieved (assigned the moniker zero Brillouin activity condition, or ZeBrA) [16].

As with the previously noted nonlinearity, the use of compositional tuning can also represent a double-edged metaphorical sword. For example, while a fiber can be designed to be immune to temperature, but not strain, giving rise to athermal distributed strain sensors, the requisite composition may have a greatly reduced BGC, thus necessitating the use of more power in such systems. As a result, the use of compositional tailoring to control Brillouin scattering requires knowledge of how the constituents of a multi-component glass influence all of the Brillouin and optical properties, and not just the refractive or acoustic [9] indices. A summary of several oxide materials and their effects on a number of Brillouin and optical properties can be found in [18,19]. Of particular interest toward SBS is the existence of materials with a negative photoelastic constant ($p_{12}$), which appears to be characteristic of alkaline earth metal oxides, lanthanoid oxides, and some post-transition metal oxides (e.g., alumina, ${\mathrm{Al}}_2{\mathrm{O}}_3$). These generalizations, however, must be validated with direct measurements of material characteristics for a broader palette of potential glass constituents [20].

In an effort to further understand the influence of rare-earth oxides, measurements are presented here that detail the effect of lanthana (${\mathrm{La}}_2{\mathrm{O}}_3$) on the Brillouin characteristics of silica-based oxide glass optical fibers. Lanthana is an interesting species to investigate since it possesses a wide transparency window covering the common fiber laser and telecom system wavelengths. Further, lanthana can be a surrogate glass component for the subsequent doping of active light absorbing or emitting species. As might be expected, it is found that the properties of lanthana are very similar to those of ytterbia (${\mathrm{Yb}}_2{\mathrm{O}}_3$) [21] in nearly all characteristics, namely, relatively low acoustic velocity, wide Brillouin spectral width, and a negative photoelastic constant.

However, lanthana is found herein to possess thermo-acoustic and strain-acoustic coefficients (acoustic velocity versus temperature or strain, and thermo-acoustic coefficients, TAC, and strain-acoustic coefficients, SAC, respectively) with signs that are opposed to those of ytterbia. The lanthano–aluminosilicate (SAL) fiber utilized in this study is athermal, but is not independent of the strain, which is believed to be, to the best of our knowledge, the first demonstration of such a glass fiber utilizing a compositional engineering approach. Ultimately, as will be shown with a diffusion analysis, the resultant fiber compositions are driven by the cladding material (silica) dissolving into the core (although the exact mechanism, diffusion versus dissolution versus mixing, etc., is not yet fully understood). Therefore, careful control of the fiber fabrication process is critical in achieving low-loss optical fiber with the requisite composition.

2. Optical Fiber

A. Fabrication and Basic Characterization

Based on the starting composition of 65 ${\mathrm{SiO}}_2$, 20 ${\mathrm{Al}}_2{\mathrm{O}}_3$, and 15 ${\mathrm{La}}_2{\mathrm{O}}_3$ (all in mole percent), SAL glass monoliths were prepared in 500 g batches using conventional melting approaches. Highly pure raw materials were crucible-melted and, for better homogeneity, initially processed into a fritted, or water quenched, glass. The re-melted frit and additional stirring produce a homogeneous, bubble-free glass. The thermal and optical properties of bulk SAL glass compared with commercial silica glass are described in [22]. The bulk glass samples were ground and polished into cylindrical preforms with a diameter of about 20 mm and a length of about 110 mm, to be used for single-material and core/clad optical fibers.

Bulk glass fibers with a diameter of about 125 μm were drawn from SAL preforms for basic optical characterization. Core/clad fibers were composed of a SAL core glass core and high-purity silica cladding. The major challenge is the adaption of highly lanthanum-doped glasses to the properties of silica concerning the glass transition temperature, thermal expansion coefficient, and refractive index. Due to their high glass transition temperatures and their moderate expansion coefficients, SAL glasses are suitable candidates for such a combination. They have been produced in a two-step drawing process using rod-in-tube (RIT) technology. In a first drawing step, the polished preform ($65{\mathrm{SiO}}_2–20{\mathrm{Al}}_2{\mathrm{O}}_3–15{\mathrm{La}}_2{\mathrm{O}}_3$) was stretched at a temperature of about 1300°C to SAL core rods, with an outer diameter of about 1 mm. In a second step, a structured preform consisting of SAL core rod and a double silica cladding (Heraeus, Suprasil F300 tubes with inner/outer diameter of $1.2/3.3\mathrm{mm}$ and $3.5/14\mathrm{mm}$, respectively) was drawn at a temperature of about 2000°C into the core/clad optical fiber. Finally, the fiber was coated with a single layer of high-index acrylate. The core/clad fiber investigated here possessed an outer diameter of about 125 μm and core diameter of about 10 μm. Figure 1 shows the attenuation spectrum of the core/clad optical fiber. The relatively high losses (minimum of about $0.7\mathrm{dB}/\mathrm{m}$ at 1200 nm) are based on the fabrication technology of the core glass. Despite the usage of high-purity raw materials, the losses in melt glasses are generally higher compared with vapor-phase technologies, due to impurities and structure imperfections (micronscale striae, scattering centers caused by surface treatment, etc.). The peak at 1413 nm is related to OH vibrations and slightly redshifted compared with low-doped silica or undoped quartz glasses.

 

Fig. 1. Loss spectrum of the hybrid fiber with SAL glass core and silica cladding.

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The refractive index profile (RIP) was measured by Interfiber Analysis (Livingston, New Jersey) at wavelengths of both 1000 and 1550 nm, using a spatially resolved Fourier-transform technique [23]. For the subsequent calculations derived from Brillouin scattering measurements at 1534 nm, the measured refractive index difference $\mathrm{\Delta }n$ at the wavelength 1550 nm was utilized. The measurement at 1000 nm was principally performed to get a sense of the dispersion in $\mathrm{\Delta }n$.

Compositional electron probe microanalyses (EPMA) of the fiber cross sections were performed using both energy dispersive (EDX) as well as wavelength dispersive (WDX) x-ray spectrometry on a JEOL microprobe JXA8800L. The energy of the exciting electrons was mostly set to 20 keV. The elemental uncertainty is around $\pm 0.01\mathrm{wt}\%$ in the case of the standard related quantitative WDX measurements. All fiber samples were coated with approximately 20 nm carbon prior to analysis to provide a conductive layer to mitigate charging effects from the glass. For high-resolution imaging of the fiber cross sections, a field emission scanning electron microscope JEOL JSM-6300F, with a YAG-detector for the backscattered electrons (BSE; compositional contrast), was used. Throughout the remainder of this paper, [${\mathrm{La}}_2{\mathrm{O}}_3$] is defined to be the ${\mathrm{La}}_2{\mathrm{O}}_3$ concentration in units of mole percent, unless otherwise stated. Figure 2 provides the results of the RIP measurements and the measured alumina and lanthana concentrations at the core center are provided in Table 1. In the final fiber, the alumina-to-lanthana ratio was approximately 1.21, and a slight decrease is observed in $\mathrm{\Delta }n$ with a decrease in the wavelength.

 

Fig. 2. RIPs measured at 1000 and 1550 nm (the latter has a slightly higher peak $\mathrm{\Delta }n$). The data taken at 1000 nm is shown with the dashed line.

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Table 1. Summary of Physical Properties of the SAL Fiber

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B. Diffusion Experiments

Based on observations of the decreasing numerical aperture during fiber fabrication, an investigation on diffusion and phase segregation was performed. These investigations were helpful for the understanding of compositional effects (e.g., refractive index, expansion coefficient, etc.) and the limits of hot processing steps during the fiber fabrication. Initially, hybrid fibers with a core composition of $65{\mathrm{SiO}}_2–20{\mathrm{Al}}_2{\mathrm{O}}_3–{15\mathrm{La}}_2{\mathrm{O}}_3$ (for active fibers lanthanum oxide can be partially substituted by ytterbium or other rare-earth oxides) and a silica cladding were fabricated. Calculating the NA from the starting materials, a value of 0.65 was expected; however, a value of 0.32 was instead measured by the refracted near-field (RNF) method. A RIP measurement on the neck down region showed the already strongly decreased refractive index (Figs. 3 and 4). This suggests a significant change of the core glass composition during the hot processing steps. The reduction in the core refractive index is likely due to the dissolution of lower-index silica from the cladding into the higher-index core. A loss measurement substantiated the idea of a “dilution” of the core material by high-purity silica. The change of the drawing conditions (changing of the preform feed rate from 1 to $0.4\mathrm{mm}/\min$) has shown a significant reduction of the fiber attenuation from 2.1 to $0.7\mathrm{dB}/\mathrm{m}$ at 1200 nm (Fig. 5). The results are in good agreement with diffusion investigations made by Dejneka et al. [24] and Cheng and Dejneka [25].

 

Fig. 3. Refractive index profile in the neck-down region of the hybrid fiber preform.

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Fig. 4. Refractive index profile of the hybrid fiber (RNF).

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Fig. 5. Comparison of fiber attenuation with identical starting compositions, but different drawing conditions (Note that an Yb-doped sample was used for the loss investigations to demonstrate the dwell time influence on loss properties).

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To investigate the influence of time and temperature on the core material, a cane ($\varnothing =2\mathrm{mm}$) was drawn from a SAL core glass and silica glass cladding. The core diameter was measured to be 0.1 mm. A sample with an initial core composition of $65{\mathrm{SiO}}_2–20{\mathrm{Al}}_2{\mathrm{O}}_3–15{\mathrm{La}}_2{\mathrm{O}}_3$ was utilized for these tests. It was not possible to simulate the drawing conditions regarding the temperature due to expected strong mechanical deformations and an unknown exact temperature inside of the drawing furnace. Therefore, the samples were tempered for 200 h at 1100°C. The temperature of 1100°C was chosen as one between the transformation temperature of the core glass (840°C) and the cladding material (1170°C) to prevent significant sample deformation. Before and after tempering, EDX spot analyses were made to determine the core composition.

The FE-SEM BSE images (Fig. 6) show the core clad interface of the hybrid fiber sample before and after tempering. Phase separation clearly occurs in the core region and consists of two layers. The outer layer (L1) is about 0.4 μm thick and ${\mathrm{SiO}}_2$ rich. The inner layer (L2) has a thickness of about 15 μm and shows drop-shaped phase separations embedded in a homogeneous matrix. Results of the EPMA investigations are shown in Table 2. A clear elemental analysis was only possible for the inner layer L2. The analysis shows a strong compositional change already occurring during the cane fabrication (${\mathrm{SiO}}_2$ enrichment). However, the additional tempering step (1100°C/200  h) initializes the phase separation mainly in the core periphery. As there was no phase separation observed in pure core glasses (even after several drawings), besides a certain temperature and time, a surplus of ${\mathrm{SiO}}_2$ is needed to initialize this process. This surplus of ${\mathrm{SiO}}_2$ can trace its origins to the cladding glass.

 

Fig. 6. BSE images showing the compositional contrast of an untreated sample and a tempered sample, initial core composition $65{\mathrm{SiO}}_2–20{\mathrm{Al}}_2{\mathrm{O}}_3–15{\mathrm{La}}_2{\mathrm{O}}_3$. Above: untreated sample. Below: 1100°C, 200 h.

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Table 2. EPMA Analysis of Different Core Areas of an Untreated and Tempered Sample (Canes)

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Yb-doped samples were also investigated (the majority of this data will be presented elsewhere). What was found is that the phase separation tendency is strongly dependent on the co-doping of the material. Measurements indicate that the phase separation tendency is much higher in ${\mathrm{Yb}}^{3+}$-rich samples.

3. Experimental Configurations

The experimental methodologies utilized in this study are identical to those found elsewhere [16,17,21] and, therefore, detailed descriptions will not be provided here. Instead, they are only briefly outlined as follows. The spontaneous BGS was measured utilizing a heterodyne approach. More specifically, it was measured by launching pump power into the fiber and the resulting backscattered signal was mixed with the pump on a fast square-law detector. The detector signal was subsequently investigated with an electrical spectrum analyzer to retrieve the BGS. Fibers were kept short ($\sim 2\mathrm{m}$) to minimize Brillouin spectral broadening due to any length-wise variations in the fiber. The dependence of the Brillouin frequency on temperature was measured by immersing the fiber in a thermally controlled heated water bath. The dependence of the Brillouin frequency on strain was measured by securing a segment of the SAL fiber at both ends with epoxy. One end was affixed to a micrometer-based precision translation stage (with the other end held rigidly), from which the applied strain could be carefully measured.

The measurements of the thermo-optic and strain-optic coefficients (TOC and $\epsilon \mathrm{OC}$, respectively) were achieved through the use of a ring laser configuration. In short, the SAL fiber was placed into the cavity of a fiber ring laser operating at 1550 nm (utilizing an erbium-doped fiber as the gain medium). Since the free spectral range of the laser is inversely proportional to the refractive index (modal), and the refractive index is a function of temperature or strain, measurements of the laser free spectral range as a function of temperature or strain disclose the TOC and $\epsilon \mathrm{OC}$. Clearly this assumes that the change in length (due to thermal expansion or strain) for each of the measurements is known and measured.

The Brillouin gain coefficient is estimated by comparing the strength of the spontaneously generated Stokes signal with that of a fiber of known Brillouin gain. The fiber was selected so that its BGS overlaps with neither that of the apparatus (Corning SMF-28 fiber) nor that of the SAL fiber. As a result, a ${\mathrm{P}}_2{\mathrm{O}}_5$-doped fiber (described in [26,27]) was utilized. The optical mode from this fiber also had good spatial overlap with the fundamental mode of the SAL fiber, such that mainly the fundamental optical mode could be excited. In mathematical terms, the Brillouin reflectivities for the two fibers were compared using the analysis in [28]:

4. Modeling Methodology

The models utilized in this work are also provided elsewhere [18–20] and will only be briefly discussed here. First, the assumption is made that the constituents in the present work (silica, alumina, and lanthana) are well-mixed and that the aggregate glass follows a law of mixtures (or additive model) of the form

The photoelastic constant ($p_{12}$) is calculated in a slightly different way. The Pockels coefficients (both $p_{11}$ and $p_{12}$) are assumed to carry through Eq. (5) via the dependence of the refractive index on the strain and stress-optic coefficients [30]. More specifically, the model assumes that the individual constituent strain optic coefficients ($\epsilon \mathrm{OC}=p_{12}-{\nu }_{\text{Poisson}}(p_{11}+p_{12})$, where ${\nu }_{\text{Poisson}}$ is the Poisson ratio, which is also assumed to obey Eq. (5), and stress optic coefficients ($\sigma \mathrm{OC}=(p_{11}-p_{12})/2$) are additive with an $n_i^3$ scaling factor, giving rise to the aggregate strain- and stress-optic coefficients. Once these aggregate coefficients are determined, the system of two equations can be used to find both $p_{11}$ and $p_{12}$ for the aggregate.

The optical fiber is not a homogeneous medium and, therefore, the measured data are that for the optical or acoustic mode (it will be shown later that this optical fiber acts as an acoustic waveguide). To calculate the effects of wave guidance on the data, a five-layer approximation is made to the profiles in Fig. 2. Each layer possesses a unique composition and, therefore, the aggregate physical characteristics (such as refractive index or acoustic velocity) calculated using Eq. (5) are unique for each layer of the approximation. Once the optical and acoustic index profiles are determined (and letting $n_i\to n_i(\epsilon ,T)$ and $V_i\to V_i(\epsilon ,T)$) the various physical quantities of the optical or acoustic mode can be determined. Since the present ternary system possesses silica, alumina, and lanthana, the assumption is made that the physical characteristics of the silica and alumina are those in Ref. [31], with the data for the sapphire-derived fiber [16] being utilized due to relatively large alumina content in the present SAL fiber. These values are reproduced later in the paper for the benefit of the reader. As a result, the physical properties of the lanthana component can be used as fit parameters with these values iterated until the calculation matches the measured data for the optical or acoustic modes. More detail can be found in [26,27].

5. Experimental Results and the Physical Characteristics of Lanthana

A. Refractive Index

The five-layer approximation to the RIP provided in Fig. 2 includes a central layer (2.5 μm radius) with a maximum $[{\mathrm{Al}}_2{\mathrm{O}}_3,{\mathrm{La}}_2{\mathrm{O}}_3]=[11.33,9.38]\mathrm{mol}\mathrm{.}\%$ (or about 4% lower than the measured value at the very peak). The refractive index difference of this layer of the approximation is 0.10 (the physical characteristics of the fiber, including measured ones, are tabulated in Table 1). The additive model is utilized in practice by specifying the volume occupied by a constituent in terms of the molar quantity and molar volume of the constituent. Since the molar volume is the molar mass divided by the mass density, the mass density of the lanthana constituent is required for the calculation. However, utilizing the observation that the mass density of a constituent of a glass fiber seems to be about 20% lower than its bulk crystalline value (assuming that a stable form exists) [18], this value can be estimated for the glassy lanthana component. Here, it is assumed to decrease in value in equivalent proportion to that observed for the bulk-to-glassy transition of ${\mathrm{Al}}_2{\mathrm{O}}_3$ for the sapphire-derived fibers [16], thus giving rise to a density of $5676\mathrm{kg}/{\mathrm{m}}^3$. These quantities, along with those of silica and alumina, are listed in Table 3. The use of the additive model, with the refractive index of lanthana used as a fit parameter, gives rise to $n_{\text{lanthana}}$ of approximately 1.877. At low concentrations, this results in a change in the refractive index of about $8.2\times {10}^{-3}$ per mol. % of lanthana in the silica–lanthana binary system. Utilizing the step-index approximation, the modal index at 1534 nm is calculated to be 1.5371.

Table 3. Summary of Physical Properties of the Various Glass Constituents

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B. Thermo-Optic and Strain- and Stress-Optic Coefficients

Figure 7 shows the data for the measurement of the strain-optic coefficient along with the model fit. The data for temperature are similar (very linear) and will not be reproduced here. An approximately 2 m segment of the SAL fiber was used as part of the ring laser cavity described above. The equation that governs the change in the free spectral range is

 

Fig. 7. Change in the free spectral range versus strain of a ring laser utilizing $\sim 2\mathrm{m}$ of SAL fiber. The strain-optic coefficient is determined from this data.

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The TOC is higher than that of pure silica ($1.042\times {10}^{-5}{\mathrm{K}}^{-1}$, [17]) and the $\epsilon \mathrm{OC}$ is lower (0.174 for pure silica, see Table 3). One interpretation of these results is that, if the threshold for the onset of modal instabilities observed in high-power large mode area fiber lasers is indeed proportional to the TOC [33,34], then the threshold will be lower in the SAL fibers relative to more conventional fibers (which generally possess more silica than the present SAL fibers). Regarding the $\epsilon \mathrm{OC}$, mode coupling in multimode fibers is known to be a function of bend-induced strain in the fiber, causing changes to the RIP locally along a fiber [35]. The change in refractive index due to strain is proportional to $n^3\times \epsilon \mathrm{OC}$. In the SAL fiber of the present investigation, the relative decrease in the $\epsilon \mathrm{OC}$ exceeds the relative increase in $n^3$ and, thus, one may conclude that this fiber will be less sensitive to bend-induced mode coupling than conventional fibers possessing more silica.

Next, assuming the TOC and $\epsilon \mathrm{OC}$ for pure silica and alumina found in Table 3, and utilizing the five-layer step index approximation to the RIP, the optical mode index may be calculated as a function of strain and temperature. This requires letting $n$ become a function of $\epsilon$ and $T$ for each layer in the additive model. The $\epsilon \mathrm{OC}$ and TOC for pure lanthana were used as fit parameters in the model and were iterated until the calculated $\epsilon \mathrm{OC}$ and TOC for the fiber matched the experimental data. It is determined that $\epsilon {\mathrm{OC}}_{\text{lanthana}}=+0.019$ (similar in magnitude to ytterbia [21]) and that ${\mathrm{TOC}}_{\text{lanthana}}=2.037\times {10}^{-5}{\mathrm{K}}^{-1}$ (about twice that of silica).

The stress-optic coefficient ($\sigma \mathrm{OC}$) can be measured by applying a twist to the fiber. A measurement of the rotation of the polarization of a linearly polarized light source launched into the twisted fiber (versus the number of twists) gives the stress-optic coefficient [36]. However, for this technique to work, the fiber must be free from birefringence. That was not the case in the fiber employed, as it was found to be polarization-maintaining (PM) and, therefore, the $\sigma \mathrm{OC}$ could not be measured directly. This can be explained by the presence of residual stress in the core, which probably also resulted in the slightly asymmetric shape of the core (see Fig. 8). It should be pointed out that the RIP in Fig. 2 represents an azimuthal average of the refractive index. Knowledge of both the $\sigma \mathrm{OC}$ and $\epsilon \mathrm{OC}$ enables the determination of $p_{11}$ and $p_{12}$ and, without the $\sigma \mathrm{OC}$, an alternative method to determine $p_{12}$ is needed. The approach here will be to estimate the BGC to determine $p_{12}$ (since the BGC is proportional to the square of this value). Given this and the $\epsilon \mathrm{OC}$, $p_{11}$ may also be determined. This will be shown later.

 

Fig. 8. SEM image of the SAL fiber core. A slight ellipticity is observed.

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C. Brillouin Gain Spectrum

The normalized BGS for the SAL fiber of this investigation is provided in Fig. 9. The slight asymmetry in the spectrum is attributed to minor excitation of higher-order optical modes [16] in the transition between the measurement apparatus fiber (Corning SMF-28) and the SAL fiber. No efforts were made to optimize the splice, with losses routinely falling well below 10% utilizing a standard telecommunications fusion splicer. A Lorentzian fit to the data is also provided (dashed line) with the result that the peak frequency $\nu =11.476\mathrm{GHz}$ and the Brillouin spectral width $\mathrm{\Delta }{\nu }_B=82.0\mathrm{MHz}$. With the knowledge that the Brillouin frequency is determined by the Bragg condition ($\nu =V/{\lambda }_a=2nV/{\lambda }_o$, where $n$ is the modal index and ${\lambda }_{a,o}$ are the acoustic and optical wavelengths, respectively) the acoustic velocity of the acoustic mode is found to be $5727\mathrm{m}/\mathrm{s}$. Since this velocity is less than that of the pure silica cladding, it may be concluded that the SAL fiber is acoustically guiding [37]. Finally, again invoking the five-layer approximation and calculating the acoustic mode velocity utilizing that of bulk lanthana as a fit parameter, the best fit value is $V_{\text{lanthana}}=3979\mathrm{m}/\mathrm{s}$; a value again very similar to that of ytterbia [21].

 

Fig. 9. Measured BGS for the SAL fiber at 1534 nm. The small peak near 11.1 GHz is identified as the $L_{02}$ acoustic mode in the apparatus fiber. The remaining structure to the red side of the peak is due to higher-order optical modes in the SAL fiber [16]. The dashed line is a Lorentzian fit to the data.

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The spectral width for the measured acoustic mode was modeled using the following expression [38]:

D. Strain and Thermo-Acoustic Constants

Figure 10 shows the dependence of the Brillouin frequency shift on the applied strain. The dashed line is a linear fit to the data, possessing a slope of 164 MHz/% (or $16.4\mathrm{GHz}/\mathrm{\epsilon }$, if strain is fractional elongation). This is considerably lower than typically observed for conventional fiber [39], but is not negligibly small. To determine the influence of lanthana, once again the data for the alumina and silica components shown in Table 3 are used. Since the SAC was not measured for the sapphire-derived fiber [16], and it appears to be independent of the density of the alumina constituent [31], the average of the measured values in [31] is listed in Table 3, and this was used for the calculations. Since straining the fiber not only changes the acoustic velocity but also the refractive index, the $\epsilon \mathrm{OCs}$ in Table 3 must be utilized in the simulation [27]. Again, utilizing the value for lanthana as a fit parameter, it is found that ${\mathrm{SAC}}_{\text{lanthana}}=-19.7\mathrm{km}/\mathrm{s}/\mathrm{\epsilon }$ (where $\epsilon$ is the fractional elongation).

 

Fig. 10. Brillouin frequency shift (GHz) as a function of the applied strain (in %). The linear fit (dashed line) shows the data (points) are very linear.

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Silica is known to have an anomalous dependence of the acoustic velocity on pressure, in that it decreases rather than increases with increasing pressure [40–42] and, thus, application of strain causes the velocity to increase. Both alumina and lanthana appear to have a dependence of the acoustic velocity on strain that is “normal,” or negative, and all appear to have values that are roughly similar in magnitude. In addition, lanthana was found to have a dependence that has a sign opposite to that of ytterbia [21]. Since both species are rare-earth oxides, one would have expected them to behave similarly; unfortunately, the origin of this difference is currently not understood. Finally, the results may be extrapolated to determine the atensic composition to be $1.21{\mathrm{Al}}_2{\mathrm{O}}_3–1.00{\mathrm{La}}_2{\mathrm{O}}_3–4.60{\mathrm{SiO}}_2$ (molar), if one assumes that the alumina content is 1.21 times the lanthana content as in the present fiber. As will be discussed in the next paragraph, this composition is expected to have a thermal dependence of the Brillouin frequency that is negative. In general, if $[{\mathrm{Al}}_2{\mathrm{O}}_3]\ne 1.21\times [{\mathrm{La}}_2{\mathrm{O}}_3]$, then a wide range of ternary lanthanum aluminosilicate compositions satisfy the atensic condition.

The data for the dependence of the Brillouin frequency as a function of temperature are also linear and, therefore, the data will not be reproduced here. However, the slope of the linear fit to the data ($0.037\mathrm{MHz}/\mathrm{K}$) represents a fiber that is nearly athermal. Such as observed with the aluminosilicate [16] and bariosilicate [17] systems, combining materials whose dependencies of the Brillouin frequency on temperature have opposite signs can give rise to athermal fibers. Performing an analysis similar to that for the SAC, the TAC for lanthana can be determined to be ${\mathrm{TAC}}_{\text{lanthana}}=-0.80\mathrm{m}/\mathrm{s}/\mathrm{K}$. Utilizing this value, the thermal coefficient for the atensic composition identified in the previous paragraph is extrapolated to be $-0.48\mathrm{MHz}/\mathrm{K}$.

The TAC values for alumina were determined in [31] to be much larger than would be expected given data published in the literature [43,44]. Specifically, the TAC for glassy alumina in fibers derived from sapphire was roughly five times or more (depending on the composition) larger than what is seen for its crystal counterparts. When comparing the glassy and crystalline states, it is reasonable to expect physical properties should be at least similar. Thus, such as with the refractive index or density, changes of more than 10%–20% to the magnitude of a particular value would be unexpected. In [31] it was postulated that a thermal expansion (${\alpha }_L$) mismatch between the core and cladding may be the culprit for the enhanced TAC. Thus, if the thermal expansion of the glass is known, it is possible to estimate its contribution to the thermal coefficient of Brillouin scattering.

Fortunately, for the SAL network, the thermal expansion characteristics have been measured for some compositions of the bulk glass precursors [45]. Thus, the thermal expansion coefficient of the core can be estimated. Figure 11 shows a graph of the influence of the addition of lanthana to the thermal expansion coefficient of the lanthanum aluminosilicate glass (bulk) at constant $[{\mathrm{Al}}_2{\mathrm{O}}_3]=20\mathrm{mol}\mathrm{.}\%$ (along with a linear fit). Assuming that the extrapolated zero-lanthana thermal expansion ($2.08\times {10}^{-6}{\mathrm{K}}^{-1}$) is that of a $20{\mathrm{Al}}_2{\mathrm{O}}_3–80{\mathrm{SiO}}_2$ glass, alumina has a thermal expansion coefficient of about $7.7\times {10}^{-6}{\mathrm{K}}^{-1}$ [assuming that ${\alpha }_L$ obeys the additive model given by Eq. (5) and taking ${\alpha }_L$ for silica to be $0.54\times {10}^{-6}{\mathrm{K}}^{-1}$, as before]. Fitting to the slope in Fig. 11, the thermal expansion coefficient of lanthana is approximated to be $12.3\times {10}^{-6}{\mathrm{K}}^{-1}$. Thus, using the concentrations at the center of the fiber (in the first layer of the step-wise approximation), the thermal expansion coefficient of the core is estimated to be $3.42\times {10}^{-6}{\mathrm{K}}^{-1}$.

 

Fig. 11. Thermal expansion coefficient of lanthanum aluminosilicate glass as a function of lanthana concentration at constant ${\mathrm{[Al}}_2{\mathrm{O}}_3]=20\mathrm{mol}\mathrm{.}\%$.

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Hence, the core will tend to thermally expand at a rate of $3.42\times {10}^{-6}{\mathrm{K}}^{-1}$; however, it is held rigidly in place by the cladding (pure silica) of the fiber, which has a thermal expansion coefficient of $0.54\times {10}^{-6}{\mathrm{K}}^{-1}$. Thus, in the longitudinal direction, this restricted expansion could be likened to a negative strain or positive compression of the fiber. Since most of the core is silica, and silica has an anomalous response to pressure (velocity decreases with increasing pressure), the thermally induced negative strain will counterbalance the positive TAC and result in a reduced sensitivity to temperature. To estimate this, the negative strain imparted due to the thermal expansion mismatch is taken to be $\epsilon =-({\alpha }_L^{\text{core}}-{\alpha }_L^{\text{cladding}})\mathrm{\Delta }T$, where $\mathrm{\Delta }T$ is the change in temperature. The temperature derivative can then be found as $d\epsilon /dT=-({\alpha }_L^{\text{core}}-{\alpha }_L^{\text{cladding}})$. Multiplying this by the SAC ($dV/d\epsilon$) of silica then gives $dV/dT=-0.08\mathrm{m}/\mathrm{s}/\mathrm{K}$ due to the negative strain induced by the thermal expansion mismatch. Thus, the negative strain decreases the effective TAC of the fiber (if it is mostly silica) by about 15%; not an insignificant value, and this is only in the longitudinal direction.

In the radial direction, the thermal expansion mismatch imparts a positive stress on the fiber. This stress can be approximated from the compressibility (inverse of the bulk modulus $G$ for the glass). The change in volume is the pressure (stress) divided by $G$. Since the stress is in the radial direction, the change in volume can only be attributed to a stress-induced change in the cross sectional area. If the core were loose (not cladded), then its area as a function of the change in temperature would be $A_{\text{core}}=\pi {(a+{\alpha }_L^{\text{core}}a\mathrm{\Delta }T)}^2$, where $a$ is the core diameter. Pulling the core diameter out of this equation, then expanding this expression and assuming that $2{\alpha }_L^{\text{core}}\mathrm{\Delta }T\gg {({\alpha }_L^{\text{core}}\mathrm{\Delta }T)}^2$, the change in volume versus temperature is approximately $(d\text{Volume}/dT)(1/\text{Volume})\approx 2{\alpha }_L^{\text{core}}$. Since this expansion is restricted by the cladding, the pressure (stress) on the core as a function temperature is $(dP/dT)=2({\alpha }_L^{\text{core}}-{\alpha }_L^{\text{cladding}})G$. Using $G$ for pure silica ($\sim 37\mathrm{GPa}$), $dP/dT$ is estimated to be about $210\mathrm{kPa}/\mathrm{K}$. Taking the pressure dependence of the acoustic velocity to be $-7.5\times {10}^{-4}\mathrm{m}/\mathrm{s}/\mathrm{k}\mathrm{Pa}$ (silica) [42], the change in acoustic velocity as a function of temperature is calculated to be $-0.16\mathrm{m}/\mathrm{s}/\mathrm{K}$. Combining this result with that of the negative longitudinal strain, and keeping in mind that several assumptions were made here, the thermal expansion mismatch alone in the current composition has offset the bulk thermal response of Brillouin scattering by more than 40%. An underestimated thermal expansion coefficient for the core of the SAL fiber means that this value is higher (and vice versa).

Thus, the effect of the thermal expansion mismatch is not one that enhances the influence of alumina or lanthana. The expansion of these species results in a positive pressure that decreases the acoustic velocity of silica (the elastically anomalous material) with increasing temperature and, thus, offsets the increased velocity due to the positive TAC. This has the effect of decreasing the sensitivity of the peak Brillouin frequency to temperature. Or, stated another way, the alumina and lanthana content required for an athermal fiber is less for the core of a silica-clad fiber than what is needed for the bulk material. It is noted that there is also an influence of these pressures on the refractive index, but it is far less significant than their influence on the acoustic velocity. Although the discussion provided here has been relatively short, a more rigorous analysis will follow in [46].

E. Pockels Coefficients and Brillouin Gain

Next, the Pockels photoelastic coefficients are investigated. The Pockels coefficients ($p_{11}$ and $p_{12}$) may be determined from knowledge of both the $\epsilon \mathrm{OC}$ and $\sigma \mathrm{OC}$. However, since the latter could not be measured, $p_{12}$ is instead determined from an estimate of the Brillouin gain coefficient. The BGC is estimated by comparing the Brillouin reflectivity (see “Experimental Configurations”) of the SAL fiber to that of one with a measured gain. Two meters of fiber are used in both cases. The relevant properties for the ${\mathrm{P}}_2{\mathrm{O}}_5$-doped fiber [27] used here, along with those of the SAL fiber, are provided in Table 4. Since the SAL fiber has a loss that is significant on the order of the test fiber length (2 meters), an effective length is determined by integrating a double-pass loss (forward propagating loss to the pump combined with the backward propagating loss to the Stokes wave) to obtain

Table 4. Parameters Used for Estimating the Brillouin Gain of the SAL Fiber

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To make the comparison, the ${\mathrm{P}}_2{\mathrm{O}}_5$-doped fiber was first spliced to the Brillouin measurement apparatus and the SAL fiber was then spliced to that. Using cutback, the splice loss was determined to be about 0.3 dB. Any splice loss between the ${\mathrm{P}}_2{\mathrm{O}}_5$-doped fiber and apparatus fiber is irrelevant since signals from both fibers experience this same loss and a relative measurement of $R_B$ [see Eq. (1)] is the goal here. The Brillouin scattered signal is shown in Fig. 12 with the key features identified. $R_B$ for the SAL fiber is about $0.48\times$ that of the ${\mathrm{P}}_2{\mathrm{O}}_5$-doped fiber, leading to a fitted $g_B$ for the SAL fiber of $0.26\times {10}^{-11}\mathrm{m}/\mathrm{W}$. It is noted that, in [27], the measured gain was lower than what is specified in Table 4. That measurement was for a much longer fiber ($\sim 50\mathrm{m}$) and, therefore, lengthwise fiber variations resulted in a reduction in the peak gain. This was corrected for the Table 4 data.

 

Fig. 12. Brillouin spectrum for 2 m of SAL fiber spliced to 2 m of ${\mathrm{P}}_2{\mathrm{O}}_5$ fiber spliced to the Brillouin apparatus. The small peaks associated with the ${\mathrm{P}}_2{\mathrm{O}}_5$ fiber are higher-order acoustic modes. Similarly, the small peak near 11.15 GHz is the $L_{02}$ acoustic mode in the apparatus fiber co-contributing with a higher-order mode from the ${\mathrm{P}}_2{\mathrm{O}}_5$ fiber. The structure just to the red side (near 11.35 GHz) of the SAL fiber feature is scattering from the cladding of the ${\mathrm{P}}_2{\mathrm{O}}_5$ fiber.

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Utilizing the fiber parameters provided in Table 1, Eq. (6) can be solved for $p_{12}$ for the SAL fiber, giving rise to a value of 0.141. Again, using the lanthana Pockels coefficients as fit parameters in the model described above, and assuming the silica and alumina values in Table 3, it is obtained that $p_{11}=-0.162$ and $p_{12}=-0.027$. These values are smaller in magnitude to those deduced for ytterbia [21] but, as with ytterbia, are both negative. Propagating the uncertainty associated with the alumina constituent gives rise to an uncertainty of $\pm 0.009$ in $p_{11}$ and $\pm 0.005$ in $p_{12}$ (this is listed in Table 3). Furthermore, if $g_B$ (as determined here) for the SAL fiber were underestimated by 10%, then the resultant $p_{12}$ (for the lanthana component) would take on a value of 0.018. It is interesting to note that the magnitude of $p_{12}$ for lanthana reported here is very similar to that reported for yttria [47]. Finally, the Poisson ratio for ytterbia was utilized in these calculations (as listed in Table 3), since this value could not be identified for lanthana.

Finally, and for completeness, a plot of the Brillouin gain coefficient for the lanthanum aluminosilicate system is provided in Fig. 13, with the condition that $[{\mathrm{Al}}_2{\mathrm{O}}_3]=1.21\times [{\mathrm{La}}_2{\mathrm{O}}_3]$ and using the data from Table 3. It is on a log scale and plotted relative to a value of $2.5\times {10}^{-11}\mathrm{m}/\mathrm{W}$ (a value typical of conventional telecom-grade single-mode fiber [18]). For both athermal and atensic compositions (identified in Fig. 13), the Brillouin gain coefficient is decreased significantly (about 10 dB and 13 dB, respectively) relative to fibers typically utilized in sensor applications. Clearly, this introduces a trade-off in distributed sensing applications in that atensic or athermal fiber systems require more optical power. On the other hand, these decreases in the Brillouin gain are significant and desirable for high-power narrow-linewidth systems. Along this line of thinking, the negative $p_{12}$ contribution by lanthana gives rise to a ZeBrA composition ($[{\mathrm{Al}}_2{\mathrm{O}}_3]\approx 44.9\mathrm{mol}\mathrm{.}\%$ and $[{\mathrm{La}}_2{\mathrm{O}}_3]\approx 37.1\mathrm{mol}\mathrm{.}\%$). Again, if $[{\mathrm{Al}}_2{\mathrm{O}}_3]\ne 1.21\times [{\mathrm{La}}_2{\mathrm{O}}_3]$, then a wide range of ternary lanthanum aluminosilicate compositions give rise to the ZeBrA condition.

 

Fig. 13. Brillouin gain coefficient calculated for the lanthanum aluminosilicate system (bulk) utilizing the parameters in Table 3 and assuming that ${\mathrm{[Al}}_2{\mathrm{O}}_3]=1.21\times [{\mathrm{La}}_2{\mathrm{O}}_3]$.

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6. Conclusion

A lanthanum aluminosilicate optical fiber was characterized for various Brillouin-specific parameters. Assuming previously determined characteristics of silica and alumina, the physical properties of lanthana were determined. Its large mass density, wide Brillouin spectral width, and negative Pockels coefficient cause a reduction in the Brillouin gain relative to pure silica, whereas its large refractive index and low acoustic velocity (lower than silica) cause a relative increase in $g_B$. Lanthana is found to have a thermo-optic coefficient larger than silica and a strain-optic coefficient that is lower. Both the dependence of the acoustic velocity on strain and temperature are negative-valued, with the former of similar magnitude to that of silica (but with opposite sign). The acoustic velocity versus temperature is probably overestimated due to a positive pressure imparted on the core, resulting from a thermal expansion coefficient mismatch (core having a higher value than that of the cladding). This overestimate has its origins in the acoustic velocity of the dominant silica component being decreased with increasing pressure. This is in opposition to the increase in velocity due to the thermo-acoustic coefficient. This phenomenon is, therefore, found to give rise to a lower required content of lanthana and alumina to achieve an athermal composition.