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We present measurements and modeling of the effect of P2O5 doping on the acoustic damping and temperature sensitivity coefficients of silica fibers. In particular, the Brillouin gain spectrum of a highly P-doped fiber is measured and investigated at different temperatures. It is found that the acoustic damping coefficient (proportional to the Brillouin spectral width) of phosphorus pentoxide (1.41 × 105 m−1 for bulk P2O5 at 11 GHz) is similar to, but larger than, that of germanium dioxide. Additionally, the acoustic velocity (and thereby the Stokes’ shift) is found to be much less dependent on temperature in P2O5 ( + 0.12 m/s/°C) than in SiO2 ( + 0.56 m/s/°C). Using these coefficients (the thermo-acoustic coefficients), the modeled and unique slopes of the Stokes’-shift-versus-temperature curves for the four observed acoustic modes each lie within 3% of the measured values. Finally, utilizing both the thermo-optic and thermo-acoustic coefficients, a design example is presented where a composition is determined for which the dependence of the Brillouin frequency shift on temperature is minimized. In this example, the calculated temperature sensitivity is less than 5 kHz/°C over the temperature range −100 °C < T < 100 °C for the molar composition 0.54P2O5:0.46SiO2.
©2011 Optical Society of America
1. Introduction
Recently, novel optical fibers with tailored acoustic profiles have been designed specifically for applications where Brillouin scattering may be encountered, whether requiring its suppression [1,2] or utilization [3]. Clearly imperative in the process of designing an acoustic profile is the knowledge of how potential dopants influence the relevant acoustic parameters in a ‘host’ material, such as silica. While there can be found publications in the literature that show how some common dopants influence the acoustic velocity in SiO2 [4], much less has been done to characterize how the dynamic viscosity (Brillouin spectral width) and both the fiber thermo-acoustic and thermo-optic coefficients (TAC and TOC, respectively) are influenced by these dopants [5]. This data is particularly important, for example, when the large acoustic damping coefficients associated with some dopants can be used as a degree-of-freedom in the design of a specialty fiber. Furthermore, understanding the TACs and TOCs of the various dopants can lead to the development of novel optical fibers for distributed sensor applications.
Recently [6], we presented an analysis that began with the measurement of the Brillouin gain spectrum (BGS) of a heavily germanium-doped optical fiber. From the measured BGS, and a fit to a simple materials model [7] for Ge-doped fibers, we were able to extract an acoustic attenuation coefficient of about 1.38 × 105 m−1 for bulk GeO2 at an acoustic frequency of 11 GHz [7]. This value was later revised downward to 1.11 × 105 m−1 after a more accurate compositional profile was obtained [8]. This measurement was partly enabled by a large variation in the viscosity profile due to a central burn-out characteristic of fibers manufactured via the modified chemical vapor deposition (MCVD) process. This acoustic attenuation value increases with acoustic-frequency-squared for Ge-doped fibers [9]. This data point, in addition to other basic materials parameters (including the acoustic velocity and mass density) for the bulk material, can then be used to calculate and design sets of relevant acoustic profiles for arbitrary compositional profiles, and can be extended to include systems of multiple dopants.
Here, we present a similar analysis for a P2O5-doped optical fiber, and refer the reader to [6] for details such as the model and experimental setup. The present fiber preform is characterized by a similar central index dip, and has fairly sizeable doping (~14.3 mol% oxide at the peak). However, in contrast to the Ge-doped fiber studied previously, it will be shown that the central burn-out becomes ‘filled-in’ due to phosphorus diffusion during the fiber draw process. Thus, the test fiber does not contain a significant index dip. It is found that, for the purposes of modeling P-doped fibers, including P-doped inner cladding layers, bulk phosphorus oxide has an acoustic attenuation coefficient (αP) of about 1.411 × 105 m−1 at an acoustic frequency of 11 GHz. This value is similar to, but larger than, that of bulk GeO2. Both of these values are approximately 10 times larger than of silica [8]. The spectral width of the L01 acoustic mode is found to decrease at a rate of ~−77.7 kHz/°C.
In addition, the results show for the L01 acoustic mode that the Stokes’ shift increases at a rate of ~ + 0.74 MHz/°C. Utilizing similar measurements on each of the four observed acoustic modes located in the core, we find the TAC (of the bulk acoustic velocity) of P2O5 to be about + 0.119 m/sec/°C; a value less than that of silica. In order to obtain this coefficient from the P and Si co-doped oxide glass, a pure SiO2-core (Sumitomo Z-FiberTM) is first tested in order to obtain the temperature-dependence of the pure silica component. Then, the SiO2-P2O5 system is modeled in an additive way to determine the P2O5 value. The TAC value for silica, + 0.555 m/sec/°C, is in good agreement with previous measurements on the bulk material [10,11], providing a large degree of confidence in the P2O5 value. We will also show that each acoustic mode has a unique dependence of the Brillouin frequency shift on temperature due to differing spatial distributions. Finally, using the determined coefficients, a design example is presented where a composition is found whose Stokes’ shift dependence on temperature is minimized in the range −100 °C < T < 100 °C.
2. Experimental details and optical fiber
A. Optical Fiber
The optical fiber used in this set of experiments is one with a P2O5-doped silica core fabricated by drawing the corresponding optical preform. The fiber was then coated on-line by standard UV-cured acrylate. The silica fiber preform was fabricated by the MCVD process. A combination of a small MCVD substrate tube, some matched cladding layers, and a large preform core size helped lower the final collapse temperature which maximized the P2O5 content in the silica core to 14.3 mol %.
The tube outside diameter was 18 mm and the inside diameter was 15 mm. The matched cladding consisted of 21 P2O5 and F co-doped silica layers. The core consisted of 12 P2O5 doped silica layers. These layers are visible in the refractive index profile (RIP) provided in Fig. 1 , which was measured using a York P102 preform analyzer.
The preform was drawn into a cane for reducing the core size. The chemical compositional profiles of the cane were measured by the electron probe microanalysis (EPMA) technique. The resulting data is provided in Fig. 2 . The EPMA data is expressed in units of measurement X-ray counts, and for P corresponds to a peak P2O5 concentration of roughly 14.3 mol% in the core. The final preform was obtained by over-cladding the cane with a silica tube. The final preform was then drawn into a fiber.
Fig. 2 Measured compositional profile of the cane precursor to the P2O5-doped silica optical fiber. Fluorine (F) is present only in the inner cladding with a small amount of phosphorus (P) to create an index-matched inner cladding. The MCVD layers are visible in the P2O5 data.
Figures 1 and 2 reveal a significant index dip in the central region of the fiber perform core, which can have a significant impact on the analysis presented in this paper. Therefore, the RIP of the final fiber is measured to determine if this feature was preserved through the draw process. The refractive index profile was measured using two methods: 1) the well-known refracted near field (RNF) technique and 2) a high-resolution spatially-resolved Fourier transform technique [12], with measurement wavelengths at 670 nm and 1000 nm, respectively. The results are shown in Fig. 3 . EPMA was not used on the small fiber due to the limited spatial resolution of the available system (~1 μm).
Fig. 3 Refractive index profile of the final P-doped silica fiber measured at 670 nm (dashed line) and 1000 nm (solid line).
The RIP measurements suggest that the P2O5 dopant experienced significant diffusion during the fiber drawing stage. While a slight central dip can be seen in the 1000 nm high-resolution data, the large burnout observed in both Figs. 1 and 2 appears to have been completely filled-in. In addition, a slight decrease in the refractive index difference is observed in going from 670 nm to 1000 nm, which can be attributed to either 1) chromatic dispersion of the P2O5 dopant or 2) a slight variation in the dopant concentration between the two measurement samples.
In order to obtain the dopant concentration in the core, we rely on the RIP measurements of the fiber and the results found in [13]. In [13], preform RIPs were measured using a York P102 preform analyzer leading to a molar refractivity determined to be 0.88 × 10−3 per mol%. Due to the proximity of the wavelength of the York system (632 nm) to that of the RNF method employed here (670 nm), we use the RNF data to determine the P2O5 concentration (represented here as [P2O5] in mol%). Therefore, based on the measured RIP, we determine that [P2O5] in the center of the fiber is 12.2 mol%.
Integrating the compositional profile in Fig. 2 and that derived from the RNF RIP across the core region give rise to average P2O5 concentrations of 10.5 mol% and 10.3 mol%, respectively (note that this is also a representation of the total number of P atoms in the core). This finding supports the conclusion that the fiber of the present study does not possess a central dip since P2O5 diffused into this region during the drawing process.
The results in [13] suggest a linear relationship between [P2O5] and Δn. Throughout this paper, however, we will be using a simplified additive model in representing the refractive index of the P2O5-SiO2 mixture. Within this model, the refractive index is represented as
where the subscripts P and S refer to bulk (and quenched) P2O5 and SiO2, respectively. m is the additivity variable and is a function of the molar volume and [P2O5] bywhere Mi is the molar mass, ρi is the mass density, and i = P or S. Within the scope of this model, since (ρs/Ms)(Mp/ρp) > 1 the molar refractivity curve (index versus [P2O5]) is sub-linear. This is shown in Fig. 4 where the additive model is plotted using bulk parameters (to be provided in the next section) and is shown with the linear fit from [13]. The two models are very close over the range 0 < [P2O5] < 15 mol%, with a maximum difference of about 6%.Fig. 4 Index difference vs. [P2O5] calculated via 1) the additive model (solid line) using the bulk parameters and 2) the linear approximation in [13].
The additive model assumes that the glass constituents are independent of each other, and thus requires modification in cases where dopants interact in a non-additive way. A well-known example is the P2O5-Al2O3 co-doped silica system. While doping P or Al alone into silica increases the refractive index (and mass density), adding P to an Al-doped fiber (or Al to a P-doped fiber) decreases the refractive index (and mass density) relative to the binary system [14], suggesting an interaction that leads to structural modifications. In this case, one would treat the AlPO4 unit cell as an independent species within the additive model.
As a side note, an interesting special case arises in the additive model when the molar volumes of the dopant and host oxides are identical ((ρs/Ms)(Mp/ρp) ≈1). In that case, the refractive index difference reduces to a linear function of the dopant concentration (in mol%). Finally, since our Brillouin scattering measurements are performed at 1534 nm (λo), we assume that the RIP measurement at 1000 nm is a good approximation to the RIP at 1534 nm, and thus from this data the optical mode will be calculated.
B. Experimental Details
The experimental configuration used to acquire the BGS is identical to that described in [6]. In short, a continuous-wave, narrow-linewidth (< 100 kHz) external-cavity laser diode signal is amplified and passed through a circulator to a 3.5-meter test fiber. The back-scattered Stokes’ signal and a small quantity of pump signal (courtesy of a carefully-controlled angled cleave at the test fiber output) then propagate back through the third port of the circulator, are optically pre-amplified, and finally heterodyned onto a fast detector and the resulting spectrum is captured with a signal analyzer. Caution was exercised to ensure that Brillouin gain did not narrow the spectra and obscure the measurements [15].
3. Room temperature measurements and the acoustic attenuation coefficient of P2O5
The measured BGS of the P-doped fiber is shown in Fig. 5 . A total of eleven acoustic modes is observed. Four peaks (L01 to L04) can be attributed to modes that reside primarily in the core of the test fiber. Two (near 11 GHz) are from the optical circulator in the test apparatus (which includes a Corning SMF-28TM leader attached to the circulator fiber). The remaining five peaks (L05 to L09) result from acoustic modes that reside primarily in the matched-index deposited inner cladding. A best-fit consisting of a summation of eleven Lorentzian profiles was performed, with the results of the measured values summarized in Table 1 (columns 2 and 4).
Fig. 5 Brillouin spectrum of a 3.5 m segment of the P2O5-doped optical fiber at 1534 nm and at two different temperatures, 21.5°C and 118.3°C. Four acoustic modes primarily located in the core are observed, while five reside mainly in the inner cladding. The two peaks near 11 GHz are due to the measurement apparatus.
Table 1. Measured Parameters for All Observed Acoustic Modes m
Also shown in Fig. 5 is a spectrum of the fiber when heated from room temperature (21.5 °C) to a temperature of 118.3 °C (only the P-doped fiber was heated). Much like is observed with germanium-doped fibers [16], the spectrum narrows and the Stokes’ frequency increases with increasing temperature. The fibers associated with the measurement apparatus were not heated, as indicated by the data. Therefore, these peaks are fixed in frequency while the test fiber was heated. In addition, the L04 mode almost merges into the peak of the test apparatus at 21.5 °C due to the relative strength and position of this peak. At 118.3 °C the L04 mode becomes hidden underneath the peaks of the test apparatus due to an increasing frequency shift with increasing temperature. Even though the peaks of the test apparatus get smaller, the L04 mode is essentially invisible under them. This introduces increased measurement error for the L04 mode at elevated temperatures.
Finally, a fit to the data was performed using a model in much the same way as found in [6]. In the present case, a six-layer step-wise approximation was made to the profile. The first four layers correspond to the core, one to the inner cladding, and one to the outer pure silica cladding. The velocity of the deposited inner cladding was deduced from the measured data since it had a fairly complicated profile owing to the deposited MCVD layers (see Fig. 1) and presence of fluorine. To formulate the fit, the approximate compositional profile was first determined with the help of the data in Fig. 3 (this is the second column in Table 2 and the method is described in Section 2.A). Then, invoking the materials model in [7], here VL = 1/[m/VP + (1-m)/VS] and α(νB) = mαP(νB) + (1-m)αS(νB) are used to calculate the bulk values in each layer, with the acoustic velocity (VP and VS) and attenuation coefficients (αP and αS) for the bulk materials (P2O5 and SiO2) used as fit parameters. More specifically, these values were iterated until the differences (see the footnotes in Table 1) between the measured and calculated modal frequencies (columns 2 and 3 in Table 1, respectively) and spectral widths (columns 4 and 5 in Table 1, respectively) were minimized for all modes simultaneously. Given the close proximity of the modeled and measured modal frequencies, we are confident in our assignment of the acoustic modes.
Table 2. Approximation (best-fit) to the Profile Provided in Fig. 3
The results of this fit are shown in Table 2 for the bulk materials, as well as the resulting values for each of the layers. We reiterate that since the fifth layer has a very small amount of fluorine to make this layer a matched-index cladding (fluorine reduces the acoustic velocity), we use the measured acoustic velocity value via the fundamental cladding mode frequency (L05) and ν L05 = 2Vm n modal /λo (Vm is acoustic velocity of L05, n modal is modal index of the optical mode in the P-doped fiber, and λo is optical wavelength). For completeness, Fig. 6 shows the bulk acoustic velocity as a function of P2O5 concentration (mol%). The plot suggests that the acoustic velocity has a nonlinear, monotonically decreasing relationship with increasing P2O5 concentration. For a very small range of concentrations one can assume that the curve is approximately linear, but not simply linear as a whole (~−0.90%/mol% up to 10mol %).
Fig. 6 The bulk acoustic velocity nonlinearly decreases with increasing P2O5 concentration (mol%). It is not simply linear as a whole according to the additive model.
The modeled Stokes’ shifts and Brillouin spectral widths (i.e. the result of the fit) are shown in Table 1 in order to compare with the measured data. The largest error in the Brillouin spectral width is for the L02 mode. The difference in the Stokes’ shift for this mode, 12MHz, corresponds to about 6.21 m/s, which is roughly 1.27% of the difference between the L02 modal velocity and that of bulk silica. The uncertainties of the measured Brillouin spectral widths for the L01, L02, L03, and L04 modes are ± 0.5MHz, ± 1.0MHz, ± 1.0MHz, and ± 3.0MHz, respectively. As described above, because the L04 mode is close to the peaks of the test apparatus, their overlapping nature introduces more spectral width uncertainty for this mode. The spectral width of each mode was modeled using the following expression [8]
where ΔνB (νm, r) is the bulk spectral width at the modal frequency νm at the spatial position r and u(r) is the power-normalized acoustic displacement, which is dependent on the acoustic mode. Equation (3) essentially represents an average spectral width, weighted by the power distribution of the acoustic mode within the dynamic viscosity profile. Details on how the acoustic mode was calculated, using a simplified eigenvalue problem, are also provided in [6]. We point out that the fitted spectral widths provided in Table 2 for each layer are listed utilizing the L01 parameters, in particular the modal acoustic velocity (which together with the optical modal index defines the acoustic mode frequency) is used in ∆νm = αVm /π with Vm = νmλo/2n modal and α ∝ ν 2 m. Since these modal values are different for the higher-order acoustic modes, these spectral widths are also different for each mode.The L01 parameters provide for a point-of-reference from which the spectral widths of the other modes can be calculated by invoking the frequency-squared dependence of the Brillouin spectral width. Since the Brillouin spectral width of the fiber of this study decreases with increasing temperature, we have made the assumption that the spectral width of P-doped silica fibers obeys the simple frequency-squared law, in contrast to some materials such as boron doped germanosilicate optical fibers [17]. These effects were taken into consideration for the final fit. We point out that the best-fit spectral width for phosphorus oxide (177 MHz at 11 GHz) in Table 2 is the bulk-material value. This corresponds to an acoustic attenuation coefficient of about 1.411 × 105 m−1 at 11 GHz. This frequency was previously taken as a convenient reference since it is the Brillouin frequency shift of the L01 acoustic mode of Corning SMF-28TM at 1534 nm.
We point out that as with the Ge-doped fiber of [6], the spectral width decreases with increasing mode number. In [6], this resulted from the fact that as the mode number increases, the proportion of acoustic power occupying the inner dip region of the core increases. The fact that the central dip region in the Ge fiber contained a lower GeO2 concentration (which has a larger acoustic attenuation coefficient than SiO2) leads to narrower Brillouin spectra for modes occupying a greater proportion of that region. In the present case, however, the P concentration is decreasing radially-outward in the region between 1 and 3 μm. Since the inner-core region has more P and less Si than the outer-core region, and since SiO2 has a much lower attenuation coefficient than P2O5, the spectral widths of the higher-order acoustic modes (HOAMs) are smaller than that of the fundamental mode. This results from the increasing mode diameter of the HOAMs with increasing mode number (i.e. the fundamental acoustic mode is most tightly confined to the center of the fiber where the acoustic attenuation is the largest). To illustrate this point, Fig. 7 shows a plot of normalized longitudinal acoustic modes L01 and L04 together with the measured RIP.
Fig. 7 Normalized longitudinal acoustic modes L01 and L04 with the measured RIP. The red-dashed curve is the RIP. The green curve is the spatial distribution of L01 and the brown curve is the spatial distribution of L04. The spatial distribution of L04 has a bigger mode diameter than that of L01 and occupies more of the outer-core region.
4. Temperature-dependent measurements and analysis
In addition to room temperature (21.5°C) measurements, we study the acoustic properties of the P2O5 fiber at higher temperatures. Figure 5 shows the differences in the Brillouin gain spectrum between 21.5°C and 118.3°C for all acoustic modes in the core and cladding of the fiber. We first investigate the spectral widths and frequency shifts of the fundamental mode (L01) over the measured temperature range. In Fig. 8 we find that the trends are both approximately linear, with the Stokes’ shift increasing at a rate of ~ + 0.74 MHz/°C and the spectral width decreasing at a rate of ~−77.7 kHz/°C. Interestingly, in Fig. 5 we see that the inner cladding has a Stokes’ shift rate (~1.22MHz/°C) that is larger than that in the core of the fiber. This can mainly be attributed to the fact that the inner cladding is lightly doped (i.e. mainly silica). This will be described in more detail later in the paper.
Fig. 8 The trends are both approximately linear in the available measurement range, with the Stokes’ shift increasing at a rate of ~ + 0.74 MHz/°C and the spectral width decreasing at a rate of ~−77.7 kHz/°C for the main mode.
Next, we study the HOAMs in the core at different temperatures. We follow the modeling method of Section 3 to investigate the temperature dependency of the Stokes’s shift of all the acoustic modes in the core. To first order, we only consider that the acoustic velocity and refractive index are temperature-dependent. Referring to Table 2, this means that the values in columns 3, 4, and 7 vary with the temperature, and this gives rise to a temperature-dependence of the calculated Stokes’ shift and modal index. As a side note, since the P2O5 concentration is different for each layer, each layer will have a unique TAC and TOC. Other relevant quantities (namely the mass density) are assumed to be negligibly dependent on the temperature.
Meanwhile, we use the temperature-dependent Sellmeier model [18] for the refractive index of bulk pure silica and offset the refractive index in [18] at 21.5 °C to 1.443 in accordance with the Table 2 assumption. With the refractive index of bulk P2O5 found in Table 2 and using the TOC (−9.22 × 10−5/°C) found in [19,20], one can gain the refractive index of P2O5 at different temperatures and different concentrations for simulating the P2O5-SiO2 mixture via the additive model. In addition, we assume the thermo-optic effect of phosphorus dominates fluorine in the fifth layer in Table 2 at higher temperatures due to the insignificant thermo-optic effect of fluorine in a silica glass with very low concentration as in the present case [21]. Then, from the data, we will deduce an analogous TAC for the acoustic velocity.
To model the acoustic velocities, we first assume that the bulk acoustic velocity of both P2O5 and SiO2 are linearly related to the temperature (over a small temperature range). For P2O5 we can write
where Cp is a constant (units of m/s/°C). In addition, a similar temperature-dependent linear equation of acoustic velocity for SiO2 is extracted from measurements of frequency vs. temperature on a commercial pure silica core fiber (Sumitomo Z-fiberTM). Figure 9 shows the best linear fit to the measured Brillouin frequency shift of the Z-fiberTM as a function of temperature.Utilizing the calculated temperature-dependent modal index [18] for the Z-fiberTM, we convert this linear equation to obtain the temperature-dependent linear equation of acoustic velocity asFig. 9 The linear equation for the frequency shift of Z-fiberTM as a function of fiber temperature as a best fit to the measured data. We use this linear equation to obtain the temperature-dependent acoustic velocity of SiO2.
The fundamental mode of the Z-fiberTM dominates the Brillouin gain spectrum, and the acoustic and optical index profiles of the Z-fiberTM in the core are almost uniform. Furthermore, our calculations indicate that the fundamental acoustic mode velocity is very similar to the value of the core material, and thus we assume that its modal acoustic velocity can represent the acoustic velocity of bulk pure silica (core material) without introducing significant error. The slope found here, + 0.555 m/sec/°C, is in good agreement with previous measurements on the bulk material over our measured temperature range [10,11], leading to a high degree of confidence in Eq. (5).
However, we indicate that the P2O5-doped fiber and the Z-fiberTM are produced by different manufacturers with different conditions. It is therefore not surprising that the room temperature acoustic velocity of pure bulk silica derived from the Sumitomo fiber (5969 m/s) differs (although only slightly) from that provided in Table 2 (5970 m/s). This may be due to different thermal histories [22] or draw tensions [23]. For consistency with the present analysis, we offset the interception in Eq. (5) of the linear equation of the acoustic velocity to 5970 m/s for the temperature-dependent acoustic velocity of SiO2 (~1.4 m/s or 0.02%).
Next, we determine Cp in Eq. (4) by performing a best-fit to measured data. To do this, each acoustic mode is calculated using the temperature-dependent six-layer approximation (see Table 2 showing room temperature values, with each layer possessing a unique TAC and TOC) and Cp is iterated until the error across all acoustic modes is minimized. While Cp is the only remaining unknown here, fitting simultaneously to all four acoustic modes increases the confidence in the determined value and leads to interesting physical conclusions. The result is shown in Fig. 10 , where the solid lines correspond to the modeled data and the circles to the measured data (Brillouin frequency vs. temperature). Figure 10 shows that modes 1, 2, 3, and 4 have maximum errors 0.161%, 0.150%, 0.156%, and 0.078% in the acoustic frequencies, respectively. The slight error in frequency may be attributable to uncertainty in the dopant profile. Table 3 provides the temperature-dependent linear equations of measured and modeled frequency shifts for the first four modes. The best-fit value for Cp is found to be + 0.119 m/sec/°C. Using this value, the measured and modeled slopes of the frequency versus temperature curves are all within 3.02% of each other, indicating that this is a very good fit to data.
Fig. 10 The modeled frequency shift (solid line) and the measured frequency shift (circle) vs. temperature. All the trends are approximately linear in the available measurement range. The modeled data of each of the modes are very close to the measured points.
Table 3. The Comparison of Measured and Modeled Linear Equations of the Temperature-Dependent Frequency Shift
Interestingly, the slope of the temperature curves is increasing with increasing mode number. As with the spectral width, since the HOAMs occupy proportionally more space in the outer region of the core than the fundamental mode, where there is a lesser content of P2O5, the slope is expected to increase with increasing mode number. This is because the acoustic velocity of silica has a larger dependence on temperature than P2O5 (comparing Eq. (4) given that Cp = + 0.119 m/sec/°C and Eq. (5)), and P2O5 has a relatively large negative thermo-optic coefficient.
Finally, a design example is presented where a composition is determined for which the dependence of the Brillouin frequency shift on temperature is minimized. Given the large negative TOC of P2O5 and its smaller TAC than silica, a composition may be found that will minimize the slope of the frequency shift versus temperature curve. The acoustic velocity as a function of temperature may be written
Taking the derivative with respect to T and setting it to zero yields the following condition for a temperature-independent νa (T)
In the present case, P2O5 has negative TOC, dn(T)/dT < 0, but positive TAC, dVL(T)/dT > 0. Assuming that the derivatives are approximately constant over a small temperature range, Eq. (7) is simplified to
In general, this expression is difficult to satisfy for all T due to the dissimilarity in the functional forms of n and VL. However, a range of temperatures can be found where the slope, dν(T)/dT, is minimized. First, we assume the temperature-dependent forms of the refractive index [18,19] and acoustic velocity (Eq. (4) and Eq. (5)) as described previously in this paper. We also assume that over a small temperature range, dn(T)/dT for pure P2O5 is a constant. The dν(T)/dT of P2O5-doped silica fiber (Table 3) is lower than that of pure silica (Fig. 9), therefore, this suggests that pure P2O5 possesses a ν(T) that has a negative slope through Eq. (6), while that of pure silica has a positive slope. Thus, one can conclude that there is a composition of P2O5-SiO2 mixture where the slope is balanced to some minimized value and the acoustic frequency of the composition is independent of temperature.
To determine the temperature dependencies of the P2O5-SiO2 mixture, the temperature-dependent physical properties of the individual pure constituents are used in the additive model as described above, and the composition is adjusted until the frequency shift variation is minimized over the region −100 °C < T < 100 °C. The result is shown in Fig. 11 for the molar composition 0.54P2O5:0.46SiO2. The inset shows the derivative of the frequency-versus-temperature curve. In the range −100 °C < T < 100 °C, the Brillouin frequency shift changes by less than 5 kHz per °C. Therefore, this composition is calculated to be more than 200 times less sensitive to temperature changes than the pure silica core fiber (see Fig. 9). An absolute minimum in the slope of frequency shift-versus-temperature curve is observed near 10°C. While this result is for the bulk material, wave-guiding properties should be taken into consideration when designing a fiber.
Fig. 11 Brillouin frequency shift versus temperature for the molar composition 0.54P2O5:0.46SiO2. In the range, −100 °C < T < 100 °C, the Stokes’ shift changes by less than 5 kHz per °C. An absolute minimum frequency shift variation is around 10°C where the constant ratio of VL to n is about 2946 m/sec. The inset shows the derivative of the frequency-versus-temperature curve.
5. Conclusion
From BGS measurements of a fiber with a large P content in the core, we have provided an acoustic damping coefficient for bulk P2O5 that is suitable for modeling purposes. We find that this coefficient is about 1.411 × 105 m−1, which is similar to, but larger than, that of GeO2. Furthermore, our results for the effect of phosphorus oxide on the longitudinal velocity (~−0.90%/mol% up to 10mol %) are similar to those (~−0.71%/mol% up to 5.5mol%) found in [4]. We have found that the temperature coefficients for the Stokes’ frequency shift and Brillouin spectral width for the L01 acoustic mode of the fiber are ~ + 0.74 MHz/°C and ~−77.7 kHz/°C, respectively. In addition, we also determine the thermo-acoustic coefficient of bulk P2O5 to be about + 0.119 m/sec/°C, which is much lower than that of pure silica ( + 0.555 m/sec/°C). We have shown that the use of these bulk coefficients can be used to predict the temperature-dependent Stokes’s shift of the higher-order acoustic modes with a high degree of accuracy. This data is also useful for the design of acoustic profiles of optical fiber for applications where Brillouin scattering is significant. To illustrate this point, a design example was presented where the slope of the Stokes’ shift versus temperature curve was compositionally minimized to be less than 5 kHz per °C over the temperature range −100 °C < T < 100 °C.
Acknowledgments
This work was supported in part by the Joint Technology Office (JTO) through their High Energy Laser Multidisciplinary Research Initiative (HEL-MRI) program entitled “Novel Large-Mode-Area (LMA) Fiber Technologies for High Power Fiber Laser Arrays” under ARO subcontract # F014252. P.-C. Law would like to acknowledge Professor Gary R. Swenson for his full support. The RIP measurements at 1000nm were performed by A. Yablon at Interfiber Analysis.
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