## Abstract

We present measurements and modeling of the effect of P_{2}O_{5} doping on the strain sensitivity coefficients of silica fibers. In particular, the Brillouin gain spectrum of a heavily P_{2}O_{5}-doped fiber is measured and investigated at different strains. We provide measurements of the strain-optic coefficient (SOC) and the strain-acoustic coefficient (SAC), obtained to be + 0.139 and + 9854m/sec/*ε*, respectively, both of which are less than the pure silica values. The Pockels’ coefficients *p*_{11} and *p*_{12} for bulk P_{2}O_{5} are also estimated via Brillouin gain measurements. Using the strain coefficients, the modeled and unique slopes of the Stokes’-shift-versus-strain curves for the four observed acoustic modes in the fiber each lie within 2% of the measured values.

©2012 Optical Society of America

## 1. Introduction

Recently, novel optical fibers with tailored acoustic profiles have been designed specifically for applications where Brillouin scattering may occur. For example, stimulated Brillouin scattering (SBS) suppression in high-power narrow linewidth systems may be achieved by reducing the overlap between the optical and acoustic fields [1] or by broadening the Brillouin spectral width via a large acoustic damping coefficient [2]. Alternatively, SBS may be utilized for distributed sensing applications via a strain-dependent Brillouin frequency [3]. Imperative in the process of designing an appropriate acoustic profile is knowledge of how potential dopants influence the relevant acoustic parameters in a host material, such as silica.

Previously [4], utilizing a simple additive materials model [5] fitted to a series of measurements of the Brillouin gain spectrum (BGS) of a heavily P_{2}O_{5}-doped fiber, we provided the acoustic velocity (*V _{A}*), acoustic attenuation (

*α*), and thermo-acoustic coefficient (

*C*) of bulk P

_{P}_{2}O

_{5}. Table 1 reproduces the room-temperature measured and modeled acoustic (Brillouin) frequencies (

*ν*

_{B}), spectral widths (Δ

*ν*

_{B}), and relative amplitudes of the observed acoustic modes from [4]. Four acoustic modes of the acoustic waveguides were observed. These modes are the azimuthally-symmetric L

_{0}

*modes [6] of the cylindrical waveguide. The P*

_{m}_{2}O

_{5}-doped fiber is acoustically waveguiding since P is known to lower the acoustic velocity when added to silica [7]. Additionally, the observed spectral width of a mode depends on the spatial overlap with the acoustic damping profile [8], while the relative amplitude is a function of the spatial overlap with the optical mode [9].

The bulk values (reproduced from [4] here in Table 2
), including mass density (*ρ*), refractive index difference (Δ*n*) and the shear (*V _{S}*) and longitudinal (

*V*) acoustic velocities, can then be used to calculate and frame sets of designer acoustic profiles for specialty fibers for distributed temperature sensing applications. More specifically, the additive model used here essentially provides an average of optical and acoustic parameters for silica fibers containing P

_{L}_{2}O

_{5}, with the additivity parameter being the molar volume [5]. In this paper, we continue this work and present a similar analysis for the tensile strain effect on the same P

_{2}O

_{5}-doped optical fiber, and refer the reader to [4] for details such as the model, fiber characteristics, and experimental setup for the measurement of the BGS. We derive and present a strain-optic coefficient (SOC) and strain-acoustic coefficient (SAC) for bulk P

_{2}O

_{5}that can be utilized in designing specialty fibers for distributed strain sensing applications.

To summarize the previous [4] modeling procedure, fits to measured data were performed by first making a six-layer step-wise approximation to the P_{2}O_{5} compositional profile [4]. The additive model was then used to determine the bulk physical parameters in each layer. A simplified eignvalue method is used to calculate the acoustic modes of resulting six-layer structure. This gives rise to optical and acoustic parameters and mode fields of P_{2}O_{5}-doped silica fiber which can then be compared with measured data. Finally, the bulk values of P_{2}O_{5} were iterated to gain the best-fit to the measured data and thus the desired coefficients of bulk P_{2}O_{5} were determined.

To first order, we assume that only the acoustic velocity and refractive index are strain-dependent. Thus, the strain (*ε*) dependence of the Brillouin frequency (*ν _{B}*) on optical wavelength (

*λ*), acoustic velocity (

_{o}*V*), and refractive index (

_{A}*n*) is given by

*p*

_{11}and

*p*

_{12}, the Poisson ratio,

*σ*, and the zero-strain refractive index

*n*

_{0}. In this paper, we define the SOC to be SOC =

*p*

_{12}−

*σ*·(

*p*

_{11}+

*p*

_{12}). Despite an extensive literature search, we could not find a reported SOC for bulk P

_{2}O

_{5}. This can be attributed to the difficulty in working with the bulk material due to its being highly hygroscopic [11]. On the other hand, the SOC values of bulk SiO

_{2}and silica fiber are well-known (SOC = 0.1936 if

*p*

_{11}= 0.113,

*p*

_{12}= 0.252,

*σ*= 0.16) [12].

In order to address the lack of SOC values for bulk P_{2}O_{5}, we first present a measurement of the SOCs of the heavily-P_{2}O_{5}-doped fiber and a pure silica fiber utilizing the strain-dependent free spectral range (FSR) of a fiber ring laser that is constructed using a segment of the test fibers. From these measurements, the bulk P_{2}O_{5} SOC value is deduced. Then, measurements of the BGS at various strains enable a determination of the SAC ( = *dV _{A}/dε*). As done before [4], the SiO

_{2}-P

_{2}O

_{5}system is modeled in the same additive way [5] and we find that the SOC and SAC of bulk P

_{2}O

_{5}are ~ + 0.139 and + 9854 m/s/

*ε*, respectively. The maximum uncertainties are 0.007 and 1326m/sec/

*ε*for the SOC and SAC, respectively. Both of these are lower than the pure silica values (SOC = + 0.194 and SAC = ~ + 29240 m/s/

*ε*), which are confirmed via measurements on a Sumitomo Z-Fiber

^{TM}(pure silica core and F-doped cladding). Utilizing the determined SOC and SAC of bulk P

_{2}O

_{5}, the modeled and unique slopes of the Stokes’-shift-versus-strain curves for the four observed acoustic modes each lie within 1.974% of the measured values.

Finally, utilizing Brillouin gain measurements and the measured SOC for bulk P_{2}O_{5} the Pockels’ coefficients are estimated. Although this estimate may have considerable error associated with it, to the best of our knowledge, we believe this to be the first attempt at elucidating these values for P_{2}O_{5}.

## 2. Optical fiber and experimental details

#### A. Optical fiber

The optical fiber used in this set of experiments is one with a heavily P_{2}O_{5}-doped silica core fabricated by the modified chemical vapor deposition (MCVD) process at INO of Canada. The core glass of the resulting fabricated fiber is in an amorphous state, while the additive model averages any local structure in the P_{2}O_{5}-dopant, resulting from the MCVD process, into ‘net’ values for bulk P_{2}O_{5}. Detailed information regarding its refractive index and compositional profiles are described in [4]. In short, based on the measured refractive index profile (RIP), [P_{2}O_{5}] in the center of the fiber is approximately 12.2 mol% and it does not possess an index dip.

#### B. SOC Measurement

We utilized the fiber ring-based strain sensor [13,14] as the apparatus to measure the SOC of the P_{2}O_{5}-doped fiber. In this configuration, the test fiber becomes part of the laser cavity, and any strain imparted on this fiber will result in a change in the cavity free-spectral range (FSR). The change in FSR can be linked to the strain-induced change in fiber length and refractive index.

Figure 1 shows the experimental apparatus. A fiber ring laser is constructed from a commercial erbium-doped fiber amplifier (EDFA, JDSU MicroAmp) that serves as the gain block, and a tap coupler that serves as the output port and feedback mechanism. Unidirectional laser operation is ensured by isolators built into the EDFA, and no attempt to otherwise stabilize the cavity was made since, as will be shown later, the presence of a large number of cavity modes is a desirable laser attribute. Incorporation of the test fiber into the laser cavity is facilitated by splicing FC/APC-connectorized pigtails to its ends.

The output of the laser is sent directly to a P*i*N detector (New Focus 1534) and the spectrum of the resulting beat signal is then measured with an electrical spectrum analyzer (ESA, Agilent 8560). The beat frequencies measured by the ESA represent the cavity mode spacing (FSR) and its harmonics resulting from the presence of multiple evenly-spaced cavity modes. A strain on the test fiber (the P_{2}O_{5}-doped fiber) then results in a change in FSR that is observed as a frequency shift at the ESA. From the change in FSR as a function of strain, the SOC can be determined.

#### C. Strain-acoustic frequency measurements

The experimental configuration used to acquire the BGS is identical to that described in [4,15], except that the fiber is under tension. In short, the BGS is measured utilizing spontaneous Brillouin scattering with the fiber under test (FUT) spliced to the output of the measurement system. The output end of the FUT is secured to a metal plate via an epoxy and the other end is attached to a linear translation stage with a strain-gauge. A linear strain is then applied to the whole length of fiber. The BGS is measured to gain the Stokes’ frequency shifts up to around 1% elongation (linear and elastic region) at 1534nm.

#### D. Estimate of the Pockels’ coefficients

A block diagram of the experimental apparatus used to measure the Brillouin gain coefficient, *g _{B}*, is shown in Fig. 2
utilizing SBS. As will be described later,

*g*can be used to estimate

_{B}*p*

_{12}. We start with a narrow-linewidth external-cavity diode laser (ECDL) (1550 nm) seed source that is pre-amplified in a first erbium doped fiber amplifier (EDFA). The signal passes through a fiber-coupled acousto-optic modulator (AOM) and is pulse-amplified by a second EDFA. The output of the second EDFA is then sent into port 1 of a circulator. The signal is then launched into the fiber under test (FUT) via port 2. Any scattered light then passes back through the circulator and emerges from port 3, and into a pulse-energy meter.

A pulse length of 347 ns is utilized in the measurements to match the fiber length. Pulses shorter than this value result in a reduced effective SBS interaction length, decreasing the Brillouin gain [16]. Pulses much longer than this, on the other hand, were found to have a considerably altered shape such that much of the pulse did not contribute to the SBS process [17].

## 3. Experimental results

#### A. Strain-optic coefficient

The FSR of a unidirectional traveling-wave ring laser is well-known to be

where*n*is the refractive index (modal index) and

*l*is the cavity length. The refractive index is assumed to have the linear form

*n*=

*n*

_{0}+

*ε·Q*, and similarly the length has

*l*=

*l*

_{0}+

*ε·l*

_{0}, with the subscript ‘0’ representing the zero-strain values. Throughout this paper, unless otherwise stated, the strain

*ε*is taken as a fractional elongation. Since only a small portion of the ring laser is to experience strain, the

*FSR*is rewritten to explicitly contain contributions from both the unstrained and strained fiber components asHere,

*nl*is now the contribution from the segment of fiber that will be strained, and

*NL*is due to the remainder of the ring cavity. Next, in order to determine how strain influences the FSR, we take the derivative of this equation to find

This can be represented in differential form in the following way

giving rise toEquation (7) describes the change in cavity FSR (Δ*FSR*) from the zero-strain value for an induced strain *ε* on fiber segment *l* (the subscript ‘0’ again represents the zero-strain value). The (beat) frequencies (Δ*ν _{ESA}*), which are measured by the ESA in Fig. 1, correspond to Δ

*FSR*(first harmonic) and higher-order harmonics due to the presence of a multiplicity of cavity modes. Hence, to improve the resolution of the measurement, a higher-order harmonic is utilized since a larger frequency shift is measured than just Δ

*FSR*, thereby greatly reducing the measurement uncertainty. In fact, the measured frequency becomes a multiple of Δ

*FSR*and is related to the harmonic number

*M*in the following way

*Q*is the only unknown in Eq. (8), and its value can be determined from a set of measurements. We arbitrarily select

*M*= 86 for the measurements presented here.

Two fibers were tested for their SOC utilizing the ring laser apparatus in Fig. 1. These are the Sumitomo Z-Fiber^{TM} (pure silica core) and the P_{2}O_{5}-doped fiber. The Z-Fiber^{TM} was tested in order to validate the SOC values assumed above for pure silica [12], while measurements on the P_{2}O_{5}-doped fiber were used to determine the SOC of bulk P_{2}O_{5}. Some specifications and measurement results are provided in Table 3
. The modal indices were calculated from the refractive index profiles (RIP), and the strained fiber length was measured. *NL* was determined by measuring the FSR of the unstrained cavity and utilizing Eq. (4) to solve for *NL* by subtracting *n*_{0}*l*_{0}.

Figure 3
provides the measured change in FSR (∆*FSR*) as a function of the strain (%) for (a) the Z-Fiber^{TM} and (b) the P_{2}O_{5}-doped fiber. The 86th harmonic was utilized for the measurements, and therefore this value was applied as a divisor to the data in order to determine ∆*FSR* via Eq. (8). As a representative example, Fig. 4
shows the actual measured data from the ESA for each of the Z-Fiber^{TM} data points provided in Fig. 3(a).

A least-squares fit of Eq. (8) to the measured data was performed and is provided as the dashed line in Fig. 3. In both cases the R-squared value is greater than 0.999. Such R^{2} values show that the measured values can be well-fitted by the model, which is found to be very linear in the measured strain range. Since the only unknown in Eq. (8) is the SOC (through *Q* by Eq. (2), where the refractive index *n* is taken from Table 3), the SOC becomes the only fitting parameter. The resulting SOCs for the two fibers are provided in Table 3. The value for the Z-Fiber^{TM} is very close to the value (0.1936) provided in the Introduction for pure silica with *n _{0}* = 1.443, offering confidence in our use of that value (0.1936) in the simulations.

In order to calculate the SOC of bulk P_{2}O_{5} from the fiber data, we utilize a step-wise approximation to the refractive index profile, and the additive model is applied to each layer to determine the index of the doped material. The approach is identical to that in [4], except that the change in bulk refractive index values due to strain of each of the P_{2}O_{5} and SiO_{2} constituents now takes the form of Eq. (2). Thus, the refractive index of each layer can be determined as a function of strain, and from this the modal index can also be calculated as a function of strain. Since the SOC of pure silica has already been assumed, the SOC of the bulk P_{2}O_{5} component in the additive model is the only remaining unknown, and is adjusted until the calculated SOC of the P_{2}O_{5}-doped fiber (modal value) matches that of the measurement (see Table 3). The best-fit to the measured data is an SOC of bulk P_{2}O_{5} of + 0.139 via the additive model [5]. We will utilize a similar approach for the SAC later in the paper. We point out that the values of *p*_{11}, *p*_{12}, *σ* of silica reported in [12] had associated measurement errors and we will use these in an error analysis here. Meanwhile, the SOC measurement error of the P_{2}O_{5}-doped silica fiber, ~1.93% (See footnote in Table 3), corresponds to an SOC error of ± 0.004. Thus propagating the silica and P_{2}O_{5}-doped fiber uncertainties through the additive model, the deduced error for the bulk P_{2}O_{5} SOC is around ± 0.007.

The measured SOC value is less than that of both glassy GeO_{2} [18] and bulk silica [12], and will be utilized throughout the remainder of this paper. The parameters used for the step-wise approximation can be found in Table 2 in [4]. As a convenience, Table 4
provides the refractive index values utilized in the calculation at 0.80% strain for the six layers, compared with the zero-strain case, since this was the largest applied strain during the SOC measurements on the P_{2}O_{5}-doped fiber.

#### B. Strain-acoustic frequency measurements and analysis

We measured the Stokes’s shift of the fundamental acoustic modes in the P_{2}O_{5}-doped fiber, and for standard Ge-doped SMF-28^{TM} fiber and Z-Fiber^{TM} as reference fibers. The Brillouin spectra were measured up to around 1% elongation (linear and elastic region) at 1534nm with the results shown in Fig. 5
. The experimental results show that these fiber samples are linear and elastic systems under the tension range and the acoustic frequency increases with increasing tension (strain), but at different rates for the various fibers.

We find that the Stokes’ frequency shift increases at a rate of ~ + 506 MHz/% for SMF-28^{TM} fiber (blue dots) and at a rate of ~ + 407 MHz/% for the P_{2}O_{5}-doped fiber (red dots) at 1534nm. Both of these values are less than that (525 MHz/%) of pure silica fiber (Z-Fiber^{TM}, green dots). We also point out that the rate-of-change for standard Ge-doped SMF-28^{TM} fiber is much higher than for P_{2}O_{5}-doped silica fiber. The results indicate that the dopants each significantly, but differently, influence the strain sensitivity. The data provide a direction for the design of specialty fibers for strain sensor systems. Interestingly, the pure silica fiber data suggest that silica may have the largest strain sensitivity of all the ‘common’ fiber materials. In order to verify this, it is worth investigating other dopants and concentrations.

Next, in order to extract the SAC of bulk P_{2}O_{5}, we study the fundamental and higher-order acoustic modes (HOAMs) in the core at different strains. To the best of our knowledge, there has been no attempt to determine this coefficient described in the literature. We follow the same modeling method as in [4] and Section 3*A* to investigate the strain dependency of the Stokes’s shift of all the acoustic modes in the core. In short, the compositional profile is approximated step-wise with six layers. Each layer has a unique SAC and SOC, and the various modal (optical and acoustic) quantities are calculated for each strain. Investigating all four observed acoustic modes simultaneously increases the confidence in the determined value.

With the strain dependency of the refractive index known, we next assume that the bulk acoustic velocity of both P_{2}O_{5} and SiO_{2} are linearly related to the strain (over the measured range) for modeling the acoustic velocities. For P_{2}O_{5} we can write [4]

*R*is strain-acoustic coefficient (

_{p}*R*≡ SAC

_{p}*) of bulk P*

_{p}_{2}O

_{5}(units of m/s/

*ε*) and

*ε*is fractional strain. Similarly, we write for pure silica the strain-dependent linear equation of acoustic velocity aswhere

*R*is strain-acoustic coefficient (

_{s}*R*≡ SAC

_{s}*) of bulk SiO*

_{s}_{2}(units of m/s/

*ε*).

The fit-to-data for the pure silica fiber in Fig. 5 is given as

*ε*is percentage strain. Assuming that the fundamental acoustic mode velocity is very similar to the value of the core material in the Z-Fiber

^{TM}, and knowing the SOC for silica [12], we find from Eqs. (10) and (11) that

*R*= + 29240m/sec/

_{s}*ε*, where the maximum best-fit error between the theoretical and the measured points in the acoustic frequency vs. strain curve (Fig. 5) is negligible.

Next, we determine *R _{p}* in Eq. (9) by performing a best-fit to measured data. To do this, each acoustic mode is calculated using the same strain-dependent six-layer approximation as before with each layer possessing a unique SAC and SOC.

*R*is iterated until the error across all acoustic modes is minimized. While

_{p}*R*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. 6 where the solid lines correspond to the modeled data and the circles to the measured data (Brillouin frequency vs. strain). Investigating Fig. 6, modes 1, 2, 3, and 4 have maximum errors 0.166%, 0.184%, 0.207%, and 0.161% in the acoustic frequencies, respectively. We point out that frequency increases with increasing mode number. The missing data points in the fourth mode results from spectral overlap with the apparatus BGS obscuring the data.

_{p}Table 5
provides the strain-dependent linear equations of the measured and modeled frequency shifts for the four acoustic modes in the core. Again, the R^{2} values show that the measured values can be well-represented by a linear model. The best-fit value for *R _{p}* is found to be + 9854 m/sec/

*ε*. Using this value, the measured and modeled slopes of the frequency versus strain curves are all within 1.974% of each other, indicating that this is a very good fit to data with 0.151% intercept-errors at the zero-strain acoustic frequency. In addition, the Chi-Square (χ

^{2}) test for goodness-of-fit was performed. The Chi-Square values of the four acoustic modes and all the data are much smaller than their critical values with adequate percentile points and degrees of freedom. All the test results support that there is extremely little difference between measured data and modeled data. This conclusion also confirms the previous error analysis of the four modes and the R

^{2}values with very low error for each mode.

Next, an estimate of the uncertainty in the SAC of P_{2}O_{5} is presented. The reported errors in the SOC of silica [12], along with the SOC measurement error of P_{2}O_{5} as described above, will propagate throughout all the SAC simulations of bulk P_{2}O_{5}. However, the SAC term dominates the SOC term in Eq. (1) and we therefore neglect the SOC error contribution. The maximum error associated with the slope (see Table 5), 1.974% for the L_{04} mode, is assumed to be only due to error in the SAC of P_{2}O_{5}. Propagating this through the models gives rise to an uncertainty in the SAC of bulk P_{2}O_{5} to be around 1326m/sec/*ε*. This represents a worst-case estimate with the assumption that the SAC of silica has no uncertainty.

Interestingly, as with the temperature-dependent slope [4], the slope of the strain curves is increasing with increasing mode number. 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 P_{2}O_{5}, the slope is expected to increase with increasing mode number. This is because the acoustic velocity of silica has a much larger dependence on strain than P_{2}O_{5} (comparing *R _{s}* = + 29240m/sec/

*ε*in Eq. (10) with

*R*= + 9854m/sec/

_{p}*ε*in Eq. (9)), and P

_{2}O

_{5}has a relatively small strain-optic coefficient (SOC), + 0.139 compared to + 0.194 of SiO

_{2}, although the SAC term dominates Eq. (1).

#### C. Estimate of the Pockels’ coefficients

In order to estimate the photoelastic constant *p*_{12}, Brillouin gain measurements are performed with narrow-linewidth power transmission testing. The apparatus utilized for this measurement is provided in Section 2*D*. The Brillouin gain is calculated using the following well-known equation [16]

*c*is speed of light and the other quantities are defined in Table 6 .

*g*can be determined from measuring the back-scattered SBS power as a function of launched power in a test fiber, which has the form given by (assuming an un-depleted pump) [19]

_{B}*α*is the optical attenuation coefficient (units of m

_{o}^{−1}) and

*L*is the actual fiber length.

*P*is the launched pump power, ${P}_{s,in}^{eff}$ is an effective Stokes’ input power [16], and

_{p}*A*is the effective area of the optical mode. Finally, Γ describes a normalized overlap integral between the optical and acoustic waves and is calculated for the fundamental acoustic mode (L

_{eff}_{01}) using

*E*E*is the normalized optical power distribution and

*u*is the power-normalized acoustic displacement vector. Γ takes on values between zero and one.

Figure 7
shows the results of the power transmission measurements for a fiber of length *L* = 54 m and the fit to data. Some depletion of the Stokes’ signal is found at the higher powers, and thus the fit is limited to the lower-power end of the data set. Additionally, at higher powers when significant SBS is present, the power readings contain a significantly increased instability, further justifying limiting the fit to the lower-power data points.

The mode field diameter (MFD) of the P_{2}O_{5}-doped silica fiber was measured to be 7.6 μm at 1550 nm by the transmitted near-field method on an EXFO NR-9200 Optical Fiber Analyzer. A transmitted near-field scan directly provided the light intensity distribution at the output of the fiber. The MFD was calculated with the well-known Petermann II equation, which relies on the far-field intensity distribution. The spatial resolution is 0.2 µm and the uncertainty on the MFD is 0.5 µm. The effective area is then calculated from this measurement of the MFD. Owing to the low-loss nature of the fiber, the effective length is calculated to be 53.2 m. The effective input Stokes’ power is calculated to be ~4.4 nW [16]. The resulting fitted gain coefficient is consequently found to be ~(0.50 ± 0.03) × 10^{−11} m/W, with the uncertainty arising from that of the MFD. With *g _{B}* estimated, Eq. (12) is utilized to estimate

*p*

_{12}.

The assumptions made for this calculation are outlined in Table 6. Note that we have made these measurements at 1550 nm, slightly altering the values found in [4]. Since the dominant acoustic mode is the fundamental mode (L_{01}), values that are characteristic of this mode are utilized [4]. We obtain a value of around 0.255 for *p*_{12} for the P_{2}O_{5}-doped fiber. Next, we describe two more assumptions used for this analysis.

First, since the fiber utilized in the power transmission tests was much longer (54 m) than that utilized in previous measurements (~3 m) [4], length-wise fiber compositional variations may contribute to a broadening of the BGS, and influence our measurement of *g _{B}*. Thus to remove this as a source of uncertainty, the BGS was re-measured for the longer segment of fiber, and the resulting L

_{01}acoustic mode values were utilized in the

*p*

_{12}calculation. The new spectrum is shown in Fig. 8 along with that measured in [4]. The spectral width of the L

_{01}acoustic mode is found to be somewhat broader (69 MHz) in the longer fiber, compared with 56.5 MHz for the short fiber segment [4].

Second, the overlap integral is determined by first fitting a series of Lorentzian functions to the spectrum. Then, we assume that the total integrated Brillouin gain is unity [1], and use the following expression for Γ with a total of *N* modes

*A*

_{0}

*’s represent the relative amplitudes of mode*

_{m}*m*. The ‘wings’ that appear in the spectrum of the longer fiber near 10.03 GHz and 10.23 GHz, to the red and blue of the L

_{01}peak, respectively, are treated as independent acoustic modes, and their origin is currently not known, but may be due to an induced birefringence in the coiled (standard 15 cm diameter spool) fiber.

Given the similarity of *p*_{12} of silica [12] to that of the P_{2}O_{5}-doped fiber, we approximate that *p*_{12} for silica and phosphorus pentoxide are approximately equivalent. Thus, utilizing *p*_{12} of 0.252 ± 0.008, a Poisson ratio (*σ*) of 0.294 [20], and an SOC of + 0.139 ± 0.007 for bulk P_{2}O_{5} (see Section 3*A*), we obtain a value of 0.132 ± 0.043 for *p*_{11}. We have performed an extensive search on the photoelastic constants of phosphorus pentoxide, phosphate glasses, and silicophosphate (phosphosilicate) glasses, and have not been able to reliably estimate the Pockels’ coefficients for P_{2}O_{5} from the literature in order to compare with our data. However, two pieces of evidence are consistent with our observations and findings. First, our *p*_{12} - *p*_{11} has a positive value, consistent with the report in [21]. Second, the *p*_{12} - *p*_{11} value (0.120) of our findings is slightly less than these (0.124~0.158) of GeO_{2} [18,22] and (0.134~0.149) of SiO_{2} [12,22,23]. This therefore suggests that our estimate of *p*_{11} falls within a reasonable range.

## 4. Conclusion

We extend the work presented in [4] to an investigation of the strain effect in P_{2}O_{5}-doped fiber. From the strain coefficient measurements of a fiber with a large P content in the core, we have provided strain-optic and strain-acoustic coefficients for bulk P_{2}O_{5} that are suitable for modeling purposes. We found that the strain-optic coefficient is about + 0.139 ± 0.007, a value that is less than that of glassy GeO_{2} [18] and bulk silica [12].

In strain-acoustic frequency measurements, the trends of frequency shift vs. strain for the P_{2}O_{5}-doped fiber, a sample of standard Ge-doped SMF-28, and pure silica fiber (Z-Fiber^{TM}) are approximately linear in the available strain range. The Stokes’ frequency shifts are highly sensitive to the tensile strain, but less so in the P_{2}O_{5}-doped fibers. The results show that the strain dependencies for the L_{01} acoustic mode are at rates of ~ + 506 MHz/% for SMF-28^{TM} fiber, ~ + 407 MHz/% for P_{2}O_{5}-doped fiber, and ~ + 525 MHz/% for Z-Fiber^{TM}. From fitting a simple additive model to the data for the four observed acoustic modes, we determine the strain-acoustic coefficient of bulk P_{2}O_{5} to be about + 9854m/sec/*ε* with maximum uncertainty of 1326m/sec/*ε*, which is much lower than that of pure silica ( = + 29240m/sec/*ε*). We showed that these bulk coefficients can be used to predict the strain-dependent Stokes’s shift of the higher-order acoustic modes with a high degree of accuracy.

Finally, the Pockels’ coefficients for bulk P_{2}O_{5} are estimated. Using the measured SOC of of bulk P_{2}O_{5}, the reported *p*_{12} of bulk SiO_{2} and pure silica fiber [12], the measured *p*_{12} of the P_{2}O_{5}-doped silica fiber, and a Poisson ratio of bulk P_{2}O_{5} from the literature [20], we obtain *p*_{12} of 0.252 ± 0.008 and a value of 0.132 ± 0.043 for *p*_{11} for bulk P_{2}O_{5}. This data are useful for the design of acoustic profiles of optical fiber for applications where Brillouin scattering is encountered.

## 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.

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