## Abstract

Residual stresses inside optical fibers can impact significantly on Brillouin spectrum properties. We have analyzed the importance of internal stresses on the Brillouin Gain Spectrum (BGS) for a conventional G.652 fiber and compared modeling results to measurements. Then the residual internal stresses have been investigated for a set of trench-assisted fibers: fibers are coming from a single preform with different draw tensions. Numerical modeling based on measured internal stresses profiles are compared with corresponding BGS experimental results. Clearly, Brillouin spectrum is shifted linearly versus draw tension with a coefficient of −20MHz/100g and its linewidth increases.

© 2012 OSA

## 1. Introduction

Fabrication of preforms and optical fibers always generates intrinsic residual stresses. The difference of doping compositions between core and cladding results in a mismatch of thermal expansion coefficients and viscosities. Thermal internal stresses are created during the cooling from fusion temperature to room temperature. Moreover, drawing conditions can modify significantly the internal frozen-stresses [1]. These stresses introduce changes in the refractive optical index profile and the acoustic velocity profile. So, Brillouin frequency shift is modified. While the dependence of the Brillouin Gain Spectrum (BGS) with the doping composition profile is already well-known [2,3], only recently the influence of internal frozen- stresses in the BGS has been investigated [4,5].

In this paper, we propose a 2D-FEM model to predict BGS of optical fibers considering the corresponding measured frozen-stresses profiles. Optical and acoustic properties are calculated from measured doping profiles of fibers [6,7]. This model is able to predict the BGS’s behaviors of any optical fiber with mono- or multi-doping composition, considering any residual stresses profile. In order to validate the inclusion of residual stresses in our model, we consider fibers with different stresses profiles and we compare our Brillouin gain simulations to experimental measurements. We present the BGS of a conventional G.652 step-index fiber including residual internal stresses profile. Then, we analyze a set of G.657 single-trench-assisted Bend Insensitive SMFs (BI-SMF). All these fibers are coming from a single preform but are drawn with different tensions. The calculated BGS spectra show very good agreement with corresponding measurements. These results confirm the influence of internal composition and residual frozen-in stresses on the BGS properties.

## 2. Theoretical background and 2D-FEM model of simulation

BGS calculation requires a rigorous determination of the acoustic-optical interaction (i.e. overlap integral between optical and acoustic modes). The modeling of optical and longitudinal acoustic waves was performed, neglecting the contribution of torsional modes in the BGS [8]. The properties of the optical mode $E(x,y)$ and the longitudinal acoustic m^{th}-order L_{0m} mode ${u}_{m}(x,y)$are determined by solving the 2D scalar-wave propagation Eqs. (1) and (2) respectively [6,9,10]:

_{01}optical mode, ${\beta}_{acoust}$ is the acoustic propagation constant, ${u}_{m}(x,y)$ is the longitudinal displacement field of the L

_{0m}acoustic mode and ${\Omega}_{m}$ is the acoustic Brillouin resonance angular frequency. Assuming that ${\Omega}_{m}$ is much smaller than the optical frequency, the phase matching (i.e. Bragg condition) with the optical wave leads to the relation ${\beta}_{acoust}=\text{\hspace{0.17em}}2{\beta}_{opt}$ where ${\beta}_{opt}=2\pi {n}_{eff}/\lambda $ is the propagation constant of the optical mode. The Brillouin frequency shift is then ${\Omega}_{m}=4\pi {n}_{eff}{V}_{eff}/\lambda $where ${V}_{eff}$is the effective velocity of the longitudinal acoustic mode.

It is well-known that the refractive index *n* and the longitudinal acoustic velocity _{${V}_{L}$} vary on the cross-section of the fiber according to the type and concentration of doping materials [11]. The refractive index profile is measured on optical fibers with a near-field measurement technique (EXFO NR-9200). The relative contributions of the different dopants are measured by chemical analysis on the preform. The fractional dependence for unit doping concentration on the optical and acoustic properties of pure silica is listed in Table 1
for used dopants. Propagation equations were solved using COMSOL Multiphysics, a FEM-2D commercial solver. The model and the parameters of resolution are precisely described in [6].Due to the exponential decay of acoustic wave magnitude, the spontaneous Brillouin spectrum of each acoustic mode has a Lorentzian shape. As each acoustic mode contribution adds up in an incoherent way, the BGS is then computed by adding Lorentzian curves centered at each mode’s Brillouin frequency shift ${\nu}_{B}^{m}={\Omega}_{m}/2\pi $ weighted by the acousto-optic overlap integral _{${I}_{m}^{ao}$} [2] (see Eq. (3) and Eq. (4)). In this work, the full width at half maximum (FWHM), noted Γ, of Lorentzian curves is assumed to be identical for all the modes and depends of the fiber-under-test.

The residual internal stresses in an optical fiber not only induce changes in optical refractive index [12] but also modify the acoustic velocity behavior. The acoustic velocity of a homogenous and isotropic infinite material subjected to a uniaxial longitudinal stresses can be expressed as [13]:

where_{${V}_{L}^{0}(x,y)$}is the longitudinal acoustic velocity profile on the fiber cross-section without stresses, $\sigma (x,y)$is the total axial stresses (both thermal and draw-induced stresses), varying on the fiber cross-section. The parameter

_{${K}_{L}$}is the acousto-elastic coefficient and represents the velocity variation of the acoustic wave with stresses. Second-order expression of

_{${K}_{L}$}(Eq. (6)) can be found in [13]. The Lame’s constants $\lambda $(Eq. (7)) and

_{$\mu $}(Eq. (8)) are calculated from the profiles of the stress-free longitudinal and shear wave velocities ${V}_{L}^{0}$and

_{${V}_{T}^{0}$}respectively.

_{${V}_{T}^{0}$}are computed considering the doping composition [11] and the acoustic velocities of pure fused silica ${V}_{L}^{silica}$and

_{${V}_{T}^{silica}$}. Note, that glass material’s has been measured on used preforms in the [5940m/s - 6000m/s] range depending of the fictive temperature during the drawing process [14]. The typical value of 3750m/s has been taken for

_{${V}_{T}^{silica}$}[15]. Lame’s silica constants are typically $\lambda $ = 16GPa and

_{$\mu $}= 31GPa, that results ${K}_{L}$ = 3.4 10

^{−11}Pa

^{−1}.

The calculated BGS is compared to experimental results that have been obtained using a ~1km fiber. The BGS measurement has been performed using the well-known self-heterodyne technique [16] (see experimental set-up in Fig. 1 ). A 1560nm DFB laser is used as input signal. The heterodyne detection of both the Brillouin frequency shift back-propagated Stokes wave in the fiber-under-test and the injected input signal allows to direct measurement of the BGS. A polarization scrambler is used for polarization-insensitive measurement. The laser linewidth is of the order of 1MHz, compared to 30-40MHz for the acoustic modes, so the detected electrical spectrum corresponds directly to the spontaneous BGS.

## 3. Validation of stresses profile impact for a step-index fiber

The 2D-FEM model including stress dependence has been validated on a conventional G.652 step-index fiber sample drawn at a typical 90g tension. The fiber consists of a GeO_{2}-doped core fiber with a radius of 5µm and a 6% [wt%] concentration, and a pure silica cladding with radius 62.5µm. The measured index and stress profiles are given in Figs. 2(a)
and Fig. 2(b) respectively.The BGS of the characterized step-index fiber has four significant L_{0m} modes. The L_{01} mode is dominating because its acousto-optic overlap integral${I}_{1}^{ao}$~0.9. The acousto-optic integrals are different for simulated BGS with and without stresses. We can observe a difference of 7MHz in the Brillouin frequency shifts if we don’t account for the residual stresses. Figure 3(a)
plots acousto-optic overlap integrals_{${I}_{m}^{ao}$}.We observe that the L_{04} mode is not excited if we compute the BGS without stresses. Modal repartitions of L_{01} and L_{02} acoustic modes are plotted in Fig. 3(b). Clearly, the spatial distributions of acoustic modes don’t change, so the acousto-optic overlap integral values are similar as shown in Fig. 3(a). However, the Brillouin frequencies vary significantly with the stresses consideration. It can be explained by the great guiding of acoustic modes in the core where the acoustic velocity dependence with stress is almost uniform. Finally, the computed BGS is compared to the experimental result as shown in Fig. 4
. We observe a slightly different overlap integral for L_{02} mode between the experiments and the simulations that is probably due to index measurement sensitivity around 10^{−4}. Even if residual stresses are not very high, their consideration make the modeling even more accurate.

## 4. Draw tension impact for trench-assisted profile

Single-trench-assisted BI-SMFs fibers [17] corresponding to the recent G.657 fibers designed for small bending radius applications. These properties let BI-SMFs be also interesting for sensing applications especially for distributed stress and temperature sensors based on Brillouin effect. For our knowledge, the impact of the trench to the Brillouin shift had never been evaluated. A schematic trench-assisted step index profile shape is represented in Fig. 5 .

During fabrication process, with or without applied external forces, stresses develop inside the fibers during cooling at room temperature. The draw-induced stresses are due to a difference of the viscoelastic properties in the core and the cladding respectively [18]. So the amplitude of residual stresses depends on the doping composition of fibers and the drawing tensions. In order to investigate the impact of draw-induced stresses on the BGS, we used optical fibers coming from a single preform (i.e. same doping composition and geometric parameters), with same melting point, but drawn with different tensions.

The draw tensions are going from very low values (20g, 40g) to very high values (120g, 160g). The internal stresses profiles have been measured based on a polarimetric technique [19]. The core stresses have negative values, corresponding to a compressive state, and decrease with increasing draw tensions. Acoustic velocities profiles have been calculated according to the measured index and stresses profiles.

An example of stresses impact on the BGS is plotted in Fig. 6(a)
. As for the step-index G.652 fiber, the L_{01} mode has the most significant contribution to BGS in the trench-assisted fibers. Pure Silica used for the preform has a refractive index *n* = 1,444 and a velocity = 5990m/s. The BGS are plotted in Fig. 6(b) for three different tensions. A very good agreement between BGS calculation and measurements is obtained when the stresses are taken into account. Brillouin spectra are moving towards decreasing frequencies when draw tension increases, so when absolute value of core stresses increase. The evolution of L_{01} mode Brillouin frequency shift is plotted in Fig. 7(a)
according to draw tension. The results are given with an uncertainty of 2MHz for the Brillouin frequency shifts. We have also plotted the evolution of the ratio R corresponding to core stresses values (σ(r)|_{r = 0}), the highest draw tension (160g) is chosen as reference (Eq. (9)). We observe that in absolute value, the core stresses increase with draw tension. Due to the uncertainty in the draw tension ( ± 10g) and the difficulty of the drawing process stabilization at low tension, we have a quite significant scattering in the data. Moreover the uncertainty of stresses measurement, in the order of 1MPa, contributes to this scattering. Amplitude of the core stresses varies linearly with draw tension with a slope of (−0.48 ± 0.08)MPa/g.

The obtained experimental results show that the Brillouin shift decreases with draw tension. A linear approximation give a −20MHz/100g dependence (accuracy ± 2.4MHz/100g.). Note that this value is about half the value found in [4]. This difference can be explained by the higher GeO_{2} concentration and the multi-doping composition of the fibers. Moreover, the BI-SMF index profiles are different from [4].

The FWHM Brillouin linewidth of L_{01} mode is shown in Fig. 7(b) for the different draw tensions. The measurement of Brillouin linewidth is estimated to ~1MHz. We have observed that the Brillouin linewidth increases significantly with draw tension. It can be probably explained by the structural order/disorder changes in the silica glass due to the increase of stresses level.

## 6. Conclusion

That is the first time to our knowledge that effective residual stresses profiles are considered in the calculation of BGS for any profile type. This approach has been successfully demonstrated for single trench-assisted G.657 fibers for extreme drawing conditions. For these fibers the increase of drawing tension, thus an increase in absolute value of internal residual stresses induces a decrease of Brillouin frequency shift and an increase of Brillouin linewidth. The stresses variation not only induces a BGS frequency shift, but it changes BGS shape through its impact on the acousto-optic integrals. Next step will consist in analyzing a larger number of fibers, especially highly doped fibers or fibers with very high draw tensions for which residual stress amplitudes are much more important.

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