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Sánchez, L.A.; Díez, A.; Cruz, J.L.; Andrés, M.V. Sensor Applications of Forward Brillouin Scattering. Encyclopedia. Available online: https://encyclopedia.pub/entry/41299 (accessed on 20 June 2024).

Sánchez LA, Díez A, Cruz JL, Andrés MV. Sensor Applications of Forward Brillouin Scattering. Encyclopedia. Available at: https://encyclopedia.pub/entry/41299. Accessed June 20, 2024.

Sánchez, Luis A., Antonio Díez, José Luis Cruz, Miguel V. Andrés. "Sensor Applications of Forward Brillouin Scattering" *Encyclopedia*, https://encyclopedia.pub/entry/41299 (accessed June 20, 2024).

Sánchez, L.A., Díez, A., Cruz, J.L., & Andrés, M.V. (2023, February 16). Sensor Applications of Forward Brillouin Scattering. In *Encyclopedia*. https://encyclopedia.pub/entry/41299

Sánchez, Luis A., et al. "Sensor Applications of Forward Brillouin Scattering." *Encyclopedia*. Web. 16 February, 2023.

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In-fiber opto-mechanics based on forward Brillouin scattering enables sensing the surrounding of the optical fiber. Optical fiber transverse acoustic resonances are sensitive to both the inner properties of the optical fiber and the external medium. A particularly efficient pump and probe technique—assisted by a fiber grating—can be exploited for the development of point sensors of only a few centimeters in length. When measuring the acoustic resonances, this technique provides the narrowest reported linewidths and a signal-to-noise ratio better than 40 dB. The longitudinal and transverse acoustic velocities—normalized with the fiber radius—can be determined with a relative error lower than 10^{−4}, exploiting the derivation of accurate asymptotic expressions for the resonant frequencies.

forward Brillouin scattering
opto-mechanics
acoustic transverse resonances
temperature ans strain sensors
Poisson's ratio

The study of forward Brillouin scattering (FBS) in optical fibers, i.e., forward scattering of a guided optical wave by transverse acoustic resonances of the fiber itself, started in the 1980s ^{[1]} and is attracting increasing interest in recent years ^{[2]}. Early studies employed optical heterodyne detection to resolve the fine structure of thermally excited acoustic modes. Since then, continuous improvements in the excitation and detection approaches have impelled both fundamental studies and sensor applications. Preferable excitation schemes use either a simple optical pulse to excite a broadband of acoustic frequencies simultaneously ^{[3]}, or a dual-frequency laser source for the selective excitation of acoustic resonances matching the frequency difference ^{[4]}. In both cases, electrostriction is the dominant physical effect responsible for the optical excitation of transverse acoustic resonances. Although heterodyne detection is always an option ^{[5]}, detection has been carried out frequently using a Sagnac interferometer driven by an auxiliary probe signal.

The idea of emulating the success of distributed fiber sensing based on backward Brillouin scattering has driven some recent developments, bearing in mind that FBS would enable measuring properties of the fiber surrounding. The need for removing the coating of a standard optical fiber is certainly a severe drawback for practical applications of distributed sensors ^{[6]}^{[7]}. Typically, only some sections of the fiber are uncoated, and the reported spatial resolutions are higher than 2 m. One approach to overcome this limitation is the use of optical fibers coated with a thin layer of polyimide, since the mechanical properties of this material significantly reduce the attenuation of acoustic waves in silica fibers ^{[8]}^{[9]}. Thus, it is possible to distinguish between air, water, and ethanol outside a fiber coated with polyimide ^{[10]}, with a reported resolution of 50 m. It is worth mentioning that FBS has been demonstrated to be a useful tool for the characterization of elastic properties of fiber coatings ^{[11]}.

Sensor applications based on FBS would appear to be doomed to large spatial resolutions of the order of meters. Thus, sensing liquids with a simple drop would be beyond the achievable. The development of point sensors in which the physical mechanism for sensing the external medium is the acoustic field, but not the optical field, can give rise to a range of applications parallel and complementary to the more conventional fiber sensors based on optical mechanisms.

$$\delta {\lambda}_{LPG}=\Lambda \delta {n}_{co},\delta {n}_{co}=\frac{\delta T}{s\Lambda}.$$

In addition to a good SNR, the present technique provides the narrowest reported linewidths for the transverse acoustic resonances. The series of resonances observed in **Figure 3** are the radial resonances, R_{0,m}. Accurate measurement of each resonance with a RF signal analyzer permits to determine its linewidth. **Figure 3** shows the spectra of resonances R_{0,5} and R_{0,10}—experimental points and fitted Breit–Wigner–Fano function—and the linewidth of R_{0,m} modes versus their resonance frequencies.

The characteristic equations for R_{0,m} and TR_{2,m} resonances are:

$${\mathrm{R}}_{0,m}\mathrm{resonances}:\left(1-{\alpha}^{2}\right){J}_{0}\left(\alpha z\right)-{\alpha}^{2}{J}_{2}\left(\alpha z\right)=0,$$

$${\mathrm{TR}}_{2,m}\mathrm{resonances}:\left|\begin{array}{cc}\left(3-{z}^{2}/2\right){J}_{2}\left(\alpha z\right)& \left(6-{z}^{2}/2\right){J}_{2}\left(z\right)-3z{J}_{3}{\left(z\right)}^{}\\ {J}_{2}\left(\alpha z\right)-\alpha z{J}_{3}\left(\alpha z\right)& \left(2-{z}^{2}/2\right){J}_{2}\left(z\right)+z{J}_{3}{\left(z\right)}_{}\end{array}\right|=0,$$

where z is the normalized frequency given by z = 2πa f /V_{S}, and α = V_{S}/V_{L}, with f being the frequency, a the fiber radius, V_{S} and V_{L} the shear and longitudinal acoustic wave velocities, and J_{m} the Bessel functions of the first kind of order, m.

Having in mind the idea of extracting the information from the whole spectrum of acoustic resonances, better than from one or two resonances, as it is usual in sensor applications developed so far, we found it very useful to derive accurate asymptotic expressions for the resonance frequencies determined by Equations (2) and (3). Using Hankel’s asymptotic expansions of Bessel functions for large arguments ^{[15]}, and following the procedure outlined in ^{[16]}:

$${\mathrm{R}}_{0,m}\mathrm{resonances}:{f}_{R,m}=\frac{{V}_{L}}{2\pi a}\left[{c}_{m}-\frac{16{\alpha}^{2}-1}{8{c}_{m}}\right],$$

$${\mathrm{TR}}_{2,m}^{\left(1\right)}\mathrm{resonances}:{f}_{TR,m}^{\left(1\right)}=\frac{{V}_{S}}{2\pi a}\left[{c}_{m+1}-\frac{15}{8{c}_{m+1}}\right],$$

$${\mathrm{TR}}_{2,m}^{\left(2\right)}\mathrm{resonances}:{f}_{TR,m}^{\left(2\right)}=\frac{{V}_{L}}{2\pi a}\left[{c}_{m+1}-\frac{15}{8{c}_{m+1}}\right],$$

where ${c}_{m}=m\pi -\pi /4$ , m = 1, 2, 3, etc. These expressions retain the first two dominant terms for high-order resonances and provide high-accuracy numerical values for the frequencies of resonances, provided there is no degeneracy between ${\mathrm{TR}}_{2,m}^{\left(1\right)}$ ,m and ${\mathrm{TR}}_{2,m}^{\left(2\right)}$ ,m resonances.

An accurate determination of Poisson’s ratio (ν) of optical fibers is an evasive issue that has been unattainable for many years. A value ranging between 0.16 and 0.17 is assumed, with a relative error of 6% ^{[17]}^{[18]}. The determination of ν is carried out typically by combining interferometric and polarimetric measurements. Using the pump & probe technique reporterd here and the asymptotic expressions (4) and (5), it is possible to achieve an accuracy improvement of about two orders of magnitude, pushing the relative error down to 10^{−3}. This result proves the potential of the pump and probe approach to develop point sensors based on FBS with low detection limits ^{[19]}.
### 4.2. Simultaneous Strain and Temperature Measurement with a Single-Point Sensor

**Figure 4.** Relative frequency shift of resonances ${\mathrm{R}}_{0,20}$ and ${\mathrm{TR}}_{2,24}^{\left(1\right)}$ versus temperature (**a**) and strain (**b**). Both figures include the averaged values of Δf/f over all the resonances, ${\mathrm{R}}_{0,m}$ and ${\mathrm{TR}}_{2,m}^{\left(1\right)}$ , of each series (solid lines).

Here, the implementation of simultaneous and discriminative measurements of strain (ε) and temperature using a single-point sensor that exploits the FBS pump and probe technique is discussed. The proposed approach exploits the different sensitivities of radial, ${\mathrm{R}}_{0,m}$ , and torsional-radial, ${\mathrm{TR}}_{2,m}^{\left(1\right)}$ , resonances with strain and temperature, generated by the different temperature and strain coefficients of the longitudinal and shear acoustic wave velocities (∂V_{L}/∂T ≠ ∂V_{S}/∂T and ∂V_{L}/∂ε ≠ ∂V_{S}/∂ε). In addition, for large values of the order m and according to the asymptotic expressions (4) and (5), we found that the relative shift of all the resonance frequencies of radial modes versus temperature and strain, $\Delta {f}_{R,m}/{f}_{R,m}$ , will be independent of the order m, and the same happens for the relative shift of the torsional-radial resonances, $\Delta {f}_{TR,m}^{\left(1\right)}/{f}_{TR,m}^{\left(1\right)}$ . Thus, instead of measuring only one radial resonance and one torsional-radial resonance, it can be more robust to measure several of them, or even the whole spectrum. **Figure 4** shows the relative frequency shift of two specific resonances and the averaged value obtained using the whole spectrum, showing that there is a perfect agreement. From these measurements, one can calibrate the sensor and obtain the temperature and strain coefficients for $\Delta {f}_{R}/{f}_{R}$ and $\Delta {f}_{TR}/{f}_{TR}$ defined by the elements ${c}_{R,TR}^{\epsilon ,T}$ of the following 2 × 2 matrix:

$$\left[\begin{array}{c}\Delta {f}_{R}^{}/{f}_{R}\\ \Delta {f}_{TR}^{}/{f}_{TR}\end{array}\right]=\left[\begin{array}{cc}{c}_{R}^{\epsilon}& {c}_{R}^{T}\\ {c}_{TR}^{\epsilon}& {c}_{TR}^{T}\end{array}\right]\left[\begin{array}{c}\Delta \mathsf{\epsilon}\\ \Delta T\end{array}\right].$$

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*Optics Express***2022**,*30*, 42, https://doi.org/10.1364/OE.442295.

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