## Acoustic impedance matching condition

The transverse acoustic waves are activated to steady-state by launching an intense activating pulse via electrostriction effect. Once the pulse is interrupted, the transverse acoustic waves can still maintain the phase continuity with the driven acoustic field and decay exponentially [19]. The reflection and transmission of the transverse acoustic waves which propagate within multi-layered structures are determined by the acoustic impedances of each layer. Assuming that the reflection and transmission of the wave at the interface are energy-conserving, the acoustic wave reflectivity at the boundary is expressed as [20]

$$r=\left| {\frac{{{Z_s} - {Z_i}}}{{{Z_s}+{Z_i}}}} \right|,$$

1

where *Z**s* and *Z**i* are the acoustic impedance of silica and coating respectively. Note that when the impedance of the cladding and coating layers are equal (*Z**s* = *Z**i*), the acoustic wave reflectivity is zero (*r* = 0), which means that the transmitted power of the acoustic wave is maximum, i.e., the acoustic impedance matching condition. Concurrently, the acoustic impedance of the outer material of the fiber is directly related to the resonance linewidth Γ*m* of the FSBS spectra [24]

$${\Gamma _m}={\Gamma _s}+\frac{{{V_L}}}{{\pi d}}\ln \frac{1}{{\left| R \right|}},$$

2

where Γ*s* is the intrinsic linewidths of FBSS resonance, *V**L* = 5996 m/s is denoted as the longitudinal acoustic velocity, R is the total reflectance of sound waves. The boundary condition of the acoustic wave can be represented by [1,25]

$$(1 - \frac{{V_{T}^{2}}}{{V_{L}^{2}}}){J_0}(y) - \frac{{V_{T}^{2}}}{{V_{L}^{2}}}{J_2}(y)=0,$$

3

where *V**T* = 3740 m/s is the transverse acoustic velocity. *J**n* is the *n*th Bessel function of the first kind. With the *m*th-order root *y**m*, the boundary condition gives rise to an inverse relationship between the resonance frequency *f**m* and the cladding diameter d [9]

$${f_m}=\frac{{{V_L} \cdot {y_m}}}{{\pi d}}.$$

4

Fortunately, the acoustic impedance of the Aluminum (~ 17.21×106 kg/(m2⋅s)) [26] is close to that of Silica (~ 13.19×106 kg/(m2⋅s)) [20], realizing a quasi-acoustic impedance matching (*r**si − Al* = 0.13). Thus, the acoustic waves tend to oscillate within the cross-section of the coating, rather than merely cover the area of cladding. Consequently, the dispersion relation given by Eq. (4) is no longer applicable which is expected to be replaced by the more general elastic dynamic equation [3]

$$\rho \frac{{{\partial ^2}{u_i}}}{{\partial {t^2}}} - {\left[ {{c_{ijkl}}{u_{kl}}+{\eta _{ijkl}}\frac{{\partial {u_{kl}}}}{{\partial t}}} \right]_j}={\left[ {{\varepsilon _0}{\chi _{klij}}E_{k}^{{(1)}}E_{l}^{{(2) * }}} \right]_j},$$

5

where\({\chi _{klij}}={\varepsilon _{im}}{\varepsilon _{jn}}{p_{klmn}}\), \(\rho\)is the medium density, \({c_{ijkl}}\)and\({\eta _{ijkl}}\)are stiffness matrix and viscosity matrix,\(\varepsilon\)and\({p_{klmn}}\)are dielectric constant and polarization tensor, respectively. According to Eq. (5), the acoustic field distribution , along with the resonance frequency can be calculated. The exemplary intensity and displacement distribution of *R*0,6 mode in both the conventional SMF and the used Al-fiber are present in Fig. 1(a), and (b), respectively. The significant difference is that the intensity of acoustic wave distribution in the coating of Al-fiber maintains a high level while that of SMF is rapidly attenuated to a negligible level, which is also valid for acoustic modes of other orders. Under the quasi-acoustic impedance matching between the cladding and Al-coating layer, the acoustic waves have the chance to directly interact with the outside media, realizing the chemical substance identification or liquid acoustic impedance sensing.

## Experimental setup of OMTDA system

Incorporating the Al-fiber, a high resolution OMTDA configuration is used to realize distributed optomechanical sensing, which is demonstrated in Fig. 2. A narrow linewidth laser of 1550 nm is divided into two branches. In the upper branch with 90% laser, an activation pulse and a probing pulse end to end are generated to excite and probe the transverse acoustic waves successively in a coherent way using an arbitrary waveform generator (AWG). The sequential radio-frequency pulse signals are imposed onto the light wave through a double-sideband electro-optic modulator (EOM1) working under the suppressed carrier condition, and the light power is amplified by an erbium-doped fiber amplifier (EDFA). Note a frequency interval *f*1 of 1.5 GHz is added to the activation pulse to remove the backward stimulated Brillouin scattering (BSBS) interaction between the activation pulse and the detecting light. After being modulated, the activation pulse contains four frequency components distributed symmetrically in pairs with the carrier component. According to Ref. [17], the probing pulse contains two frequency components with a frequency difference around the FSBS resonance frequency. Meanwhile, in the lower branch, another EOM2 cooperating with a tunable optical filter (OF2) is employed to generate the continuous detecting light. By switching the frequency loaded on the EOM2 between the Brillouin gain regions of the two sidebands contained by the probing pulse, the intensity evolution along the sensing fiber of the upper and lower sidebands are recorded via BSBS successively. Then, with the frequency difference between the probing pulse swept around the resonance frequency of the acoustic wave, the localized FSBS spectrum can be scanned and demodulated by applying differential calculation of the energy accumulation.