High spatiotemporal resolution optoacoustic sensing with photothermally induced acoustic vibrations in optical fibres

Optoacoustic vibrations in optical fibres have enabled spatially resolved sensing, but the weak electrostrictive force hinders their application. Here, we introduce photothermally induced acoustic vibrations (PTAVs) to realize high-performance fibre-based optoacoustic sensing. Strong acoustic vibrations with a wide range of axial wavenumbers kz are photothermally actuated by using a focused pulsed laser. The local transverse resonant frequency and loss coefficient can be optically measured by an intra-core acoustic sensor via spectral analysis. Spatially resolved sensing is further achieved by mechanically scanning the laser spot. The experimental results show that the PTAVs can be used to resolve the acoustic impedance of the surrounding fluid at a spatial resolution of approximately 10 μm and a frame rate of 50 Hz. As a result, PTAV-based optoacoustic sensing can provide label-free visualization of the diffusion dynamics in microfluidics at a higher spatiotemporal resolution.

. Effect of the acoustic source size on the spatial resolution capability. The step of the resonant frequency is ∆ =100 kHz, and its width is 200 μm. The size of the optoacoustic source is 10 μm in (a) and 800 μm in (b).
Supplementary Figure 2. Dispersion curves of the torsional-radial TR2n modes. (a) Dispersion diagrams of the lowest-order torsional-radial (TR2n) modes. Inset: mode profiles of the TR21 and TR23 modes. The arrows represent the amplitudes and directions of displacements. The interplay of the TR21 and TR22 modes forms an enclosed dispersion curve in the evanescent wave regime. It also significantly changes the dispersive property of the TR21 mode. (b) Plot of kz 2 as a function of acoustic frequency. In contrast to the non-dispersive assumption (red dashed curve), the TR21 mode has a slope of -7700 rad 2 m -2 s -1 , which induces a 12-fold narrower mode width. Acoustic mode profiles. Colour scale: amplitude of local displacement. The fibre has a slow variation in cross-sectional geometry, which is characterized by the diameter H. The TR21, TR31 and TR23 modes are degenerated with a round geometry. The side polishing modifies the transverse geometry, changes the mode profiles and lifts the mode degeneracy. Here, we perform a comparative study to estimate the optical-to-acoustic efficiencies of the electrostrictive force and photothermal excitation. The acoustic wave equation with a source can be generally expressed as

Supplementary
where p denotes the acoustic wave and Fopt is the optically induced acoustic source. For electrostriction, the source is written as represents the vacuum permittivity, =1.17 is the electrostrictive constant, and is the electric field of the incident light. Substituting | | = | | , where S denotes the Poynting vector, nsi=1.45 is the refractive index of silica glass, and c is the velocity of light in vacuum, and considering that the spot size of the focused laser beam is approximately 10 μm, we can approximately obtain ∇ = ≅ photothermally induced force with a 1 W m -2 intensity can be estimated as = 2.13 × 10 J m . The above calculation suggests that the amplitude of the electrostrictive force is largely determined by the spatial gradient of the optical intensity. It depends on how tightly the light beam is focused. In contrast, photothermal excitation is highly efficient, taking advantage of the short pulse duration and the high absorptive coefficient of the gold coating. As a result, the photothermal effect can have an overwhelming optical-to-acoustic efficiency and induce a 10 5 to 10 6 times stronger force at the laser spot.
Supplementary Note 2 Modelling of the optoacoustic sensing: a matrix method.
Here, we develop a matrix method performed in the kz domain for modelling of the optoacoustic sensing. A MATLAB code based on this method is provided. Calculations in the kz domain can differentiate the mechanisms between F-SBS and the present PTAV-based sensing.
Suppose a nonuniform optical fibre to be measured is characterized by axial variation ( ); the wave equation in the ~kz space domain can be expressed as where ( ) and ( , ) are the -domain Fourier transformation of axial variation ( ) and the external source in ~kz space, ( , ′ ) denotes the solution in the ′ domain, and " * " represents the convolution operator. The first step is to convert Equation S-4 into matrix form by quantizing the kz and domains into N segments. A larger quantization number enables better frequency and spatial resolutions, but the calculation is more time consuming. Thus, ( ), ( , ) and ( , ′ ) at a selected frequency can all be written as N-element vectors. Now, the wave equation can be represented in matrix form: is a diagonal matrix with the ith element equal to , . The term " ( ) * " is expressed as a Toeplitz matrix with shifting ( ) profile among each column, and its diagonal elements are all (0). Given a known vector that describes the acoustic source, the solution can be calculated via Here, we would like to differentiate F-SBS and PTAV-based optoacoustic sensing based on the matrix method. We set a Gaussian-shaped acoustic source in the MATLAB code. Here, is the Fourier transformation of the axial profile of the source. Supplementary Figure 1 suggests that a photothermally induced acoustic source (10 μm in size) can well reconstruct the target ( ) with a step change of 100 kHz and a width of 200 μm. In contrast, when the size of the acoustic source is increased to 800 μm, the step change can hardly be reconstructed. For F-SBS, fibres tens of metres long are typically needed due to the weak electrostrictive excitation. As a result, the high-kz components of the acoustic modes cannot be effectively excited for accurate reconstruction of the microstructure in the fibre.
Supplementary Note 3 Vibrational modes of a cylindrical optical fibre.
The acoustic vibration in an elastic cylinder can be described by the scalar and vectorial potentials φ and H, which satisfy the wave equations [3] (See the definition of terms in Supplementary Table 1 Substituting the densities of silica glass and the coating material and , as well as the fibre radius r0 and coating thickness dc (dc<<r0), the coating-induced resonant frequency change can be written as With the radial and azimuthal displacements ( , ) of the TR21 mode based on the theory in Supplementary Note 3, we can calculate that a gold coating (with a density of 1.93×10 4 kg m -3 ) can shift the resonant frequency by 2 kHz nm -1 . Figure 3 shows the dispersion diagram as well as the spatial mode profile of the PTAVs in water. The TR21 mode curve significantly broadens as a result of the strong acoustic interaction with the surrounding medium. The spectral recovery though fibre-optic detection still applies, which has been verified by the calculated mode profiles. Therefore, the acoustic spectra can be calculated with a two-dimensional model, supposing an invariant acoustic structure along the axial direction. Note that the fibre vibration can exert pressure waves, which are depicted as outwards propagating cylindrical waves ( ) ( ), where ( ) represents the outwards propagating pressure wave and Cn denotes its amplitude. The interaction between the solid fibre and surrounding medium can be depicted by the following linear equations, S-12 (a-c), based on the boundary conditions. The continuity of the radial stress/pressure can be expressed as [4]

Now consider the PTAVs damped by a liquid medium. Supplementary
Considering that shear waves are not supported in the fluidic medium, the expression presenting zero shear stress at the boundary still holds and can be rewritten as ) where a23=0 and b2=0. In addition, the continuity of radial displacement demands . Assume that the radial displacement created by the acoustic dipole source can be simply written as b3cos(lθ). The amplitude of b3 is related to the optical absorption, thickness of the absorptive layer, irradiation area and elastic properties of silica glass. However, the amplitudes do not affect the acoustic spectrum and will not be analysed in detail in this context. Combining Eqs. S-18 (a-c), the coefficients Al, Bl, and Cl can be solved. The response in terms of the beat frequency variation Δfb is proportional to the birefringence change described as Δ = • ∆ . The birefringence change ΔB is determined by  Figure 5b exhibits the corresponding frequency responses, which are normalized to the peak amplitude of the TR21 mode. The acoustic signal decays faster at higher impendence (decay rate from 9.7 to 15.5 μs -1 ), resulting in broadened TR21 resonance as a result of the stronger dissipation to the surrounding medium. In contrast, the TR23 mode is hardly sensitive to the impedance due to the much weaker interaction with the surrounding medium. This is a result of the minimal radial displacement at the fibre surface, which originates from the cancellation in the radial direction (see Supplementary Figure 2). The impedance response can be understood based on a simplified resonator model. The acoustic spectrum of the damped fibre is largely determined by the term • det ( ) in the denominator, which is extended as • det( ) = ( ) ( ) + ( ) ( ). Considering the phase relation between the Hankel function and its derivative (90 degrees in advance), these two terms are analogous to detuning and damping terms, respectively. Supplementary Figure 6 shows the amplitudes of these two terms. At frequencies of 22.3 and 39.6 MHz with the detuning term approaching zero, the acoustic response reaches a maximum. The damping term contains ρwcw, suggesting that the radiation loss is proportional to the acoustic impedance of the surrounding medium. The spectral width can be approximately expressed as Δ = • (S-21) The first term of Eq. S-21 suggests that the spectral width is proportional to the impedance contrast between the two media. The second term suggests its relation to the vibration mode property, which is determined by the parameters including acoustic velocities and material densities. Comparing the TR21 and TR23 modes, we found that the effective slopes of detuning are 4.88×10 8 and 1.11×10 10 Pa MHz -1 and the damping rates are 1.95×10 9 and 6.23×10 8 Pa, respectively. As a result, the spectral width of the TR23 peak is approximately 70-fold narrower than that of the TR21 peak. This suggests a much weaker interaction between the TR23 mode and the surrounding medium. Both the detuning and damping terms approach infinity at 38.4 MHz as a result of the infinitely large Γ. This corresponds to the Fano resonance between the TR21 and TR23 modes.