## 2 Theoretical Model  
In an attempt to explore the whole process of the absorbed energy conversion, the model is focused on the interaction of the one-dimensional plane of a sample. For simplicity, acoustic heterogeneity is ignored, giving a constant acoustic propagation velocity vs. over the considered region. Effects of optical dispersion are ignored assuming excitation and probe wavelengths are the same as their respective central wavelength for their respective whole beam bandwidths. Additionally, because the phantom is covered with agar, there is no free flow, convection heat transfer can be ignored in the model. When the pulse laser irradiates the sample, according to the energy conservation formula, the temperature changes according to Equation (1) below [22; 23]:  
\[\rho\;\mathrm{Cp}\;\frac{\partial\mathrm{T}}{\partial\mathrm{t}}-(\
abla\; \mathrm{k}\;\
abla\mathrm{T})=\mathrm{Q} \tag{1}\]  
where \(\rho\) is the mass density, \(\mathrm{Cp}\) is the specific heat capacity at constant pressure, \(\mathrm{k}\) is the thermal conductivity, \(\mathrm{T}\) is the temperature, and \(\mathrm{Q}\) is the energy of the pulsed laser. Consider the following Fourier transform pairs  
\[ \begin{cases}\frac{\partial\mathrm{T}}{\partial\mathrm{t}}=\mathrm{a}^{2} \;\frac{\partial^{2}\mathrm{T}}{\partial\mathrm{x}^{2}}+\mathrm{A}(\mathrm{r}, \;\mathrm{t})\\ \mathrm{T}|_{t=0}=\varphi(\mathrm{r})\end{cases}  \tag{2}\]  
where if assuming \(\mathrm{a}^{2}=\frac{\mathrm{K}}{\rho\;\mathrm{C}\mathrm{P}}\), \(\mathrm{A}=\frac{\mathrm{Q}}{\rho\;\mathrm{Cp}}\), and \(\varphi(\mathrm{r})=\mathrm{T}_{0}=293\;\mathrm{K}\), then after Fourier transform and inverse Fourier transform,  
\[\mathrm{T}(\mathrm{x},\;\mathrm{t})=\frac{1}{2\;\mathrm{a}\;\sqrt{\pi\;t}}\; \int_{-\infty}^{\infty}\;\varphi(\mathrm{\upxi})\;\mathrm{e}^{-\frac{(\mathrm{ x}-\mathrm{\upxi})^{2}}{4\mathrm{\upxi}^{2}}}\;\mathrm{d}\mathrm{\upxi}+\frac{1}{2\; \mathrm{a}\;\sqrt{\pi}}\;\int_{0}^{\mathrm{t}}\;\int_{-\infty}^{\infty}\; \frac{\mathrm{A}(\mathrm{\upxi},\;\mathrm{\uptau})}{\sqrt{t-\tau}}\;\mathrm{ e}^{-\frac{(\mathrm{x}-\mathrm{\upxi})^{2}}{4\mathrm{\upxi}^{2}}(\mathrm{t}-\mathrm{\uptau})}\; \mathrm{d}\mathrm{\upxi}\mathrm{d}\mathrm{\uptau} \tag{3}\]  
so Equation (1) can be simplified as Equation (4) as follows:  
\[\mathrm{T}(\mathrm{x},\;\mathrm{t})\;\frac{\mathrm{T}_{0}}{2\;\mathrm{a}\; \sqrt{2\mathrm{t}}}\;\left(1+\mathrm{erf}\!\left(\frac{\mathrm{\upxi}-\mathrm{ r}}{2\;\mathrm{a}\;\sqrt{\mathrm{t}}}\right)\right)+\frac{1}{2\;\mathrm{a}\;\sqrt{\pi}}\; \frac{1}{\rho\;\mathrm{C}_{\mathrm{P}}}\;\int_{0}^{\mathrm{t}}\;\int_{-\infty} ^{\infty}\;\frac{\mathrm{Q}(\mathrm{\upxi},\;\mathrm{\uptau})}{\sqrt{t-\tau} }\;\mathrm{e}^{-\frac{(\mathrm{x}-\mathrm{\upxi})^{2}}{4\mathrm{\upxi}^{2}( \mathrm{t}-\mathrm{\uptau})}}\;\mathrm{d}\mathrm{\upxi}\mathrm{d}\mathrm{ \uptau} \tag{4}\]  
The analytical solution cannot be directly derived from Equation (3) but the numerical solution can be obtained. Therefore, as the temperature increases, the displacement \(\overset{\rightarrow}{\mathrm{u}}(\mathrm{r},\mathrm{t})\) of this region will change [24]. \[\
abla\overset{\rightarrow}{\mathrm{u}}(\mathrm{r},\mathrm{t})=\beta\mathrm{ T}(\mathrm{r},\mathrm{t})-\frac{\mathrm{P}(\mathrm{r},\mathrm{t})}{\rho\mathrm{V_{s}}^{2}} \tag{5}\]  
where \(\overset{\rightarrow}{\mathrm{u}}(\mathrm{r},\mathrm{t})\) is the displacement, \(\beta\) is the expansion coefficient at constant pressure, \(\mathrm{P}(\mathrm{r},\mathrm{t})\) is the pressure, and vs. is acoustic velocity. The first term on the right side of Equation (5) indicates the temperature expansion caused by stimulated endothermic reaction, and the second term implies the disturbance of displacement caused by pressure at this point.