## 2 Theoretical Model
The relation between displacement and pressure produced by the volume element is  
\[\rho\frac{\partial^{2}\overset{\rightarrow}{\mathrm{u}}(\mathrm{r},\mathrm{ t})}{\partial\mathrm{t}^{2}}=-\
abla\mathrm{P}(\mathrm{r},\mathrm{t}) \tag{6}\]  
Assuming that the above formula is continuously differentiable in time and space, then substituting Equation (5) into Equation (6) and calculating the divergence leads to  
\[\beta\;\rho\;\frac{\partial^{2}\mathrm{T}(\mathrm{r},\;\mathrm{t})}{\partial \mathrm{t}^{2}}-\rho\;\frac{\partial^{2}}{\partial\mathrm{t}^{2}}\;\left( \frac{\mathrm{P}(\mathrm{r},\;\mathrm{t})}{\rho\mathrm{V_{s}}^{2}}\right)=- \
abla^{2}\mathrm{P}(\mathrm{r},\;\mathrm{t}) \tag{7}\]  
Substituting Equation (1) into Equation (7) gives  
\[\
abla^{2}\mathrm{P}(\mathrm{r},\;\mathrm{t})-\frac{1}{\mathrm{V_{s}}^{2}}\; \frac{\partial^{2}}{\partial\mathrm{t}^{2}}\mathrm{P}(\mathrm{r},\;\mathrm{t}) =-\;\frac{\beta}{\mathrm{C}_{\mathrm{P}}}\;\frac{\partial}{\partial\mathrm{t} }(\mathrm{Q}(\mathrm{r},\;\mathrm{t})+\mathrm{k}\;\
abla^{2}\mathrm{T}(\mathrm{r },\;\mathrm{t})) \tag{8}\]  
Equation (8) is the prevailing photoacoustic wave equation if the term \((\frac{\beta}{\mathrm{C}_{\mathrm{P}}}\;\frac{\partial}{\partial\mathrm{t}} \mathrm{Q})\) is removed [25]. In this paper, this item is retained. Based on the [PERSON]'s function [24; 26] Equation (8) can be expressed as a unified form of the partial differential equation, \(\mathrm{L}\;\mathrm{P}(\mathrm{r},\mathrm{t})=\mathrm{F}(\mathrm{r},\mathrm{t})\),  
\[\mathrm{L}\;\mathrm{P}(\mathrm{r},\mathrm{t})=\mathrm{F}(\mathrm{r},\mathrm{t}) \tag{9}\]where  
\[L=\
abla^{2}-\frac{1}{\rm{v_{s}}^{2}}\ \frac{\partial^{2}}{\partial t^{2}},\ F(r,\ t)=- \frac{\beta}{C_{P}}\ \frac{\partial}{\partial t}(Q(r,\ t)+k\ \
abla^{2}T(r,\ t)). \tag{10}\]  
The corresponding [PERSON]'s function can be defined as G(r, t; r', t') satisfying the equation  
\[L\ G(r,t;r^{\prime},t^{\prime})=-\delta(r-r^{\prime})\ \delta(t-t^{\prime}), \tag{11}\]  
where r' and t' indicate the position and time of the source, respectively. In infinite space, and when there is no boundary, according to initial conditions,  
\[\left\{ \begin{array}{l}G(r,t;r^{\prime},t^{\prime})|_{t=t^{\prime}}=\delta(r- r^{\prime})\\ \frac{\partial}{\partial t}G(r,t;r^{\prime},t^{\prime})\Big{|}_{t=t^{\prime}}= 0\end{array} \right. \tag{12}\]  
G(r, t; r', t') can be expressed as  
\[G\big{(}r,t;r^{\prime},t^{\prime}\big{)}=\frac{\delta(t-t^{\prime}-|r-r^{ \prime}|/\rm{v_{s}})}{4\pi|r-r^{\prime}|} \tag{13}\]  
In general, assuming that the photoacoustic pressure produced by any light source in an infinite medium can be obtained  
\[P(r,t)= \int_{0}^{t^{\prime}}dt^{\prime}\int_{-\infty}^{+\infty}dr^{\prime}G \big{(}r,t;r^{\prime},t^{\prime}\big{)}\ \frac{\beta}{c_{P}}\frac{\partial}{\partial t}\Big{(}Q(r^{\prime},t^{\prime}) +k\
abla^{2}T(r^{\prime},t^{\prime})\Big{)}\bigg{|}_{t^{\prime}=t-\frac{|r-r^{ \prime}|}{\rm{v_{s}}}} \tag{14}\]  
Therefore,  
\[P(r,t)=\frac{\beta}{4\pi C_{P}}\frac{\partial}{\partial t}\int dr^{\prime} \frac{1}{|r-r^{\prime}|}\bigg{(}Q\bigg{(}r^{\prime},t-\frac{|r-r^{\prime}|}{ \rm{v_{s}}}\bigg{)}+k\
abla^{2}T\bigg{(}r^{\prime},t-\frac{|r-r^{\prime}|}{ \rm{v_{s}}}\bigg{)}\bigg{)} \tag{15}\]  
Following the elasto-optic relation, these large photoacoustic pressures can create nontrivial modulations \(\Delta n(r,t)\) in the local index of refraction following the elasto-optic relation [19]:  
\[\Delta n(r,\ t)=\frac{\varepsilon\ n_{0}^{3}(r)\ p(r,\ t)}{2\ \rho\ \rm{v_{s}^{2}}} \tag{16}\]  
where n\({}_{0}\)(r) is the local refraction index and \(\varepsilon\) is the elasto-optic coefficient. The heat-induced refractive index changes also exist, though it is a relatively weak effect. For some crystalline materials, the local temperature rise brings about a change in the refractive index [16]  
\[\Delta n=\frac{dn}{T}\ \Delta T=\alpha\ \Delta T \tag{17}\]  
and in displacement  
\[\Delta u=\frac{1}{u}\frac{du}{dT}\ r\ \Delta T=\beta\ \Delta T \tag{18}\]  
where \(\alpha\) is the thermo-optic coefficient.