#### Temperature distribution

Absorption depth of electromagnetic radiation (with the wavelength below 600 nm) in the mesoporous silicon is about a few microns. Therefore, the case of near-surface absorption has been considered. The

*Z* axis is directed in the depth of the sample, and the coordinate origin is situated on the top surface of the porous silicon layer. The temperature distribution inside the sample can be described by heat diffusion equation:

$c\rho \frac{\partial T}{\partial t}=\frac{\partial}{\partial z}\left(\chi \frac{\partial T}{\partial z}\right)\text{,}$

(1)

where *c* and *ρ* are the heat capacity and volume density, respectively; *χ*, the thermal conductivity. The boundary conditions for this equation are as follows:

, where the fluid layer is thermally thick, and thermal perturbation does not penetrate to the top boundary of the liquid;

${\left(\partial T/\partial z\right)|}_{z={l}_{\mathit{\text{Si}}}}=0$

, where thermal conductivity of the glass ceramics is much smaller than the thermal conductivity of bulk silicon, so we assumed that the heat did not penetrate in it;

${\chi \left(\partial T/\partial z\right)|}_{z=0+0}-{\chi \left(\partial T/\partial z\right)|}_{z=0-0}=I\xb7f\left(t\right)$

, where electromagnetic irradiation is absorbed at the top surface of the porous silicon and where

*I* is the irradiation intensity;

*f*(

*t*) is the function describing temporal dependence (modulation) of the irradiation intensity.

Periodic rectangular modulated irradiation with cycle duty of 0.5 was used in our experiment:

$f\left(t\right)=\{\begin{array}{cc}\hfill 1\hfill & \hfill 0<t\le {T}_{p}/2\hfill \\ \hfill 0\hfill & \hfill {T}_{p}/2<t\le {T}_{p}\hfill \end{array},\phantom{\rule{0.5em}{0ex}}f\left(t\right)=f\left(t+{T}_{p}\right)$

(2)

According to this,

$f\left(t\right)={\displaystyle \sum _{n=-\infty}^{+\infty}{f}_{n}\xb7{e}^{i{\omega}_{n}t}},{f}_{n}=\frac{1}{{T}_{p}}{\displaystyle \underset{0}{\overset{{T}_{p}}{\int}}f\left(t\right){e}^{-i{\omega}_{n}t}dt}\text{,}$

(3)

where *ω*_{
n
} =2π*n*/*T*_{
p
} (*n* is the integer number); *n* = 0 in the sum term corresponding to the averaged intensity distribution for one period has been omitted.

Thus, the depth and time-dependent temperature in the sample can be written in the following form:

$T\left(z,t\right)={\displaystyle \sum _{n=-\infty}^{+\infty}{T}_{n}\left(z\right)\xb7{e}^{i{\omega}_{n}t}}\text{.}$

(4)

The equations for temperature harmonic components can be obtained by substitution of expressions (Equations 2 and 3) in Equation

1 with the following boundary conditions:

$\begin{array}{c}\hfill \frac{\partial}{\partial z}\left(\chi \frac{\partial {T}_{n}}{\partial z}\right)-c\rho i{\omega}_{n}{T}_{n}=0\hfill \\ \hfill {{T}_{n}|}_{z=-\infty}\to 0\hfill \\ \hfill {\left(\partial {T}_{n}/\partial z\right)|}_{z={l}_{\mathit{\text{Si}}}}=0\hfill \\ \hfill {\chi \left(\partial {T}_{n}/\partial z\right)|}_{z=0+0}-{\chi \left(\partial {T}_{n}/\partial z\right)|}_{z=0-0}=I\xb7{f}_{n}\hfill \end{array}$

(5)

Solution of these equations for the simulated layer structure shown in Figure

1 can be represented as follows:

${T}_{n}\left(z\right)=\{\begin{array}{cc}\hfill {A}_{n}^{l}{e}^{{\mu}_{n}^{l}z}\hfill & \hfill z<0\hfill \\ \hfill {A}_{n}^{\mathit{\text{PS}}}{e}^{{\mu}_{n}^{\mathit{\text{PS}}}z}+{B}_{n}^{\mathit{\text{PS}}}{e}^{-{\mu}_{n}^{\mathit{\text{PS}}}z}\hfill & \hfill 0<z<{l}_{\mathit{\text{PS}}}\hfill \\ \hfill {A}_{n}^{\mathit{\text{Si}}}{e}^{{\mu}_{n}^{\mathit{\text{Si}}}z}+{B}_{n}^{\mathit{\text{Si}}}{e}^{-{\mu}_{n}^{\mathit{\text{Si}}}z}\hfill & \hfill {l}_{\mathit{\text{PS}}}<z<{l}_{\mathit{\text{Si}}}\hfill \end{array}\text{,}$

(6)

where${\mu}_{n}^{j}={\left(i{\omega}_{n}{c}_{j}{\rho}_{j},/,{\chi}_{j}\right)}^{1/2}\text{.}$

The constants${A}_{n}^{l}$,${A}_{n}^{\mathit{\text{PS}}}$,${B}_{n}^{\mathit{\text{PS}}}$,${A}_{n}^{\mathit{\text{Si}}}$ and${B}_{n}^{\mathit{\text{Si}}}$ can be obtained from the boundary conditions.

#### Thermally induced pressure

At quasi-stationary approximation, the thermally induced pressure (TIP) distribution for a liquid infiltrated in the mesoporous layer can be described by filtration equation[

6]. Considering thermal expansion of the liquid, this equation can be written as follows:

$\frac{\partial p}{\partial t}-\frac{K}{\eta \beta \epsilon}\frac{{\partial}^{2}p}{\partial {z}^{2}}=\frac{{\beta}_{\mathit{Tl}}}{\beta}\frac{\partial T}{\partial t}\text{,}$

(7)

where *ε* is the porosity, *β*_{
Tl
} is the coefficient of volume thermal expansion of the liquid, *K* is the fluid permeability of the porous silicon, *η* is the liquid’s viscosity, *β* is the liquid’s compressibility.

Boundary conditions for Equation

6 are the following:

${p|}_{z=0}=0{\frac{\partial p}{\partial z}|}_{z={l}_{\mathit{\text{por}}}}=0\text{.}$

(8)

Taking into account linear approximation for Equation

3, one can obtain expression for the liquid pressure inside the pores:

$p\left(z,t\right)={\displaystyle \sum _{n=-\infty}^{+\infty}{p}_{n}\left(z\right){e}^{i{\omega}_{n}t}}$

(9)

The equation for pressure harmonic component can be obtained by substitution of expressions (Equations 3 and 8) in Equations

6 and

7:

$\begin{array}{c}\hfill \text{i}{\omega}_{n}{p}_{n}-\frac{K}{\eta \beta \epsilon}\frac{{\partial}^{2}{p}_{n}}{\partial {z}^{2}}=\frac{{\beta}_{\mathit{\text{Tl}}}}{\beta}\text{i}{\omega}_{n}{T}_{n}\hfill \\ \hfill {{p}_{n}|}_{z=0}=0{\frac{\partial {p}_{n}}{\partial z}|}_{z={l}_{\mathit{\text{por}}}}=0\hfill \end{array}$

(10)

Solution of this equation can be represented as follows:

${p}_{n}={A}_{n}^{1}{e}^{-{\gamma}_{n}z}+{A}_{n}^{2}{\text{e}}^{+{\gamma}_{n}z}+\text{i}\omega \frac{3{\beta}_{{T}_{l}}\eta}{\kappa \left({\gamma}_{n}^{2}-{\mu}_{\mathit{\text{por}}}^{2}\right)}{T}_{n}\text{,}$

(11)

where

${\gamma}_{n}^{2}=i{\omega}_{n}\frac{\eta \beta \Pi}{\kappa}\text{.}$

(12)

The constants${A}_{n}^{1}$ and${A}_{n}^{2}$ can be obtained from the boundary conditions.

#### Photoacoustic signal formation

Elastic deformations in the nanocomposite structure porous silicon-liquid can appear under its heating by non-stationary irradiation as a result of a thermoelastic mechanism. According to the equation[

7] for quasi-stationary approximation, the source of thermoelastic force in such structures can be presented as follows:

${\sigma}_{T}\left(z,t\right)=\frac{{\alpha}_{T}\left(z\right)E\left(z\right)}{1-\nu \left(z\right)}T\left(z,t\right)+\epsilon p\left(z,t\right)\text{.}$

(13)

This source consists of two components: (1) the thermoelastic stresses of the porous matrix and monocrystalline Si wafer (first term) as well as (2) the TIPs of the liquid inside the pores (second term).

Thus, the elastic stresses in the whole sample can be represented as follows:

$\sigma \left(z,t\right)={\displaystyle \underset{0}{\overset{{l}_{\mathit{\text{Si}}}}{\int}}{\sigma}_{T}\left({z}^{\prime},t\right)G\left(z,{z}^{\prime}\right)d{z}^{\prime}}\text{,}$

(14)

where

*G*(

*z*,

*z′* ) is the elasticity Green's function depending on elasticity parameters and geometry of the investigated sample. This function can be easily obtained for the case of quasi-stationary approximation (see, for example, reference[

7]) using reciprocity theorem and Kirchhoff-Love theory:

$G\left(z,{z}^{\prime}\right)=\frac{E\left(z\right)}{1-\nu \left(z\right)}\left(\frac{{a}_{1}{z}^{\prime}-{a}_{2}+\left({a}_{1}-{a}_{0}{z}^{\prime}\right)z}{{a}_{0}{a}_{2}-{a}_{1}^{2}}\right)\text{,}$

(15)

where${a}_{i}={\displaystyle \underset{0}{\overset{l}{\int}}\frac{E\left(z\right)}{1-\nu \left(z\right)}{z}^{i}dz}\text{.}$

If the polarization axis is perpendicular to the PZT surface, then time evolution of the photoacoustic signal shape at the PZT electrodes can be presented in this form:

$\begin{array}{l}\Delta \varphi \left(t\right)\sim {\displaystyle \underset{{l}_{\mathit{\text{el}}}}{\overset{{l}_{\mathit{\text{PZT}}}}{\int}}\sigma \left(z,t\right)dz}={\displaystyle \underset{{l}_{\mathit{\text{el}}}}{\overset{{l}_{\mathit{\text{PZT}}}}{\int}}{\displaystyle \underset{0}{\overset{{l}_{\mathit{\text{Si}}}}{\int}}{\sigma}_{T}\left({z}^{\prime},t\right)G\left(z,{z}^{\prime}\right)d{z}^{\prime}}dz}\hfill \\ ={\displaystyle \underset{0}{\overset{{l}_{\mathit{\text{Si}}}}{\int}}{\sigma}_{T}\left({z}^{\prime},t\right)g\left({z}^{\prime}\right)d{z}^{\prime}}\hfill \end{array}\text{.}$

(16)

where$g\left({z}^{\prime}\right)={\displaystyle \underset{{l}_{\mathit{\text{el}}}}{\overset{{l}_{\mathit{\text{PZT}}}}{\int}}G\left(z,{z}^{\prime}\right)dz}$is the function describing the voltage excitation performance depending on a point at which the elastic force is applied.