Ultra short pulse lasers with sub-picosecond pulse duration can generate ultrasonic acoustic pulses with wavelengths in the sub-micrometer range. The reflectance modulation induced by the acoustic pulses can be detected by a probe laser. Once a method of calculating the acoustic pulse propagation and the reflectance modulation is established, it will be possible to extract information about the internal structures of a medium from measurements of the time-sequential reflectance modulation by fitting the calculations to experimental results. Thus, a multi-slicing matrix method of calculating the reflectance modulation with an arbitrary incident angle of the probe laser for s-polarization is proposed here. The method based on the Abeles transfer matrix method applicable to stratified media is mathematically validated and is shown to reproduce the analytical formula derived using the Born approximation. The method only uses matrix multiplications, which makes the calculation algorithm simple and is easy to code with matrix manipulation software. The thickness of a stratified layer of an amorphous carbon stacked on a substrate of silicon is demonstrated to be measured with the method.
1. Introduction
Ultra short pulse lasers with sub-picosecond pulse duration have proven their worth in probing sub-micrometer structures of media, because they can generate an ultrasonic acoustic pulse with wavelength of sub-micrometer range. In fact, acoustic pulses with sub-micrometer wavelengths have been detected in metallic, semiconducting, and insulating media with sub-picosecond pulse lasers [1–15]. A laser ultrasonic technique using an optical pump-and-probe method measures the time-sequential reflectance modulation with the probe laser irradiated on the top surface of a medium. The reflectance modulation is induced by the acoustic pulse that is initially generated at the surface of the medium by the pump laser. The time-sequential reflectance modulation depends on the internal structures of the medium because the acoustic pulse is reflected toward the top surface of the medium by the internal structures. Thus, the internal information of the medium can be obtained by measuring the time-sequential reflectance modulation.
The laser ultrasonic technique is a particularly powerful tool for probing a medium composed of stratified layers. The thickness of each layer in the stratified structure can be reconstructed by fitting theoretical calculations of the time-sequential reflectance modulation to an experimental measurement with the fitting parameters of the thicknesses. To calculate the time-sequential reflectance modulation, it likely follows that three calculation methods are necessary. That is, these are methods of calculating the spatial distribution of the absorbed power density of the pump laser that determines the shape of the acoustic pulse, the propagation of the acoustic pulse, and the reflectance of the probe laser. The calculations of the absorbed power density and the acoustic pulse propagation can be performed with the FDTD (Finite-Difference Time-Domain) algorithm [16,17]. For calculating the reflectance modulation, the Born approximation method is often used under the assumption of a small permittivity modulation with normal incidence on the surface of a medium. Although the Born approximation method predicts the actual measurement well, the mathematical treatment is complicated for stratified media. There is an elaborate article that describes well the mathematical treatment based on the Born approximation for the normal incidence of the probe laser on the surface of a medium [18].
Another method of calculating the reflectance modulation for the stratified medium with an arbitrary incident angle of a probe laser for s-polarization using a simple matrix multiplication algorithm is proposed here. The proposed method applies the Abeles transfer matrix to the stratified medium where the medium modulated by the acoustic pulse is assumed to consist of multiple-films, each of which has an infinitesimally small thickness of the same size. Thus, the method is here called the multi-slicing method. The method only uses matrix multiplications, which makes the calculation algorithm simple and is easy to be coded with matrix manipulation softwares. The dependencies of the reflectance modulation on an arbitrary incident angle of the probe laser can be calculated by the multi-slicing matrix method. The method is mathematically validated, and is shown to reproduce the analytical formula derived using the Born approximation. The method is also applied to an experiment to demonstrate to measure a thickness of a stratified medium.
2. Acoustic pulse and permittivity modulation in a stratified medium
The acoustic pulse can be generated by irradiation of a pump pulse laser. For a stratified medium, the propagation of the acoustic pulse in the medium can be formulated as follows. The stratified medium is assumed to consist of various layers stacked in z-direction where the refractive index of each layer is constant. The acoustic pulse propagating in the medium can modulate the refractive index of each layer. Thus, the reflectance of a probe laser incident on the surface of the medium is slightly modulated by the acoustic pulse.
Figure 1 shows the cross-sectional view of x-z plane of the stratified medium stacked with L layers, each of thickness dl. The ambient permittivity is set to ɛa. The ambient relative permittivity ${\bar{\varepsilon }_0}$ is set to 1 where the bar symbol over the variable represents a value before the elastic modulation induced by the acoustic pulse. The l-th layer (l = 1, 2, …, L) is located in the region from z = Zl-1 to z = Zl with the relative permittivity of ${\bar{\varepsilon }_l}$ that corresponds to ${\bar{n}_l}^2$ where the bar symbols over the variables also represent values before the elastic modulation. The 0-th location of Z0 is set to the top surface of the medium where Z0 is set to zero before the elastic displacement induced by the acoustic pulse. The relative permittivity of a medium of light outgoing region of z > ZL is ${\bar{\varepsilon }_{L + 1}}$. The pump laser is irradiated on the top surface of the medium with an incident angle of θ0. The probe laser is irradiated on the surface of the medium with an incident angle of θ. The reflected light from the surface propagates with the reflection angle of θ.
The electric field E and magnetic field H satisfy Maxwell’s equations assuming that there is no true charge density:
(1)$$\nabla \times {\textbf E} = - {\mu _0}\frac{{\partial {\textbf H}}}{{\partial t}},$$
(2)$$\nabla \times {\textbf H} = {\varepsilon _{\textrm{a}}}\frac{{\partial ({\bar{\varepsilon }{\textbf E}} )}}{{\partial t}},$$
(3)$$\nabla \cdot ({\bar{\varepsilon }{\textbf E}} )= 0,$$
(4)$$\nabla \cdot {\textbf B} = 0,$$
where μ0, ɛa, and $\bar{\varepsilon }$ are permeability of the vacuum, permittivity of the ambient, and relative permittivity without the modulation induced by the acoustic pulse respectively.The absorbed power density q by the material can be written as
(5)$$q = - \frac{\partial }{{\partial z}}({{{\textbf e}_z} \cdot ({{\mbox{Re}} ({\textbf E} )\times {\mbox{Re}} ({\textbf H} )} )} ),$$
where ez is a unit vector of the direction of z.The acoustic pulse is assumed to be generated by the thermal stress due to the instantaneous temperature rise without any thermal diffusion. The beam diameter of the pump laser is assumed to be sufficiently larger than the thickness of the stratified medium. Under those assumptions, z-component of the elastic displacement uz of the acoustic pulse can be described with zz-components of the stress tensor σzz by the Newtonian equation of motion:
(6)$$\rho \frac{\partial }{{\partial t}}\left( {\frac{{\partial {u_z}}}{{\partial t}}} \right) = \frac{\partial }{{\partial z}}{\sigma _{zz}}.$$
The relationship between σzz and uz can be described by Hooke’s law, including the thermal stress generated by the laser absorption with the absorbed power density q: (7)$${\sigma _{zz}} = 3\Lambda \frac{{1 - \nu }}{{1 + \nu }}\frac{{\partial {u_z}}}{{\partial z}} - \frac{{3\Lambda \gamma }}{{\rho C}}\int\limits_0^{\Delta t} q {\mbox{d}} t,$$
where Λ, ν, γ, ρ, and C are bulk modulus, Poisson’s ratio, linear expansion coefficient, density, and specific heat of the material respectively, and Δt is the time duration of the laser irradiation. Combining Eq. (6) and (7), the following equation can be derived: (8)$$\rho \frac{{{\partial ^2}{\eta _{zz}}}}{{\partial {t^2}}} = 3\Lambda \frac{{1 - \nu }}{{1 + \nu }}\frac{{{\partial ^2}{\eta _{zz}}}}{{\partial {z^2}}} - \frac{{3\Lambda \gamma }}{{\rho C}}\int\limits_0^{\Delta t} {\frac{{{\partial ^2}q}}{{\partial {z^2}}}} dt,$$
where ηzz is the zz-component of the strain tensor and defined as (9)$${\eta _{zz}} = \frac{{\partial {u_z}}}{{\partial z}}.$$
Equation (8) represents the elastic wave equation for the acoustic pulse. In this work, a velocity of the elastic wave is assumed to be sufficiently slow in comparison with that of the light.The refractive index is modulated by the elastic strain of the acoustic pulse. A beam diameter of the pump laser is assumed to be sufficiently larger than the thickness of the stratified medium. Thus, the elastic displacements in x- and y-direction can be considered to be homogeneous, and the strain tensor components derived by these displacements become zero. The refractive index modulation can therefore be written with the opto-stress coefficient K and the zz-component of the strain tensor as
(10)$$\Delta n = K{\eta _{zz}}.$$
Using Eq. (10), the modulation of the relative permittivity by the acoustic pulse can be written in the l-th layer as (11)$$\Delta \varepsilon = {({{{\bar{n}}_l} + \Delta n} )^2} - {\bar{n}_l}^2 \cong 2{\bar{n}_l}\Delta n = 2\sqrt {{{\bar{\varepsilon }}_l}} K{\eta _{zz}}.$$
Thus, the permittivity modulation over the entire region inside the medium can be obtained with the elastic strain tensor.To calculate the reflectance modulation induced by the acoustic pulse, the absorbed power density q represented by Eq. (5) should be calculated in the first place. With the absorbed power density q, the acoustic pulse can be calculated using Eq. (8). There are several methods of calculating the acoustic pulse propagation. One of the candidates is the FDTD method that can solve both the absorbed power density and the propagation of the acoustic pulse. The absorbed power density can also be solved with the Abeles matrix method for a stratified media [19–24]. For a single layer of a semi-infinite medium, there is also an analytical solution [1].
3. Multi-slicing matrix method for reflectance modulation
3.1 Matrix representation for the electro-magnetic field
When the permittivity is modulated by the acoustic pulse, Eqs. (1)–(4) for the electro-magnetic fields of E and H can be written with the time-dependent factor of the fields of exp(-iωt) and ω of an angular frequency as,
(12)$$\nabla \times {\textbf E} = i\omega {\mu _0}{\textbf H},$$
(13)$$\nabla \times {\textbf H} = - i\omega \varepsilon (z ){\varepsilon _{\textrm{a}}}{\textbf E},$$
(14)$$\nabla \cdot ({\varepsilon (z ){\textbf E}} )= 0,$$
(15)$$\nabla \cdot {\textbf H} = 0,$$
where ɛ(z) is the relative permittivity including the modulation induced by the acoustic pulse and related to the complex refractive index n(z) as, (16)$$\varepsilon (z )= {n^2}(z ).$$
The electro-magnetic fields can be written as, (17)$${\textbf E} = \left[ {\begin{array}{{c}} {{E_x}(z )}\\ {{E_y}(z )}\\ {{E_z}(z )} \end{array}} \right]\exp ({i{k_x}x - i\omega t} ),$$
(18)$${\textbf{H}} = \left[ {\begin{array}{{c}} {{H_x}(z )}\\ {{H_y}(z )}\\ {{H_z}(z )} \end{array}} \right]\exp ({i{k_x}x - i\omega t} ).$$
Note that Ex, Ey, Ez, Hx, Hy, and Hz depend on z where the argument z of these functions will be omitted hereafter. Here, kx is the wave number for the x-component, which can be written using the incident angle θ of the probe laser as (19)$${k_x} = {k_0}\sin \theta ,$$
where ${k_0}$ is the wave number in the ambient medium as (20)$${k_0} = \omega \sqrt {{\varepsilon _{\textrm{a}}}{\mu _0}} .$$
Note that the x-component of the wave number kx is constant all along z-direction owing to the parallel translation symmetry of x-direction of the system.For a stratified medium, the medium is considered to consist of multi-slicing films, with each film having the same infinitesimal thicknesses Δz as shown in Fig. 2. Each film is located in the region of ${z_{m - 1}} < z \le {z_m}$ with zm-zm-1=Δz (m = 1, …, N) where the relative permittivity ɛ(zm) of each film at z = zm is set to ɛm. The 0-th location of z0 is set to the same as the top surface of the medium of Z0 where z0 and Z0 are zero before the elastic displacement induced by acoustic pulse. The (N + 1)-th location of zN+1 in the light outgoing region is set to the same as zN where zN is set to the bottom surface of the medium of ZL.
Equation (14) can be written for each slicing film where the permittivity depends only on z as
(21)$$\nabla \cdot {\textbf E} = - \frac{1}{{\varepsilon (z )}}\frac{{\partial \varepsilon (z )}}{{\partial z}}{E_z}(z )\exp ({i{k_x}x - i\omega t} ).$$
Using Eqs. (12), (13), and (21), the following equation can be derived as (22)$$\Delta {\textbf E} + \varepsilon (z ){k_0}^2{\textbf E} = \nabla ({\nabla \cdot {\textbf E}} )= - \left[ {\begin{array}{{c}} {i{k_x}\frac{1}{{\varepsilon (z )}}\frac{{\partial \varepsilon (z )}}{{\partial z}}{E_z}}\\ 0\\ {\frac{\partial }{{\partial z}}\left( {\frac{1}{{\varepsilon (z )}}\frac{{\partial \varepsilon (z )}}{{\partial z}}{E_z}} \right)} \end{array}} \right]\exp ({i{k_x}x - i\omega t} ).$$
Note that the right-hand equation of Eq. (22) becomes zero for the incident angle θ of 0 degrees or s-polarization whose electric filed is parallel to the y-direction. The permittivity can be written with a constant $\bar{\varepsilon }$ and a permittivity modulation Δɛ induced by the acoustic pulse as (23)$$\varepsilon (z )= \bar{\varepsilon } + \Delta \varepsilon (z ).$$
Equation (23) can further be transformed for the m-th film in the l-th layer as (24)$$\begin{array}{ccccc} \varepsilon (z )& = {{\bar{\varepsilon }}_l} + \Delta \varepsilon ({{z_m}} )+ ({\Delta \varepsilon (z )- \Delta \varepsilon ({{z_m}} )} )\\ & = {{\bar{\varepsilon }}_l} + \Delta {\varepsilon _m} + \Delta \varepsilon ^{\prime}({{z_m}} )\Delta z + O({\Delta {z^2}} ), \end{array}$$
where Δz is defined as z-zm, and the dash symbol indicates differentiation with respect to z. Note that the sub-index l of ${\bar{\varepsilon }_l}$ indicates the layer number while the sub-index m of ɛm indicates the film number. Hereafter, a sub-index l of a quantity with the bar symbol indicates the layer number. Equation (22) can therefore be transformed using Eq. (24) for the m-th film in the l-th layer as (25)$$\Delta {\textbf E} + {\bar{\varepsilon }_l}{k_0}^2{\textbf E} = - \Delta {\varepsilon _m}{k_0}^2{\textbf E} - \Delta \varepsilon ^{\prime}({{z_m}} )\Delta z{k_0}^2{\textbf E} - \left[ {\begin{array}{{c}} {i{k_x}\frac{{\Delta \varepsilon^{\prime}}}{\varepsilon }{E_z}}\\ 0\\ {{{\left( {\frac{{\Delta \varepsilon^{\prime}}}{\varepsilon }{E_z}} \right)}^\prime }} \end{array}} \right]\exp ({i{k_x}x - i\omega t} ).$$
Equation (25) indicates that the second term on the right-hand equation becomes negligible in comparison with the first term when following equation is satisfied as (26)$$\Delta z \ll \frac{{\Delta {\varepsilon _m}}}{{\Delta {{\varepsilon ^{\prime}}_{}}({{z_m}} )}}.$$
The third term on the right-hand equation only appears for p-polarization whose electric field is parallel to the x-z plane, and this term should be taken into consideration when the incident angle of the light is large and the z-component of the electric filed is not negligible. In other words, the third term on the right hand of Eq. (25) is not negligible for p-polarization even though the limit of Eq. (26) is taken. In this work, the electric field is assumed to be s-polarization. Thus, Eq. (26) ensures that Eq. (25) can be written with the constant permittivity that is independent of z. Indeed, Eq. (25) for s-polarization can be written with infinitesimally small thickness of Δz as (27)$$\Delta {\textbf E} + ({{{\bar{\varepsilon }}_l} + \Delta {\varepsilon_m}} ){k_0}^2{\textbf E} = \Delta {\textbf E} + {\varepsilon _m}{k_0}^2{\textbf E} = - \Delta \varepsilon ^{\prime}({{z_m}} )\Delta z{k_0}^2{\textbf E} \cong 0.$$
Note that the left-hand of Eq. (27) includes the permittivity modulation Δɛ while the Born approximation considers Δɛ as a perturbation and set Δɛ to the right-hand equation to consider it as a perturbation term. When following equation is satisfied as (28)$$\Delta z \le \frac{{\Delta {\varepsilon _m}^2}}{{\Delta {{\varepsilon ^{\prime}}_{}}({{z_m}} )}},$$
the accuracy of the electric-field is ensured to be better than that of the Born approximation where the second term on the right-hand of Eq. (25) is less than the second order of Δɛ. This ensures that the electric fields of the first order of Δɛ converge to the Born approximation in the limit of Eq. (26).The two independent solutions of Eq. (27) for individual component, E, can be written for the m-th film as
(29)$$E = A\exp ({i{k_{mz}}({z - {z_{m - 1}}} )} ),$$
(30)$$E = B\exp ({i{k_{mz}}({{z_m} - z} )} ),$$
where A and B are constant in the film, and kmz for m-th film is defined as (31)$${k_{mz}} = {k_0}\sqrt {{\varepsilon _m} - {{\sin }^2}\theta } .$$
The electric field of the s-polarization is parallel to y-direction. Thus, in each slicing film, the electric field can be written using Eqs. (17), (29), and (30) with y-component of electric field Es as, (32)$$\left[ {\begin{array}{{c}} {{E_x}}\\ {{E_y}}\\ {{E_z}} \end{array}} \right] = \left[ {\begin{array}{{c}} {{E_x}}\\ {{E_s}}\\ {{E_z}} \end{array}} \right] = \left[ {\begin{array}{{c}} 0\\ {{a_{sm}}\exp ({i{k_{mz}}({z - {z_{m - 1}}} )} )+ {b_{sm}}\exp ({i{k_{mz}}({{z_m} - z} )} )}\\ 0 \end{array}} \right].$$
The magnetic field can also be derived using Eq. (12) with x-component of magnetic field defined as -Hs, which can be written as (33)$$\left[ {\begin{array}{{c}} {{H_x}}\\ {{H_y}}\\ {{H_z}} \end{array}} \right] = \left[ {\begin{array}{{c}} { - {H_s}}\\ {{H_y}}\\ {{H_z}} \end{array}} \right] = \frac{1}{{i\omega {\mu _0}}}\left[ {\begin{array}{{c}} { - i{k_{mz}}({{a_{sm}}\exp ({i{k_{mz}}({z - {z_{m - 1}}} )} )- {b_{sm}}\exp ({i{k_{mz}}({{z_m} - z} )} )} )}\\ 0\\ {i{k_x}({{a_{sm}}\exp ({i{k_{mz}}({z - {z_{m - 1}}} )} )+ {b_{sm}}\exp ({i{k_{mz}}({{z_m} - z} )} )} )} \end{array}} \right].$$
Using Eqs. (32) and (33), a field vector consisting of two components of Es and Hs can be written at z = zm-1 and zm respectively as (34)$$\left[ {\begin{array}{{c}} {{E_s}({{z_{m - 1}}} )}\\ {{{{H_s}({{z_{m - 1}}} )} / {{Y_0}}}} \end{array}} \right] = \left[ {\begin{array}{{lc}} 1&{\exp ({i{k_{mz}}\Delta z} )}\\ {{\beta_m}}&{ - {\beta_m}\exp ({i{k_{mz}}\Delta z} )} \end{array}} \right]\left[ {\begin{array}{{c}} {{a_{sm}}}\\ {{b_{sm}}} \end{array}} \right],$$
(35)$$\left[ {\begin{array}{{c}} {{E_s}({{z_m}} )}\\ {{{{H_s}({{z_m}} )} / {{Y_0}}}} \end{array}} \right] = \left[ {\begin{array}{{cc}} {\exp ({i{k_{mz}}\Delta z} )}&1\\ {{\beta_m}\exp ({i{k_{mz}}\Delta z} )}&{ - {\beta_m}} \end{array}} \right]\left[ {\begin{array}{{c}} {{a_{sm}}}\\ {{b_{sm}}} \end{array}} \right],$$
where Δz is defined as zm-zm-1, Y0 and βm are defined as (36)$${Y_0} = \frac{{{k_0}}}{{\omega {\mu _0}}},$$
(37)$${\beta _m} = \frac{{{k_{mz}}}}{{{k_0}}} = \sqrt {{\varepsilon _m} - {{\sin }^2}\theta } .$$
Using Eqs. (34) and (35), the transfer of the field vector from z = zm-1 to zm can be written with a transfer matrix Ms as, (38)$$\left[ {\begin{array}{{c}} {{E_s}({{z_{m - 1}}} )}\\ {{{{H_s}({{z_{m - 1}}} )} / {{Y_0}}}} \end{array}} \right] = {{\textbf M}_s}({{z_{m - 1}},{z_m}} )\left[ {\begin{array}{{c}} {{E_s}({{z_m}} )}\\ {{{{H_s}({{z_m}} )} / {{Y_0}}}} \end{array}} \right],$$
where Ms is defined for s-polarization in the form of the Abeles transfer matrix as (39)$${{\textbf M}_s}({{z_{m - 1}},{z_m}} )= \left[ {\begin{array}{{cc}} {\cos ({{k_{mz}}\Delta z} )}&{ - i{\beta_m}^{ - 1}\sin ({{k_{mz}}\Delta z} )}\\ { - i{\beta_m}\sin ({{k_{mz}}\Delta z} )}&{\cos ({{k_{mz}}\Delta z} )} \end{array}} \right].$$
Using Eq. (38) iteratively, the following equation can be derived as (40)$$\left[ {\begin{array}{{c}} {{E_s}({{z_0}} )}\\ {{{{H_s}({{z_0}} )} / {{Y_0}}}} \end{array}} \right] = {{\textbf M}_s}({{z_0},{z_1}} )\cdots {{\textbf M}_s}({{z_{N - 1}},{z_N}} )\left[ {\begin{array}{{c}} {{E_s}({{z_N}} )}\\ {{{{H_s}({{z_N}} )} / {{Y_0}}}} \end{array}} \right].$$
Equation (40) is the multi-slicing matrix transfer formula for electro-magnetic field where the stratified medium is assumed to be consisting of multi-slicing films, with each film having the same thickness Δz that satisfies Eq. (26).3.2 Electric field of reflected light in an ambient medium
The acoustic pulse modulates the refractive index along z-direction in the medium, which means that the refractive index may vary continuously in z-direction. The acoustic pulse also causes the elastic displacements of the thicknesses of layers of the stratified medium. Note that the elastic displacements of the thicknesses of the layers change positions of the boundaries of the layers. The displacement at z = Zl of the l-th layer can be written with the elastic strain tensor of the acoustic pulse as
(41)$${u_l} = - \int\limits_{{Z_l}}^{{Z_L}} {{\eta _{zz}}} ({z^{\prime}} )dz^{\prime}.$$
The relative permittivity along z-direction can be written with the relative permittivity modulation Δɛ induced by the acoustic pulse for l-th layer (l = 1, 2, …, L) as (42)$$\varepsilon (z )= {\bar{\varepsilon }_l} + \Delta \varepsilon (z ),$$
with (43)$${Z_{l - 1}} = {\bar{Z}_{l - 1}} + {u_{l - 1}} < z \le {\bar{Z}_l} + {u_l} = {Z_l},$$
where the bar symbols over the quantities indicate values before the elastic modulation induced by the acoustic pulse.The multi-slicing matrix method assumes that the stratified medium consists of infinitesimal thin films. Thus, permittivity ɛm defined as ɛ(zm) for the m-th film (m = 1, 2, …, N) in the l-th layer is defined as
(44)$${\varepsilon _m} = {\bar{\varepsilon }_l} + \Delta \varepsilon ({{z_m}} )= {\bar{\varepsilon }_l} + \Delta {\varepsilon _m},$$
where the region of the m-th film is defined as (45)$${z_{m - 1}} < z \le {z_m},$$
with the infinitesimal thickness with z0=Z0 and zN=ZL as (46)$$\Delta z = {z_m} - {z_{m - 1}} = \frac{{{Z_L} - {Z_0}}}{N}.$$
For s-polarization, in the ambient medium from which the light is incident, the electric field can be written as (47)$$\left[ {\begin{array}{{c}} {{E_x}}\\ {{E_y}}\\ {{E_z}} \end{array}} \right] = \left[ {\begin{array}{{c}} {{E_x}}\\ {{E_s}}\\ {{E_z}} \end{array}} \right] = \left[ {\begin{array}{{c}} 0\\ {{a_{s0}}\exp ({i{k_{0z}}z} )+ {b_{s0}}\exp ({ - i{k_{0z}}z} )}\\ 0 \end{array}} \right].$$
The magnetic field can also be derived using Eq. (12) with x-component of magnetic field defined as -Hs, which can be written as (48)$$\left[ {\begin{array}{{c}} {{H_x}}\\ {{H_y}}\\ {{H_z}} \end{array}} \right] = \left[ {\begin{array}{{c}} { - {H_s}}\\ {{H_y}}\\ {{H_z}} \end{array}} \right] = \frac{1}{{i\omega {\mu _0}}}\left[ {\begin{array}{{c}} { - i{k_{0z}}({{a_{s0}}\exp ({i{k_{0z}}z} )- {b_{s0}}\exp ({ - i{k_{0z}}z} )} )}\\ 0\\ {i{k_x}({{a_{s0}}\exp ({i{k_{0z}}z} )+ {b_{s0}}\exp ({ - i{k_{0z}}z} )} )} \end{array}} \right].$$
Using Eqs. (47) and (48), the field vector at z = Z0=0 + u0 of the top surface of the medium can be written with ${\bar{\beta }_0} = \sqrt {{{\bar{\varepsilon }}_0} - {{\sin }^2}\theta } $ as (49)$$\left[ {\begin{array}{{c}} {{E_s}({{u_0}} )}\\ {{{{H_s}({{u_0}} )} /{{Y_0}}}} \end{array}} \right] = \left[ {\begin{array}{{cc}} {\exp ({i{k_{0z}}{u_0}} )}&{\exp ({ - i{k_{0z}}{u_0}} )}\\ {{{\bar{\beta }}_0}\exp ({i{k_{0z}}{u_0}} )}&{ - {{\bar{\beta }}_0}\exp ({ - i{k_{0z}}{u_0}} )} \end{array}} \right]\left[ {\begin{array}{{c}} {{a_{s0}}}\\ {{b_{s0}}} \end{array}} \right].$$
In the region of the transmitted light where there is no light propagating in negative z-direction, the electric field can be written with the coefficient asN+1 and ${k_{0z}} = {k_0}{\bar{\beta }_0}$ as (50)$$\left[ {\begin{array}{{c}} {{E_x}}\\ {{E_y}}\\ {{E_z}} \end{array}} \right] = \left[ {\begin{array}{{c}} {{E_x}}\\ {{E_s}}\\ {{E_z}} \end{array}} \right] = \left[ {\begin{array}{{c}} 0\\ {{a_{sN + 1}}\exp ({i{k_0}{{\bar{\beta }}_{L + 1}}({z - {Z_L}} )} )}\\ 0 \end{array}} \right].$$
Using Eqs. (12) and (50), the field vector at z = ZL of the bottom surface of the medium can be written as (51)$$\left[ {\begin{array}{{c}} {{E_s}({{Z_L}} )}\\ {{{{H_s}({{Z_L}} )} / {{Y_0}}}} \end{array}} \right] = \left[ {\begin{array}{{c}} 1\\ {{{\bar{\beta }}_{L + 1}}} \end{array}} \right]{a_{sN + 1}}.$$
The reflectance defined as the ratio of the bs0 to as0 of Eq. (49) in the ambient medium can be derived using Eqs. (40), (49) and (51) as (52)$$\left[ {\begin{array}{{c}} {{{{a_{s0}}} / {{a_{sN + 1}}}}}\\ {{{{b_{s0}}} / {{a_{sN + 1}}}}} \end{array}} \right] = \frac{1}{2}\left[ {\begin{array}{{lc}} {{\zeta^{ - 1}}}&{{{\bar{\beta }}_0}^{ - 1}{\zeta^{ - 1}}}\\ \zeta &{ - {{\bar{\beta }}_0}^{ - 1}\zeta } \end{array}} \right]{{\textbf M}_s}({{z_0},{z_1}} )\cdots {{\textbf M}_s}({{z_{N - 1}},{z_N}} )\left[ {\begin{array}{{c}} 1\\ {{{\bar{\beta }}_{L + 1}}} \end{array}} \right],$$
where ζ is defined as (53)$$\zeta = \exp ({i{k_{0z}}{u_0}} ).$$
Before the elastic modulation is occurred (e.g., Δɛ=0, ul=0, and ζ=1), Eq. (52) can be written as (54)$$\left[ {\begin{array}{{c}} {{{{{\bar{a}}_{s0}}} / {{{\bar{a}}_{sL + 1}}}}}\\ {{{{{\bar{b}}_{s0}}} / {{{\bar{a}}_{sL + 1}}}}} \end{array}} \right] = \frac{1}{2}\left[ {\begin{array}{{lc}} 1&{{{\bar{\beta }}_0}^{ - 1}}\\ 1&{ - {{\bar{\beta }}_0}^{ - 1}} \end{array}} \right]{{\bar{\textbf M}}_s}({{{\bar{Z}}_0},{{\bar{Z}}_1}} )\cdots {{\bar{\textbf M}}_s}({{{\bar{Z}}_{L - 1}},{{\bar{Z}}_L}} )\left[ {\begin{array}{{c}} 1\\ {{{\bar{\beta }}_{L + 1}}} \end{array}} \right],$$
where the bar symbols over the quantities of a, b, β, Z, and M are defined as values before the elastic modulation induced by the acoustic pulse where each layer is not elastically displaced by the acoustic pulse. Note that a sub-index number of a quantity with the bar symbol indicates a layer number.4. Results and discussions
4.1 Reflectance modulation using the multi-slicing matrix method
Using Eqs. (52)–(54), the reflectance modulation induced by the acoustic pulse with respect to the reflectance before the elastic modulation can be derived as
(55)$$\Delta {r_s} = \frac{{{{{b_{s0}}} / {{a_{sN + 1}}}}}}{{{{{a_{s0}}} / {{a_{sN + 1}}}}}} - \frac{{{{{{\bar{b}}_{s0}}} / {{{\bar{a}}_{sL + 1}}}}}}{{{{{{\bar{a}}_{s0}}} / {{{\bar{a}}_{sL + 1}}}}}} = \frac{{{b_{s0}}}}{{{a_{s0}}}} - \frac{{{{\bar{b}}_{s0}}}}{{{{\bar{a}}_{s0}}}},$$
where the sub-index symbol s indicates s-polarization, the bar symbols over the quantities indicate values before the elastic modulation induced by the acoustic pulse. Equation (55) can be transformed using Eqs. (52) and (54) as (56)$$\Delta {r_s} = {\zeta ^2}\frac{{{{\textbf c}_{s1}}^{\textrm{t}}{{\textbf M}_s}({{z_0},{z_1}} )\cdots {{\textbf M}_s}({{z_{N - 1}},{z_N}} ){{\textbf f}_s}}}{{{{\textbf c}_{s0}}^{\textrm{t}}{{\textbf M}_s}({{z_0},{z_1}} )\cdots {{\textbf M}_s}({{z_{N - 1}},{z_N}} ){{\textbf f}_s}}} - \frac{{{{\textbf c}_{s1}}^{\textrm{t}}{{{\bar{\textbf M}}}_s}({{{\bar{Z}}_0},{{\bar{Z}}_1}} )\cdots {{{\bar{\textbf M}}}_s}({{{\bar{Z}}_{L - 1}},{{\bar{Z}}_L}} ){{\textbf f}_s}}}{{{{\textbf c}_{s0}}^{\textrm{t}}{{{\bar{\textbf M}}}_s}({{{\bar{Z}}_0},{Z_1}} )\cdots {{{\bar{\textbf M}}}_s}({{{\bar{Z}}_{L - 1}},{{\bar{Z}}_L}} ){{\textbf f}_s}}},$$
where the c and f are defined as (57)$$\left[ {\begin{array}{{c}} {{{\textbf c}_{s0}}^{\textrm{t}}}\\ {{{\textbf c}_{s1}}^{\textrm{t}}} \end{array}} \right] = \frac{1}{2}\left[ {\begin{array}{{cc}} {\left( {\begin{array}{{cc}} 1&{{{\bar{\beta }}_0}^{ - 1}} \end{array}} \right)}\\ {\left( {\begin{array}{{cc}} 1&{ - {{\bar{\beta }}_0}^{ - 1}} \end{array}} \right)} \end{array}} \right],$$
(58)$${{\textbf f}_s} = \left[ {\begin{array}{{c}} 1\\ {{{\bar{\beta }}_{L + 1}}} \end{array}} \right].$$
Equation (56) represents the reflectance modulation due to the acoustic pulse generated by the irradiation of the pump laser. The first term on the right-hand of Eq. (56) represents the reflectance from the elastically modulated medium using the infinitesimal transfer matrix of the multi-slicing films. Equation (56) is mathematically validated for arbitrary inclination angles for s-polarization. Note that Eq. (56) only uses the matrix multiplications, which makes it easy to be coded with several matrix manipulation softwares. The multi-slicing matrix method is summarized in Appendix that contains only quantities and equations that need to be coded.4.2 Reproduction of the Born approximation using the multi-slicing matrix method
There is an analytical approximation formula for the limit of the small permittivity modulation using the Born approximation with s-polarization for a single layer of a semi-infinite medium. The multi-slicing matrix method is able to reproduce the Born approximation as follows. For the limit of the small permittivity modulation, Eq. (56) for s-polarization can be transformed by taking the first order of Δɛ where ZL is assumed to be sufficiently large with L = 1 and ${\bar{\beta }_{L + 1}} = {\bar{\beta }_L} = {\bar{\beta }_1}$ as
(59)$$\begin{aligned}\left[ {\begin{matrix}{{{{a_{s0}}} / {{a_{sN + 1}}}}}\\ {{{{b_{s0}}} / {{a_{sN + 1}}}}} \end{matrix}} \right] & =\frac{1}{2}\left[ {\begin{matrix}{{\zeta^{ - 1}}}&{{\zeta^{ - 1}}{{\bar{\beta }}_0}^{ - 1}}\\ \zeta & { - \zeta {{\bar{\beta }}_0}^{ - 1}} \end{matrix}} \right]{{\textbf M}_s}({{Z_0},{z_1}} )\cdots {{\textbf M}_s}({{z_{N - 1}},{Z_1}} )\left[ {\begin{matrix} 1\\ {{{\bar{\beta }}_1}} \end{matrix}} \right]\\ & \cong \frac{1}{2}\left[ {\begin{matrix}{1 - i{k_{0z}}{u_0}}&{({1 - i{k_{0z}}{u_0}} ){{\bar{\beta }}_0}^{ - 1}}\\ {1 + i{k_{0z}}{u_0}}&{ - ({1 + i{k_{0z}}{u_0}} ){{\bar{\beta }}_0}^{ - 1}} \end{matrix}} \right]{ {{{\textbf M}_s}} |_{\Delta \varepsilon = 0}}({{Z_0},{z_1}} )\cdots { {{{\textbf M}_s}} |_{\Delta \varepsilon = 0}}({{z_{N - 1}},{Z_1}} )\left[ {\begin{matrix} 1\\ {{{\bar{\beta }}_1}} \end{matrix}} \right]\\& + \frac{1}{2}\left[ {\begin{matrix}1&{{{\bar{\beta }}_0}^{ - 1}}\\ 1&{ - {{\bar{\beta }}_0}^{ - 1}} \end{matrix}} \right]\left( {\sum\limits_{j = 1}^N {\frac{\partial }{{\partial {\varepsilon_j}}}({{{{\bar{\textbf M}}}_s}({{{\bar{Z}}_0},{z_1}} )\cdots {{{\bar{\textbf M}}}_s}({{z_{N - 1}},{{\bar{Z}}_1}} )} )\Delta {\varepsilon_j}} } \right)\left[ {\begin{matrix} 1\\ {{{\bar{\beta }}_1}} \end{matrix}} \right] + O({\Delta {\varepsilon^2}} ), \end{aligned}$$
where u02 and u0Δɛ are assumed to be small enough to be neglected in the right-hand equation of Eq. (59). Equation (59) can further be transformed as (60)$$\begin{aligned}\left[ {\begin{matrix}{{{{a_{s0}}} /{{a_{sN + 1}}}}}\\ {{{{b_{s0}}} / {{a_{sN + 1}}}}} \end{matrix}} \right] & =\frac{1}{2}\left[ {\begin{matrix}{1 - i{k_{0z}}{u_0}}&{({1 - i{k_{0z}}{u_0}} ){{\bar{\beta }}_0}^{ - 1}}\\ {1 + i{k_{0z}}{u_0}}&{ - ({1 + i{k_{0z}}{u_0}} ){{\bar{\beta }}_0}^{ - 1}} \end{matrix}} \right]{ {{{\textbf M}_s}} |_{\Delta \varepsilon = 0}}({{Z_0},{z_1}} )\cdots { {{{\textbf M}_s}} |_{\Delta \varepsilon = 0}}({{z_{N - 1}},{Z_1}} )\left[ {\begin{matrix}1\\ {{{\bar{\beta }}_1}} \end{matrix}} \right]\\ & \quad+ \frac{1}{2}\Delta z\left[ {\begin{matrix} 1&{{{\bar{\beta }}_0}^{ - 1}}\\ 1&{ - {{\bar{\beta }}_0}^{ - 1}} \end{matrix}} \right]\\ &\quad\times \left( {\sum\limits_{j = 1}^N {{{{\bar{\textbf M}}}_s}({{{\bar{Z}}_0},{z_1}} )\cdots {{{\bar{\textbf M}}}_s}({{z_{j - 1}},{z_j}} ){{\textbf X}_s}{{{\bar{\textbf M}}}_s}({{z_j},{z_{j + 1}}} )\cdots {{{\bar{\textbf M}}}_s}({{z_{N - 1}},{{\bar{Z}}_1}} )\Delta {\varepsilon_j}} } \right)\left[ {\begin{matrix}1\\ {{{\bar{\beta }}_1}} \end{matrix}} \right]\\&\quad + O({\Delta \varepsilon \Delta {z^2}} )+ O({\Delta {\varepsilon^2}} ), \end{aligned}$$
where Xs is defined as (61)$${\textbf X}_s^{} = - i{k_0}\left[ {\begin{array}{{cc}} 0&0\\ 1&0 \end{array}} \right].$$
For the single layer of the semi-infinite medium, the following equation can be derived for the m-th film as (62)$${\beta _m} \cong {\bar{\beta }_1} + \frac{{\Delta \varepsilon {}_m}}{{2\sqrt {{{\bar{\varepsilon }}_1} - {{\sin }^2}\theta } }} + O({\Delta {\varepsilon^2}} )= {\bar{\beta }_1} + \frac{{\Delta \varepsilon {}_m}}{{2{{\bar{\beta }}_1}}} + O({\Delta {\varepsilon^2}} ).$$
In the region from zi to zj, the following equation can also be derived with Eq. (62) as (63)$$\begin{aligned}{ {{{\textbf M}_s}} |_{\Delta \varepsilon = 0}} & ({{z_i},{z_{i + 1}}} )\cdots { {{{\textbf M}_s}} |_{\Delta \varepsilon = 0}}({{z_{j - 1}},{z_j}} )\\ & = \left[ {\begin{array}{{cc}} {\cos ({{k_0}{{\bar{\beta }}_1}({{z_j} - {z_i}} )} )}&{ - i{{\bar{\beta }}_1}^{ - 1}\sin ({{k_0}{{\bar{\beta }}_1}({{z_j} - {z_i}} )} )}\\ { - i{{\bar{\beta }}_1}\sin ({{k_0}{{\bar{\beta }}_1}({{z_j} - {z_i}} )} )}&{\cos ({{k_0}{{\bar{\beta }}_1}({{z_j} - {z_i}} )} )} \end{array}} \right] = { {{{\textbf M}_s}} |_{\Delta \varepsilon = 0}}({{z_i},{z_j}} ).\end{aligned}$$
Inserting Eqs. (62) and (63) into Eq. (60), the following equation can be derived: (64)$$\begin{aligned}\left[ {\begin{array}{{c}} {{{{a_{s0}}} / {{a_{sN + 1}}}}}\\ {{{{b_{s0}}} / {{a_{sN + 1}}}}} \end{array}} \right] & =\frac{1}{2}\left[ {\begin{array}{{cc}} {1 - i{k_{0z}}{u_0}}&{({1 - i{k_{0z}}{u_0}} ){{\bar{\beta }}_0}^{ - 1}}\\ {1 + i{k_{0z}}{u_0}}&{ - ({1 + i{k_{0z}}{u_0}} ){{\bar{\beta }}_0}^{ - 1}} \end{array}} \right]{ {{{\textbf M}_s}} |_{\Delta \varepsilon = 0}}({{Z_0},{Z_1}} )\left[ {\begin{array}{{c}} 1\\ {{{\bar{\beta }}_1}} \end{array}} \right]\\ & \quad + \frac{1}{2}\Delta z\left[ {\begin{array}{{cc}} 1&{{{\bar{\beta }}_0}^{ - 1}}\\ 1&{ - {{\bar{\beta }}_0}^{ - 1}} \end{array}} \right]\left( {\sum\limits_{j = 1}^N {{{{\bar{\textbf M}}}_s}({0,{z_j}} ){{\textbf X}_s}{{{\bar{\textbf M}}}_s}({{z_j},{{\bar{Z}}_1}} )\Delta {\varepsilon_j}} } \right)\left[ {\begin{array}{{c}} 1\\ {{{\bar{\beta }}_1}} \end{array}} \right]. \end{aligned}$$
Equation (64) can further be transformed as (65)$$\begin{aligned} \left[ {\begin{matrix}{{{{a_{s0}}} / {{a_{sN + 1}}}}}\\ {{{{b_{s0}}} / {{a_{sN + 1}}}}} \end{matrix}} \right] & =\frac{{\exp ({ - i{k_0}{{\bar{\beta }}_1}({{Z_1} - {Z_0}} )} )}}{2}\left[ {\begin{matrix} {1 - i{k_{0z}}{u_0}}&{({1 - i{k_{0z}}{u_0}} ){{\bar{\beta }}_0}^{ - 1}}\\ {1 + i{k_{0z}}{u_0}}&{ - ({1 + i{k_{0z}}{u_0}} ){{\bar{\beta }}_0}^{ - 1}} \end{matrix}} \right]\left[ {\begin{matrix} 1\\ {{{\bar{\beta }}_1}} \end{matrix}} \right]\\ & \quad + \frac{{ - i{k_0}\Delta z}}{2}\left[ {\begin{matrix} 1&{{{\bar{\beta }}_0}^{ - 1}}\\ 1&{ - {{\bar{\beta }}_0}^{ - 1}} \end{matrix}} \right]\left( {\sum\limits_{j = 1}^N {\left[ {\begin{matrix} { - i{{\bar{\beta }}_1}^{ - 1}\sin ({{k_0}{{\bar{\beta }}_1}{z_j}} )}\\ {\cos ({{k_0}{{\bar{\beta }}_1}{z_j}} )} \end{matrix}} \right]\exp ({ - i{k_0}{{\bar{\beta }}_1}({{{\bar{Z}}_1} - {z_j}} )} )\Delta {\varepsilon_j}} } \right). \end{aligned}$$
Inserting Eqs. (65) and Eq. (54) into Eq. (56), the reflectance modulation can be written as (66)$$\Delta {r_s} = 2i{k_{0z}}{u_0}\frac{{{{\bar{\beta }}_0} - {{\bar{\beta }}_1}}}{{{{\bar{\beta }}_0} + {{\bar{\beta }}_1}}} + \sum\limits_{j = 1}^N {\left( {i{k_0}\exp ({2i{k_0}{{\bar{\beta }}_1}{z_j}} )\frac{{2{{\bar{\beta }}_0}}}{{{{({{{\bar{\beta }}_0} + {{\bar{\beta }}_1}} )}^2}}}\Delta {\varepsilon_j}\Delta z} \right)} .$$
For the permittivity before the elastic modulation, the reflectance can be derived as (67)$$\bar{r} = \frac{{\bar{b}}}{{\bar{a}}} = \frac{{{{\bar{\beta }}_0} - {{\bar{\beta }}_1}}}{{{{\bar{\beta }}_0} + {{\bar{\beta }}_1}}}.$$
Inserting Eq. (67) into Eq. (66), the following equation can be derived: (68)$$\Delta {r_s} = 2i{k_{0z}}{u_0}\bar{r} + \bar{t}\bar{\tilde{t}}\frac{{i{k_0}^2}}{{2{{\bar{k}}_{1z}}}}\sum\limits_{j = 1}^N {({\exp ({2i{{\bar{k}}_{1z}}{z_j}} )\Delta {\varepsilon_j}\Delta z} )} .$$
where the coefficients are defined as (69)$${k_{0z}} = {k_0}{\bar{\beta }_0} = {k_0}\sqrt {{{\bar{\varepsilon }}_0} - {{\sin }^2}\theta } = {k_0}\sqrt {1 - {{\sin }^2}\theta } ,$$
(70)$${\bar{k}_{1z}} = {k_0}{\bar{\beta }_1} = {k_0}\sqrt {{{\bar{\varepsilon }}_1} - {{\sin }^2}\theta},$$
(71)$$\bar{t} = \frac{{2{k_{0z}}}}{{{k_{0z}} + {{\bar{k}}_{1z}}}},$$
(72)$$\bar{\tilde{t}} = \frac{{2{{\bar{k}}_{1z}}}}{{{k_{0z}} + {{\bar{k}}_{1z}}}}.$$
Taking the limit of Eq. (26), Eq. (68) can be written in the integral form over z as (73)$$\Delta {r_s} = 2i{k_{0z}}{u_0}\bar{r} + \bar{t}\bar{\tilde{t}}\frac{{i{k_0}^2}}{{2{{\bar{k}}_{1z}}}}\int\limits_0^\infty {\exp ({2i{{\bar{k}}_{1z}}z^{\prime}} )\Delta \varepsilon } ({z^{\prime}} ){\mbox{d}} z^{\prime}.$$
The reflectance modulation of Eq. (73) agrees with the analytical formula described in Ref. [1,18], which shows the multi-slicing matrix method can reproduce the Born approximation in a different way.4.3 Measuring thickness with the multi-slicing matrix method
The multi-slicing matrix method can be applied to measuring thicknesses of any stratified layers. To demonstrate this, a medium is assumed to be a layer of amorphous carbon of the thickness of 1680 nm that is stacked on a substrate of silicon. The reflectance modulation can be calculated using the multi-slicing matrix method. The complex refractive index of the amorphous carbon is na of Re(na)+iIm(na). A pump pulse laser (Fidelity-2, COHERENT) is irradiated with an incident angle of 0 degrees and the wavelength of 1070 nm with the repetition frequency of 70 MHz. The pulse duration is about 55 femto-seconds. A probe laser and the pump laser is set to coaxial in the experiment. Thus, an incident angle of the probe laser is subject to 0 degrees. The probe pulse laser generated using the optical second-harmonic generation (SHG-fs Harmonics, COHERENT) for the pump laser is irradiated with wavelength of 535 nm.
The temperature rise due to the absorbed power density can be derived using Eq. (5) following the Ref. [1] as
(74)$$\Delta T = \frac{q}{{\rho C}} = \frac{{Q\Delta {t_{in}}}}{{\rho C{z_a}}}\exp \left( { - \frac{z}{{{z_a}}}} \right),$$
where Q represents the absorbed power per area on the surface of the medium, and za is the light penetration length that can be written with the wavelength of the pump laser λ and extinction coefficient of Im(na) of the complex refractive index as (75)$${z_a} = \frac{\lambda }{{4\pi {\mbox{Im}} ({{n_a}} )}},$$
where the light penetration length za is 75.3 nm with the refractive index of the amorphous carbon of 1.96 + 0.56i.The thermal stress can be written with the temperature rise as
(76)$$P = - 3\Lambda \gamma \Delta T.$$
The elastic strain pulse propagation from initial generation until the reflection by the boundary between the amorphous carbon and the substrate can be derived from Eq. (8) with the velocity of the longitudinal acoustic wave V and a certain coefficient G0 as (77)$${\eta _{zz}} = {G_0}\left( {\exp \left( { - \frac{z}{{{z_a}}}} \right) - \frac{1}{2}\exp \left( { - \frac{{z + Vt}}{{{z_a}}}} \right) - \frac{1}{2}\exp \left( { - \frac{{|{z - Vt} |}}{{{z_a}}}} \right){\mbox{sgn}} ({z - Vt} )} \right).$$
Once the elastic strain pulse represented by the third term on the right-hand of Eq. (77) reaches the boundary, the elastic strain pulse is reflected toward the top surface of the medium with the reflectance Raco.Inserting Eq. (77) into Eq. (11), the permittivity modulation can be obtained. With the permittivity modulation, the reflectance modulation can be calculated using the multi-slicing matrix method with Eq. (56). Note that the matrix multiplication is only used to perform the multi-slicing method with Eq. (56), which makes the calculation algorithm simple especially in the matrix based numerical calculation softwares such as MATLAB and python [25]. Figure 3 shows the reflectance modulation calculated by the multi-slicing matrix method with respect to the time duration with a solid line. The experimental result is also plotted with respect to time duration with a gray solid line.
The infinitesimal thickness Δz of each film is taken as 1 nm in the multi-slicing matrix method except for the N-the layer, which satisfies the condition represented by Eq. (26) as
(78)$$\Delta z \ll \frac{{\Delta {\varepsilon _m}}}{{\Delta {{\varepsilon ^{\prime}}_{}}({{z_m}} )}} = {z_a},$$
where the light penetration length za is 75.3 nm. The thickness of N-th layer is set to an arbitrary value under the condition of less than 1 nm.The normalized opto-stress coefficient, κ=K/|K|, is determined by the fitting of the calculation to the experiment to be -0.961 + 0.274i. The reflectance Raco of the acoustic pulse is also determined by the fitting to be 0.3026. The calculation agrees well with the experiment. The specific values of the amorphous carbon used for the calculation and the normalized opto-stress coefficient κ are listed in Table 1.
Table 1. Specifications for amorphous carbon and fitting parameters
The thickness of the amorphous carbon can be derived by fitting the calculation to the experiment, giving a value of 1680.89 nm, which is almost the same as the actual thickness of 1680 nm.
5. Conclusions
A multi-slicing matrix method for calculating reflectance modulation induced by picoseconds acoustic pulse is proposed here for a stratified medium. The method based on the Abeles transfer matrix method applicable to the stratified media is mathematically validated. The method only uses matrix multiplications, which makes the calculation algorithm simple and is easy to be coded with matrix manipulation softwares. The multi-slicing matrix method is summarized in Appendix that contains only quantities and equations that need to be coded.
The multi-slicing matrix method is shown to reproduce the analytical formula derived using the Born approximation for s-polarization with semi-infinite single layer, which supports the validity of the method. The thickness of the amorphous carbon can be derived by fitting the reflectance modulation calculated by the multi-slicing matrix method to the experiment, giving a value of 1680.89 nm, which is almost the same as the actual thickness of 1680 nm. The infinitesimal thickness of each multi-slicing film is taken to satisfy Eq. (26).
The method is applicable to any stratified media just using the matrix multiplications of Eq. (56). The multi-slicing matrix method is a promising method for calculating the time-sequential reflectance modulations in the case of the laser ultrasonic technique.
Appendix
In this appendix, the algorithm for calculating reflectance modulation with the multi-slicing matrix method is summarized. The stratified medium is assumed to be consisting of L-th layers with the layer index of l (l = 0, 1, 2, …, L) while the multi-slicing films are assumed to be consisting of N-th films with the film index of m (m = 0, 1, 2, …, N). Here, the elastic displacement ul at z = Zl of the l-th layer and the zz-component of the strain tensor ηzz in the stratified medium are assumed to be given.
The relative permittivity modulation induced by the acoustic pulse in the l-th layer can be written in Eq. (11) with the opto-stress coefficient K and the zz-component of the strain tensor as
(A1)$$\Delta \varepsilon = 2\sqrt {{{\bar{\varepsilon }}_l}} K{\eta _{zz}},$$
where the bar symbol over the relative permittivity indicates a value before the elastic modulation, and the sub-index
l of the relative permittivity with the bar symbol indicates the layer number.
The z-component of the wave vector kmz for m-th film in the l-th layer is defined in Eq. (31) as
(A2)$${k_{mz}} = {k_0}{\beta _m},$$
where
βm is defined as
(A3)$${\beta _m} = \sqrt {{\varepsilon _m} - {{\sin }^2}\theta } = \sqrt {{{\bar{\varepsilon }}_l} + \Delta {\varepsilon _m} - {{\sin }^2}\theta } ,$$
with the
Δɛm calculated by Eq. (A1). Note that the sub-index
l of
${\bar{\varepsilon }_l}$ with the bar symbol indicates the layer number while the sub-index
m of
ɛm without the bar symbol indicates the film number.
The reflection modulation generated by the acoustic pulse can be written in Eq. (56) as
(A4)$$\Delta {r_s} = {\zeta ^2}\frac{{{{\textbf c}_{s1}}^{\textrm{t}}{{\textbf M}_s}({{z_0},{z_1}} )\cdots {{\textbf M}_s}({{z_{N - 1}},{z_N}} ){{\textbf f}_s}}}{{{{\textbf c}_{s0}}^{\textrm{t}}{{\textbf M}_s}({{z_0},{z_1}} )\cdots {{\textbf M}_s}({{z_{N - 1}},{z_N}} ){{\textbf f}_s}}} - \frac{{{{\textbf c}_{s1}}^{\textrm{t}}{{{\bar{\textbf M}}}_s}({{{\bar{Z}}_0},{{\bar{Z}}_1}} )\cdots {{{\bar{\textbf M}}}_s}({{{\bar{Z}}_{L - 1}},{{\bar{Z}}_L}} ){{\textbf f}_s}}}{{{{\textbf c}_{s0}}^{\textrm{t}}{{{\bar{\textbf M}}}_s}({{{\bar{Z}}_0},{Z_1}} )\cdots {{{\bar{\textbf M}}}_s}({{{\bar{Z}}_{L - 1}},{{\bar{Z}}_L}} ){{\textbf f}_s}}},$$
where
ζ,
cs0,
cs1, and
f are defined in Eq. (
53), (57), and (58) respectively as
(A5)$$\zeta = \exp ({i{k_{0z}}{u_0}} ).$$
(A6)$${{\textbf c}_{s0}} = \frac{1}{2}\left[ {\begin{array}{{c}} 1\\ {{{\bar{\beta }}_0}^{ - 1}} \end{array}} \right],$$
(A7)$${{\textbf c}_{s1}} = \frac{1}{2}\left[ {\begin{array}{{c}} 1\\ { - {{\bar{\beta }}_0}^{ - 1}} \end{array}} \right],$$
(A8)$${{\textbf f}_s} = \left[ {\begin{array}{{c}} 1\\ {{{\bar{\beta }}_{L + 1}}} \end{array}} \right].$$
The transfer matrix
Ms from
z =
zm-1 to
zm for the
m-th film is represented in Eq. (
39) as
(A9)$${{\textbf M}_s}({{z_{m - 1}},{z_m}} )= \left[ {\begin{array}{{cc}} {\cos ({{k_{mz}}\Delta z} )}&{ - i{\beta_m}^{ - 1}\sin ({{k_{mz}}\Delta z} )}\\ { - i{\beta_m}\sin ({{k_{mz}}\Delta z} )}&{\cos ({{k_{mz}}\Delta z} )} \end{array}} \right],$$
with the infinitesimal thickness as
(A10)$$\Delta z = {z_m} - {z_{m - 1}} = \frac{{{Z_L} - {Z_0}}}{N}.$$
The transfer matrix
${{\bar{\textbf M}}_s}$ that represents a transfer matrix before the elastic modulation can also be written in the same form as
(A11)$${{\bar{\textbf M}}_s}({{{\bar{Z}}_{l - 1}},{{\bar{Z}}_l}} )= \left[ {\begin{array}{{cc}} {\cos ({{{\bar{k}}_{lz}}({{{\bar{Z}}_l} - {{\bar{Z}}_{l - 1}}} )} )}&{ - i{{\bar{\beta }}_l}^{ - 1}\sin ({{{\bar{k}}_{lz}}({{{\bar{Z}}_l} - {{\bar{Z}}_{l - 1}}} )} )}\\ { - i{{\bar{\beta }}_l}\sin ({{{\bar{k}}_{lz}}({{{\bar{Z}}_l} - {{\bar{Z}}_{l - 1}}} )} )}&{\cos ({{{\bar{k}}_{lz}}({{{\bar{Z}}_l} - {{\bar{Z}}_{l - 1}}} )} )} \end{array}} \right].$$
With these quantities mentioned above, the reflectance modulation represented by Eq. (A4) can be rewritten as
(A12)$$\Delta {r_s} = {\zeta ^2}\frac{{{G_1}}}{{{G_2}}} - \frac{{{G_3}}}{{{G_4}}},$$
(A13)$${G_1} = \left[ {\begin{array}{{cc}} {\frac{1}{2}}&{\frac{{-{{\bar{\beta }}_0}^{ - 1}}}{2}} \end{array}} \right]{{\textbf M}_s}({{z_0},{z_1}} )\cdots {{\textbf M}_s}({{z_{m - 1}},{z_m}} )\cdots {{\textbf M}_s}({{z_{N - 1}},{z_N}} )\left[ {\begin{array}{{c}} 1\\ {{{\bar{\beta }}_{L + 1}}} \end{array}} \right],$$
(A14)$${G_2} = \left[ {\begin{array}{{cc}} {\frac{1}{2}}&{\frac{{{{\bar{\beta }}_0}^{ - 1}}}{2}} \end{array}} \right]{{\textbf M}_s}({{z_0},{z_1}} )\cdots {{\textbf M}_s}({{z_{m - 1}},{z_m}} )\cdots {{\textbf M}_s}({{z_{N - 1}},{z_N}} )\left[ {\begin{array}{{c}} 1\\ {{{\bar{\beta }}_{L + 1}}} \end{array}} \right],$$
(A15)$${G_3} = \left[ {\begin{array}{{cc}} {\frac{1}{2}}&{\frac{{-{{\bar{\beta }}_0}^{ - 1}}}{2}} \end{array}} \right]{{\bar{\textbf M}}_s}({{{\bar{Z}}_0},{{\bar{Z}}_1}} )\cdots {{\bar{\textbf M}}_s}({{{\bar{Z}}_{l - 1}},{{\bar{Z}}_l}} )\cdots {{\bar{\textbf M}}_s}({{{\bar{Z}}_{L - 1}},{{\bar{Z}}_L}} )\left[ {\begin{array}{{c}} 1\\ {{{\bar{\beta }}_{L + 1}}} \end{array}} \right],$$
(A16)$${G_4} = \left[ {\begin{array}{{cc}} {\frac{1}{2}}&{\frac{{{{\bar{\beta }}_0}^{ - 1}}}{2}} \end{array}} \right]{{\bar{\textbf M}}_s}({{{\bar{Z}}_0},{{\bar{Z}}_1}} )\cdots {{\bar{\textbf M}}_s}({{{\bar{Z}}_{l - 1}},{{\bar{Z}}_l}} )\cdots {{\bar{\textbf M}}_s}({{{\bar{Z}}_{L - 1}},{{\bar{Z}}_L}} )\left[ {\begin{array}{{c}} 1\\ {{{\bar{\beta }}_{L + 1}}} \end{array}} \right].$$
As shown in Eq. (
A13)–(
A16), the calculation of the reflectance modulation only uses matrix multiplications, which makes the calculation algorithm simple and is easy to be coded with matrix manipulation softwares.
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