# Inverse problem-solving method applied to acoustic characterisation of a thin film using Debye series model

Pooyan Manoochehrnia, Pierre Marechal, Damien Leduc, Mounsif Ech-Cherif

El-Kettani

► To cite this version:

Pooyan Manoochehrnia, Pierre Marechal, Damien Leduc, Mounsif Ech-Cherif El-Kettani. Inverse problem-solving method applied to acoustic characterisation of a thin film using Debye series model. Anglo-French Physical Acoustics Conference (AFPAC 2024), Jan 2024, Loch Lomond, United Kingdom. pp.012001, 10.1088/1742-6596/2904/1/012001. hal-04917549

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# Inverse problem-solving method applied to acoustic characterisation of a thin film using Debye series model

To cite this article: Pooyan Manoochehrnia et al 2024 *J. Phys.: Conf. Ser.* **2904** 012001

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# Inverse problem-solving method applied to acoustic characterisation of a thin film
using Debye series model

Pooyan MANOOCHEHRNIA, Pierre MARECHAL, Damien LEDUC, Mounsif ECH-CHERIF EL-KETTANI

LOMC, UMR 6294 CNRS, University of Le Havre Normandy, 75 Rue de Bellot, 76600 Le Havre, France

pooyan.manoochehrnia@gmail.com

## Abstract

In this paper, it is proposed to solve an inverse problem with the perspective of determining the ultrasonic properties (thickness, longitudinal wave velocity, density and attenuation coefficient) of a thin epoxy resin film deposited on a glass substrate with well-known properties. The question focuses on finding acoustic parameters of the structure and the transducer thanks to a numerical implementation of a physical model matching the experimental ultrasonic response. The Debye Series Model (DSM) has been used to model the structure. A step by step procedure is implemented where at each step one parameter is estimated by minimizing the MSE (Mean Square Error) between simulated model and experimental signal. Estimated parameters are checked to be physically viable. At beginning an initial set of parameters are announced together with related waveforms and at the end the final adjusted parameters are presented beside finally obtained waveforms.

## 1. Introduction

In recent years, the analysis of the non-uniqueness and uncertainty of solutions to inverse problems has played an increasing role in physics modelling. Linear problem-solving analysis has been well established for many years, while non-linear problem analysis methods have been developed more recently. Even if best-fit methods give solutions, the analysis of uncertainty and uniqueness is very important for the evaluation of scientific conclusions based on inverse calculations. The inverse problem resolution can be carried out methodically by an a priori step-by-step procedure, leading to a unique set of parameters. As a general description of the problem, a modelling method is used as a description of the physical problem, on the basis of a given set of input parameters. In ultrasound modelling of multi-layered structures, the studied structure could be either modelled in oblique incidence using the stiffness matrix model (SMM) as mentioned in [1], [2], [3], [4] or in normal incidence using Debye series model (DSM) as mentioned in [5], [6], [7]. In this paper, the structure has been modelled using DSM which is unconditionally stable and for any type of configuration as first described in [5], [8], [9], [10].

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The reflection spectrum of the multilayer structure calculated via DSM is combined to the impulse response spectrum of the transducer [11], [12], [13], by a multiplication in the frequency domain.

In summary, DSM involves calculation of the global reflection coefficient as a function of frequency, R<sub>g</sub>(f) (1), of the multilayer structure based on global transmission coefficient, T<sub>g</sub>(f) (2), iterative calculation of local reflection, R<sub>p,q</sub>(f), and transmission, T<sub>p,q</sub>(f), coefficients of each layer, (3) and (4). Equations (1) to (4), are calculated using equations (5) to (7). Once the global reflection coefficient, R<sub>g</sub>, has been calculated, it will be multiplied by spectrum of incident signal which is usually in form of a gaussian function. Finally, the inverse Fourier transform of the later function is calculated which in turn results in time-based echo signal addressing the structure response to ultrasonic acoustic solicitation.

$$R_g = R_{12} + \frac{T_{12} \rho_{23} T_{21}}{1 - R_{21} \rho_{23}} \qquad (1)$$

$$T_g = T_{12} \prod_{n=2}^{n=N} \frac{T_{n,n+1}}{1 - R_{n,n+1} \rho_{n,n-1}} \qquad (2)$$

$$\left\{ \begin{aligned} R_{n,n+1} &= \frac{Z_{n+1} - Z_n}{Z_{n+1} + Z_n} e^{2j\varphi_n} \\ T_{n,n+1} &= \frac{2Z_{n+1}}{Z_{n+1} + Z_n} e^{-j(\varphi_{n+1} - \varphi_n)} \end{aligned} \right. \qquad (3)$$

$$\left\{ \begin{aligned} R_{n+1,n} &= \frac{Z_n - Z_{n+1}}{Z_n + Z_{n+1}} e^{-2j\varphi_{n+1}} \\ T_{n+1,n} &= \frac{2Z_n}{Z_{n+1} + Z_n} e^{-j(\varphi_{n+1} - \varphi_n)} \end{aligned} \right. \qquad (4)$$

$$\left\{ \begin{aligned} \varphi_n &= k_n T h_n = \frac{\omega T h_n}{C_n} \\ \varphi_{n+1} &= k_{n+1} T h_n = \frac{\omega T h_n}{C_{n+1}} \end{aligned} \right. \qquad (5)$$

$$\left\{ \begin{aligned} \rho_{n+1,n} &= R_{n+1,n} + \frac{T_{n+1,n} \rho_{n,n-1} T_{n,n+1}}{1 - R_{n,n+1} \rho_{n,n-1}} & \text{for } 2 \le n \le N \\ \rho_{21} &= R_{21} \end{aligned} \right. \qquad (6)$$

$$\left\{ \begin{aligned} \rho_{n,n+1} &= R_{n,n+1} + \frac{T_{n,n+1} \rho_{n+1,n+2} T_{n+1,n}}{1 - R_{n,n+1} \rho_{n+1,n+2}} & \text{for } 2 \le n \le N \\ \rho_{N+1,N+2} &= R_{N+1,N+2} \end{aligned} \right. \qquad (7)$$
Two sets of parameters are dealt with in this paper, namely transducer envelope parameters and structure acoustic parameters. The parameters are identified and classified in such a way that their effect on the final simulation result is isolated. The mean square error (MSE) between simulated and experimental signals is taken as an index of similitude between two signals, before passing from one set of parameters to the next [14], [15]. Above mentioned methods lead to estimation of acoustic parameters like thickness, speed, density and other elastic constants of various layers. Although in this study scanning acoustic microscopy (SAM) has been used in reflection mode, its usage has been concentrated only on punctual measurements (A-Scan).

## 2. Experiment

An 85 µm thick layer of epoxy resin has been deposited on a glass plate of 1.66 mm thick to form a two-layered structure. The thin film of epoxy-resin is deposited on the glass plate using a spin-coater with a pre-set turning speed of 2000 rpm and duration of 20 s [21], [22], [23]. The thickness of deposit is measured using a precise roughness meter device (Mitutoyo Surftest SV-2100). Surface profile obtained from roughness meter device illustrates that the thickness profile of the deposit is not uniform. Thus, a relatively uniform part of deposit has been marked to be characterised using SAM. This thickness value is then used in procedure of calculations. In addition, on the basis of several sources, different values for density of epoxy resin are reported and vary within a very large range. Nevertheless, considering the specific combination of resin and epoxy hardener that is used in this experiment, the value of the density of the actual epoxy resin was estimated at 980 ± 2% [kg/m3]. Each layer of the structure is to be defined by its geometrical and acoustical parameters including the thickness $Th$, the longitudinal acoustic wave velocity $C$, the density $ρ$ and the acoustic attenuation coefficient $α$. The structure is characterised using the SAM equipped with a plane wave transducer with a central frequency of 50 MHz working as emitter and receiver. The structure is immersed in water together with the transducer which measures the acoustic echo reflected at normal incidence from the structure. A dedicated software has been developed by SAM manufacturer (WinSAM v8) for the driving of the translations along the (x, y, z) axes with the declared precision of 1 µm, as well as for the capture of the ultrasonic echoes. The typical configuration setup is composed of a plane wave transducer with central frequency of 50 [MHz] driven with an integrated electric impulse of 5 [ns] duration, has been positioned within a distance equivalent to $ToF$ (Time of Flight) of 8.9 [µm] from the structure under test. An oscilloscope with sampling frequency of 25 [GHz] and quantification of 8 [Bits] is used for this measurement.
## 3. Method

### 3.1. Model Calculation process

The electroacoustic response of the studied structure is simulated via an efficient numerical model based on the Debye series decomposition and named as DSM [6, 7, 8]. This DSM tool calculates the reflection coefficient of the whole structure on the basis of series of the local transmission and reflection coefficients at each interface between the layers, and as function of the frequency [6], [12]. The local reflection $R_{mn}$ and transmission $T_{mn}$ coefficients are calculated based on expressions defined in the DSM formalism. Those expression are based on the properties of the multi-layer structure, i.e. acoustic impedance $Z_n$, and the Fresnel phase parameters. Finally, the total reflection coefficient $R_g$ is calculated. Then the resulting reflection spectrum $R_g$ is then multiplied by a gaussian envelope $G_e$ representing the electroacoustic response of the transducer as a symmetric envelope. The time-domain electroacoustic response $s(t)$ is obtained as a result of the numerically implemented inverse Fourier transform.

### 3.2. Inverse problem solving by parameter matching process

The DSM has been implemented using MATLAB and it results in the electroacoustic response of a given transducer ($G_e$) on a given multilayer structure ($R_g$ and $T_g$). The resulting modelled signal $s(t)$ is then compared to the experimental one. The global electroacoustic response includes each reflected echo in the bilayer. This imposes simultaneous characterisation of acoustic parameters of both substrate (variables indexed $s$) and deposit (variables indexed $d$). Nevertheless, as we are only interested by the properties of the thin layer (those of the glass substrate are previously well identified), only the first package of echoes due to multiple reflections in the thin film are considered by the process of matching between the model and the experiment.

The first step of the inverse problem concerns the transducer's response, which is modelled by the central frequency $f_0$ and the relative bandwidth $\Delta f_{6,r}$, corresponding (in a first approach in the time domain) to the pseudo-period and the echo duration. In addition, the phase shift $\phi_s$ of the transducer's response is updated to match to the local minima and maxima of the echoes. The second step consists in the syncing of time of flight $ToF$ inside the deposit layer ($ToF_d = Th_d / C_d$), between experimental signal and DSM resulting in, in one hand inserting a reference point in signals for further adjustments of two signals and on the other hand resulting in determination of parameter pair of $C_d$ and $Th_d$.
The third step is the determination of the acoustic impedance of the deposited thin film $Z_d = \rho_d C_d$ that allows the better matching of the modelled and experimental signals amplitudes. In the fourth step, the attenuation in the thin film is adjusted so that the mismatch is minimum. The amplitudes of the subsequent echoes are directly related to the attenuation of the deposit layer $\alpha_d$, which can be finally adjusted.

The objective is to start from the above defined configuration with the input set of parameters {$\rho_d, C_d, \delta_d, Th_d$} for the deposit and modify it in such a way that the DSM matches to the experimental signal. To do this, a step-by-step method have been set up to provide a robust and reproducible result. Knowing with a good confidence some of the parameters (substrate properties), the described step-by-step preliminary approach ensures the unicity of the initial set of physical parameters. This set of initial values of properties are summarized in Table 1.

<table><caption>Table 1: Initial parameters: Fixed properties of the substrate and initial values of the deposit.</caption><thead><tr><td>Parameter</td><td>ρ [kg/m³]</td><td>C [m/s]</td><td>δ [%]</td><td>Th [μm]</td></tr></thead><tbody><tr><td>Substrate</td><td>2500</td><td>5800</td><td>0</td><td>1.6</td></tr><tr><td>Deposit</td><td>1300</td><td>2500</td><td>1.5×10⁻³</td><td>85</td></tr></tbody></table>

A zoom on the first package of echoes which illustrates the signal resulting from the simulation of the studied structure with above parameters, and experimental signal is shown in Fig. 1.

<figure><img src="image_6.png" /><figcaption>Fig. 1: DSM modelled and SAM experimental comparison between the first package of echoes for initial state.</figcaption></figure>

During the procedure following steps are followed:
### 3.2.1. Step 1: $ToF_d$

The time-of-flight corresponding to above simulated parameters is $ToF_d = 2Th_d / C_d = 68$ ns, whereas experimentally, this $ToF_d$ is clearly lower. A time delay estimation leads to reduce this $ToF_d$ by 15% of its initial value. This reduction of 15% could be done either by reducing the thickness $Th_d$ or by increasing $C_d$ or any linear combination of both. Therefore, the thickness at this step is multiplied by 0.85. At this step, the reduction of -15% in TOF is fixed and the algorithm is followed to the next step.

### 3.2.2. Step 2: $\varphi$

From the result of previous step, it is observed that even if the envelopes of the echoes are in phase, the phase shift of the 1st roundtrip echo of the DSM signal is unlike the experiment. This could be corrected throughout the transducer's impulse response $G_e$ by introducing phase shift $\varphi$ to the gaussian envelope used in DSM. Indeed, the desired phase shift inducing the required phase shift should be between 0 and $\pi$ or between $\pi$ and $2\pi$. Therefore, the value of $3\pi/2$ is tested which results in better match between two signals. With an additional refinement, the adjusted phase shift would be equivalent to $3\pi/2 \times 1.08$, which results in an excellent match between two signals. Therefore, the phase shift is fixed to above value and the algorithm is followed to the next step.

### 3.2.3. Step 3: $C_d$

It should be noted that the notion of acoustic impedance plays an important role here. It is possible to keep the $ToF_d$ fixed (as set in step 1) but to change the amplitude of signal in order to get a better match between simulated model and experimental signal. This can be done by multiplying both thickness and celerity to the same value. For example, thickness could be multiplied to 0.85 (previously set) and then again multiplied to 0.9. In order to keep $ToF_d$ fixed, at the same time, $C_d$ should be multiplied by 0.9. This will result in a better match between experimental and simulated signals.

### 3.2.4. Step 4: $Z_d$

From the above statement, it could be deduced that in order to get a better match between the amplitudes, the acoustic impedance should be changed. This means that the change in $ToF_d$ affects the acoustic impedance $Z_d$ via the velocity $C_d$. Therefore, it should be compensated by the density $\rho_d$ to reach to the desired overall change in $Z_d$ and $ToF_d$ at the same time. For example, one compromise would be to evenly distribute the 15% reduction in $ToF_d$, that has been already fixed in step 1, as a linear combination between the velocity $C_d$ and the thickness
$Th_d$, e.g. -8% on $Th_d$ and +7% on $C_d$. Above discussion leads us obligatory to a +7% increase in $Z_d$ that is compensated by -10% change in $\rho_d$ resulting in total -3% change in $Z_d$. Although this could result in good numerical match between curves, it may not result in satisfactory estimated physical values for the materials. Therefore, we need some fine adjustments as follows.

### 3.2.5. Step 5: Refinement

As a refinement of step 4, the set of parameters {$\rho_d, C_d, \delta_d, Th_d$} for the deposit improves the fit in the numerical sense, but is not satisfying physically. A fine-tuned adjustment around the previous values is necessary by modifying the results as in Table 2. It should be noted that the key for adjustment of these parameters is the known physical value of density $\rho = 980 \pm 2\%$ [kg/m3], as previously mentioned in section 2. Observing the resulting adjustment, a bigger mismatch between amplitudes is observed. This mismatch is the price to be paid for physically acceptable results. Nevertheless, the mismatch could be also due to inappropriate attenuation factor.

<table><caption>Table 2: Parameters values used in Step 5.</caption><thead><tr><td>Parameter</td><td>ρ [kg/m³]</td><td>C [m/s]</td><td>δ [%]</td><td>Th [μm]</td></tr></thead><tbody><tr><td>Step 5 values</td><td>1300×0.75</td><td>2500×1.07</td><td>1.5×10⁻³</td><td>85×0.93</td></tr></tbody></table>

### 3.2.6. Step 6: $\delta_d$

In this step, the loss coefficient $\delta_d$ in the deposit is adjusted to fit the third echo, so that its order of magnitude is between 1% and 10%. As a result, the loss coefficient is found at $\delta_d = 1.5\%$. It should be noted that the attenuation value $\alpha_d$ is directly related to the loss coefficient $\delta_d$ that is the relative imaginary part of the velocity $C_d$. It is usually written as $C_d = C_{0d} (1 + j\delta_d)$. This velocity loss coefficient $\delta_d$ can be directly related to the attenuation $\alpha_d = \alpha_{0d} f$, implicitly considered as a frequency dependent coefficient. As a result, the complex wavenumber in the deposit can be written as: $k = \omega / C_d + i \alpha_d$ and the slope of the attenuation coefficient has been identified to be $\alpha_{0d} = 3.5 \times 10^{-5} \text{ Np.m}^{-1}.\text{Hz}^{-1}$.

### 3.2.7. Step 7: $\varphi$ refinement

The previous changes result in a time shift between the 1st peaks of 0.5 ns. Therefore, an additional phase-shift correction is applied to the gaussian envelope, therefore multiplying the
phase-shift by a further 1.04 coefficient resulting in $\phi_d = 3\pi/2 \times 1.08 \times 1.04$. As illustrated in Fig. 2, it gives a much better match than all previous results.

<figure><img src="image_7.png" /><figcaption>Fig. 2: DSM modelled and SAM experimental signals using parameters in Step 7.</figcaption></figure>

## 4. Results & Discussion

In summary, the developed algorithm is divided in following two parts:

A. Initialization: in this section the maximum peaks are synchronized and then an initial set of 11 parameters is set. Then the algorithm is focused on the 1st package of echoes bypassing the subsequent return echoes of substrate momentarily at this stage.

B. Calculating the change percentage needed to apply on each of the deposit parameters {$\rho_d, C_d, \delta_d, Th_d$} and on gaussian frequency envelope representing the transducer bandwidth {$f_0, \varphi, \Delta f_0$}. Continuing to update above parameters as long as the mean square error (MSE) has not been decreased by a specified percentage.

The final matched parameters are summarized in Table 3.

<table><caption>Table 3: Finally matched set of parameters.</caption><thead><tr><th>Parameter</th><th>φ [Rad]</th><th>ρ [kg/m³]</th><th>C [m/s]</th><th>δ [%]</th><th>Th [µm]</th></tr></thead><tbody><tr><td>Final value</td><td>3π/2×1.08×1.04</td><td>1300×0.75</td><td>2500×1.07</td><td>1.5</td><td>85×0.93</td></tr></tbody></table>

As a summary, film thickness and other viscoelastic parameters of a deposited thin film on a known substrate could be deduced by the step-by-step method presented in this paper. These
parameters are occasionally inferred jointly due to their joint physical constraints.

**5. Conclusion**

A very good agreement between form of signals is achieved, since the time shift between echoes has been corrected, and the amplitudes have been matched. This results in a very good match between DSM simulation results and SAM experimental signals which presents a promising methodology for solving the inverse-problem of characterisation of thin layer using SAM.

As a perspective similar analytical studies are being explored with higher order reflections considered. Additionally, these data could be exploited as additional information for the viscoelastic behaviour, i.e. frequency dependence of the wave velocity and attenuation.

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