# Measuring Maritime Paint Thickness under Water Using THz Cross-Correlation Spectroscopy
## 2 Materials and Methods
### Spectroscopic Model  
As discussed in Section 2.2, reflections from different interfaces are separated in the time domain as seen in Figure 4A. With a clear separation of the reflection peaks in the time domain, the thickness can be found by a simple calculation through Equation (1). It should be noted that the refractive indices of the media need to be known to use this method. However, when the media thicknesses become smaller, the different reflection  
Figure 2: (**A**) Traditional reflection setup with the paint placed at the focal point (4) of the transmitting (2) and receiving (3) antennae. The antennae are set at an angle of incidence of 20\({}^{\circ}\) (1) and use the s-polarization. (**B**) The silicone and TOPAS plastic contact patch on top of the paint sample. In the measurements, the contact patch was encased in a 3D-printed device, which enabled pressure to be applied to the silicone. (**C**) Zoomed-in version of the illustration on how the thickness can be derived from the difference in arrival time (\(\Delta t\)) of the two reflections, described by Equation (1). (**D**) The TOPAS and silicone layers on the paint, without the 3D-printed holder. (**E**) A simplification of the physical model was made to reduce the complexity. peaks overlap significantly, and the location of the peaks in the time domain can no longer be reliably determined. This problem can be solved by using a more advanced spectroscopic model, where both an unknown thickness and refractive index can be determined. This procedure is now introduced. The transfer function of a three-layer system is considered. When a plane wave \(E_{0}\) coming from medium 1 is incident on medium 2, the total reflected field is the sum of the initial reflection and all internal reflections between medium 2 and 3, i.e.,  
\[\bar{E}_{r}(\omega) = \bar{E}_{0}\Big{(}r_{12}+t_{12}e^{-i\phi}r_{23}e^{i\phi}t_{21}+...\Big{)}\] \[= \bar{E}_{0}\Bigg{(}r_{12}+t_{12}t_{21}r_{23}e^{-2i\phi}\sum_{n=0} ^{\infty}\Big{(}e^{-2i\phi}r_{23}r_{21}\Big{)}^{n}\Bigg{)}\]  
where \(r_{ij}\) are the appropriate Fresnel reflection coefficients and \(\phi\) is the acquired phase upon one traversal of medium 2 given by  
\[\phi=\frac{\omega\cdot n(\omega)\cdot d}{\cos\theta\cdot c}, \tag{3}\]  
where \(\omega\) is the angular frequency, \(d\) is the layer thickness, and \(\theta\) is the angle of incidence. Simplifying Equation (2), we find that  
\[\bar{E}_{r}(\omega)=\bar{E}_{0}(\omega)\frac{r_{12}+r_{23}e^{-2i\phi}}{1+r_{1 2}r_{23}e^{-2i\phi}}. \tag{4}\]  
If an additional medium is present behind medium 3, the total reflected field from medium 3 to medium 2 is calculated by applying Equation (4) again to determine an effective expression for \(r_{23}\) labeled \(H_{3}(\omega,\,d,\,n)\). This procedure is recursively followed until all layers are accounted for [30]. With a reference measurement \(\bar{E}_{ref}(\omega)\) from a highly reflective metal plate with Fresnel coefficient \(r_{m}\)  
\[\bar{E}_{ref}(\omega)=\bar{E}_{0}(\omega)r_{m} \tag{5}\]  
and a model for the refractive index \(n\), the thickness \(d\), and refractive index parameters can be determined for each layer by minimizing  
\[L(d_{1},n_{1},...,d_{n},n_{n})=\left|\frac{\bar{E}_{sample}}{\bar{E}_{ref}}- \frac{r_{12}+H_{3}e^{-2i\phi}}{r_{m}\big{(}1+r_{12}H_{3}e^{-2i\phi}\big{)}} \right|^{2}. \tag{6}\]  
The silicone layer is modeled as having a constant refractive index \(n\) and the paint layer is modeled as a Debye medium with a permittivity function given by  
\[\epsilon(\omega,\,e_{\infty},\,e_{s},\,\tau)=\epsilon_{\infty}+\frac{ \epsilon_{s}-\epsilon_{\infty}}{1-i\omega\tau}, \tag{7}\]  
where \(\epsilon_{\infty}\) is the high-frequency limit of the permittivity, \(\epsilon_{s}\) is the static permittivity, and \(\tau\) is a material-dependent relaxation time. Even for a few layers, Equation (6) quickly becomes highly nonconvex, and therefore a local minimization algorithm is inappropriate for the parameter extraction.