# THz-TDS Techniques of Thickness Measurements in Thin Shim Stock Films and Composite Materials
## 2 Fundamental Theory  
### Measurement of Refractive Index  
Using the refractive index measurement technique, the reflection mode was applied in the time domain of the T-ray, and the refractive index was induced by picking up a signal reflected through the specimen. The progress direction of the T-ray signal is shown in Figure 1. Here, \(T\) is the transmitter of T-ray and \(R\) is the receiver of T-ray.  
The refractive index can be obtained by calculating the time of the T-ray, which is reflected from the terahertz pulsed emitter's arrival at the pulsed receiver and the time-of-flight (TOF) when the T-ray passes through a specimen of a certain thickness [1].  
This reflection mode obtains the refractive index by calculating each length of optical fiber reflected on the top and bottom of the specimen in the T-ray time-of-flight (TOF). Figure 1 shows the shape and path of T-ray. At first, if it is assumed that T-ray is projected on the specimen vertically, a time difference (\(\Delta t\)) can be obtained as follows:  
\[\Delta t=\frac{2d}{v} \tag{1}\]  
In consideration of the path of the oblique T-ray and the shape delay time in the reflection mode, as shown in Figure 1, a time difference (\(\Delta t\)) between the surface-reflected wave and back-reflected wave on the specimen can be obtained as follows:  
\[\Delta t=\frac{2l}{v}-\frac{\delta}{C_{a}} \tag{2}\]  
Here, \(l=\frac{d}{cos\theta_{r}}\), \(\delta=2 lsin^{2}\theta_{a}=2\frac{d}{cos\theta_{r}}sin^{2}\theta_{a}\), \(C_{a}\) is the velocity in air, \(d\) is the sample thickness, \(v\) is the sample velocity, \(\theta_{a}\) is the angle of inclination in the reflection mode, \(\theta_{r}\) is the refractive angle in the sample, and \(n\) is the refractive index. When the shape delay time and the path of oblique T-ray are traced, both the time difference (\(\Delta t\)) and resonance frequency (\(\Delta f\)) can be expressed as follows [13]:  
\[\Delta t=\left(\frac{2d}{v\ cos\theta_{r}}-\frac{\delta}{C_{a}}\right)=\frac{2 d}{cos\theta_{r}}\left(\frac{1}{v}-\frac{sin^{2}\theta_{a}}{C_{a}}\right) \tag{3}\]  
\[\Delta f=\frac{1}{\left(\frac{2d}{v\ cos\theta_{r}}-\frac{\delta}{C_{a}}\right) }=\frac{1}{\frac{2d}{cos\theta_{r}}\left(\frac{1}{v}-\frac{sin^{2}\theta_{a} }{C_{a}}\right)} \tag{4}\]  
Here, \(l\) is the refracted length in the sample, \(d\) is the thickness of specimen, \(v\) is the velocity in the air, \(l\) is the refracted length in the sample, \(\delta\) is the skip length of refractive waves in the sample, \(\theta_{r}\) is the refraction angle in the  
Figure 1: Diagram showing the geometry of the reflection mode [2].  
specimen, and \(\theta_{a}\) is the refraction angle in the air. The refractive index, which is one of the electromagnetic properties, can be calculated by following the steps above.  
The refractive index can be obtained with the approximate solution as follows:  
\[n^{4}-An^{4}-Asin^{2}\theta_{p1}=0 \tag{5}\]  
where \(d\) is the sample thickness, \(V_{air}\) is the light speed in air, and \(V_{s}\) is the light speed in the sample. \(\Delta t\) (\(T\)) is the difference time between with sample and without sample, and \(A=\frac{T^{2}V_{air}^{2}}{4d^{2}}\). Here, assuming that the normal reflection mode is vertical on the sample, the refractive index (\(n\)) should be \(v\Delta t/2d\). However, this reflection mode was composed with some angles. Therefore, a correction factor needs to be considered to obtain a better solution, as shown in Equation (5).  
### Measurement of Refractive Index  
In through-transmission mode, the index of refraction (\(n\)) can be calculated using the following equation, according to [2]:  
\[\therefore n=1+\frac{\Delta t\ v_{air}}{d} \tag{6}\]  
where \(\Delta t\) is the time cap between with sample and without sample, \(d\) is the sample thickness, \(V_{air}\) is the light speed in air, and \(L\) is the distance between the pulsed emitter and pulsed receiver.