# Measurement System for Finger Skin Displacement on a Textured Surface Using Index Matching
## 3 Measurement System  
### Principles  
We use the index-matching method [21; 22] from the field of hydrodynamics to enable direct skin displacement measurements on textures. Normally, when observing the skin surface through a texture, light is refracted at the textured surface because of the large difference in the refractive index between the plate and the air. This prevents a clear observation. The index-matching method is used to solve this problem. It is a technique for making a texture optically transparent by submerging a transparent object in a liquid that has the same refractive index. See Figure 1.  
Figure 1: Top: fingertip with markers; middle: fingertip viewed through the textured surface without oil; bottom: fingertip viewed through the textured surface submerged in oil. Using this method, it is very important to match the refractive indices of the sample and the medium, because when the refractive indices differ, optical deviations will occur. Here, we simulate the optical misalignment of the skin displacement measurement over a 1D triangular texture having three variables: the refractive index, angle of the triangular textured surface, and distance from the refractive point to the fingertip--defined as \(n\), \(\theta\), and \(h\), respectively. In this system, an acrylic plate (refractive index: 1.490) is used. Figure 2 illustrates the angle of refraction used in the simulation. As temporary variables, the angle of incidence and the amount of optical deviation from the actual dot are set to \(\varphi\) and \(x\), respectively. Using the refractive-angle principle, the relation between the angle of incidence and the refractive angle is determined by the following equation based on [PERSON]'s law:  
\[\varphi=\sin^{-1}\left(\sin\left(90^{\circ}-\frac{\theta}{2}\right)\times \frac{1.49}{n}\right) \tag{1}\]  
The optical misalignment, \(x\), can be expressed as follows, using the angle of incidence from the dot \(\varphi\), the angle of the texture \(\theta\), and the distance to the fingertip \(h\):  
\[x=h\times\tan\left(90^{\circ}-\frac{\theta}{2}-\varphi\right) \tag{2}\]  
When the refractive indices are in perfect agreement (\(n=1.490\)), no refraction occurs. Thus, the optical misalignment, \(x\), is zero. Here, we estimate the degree of optical misalignment that occurs when we use silicone oil (\(n=1.485\)) for the measurement system and water (\(n=1.333\)) for comparison. Figure 2: Overview of light refraction and the amount of optical misalignment caused by it. The refractive index of the medium, the angle of the texture, and the distance to the fingertip are defined as \(n\), \(\theta\), and \(h\), respectively. As temporary variables, the angle of incidence and the amount of optical deviation from the actual dot are set to \(\varphi\) and \(x\), respectively. Figure 3 shows the correspondence between the distance from the refractive point to the fingertip and the amount of optical misalignment. Figure 1 shows the correspondence between the distance from the refractive point to the fingertip and the amount of optical misalignment. Each line in the graph shows the amount of displacement when the texture surface angle, \(\theta\), is changed. From each graph, it can be seen that the optical deviation becomes smaller when the refractive index is close to that of acrylic. The optical misalignment in silicone oil becomes smaller with smaller texture angles. When compared with a two-point discrimination threshold of 3 mm [23], which is considered to be the basic human spatial resolution, it is found to be very small. Next, we compare the vibration detection thresholds of each receptor [24; 25] (SAI: 100 \(\upmu\)m; RAI: 10 \(\upmu\)m; PCs: 1 \(\upmu\)m) with the optical misalignment when silicone oil is used. When the height is less than 5 mm, and the texture angle is less than 30\({}^{\circ}\), it is smaller than the threshold of SAI. Additionally, when the texture angle is large (about 90\({}^{\circ}\)), it is possible to measure the optical misalignment below the RAI threshold.