# Quantitative Evaluation of Subsurface Damage by Improved Total Internal Reflection Microscopy
## 2 Theories and Simulation Analysis
Figure 3 shows two point defects at different depth under surface. The size in depth direction of the point defect is smaller than the DOF of the objective lens. Both of the two are a single slice, and have undergone a process of 'defocus-focus-defocus'. The definition of all images achieved through the system are calculated using Equation (1). Right graph of Figure 3 is the definition curve of the two point defects. Horizontal axis represents images' sequence corresponding to depth position. Vertical axis represents the normalized E. Two peaks are shown in the curve. The image position of the two peaks are exactly the corresponding defects on the objective plane. When the point defect is out of focus, the value of normalized E is decreased sharply. Since the lateral size of the first point defect is larger than the second point defect, its normalized E is greater than the second point defect. Subsurface cracks are usually V-shaped [27; 28], as shown in Figure 4a. Its width is always on the order of microns or sub-micron. The width and depth of cracks are initially 4 \(\upmu\)m and 13 \(\upmu\)m, respectively. The crack is divided into 13 slices by a step of 1 \(\upmu\)m along the depth direction. With a micro-focusing operation, the microscope is focused on different depths. The image captured by the camera is the integral of all 13 slices imaged through the microscopic system. Its definition curve is shown in Figure 4b, where there are two obvious turning points. Between the two turning points, the normalized E drops slowly due to the V-shaped crack becoming narrow. On the contrary, the normalized E rises or falls rapidly before or after the turning points. There are 12 points including the turning points, which means that the depth is about 12 \(\upmu\)m. It is close to the presupposed depth. As a comparison, the crack definition curves of 1 \(\upmu\)m wide and 13 \(\upmu\)m deep, 4 \(\upmu\)m wide and 18 \(\upmu\)m deep, are also given in Figure 4c,d, respectively. The two definition curves are similar to the curve in Figure 4b. There are 11 points between the two turning points in Figure 4c. The crack depth obtained from the definition curve is 13 \(\upmu\)m, which is consistent with the presupposed depth. The normalized E fluctuation of the definition curve in Figure 4c is smaller than the normalized E fluctuation of the definition curve in Figure 4b. It may be caused by a smaller change in the crack width of the adjacent slice. In Figure 4d, 15 points exist between the two turning points. The crack depth obtained from the definition curve is 17 \(\upmu\)m, which is also close to the presupposed depth. Figure 3: Definition curve of point defects. Figure 2: Microscopic imaging of SSD. The condition of total internal reflection illumination is ignored in the simulations. In the total internal reflection illumination, the incident field and reflected field are superimposed on the illumination region to form a standing wave field [21]. The standing wave field period \(\Gamma\), independent of the polarization of the incident light, is expressed as:  
\[\Gamma=\frac{\lambda}{2n\cos\theta} \tag{2}\]  
where, \(n\) is refractive index; \(\theta\) is incident angle. \(n\) = 1.46 and \(\theta\) = 45\({}^{\circ}\) are chosen, \(\Gamma\) = 0.258 \(\upmu\)m. \(\Gamma\) is smaller than the DOF, 0.84 \(\upmu\)m. The intensity of a standing wave field for s-polarized light is shown in Figure 5. When performing a micro-focusing operation, there are the same number of cycles within the DOF. This means the illumination intensity is identical in each micro-focusing operation. Therefore, total internal reflection illumination doesn't affect the effectiveness of the simulation results. Figure 4: Definition curves of subsurface crack. (**a**) Schematic diagram of V-shaped crack; (**b–d**) Crack definition curves of 4 \(\upmu\)m wide and 13 \(\upmu\)m deep, 1 \(\upmu\)m wide and 13 \(\upmu\)m deep, 4 \(\upmu\)m wide and 18 \(\upmu\)m deep, respectively. Figure 5: Distribution of the standing wave field.