# Quantitative Evaluation of Subsurface Damage by Improved Total Internal Reflection Microscopy
## 2 Theories and Simulation Analysis  
Traditional TIRM is designed based on the principle of total reflection. As shown in Ref. [20], the upper surface of an optical element under test is illuminated from within by a collimated laser, which has an incident angle that is equal to or slightly greater than the critical angle. If both the surface and subsurface of the illuminated area are defect-free, all incident light will reflect off the glass/air interface, and no light will scatter into the microscope. A camera captures a dark-field image called a background. On the contrary, incident laser will be scattered by surface defect or subsurface damage. A portion of scattering light transmits through the surface under test, and is collected by the microscope. An image composed of bright defects and dark background is obtained. Figure 1 is the conceptual illustration of subsurface damage [6]. SSD locates in the defect layer, and is hidden by the polished layer. It has a certain depth, ranging from 1 \(\upmu\)m to 100 \(\upmu\)m in the longitudinal direction. However, according to common sense, an objective lens in a microscope has a limited depth of field (DOF). Generally, along with the magnification of the objective lens increasing,  
the DOF dramatically decreases. The DOF of a high-magnification objective whose magnification no less than 50\(\times\) is not greater than 1 \(\upmu\)m. This means the greater part of the SSD is out of focus and blurred in the image, when the objective is focused on the top of the defect layer. Nevertheless, this feature is just right for our purpose in evaluating SSD depth. The micro-focusing control technique is adopted. High-magnification objective with small DOF is controlled by a one-dimensional precision translation stage to move longitudinally and can be focused on different depths under the surface. At each depth position, the microscope collects scattering light emitted from the SSD. Only the part located in the range of DOF could be imaged clearly, since other parts are out of focus and blurred. Accordingly, a series of images of different parts of the SSD are obtained. Since the TIRM image is composed of a relatively simple dark background and bright defects, a definition evaluation function is designed based on single-level wavelet transformation [24; 25], with a good balance between image processing speed and accuracy to calculate the definition of these images. The original image is divided into four sub-bands after wavelet transformation: One low-frequency sub-band and three high-frequency sub-bands. The low-frequency component is the outline of the original image, its approximation coefficients matrix is \(c_{LL}\), while the three high-frequency sub-bands represent the image details in three orientations (horizontal, vertical, and diagonal), and their details coefficients matrices are \(c_{LH}\), \(c_{HL}\), and \(c_{HH}\), respectively. The image is clearer, the summation of the absolute values of the three details coefficients matrices (\(|c_{LH}|\), \(|c_{HL}|\), and \(|c_{HH}|\)) is larger. Based on this, the definition evaluation function \(E\) is expressed as:  
\[E=\sum_{i,j}\left(|c_{LH}|+|c_{HL}|+|c_{HH}|\right) \tag{1}\]  
The clearer the image, the larger the value of \(E\). The simplified structure of the video microscope is shown in Figure 2. SSD is sliced in the depth direction. Through the micro-focusing operation, different slices of the SSD are generally located on the object plane. Scattered light emitted by SSD is collected by the objective lens, and passes through the projection eyepiece. The SSD slice in focus is imaged clearly on the camera, and other slices out of focus are blurred. The imaging process is simulated in the former paper [26]. The numerical aperture of the objective lens is N.A. = 0.80 (magnification is 50\(\times\)). The wavelength of incident light in vacuum is \(\lambda\) = 532 nm. Two types of SSD are used in simulation: Point defect and crack. Figure 1: Schematic diagram of subsurface damage (SSD).