## 2 Methods  
### Principle of Surface and Thickness Method  
The schematic diagram of the measurement system is illustrated in Figure 1. The incident light from a light-emitting diode (LED) light source was modulated by a digital micro-mirror device (DMD), then collimated by the tube lens before being reflected by the splitter mirror into the objective microscope lens. The fringe patterns on the DMD chip were projected onto the sample located in the focal plane. The charge-coupled device (CCD) camera behind the other tube lens captured the signal containing reflections taking place in the top surface of the film and the substrate. While the specimen is scanned in a \(z\)-axis direction by the piezoelectric transducer (PZT) stage, the fringe contrast response of modulated images can be used to characterize the thickness profile of the specimen. If we suppose that projected patterns are sinusoidal gratings, the light intensity distribution on the captured images can be written as:  
\[I(x,y,z)\ =A(x,y,z)[1+M(x,y,z)\ cos(2\pi fx+\phi_{0})], \tag{1}\]  
where \(A(x,y,z)\) illustrates the background irradiance and \(M(x,y,z)\) describes the modulation distribution of the projected sinusoidal fringe pattern, of which the spatial frequency is \(f\) and the initial phase is \(\phi_{0}\). The modulation distribution of each pixel can be determined by a phase-shifting algorithm [21; 22]. In order to minimize the uncertainty caused by the process of modulation evaluation, the fringe for each scanning step was laterally shifted eight times in this study. When measuring a sample with a single film layer as shown in Figure 2a, the projected fringe pattern was reflected from the film surface and the substrate. Suppose that \(j\) presents the film layer and the substrate and \(j=1,2\). Consequently, the light intensity distribution is given as:  
\[I(x,y,z)=\sum_{j=1}^{2}A(x,y,z)\big{[}1+M_{j}(x,y,z)cos(2\pi fx+\phi_{0})\big{]} \tag{2}\]  
Figure 1: Schematic of the measurement system. Figure 2: Principle of surface and thickness measurement: (**a**) the incoherent model of light reflection in a single-layered film structure; (**b**) the simulated modulation depth response (MDR) for a pixel \((x_{0},y_{0})\) generated during the \(z\)-axis scanning. Hence, the fringe modulation distribution \(M(x,y,z)\) in Equation (1) is composed of two corresponding component signals and can be written as:  
\[M(x,y,z)=M_{1}(x,y,z)+M_{2}(x,y,z) \tag{3}\]  
As determined in the study of [PERSON] [23], the normalized modulation depth response (MDR) of the individual surface is approximately a Gaussian function with an independent variable scanning position \(z\), which can be described as:  
\[M(x,y,z)\ =Rexp\left[-\left(\frac{z-p}{kFWHM}\right)^{2}\right] \tag{4}\]  
Here, \(R\) indicates the reflectivity of the thin film; \(p\) describes the focal position; \(z\) presents the scanning distance--i.e., the defocus distance from the focal position; and \(k=1/2\sqrt{\ln 2}\). \(FWHM\) is an acronym for the full width at half maximum and can be determined by [23]:  
\[FWHM=\frac{0.4470\lambda}{v(1-v)\sin^{2}[0.5 arcsin(NA/n)]}, \tag{5}\]  
where \(\lambda\) denotes the wavelength of the light source; \(n\) indicates the refraction index; \(v\) is the normalized spatial frequency of the sinusoidal fringe pattern--i.e., \(v=\lambda/(2 NA\cdot T)\); \(T\) describes the period of the sinusoidal fringe pattern; and \(NA\) presents the numerical aperture of the objective lens.