# Warpage Behavior on Silicon Semiconductor Device:
## 3 Warpage Measurements and Physical Analysis  
### Method Description for Warpage Measurements  
A method based on Moire Phase Shift is considered to experimentally quantify the warpage, using the commercial equipment \"TDM (Topography and Deformation Measurements) Compact\" by Insidix [23] presented in Figure 2.  
Analyzed sample is illuminated by a stripe pattern, which interacts with sample's surface structure. This image is captured by a Charge-Coupled Device (CCD) camera, that correlates the resulting pattern modification with surface out-of-plane deformation of each sample point. The equipment is made by two main parts: the projection (composite light source) and the viewing (CCD camera + optics) systems. The first is a common projector, which projects strip patterns piloted by software. The viewing system includes a CCD camera and the viewing lens. When using phase-shift projection Moire, the camera grating is phase-shifted against the projection grating. This technique can be applied for accurate 3D reconstruction. The equipment can regulate chamber temperature by infrared lamps and air cooler by a proportional-integral-derivative (PID) controller, in which the feedback is given by the temperature measured in real time by a thermocouple directly attached to measurement stage. The schematic of experimental equipment is reported in Figure 3.  
Figure 2: Experimental equipment TDM for warpage measurements.  
In the digital projection Moire option, the considered methodology for the presented analysis, the camera grating is imaged directly by software. The acquired image is numerically demodulated in order to make the sample profile reconstruction. The transmission of the square wave gratings through the optical system can be approximated as a cosine function of the profile number. Thus, the intensity pattern projected onto the object surface through the projection lens (\(I_{p}\)) is described as:  
\[I_{p}(x,y)=I_{s}R(x,y)A\big{[}1+cos\big{(}2\pi n_{p}(x,y)\big{)}\big{]}, \tag{1}\]  
where \(I_{s}\) is the intensity of the light source, \(R(x,y)\) is the reflectance, and \(A\) and \(n_{p}\) are, respectively, amplitude and profile number of the projection grating. Meanwhile, the virtual intensity pattern on the sample (\(I_{v}\)), coming from viewing system, could be described by:  
\[I_{v}(x,y)=A[1+cos(2\pi n_{v}(x,y))], \tag{2}\]  
where \(n_{v}\) is the profile number of the viewing grating line that virtually illuminates the point (\(x,y\)). This described overlap generates an interference pattern, which can be expressed [24]:  
\[I_{k}(x,y)=A(x,y)+B(x,y)\cdot cos\bigg{(}\frac{2\pi d(z+\delta_{k})}{p(h+z+ \delta_{k})}\bigg{)}\qquad\delta_{k}=\frac{k-1}{m}\frac{ph}{d}, \tag{3}\]  
in which \(I_{k}\) is the recorded fringe interference pattern for the \(k\)-th shift (\(k=1,2,\dots,m\)), \(d\) is the distance between the light source and the observer, \(h\) is the distance from the grating to the light source and observer, \(z\) is the out-of-plane deformation of measured surface, \(p\) is the pitch of the Moire grating, and \(A(x,y)\) and \(B(x,y)\) are, respectively, the background and local contrast. In order to find out \(z\), deleting experimental noise, measurements are performed with different \(m\) grating pitches, as explicated in Equation (3) by terms \(\delta_{k}\). Considering a Gaussian least squares approach, it is possible to solve iteratively the m equations (one for each considered phase-shift), non-linear in the coordinate \(z\). The described approach is practically implemented by the used equipment considering a \"virtual\" grating, according to Reference [24].