# Metrology for Measuring Bumps in a Protection Layer Based on Phase Shifting Fringe Projection
## 3 Theoretical Model of Fringe Projection Profilometry
### Bump Height Correction in Polymer Layers  
A PL is used in an UFI fan-in WLCSP to help secure the solder pumps in position, as shown in Figure 5a. The solder balls are directly mounted on the Cu redistribution layer (RDL), which routes electrical signals from the original pads of the chip to the desired area array pads. When the triangulation method attempts to detect the bottom of the PL (blue line assumed), which has a semitransparent medium and larger index of refraction, the transmitted and reflected light from the bottom of the PL (green line) captured by the camera creates a lateral fringe shift that causes a perceived height recession  
Figure 4: Schematic of the phase-height conversion based on the fringe projection method.  
in the calculation, as shown in Figure 5b. The PL thickness varies across the wafer, which introduces varying errors in the measured bump height from site to site. Thus, the in situ thickness of the PL layer needs to be considered to provide an accurate bump height correction for precise 3D measurement.  
In order to quantitatively analyze the influence of the PL optical properties on the measurement accuracy, we considered a new approach where a reflectometric sensor is used to accurately measure the PL thickness at representative locations across the wafer, and the bump height offset to apply to fringe projection measurements is simultaneously calculated. The reflectometric sensor is capable of nanometer-scale accuracy and repeatability; in particular, its sensing time within 10 ms at each sampling site is fast enough to provide the correct offset as a function of the PL thickness.  
The fringe refraction inside the PL layer is based on [PERSON]'s law:  
\[\sin\theta_{i}=n_{PL}\sin\theta_{i}^{\prime}\ \ ;\ \ \ n_{PL}\sin\theta_{c}^{ \prime}\ \ =\sin\theta_{c} \tag{13}\]  
where \(n_{PL}\) is the refraction index property of the PL and \(\theta_{i}^{\prime}\) and \(\theta_{c}^{\prime}\) are the refracted projection and collection angles, respectively. Because we used a specular setup where the incident projection angle \(\theta_{i}^{\prime}\) is equal to the collecting angle \(\theta_{c}^{\prime}\), the fringe displacement \(u_{PL}\) in the lateral direction due to traveling around the PL layer is given by  
\[u_{PL}(x)=t_{PL}(x)(\tan\theta_{i}^{\prime}+\tan\theta_{i}^{\prime})=2t_{PL}(x )\frac{\sin\theta_{i}}{\sqrt{n_{PL}{}^{2}-\sin^{2}\theta_{i}}} \tag{14}\]  
where \(t_{PL}(x,y)\) is the thickness of the PL at the representative location \((x,y)\) across the wafer. The offset of the phase value \(\Delta\varphi(x,y)\) due to PL refraction acquired in the image plane is given by  
\[\Delta\phi(x)=\frac{2\pi}{d_{i}}\times 2t_{PL}(x)\times(\tan\theta_{i}-\tan \theta_{i}^{\prime})=\frac{2\pi}{d_{i}}\times 2t_{PL}(x)\times(\tan\theta_{i}- \frac{\sin\theta_{i}}{\sqrt{n_{PL}{}^{2}-\sin^{2}\theta_{i}}}) \tag{15}\]  
When combined with Equation (12), the corrected bump height to phase conversion equation is given by  
\[h(x)=\frac{d}{2\pi m}\frac{\cos\theta_{c}}{\sin(\theta_{i}+\theta_{c})}[\phi( x)+\Delta\phi(x)] \tag{16}\]  
The real bump height is defined as from the bottom of the PL to the top of the bump instead of only the bump height above the PL. Because of the thickness variation of the PL  
Figure 5: (**a**) UFI process uses a protective layer that is semitransparent and varies in thickness. This introduces a significant error in the bump height/coplanarity measurement. (**b**) Spatial shift of fringes due to traveling around the PL layer.  
across the wafer, the PL thickness should be considered in the bump height measurement to improve the accuracy of the coplanarity measurement.