# Metrology for Measuring Bumps in a Protection Layer Based on Phase Shifting Fringe Projection
## 5 Experimental Results and Discussion
### Height Measurement of Bumps in a Polymer
#### 5.2.3. Bump Height Correction for Polymer Layers  
Figure 13a,b show modified versions of Figure 5a,b, respectively, with an additional oxide layer between the PI and Si substrate based on the reflectance spectrum presented in Section 5.2.1. When the fringe projection method was used to detect the bottom of PI, the transmitted and reflected light passing through the PI (green line assumed) and oxide (red line assumed) and captured by the camera both created lateral fringe shifts that caused perceived height recessions in the calculation. Equations (13)-(16) were modified to Equations (17)-(20) to eliminate the influence of the PI and oxide on the bump height measurement. The fringe refraction inside the PI and oxide layer is based on [PERSON]'s law:  
\[\sin\theta_{i}=n_{PL}\sin\theta^{\prime}_{i}=n_{ox}\sin\theta^{\prime\prime}_{ i}\ \ ;\ \ \ n_{ox}\sin\theta^{\prime\prime}_{c}=n_{PL}\sin\theta^{\prime}_{c}\ =\sin\theta_{c} \tag{17}\]  
where \(n_{PL}\) and \(n_{ox}\) are the refraction index properties of PI and oxide, respectively, and the refracted angles of projection and collection in the PI and oxide layers are \(\theta^{\prime}_{i}\), \(\theta^{\prime}_{c}\) and \(\theta^{\prime\prime}_{i}\), \(\theta^{\prime\prime}_{c}\), respectively. Because we used the specular setup, where the incident projection angles \(\theta^{\prime\prime}_{i}\) and \(\theta^{\prime\prime}_{i}\) are equal to the collection angles \(\theta^{\prime}_{c}\) and, \(\theta^{\prime\prime}_{c}\), the fringe displacements \(u_{PL}\) and \(u_{ox}\) in the lateral direction due to traveling around the PI and oxide layers are given by  
\[u_{PL}(x)=t_{PL}(x)(\tan\theta^{\prime}_{i}+\tan\theta^{\prime}_{c})=2t_{PL}(x )\frac{\sin\theta_{i}}{\sqrt{{n_{PL}}^{2}-\sin^{2}\theta_{i}}} \tag{18}\]  
\[u_{ox}(x)=t_{ox}(x)(\tan\theta^{\prime\prime}_{i}+\tan\theta^{\prime\prime}_{ c})=2t_{ox}(x)\frac{\sin\theta_{i}}{\sqrt{{n_{ox}}^{2}-\sin^{2}\theta_{i}}} \tag{19}\]  
where \(t_{PL}(x)\) and \(t_{ox}(x)\) are the thicknesses of PI and oxide, respectively, at the representative location x across the wafer. The offsets of the phase value \(\Delta\phi_{PL}(x)\) and \(\Delta\phi_{ox}(x)\) due to PI and oxide refraction were acquired in the image plane:  
\[\Delta\phi_{PL}(x)=\frac{2\pi}{d_{i}}\times 2t_{PL}(x)\times(\tan\theta_{i}- \tan\theta^{\prime}_{i})=\frac{2\pi}{d_{i}}\times 2t_{PL}(x)\times(\tan \theta_{i}-\frac{\sin\theta_{i}}{\sqrt{{n_{PL}}^{2}-\sin^{2}\theta_{i}}}) \tag{20}\]  
\[\Delta\phi_{ox}(x)=\frac{2\pi}{d_{i}}\times 2t_{ox}(x)\times(\tan\theta_{i}- \tan\theta^{\prime\prime}_{i})=\frac{2\pi}{d_{i}}\times 2t_{ox}(x)\times(\tan \theta_{i}-\frac{\sin\theta_{i}}{\sqrt{{n_{ox}}^{2}-\sin^{2}\theta_{i}}}) \tag{21}\]  
When combined with Equation (16), the corrected bump height to phase conversion is given by  
\[h(x)=\frac{d}{2\pi m}\frac{\cos\theta_{c}}{\sin(\theta_{i}+\theta_{c})}[\phi( x)+\Delta\phi_{PL}(x)+\Delta\phi_{ox}(x)]-t_{ox}(x) \tag{22}\]  
Figure 12: (**a**) Scanning white light interferometer model for measuring the PI thickness. (**b**) Experimental results showing the formation of interference fringes at the air–PI and oxide–Si substrate interfaces. The intensity corresponding to the PI–oxide interface was too weak to be sensed. The BGA bump array in the PI sample as shown in Figure 14a was analyzed in detail here. According to Equation (22), \(\theta_{i}=\theta_{c}\ \theta_{c}=19.6^{\circ}\), \(d=250\ \upmu\)m, m = 1.048\(\times\), and the conversion factor \(\frac{d}{2\pi m}\frac{\cos\theta_{c}}{\sin(\theta_{i}+\theta_{c})}\) has a constant value of 56.6.