# Metrology for Measuring Bumps in a Protection Layer Based on Phase Shifting Fringe Projection
## 5 Experimental Results and Discussion  
### Measurement of Double Peripheral Solder Bumps  
A BGA sample of 6 mm \(\times\) 6 mm size containing double peripheral solder bumps was measured. With samples of the size of 6 mm \(\times\) 6 mm, more than two sets of matrix arrays can be viewed along the scanning direction. The measurement process can be adjusted and performed fully automatically. The single exposure time to acquire the image data is 70 \(\upmu\)s. By coping with the selected ROI sensor area of 4096(H) \(\times\) 50(V), sufficient information of three fringe images with 120\({}^{\circ}\) phase shifts are retained to reconstruct a 3D model in a continuous scan mode. Figure 7 shows the result for the BGA sample measurements. Figure 7a shows a photo of the measurement area. Figure 7b-d shows three acquired  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Site**} & **Nominal** & **Calibrated(\(\upmu\)m)** & **Measured(\(\upmu\)m)** & **Discrepancy(\(\upmu\)m)** \\  & **(\(\upmu\)m)** & **Stylus** & **Fringe Projection** & **\%** \\ \hline
1 & 5 & 6.76 & 6.94 & \(-\)0.19 (2.75) \\
2 & 15 & 16.92 & 16.36 & 0.57 (3.36) \\
3 & 25 & 27.16 & 26.70 & 0.46 (1.69) \\
4 & 35 & 37.01 & 36.92 & 0.09 (0.24) \\
5 & 45 & 46.20 & 46.86 & \(-\)0.66 (1.42) \\
6 & 55 & 55.69 & 55.95 & \(-\)0.26 (0.47) \\
7 & 65 & 64.98 & 65.97 & \(-\)0.99 (1.52) \\
8 & 75 & 74.39 & 74.86 & \(-\)0.47 (0.63) \\
9 & 85 & 86.66 & 87.46 & \(-\)0.79 (0.92) \\ \hline \end{tabular}
\end{table}
Table 1: Calibrated step heights of the reference standard.  
Figure 6: (**a**) Measurement of reference step heights with the fringe projection system. (**b**) Comparison of fringe projection system measurement results with certified values (nominal step heights: 5, 15, 25, 35, 45, 55, 65, 75, and 85 \(\upmu\)m).  
images with a 120\({}^{\circ}\) phase shift. The algorithm with three-step phase-shifting algorithm was then applied to compute the phase map of the scan area, as indicated within the red frame area in Figure 8a. The wrapped phase map calculated by Equation (5) and the 3D reconstructed topography that was converted from the wrapped phase map are shown in Figure 8b,c respectively. It is possible to take profiles in any direction from the 3D measurement. For example, the red lines in Figure 8b denote the measurement data along the \(x\) and \(y\) axes used for the bump height calculation. A Gaussian filter with (5 \(\times\) 5) radius was used to smooth the random noise.  
Figure 9 shows the bump height calculation results along the red lines shown in the 2D phase map of Figure 8b. Figure 9a,b show the distribution of bump heights along the \(x\) and \(y\) axes, respectively. The bump heights were calculated by taking the difference between each bump's peak and valley. The average bump heights were 23.65 and 23.12 \(\upmu\)m along the \(x\) and \(y\) axes, direction respectively.  
The algorithm for determining the bump heights is similar to that for characterizing the step heights, which subtracts the average height of the bottom area from the height of the top of each bump. In a repeatability test, measurements taken 30 times showed a standard deviation of less than 0.40 \(\upmu\)m (1\(\sigma\)). Thus, the repeatability deviation was about 50 times less than the designed bump height.  
Figure 8: Reconstruction of the 3D BGA sample: (**a**) photograph of the scan area, (**b**) wrapped phase map, (**c**) 3D reconstructed topography.  
Figure 7: Example measurement of a BGA sample: (**a**) photograph of the BGA sample, (**b**) \(I_{\_}[0]\), (**c**) \(I[2\pi/3]\), and (**d**) \(I[4\pi/3]\).  
Figure 9: Bump height measurement results: distributions along the (**a**) \(x\) axis and (**b**) \(y\) axis.