# 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
#### 5.2.1 Reflectrometric Spectrum Model Fit  
Figure 10a shows a reflectrometric spectrum for the measured PL on the BGA sample. The illumination area on the sample surface was around 13 \(\upmu\)m, which was at nearly the center of the fringe projection position. Two distinct frequencies of oscillations were observed, which indicated more than one layer on top of the substrate. We used Fourier transform (FT) analysis to calculate the optical path difference (OPD) along the top surface to the bottom substrate, as shown in Figure 10b. The frequencies for the two-layer thickness were well separated from each other. OPD 1 and OPD 2 needed to be divided by the layer medium's refractive index to obtain the geometric thickness. Figure 11a shows the simulation model for this case, in which a silicon substrate had an insulating oxide layer with the thickness \(t_{ox}\) above and a PI layer with a thickness \(t_{PI}\) on the top. The three reflected light waves (A, B, and C) could interfere with each other. The reflecting media 1-4 were defined as air, PI, oxide, and silicon, respectively. The refractive indices were constants for air, PI, and oxide at 1, 1.653, 1.47, respectively, and varied with the wavelength for silicon: 6.706 at 375 nm and 3.696 at 780 nm. According to the Fresnel equation, light path A in air reflects off the top of the PI layer to shift 180\({}^{\circ}\), and its electrical field reflectance is multiplied by the reflecting factor \(r_{12}\)[21]. Light path B off the oxide surface is transmitted through the PI layer and undergoes the phase shift \(e^{\frac{-2 mnp_{PI}}{\lambda}}\); its electrical field reflectance is multiplied by the transmitting factor \(t_{12}\) (transmission from air into PI), reflecting factor \(r_{21}\) (reflection from oxide in the PI medium), and transmitting factor \(t_{21}\) (transmission from PI to air). Light path C off the silicon surface shifts 180\({}^{\circ}\) and transmits through the oxide film and PI layer to undergo the phase shift \(e^{\frac{-2 mnp_{PI}\sigma_{12}}{\lambda}}\times e^{\frac{-2 mnp_{PI}\sigma_{2 }}{\lambda}}\); its electrical field reflectance is multiplied by the transmitting factor \(t_{12}\) (transmission from air to PI), transmitting factor \(t_{23}\) (transmission from PI to oxide), reflecting factor \(r_{34}\) (reflection from silicon surface in the oxide medium), transmitting factor \(t_{32}\) (transmission from oxide to PI), and transmitting factor \(t_{21}\) (transmission from PI into air). The combined reflectance spectrum can be calculated by adding the interference spectrum of each pair of reflecting light paths A-C, as shown in Figure 11b. Because the oxide-silicon surface acts like a mirror, a stronger signal at the corresponding OPD 2 (A and C interference) can be acquired compared to the detection of the light reflected by the interface between the oxide film and PI layer (A and B interference). The amplitude of the interference A and B is too weak and has ignorable contribution to the total reflectance spectrum. A modulation signal at the corresponding OPD 1 comes from the light bouncing from the oxide film with the light reflected from the silicon substrate. Thus, the DFT results for OPD 1 and OPD 2 in Figure 10b correspond to \(t_{ox}\) and (\(t_{PI}+t_{ox}\)), respectively. The optical thickness of the oxide film is given by OPD 1 = \(n_{ox}(\lambda)\times t_{ox}\) which was 7.8 \(\upmu\)m (= 1.47 \(\times\) 5.31 \(\upmu\)m). The optical thickness of the additional PI and oxide film is given by OPD 2 = \(n_{PL}(\lambda)\times t_{PI}+n_{ox}(\lambda)\times t_{ox}\), which was 49.5 \(\upmu\)m (= 1.653 \(\times\) 25.34 + 1.47 \(\times\) 5.31 \(\upmu\)m). In addition to the film thickness and optical constants, other properties such as the roughness of the PI-oxide and oxide-silicon interface also influenced the amplitude of the reflectance, especially at shorter wavelengths. Thus, the reflectance spectrum in the range of 600-780 \(\upmu\)m was selected for the comparison [22].