# Measurement of Wafer Focus by Grating Shearing Interferometry
## 3 Experiment  
An experiment is proceeded to verify the feasibility and calculate the measurement precision of this technology, which is conducted by the measurement system shown in Figure 6.  
When analyzed by Equation (6), the defocus of wafer can be calculated as (light way \(\theta\) is 30\({}^{\circ}\)):  
\[\mathrm{h}=\frac{df\delta}{2\sqrt{3}\pi L_{c}} \tag{7}\]  
The measuring uncertainty can be calculated by:  
\[\mathrm{u}_{h}=\sqrt{\left(\frac{\partial h}{\partial d}u\right)^{2}+\left( \frac{\partial h}{\partial\delta}u_{\delta}\right)^{2}+\left(\frac{\partial h }{\partial L_{c}}u_{Lc}\right)^{2}+\left(\frac{\partial h}{\partial\tilde{f} }u_{f}\right)^{2}} \tag{8}\]  
In this equation, the measuring uncertainty is impacted by the phase extraction precision and the parameters of the measuring system (focal length of lens \(f\), distance between the grating and CCD sensor \(L_{c}\), and the period of grating _d_). The measurement parameter and uncertainty factor in this system are analyzed in Table 1.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Uncertainty Factor** \\ \hline _d_: 16.67 μm period of grating measured by microscope U\({}_{d}\) as 0.1 μm & 0.06 \\ _\(\delta\)_: 1.1 rad (1 μm defocus) phase extracted by wavelet transform \(U_{\delta}\) as 0.02 rad & 0.92 \\ _\(L_{c}\)_: 50 mm distance is measured by rangefinder \(U_{Lc}\) as 0.01 mm & 0.02 \\ _f_: 30 mm focal length is calibrated \(U_{\mathrm{f}}\) as 0.1 mm & 0.03 \\ Complex measuring uncertainty U\({}_{h}\) is calculated as 19.7 nm & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Measurement uncertainty in the system.  
Figure 6: Measurement system for wafer focusing technique (WFT).  
The main measuring uncertainty is impacted by phase extraction precision. Instead of the wafer, a reflex lens is applied and controlled by an electronic driver and an interferometer is also applied for reference. As the Reflex lenses are driven, they form a 1000-nm defocus to focus position; the interferometry pattern detected by CCD is shown in Figure 7.  
To indicate the precision of this technique, the reflex lens, instead of the wafer, is driven by a piezo driver 1 \(\upmu\)m from the focus to defocus position detected by the interferometer. A method based on the spatial domain phase shift technique is applied for the phase parsing of the shearing interferometry fringe.  
The phase distribution is calculated by the light intensity of a nearby pixel (\(n\) = 5 in Figure 8):  
\[\delta(x,y)=\arctan(\frac{2(I_{1}-I_{3})}{2I_{2}-I_{0}-I_{4}}\sqrt{1-\left[ \frac{I_{0}-I_{4}}{2(I_{1}-I_{3})}\right]^{2}}) \tag{9}\]  
We chose ten optional pixels to calculate the phase difference in the interferometry fringe between the focus and defocus positions of the wafer.  
Tens of measurement results are calculated by a single shot of CCD; this method ensures measurement precision and efficiency. The measurement result is shown in Table 2.  
\begin{table}
\begin{tabular}{c} Shearing: 988.2 995.3 1013.5 996.3 1003.5 985.4 991.4 986.2 1002.8 1009.7 \\ Interferometer: detected as 1000 nm \\ \hline Statistics: PV 28.1 nm, RMS: 9.8 nm \\ \end{tabular}
\end{table}
Table 2: Measurement result contrasted with interferometer.  
Figure 8: Phase parsing by spatial domain phase shift technique.  
Figure 7: Distribution of interferometry pattern detected by CCD sensor: (**a**) wafer at focus position; (**b**) wafer at defocus position.  
After analyzing and contrasting these measurement errors using the interferometer, the final measurement precision precedes 30 nm.