# Measurement of Wafer Focus by Grating Shearing Interferometry
## 2 Methodology  
The measurement system for the WFT introduced in this paper is simply composed of a laser, beam expander and collimator, 4f optical lenses, a diffractive grating and CCD (charge coupled device), as shown in Figure 1.  
The plane wavefront formed by the beam expander and collimator is transformed by lenses and sheared by the diffractive grating. The diffractive wavefront of the grating and the coordinates of the measuring system are illustrated in Figure 2.  
When the wafer under investigation is positioned at the co-focal plane lenses, the distribution of diffractive light intensity in the CCD sensor can be calculated as:  
\[E_{0}\Big{(}x_{g},y_{g}\Big{)}=E_{0}c_{0}+E_{0}c_{1}\exp\Big{(}j2\pi\frac{x_{g }}{d}\Big{)}+E_{0}c_{1}\exp\Big{(}-j2\pi\frac{x_{g}}{d}\Big{)} \tag{1}\]  
In this equation, \(d\) is the period of the Ronchi grating, \(E_{0}\) is the plural amplitude of the plane wavefront and \(c_{0}/c_{1}\) is the diffractive factor of the grating. The distribution of plural amplitude, imaged by the CCD sensor, can be calculated as:  
Figure 1: Illustration of focus measurement system (CCD: charge coupled device). Figure 2: Illustration of grating diffractive wavefront. \[ \begin{split}& E_{0}(x_{c},y_{c})=\frac{\exp(ikL_{c})}{iL_{c}}\exp \left[\frac{i\hbar\left(x_{c}^{2}+y_{c}^{2}\right)}{2L_{c}}\right]\iint\limits_ {\sum\limits_{i}E\left(x_{g},y_{g}\right)}\exp\left[-ik\left(\frac{x_{g}x_{c}+ y_{g}y_{c}}{L_{c}}\right)\right]dx_{g}dy_{g}\\ &\approx E_{0}c_{0}cir\left(\frac{\sqrt{x_{g}^{2}+y_{g}^{2}}}{r} +E_{1}c_{1}cir\left(\frac{\sqrt{\left(x_{g}-iL_{c}/d\right)^{2}+y_{g}^{2}}}{r} \right)\exp\left(2\pi i\frac{x_{g}-iL_{c}/d}{d}\right)\\ &+E_{1}c_{1}cir\left(\frac{\sqrt{\left(x_{g}+iL_{c}/d\right)^{2} +y_{g}^{2}}}{r}\right)\exp\left(-2\pi i\frac{x_{g}+iL_{c}/d}{d}\right)\end{split}  \tag{2}\]  
In Equation (2), \(r\) is the semi-diameter of diaphragm and \(L_{c}\) is the distance between the grating and the CCD sensor when the wafer is placed at a defocus position in relation to the projection lens, as shown in Figure 3.  
Based on the triangle principle, the defocus of wafer can be calculated in measuring system coordinates:  
\[\left\{ \begin{array}{l}s_{1}=2h\sin\theta\\ s_{2}=2h\cos\theta\end{array} \right. \tag{3}\]  
In this equation, \(s_{1}\) and \(s_{2}\) are axis defocus and vertical axis defocus, respectively. Defocus \(s_{1}\) transforms the plane wavefront into a spherical wavefront in the 4f optical system and defocus \(s_{2}\) engenders a phase shift in the measurement system, as shown in Figure 4.  
To simplify the analysis, the spherical wavefront is approximated as plane wavefront (\(s_{1}<<f\)). We calculated the intensity of the shearing interference area between the 0-order and \(\pm 1\)-order wavefronts of the grating diffractive:  
\[ \begin{split} E_{\text{s}}\left(x_{g},y_{g}\right)=E_{0}c_{0}\exp \left(i2\pi\frac{s_{2}}{f}\right)+E_{1}c_{1}\exp\left[i2\pi\frac{\left(x_{g}+ d\frac{s_{2}}{f}\right)}{d}\right]\\ +E_{1}c_{1}\exp\left[-i2\pi\frac{\left(x_{g}-d\frac{s_{2}}{f} \right)}{d}\right]\end{split}  \tag{4}\]  
Figure 3: Illustration of defocus measurement system. The distribution of plural amplitude, imaged by the CCD sensor, can be calculated as:  
\[ \begin{array}{l}E_{\rm s}(x_{c},y_{c})=E_{0}c_{0}c\dot{c}\left[\frac{\sqrt{\left( x_{c}+\omega_{c}/f\right)^{2}+y_{c}^{2}}}{r}\right]+\\ E_{0}c_{1}c\dot{c}\left[\frac{\sqrt{\left(x_{c}-\omega_{c}/f\right)+\omega_{c}/ f\right)^{2}+y_{c}^{2}}}{r}\right]\exp\left[2\pi\frac{\left(x_{c}-\omega_{c}/f +d\frac{2}{f}L_{c}\right)}{d}\right]\\ +E_{0}c_{1}c\dot{c}\dot{c}\left[\frac{\sqrt{\left(x_{c}+\omega_{c}/f\right)^{2} +y_{c}^{2}}}{r}\right]\exp\left[-i2\pi\frac{\left(x_{c}+\omega_{c}/f+d\frac{2} {f}L_{c}\right)}{d}\right]\end{array}