# Digital Shearing Speckle Pattern Interferometry Based on Rochon Prism and Its Application
## 3 Comparing RP with WP  
The RP and WP schemes are shown in Figure 1. The difference between RP and WP is that their optical axes of the crystal in the first prism (P1) are mutually perpendicular. In Figure 1a, the unpolarized incident beam is divided into o-light and e-light, where the o-light travels along the original optical axis of the measurement system (x-axis) without deviation and the e-light deviates from the optical axis with an angle of \(\alpha\). However, in Figure 1b, the incident beam is divided into o-e light and e-o light, neither of which remain along the x-axis. This difference produces RP's two advantages. First, maintaining the o light along the optical axis improves the SNR. As shown in Figure 2b, the sheared images generated by WP produce three overlapping images: the e-o light, the o-e light, and the transmission light, which is along the x-axis in Figure 1b. Due to unexpected transmission light, there are three overlapping fringe patterns rather than one when the WP is applied in the shearography system. Contrary to WP, RP solves this problem automatically because of its optical path structure shown in Figure 1a. Figure 2a shows that there are only two sheared images (the o-light and the e-light) and no transmission light, which protects the fringe pattern from background noise. Second, RP provides a more suitable angle of divergence. As shown in Figure 1a, the optical path difference (OPD) is composed of the following two parts: the OPD in P2 where o-light and e-light are separated, and the optical path difference in the air caused by the angle of emergence between the o-light and e-light. Next, we deduced the OPD in an RP. In the P1 of an RP, both the beams have the same refractive index and are not affected by the birefringence effect, which is different from the WP. When the beam enters the second prism (P2), the o-light has the same refractive index and is undeviated. The e-light has a different refractive index and bends away from the x-axis in Figure 1a, causing the o-light and e-light to separate after their emergence. If the angle between the o-light and e-light is \(\theta^{\prime}\), the OPD in P2 is:  
\[\Delta[l]_{1}=y^{\prime}\left(\frac{n_{e}}{\cos\theta^{\prime}}-n_{o}\right), \tag{6}\]  
where \(y^{\prime}\) is the distance that the o-light travels in P2, which can be described as:  
\[ \begin{array}{ll}y^{\prime}&=L-\left(\frac{L}{2\tan\theta}-x_{1}\right)\tan \theta\\ &=\frac{L}{2}+x_{1}\tan\theta\end{array}  \tag{7}\]  
Figure 1: Schemes of (**a**) a Rochon Prism and (**b**) a Wollaston Prism, where the line of dashes is the optical axis of the system. Figure 2: Images after shearing devices under 150-mW power laser illumination: (**a**) sheared by Rochon Prism (RP) and (**b**) sheared by Wollaston Prism (WP). where \(x_{i}\) is the distance to the geometric axis of the prism (positive above the central axis and negative below) and \(\theta\) is the prisms' wedge angle. The relationship between \(\theta^{\prime}\) and \(\theta\) is:  
\[\frac{\cos\theta}{\sin(\theta-\theta^{\prime})}=\frac{n_{o}}{n_{e}}, \tag{8}\]  
After emergence, the angle of divergence between o- and e-light is \(\alpha\). As o- and e-light have the same refractive index, the OPD in the air can be described as:  
\[\Delta[l]_{2}=y\bigg{(}\frac{1}{\cos\alpha}-1\bigg{)}, \tag{9}\]  
where \(y\) is the distance between the prism and the CCD. As most of the parameters are determined in a specific system, the general form of the phase difference can be obtained after combining the OPD of two parts, that is:  
\[\varphi_{\Delta}=\frac{\Delta[l]_{1}+\Delta[l]_{2}}{\lambda}=c+\frac{\tan \theta}{\lambda}\cdot x_{1}, \tag{10}\]  
where \(c\) is constant and \(\lambda\) is the wavelength. In a particular measuring system, the phase difference at a certain point only depends on the distance from the axis of the prism.