# Novel Generalized Three-Step Phase-Shifting Interferometry with a Slight-Tilt Reference
## 2 Basic Principle
\begin{array}{l}I_{3}=A_{o}^{2}+A_{r}^{2}\\ \qquad+A_{o}A_{r}\exp(-i\varphi_{o})\exp(i\varphi_{r})\exp(i\Delta\varphi_{r1 })\exp(i\Delta\varphi_{r2})\\ \qquad\qquad+A_{o}A_{r}\exp(i\varphi_{o})\exp(-i\varphi_{r})\exp(-i\Delta \varphi_{r1})\end{array} \, \tag{16}\]  
in order to obtain phase shifts \(\Delta\varphi_{r1}\) and \(\Delta\varphi_{r2}\), Fourier transform (FT) operations on Equations (14)-(16) are needed, and they are  
\[F_{1}(u,v)=F_{A}(u,v)+A_{r}^{2}\delta(u,v)+A_{r}\Big{[}F_{O}\Big{(}u+u_{p},v +v_{p}\Big{)}+F_{O}\Big{(}-u+u_{p},-v+v_{p}\Big{)}\Big{]}, \tag{17}\]  
\[ \begin{array}{l}F_{2}(u,v)=F_{A}(u,v)+A_{r}^{2}\delta(u,v)+A_{r}F_{O}\Big{(} u+u_{p},v+v_{p}\Big{)}\exp(i\Delta\varphi_{r1})\\ \qquad\qquad+A_{r}F_{O}^{*}\Big{(}-u+u_{p},-v+v_{p}\Big{)}\exp(-i\Delta\varphi _{r1})\end{array} \, \tag{18}\]  
\[ \begin{array}{l}F_{3}(u,v)=F_{A}(u,v)+A_{r}^{2}\delta(u,v)\\ \qquad\qquad+A_{r}F_{O}\Big{(}u+u_{p},v+v_{p}\Big{)}\exp(i\Delta\varphi_{r1}) \exp(i\Delta\varphi_{r2})\\ \qquad\qquad+A_{r}F_{O}^{*}\Big{(}-u+u_{p},-v+v_{p}\Big{)}\exp(-i\Delta\varphi _{r1})\exp(-i\Delta\varphi_{r2})\end{array} \, \tag{19}\]  
where symbol \"*\" presents the conjugate operation. \(F_{1}(u,\,v)\), \(F_{2}(u,\,v)\) and \(F_{3}(u,\,v)\) are the Fourier transforms of \(I_{1}\), \(I_{2}\) and \(I_{3}\), respectively. The first term \(F_{A}(u,\,v)\) in the right side of Equations (17)-(19) is the spectrum of the object intensity, distributing in the central part of the spectrum plane. Because the reference intensities \(I_{r}\) is a constant, the second term of the three equations give a \(\delta\)-function, showing a bright point at the origin (0, 0) on the spectrum plane. The last two terms in the right side of Equations (17)-(19) are the shifted spectra of the object wave and its conjugate, where \(F_{O}(u+u_{p},\,v+v_{p})\) is the Fourier transform of \(A_{o}\textit{exp}(-i\varphi_{o})\textit{exp}(i\varphi_{r})\) and \(F_{O}\ast(-u+up,-v+vp)\) is the Fourier transform of \(A_{o}\textit{exp}(i\varphi_{o})\textit{exp}(-i\varphi_{r})\). They are confined in two relatively small areas, but the centers are shifted to points \((-u_{p},\,-v_{p})\) and \((u_{p},\,v_{p})\) on the spectrum plane due to the tilt reference employed. In some cases of complicated object wave, points \((-u_{p},\,-v_{p})\) and \((u_{p},\,v_{p})\) may be difficult to locate as they are hidden within the spectra of \(F_{O}(u+u_{p},\,v+v_{p})\) or \(F_{O}\ast(-u+u_{p},\,-v+v_{p})\) due to the slight-tilt reference angle (fraction of a degree). However, we can perform interference of two plane waves as the spectrum of such interference contains \(\delta\)-functions centered at (0,0), (\(-u_{p}\), \(-v_{p}\)), and (\(u_{p}\), \(v_{p}\)), which can be found easily. After (\(-u_{p}\), \(-v_{p}\)), and (\(u_{p}\), \(v_{p}\)) are determined, the spectrum at these points can be used to extract unknown phase shifts \(\Delta\varphi_{r1}\) and \(\Delta\varphi_{r2}\), to be explained below. Indeed, if we isolate the spectra at points (\(-u_{p}\), \(-v_{p}\)) and (\(u_{p}\), \(v_{p}\)) in the third term of Equations (17) and (18), we have \(A_{r}F_{O}\big{(}u+u_{p},v+v_{p}\big{)}\) and \(A_{r}F_{O}\big{(}u+u_{p},v+v_{p}\big{)}\exp(i\Delta\varphi_{r1})\), respectively \(\Delta\varphi_{r1}\) can be extracted by subtracting the argument of the above two terms and by evaluating at \(u=-u_{p}\) and \(v=-v_{p}\) to give:  
\[\Delta\varphi_{r1}=\arg[A_{r}F_{O}(0,0)\exp(i\Delta\varphi_{r1})]-\arg[A_{r}F _{O}(0,0)]. \tag{20}\]  
Alternatively, we can also use their conjugate terms, i.e., \(A_{r}F_{O}^{*}\big{(}-u+u_{p},-v+v_{p}\big{)}\) and \(A_{r}F_{O}^{*}\big{(}-u+u_{p},-v+v_{p}\big{)}\exp(-i\Delta\varphi_{r1})\). However, in this case, we need to evaluate the terms at \(u=u_{p}\) and \(v=v_{p}\) to give \(\Delta\varphi_{r1}=\arg[A_{r}F_{O}^{*}(0,0)]-\arg[A_{r}F_{O}^{*}(0,0)\exp(-i \Delta\varphi_{r1})]\). Similarly, we extract \(\Delta\varphi_{r2}\) from the third terms of Equations (18) and (19).