# Novel Generalized Three-Step Phase-Shifting Interferometry with a Slight-Tilt Reference
## 2 Basic Principle  
In the TGPSI method, the complex amplitudes of object wave \(\mathbf{O}(x,y)\) and reference wave \(\mathbf{R}(x,y)\) on the recording plane \(\mathrm{P}_{\mathrm{H}}\) can be written as  
\[\mathbf{O}(x,y)=A_{o}(x,y)\exp[-i\varphi_{o}(x,y)], \tag{1}\]  
\[\mathbf{R}(x,y)=A_{r}\exp[-i\varphi_{r}(x,y)], \tag{2}\]  
where for brevity, we have assumed \(A_{r}\) to be constant. Obviously, in Equation (2) the phase of the tilt reference plane wave \(\varphi_{r}(x,y)\) is not a constant but a linear distribution on recording plane \(\mathrm{P}_{\mathrm{H}}\) given by  
\[\varphi_{r}(x,y)=\frac{2\pi}{\lambda}(x\cos\theta_{x}+y\cos\theta_{y}), \tag{3}\]  
where \(\lambda\) is the wavelength of the laser and \(\theta_{x}\) and \(\theta_{y}\) are the off-axis angles along the \(x\) and \(y\) directions. From Equations (1) and (2), the intensities for the three interferograms are described by  
\[I_{1}=A_{o}^{2}+A_{r}^{2}+2A_{o}A_{r}\cos(\varphi_{o}-\varphi_{r}), \tag{4}\]  
\[I_{2}=A_{o}^{2}+A_{r}^{2}+2A_{o}A_{r}\cos(\varphi_{o}-\varphi_{r}-\Delta\varphi _{r1}), \tag{5}\]  
\[I_{3}=A_{o}^{2}+A_{r}^{2}+2A_{o}A_{r}\cos(\varphi_{o}-\varphi_{r}-\Delta \varphi_{r1}-\Delta\varphi_{r2}), \tag{6}\]  
where \(\Delta\varphi_{r1}\) and \(\Delta\varphi_{r2}\) are unknown phase shifts that need to be determined. Subtracting Equations (5) and (6) from Equation (4) respectively, we have  
\[I_{2}-I_{1}=4A_{o}A_{r}[\sin(\varphi_{o}-\varphi_{r}-\Delta\varphi_{r1}/2)\sin (\Delta\varphi_{r1}/2)], \tag{7}\]  
\[I_{3}-I_{1}=4A_{o}A_{r}[\sin(\varphi_{o}-\varphi_{r}-\Delta\varphi_{r1}/2- \Delta\varphi_{r2}/2)\sin(\Delta\varphi_{r1}/2+\Delta\varphi_{r2}/2)]. \tag{8}\]  
From Equations (7) and (8), we derive the following expressions:  
\[A_{o}\sin(\varphi_{o}-\varphi_{r}) =\frac{1}{4A_{r}\sin(\Delta\varphi_{r2}/2)}\times \tag{9}\] \[A_{o}\cos(\varphi_{o}-\varphi_{r}) =\frac{1}{4A_{r}\sin(\Delta\varphi_{r2}/2)}\times\] (10) \[\mathbf{R}(x,y)=A_{r}\exp[-i\varphi_{r}(x,y)], \tag{2}\]0  
which are the imaginary and the real parts of the following complex amplitude:  
\[\mathbf{R}(x,y)=A_{r}\exp[-i\varphi_{r}(x,y)], \tag{2}\]1 \[\mathbf{R}(x,y)=A_{r}\exp[-i\varphi_{r}(x,y)], \tag{2}\]2Substituting Equations (9) and (10) into (11), we have  
\[ \begin{array}{l}\mathbf{O}_{1}=\frac{1}{4A_{r}\sin(\Delta\varphi_{r2}/2)}\times\\ \left\{\frac{\exp(i\Delta\varphi_{r1}/2)}{\sin((\Delta\varphi_{r1}+\Delta\varphi _{r2})/2)}(I_{1}-I_{3})-\frac{\exp[i(\Delta\varphi_{r1}+\Delta\varphi_{r2})/2] }{\sin(\Delta\varphi_{r1}/2)}(I_{1}-I_{2})\right\}\.\end{array}  \tag{12}\]  
Note that from Equation (11), we can see that this complex field is related to the original object wave \(\mathbf{O}\) (see Equation (1)) as follows:  
\[\mathbf{O}=\mathbf{O}_{1}\exp(-i\varphi_{r}). \tag{13}\]  
In order to extract \(\mathbf{O}\), we, therefore, need to know \(\mathbf{O}_{1}\) and \(\varphi_{r}\). Let us first concentrate on finding \(\mathbf{O}_{1}\). According to Equation (12), \(\mathbf{O}_{1}\) can be found with known intensities \(I_{1}\), \(I_{2}\), \(I_{3}\) along with unknown, to be determined, phase shifts \(\Delta\varphi_{r1}\) and \(\Delta\varphi_{r2}\). It is known that the precise extraction of phase shifts \(\Delta\varphi_{r1}\) and \(\Delta\varphi_{r2}\) is critical to the reconstruction of the object wave. In TGPSI, many time-consuming iterative algorithms are used to search for the actual values of the phase shift. We propose a powerful and reliable method to extract phase-shift values based on the theory developed so far. To this end, we first analyze the spectrum of the interferograms. Equations (4)-(6) can be rewritten as  
\[I_{1}=A_{o}^{2}+A_{r}^{2}+A_{o}A_{r}\exp(-i\varphi_{o})\exp(i\varphi_{r})+A_{ o}A_{r}\exp(i\varphi_{o})\exp(-i\varphi_{r}), \tag{14}\]  
\[ \begin{array}{l}I_{2}=A_{o}^{2}+A_{r}^{2}+A_{o}A_{r}\exp(-i\varphi_{o})\exp( i\varphi_{r})\exp(i\Delta\varphi_{r_{1}})\\ \qquad\qquad+A_{o}A_{r}\exp(i\varphi_{o})\exp(-i\varphi_{r})\exp(-i\Delta \varphi_{r1})\end{array} \, \tag{15}\]  
\[