# Novel Generalized Three-Step Phase-Shifting Interferometry with a Slight-Tilt Reference
## 2 Basic Principle
Again, we use \(u=-u_{p}\) and \(v=-v_{p}\), and we find  
\[\Delta\varphi_{r2}=\arg[A_{r}F_{O}(0,0)\exp(i\Delta\varphi_{r1})\exp(i\Delta \varphi_{r2})]-[\arg[A_{r}F_{O}(0,0)\exp(i\Delta\varphi_{r1})]]\quad, \tag{21}\]  
where \(\arg\) [. ] in the above equations denotes taking the argument of a complex value in the square bracket. Equations (20) and (21) use only a simple subtraction operation to extract unknown phase shifts \(\Delta\varphi_{r1}\) and \(\Delta\varphi_{r2}\) without any iteration. This is the first contribution of using the proposed small-tilt reference. Our next goal is to find object wave \(O\). Now with \(\Delta\varphi_{r1}\) and \(\Delta\varphi_{r2}\) already extracted, we can calculate \(O_{1}\) from Equation (12). \(O\) and \(O_{1}\) are related by Equation (13) through \(\varphi_{r}\) as follows:  
\[O=O_{1}\exp(i\varphi_{r})=O_{1}\exp\biggl{[}i\frac{2\pi}{\lambda}\Bigl{(}x\sin \theta_{x}+y\sin\theta_{y}\Bigr{)}\biggr{]}. \tag{22}\]  
We can calculate off-axis angles \(\theta_{x}\) and \(\theta_{y}\) by the following relationships [1]  
\[\sin\theta_{x}=\lambda\frac{u_{p}}{Md_{x}},\;\sin\theta_{y}=\lambda\frac{v_{p} }{Nd_{y}}, \tag{23}\]  
where \(M\) and \(N\) are the total pixel numbers along the horizontal and vertical directions of the hologram. \(d_{x}\) and \(d_{y}\) are the corresponding pixel size in the two directions. Note that we have been using normalized spatial frequencies, \(u\) and \(v\). Hence \(u_{p}\) and \(v_{p}\) are divided by the total length of the Charge-coupled Device (CCD) linear dimension, \(Md_{x}\) and \(Md_{y}\). Since \(u_{p}\) and \(v_{p}\) have already been located, we can determine the off-axis angles. Once the angles are found, object wave \(O\) is completely determined from Equation (22), and finally the complex amplitude of the original object can be calculated by performing backward Fresnel diffraction [1]. The whole process of the proposed new algorithm for finding the unknown phase shifts and object wave reconstruction can be summarized in the following steps:  
Step 1: Set and determine the reference tilt angle before the test object is put in the optical path. Experimentally, we fix the tilt angle when dozens of fringes appear on the CCD from the interference of two plane waves. We label \(I_{0}\) as the intensity pattern of the interference between the two plane waves. We then put in the object and record \(I_{1}\), \(I_{2}\) and \(I_{3}\). Step 2: Perform FT operations on the fringe pattern from the plane wave interference in Step 1 and search the brightest points on the spectral plane (not the point at the origin of the spectral plane) and locate the coordinates of the found points as (\(-u_{p}\), \(-v_{p}\)) or (\(u_{p}\),\(v_{p}\)). Perform FT operations on \(I_{1}\), \(I_{2}\) and \(I_{3}\) and store the complex values of these two points. In most practical situations, (\(-u_{p}\), \(-v_{p}\)) or (\(u_{p}\), \(v_{p}\)) can also be found from the spectrum of either one of \(I_{1}\), \(I_{2}\) and \(I_{3}\) without the need of two-plane wave interference. Step 3: Calculate phase shift \(\Delta\varphi_{r1}\) and \(\Delta\varphi_{r2}\) by Equations (20) and (21) using the values stored in step 2. Step 4: Reconstruct \(\mathbf{O}_{1}\) by Equation (12). Step 5: Find the tilt angle by Equation (23) from the located coordinates of \((-u_{p},-v_{p})\) or \((u_{p},v_{p})\). Step 6: Find \(\mathbf{O}\) by using Equation (22). Step 7: Recover the original object image by backward Fresnel diffraction of \(\mathbf{O}\).