# Novel Generalized Three-Step Phase-Shifting Interferometry with a Slight-Tilt Reference
## 3 Computer Simulation and Optical Experiments  
### Computer Simulation  
We have carried out a series of computer simulations to verify the effectiveness of the proposed method before any optical experiments. In these simulations, many elliptical surfaces with different parameters have been used as reflecting target objects. A plane wave of 532 nm illuminates these surfaces and is then reflected back onto the CCD in the recording setup illustrated in Figure 1.  
In the simulation setup in Figure 1[20], there were \(512\times 512\) pixels with \(15\ \upmu\times 15\ \upmu\)m size on the CCD and the recording distance was set as z = 216.5 mm (all the parameters satisfy the sampling theory to avoid aliasing [1]). With the paraxial approximation of an elliptical wave, we set the phase distribution on the original object plane Po, which is tangential to the surface at the vertex, as \(4\pi h(x,y)/\lambda\), with \(h\) the surface depth determined by radius of curvature \(R\). We used  
\[h(x,y)=\frac{8x^{2}}{25R}+\frac{y^{2}}{2R}. \tag{24}\]  
Accounting for the possible amplitude variation of an object wave, a Gaussian amplitude intensity distribution of the object wave in plane Po was introduced, decreasing gradually from the center of maximum 1 to the edge of minimum 0.7. Two-dimensional Fresnel diffraction makes the complex amplitude in plane Po give the object wave \(\mathbf{O}(x,y)\), in the recording plane \(\mathrm{P}_{\mathrm{H}}\). Three different interferograms by two reference phase shifts are computer-generated. In this simulation we set two phase-shift values of 1 and 0.6 rad respectively, and we give only a few results with \(R\) = 2000 mm as example. The reference wave is a plane wave with a slight tilt angle along both the \(x\)-axis and the \(y\)-axis. As for optical experimental procedures, in order to accurately determine the small tilt angle of the reference light, we introduce a plane wave as the object wave to obtain \(I_{0}\). We control the tilt angle of the reference until there are dozens of fringes appearing on the CCD. We then obtain the corresponding spectrum to find \((-u_{p},-v_{p})\) and \((u_{p},v_{p})\). Fast Fourier Transform (FFT) operations are carried out on the interferograms on the computer. Figure 2a shows the pattern of \(I_{0}\) and its corresponding spectrum is shown in Figure 2d. The two delta functions are located at \((-u_{p},-v_{p})\) and \((u_{p},v_{p})\), which have been zoomed in for visual inspection. Figure 2b shows the pattern of \(I_{1}\) and its corresponding spectrum is shown in Figure 2e. Note that the  
Figure 1: Simulation setup in phase-shifting digital holography with the tilted reference wave. zero frequency has been suppressed for display purposes in Figure 2d,e. Figure 2c,f show the pattern of \(I_{2}\) and \(I_{3}\), respectively. By searching the maximum value of the spectrum intensity from the FT of \(I_{0}\), coordinates (\(-u_{p}\), \(-v_{p}\)) and (\(u_{p}\), \(v_{p}\)) are found to be (\(-\)45, 45) and (45, \(-\)45). Substituting \(u_{p}\) = 45, \(v_{p}\) = 45, \(\lambda\) = 0.532 \(\upmu\)m, \(d_{x}\) = 15 \(\upmu\)m, \(d_{y}\) = 15 \(\upmu\)m and \(M\) = \(N\) = 512 into Equation (23), the reference wave angles in the \(x\) and \(y\) directions are found to be both approximately 0.18 degrees. After the spectrum coordinates (\(-u_{p}\), \(-v_{p}\)) and (\(u_{p}\), \(v_{p}\)) are found, the phase angle of the complex values of the spectra at pixel (\(-\)45, 45) or (45, \(-\)45) can be calculated and then the two phase shifts are extracted by subtraction operation using Equations (20) and (21). The phase shifts, \(\Delta\varphi_{r1}\) and \(\Delta\varphi_{r2}\), calculated by our proposed method are 1.0002 rad and 0.6003 rad, respectively. The two phase shifts with errors of 0.0002 and 0.0003 rad are accurate enough in the non-iterative GPSI method. These phase shift errors come from the digital resolution on the FT spectral plane.