# Robust Evaluation of Reference Tilt in Digital Holography
## 3 Numerical Simulations  
To investigate the performance of the local fitting effects of our method in detail, a series of numerical simulations are carried out, and some RTA retrieving results are given. Figure 2: Flowchart for the robust RTA evaluation method. In these simulations, three cases with different simulated objects are introduced to check the applicability of the proposed method. In the first case, the configuration [26] reported before is used to detect RTA and improve the precision of the RTA calculation. In this circumstance, the other fringe by the interference of two plane waves is used to avoid the disturbance of another object spectrum so that the effects of the method can be observed more thoroughly and carefully. For comparison, the RTA searching results by direct searching method (DSM) without any more operations, local cubic spline interpolation (LCSI), and Local quintic polynomial fitting (LQPF) are also studied in this work. During the simulation, there are 1024 \(\times\) 1024 pixels with 5 \(\upmu\)m \(\times\) 5 \(\upmu\)m size on the CCD recording chip and the recording distance is set to be \(z\) = 48 mm to satisfy the requirement of the sampling theory [11]. The laser source with the wavelength of \(\lambda\) = 532 nm is used as the illumination. To exclude the interference of other factors, the configuration is equipped with another CCD2 to record the fringes carrying the reference tilt information [26]. The real amplitudes of the two plane waves are equal and the tilt angle of one plane wave is set to be smaller than 0.05 rad (about 3 degrees which is always used in digital holography) across \(x\) and \(y\) directions. The interference fringes on the CCD2 recording plane are shown in Figure 2(a). Figure 2(b) gives the spectrum distribution from the Fourier transform of Figure 2(a) with the zero frequency spectrum depressed. The two delta functions in Equation (6) are obvious since there is no disturbance from other objective spectra. The spectra distribution in adjacent ten \(\Delta u\)s and ten \(\Delta v\)s around one delta function center on the SSP is drawn in Figure 2(c), where the spectra distributions are discrete, and the values are averaged in one unit \(\Delta u\)\(\times\)\(\Delta v\). The curves for these distributions in two directions are plotted in Figure 2(d), where the center (\(u_{0}\), \(v_{0}\)) of the frequency peak before fitting deviates from the fitted Gaussian function center (\(u_{0}\), \(v_{0}\)) and the maximum offset is (0.5\(\Delta u\), 0.5\(\Delta v\)). Because the fast Fourier transform and other factors incur diffusion, neither of the two curves is a theoretical delta function but a distribution similar to the Gaussian function. So the Gaussian function is chosen for fitting as analyzed before. To fit the spectra distribution, the small area around point (\(u_{0}\), \(v_{0}\)) is emnesed again with a smaller grid of 0.2\(\Delta u\) and 0.2\(\Delta v\). A target Gaussian function is established in the computer and the new function center (\(u_{0}\), \(v_{0}\)) is determined by Equation (11), and then, RTAs are calculated by Equation (13). By recording theory and applications in digital holography, RTA values should be smaller than the value of \(\lambda/(2\Delta x)\) and \(\lambda/(2\Delta y)\), which is about 0.05 rad in the simulation conditions here. Considering this selectable scope, to investigate the performance of the method in the practice applications, 20 RTAs are chosen from 0.0025 rad to 0.05 rad with intervals of 0.0025 rad and corresponding errors are calculated by DSM. Meanwhile, the calculation results from Equation (13), interpolation method, and polynomial method are also carried out for comparison. The errors are calculated by the subtraction of the preset value from the retrieved one of RTA.