# Robust Evaluation of Reference Tilt in Digital Holography
## 3 Numerical Simulations
The error distributions from different methods are shown by four curves in Figure 2(e),f. The curves marked by the acronyms of LGF, LQFP, LCSI, and DSM are calculating results from the corresponding methods of local Gaussian fitting, local quintic polynomial fitting, local cubic spline interpolation, and direct searching method, respectively. Figure 2(e) depicts the calculation results of the errors \(\Delta\theta_{x}\) when \(\theta_{x}\) varies from 0.0025 rad to 0.05 rad while \(\theta_{y}\) = 0.025 rad. In Figure 2(e), the error distribution curve from LGF denoted by symbol '\(-\)\" keeps stable and is almost the lowest error value, so the method proposed here performs well. Meanwhile, the LQPF presented by '\(-\)\" works slightly worse and LCSI performs slightly poorly. The maximum error for LGF is about 2 \(\times\) 10\({}^{-5}\) rad, whereas the maximum errors from both interpolation and polynomial fitting reach about 2.5 \(\times\) 10\({}^{-5}\) rad. The errors in Figure 2(f) show a similar performance as that in Figure 2(e) under the same conditions. The trend that the errors \(\Delta\theta_{y}\) do not fluctuate violently with the varying of \(\theta_{x}\) shows that the spectra distribution is a function with variable separable as Gaussian function. The maximum value of the tilt angle error by DSM reaches 5 \(\times\)10\({}^{-5}\) rad, which conforms to the theoretical prediction. In theory, this threshold value can be calculated by Equation (9) when \(m_{0}\) or \(n_{0}\) is replaced by 0.5 under the condition that the spectrum falls at the edge of one FP, which is the worst situation for DSM. The biggest theoretical error is \(0.5\times\lambda/(M\Delta x)\) = \(5.2\times 10^{-5}\) rad, which is consistent with the simulation results. To test the availability of the local fitting method more comprehensively, a portrait figure [PERSON] in Figure 4a is used as the object in the holography recording, and another plane wave is introduced to enhance the zero-frequency component in the objective wave. Both the set up used here and the reconstructed object image have been reported earlier [24]. The amplitudes of the two plane waves are set to be equal and other parameters are similar to the first case in Figure 3. The RTAs are set as \(\theta_{x}\) = 0.025 rad and \(\theta_{y}\) = 0.025 rad, which is consistent with the simulation results. Figure 3: Analysis of one frequency peak on spatial spectrum plane and comparison of the result from LGF with that from other RTA evaluation methods (**a**) interference fringe by two plane waves (**b**) one frequency spectrum distribution on SSP (**c**) three-D frequency spectrum distribution and (**d**) error source for the location of one frequency peak center, (**e**) \(\Delta\theta_{x}\) and (**f**) \(\Delta\theta_{y}\) distributions from four methods when \(\theta_{y}\) = 0.025 rad. rad. The same operations are conducted and the results are shown in other sub-figures in Figure 4. Figure 4b is one of the multiplexing interferometry fringes and Figure 4c is its spectra distribution in the Fourier domain. The zero and extreme low frequency spectra are removed for proper display and frequency peak searching. The two spots do not distribute regularly because of the disturbance of the spectra of the portrait, but their maximum centers locate symmetrically about the origin in Figure 4c. The errors from RTA detecting results for varying values of \(\theta_{x}\) from 0.0025 to 0.05 rad and \(\theta_{y}\) = 0.025 rad are displayed in Figure 4d,e, respectively. The biggest error for \(\theta_{x}\) is still \(5\times 10^{-5}\) rad by DSM. Although the LQPF and LCSI methods work better for certain \(\theta_{x}\) values, the LGF proposed here performs well because the error distribution keeps low and stable, which can ensure the precision of the retrieved value \(\theta_{xl}\) and \(\theta_{y}\). Although LGF does not perform well for very few specific values of RTA in Figure 4d, the error curve of LGF keeps stable with small values in the full RTA range shown.