# Robust Evaluation of Reference Tilt in Digital Holography
## 3 Numerical Simulations
The root mean square (RMS) of RTA errors in Figure 4d for LGF, LQPF, LCSI, and DSM are \(8.6\times 10^{-6}\) rad, \(8.6\times 10^{-6}\) rad, \(1.9\times 10^{-5}\) rad, and \(2.9\times 10^{-5}\) rad, respectively. RMS of the errors in Figure 4e for the same methods are \(4.0\times 10^{-6}\) rad, \(1.3\times 10^{-5}\) rad, \(2.3\times 10^{-5}\) rad, and \(4.4\times 10^{-5}\) rad consequently. The results from these RMS calculations show that the LGF methods are more stable than methods LQPF, LCSI, and DSM. It should be noted that the effect of the RTA error on this amplitude image is not easy to observe, but the RTA error affects the location precision and background of the reconstructed image. To improve the precision of the retrieved RTA, the zero-padding method (ZPM) is used first before local fitting or interpolating. The interferometry fringe in Figure 3 of case one is zero-padded from 1024 \(\times\) 1024 to 2048 \(\times\) 2048 pixels. After the Fourier transform, the small area in Figure 3d is further refined from 11 \(\times\) 11 to 55 \(\times\) 55 FPs. The preset values of RTA \(\theta_{x}\) are chosen from 0.0025 rad to 0.05 rad with the intervals of 0.0025 rad and \(\theta_{y}\) is fixed as 0.025 rad. The corresponding errors are shown in Figure 5a,b. Because of the  
Figure 4: Spectra distribution of one real hologram and comparison of the result from LGF with that from other RTA evaluation methods (**a**) original object (**b**) hologram (**c**) corresponding spectrum distribution. (**d**) \(\Delta\theta_{x}\) and (**e**) \(\Delta\theta_{y}\) distributions from four methods when \(\theta_{y}\) = 0.025 rad. zero-padding, the data array of the interference fringe across one direction is enlarged double and the maximum error should be decreased by half in theory. The errors for \(\theta_{x}\) and \(\theta_{y}\) in Figure 5a,b exhibit that the biggest error by DSM is about 2.5 \(\times\) 10\({}^{-5}\) rad and LGF method works best with the LCSI and LQPF following. The RMS of RTA errors in Figure 5a for LGF, LQPF, LCSI, and DSM are 1 \(\times\) 10\({}^{-6}\) rad, 6.9 \(\times\) 10\({}^{-6}\) rad, 3.1 \(\times\) 10\({}^{-6}\) rad, and 1.5 \(\times\) 10\({}^{-5}\) rad respectively. RMS of the errors for them in Figure 5b are 1.1 \(\times\) 10\({}^{-6}\) rad, 4.1 \(\times\) 10\({}^{-6}\) rad, 2.4 \(\times\) 10\({}^{-6}\) rad, and 7.9 \(\times\) 10\({}^{-6}\) rad, consequently. From the whole range of the RTA selection scope, the LGF and LCSI methods are more stable than LQPF and DSM with LGF works a little better based on accuracy. To inspect the effectiveness of RTA precision improving on the imaging quality, a pure phase object with assumed phase in Figure 6a is employed in phase-shifting digital holography (PSDH) simulations with slight RTAs \(\theta_{x}=\theta_{y}=0.025\) rad. To simulate micro-organism microscopy, there are sharp phase changes in the center area of Figure 6a and the phase around is a constant. The algorithm of the PSDH technique [26] is used to retrieve the object wave on the recording plane, and the RTA values are detected to correct this object wave affected by the reference tilt. Figure 5: Comparison of the result from LGF with that from other RTA evaluation methods after zero-padding operation. (**a**) \(\Delta\theta_{x}\) and (**b**) \(\Delta\theta_{y}\) distributions from four methods when \(\theta_{y}=0.025\) rad. Figure 6: Phase object reconstruction results with different RTA errors (**a**) Ground truth, and by (**b**) DSM method, (**c**) LGF method, (**d**) LGF with ZPM. The RTA errors calculated by DSM are \(\Delta\theta_{x}=\Delta\theta_{y}=4.4\ \times 10^{-5}\) rad and the object phase is shown in Figure 6b. Because of the RTA error, the reconstructed phase is disturbed by square grid at the edge. Because of the residual RTA error, the lower right corner of Figure 6b tilts up, showing that the constant phase background is spoiled. The errors of RTA by LGF are \(\Delta\theta_{x}=\Delta\theta_{y}=5\ \times 10^{-6}\) rad, and the precision is improved by one order of the magnitude.