# Robust Evaluation of Reference Tilt in Digital Holography
## 2 Principles  
The hologram recording setup for both OADH and PSDH have been reported [24,26], respectively. Supposing the reference tilt is determined by the two angles \(\theta_{x}\) and \(\theta_{y}\) across \(x\) and \(y\) directions, respectively, the reference wave can be written as  
\[\mathbf{R}(x,y)=A_{r}\exp\left[-i\frac{2\pi}{\lambda}(x\sin\theta_{x}+y\sin\theta_ {y})\right] \tag{1}\]  
where \(A_{r}\) and \(\lambda\) are the real amplitude and the wavelength of the reference wave, respectively, and \(i\) is the imaginary unit. Commonly the real amplitude \(A_{r}\) is always chosen as a constant in DH. In Equation (1), \(\theta_{x}\) and \(\theta_{y}\) represent RTA and bring about the periodic stripes on the fringes after its interference with the zero frequency part of the objective wave. For the object wave, there must exist a sufficient zero frequency component so that the corresponding spectrum can be detected in the Fourier transform domain. The transparent sample with low diffraction can satisfy this requirement naturally. However, for the object with strong diffraction, the energy of the zero frequency spectrum is not so powerful to find the carrier frequency after the Fourier transform. So a plane wave on-axis is always necessary to supply the zero frequency ingredient in the object wave [24] or an additional frame of a two-plane wave interference is captured for RTA detection [25,26] under the condition of strong diffraction. To make the zero frequency component of the Fourier transform clear, the object wave is expressed separately as  
\[\mathbf{O}(x,y)=\mathbf{O}_{1}(x,y)+\mathbf{O}_{2}(x,y) \tag{2}\]  
where \(O_{1}(x,y)\) is the zero frequency component in the objective wave and can be seen as an on-axis plane wave with constant amplitude \(A_{1}\) and phase \(\varphi_{1}\)  
\[\mathbf{O}_{1}=A_{1}\exp[-i\varphi_{1}] \tag{3}\]The other part of Equation (2) \(O_{2}(x,y)\) is the non-zero frequency wave with object information in the complex amplitude  
\[O_{2}(x,y)=A_{2}(x,y)\exp[-i\varphi_{2}(x,y)] \tag{4}\]  
The supposing three waves interfere on the recording plane and the hologram is  
\[I(x,y)=[R(x,y)+O_{1}+O_{2}(x,y)][R(x,y)+O_{1}+O_{2}(x,y)]^{*} \tag{5}\]  
where \"*\" represents the conjugation. After the Fourier transform is operated on the hologram of Equation (5), the frequency spectrum distribution can be written in [26]  
\[F(u,v)=A_{r}A_{1}[\delta(u+u_{0},v+v_{0})\exp(i\varphi_{1})+\delta(-u+u_{0},-v +v_{0})\exp(-i\varphi_{1})]+F_{h}(u,v) \tag{6}\]  
where \(u\) and \(v\) are frequency-pixelated coordinates on SSP. Obviously, the first term in Equation (6) is the spectrum with distributions like two delta functions from the interference of two plane waves with the tilt angles \(\theta_{x}\) and \(\theta_{y}\). The second term \(F_{h}\left(x,y\right)\) on the right side of Equation (6), includes the other spectra from the Fourier transform of Equation (5). Most of the intensity distribution of \(F_{h}\left(x,y\right)\) is weak compared with the first terms of Equation (6), so the center of the spectra can be detected by searching the strongest point on the SSP in the Fourier frequency domain. The coordinates \(u_{0}=\sin\theta_{x}/\lambda\), \(v_{0}=\sin\theta_{y}/\lambda\) are the centers of the two delta functions in Equation (6) [24]. It should be two ideal spots with the maximum value at their centers on SSP in theory. Because of the disturbance of the other part in Equation (6), the spots always become irregular. Since fast Fourier transforms are commonly carried out digitally in a computer, \(u\) and \(v\) are not continuous and Equation (6) is a discrete function. Naturally, \(u_{0}\) and \(v_{0}\) are not an exact point on SSP either.