# Calibrated Phase-Shifting Digital Holographic Microscope Using a Sampling Moire Technique
## 3 The Sampling Moire Technique  
The principle of the sampling Moire technique [17] is represented in Figure 2. If the pitch of the captured grating pattern in the image sensor plane is supposed as \(P\), then the recorded intensity of the grating can be described as  
\[\begin{array}{rl}f(x,y)&=A_{g}\cos\left\{2\pi\frac{x}{P}+\phi_{g0}\right\}+B_ {g}\\ &=A_{g}\cos\left\{\phi_{g}(x,y)\right\}+B_{g}\end{array}. \tag{2}\]  
Here, \(A_{g}\) is the amplitude distribution of the grating pattern, \(B_{g}\) is the intensity of the background, \(\phi_{g0}\) is the initial phase value at position \(x\), and \(\phi_{g}\) is the phase distribution of the grating pattern. Multiple phase-shifted Moire fringe patterns can be obtained through down-sampling and intensity interpolation processing. In general, an integer \(T\) that is close to \(P\) is applied for down-sampling. \(T\) must be larger than or equal to 3 to calculate the phase distribution of the Moire fringe patterns. The beginning of the down-sampling position is set to the first row of the recorded grating, the pixels  
Figure 2: Principle of the sampling Moiré technique to calculate the phase distribution from a single grating pattern. FFT denotes the fast Fourier transform and DFT denotes the discrete Fourier transform.  
at intervals of \(T\)-1 rows are extracted, and the vacant pixels are then interpolated using adjacent sampled pixels. \(T\)-phase-shifted Moire fringe patterns can be obtained when the beginning of the down-sampling position increases from the first to the \(T\)-th row. The intensity of the phase-shifted Moire fringe patterns can be represented as  
\[\begin{array}{ll}f_{m}(x,y;k)&=A_{m}\cos\Bigl{\{}2\pi(\frac{1}{P}-\frac{1}{T} )x+2\pi\frac{k}{T}+\phi_{g0}\Bigr{\}}+B_{m}\\ &\qquad=A_{m}\cos\{\phi_{m}(x,y)\}+2\pi\frac{k}{T}+B_{m}\end{array} \tag{3}\]  
where \(A_{m}\), \(B_{m}\) and \(\phi_{m}\) are the amplitude distribution, the intensity of the background, and the phase distribution of the Moire fringe pattern, respectively. \(k\) is the number of the Moire fringe patterns. The phase distribution \(\phi_{m}\) of the phase-shifted Moire fringe patterns can be obtained by a phase-shifting method [17; 18], expressed as  
\[\phi_{m}(x,y)=-\tan^{-1}\frac{\sum_{k=0}^{T-1}f_{m}(x,y;k)\sin(2\pi k/T)}{\sum _{k=0}^{T-1}f_{m}(x,y;k)\cos(2\pi k/T)}. \tag{4}\]  
Figure 3 illustrates the principle of the sampling Moire technique to determine the phase-shifting errors. In the calibrated phase-shifting digital holography system, several interferograms with a periodic repetitive fringe pattern and phase-shifted holograms including the object information are captured by two synchronized cameras. The phase-shifting amount of two neighboring holograms can be calculated from the phase difference of the Moire fringe patterns. Therefore, if the phase-shifting errors are occurred in the recorded holograms, they will be accurately calculated by  
\[\Delta\delta_{j}=\sum_{n=1}^{j}\Delta\phi_{n}-j\varphi_{s}\;(j\;=\;1,\;2,\;3, \;\ldots) \tag{5}\]  
Figure 3: Principle of the sampling Moiré technique to determine the phase-shifting errors.  
Here, \(\Delta\varphi_{n}\) is the phase-shifting amount calculated from the phase difference of two neighboring interference fringe patterns. High-quality object images can be reconstructed using the phase-shifting error compensation algorithm [15].