# Computer-Generated Holography Methods for Data Page Reconstruction Using Phase-Only Medium
## 3 Methods for Synthesizing Phase Fourier CGH of Data Page
### Numerical Modeling  
Image formation of a complex-modulated data page was numerically simulated with the application of both methods of phase-only fringe pattern synthesis under consideration. To demonstrate the quality of reconstructed complex values, the constellation diagram is useful for quadrature amplitude modulation (QAM). The diagram points are usually arranged in a square grid. The 16-QAM digital format was used in numeric modeling. This modulation format allows representing 4 bits of a binary code using one element of the data page. The dimension of the simulated data page was 200 \(\times\) 200 pixels. The Fourier CGH of the holographic data page images was calculated using the two investigated methods. The hologram size was equal to 1024 \(\times\) 1024 pixels. The Fourier CGH has 256 phase shift levels at a 2\(\pi\) phase modulation depth. Images of complex-modulated data pages were numerically reconstructed. These images are matrices of values of the complex amplitude distribution function of the light field in space. After the numeric reconstruction, the in-phase and quadrature components were demodulated and normalized. The SNR values for these components were estimated, and the obtained constellation diagrams for both methods are shown in Figure 3. Red dots correspond to ideal constellation points. Blue dots correspond to reconstructed constellation points.  
It can be seen that data page images for the bipolar intensity method have a normal error distribution. The dispersion values (normalized rooted mean square error, NRMSE) in both in-phase and quadrature components of complex amplitude and amplitude modulus were obtained at about 0.12, corresponding to a BER of 0.001.  
In the double-phase coding method, the distortions have a predominant phase noise. It leads to the variance of the in-phase and quadrature components at 0.16. Obtained BER value is 0.006. However, the variance of the normalized amplitude modulus was obtained at about 0.04. Additionally, it was observed that different parts of the data page produce different phase errors in the case of double-phase Fourier CGH. The larger the distance of the data page element from the vertical axis of the data page, the bigger the error obtained. The shift to the left from the vertical axis produces a negative phase error, while the shift to the right from the vertical axis produces a positive phase error. It is worth noting that the CGH superpixel was contained by two phasor subpixels aligned horizontally.  
Figure 3: Numerical simulation: the reconstructed data pages using phase Fourier CGH for bipolar intensity (**a**) and double-phase (**b**) methods.  
A numeric experiment with an enlarged data page was also carried out. Doubling the data page size resulted in the rise of NRMSE to 0.27 and 0.29 for bipolar intensity and double-phase cases, respectively. Such an error does not allow for the satisfactory reconstruction of the data page in a 16-QAM format. Even in the case of 4-level phase modulation, the value of BER was obtained at about 0.02. It should be added that the modulus-of-amplitude NRMSE in the case of double-phase CGH appeared to be 0.08, enough to reconstruct 8-level bipolar amplitude modulated data pages with BER value on the order of \(10^{-7}\).  
Thus, the bipolar intensity method allows the forming of quadrature-modulated data page images with better quality than the double-phase method. The reconstructed data pages for the double-phase coding method have higher phase noise. However, the amplitude noise is lower for the double-phase coding method. Therefore these holograms are promising for the multilevel amplitude modulation types of the data pages. Ternary, bipolar amplitude, and unipolar amplitude (intensity) modulation are examples of these modulation types.