## 2 Proposed Method
### Proposed Method 1: Scaled Diffraction-Based Hologram Optimization  
A conventional algorithm can optimize a hologram. However, by introducing the blank area, it requires an expanded area that is \(m\)-times the size of the original hologram. In this section, we describe the proposed algorithm, which uses scaled diffraction to introduce a virtual blank area, to solve this problem. Several scaled diffraction calculations have been previously proposed [17; 18; 19; 20; 21; 22]. In this study, we used aliasing-reduced shifted and scaled (ARSS)-Fresnel diffraction [22] as the scaled diffraction. The ARSS-Fresnel diffraction can perform Fresnel diffraction with different sampling pitches on the source and destination planes. Note that the number of pixels on the source and destination is \(N\times N\) pixels. As with other diffraction calculations, the ARSS-Fresnel diffraction can be acceleration using fast Fourier transforms (FFTs). In this study, we denote the operator of the scaled diffraction as  
\[u_{2}(x,y)=\mathcal{P}_{p1,p2}^{z}\{u_{1}(x,y)\}, \tag{1}\]  
where \(u_{1}(x,y)\) and \(u_{2}(x,y)\) are the source plane and destination plane with sampling pitches \(p_{1}\) and \(p_{2}\), respectively, and \(z\) is the distance between the planes. For simplicity, we only show the one-dimensional ARSS-Fresnel diffraction, but two-dimensional ARSS-Fresnel diffraction can be simply derived using separation of variables in Fresnel diffraction. The ARSS-Fresnel diffraction is expressed as  
\[u_{2}(x_{2})=\mathcal{P}_{p1,p2}^{z}\{u_{1}\}=C_{z}\mathcal{F}^{-1}\bigg{[} \mathcal{F}\bigg{[}u_{1}(x_{1})\exp(i\phi_{u})\bigg{]}\mathcal{F}\bigg{[}\exp (i\phi_{h})\text{Rect}(\frac{x_{h}}{2x_{max}})\bigg{]}\bigg{]}, \tag{2}\]  
where \(x_{1}=p_{1}m\) and \(x_{2}=p_{2}n\), and where \(-N/2\leq m,n<N/2\) are the integer coordinates, \(\mathcal{F}\big{[}\cdot\big{]}\) denotes the Fourier transform, \(\text{Rect}(\cdot)\) is the rectangular function for reducing aliasing error, \(x_{h}\) is a variable for the generation of \(\exp(i\phi_{h})\), \(x_{max}\) is the aliasing-free area and \(\exp(i\phi_{u})\), \(\exp(i\phi_{h})\) and \(C_{z}\) are defined by \(\exp(i\phi_{u})=\exp(i\pi((s^{2}-s)x_{1}^{2})/(\lambda z))\), \(\exp(i\phi_{h})=\exp(i\pi sx_{h}^{2}/(\lambda z))\) and \(C_{z}=\exp(i\phi_{c})/(i\lambda z)=\exp(i\hbar z+i\pi((1-s)x_{2}^{2}))/( \lambda z))/(i\lambda z)\), where \(s\) is defined by \(p_{1}/p_{2}\). The flow of the proposed algorithm is shown in Figure 2. The proposed algorithm using the scaled diffraction is performed in the following steps:  
1. We initially set random phase values in the hologram plane \(u_{h}(x,y)\), with the sampling pitch \(p_{h}\);
2. We compute the diffraction calculation from \(u_{h}(x,y)\) to the object plane \(u_{o}(x,y)\) with the sampling pitch of \(p_{o}=mp_{h}\) where \(m\) is the magnification. The calculation is performed by \[u_{o}(x,y)=\mathcal{P}_{p_{h}p_{o}}^{z}\{u_{h}(x,y)\};\] (3)  
Figure 1: Conventional algorithm. As an object plane constraint, the area where the original object in the object plane exists is replaced by down-sampled original object \(o_{d}(x,y)\) with the number of pixels \(N/m\times N/m\). The complex values in the virtual blank area remain;
4. The updated \(u_{o}(x,y)\) is back-propagated to the hologram plane \(u_{h}(x,y)\) using the same diffraction calculation: \[u_{h}(x,y)=\mathcal{P}_{p_{o},p_{h}}^{-z}\{u_{o}(x,y)\};\] (4)
5. As a constraint on the hologram plane, we calculate \(u_{h}(x,y)=u_{h}(x,y)/|u_{h}(x,y)|\), because our target is a phase-only hologram;
6. We repeat Steps 2 to 5 until the number of iterations reaches a predefined number or the image quality of the reconstructed complex amplitude reaches a predefined quality. We used scaled diffraction in Steps 2 and 4. The number of pixels in the scaled diffraction source and destination planes is \(N\times N\), unlike in the case of the conventional algorithm; therefore, the computational complexity and memory usage of the algorithm are only \(O(LN^{2}\log_{2}N)\) and \(O(N^{2})\), respectively.