## 2 Methods
### Encoding Methods for the Amplitude-Modulating Device  
In this section, we first explore the established approaches to convert the complex-valued hologram to express the amplitude-modulating SLM (AM-SLM). We then propose our novel encoding scheme, i.e., the modified amplitude-only encoding (MAO) approach. Among methods to circumvent the non-availability of commercial complex-modulating SLMs, a conventional amplitude-only encoding (CAO) [16] has been used. The CAO approach is to extract only the real part of a complex-valued optical field at the pixel coordinates of the hologram plane, and subsequently to bias the real-valued space into nonnegative, real-valued space by level shifting, which can be written as  
\[H_{CAO}(x,\,y)=H_{R}(x,\,y)+H_{off}, \tag{2}\]  
where \(H_{R}(x,\,y)=|H(x,\,y)|\cos[\Phi(x,\,y)]\) is the real part of the original complex field, and \(H_{off}\) is an offset to make \(H_{CAO}(x,\,y)\) non-negative and is equal to the absolute value of a minimum of \(H_{R}(x,\,y)\) for the CAO, i.e., \(H_{R}=|min\{H_{R}(x,\,y)\}|\). On the other hand, as an alternative representation format of computer-generated holograms (CGHs) available for the AM-SLM, there are two kinds of effective solutions from an analytic perspective: The first analytic representation scheme is [PERSON]'s method, which decomposes a complex-valued field into four real and non-negative components [13]. The decomposition by [PERSON]'s method is expressed as  
\[H_{L}(x,\,y)=L_{1}(x,\,y)e^{i0}+L_{2}(x,\,y)e^{i\pi/2}+L_{3}(x,\,y)e^{i\pi}+L_ {4}(x,\,y)e^{i3\pi/2}, \tag{3}\]  
where at least two among four coefficients \(L_{m}(x,\,y)\) (\(m=1\)\(\sim\)4) are equal to zero. The relative phase values for [PERSON]'s method are physically implemented by lateral displacement within a single macro-pixel that consists of four sub-pixels. Alternatively, [PERSON]'s method, which is a simplified version of [PERSON]'s method, uses the feature that any complex-valued function can be analytically decomposed into three real and nonnegative components. The decomposition by [PERSON]'s method can be expressed by  
\[H_{B}(x,\,y)=B_{1}(x,\,y)e^{i0}+B_{2}(x,\,y)e^{i2\pi/3}+B_{3}(x,\,y)e^{i4\pi/3}, \tag{4}\]  
Figure 4: Geometry of the optical system for each eye in the near-eye holographic 3D display. The eye of the observer lens on the \(u\)-\(v\) coordinate is located at the position of the focal length of the field lens, corresponding to the center of the Fourier plane of the holographic display. where at least one among the three coefficients \(B_{n}(x,\,y)\) (\(n=1{\sim}3\)) is equal to zero [14; 15]. The relative phase values for [PERSON]'s method are realized by lateral displacement within one macro-pixel that comprises three sub-pixels. Finally, the equational representation of the modified amplitude-only encoding (MAO) method we propose is described in the following formula, provided that the minimum of \(H_{R}(x,\,y)\) is negative:  
\[H_{MAO}(x,\,y)=H_{int}(x,\,y)-min\{H_{int}(x,\,y)\} \tag{5}\]  
where \(H_{int}(x,\,y)\) is an intermediate function and is defined by  
\[H_{int}(x,\,y)=\left\{ \begin{array}{ll}H_{R}(x,\,y)&for\,\,H_{R}\geq 0,\\ \frac{eH_{R}(x,\,y)}{min\{H_{R}(x,\,y)\}}&for\,\,H_{R}<0.\end{array} \right. \tag{6}\]  
This formula of the MAO scheme has two key features: (i) For the pixel of \(H_{R}(x,\,y)\geq 0\), the intermediate function \(H_{int}(x,\,y)\) is not changed from \(H_{R}(x,\,y)\) and retains the original form of \(H_{R}(x,\,y)\). The result of MAO encoding is thus equivalent to that of conventional amplitude-only encoding, that is, \(H_{MAO}(x,\,y)=H_{CAO}(x,\,y)\). (ii) For the pixel of \(H_{R}(x,\,y)<0\), a scale-down transform process is first performed before \(H_{R}(x,\,y)\) is divided by the minimal value of \(H_{R}(x,\,y)\). Its effect reduced with a fixed rate is determined by a parametric factor \(\varepsilon\), where \(0<\varepsilon\ll 1\) is assumed.