## 2 Theoretical Analysis
### Modified Off-Axis Reference Beam Cgh  
To record an object light field represented by a complex function \(\widetilde{U}(x,y)=A(x,y)\)\(\exp[j\varphi(x,y)]\), it can be interfered with an off-axis reference light \(\widetilde{R}(x,y)=R\exp j\varphi(2\pi\alpha x)\). The resulting off-axis hologram has a transmittance:  
\[\begin{array}{l}h(x,y)=\left|\widetilde{U}(x,y)+\widetilde{R}(x,y)\right|^{ 2}=\\ R^{2}+\left|\widetilde{A}(x,y)\right|^{2}+2 RA(x,y)\cos[2\pi\alpha x-\varphi(x,y)]\end{array} \tag{9}\]  
In this equation, \(A(x,y)\) and \(\varphi(x,y)\) represent the amplitude and phase distribution of the object optical field at position (\(x\), \(y\)), respectively, while \(R\) and \(\alpha\) denote the amplitude and carrier frequency of the reference light [17].  
[PERSON] proposed replacing \(R^{2}+\left|\widetilde{A}(x,y)\right|^{2}\) in Equation (9) with a DC bias, thereby reconstructing the transmittance function of the off-axis hologram as:  
\[\begin{array}{l}h(x,y)=K+\widetilde{U}(x,y)\times\widetilde{R}^{*}(x,y)+ \widetilde{R}(x,y)\times\widetilde{U}^{*}(x,y)\\ =K+2 RA(x,y)\cos[2\pi\alpha x-\varphi(x,y)]\end{array} \tag{10}\]  
where \(K\) is a constant, ensuring \(h(x,y)\) is a non-negative real value for all (\(x,y\)). Normalizing \(\left|A(x,y)\right|_{\max}=1\), taking \(R=1\), Equation (10) simplifies to:  
\[h(x,y)=0.5\{1+A(x,y)\cos[2\pi\alpha x-\varphi(x,y)\} \tag{11}\]  
When [PERSON] encoded and fabricated holograms using Formula (11), they directly recorded the transmittance function onto film using a microdensitometer, creating a 256-gray-level computer-generated hologram [18].  
Another commonly modified off-axis hologram is the [PERSON] delayed sampling hologram. This allows the complex function \(\widetilde{F}\) at each sampled point on the hologram to be represented by four real, non-negative values:  
\[\widetilde{F}=F_{1}+jF_{2}-F_{3}-jF_{4} \tag{12}\]  
Figure 2: Mode overflow correction algorithm for Lohmann Type III holograms.  
In the equation,  
\[F_{1}=\begin{cases}|F|\cos\phi&\cos\phi\geq 0\\ 0&\cos\phi<0\\ \end{cases}\] \[F_{2}=\begin{cases}|F|\sin\phi&\cos\phi\geq 0\\ 0&\cos\phi<0\\ \end{cases}\] \[F_{3}=\begin{cases}-|F|\cos\phi&\cos\phi\geq 0\\ 0&\cos\phi<0\\ \end{cases}\] \[F_{4}=\begin{cases}-|F|\sin\phi&\cos\phi\geq 0\\ 0&\cos\phi<0\\ \end{cases}\]  
where \(F_{1}\), \(F_{2}\)\(F_{3}\), and \(F_{4}\) are all real, non-negative functions.  
Essentially, this decomposes a complex vector in the complex plane by using four basis vectors: \(r^{+}=(1,0)\), \(r^{-}=(-1,0)\), \(j^{+}=(0,1)\), and \(j^{-}=(0,-1)\). Any complex vector can be expressed as:  
\[\widetilde{F}=F_{1}r^{+}+F_{2}j^{+}+F_{3}r^{-}+F_{4}j^{-} \tag{13}\]  
These basis vectors have fixed angular relationships in the complex plane, with adjacent vectors separated by \(\pi/2\). By applying delayed sampling technology, the original hologram plane is divided into \(N\times N\) sampling points, each further subdivided into four equally spaced sub-cells. The amplitude transmittance of each sub-cell is made proportional to \(F_{i}[mdx+(i-1)dx/4,ndy],i=1,2,3,4\). This enables the decomposition of the complex vector by using four non negative basis vectors. This method produces continuous-gray-level CGH [19].