## 2 Analysis and Methods  
Our method is based on digital holography. Figure 1 shows a sketch of the phase measurement setup. Transmissive SLM is used here as an example. Plane waves are used as the light beam. The wavefunction of the beam received at the monitor plane can be written as:  
\[U=U_{O}+U_{R} \tag{1}\]  
Here, the subscripts \(O\) and \(R\) represent the object and reference beam, respectively. The intensities of the two beams are adjusted to be the same and set to 1 in the analyses. The intensity of the obtained interferogram can be written as:  
\[I=(U_{O}+U_{R})(U_{O}^{*}+U_{R}^{*})=|U_{O}|^{2}+|U_{R}|^{2}+U_{O}U_{R}^{*}+U_{ R}U_{O}^{*} \tag{2}\]The interferogram can be considered as the hologram of SLM after diffraction. By applying Fourier transforms to the hologram, there will be three spots in the Fourier domain, namely the 0 th, +1 st and \(-1\)st order spots. They correspond to \(F\Big{\{}\big{|}U_{\text{O}}\big{|}^{2}+\big{|}U_{R}\big{|}^{2}\Big{\}}\), \(F\{U_{\text{O}}U_{R}^{*}\}\) and \(F\big{\{}U_{R}U_{O}^{*}\big{\}}\), respectively. \(F\{\}\) is the operator of the Fourier transform. Set (\(\alpha\), \(\beta\), \(\gamma\)) as the direction cosine of the beams, \(Z\) as the axis perpendicular to the monitor plane, \(X\) and \(Y\) as the axes parallel to it, (\(x\), \(y\), \(z\)) as the coordinates in the \(X\)-\(Y\)-\(Z\) coordinate system, and \(\theta\) as the phase added by SLM in the object beam, then:  
\[U_{\text{O}}=F^{-1}\bigg{\{}F\big{\{}W\exp(j\theta)\big{\}}\exp\bigg{(}jkz\sqrt {1-\big{(}\lambda f_{x}\big{)}^{2}-\big{(}\lambda f_{y}\big{)}^{2}}\bigg{)} \bigg{\}} \tag{3}\]  
We chose \(z=0\) as the position of the SLM; and the modulation of SLM is considered to be in the pure phase mode. W is the transmitting zone of SLM. It can be treated as a two-dimensional (2D) rectangular function, i.e.,  
\[W=\text{rect}\Big{(}\frac{x}{a},\frac{y}{b}\Big{)}=\text{rect}\Big{(}\frac{x}{ a}\Big{)}\text{rect}\Big{(}\frac{y}{b}\Big{)} \tag{4}\]  
Here, \(a\) and \(b\) are the size of the width and length of the transmitting zone. The wave function of \(U_{R}\) can be expressed as:  
\[U_{R}=\exp\bigg{[}j\overset{\rightarrow}{k}_{R}\overset{\rightarrow}{R} \bigg{]}=\exp[jk(\alpha_{R}x+\beta_{R}y+\gamma_{R}z)] \tag{5}\]  
where \(\overset{\rightarrow}{k}=k\Big{(}\overset{\rightarrow}{e}_{x}+\beta \overset{\rightarrow}{e}_{y}+\gamma\overset{\rightarrow}{e}_{z}\Big{)}\), and \(\overset{\rightarrow}{R}=x\overset{\rightarrow}{e}_{x}+y\overset{ \rightarrow}{e}_{y}+z\overset{\rightarrow}{e}_{z}\). Here, \(\overset{\rightarrow}{k}\) is the wave vector of the beam. \(\overset{\rightarrow}{e}_{x}\), \(\overset{\rightarrow}{e}_{y}\) and \(\overset{\rightarrow}{e}_{z}\) denote the unit vector of the axis \(X\), \(Y\), and \(Z\), respectively. The +1 order in the Fourier domain can then be written as:  
\[F\{U_{\text{O}}U_{R}^{*}\}=F\bigg{\{}F^{-1}\bigg{\{}F\big{\{}W\exp(j\theta) \big{\}}\exp\bigg{(}jkz\sqrt{1-\big{(}\lambda f_{x}\big{)}^{2}-\big{(}\lambda f _{y}\big{)}^{2}}\bigg{)}\bigg{\}}\times\exp[-jk(\alpha_{R}x+\beta_{R}y+\gamma _{R}z)]\bigg{\}} \tag{6}\]  
According to the properties of the Fourier transform [25] and considering that \(\theta\) and \(z\) are constant, Equation (6) can be calculated as:  
\[ \begin{array}{ll}F\{U_{\text{O}}U_{R}^{*}\}=&|ab|\text{sinc}\big{(}a\big{(} f_{x}+\frac{\alpha_{R}}{\Lambda}\big{)}\big{)}\text{sinc}\big{(}b\big{(}f_{y}+ \frac{\alpha_{R}}{\Lambda}\big{)}\big{)}\times\\ &\exp\bigg{(}jkz\sqrt{1-\lambda^{2}\big{(}f_{x}+\frac{\alpha_{R}}{\Lambda} \big{)}^{2}-\lambda^{2}\Big{(}f_{x}+\frac{\alpha_{R}}{\Lambda}\big{)}^{2}} \bigg{)}\exp(j\theta)\exp(-jk\gamma_{R}z)\end{array}  \tag{7}\]  
Figure 1: Sketch of the phase measurement setup. L1: collimation lens; NBS: nonpolarized beam splitter; P: Polarizer; M: Mirror. The maximum value of the function \(\mathrm{sinc}(x)\) is at \(x=0\)[25].