## 2 The Influence of the Driving Conditions on the Performance of a Phase-Only OASLM  
The phase-only OASLM is schematically depicted in Figure 1a. Considering the photoconductive characteristic and spectral absorption coefficient of Bi\({}_{12}\)SiO\({}_{20}\) (BSO) crystal [14], we use the light-emitting diode (LED) with a wavelength of 470 nm as the light source of the write light. An amplitude-modulating LCOS SLM (Rui Like Co., WXGA Active Matrix LCD, the pixel pitch is 20 \(\upmu\)m) has been used to control the collimated beam from the LED source. The bitmapped image is then projected onto the BSO layer through an imaging system. We can control the wave-front of the transmissive read light (with a wavelength of 1053 nm the photoconductive effect of the BSO crystal at 1053 nm is almost negligible) by setting an appropriate bitmap in the LCOS modulator. When the BSO layer (1 mm thickness, 20 \(\times\) 20 mm\({}^{2}\) effective area) is illuminated with the write light, and an electric field is applied across the optically addressable liquid-crystal light valve (OALCLV), the resistance of the BSO decreases and the voltage begins to act on the LC layer, so the LC director is redirected towards the applied field [15]. As the voltage increases, the director rotates further, which changes the extraordinary refractive index of the LC and results in a reduction in the phase retardance of the OASLM [16]. In this way, the voltage on the LC layer affects the phase controllability of the OASLM. Figure 1: (**a**) Working principle of the optically addressable spatial-light modulator (OASLM) and structure of the optically addressable liquid-crystal light valve (OALCLV) consisting of the glass substrate, 6-\(\upmu\)m-thick liquid crystal (LC), and BSO photoconductor, PI (polyimide, the alignment layer); a reflective liquid crystal on silicon (LCOS) type is used to control the write light, PBS refers to the polarized beam splitter. (**b**) Phase change of a 6-\(\upmu\)m-thick LC cell composed of two glass substrates. The LC has \(n_{e}=1.820\) and \(n_{o}=1.515\). The voltage response of the LC causes a phase change of the OALCLV, so first we studied the relationship between the phase change and the voltage of the LC cell without BSO. We oriented the test cell at 45\({}^{\circ}\) between crossed polarizers and applied the alternating voltage. The intensity value of the modulated 1053-nm laser was obtained using a CCD, and the average value of all pixels was taken as the effective value, which is expressed as follows [17]. \[T=\frac{I}{I_{max}}=\frac{1-cos\delta}{2} \tag{1}\]  
where \(T\) refers to the transmittance. Phase change is added to the laser beam by the OASLM according to the loaded grayscale images of the write light. The retardance \(\delta\) of the liquid crystal layer is expressed as follows:  
\[\delta=2\times sin^{-1}\Big{(}\sqrt{T}\Big{)} \tag{2}\]  
The phase change is deduced from the transmittance according to Equation (2). Because of the periodicity, \(\delta\) can only be within the range of \(0-\pi\), so in the data processing, the phase unwrapping [18] is performed to calculate the actual phase change. We measured the voltage response curve of an LC cell with 6-um-thick LC layer (HCCCH Co., Jiangsu, China, \(n_{e}=1.820\) and \(n_{o}=1.515\) at \(\lambda=589\) nm, with the same thickness as the LC layer in OALCLV) and two glass substrates. The voltage response shown in Figure 1b will be used as a reference in the phase change-voltage relationship of the OALCLV. According to this figure, we can acquire the threshold voltage of LC molecule rotation as \(V_{thr}=0.4\) V and the saturated voltage as \(V_{sat}=3\) V, and the peak-to-valley (PV) value of the phase change is approximately 1.1\(\lambda\). The OALCLV, the main component of the OASLM, can be modelled with an equivalent circuit in Figure 2a. The resistance and capacitance of the LC layer are \(R_{LC}\) and \(C_{LC}\) respectively. The dark BSO crystal resistance is \(R_{0}\).