## 3 Experimental Study
### Imaging the Object  
After the \(T_{\mathrm{O\to}E}(x,y;\xi,\eta)\) is measured, the system is ready to image the target object. In order to test the imaging quality precisely, we don't capture the image of a natural real object directly. Instead, a binary complex amplitude object which consists of the characters part and the background part, is constructed by a phase-only-modulation spatial light modulator (SLM: Meadowlark Optics, P1920-0635-HDMI, 1920 \(\times\) 1152 pixels) using two phase masks \(O^{(1)}_{\mathrm{SLM}}(\xi,\eta)\) and \(O^{(2)}_{\mathrm{SLM}}(\xi,\eta)\)[15; 16]. The mask \(O^{(1)}_{\mathrm{SLM}}(\xi,\eta)\) is a plane phase whose amplitude \(=1\) and phase \(=\uppi\). The mask \(O^{(2)}_{\mathrm{SLM}}(\xi,\eta)\) is obtained by flipping the phase of \(O^{(1)}_{\mathrm{SLM}}(\xi,\eta)\) from \(\pi\) to \(2\pi/3\) for the characters part, and from \(\pi\) to \(\cdot\pi/3\) for the background part. As a result, a virtual 3D complex amplitude object \(O^{(2)}_{\mathrm{SLM}}(\xi,\eta)-O^{(1)}_{\mathrm{SLM}}(\xi,\eta)\)  
Figure 3: Sketch map of the measured round-trip TM \(T_{\mathrm{O\to}E}(x,y;\xi,\eta)\). Amplitude (**a**) and corresponding phase (**b**), respectively. Color bar: phase in radian. Figure 2: Experimental setup. Scattering medium (a 220-grit and a 1500-grit Thorlabs Optics ground-glass diffuser are stacked together to enhance the scattering ability), DMD: digital micromirror devices (ViALUX, V-7001 VIS, 1024 \(\times\) 768 pixels), LCVR: liquid crystal variable retarder (Meadowlark Optics, D5020), CCD ([PERSON], C13440-20 CU, 2048 \(\times\) 2048 segments (pixels), the central 400 \(\times\) 300 segments are used for imaging), L#: lens, P#: polarizer, BS#: beam splitter, A: attenuation film, M: mirror. can be constructed, which corresponds to the characters part with amplitude \(=1\) and phase \(=\pi/3\) [as shown in Figure 4a], as well as the background part with amplitude \(=\sqrt{3}\) and phase \(=-\pi/6\) [as shown in Figure 4b], respectively. Next, we will perform the imaging, and the SLM is located at the same place after removing the DMD. To match the size of the DMD segment (219 \(\upmu\)m \(\times\) 219 \(\upmu\)m) used for measuring the round-trip TM of the scattering medium, the SLM is divided into 80 \(\times\) 48 segments (the size of each segment 220 \(\upmu\)m \(\times\) 220 \(\upmu\)m), and the central 18 \(\times\) 14 segments are used to construct the complex amplitude object \(O^{(2)}_{\text{SLM}}(\xi,\eta)-O^{(1)}_{\text{SLM}}(\xi,\eta)\) whose amplitude and phase are shown in Figure 5a1,a2, respectively. The corresponding output optical field \(E^{(2)}_{\text{CCD}}(x,y)-E^{(1)}_{\text{CCD}}(x,y)\) is shown in Figure 5b1,b2 for the amplitude and the phase, respectively, where \(E^{(1)}_{\text{CCD}}(x,y)\) and \(E^{(2)}_{\text{CCD}}(x,y)\) are the output optical field correspond to the \(O^{(1)}_{\text{SLM}}(\xi,\eta)\) and \(O^{(2)}_{\text{SLM}}(\xi,\eta)\), respectively. It can be seen from Figure 5b1,b2 that the information of the object \(O^{(2)}_{\text{SLM}}(\xi,\eta)-O^{(1)}_{\text{SLM}}(\xi,\eta)\) has indeed been seriously destroyed, and it is impossible to image the object directly. Figure 4: Complex amplitude object constructed by the SLM using two phase masks \(O^{(1)}_{\text{SLM}}(\xi,\eta)\) and \(O^{(2)}_{\text{SLM}}(\xi,\eta)\). (**a**) amplitude \(=1\) and phase \(=\pi/3\) for the characters part, (**b**) amplitude \(=\sqrt{3}\) and phase \(=-\pi/6\) for the background part. Figure 5: Imaging the complex amplitude object. (**a1,a2**) Amplitude and phase of the object \(O^{(2)}_{\text{SLM}}(\xi,\eta)-O^{(1)}_{\text{SLM}}(\xi,\eta)\), respectively. (**b1,b2**) Amplitude and phase of the corresponding output optical field \(E^{(2)}_{\text{CCD}}(x,y)-E^{(1)}_{\text{CCD}}(x,y)\), respectively. (**c1,c2**) Amplitude and phase of the recovered object, respectively. (**d1**) Section profiles corresponding to the lines in (**a1,c1**); (**d2**) section profiles corresponding to the lines in (**a2,c2**).