## 4 Implementation and Experimental Results
### Calibration Modeling  
The modeling procedure is examined on our testbed displayed in Figure 5, from which we gain the coefficient matrices of DM and FSM describing the mapping relationship between spots offsets and controlling voltages. On our testbed, a 800 nm wavelength laser beam with a diameter of 15 mm is used as the ideal incident light. As for the AO system, the internal topology is shown in Figure 1. Within it, the wavefront sensor is a charge coupled device (CCD) camera with a micro lens array covered in the front of it which has a focus of 18.8 mm. Moreover, the effective area of the wavefront sensor contains \(1088\times 1088\) pixels whose size is 5.5 um, while the size of a single aperture of the micro lens array is 300 um which means the wavefront sensor can capture spots in an around 19 \(\times\) 19 rectangle area. However, in our experiment environment, the detected spots on the wavefront sensor cover a circular area with a diameter of 17 apertures. Additionally, the PDM in front of the light entrance is not drived on the current stage and it is set for a future experiment. On the left of the AO system, devices are deployed for observing the emitting light.  
For DM, the calibration voltages in one period are shown in Figure 6. During the process of modeling the DM, we repeatedly configure the calibration voltages on every electrode for 10 periods. Each calibration voltage produces 20 data samples (two data per period). After receiving the 20 data samples, we use the least square method to solve pre-term coefficients for the current affecting electrode. The calibration voltages take 200 as the step and 2048 as center, traversing the voltage range of \(\pm\)1600. According to Equation (3), a digital voltage ranging from 448 to 3648 can be linearly interpreted to an analog voltage ranging from 0.273 vol  
Figure 5: photograph of the setup.  
Figure 7 displays the favorable fitting result of the first electrode for the x coordinate of the 111 st spot. It indicates that a maximum modeling error of 0.05 pixels exists in the system which is kept under an acceptable threshold.  
Different from the modeling structure of the DM, the structure of FSM contains coupling terms depicting the collaborative effects of electrodes on spots offsets. This leads to the requirement that when modeling, the two electrodes should be modeled together. Therefore, the horizontal electrode is sequentially configured with a period of calibration voltages. Whenever the horizontal electrode is configured with a calibration voltage, the vertical electrode is configured with a period of calibration voltages in turn. Furthermore, due to the large maximum stroke of FSM, the used range of it is limited to \(\pm\)400 in digital voltages, as displayed in Figure 8.  
Figure 6: (**a**) Calibration voltages for DM constructing models in one period. (**b**) Normalized calibration voltages for DM constructing models in one period.  
Figure 7: The fitting curve of the first electrode for the x coordinate of the 111 st spot.  
Figure 9 shows the fitting result of the fitting condition of the model for the spot offsets of the first spot. As is demonstrated in the figure, a maximum modeling error of 0.9 pixels shows up which is still within the acceptable range.