# Sixty-Nine-Element Voice Coil Deformable Mirror for Visible Light Communication
## 4 Discussion
### Wavefront Fitting Precision  
The fitting precision of VCDM is directly related to its phase modulation ability. The influence function of VCDM could be expressed as [25]:  
\[Z(r)=k\times\exp[\ln(\omega)\times(\frac{r}{d_{0}})^{\alpha}], \tag{8}\]  
where \(Z(r)\) is the thin mirror deformation at a distance of \(r,k\) is the deformation of the mirror at the position of the central actuator, \(\omega\) is the coupling coefficient that is the influence of the central actuator on neighboring actuators, the actuator spacing \(d_{0}\) is 2.5 mm, and \(\alpha\) is Gaussian index.  
The coupling coefficient \(\omega\), a key indicator of the DM, is mainly influenced by the thickness of the mirror and the stiffness of the actuator. There is an optimum range of  
Figure 8: Time response of the VCA.  
Figure 9: The first resonance frequency of the DM.  
coupling coefficients for a given DM, too large or too small will affect the performance of the system [26]. As shown in Figure 10, the coupling coefficient increases with mirror thickness. In addition, the larger the stiffness is, the smaller the coupling coefficient is.  
The wavefront fitting errors as a function of the coupling coefficient are shown in Figure 11 for four Zernike modes Z1, Z3, Z6, and Z10. The results indicate that with the increase of the coupling coefficient, errors decrease rapidly at first and then increase slowly. The optimal range of coupling coefficients for the compact VCDM is 23-28%, which is much smaller than that of a standard VCDM with a value larger than 60%. Generally, a relatively low coupling coefficient is preferable for high wavefront correction precision. To optimize the VCDM that meets the requirements further, we built four different VCDM models with 69 actuators as listed in Table 4. Their influence functions are shown in Figure 12.  
The MATLAB codes were programmed to complete the wavefront fitting and residual calculation according to the influence functions. The first 14 terms of the Zernike aberrations are fitted with four VCDM listed in Table 4. In the fitting experiment, four DMs are actuated to generate Zernike shapes with a PV value of 8 \(\upmu\)m. Figure 13 shows the fitted wavefront and residuals of P3 for the 4 th, 7 th, 10 th, and 13 th Zernike modes. The results indicate that the fitting effect of P3 for some typical Zernike aberrations is less than 0.016\(\lambda\). The results as shown in Figure 14 demonstrate the maximum phase stroke of P3. The tip or tilt stroke of P3 is up to 50 \(\upmu\)m and the defocus stroke is about 30 \(\upmu\)m. The experiment results are shown in Figure 15a. All four VCDMs listed in Table 4 fit the first 14 Zernike aberrations well, with a maximum RMS value of only 53 nm. P3 as the best VCDM is compared with DM69-25 from ALPAO S.A.S. The results in Figure 15b indicate that P3 has a better correction performance than the DM69-25, especially for Zernike aberrations that contain a defocus component. The fitting residuals of P3 for the first 10 Zernike modes with a PV value of 8 \(\upmu\)m are all below 10 nm (RMS). Compared with the 121-element VCDM designed by [PERSON], the wavefront fitting precision of P3 has increased by 13% [27].  
Figure 12: Influence functions of four patterns.  
Figure 13: Wavefront Fitting and error of some typical Zernike modes. (**a**) The 4 th Zernike mode Z4. (**b**) The 7 th Zernike mode Z7. (**c**) The 10 th Zernike mode Z10. (**d**) The 13 th Zernike mode Z13.