# Wide-Aperture Bimorph Deformable Mirror for Beam Focusing in 4.2 PW Ti:Sa Laser
## 3 Results  
### Investigations of the Deformable Mirror  
DM study was carried out at the diagnostic setup based on Shack-Hartmann WFS [23]. The initial surface error excluding the overall surface curvature was about 14 \(\upmu\)m after mirror assembly. To compensate for the curvature, the electric voltage of \(-34\) V was applied to all 127 electrodes. After mechanical alignment with 16 screws, a flatness (P-V) equal to 1.03 \(\upmu\)m (RMS = 0.17 \(\upmu\)m) was achieved on a diameter of 300 mm. After correction in the closed-loop system with WFS feedback, the surface aberrations were decreased to P-V = 0.07 \(\upmu\)m and RMS = 0.01 \(\upmu\)m. Such a surface quality corresponds to a Strehl number of 0.99 (for the wavelength \(\lambda\) = 837 nm). The Fizeau interferograms of the DM surface before and after correction are shown in Figure 4a,b, where the flatness values (P-V and RMS) and residual aberrations in terms of orthogonal Zernike polynomials are presented on the right side of the pictures. The local deformation of the surface was about 1 \(\upmu\)m (corresponding to the change in the wavefront of 2 \(\upmu\)m) when the electric voltage of +150 V was applied to a single electrode. For the full range of the control voltages (from \(-300\) to +600 V), deformation introduced by one electrode was expected to be at least 6 \(\upmu\)m.  
To measure the response functions, an electrical voltage of 100 V was applied sequentially to each electrode of the DM, and the change in the surface shape relative to its state in the absence of applied voltage was measured. In this case, the expansion coefficients were calculated in terms of the orthogonal Zernike polynomials, which can be used to construct a phase surface and represent it in the graphical form. The response functions of the bimorph DM are modal; when a voltage is applied to one of the electrodes, the surface deformation occurs not only in the area of the electrode location, but throughout the entire  
Figure 4: The DM surface shape in terms of interferogram and expansion in Zernike polynomials (#—ordinal number of the Zernike polynomial): (**a**) the initial surface after manual correction without taking into account the curvature, P-V = 1.03 \(\upmu\)m, RMS = 0.17 \(\upmu\)m; (**b**) surface after aberrations correction in the closed-loop, P-V = 0.07 \(\upmu\)m, RMS = 0.01 \(\upmu\)m.  
Figure 3: Photos of the front and back views of DM.  
mirror aperture. Figure 5 shows examples of the response functions of some electrodes of a 320-mm bimorph DM presented in the form of the interferograms.  
Investigations of the control frequency range of the corrector were carried out using a sound frequency generator and oscilloscope [24]. In this case, the induced piezoelectric signal was measured at one of the electrodes when a sine voltage was applied to the neighboring one. The phase shift between these signals was also studied. Figure 6 shows the change of the induced signal amplitude, as well as the phase shift versus the excitation frequency. The first resonance frequency when the response signal achieved the maximum amplitude and the phase shift was equal to \(\pi/2\) was registered at 774 Hz. The presented graphs show the stability of the amplitude and phase for frequencies in the range from 0 to 400 Hz. Thus, this mirror provides the WF control at the frequencies up to 400 Hz and could even be used to correct for slowly changing phase fluctuations of the atmosphere in the pavilion where the whole laser is installed.