In the design of the deformable mirror, the main characteristic of the atmosphere is the Fried coherence length.⁷ The Fried coherence length is calculated by the wave length, the angle with zenith, the height (altitude) and the turbulence contributions. The deformation displacement and the actuator stroke are derived from the Fried coherence length and the telescope diameter as shown in Fig. 1.

Fig. 1 
Requirements analysis of the actuator stroke considering the telescope and the adaptive optics

The Fried coherence length is the range in which the same atmospheric effect is exhibited as 0.1219 m by Hardy’s Eq. (1) The main factors for deriving above result are wave length and height. The wave length range of 550 nm - 850 nm, which is the visible light region, should be considered, and the large telescope should be located at a high altitude in consideration of light pollution and viewing angle. As a result, the Fried coherence length was calculated assuming that the wavelength was 500 nm and the height was 1000 m, and other parameters are shown in Table 1.

In the ideal case, the deformation displacement is the same as the wavefront distortion derived from Fried coherence length and the telescope diameter as shown in eq. (2) Since the wavefront distortion is a deviation, for the adaptive optics system the RMS and PTV values are often related via a scaling factor 5.⁸ Therefore, it was analyzed that the deformation displacement should be more than ±1.64 μm.

In this study, we have developed a single continuous mirror, which has a reflecting mirror and the actuators are under the mirror and the actuators pushes or pulls the mirror as shown in Fig. 2. The thickness of the reflecting mirror was set to 1 mm in consideration of the beam transmission system. In addition, the base has a sufficient rigidity to minimize the effect of the actuator being acted upon by the reaction. The actuator is bonded to the reflecting mirror using an adhesive and is also bonded to the base. For bonding, 353NDPK (Thorlabs) and TS10 (Thorlabs) are used as an adhesive, and a guide hole for fixing the position is provided on the base, so that the actuator can be fixed in place and adhered.

Fig. 2 
Schematic of the deformable mirror design

The reflecting mirror must have optically and mechanically stable properties. It should also be possible to polish the surface and to coat the required reflection coefficient with the spun material. Table 2 summarizes the materials that can be used as a material for reflecting mirrors according to theses design requirements.

The reflecting mirror must be displaced with respect to the actuation of the actuator. In addition, the deformation displacements must be deformable beyond the total required displacement analyzed in section 2.1. As a result, the reflecting mirror material is selected as the ULE which is the smallest in Young’s modulus and suitable for the reflecting mirror.

In order to confirm whether the reflecting mirror can be deformed more than the standard which already derived in section 2.1, the deformable mirror to be developed in modeled in a simplified form composed of the reflecting mirror and the actuator, and the finite element analysis is performed. The model shape including the design element is shown in Fig. 3, and the elastic modulus of the actuator is 50 GPa assuming that the piezoelectric ceramic actuator will be used.

Fig. 3 
Finite element analysis model

In the finite element analysis, 9 actuators were placed under the reflecting mirror. Among them, only the central actuator was driven. The bottom parts of the actuators were analyzed under fixed conditions and the damping of the ceramic can be neglected because it was very low. Also, the part fixed by bonding was assumed to be 1 mm in diameter. In order to confirm whether deformation exceeding the standard occurs, deformation of the reflecting mirror according to the generated force of the actuator is checked and summarized as shown in Table 3.

Based on the finite element analysis, it was confirmed that the actuator should generate a force of 240 N or more in order to satisfy the deformation standard of the reflecting mirror. Accordingly, the actuator was determined as PK4JMP1 (Thorlabs) by comparing the specifications of the actuators that can be purchased. This actuator has a maximum displacement of 8.8 μm at 150 V and a blocking force of 360 N at 150 V. The size of this actuator is 3.4 × 4.8 × 10.0 mm.

The deformable mirror designed by using the previously determined reflecting mirror and actuator is shown in Fig. 4. There is a spacer for fixing the reflecting mirror, and the reflecting mirror is pressed with a gasket. The base is designed to have sufficient rigidity, and actuators are bonded to the reflecting mirror and base. Each actuator is arranged at 5 mm distance and all the actuators are located inside the barrel. The actuator receives driving signals from the outside, thus each cable of the actuator is connected to an interface board inside the housing to receive an external signal.

Fig. 4 
Configuration of the deformable mirror

The deformable mirror consists of control software, actuator controller and deformable mirror prototype as shown in Fig. 5. The control software was developed in this study so that the driving signal can be input to the deformable mirror and the amount of change can be confirmed immediately. The actuator controller, purchased from Thorlabs, supplies voltage up to 150 V at a speed of 50 - 60 Hz to each actuator through the interface board so that the deformable mirror can be deformed. With this configuration, the surface of the deformable mirror can be deformed, and the amount of the deformation can be confirmed by using an interferometer.

Fig. 5 
Equipment setup

The surface phase data values of the reflecting mirror plane derived from the finite element analysis are shown in Fig. 6(a). when 1 μm deformation is generated on the surface of the deformable mirror, the deformation around the center of the actuator is expressed as shown in Fig 6(a).

Fig. 6 
Analysis of the influence function from finite element analysis

The surface phase data values derived from the finite element analysis can be represented by the cross-section profiles as shown in Fig. 6(b). According to the cross-section profiles, the coupling coefficient in each direction can be calculated. The deformable mirror has a coupling coefficient of 13.6% in the x-direction, 16.2% in the y-direction and 21.0% in the diagonal direction. The coupling coefficient can be changed the Gaussian model by linear fitting method as eq. (3).

In eq. (3), ω represents the coupling coefficient, d represents the actuator spacing, and α represents the Gaussian coefficient. The result of the analysis of the influence function is that each coefficient of eq. (3) is approximated by ω = 0.17, α = 2.

A Zernike polynomial is a set of polynomials defined on a unit circle. Each polynomial is expressed as the product of a radial polynomial and a circumferential trigonometric function. The radial polynomial represents how the representation changes from the center of the circular aperture to the edge. It is normalized as a special case of the Jacobi polynomial and is orthogonal on a unit circle. In this study, Zernike polynomials using the orthogonalization Noll’s method were used.^11,12

The performance analysis of the Zernike polynomial from the 4th to 11th terms of the polynomial was conducted using the design parameters of the deformable mirror prototype. The result is the same as the Zernike column in Fig. 7. In addition, the performance analysis is performed using the Gaussian model of the influence function derived in section 4.1. The result is the same as the Gaussian column in Fig. 7. When we compare the two indicators, we ca n see that they are almost similar from the 4th to 11th terms. This means that the correction performance is close to 1.

Fig. 7 
Three-dimensional shape of Zernike polynomial and Gaussian influence function

Using the ZYGO interferometer, the deformation of the deformable mirror prototype was measure as shown in Fig. 8. After the deformation, it is confirmed that the shape is deformed based on the center of the deformable mirror. Before deformation, the surface was 0.859 λ (P-V), 0.207 λ (RMS), but after deformation it was confirmed to be 2.954 λ (P-V), 0.591 λ (RMS).

Fig. 8 
Interferometer images before/after deformation of the deformable mirror