We propose a novel deformable mirror (DM) for adaptive optics in high power laser applications. The mirror is made of a Silicon carbide (SiC) faceplate, and cooling channels are embedded monolithically inside the faceplate with the chemical vapor desposition (CVD) method. The faceplate is 200 mm in diameter and 3 mm in thickness, and is actuated by 137 stack-type piezoelectric transducers arranged in a square grid. We also propose a new actuator influence function optimized for modelling our DM, which has a relatively stiffer faceplate and a higher coupling ratio compared with other DMs having thin faceplates. The cooling capability and optical performance of the DM are verified by simulations and actual experiments with a heat source. The DM is proved to operate at 1 kHz without the coolant flow and 100 Hz with the coolant flow, and the residual errors after compensation are less than 30 nm rms (root-mean-square). This paper presents the design, fabrication, and optical performance of the CVD SiC DM.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Since the first adaptive optics (AO) proposed by H. W. Babcock in 1953 , a lot of deformable mirrors (DM) have been developed. Most DMs compensate for the atmospheric turbulence through the optical path of imaging systems, and they have thin faceplates made of low expansion glass materials like fused silica or Zerodur . Nowadays, DMs are used in high power laser (HPL) applications such as free-space communication, satellite ranging, fusion energy, and defense. The faceplate for HPL should dissipate the heat upon radiation effectively to minimize the wavefront errors due to thermal distortion and to increase the lasing time. Szetela and Chalfant showed that the temperature rise of a mirror caused by high energy absorption could result in severe distortions and even in the destruction of the faceplate . Ealey and Wellman reported that water-cooling was verified as an effective method to reduce the thermal deformation and it has been widely utilized for the whole reflecting mirror . So far, many studies have reported using a glass faceplate fixed on a metallic substrate for active cooling [5, 6], and the thermal coupling between the glass faceplate and the cooling medium is crucial.
It is also important to choose a suitable material of the faceplate for HPL. Table 1 shows the figure of merit of the faceplate materials often used for DMs . Thermal stability κ/α is represented by combining thermal conductivity κ and thermal expansion α. With high thermal stability, the wavefront distortion due to temperature gradients and thermal expansion can be reduced . Thermal diffusivity is defined as κ/(ρCp), where ρ is density and Cp is specific heat capacity. With high thermal diffusivity, the thermal relaxation time will be shortened under modulated heat sources. Another figure of merit is specific stiffness E/ρ represented by elastic modulus E and density. With a high specific stiffness material, cooling channels can be integrated monolithically inside the faceplate and the optical stability can be preserved even when the coolant flows. Also the mirror surface can be polished without the print-through effect which shows the internal pattern of a substrate. Based on the figures of merit in Table 1, CVD SiC can be a good candidate for the DM faceplate in HPL.
SiC has long been recognized as an attractive optical material due to its superior optomechanical properties including extremely high specific stiffness, thermal conductivity, and little distortion by thermal energy or pressure, when compared with conventional optical materials. There are several methods for making SiC faceplates. Reaction bonding (RB) and chemical vapor deposition (CVD) processes are most often used among several methods. RB SiC is formed by casting a slurry of SiC grains in a sacrificial mold, baking the casting to burn off the mold material, fusing the grains together, and finally infiltrating the voids with molten silicon to form a solid structure that is 70~85% SiC. This produces a solid faceplate, but the surface roughness is no better than 20 Å . Heterogeneous layers are sometimes added for better optical surfaces, but this incurs optical instability due to thermal expansion mismatch. On the other hand, CVD process consists of depositing gaseous chemicals on a graphite mandrel that is subsequently leached away and often used for mirror blanks. The deposition process is relatively slow but leads to an extremely pure SiC, which can be ground and polished with diamond grits to a surface roughness of less than 5 Å. Therefore, the CVD form is more appropriate for making a mirror in HPL applications in terms of surface roughness and optomechanical properties.
We already proposed to use the SiC as a DM's faceplate material for the first time and demonstrated its possibility [9, 10]. The faceplate, of which diameter is 100 mm, was actuated by 37 piezoelectric transducers and no active cooling was integrated. In this paper, however, we propose a novel CVD SiC DM with the active cooling capability. Different from the previous approaches [3–6], where the faceplate is bonded with the metallic cooling substrate, we embedded water-cooling channels monolithically inside the faceplate by the CVD method. Therefore, the heat transfer from the mirror to the coolant is maximized without the thermal resistance of a heterogeneous coupling medium. The faceplate is 200 mm in diameter and 3 mm in thickness, and it is actuated by 137 stack-type piezoelectric transducers arranged in a square grid. We also propose a new actuator influence function (IF) optimized for modelling our DM which has a relatively stiffer faceplate and a higher coupling ratio compared with other DMs having thin faceplates. The cooling capability and optical performance of the DM are verified by simulations and actual experiments with a heat source. Section 2 explains the design and fabrication of the CVD SiC DM, and section 3 derives a new actuator IF and compares the results from simulations and experiments. Section 4 shows the optical performance of the DM using our in-house adaptive optics system, and section 5 verifies the active cooling capability. Section 6 concludes this paper.
2. Design and fabrication
2.1 Faceplate with monolithic cooling channels
Figure 1 shows the design of the proposed SiC faceplate for deformable mirrors in HPL applications. Monolithic cooling channels are embedded in the faceplate which is 200 mm in diameter and 3 mm in thickness. There are 4 inlets and 4 outlets as indicated by arrows, and each port is 8 mm in width and 1 mm in thickness. They are arranged for intensive cooling at the mirror's center considering the Gaussian irradiance from HPLs. The size and position of the channels are optimized for effective cooling from the results of computational fluid dynamics (CFD). Actuator positions, which are marked as small circles in Fig. 1, are also considered for the channel design to reduce the variation of the actuator IFs caused by the internal shape of the faceplate.
In the CFD, we assumed that the faceplate is irradiated by a Gaussian beam laser. The intensity profile is expressed as11]. Total power is 10 kW, and the beam waist is 60 mm. Only 1% of the HPL output is assumed to be absorbed for 99% reflective coating. Figure 2 shows the CFD simulation results in a steady state when the faceplate is placed in a room temperature. Figure 2(a) is the water coolant temperature inside the channel when the flowrate is 600 ml/min. The surface temperature of the faceplate is shown in Fig. 2(b). Even though the temperature variation is less than 5 °C, this induces the surface displacement of 4.6 nm rms as shown in Fig. 2(c). The Gaussian peak at the center can be easily compensated by the DM actuators during AO operation. Even though the print-through features near the cooling channels are inevitable, the amount is small considering the application requirements and they can be ignored. With the proposed cooling channels, we can reduce the surface distortion of the faceplate under thermal loads and increase the usable stroke of the actuators.
Monolithic cooling channels are made by combining two CVD SiC faceplates. First, we make two CVD SiC blanks having 2 mm thickness which is oversized for later machining. Then we use electrical discharge machining (EDM) to make grooves on a single side of each blank . After that, two blanks are combined by thermal fusion to form a monolithic faceplate with cooling channels. Diamond grinding and polishing processes are required to make the optical surface on the faceplate. Figure 3(a) is the external view of the CVD SiC faceplate after diamond grinding. Figure 3(b) is the ultrasonic scan image showing the internal shape of the cooling channels, and no void or blockage is observed.
The DM is made by assembling the faceplate with actuators on a baseplate as shown in Fig. 4. The actuators are arranged in a 13 × 13 grid except for 8 nodes at each corner of the square grid. The actuator spacing is 15 mm in the x- and y- directions. The maximum actuator drive voltage is 75 V, and the corresponding displacement is 7.5 μm under unloaded condition. The baseplate is made of invar to minimize the thermal expansion mismatch with the CVD SiC faceplate, and it has indents for actuator positioning. We used an epoxy adhesive to fix the faceplate and the actuators. Figure 4(c) shows the assembled CVD SiC DM on a 5-axis stage mount.
3. Actuator influence function
3.1 Analytical model
For DMs using piezoelectric actuators, the deformation of the faceplate due to the actuator motion is assumed to have a linear relation with the input voltage, and this relation forms a influence function (IF). Wavefront compensation is made by the superposition of the IFs of all actuators, and prior knowledge of the IFs is crucial for the fast convergence of an AO system. The shape of the IF for each actuator varies depending on the position and its neighboring actuators. But the general topology of the IF and the actuator coupling ratio are mostly dependent on the actuator spacing and material properties of the faceplate. The proposed CVD SiC DM has a faceplate with a relatively high stiffness, and the IF is also quite different from other conventional DMs’. Analytical IF to characterize the proposed DM is necessary for the parametric design and analysis.
Gaussian IF, which has been employed in early studies [13–15], is expressed as16], modified Gaussian IF (MGIF) , and Bessel Fourier IF  were introduced to fit the actual IFs more closely. The MGIF has a cosine function in the argument of a Gaussian function, and it is suitable for modelling IFs having azimuthal variations due to the square arrangement of actuators. The MGIF is expressed as
The proposed CVD SiC DM has a high coupling ratio, because the faceplate is thick to embed monolithic cooling channels and the material itself is stiff. Previous analytical IFs are found to be inaccurate for modelling our DM, and a new IF, which is more general to encompass broad ranges of coupling ratio, is required. Therefore, we propose a novel IF called optimized Gaussian IF (OGIF) in this paper. The OGIF is expressed as
Figure 5 shows the comparison of an actual IF obtained from the CVD SiC DM and its fitted analytical IF models. We used a Fizeau interferometer having a 300-mm transmission flat to measure the actual IF in Fig. 5(a), and optimized the parameters of each IF model by minimizing the fit errors. The Gaussian IF in Fig. 5(b) could not express the square shape near the actuator, and the MGIF in Fig. 5(c) could not follow the symmetric Gaussian decay at the boundary. However, the proposed OGIF in Fig. 5(d) could model the actual IF very closely by balancing between radial and azimuthal variations. Residual errors of the IF models are shown in Fig. 6 after fitting with the actual IF. For comparison, all IFs are normalized by their maximum values. The residual error of the Gaussian IF is 0.271 PV and 0.051 rms, and the residual error of the MGIF is 0.280 PV and 0.036 rms. But the residual error of the OGIF is 0.028 PV, 0.004 rms, and they are one order of magnitude smaller than the previous IF models. The actuator coupling ratio of the CVD SiC DM is relatively high, which is above 0.5-0.7 depending on the actuator position, and the proposed OGIF is proved to represent the IF successfully.
3.2 Simulation and experiment
IFs are different among the actuators positioned at the center and the boundaries, and they can be predicted easily by using a finite element analysis (FEA). We used CATIA as the FEA tool, and modeled the CVD SiC DM with parabolic tetrahedral meshes of 4,953,736 elements and 6,419,947 nodes for a realistic analysis after convergence tests . In the simulation, we applied a 7.5-μm displacement load at each actuator, and acquired the raw displacement of the faceplate. The IF was then obtained after processing with Matlab to interpolate the raw displacement data into a predefined grid. The maximum displacement varied from 5.32 μm for the actuator at the center to 8.02 μm for the actuators at the edges. Also the IFs at the edges show asymmetric shapes due to the unbalanced restraints from neighboring actuators. Figure 7 compares the IFs simulated by using the FEA and the IFs measured with the interferometer for all actuators, and each IF is downsized and placed at the corresponding actuator location. The maximum displacement difference between the simulated IFs and the measured IFs are only 0.3%. But the simulated IF is steeper than the measured IF, and it makes the actuator coupling ratio 10% higher for the measured IF. The difference can be ascribed to the statistical deviation of the adhesive layer’s thickness and the mechanical tolerance of the actuator assembly. Therefore we used the measured IFs in the actual experiment for optical performance validation.
4. Optical performance validation
4.1 Wavefront reconstruction
We examined the optical performance of the CVD SiC DM by reconstructing various Zernike modes  which are useful to compensate for the atmospheric turbulence in AO systems. Each IF is superimposed after having been multiplied by the respective coefficient to form a specific shape of the faceplate. The surface shape of the CVD SiC DM can be described asEq. (5) can be rewritten asFigures 8(a) and 8(b) show a block diagram of the measurement setup and the actual implementation, respectively. We used a Fizeau interferometer having a 300-mm transmission flat from Zygo to measure the shape of the faceplate and obtain the influence matrix.
The surface of the faceplate could change after adhesive curing in the DM assembly, and the initial surface error can be removed by offsetting all actuators to their predefined positions before the AO operation. We found the initial offset values using Eq. (7). Figure 9(a) shows the initial surface error of the faceplate before flattening and Fig. 9(b) shows the surface error of the faceplate after flattening. The rms value reduced from 496 nm to 7 nm.
We also generated various Zernike modes to examine the performance of the CVD SiC DM. We could make 21 Zernike modes except for the piston and 2 tilt modes which could be compensated by adding another tip-tilt mirror in the AO system. The Zernike modes measured with the interferometer and their residual errors with respect to theoretical ones are visualized in Fig. 10. The residual surface error of the reconstructed Zernike modes are plotted comparatively in Fig. 11. The high-order terms show increasing residual errors, because the number of actuators is insufficient to generate the sharp edges of high-order terms. However, the residual errors are lower than 30 nm rms, which is acceptable considering the visible wavelength of the AO system. Compared with the previous prototype [9, 10], the CVD SiC DM developed in this paper could generate more high-order modes with reduced residual errors.
4.2 Wavefront compensation
In this section, we examine the optical performance of the CVD SiC DM by compensating the wavefront distortion in the closed-loop AO system. The AO system consists of a light source, an aberration generator, a wavefront sensor, an off-axis mirror, a CVD SiC DM, and a control computer. Figure 12 shows the setup of the closed-loop AO system. The aberration generator deforms the wavefront from the light source. The wavefront sensor measures the distortion of the wavefront, and the control computer calculates the signal to compensate for the distortion of the wavefront. The CVD SiC DM receives the signal and compensates for the distorted wavefront by deforming the faceplate. A Multi DMTM of Boston Micromachines Corporation is used for the aberration generator, and the in-house Shack-Hartmann sensor is used as the wavefront sensor. The Shack-Hartmann sensor uses a high-speed camera having a frame rate of 2 kHz for high-speed compensation, and it has a 20 × 20 microlens array. In our experiments, we switched between Modal  and Zonal  method to reconstruct the wavefront from the slope data of the wavefront sensor. The control computer is made up of 4 FPGAs (Field-programmable gate array) from NI, and it operates at the highest speed of 1 kHz for generating the input signal from the output of the wavefront sensor. Each of 4 FPGAs calculates the slope from the sensor image, reconstructs the wavefront from the slopes, computes the signal using the measured IF to compensate for the distorted wavefront, and converts the digital signal into analog signal to control the CVD SiC DM. The frequency response of the CVD SiC DM itself was measured by a capacitive sensors under a sinusoidal sweep from 20 Hz to 2 kHz as in Ref . The 3 dB bandwidth was measured to be 1.3 kHz.
In the wavefront compensation experiments, the aberration generator simulated the atmospheric turbulence having a Fried parameter of 15 cm, and distorted the wavefront with an operating frequency from 100 Hz to 1 kHz. Figure 13 shows the results of the wavefront compensation experiments. The wavefront distorted by the aberration generator has the rms values of 65 nm ~100 nm. After starting the wavefront compensation, the rms value reduced below 30 nm even for the high frequency wavefront distortion. Therefore, we conclude the CVD SiC DM could compensate for the distorted wavefront in the closed-loop AO system with the maximum operating frequency of 1 kHz.
5. Cooling capability validation
5.1 Cooling simulation
We already examined the cooling capability of the monolithic cooling channels of the CVD SiC DM in the previous research . The faceplate was 100 mm in diameter and heat tapes were attached on the backside of the faceplate instead of piezoelectric actuators. We observed the static deformation without actuators and the transient temperature variation of the faceplate. In this paper, however, we use an infrared heat source in front of the faceplate and examine the dynamic cooling capability of the DM using the AO system. This section provides the CFD simulation results of the actual configuration. We used the same setup as in Fig. 12, and only a radiative heat source was added and skew-aligned in front of the faceplate. As shown in Fig. 14, the heat source is 250 mm apart from the faceplate, and the angle is 45° between their surface normals. The intensity of the heat source is 8.54 kW/m2, and it is calculated by using Stefan-Boltzman’s law , where is 5.67 × 10−8 W/m2K4, and is measured as 623 K. The irradiance on the faceplate can be expressed by Lambert’s law  as
The steady state simulation results are shown in Fig. 15. Without the cooling, the maximum temperature of the faceplate is 37.9 °C, and the surface error of the faceplate is 97.5 nm PV, 18.9 nm rms. With the cooling, the maximum temperature of the faceplate is 21.63 °C, and the surface error of the faceplate is 7.4 nm PV, 1.6 nm rms. Even though there are temperature variations on the faceplate due to the skew alignment of the heat source, the active cooling could dramatically reduce the temperature rise and the surface distortion.
5.2 Wavefront compensation with the coolant flow
In this section, we verify the optomechanical performance of the proposed CVD SiC DM experimentally. The faceplate of the DM is irradiated by an infrared heat source and it compensates for the wavefront distortion. We examine the optical stability of the wavefront compensation even when the coolant flows through the monolithic cooling channels. Figure 16 shows a block diagram of the AO system and a partial view of the actual setup. The temperature controller of the cooling system keeps the coolant temperature at 20 °C, and the water pump supplies the coolant without pulsation.
In the experiment, we first observed the steady-state temperature of the faceplate without cooling as shown in Fig. 17(a). The temperature increased up to 45.2 °C at the center. We then circulated the coolant with a flowrate of 600 ml/min, and operated the closed-loop AO system at 100 Hz that is the requirement specification of the project. The steady-state temperature of the faceplate dropped to 21.6 °C as shown in Fig. 17(b), which is similar to the simulation results in Fig. 15(b) and verifies the calculation. We could not measure the surface deformation of the faceplate at those temperatures due to the convection current in front of the Fizeau interferometer, but the change could be observed in the Shack-Hartmann sensor of the AO system.
Next, we repeated the same experiment with 4 thermocouples attached on the faceplate to check the non-uniformity and the transient response of the faceplate's temperature as shown in Fig. 18. At the start of the cooling system, the temperature dropped most rapidly at the center with an exponential decay. The position P4, which is away from the cooling channels, showed a slow convergence, but the faceplate reached the uniform temperature within 30 sec.
Figure 19 shows the history of the wavefront rms in the closed-loop AO system when the DM is irradiated by the heat source. Once the cooling system is turned on, rms value drops exponentially and reaches the steady-state as the faceplate recovers from the thermal deformation. When the AO system was activated at 100 Hz, the wavefront rms dropped below 30 nm and kept the stability for hours.
We presented a new deformable mirror for adaptive optics in high power laser applications. We proposed to use the CVD SiC as a faceplate material and embedded water channels inside the faceplate monolithically, which is the first to our knowledge. We also proposed a new analytical influence function optimized for modelling the DM having a high coupling ratio. We verified the optomechanical performance of the proposed DM experimentally with our in-house adaptive optics system. The CVD SiC DM and the AO system could operate at 1 kHz compensating for the wavefront distortion equivalent to the Fried parameter of 15 cm. The faceplate having monolithic cooling channels was proved to be effective for active cooling without any hindrance to the optical stability of the system. The number of the actuators and the bandwidth of the DM speed can be compromised for specific applications using high power lasers. We expect the proposed DM can be used widely in free-space communication, satellite ranging, fusion energy, and defense areas.
National Research Council of Science & Technology (NST) grant (MSIP) (No. CAP-12-04-KRISS) by the Korea government
The authors would like to thank Dr. Sang-Won Hyun for supporting the cooling experimental setup and helpful discussions, and Dr. Ho-Jae Lee for developing the in-house Shack-Hartmann sensor and technical supports.
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