Abstract

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

1. Introduction

Since the first adaptive optics (AO) proposed by H. W. Babcock in 1953 [1], 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 [2]. 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 [3]. 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 [4]. 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 [4]. 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 [7]. 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.

Tables Icon

Table 1. Faceplate material’s figure of merit

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 [8]. 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.

 

Fig. 1 Design of the CVD SiC faceplate with monolithic cooling channels. Small circles indicate the positions of 137 actuators. Blue and red arrows represent inlets and outlets respectively.

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In the CFD, we assumed that the faceplate is irradiated by a Gaussian beam laser. The intensity profile is expressed as

I(r)=I0e2r2/w02
, where the peak intensity I0 is 2P0/πw02 [11]. Total power P0 is 10 kW, and the beam waist w0 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.

 

Fig. 2 The CFD simulation results; (a) Steady state water temperature, (b) faceplate temperature with cooling, and (c) faceplate displacement due to the steady state temperature gradient.

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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 [12]. 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.

 

Fig. 3 A CVD SiC faceplate with monolithic cooling channels; (a) external view, (b) ultrasonic scan image showing the internal shape of the cooling channels.

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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.

 

Fig. 4 An assembled CVD SiC DM with 5-axis mount for the experiments.

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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 as

I(r)=exp[ln(ω)(rd)α]
, where ω is the actuator coupling ratio, d is the actuator spacing, and α is the Gaussian index. This equation models only a circularly symmetric IF, and it cannot account for the angular variation induced by the high coupling ratio from dense actuators arranged in a square grid. Later, double Gaussian IF [16], modified Gaussian IF (MGIF) [17], and Bessel Fourier IF [18] 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
I(r,θ)=exp[ln(ω){r[1+τcos(4θ)]d¯}α]+βexp[{(rd¯)γ}2]
, where d¯ is the mean actuator spacing, τ is the amplitude of period. The number in the cosine term refers to the square arrangement of actuators, β is the amplitude representing the depth pulled by the surrounding actuators, and γ is the width pulled by the surrounding actuators. The MGIF has two Gaussian terms. The first term means the azimuthal direction modification, and the second term means the radial direction modification. The azimuthal and radial direction modifications are used for expressing the influence of the surrounding actuators. The MGIF is useful for modelling actuators arranged in a square grid, but it is effective only for the DM having a relatively low coupling ratio.

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

I(r,θ)=(1β)exp[ln(ω){r[1+τcos(4(θπ4))]d¯}α]+βexp[ln(ω)(rd)2]
When compared with the MGIF, the OGIF has the azimuthal direction modification term of the MGIF, and a Gaussian IF of index 2. Two terms are multiplied by the amplitude β that is used as an optimization parameter to balance between the azimuthal and the radial modifications. High β values make the IF circularly symmetric, and low β values make the IF close to a square shape.

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.

 

Fig. 5 Comparison of an actual IF obtained from the CVD SiC DM and its fitted analytical models. (a) Actual IF measured with a 300-mm Fizeau interferometer for flatness tests, (b) the Gaussian IF, (c) the MGIF, and (d) the proposed OGIF

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Fig. 6 Residual error images (left) and their cross-sectional plots (right) of the analytical IF models after fitting with the actual IF. (a) The Gaussian IF, (b) the MGIF, and (c) the OGIF

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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 [19]. 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.

 

Fig. 7 A comparison of (a) the IFs simulated by using FEA and (b) the IFs measured by a Zygo interferometer. All IFs are downsized and placed at the corresponding actuator locations. IFs in the second quadrant are displayed due to the symmetry.

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4. Optical performance validation

4.1 Wavefront reconstruction

We examined the optical performance of the CVD SiC DM by reconstructing various Zernike modes [20] 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 s(x,y) of the CVD SiC DM can be described as

s(x,y)=i=1nviϕi(x,y)+s0
, where nis the number of the actuators, ϕi is the influence function of the ith actuator, vi is the voltage applied to the ith actuator, and s0 is the initial shape of the CVD SiC DM before actuation. In a matrix form, Eq. (5) can be rewritten as
S=VΦ+S0
By rearranging the above equation, we can obtain the voltage to the actuator for the desired shape Sd of the faceplate as
V=Φ1(SdS0)
As Φ is not a square matrix, we use the pseudo-inverse of Φ which is obtained by using the singular value decomposition. Figures 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.

 

Fig. 8 The wavefront reconstruction experimental setup. (a) is a block diagram of the experimental setup and (b) is the actual implementation

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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.

 

Fig. 9 (a) The initial surface error of the faceplate before flattening, and (b) the surface error of the faceplate after flattening. The faceplate has the clear aperture of 160 mm in diameter which is 80% of faceplate.

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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.

 

Fig. 10 The Zernike modes measured with the interferometer (experiment) and their residual errors with respect to theoretical ones. The generated Zernike modes have the PV value of 2 waves within the clear aperture which is 80% of the physical diameter.

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Fig. 11 The residual surface errors (PV and rms) of the reconstructed Zernike modes. The amplitude of the input Zernike modes is 2 waves in PV (λ = 632.8 nm).

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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 [21] and Zonal [22] 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 [10]. The 3 dB bandwidth was measured to be 1.3 kHz.

 

Fig. 12 (a) Block diagram, and (b) actual implementation of the closed-loop AO system used for wavefront compensation experiments.

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In the wavefront compensation experiments, the aberration generator simulated the atmospheric turbulence having a Fried parameter r0 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.

 

Fig. 13 History of the wavefront rms in the closed-loop AO system without coolant flow.

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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 [23]. 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 I=σT4, where σ is 5.67 × 10−8 W/m2K4, and T is measured as 623 K. The irradiance dQ˙12 on the faceplate can be expressed by Lambert’s law [24] as

dQ˙12=IdA1cosθ1dA2cosθ2R2
, where I is the intensity of the heat source, dA1 is the area of the heat source, and dA2 is the area of the faceplate. The total irradiance on the faceplate is calculated to be 103.54 W assuming only the radiative heat transfer without atmospheric effects. We used the above parametric values for the CFD simulation inputs, and could predict the temperature variation and the surface error of the faceplate while the coolant flows.

 

Fig. 14 (a) Simulation model and (b) schematic diagram of the CVD SiC DM irradiated by a heat source. The heat source is 250 mm apart from the faceplate and the angle is 45° between the heat source and the faceplate.

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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.

 

Fig. 15 2D view of the surface temperature variations (left) and the surface errors (right) from the CFD simulation results. (a) is the results without the coolant flow, and (b) is the results with the coolant flow.

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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.

 

Fig. 16 (a) Block diagram of the closed-loop AO system with the cooling system and a heat source, (b) the CVD SiC DM with a heat source and an IR camera.

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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.

 

Fig. 17 (a) A steady-state thermal image of the faceplate when irradiated by a heat source without cooling, and (b) with cooling.

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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.

 

Fig. 18 The temperature variation of the faceplate measured by 4 thermocouples. The positions of the thermocouples are indicated inside the figure.

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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.

 

Fig. 19 History of the wavefront rms in the closed-loop AO system when the DM is irradiated by a heat source and the coolant flows.

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6. Conclusions

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.

Funding

National Research Council of Science & Technology (NST) grant (MSIP) (No. CAP-12-04-KRISS) by the Korea government

Acknowledgment

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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22. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70(8), 998–1006 (1980). [CrossRef]  

23. S. Hyun, K. H. Kim, J. Y. Bae, and H. C. Kang, “Development of a water-cooling system for the 100 mm diameter deformable mirror,” in Proceeding of KSPE 2015 Autumn Conference (Korean Society for Precision Engineering, 2015), pp. 106–106.

24. A. F. Mills, Basic Heat and Mass Transfer (IRWIN, 1995), Chap. 6.

References

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  1. H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65(386), 229–236 (1953).
    [Crossref]
  2. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998), Chap. 6.
  3. E. J. Szetela and A. I. Chalfant, “Thermal distortion of mirrors,” Thermochim. Acta 26(1–3), 191–197 (1978).
    [Crossref]
  4. M. A. Ealey and J. A. Wellman, “Cooled ISOFLOW laser mirrors,” Proc. SPIE 1739, 374–382 (1992).
    [Crossref]
  5. G. Rabczuk and M. Sawczak, “Study on the possibilities of controlling the laser output beam properties by an intracavity deformable mirror,” Opto-Electron. Rev. 14(2), 141–147 (2006).
    [Crossref]
  6. A. V. Kudryashov and V. V. Samarkin, “Control of high-power CO2-laser beam by adaptive optical elements,” Opt. Commun. 118(3–4), 317–322 (1995).
    [Crossref]
  7. R. H. Freeman and H. R. Garcia, “High-speed deformable mirror system,” Appl. Opt. 21(4), 589–595 (1982).
    [Crossref] [PubMed]
  8. P. Y. Bely, The Design and Construction of Large Optical Telescopes (Springer, 2002), Chap. 4.
  9. K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
    [Crossref]
  10. K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
    [Crossref]
  11. E. Hecht, Optics, 4th ed. (Pearson, 2014), Chap. 13.
  12. C. J. Luis, I. Puertas, and G. Villa, “Material removal rate and electrode wear study on the EDM of silicon carbide,” J. Mat. Proc. Technol. 164, 889–896 (2005).
    [Crossref]
  13. W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
    [Crossref]
  14. J. Alda and G. D. Boreman, “Zernike-based matrix model of deformable mirrors: optimization of aperture size,” Appl. Opt. 32(13), 2431–2438 (1993).
    [Crossref] [PubMed]
  15. B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
    [Crossref]
  16. M. A. van Dam, D. Le Mignant, and B. A. Macintosh, “Performance of the Keck Observatory adaptive-optics system,” Appl. Opt. 43(29), 5458–5467 (2004).
    [Crossref] [PubMed]
  17. L. Huang, C. Rao, and W. Jiang, “Modified Gaussian influence function of deformable mirror actuators,” Opt. Express 16(1), 108–114 (2008).
    [Crossref] [PubMed]
  18. S. Li and S. Zhang, “Numerical model of the influence function of deformable mirrors based on Bessel Fourier orthogonal functions,” Res. Astron. Astrophys. 14(11), 1504–1510 (2014).
    [Crossref]
  19. H. Kihm and H.-S. Yang, “Design optimization of a 1-m lightweight mirror for a space telescope,” Opt. Eng. 52(9), 091806 (2013).
    [Crossref]
  20. R. R. Shannon and J. C. Wyant, Applied Optics and Optical Engineering (Academic, 1992).
  21. J. Hermann, “Least-squares wave-front errors of minimum norm,” J. Opt. Soc. Am. 70(1), 28–35 (1980).
    [Crossref]
  22. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70(8), 998–1006 (1980).
    [Crossref]
  23. S. Hyun, K. H. Kim, J. Y. Bae, and H. C. Kang, “Development of a water-cooling system for the 100 mm diameter deformable mirror,” in Proceeding of KSPE 2015 Autumn Conference (Korean Society for Precision Engineering, 2015), pp. 106–106.
  24. A. F. Mills, Basic Heat and Mass Transfer (IRWIN, 1995), Chap. 6.

2016 (1)

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

2015 (1)

K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
[Crossref]

2014 (1)

S. Li and S. Zhang, “Numerical model of the influence function of deformable mirrors based on Bessel Fourier orthogonal functions,” Res. Astron. Astrophys. 14(11), 1504–1510 (2014).
[Crossref]

2013 (1)

H. Kihm and H.-S. Yang, “Design optimization of a 1-m lightweight mirror for a space telescope,” Opt. Eng. 52(9), 091806 (2013).
[Crossref]

2008 (1)

2006 (1)

G. Rabczuk and M. Sawczak, “Study on the possibilities of controlling the laser output beam properties by an intracavity deformable mirror,” Opto-Electron. Rev. 14(2), 141–147 (2006).
[Crossref]

2005 (1)

C. J. Luis, I. Puertas, and G. Villa, “Material removal rate and electrode wear study on the EDM of silicon carbide,” J. Mat. Proc. Technol. 164, 889–896 (2005).
[Crossref]

2004 (1)

1997 (1)

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

1995 (1)

A. V. Kudryashov and V. V. Samarkin, “Control of high-power CO2-laser beam by adaptive optical elements,” Opt. Commun. 118(3–4), 317–322 (1995).
[Crossref]

1993 (1)

1992 (1)

M. A. Ealey and J. A. Wellman, “Cooled ISOFLOW laser mirrors,” Proc. SPIE 1739, 374–382 (1992).
[Crossref]

1991 (1)

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

1982 (1)

1980 (2)

1978 (1)

E. J. Szetela and A. I. Chalfant, “Thermal distortion of mirrors,” Thermochim. Acta 26(1–3), 191–197 (1978).
[Crossref]

1953 (1)

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65(386), 229–236 (1953).
[Crossref]

Ahn, K.

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
[Crossref]

Alda, J.

Babcock, H. W.

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65(386), 229–236 (1953).
[Crossref]

Boreman, G. D.

Chalfant, A. I.

E. J. Szetela and A. I. Chalfant, “Thermal distortion of mirrors,” Thermochim. Acta 26(1–3), 191–197 (1978).
[Crossref]

Dekany, R.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Ealey, M.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Ealey, M. A.

M. A. Ealey and J. A. Wellman, “Cooled ISOFLOW laser mirrors,” Proc. SPIE 1739, 374–382 (1992).
[Crossref]

Freeman, R. H.

Garcia, H. R.

Hermann, J.

Huang, L.

Jiang, W.

Jiang, W. H.

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Kihm, H.

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
[Crossref]

H. Kihm and H.-S. Yang, “Design optimization of a 1-m lightweight mirror for a space telescope,” Opt. Eng. 52(9), 091806 (2013).
[Crossref]

Kudryashov, A. V.

A. V. Kudryashov and V. V. Samarkin, “Control of high-power CO2-laser beam by adaptive optical elements,” Opt. Commun. 118(3–4), 317–322 (1995).
[Crossref]

Le Mignant, D.

Lee, H. J.

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

Lee, J. H.

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

Li, S.

S. Li and S. Zhang, “Numerical model of the influence function of deformable mirrors based on Bessel Fourier orthogonal functions,” Res. Astron. Astrophys. 14(11), 1504–1510 (2014).
[Crossref]

Ling, N.

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Luis, C. J.

C. J. Luis, I. Puertas, and G. Villa, “Material removal rate and electrode wear study on the EDM of silicon carbide,” J. Mat. Proc. Technol. 164, 889–896 (2005).
[Crossref]

Macintosh, B. A.

Oppenheimer, B. R.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Palmer, D.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Price, T.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Puertas, I.

C. J. Luis, I. Puertas, and G. Villa, “Material removal rate and electrode wear study on the EDM of silicon carbide,” J. Mat. Proc. Technol. 164, 889–896 (2005).
[Crossref]

Rabczuk, G.

G. Rabczuk and M. Sawczak, “Study on the possibilities of controlling the laser output beam properties by an intracavity deformable mirror,” Opto-Electron. Rev. 14(2), 141–147 (2006).
[Crossref]

Rao, C.

Rao, X. J.

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Rhee, H. G.

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
[Crossref]

Samarkin, V. V.

A. V. Kudryashov and V. V. Samarkin, “Control of high-power CO2-laser beam by adaptive optical elements,” Opt. Commun. 118(3–4), 317–322 (1995).
[Crossref]

Sawczak, M.

G. Rabczuk and M. Sawczak, “Study on the possibilities of controlling the laser output beam properties by an intracavity deformable mirror,” Opto-Electron. Rev. 14(2), 141–147 (2006).
[Crossref]

Shi, F.

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Sivaramakrishnan, A.

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

Southwell, W. H.

Szetela, E. J.

E. J. Szetela and A. I. Chalfant, “Thermal distortion of mirrors,” Thermochim. Acta 26(1–3), 191–197 (1978).
[Crossref]

van Dam, M. A.

Villa, G.

C. J. Luis, I. Puertas, and G. Villa, “Material removal rate and electrode wear study on the EDM of silicon carbide,” J. Mat. Proc. Technol. 164, 889–896 (2005).
[Crossref]

Wellman, J. A.

M. A. Ealey and J. A. Wellman, “Cooled ISOFLOW laser mirrors,” Proc. SPIE 1739, 374–382 (1992).
[Crossref]

Yang, H. S.

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
[Crossref]

Yang, H.-S.

H. Kihm and H.-S. Yang, “Design optimization of a 1-m lightweight mirror for a space telescope,” Opt. Eng. 52(9), 091806 (2013).
[Crossref]

Zhang, S.

S. Li and S. Zhang, “Numerical model of the influence function of deformable mirrors based on Bessel Fourier orthogonal functions,” Res. Astron. Astrophys. 14(11), 1504–1510 (2014).
[Crossref]

Appl. Opt. (3)

J. Korean Phys. Soc. (1)

K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015).
[Crossref]

J. Mat. Proc. Technol. (1)

C. J. Luis, I. Puertas, and G. Villa, “Material removal rate and electrode wear study on the EDM of silicon carbide,” J. Mat. Proc. Technol. 164, 889–896 (2005).
[Crossref]

J. Opt. Soc. Am. (2)

Korean J. Opt. Photonics (1)

K. Ahn, H. G. Rhee, H. J. Lee, J. H. Lee, H. S. Yang, and H. Kihm, “Wave-front compensation using a silicon carbide deformable mirror with 37 actuators for adaptive optics,” Korean J. Opt. Photonics 27(3), 106–113 (2016).
[Crossref]

Opt. Commun. (1)

A. V. Kudryashov and V. V. Samarkin, “Control of high-power CO2-laser beam by adaptive optical elements,” Opt. Commun. 118(3–4), 317–322 (1995).
[Crossref]

Opt. Eng. (1)

H. Kihm and H.-S. Yang, “Design optimization of a 1-m lightweight mirror for a space telescope,” Opt. Eng. 52(9), 091806 (2013).
[Crossref]

Opt. Express (1)

Opto-Electron. Rev. (1)

G. Rabczuk and M. Sawczak, “Study on the possibilities of controlling the laser output beam properties by an intracavity deformable mirror,” Opto-Electron. Rev. 14(2), 141–147 (2006).
[Crossref]

Proc. SPIE (3)

M. A. Ealey and J. A. Wellman, “Cooled ISOFLOW laser mirrors,” Proc. SPIE 1739, 374–382 (1992).
[Crossref]

B. R. Oppenheimer, D. Palmer, R. Dekany, A. Sivaramakrishnan, M. Ealey, and T. Price, “Investigating a Xinetics Inc. deformable mirror,” Proc. SPIE 3126, 569–579 (1997).
[Crossref]

W. H. Jiang, N. Ling, X. J. Rao, and F. Shi, “Fitting capability of deformable mirror,” Proc. SPIE 1542, 130–137 (1991).
[Crossref]

Publ. Astron. Soc. Pac. (1)

H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65(386), 229–236 (1953).
[Crossref]

Res. Astron. Astrophys. (1)

S. Li and S. Zhang, “Numerical model of the influence function of deformable mirrors based on Bessel Fourier orthogonal functions,” Res. Astron. Astrophys. 14(11), 1504–1510 (2014).
[Crossref]

Thermochim. Acta (1)

E. J. Szetela and A. I. Chalfant, “Thermal distortion of mirrors,” Thermochim. Acta 26(1–3), 191–197 (1978).
[Crossref]

Other (6)

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998), Chap. 6.

P. Y. Bely, The Design and Construction of Large Optical Telescopes (Springer, 2002), Chap. 4.

E. Hecht, Optics, 4th ed. (Pearson, 2014), Chap. 13.

R. R. Shannon and J. C. Wyant, Applied Optics and Optical Engineering (Academic, 1992).

S. Hyun, K. H. Kim, J. Y. Bae, and H. C. Kang, “Development of a water-cooling system for the 100 mm diameter deformable mirror,” in Proceeding of KSPE 2015 Autumn Conference (Korean Society for Precision Engineering, 2015), pp. 106–106.

A. F. Mills, Basic Heat and Mass Transfer (IRWIN, 1995), Chap. 6.

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Figures (19)

Fig. 1
Fig. 1 Design of the CVD SiC faceplate with monolithic cooling channels. Small circles indicate the positions of 137 actuators. Blue and red arrows represent inlets and outlets respectively.
Fig. 2
Fig. 2 The CFD simulation results; (a) Steady state water temperature, (b) faceplate temperature with cooling, and (c) faceplate displacement due to the steady state temperature gradient.
Fig. 3
Fig. 3 A CVD SiC faceplate with monolithic cooling channels; (a) external view, (b) ultrasonic scan image showing the internal shape of the cooling channels.
Fig. 4
Fig. 4 An assembled CVD SiC DM with 5-axis mount for the experiments.
Fig. 5
Fig. 5 Comparison of an actual IF obtained from the CVD SiC DM and its fitted analytical models. (a) Actual IF measured with a 300-mm Fizeau interferometer for flatness tests, (b) the Gaussian IF, (c) the MGIF, and (d) the proposed OGIF
Fig. 6
Fig. 6 Residual error images (left) and their cross-sectional plots (right) of the analytical IF models after fitting with the actual IF. (a) The Gaussian IF, (b) the MGIF, and (c) the OGIF
Fig. 7
Fig. 7 A comparison of (a) the IFs simulated by using FEA and (b) the IFs measured by a Zygo interferometer. All IFs are downsized and placed at the corresponding actuator locations. IFs in the second quadrant are displayed due to the symmetry.
Fig. 8
Fig. 8 The wavefront reconstruction experimental setup. (a) is a block diagram of the experimental setup and (b) is the actual implementation
Fig. 9
Fig. 9 (a) The initial surface error of the faceplate before flattening, and (b) the surface error of the faceplate after flattening. The faceplate has the clear aperture of 160 mm in diameter which is 80% of faceplate.
Fig. 10
Fig. 10 The Zernike modes measured with the interferometer (experiment) and their residual errors with respect to theoretical ones. The generated Zernike modes have the PV value of 2 waves within the clear aperture which is 80% of the physical diameter.
Fig. 11
Fig. 11 The residual surface errors (PV and rms) of the reconstructed Zernike modes. The amplitude of the input Zernike modes is 2 waves in PV (λ = 632.8 nm).
Fig. 12
Fig. 12 (a) Block diagram, and (b) actual implementation of the closed-loop AO system used for wavefront compensation experiments.
Fig. 13
Fig. 13 History of the wavefront rms in the closed-loop AO system without coolant flow.
Fig. 14
Fig. 14 (a) Simulation model and (b) schematic diagram of the CVD SiC DM irradiated by a heat source. The heat source is 250 mm apart from the faceplate and the angle is 45° between the heat source and the faceplate.
Fig. 15
Fig. 15 2D view of the surface temperature variations (left) and the surface errors (right) from the CFD simulation results. (a) is the results without the coolant flow, and (b) is the results with the coolant flow.
Fig. 16
Fig. 16 (a) Block diagram of the closed-loop AO system with the cooling system and a heat source, (b) the CVD SiC DM with a heat source and an IR camera.
Fig. 17
Fig. 17 (a) A steady-state thermal image of the faceplate when irradiated by a heat source without cooling, and (b) with cooling.
Fig. 18
Fig. 18 The temperature variation of the faceplate measured by 4 thermocouples. The positions of the thermocouples are indicated inside the figure.
Fig. 19
Fig. 19 History of the wavefront rms in the closed-loop AO system when the DM is irradiated by a heat source and the coolant flows.

Tables (1)

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Table 1 Faceplate material’s figure of merit

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

I(r)= I 0 e 2 r 2 / w 0 2
I(r)=exp[ ln(ω) ( r d ) α ]
I(r,θ)=exp[ ln(ω) { r[1+τcos(4θ)] d ¯ } α ]+βexp[ { (r d ¯ ) γ } 2 ]
I(r,θ)=(1β)exp[ ln(ω) { r[1+τcos(4(θ π 4 ))] d ¯ } α ]+βexp[ ln(ω) ( r d ) 2 ]
s(x,y)= i=1 n v i ϕ i (x,y)+ s 0
S=VΦ+ S 0
V= Φ 1 ( S d S 0 )
d Q ˙ 12 = Id A 1 cos θ 1 d A 2 cos θ 2 R 2

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