1. Introduction

At present, adaptive optics (AO) has been the key technology in Inertial Confinement Fusion (ICF) [1], high power laser [2,3], high resolution imaging [4,5] and atmosphere optics [6]. The deformable mirror (DM) is used to correct the wavefront distortion in an AO system. Its correction ability is influenced by the stability of the surface shape under different environment and the local high frequency distortion (HFD). The adverse impacts of the high frequency distortion in fitting residual have been reported. For example, the far field pattern is far from smooth with plenty of discrete minor spots caused by the uncorrected high-order aberrations in a high-power laser system such as the ShenGuang-III Facility (SG-III) and the National Ignition Facility (NIF), which are used for the inertial confinement fusion in China and the United States, respectively [7]. Many manufacture techniques have been adopted to restrain the static and dynamic HFD in the production stage [8,9]. For example, the dynamic HFD is eliminated by keeping the actuators strictly perpendicular to the steel base [8]. The adhesive curing induced static HFD is eliminated by inserting the transition layer between the mirror and the actuators in the DM [9].

However, the surface distortion could still occur in a well-manufactured DM when the temperature of the DM in the working environment ($T_e$) is not equal to the temperature in the manufacturing environment ($T_m$). The TID of the DM has been investigated by numerical analyses and experiments [10–14]. For the bimorph DM, the temperature gradient appearing in the mirror due to low thermal conductivity of the mirror results in the surface shape distortion. The distortion could be depressed by applying a Safronov’s cooling system, and the depress effect could be improved by inserting a middle metal coating conduction layer [10]. Investigation shows that the thermal induced deformation of a well-manufactured screen-printed unimorph DM varies with the load method. A homogeneous load induces the focusing of the mirror while an inhomogeneous load will induce the defocusing of the mirror [11]. The deformation induced by the homogeneous load is influenced by the mounting materials, while the deformation induced by the inhomogeneous load depends on the copper-layer thickness. For the membrane DM, the surface shape of the mirror could be affected by the convection when the membrane temperature is above 35 $℃$ [12]. To withstand 10 kW laser power and maintain the thermal characterization, a configuration of a unimorph DM is developed by mounting a cooling cavity on the back side of the DM [13].

The TID existing on the surface of the DM is one type of the actuator-corresponding dynamic HFD [14,15]. As reported, when the ambient temperature rises/falls 2$℃$, the PV value of the surface shape of a well-mounted DM would increase 1.4 μm [15], which could not be acceptable in a high-power laser system (e.g. the SG-III and the NIF). Restricted by the structure and the mechanism, the DM with limited stacked array actuators is only used to correct the low order wavefront aberrations and has limitations to correct the TID effectively [8]. Thus, in the application of an AO system, the TID will remain in the residual surface shape of the DM after correction, which obviously limits the correction ability of the DM. The uncorrectable TID will cause the degeneration of the beam quality of the optics system and the energy dispersion of the far-field spot. Moreover, under certain condition, the TID could even result in the unexpected shift of the focal point when the laser is in operation, which is very dangerous for the optical elements [9]. Thus, the application field of a DM with TID is strongly restricted by the working temperature and could not be extended to the working environment with a wider ambient temperature range.

Many techniques have been proposed to reduce the TID in the structural design, such as increasing the thickness of the base and the mirror, increasing the actuator numbers and choosing the materials of base and mirror with small thermal expansion coefficient difference [14]. The goal of all the techniques is to optimize the structure parameters under the design stage to weaken the TID, as the essential reason of the TID of the NIF-type DM hasn’t been investigated systematically. In fact, if $T_e$ is not equal to $T_m$, the TID always exists in the DM even the PV value of the TID is depressed by the optimization skills mentioned above. Therefore, proposing a disjunctive TID compensation method, which could be applicable for the manufactured DM, is of vital.

In this paper, the essential reason of the TID of the NIF-type DM is investigated systematically. Based on the analysis result, an efficient method based on an auxiliary TCM and a hybrid close-loop control algorithm is presented accordingly. In section 2, a simplified structure model of the DM is established. The mechanism of the TID is analyzed through the derivation of the residual thermal stress according to the simplified model. Simulation results show that the tilt of the actuators caused by the thermal stress difference of the mirror and the steel base is the essential reason of the TID. Finally, according to the TID’s mechanism, the mathematical equations of a disjunctive TCM is acquired for proving the feasibility to compensate the TID. In section 3, a finite element model is built to investigate the influence of various parameters on the TID, including the ambient temperature, the thermal expansion coefficient difference between the mirror and steel base, and the TCM temperature. The simulation results indicate the TID could be effectively eliminated by the TCM. In section 4, the experiment results show that the TID is well depressed and almost vanishes in the self-compensation residues with the TCM working.

2. NIF-type DM structure and TID mechanism

Various DMs have been proposed to meet different application requirements [16–22], such as the liquid crystal spatial light modulator [19,20], the MEMS DM [21,22] and the traditional NIF-type DM [7]. Due to the advantages of large aperture, wide dynamic range and high damage threshold, the NIF-type DM is widely used in high power laser system and ICF system.

A NIF-type DM is manufactured to analyze the mechanism of the TID, and the simplified structure is shown Fig. 1. The DM, which is a typical discrete PZT driven, continuous mirror DM, consists of an 84mm $\times$ 84mm $\times$ 1mm plate BK7 mirror, 49 stacked PZT actuators and an 84mm $\times$ 84mm $\times$ 40mm stainless-steel base. The diameter and height of actuators are 5mm and 36mm, respectively. The 49 actuators are square array distributed with 12mm spacing. The optical mirror is bonded to the PZT actuators, which are rigidly connected to the steel base. The thermal expansion coefficients of the mirror and the steel base are 7.1 $\times$ 10⁻⁶/K and 17.2 $\times$ 10⁻⁶/K, respectively. The stress of the mirror and the steel base varying with the change of the working environment temperature is different. As a result, the residual thermal stress at the joints between the actuators and the mirror might lead to the unwanted TID.

 

Fig. 1 (a) Simplified schematic of the NIF-type deformable mirror structure. (b) The XOY coordinates and the distribution of the actuators.

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Two coordinate systems shown in the Fig. 1(b) and Fig. 2 are established to analyze the TID. Considering the symmetry of the structure of the DM [Fig. 1], the local surface of the actuator-mirror contact surface in the first octant could be used to represent other three octants when $T_e$ is not equal to $T_m$. An actuator in the first octant, the ${\left(i, j\right)}^{th}$ actuator, is selected as the analysis object. Where i represents the row number of the selected actuator along the X-direction, and j represents the column number of the selected actuator along the Y-direction, while the number of the central actuator is set (0,0). Point $A_{i,j}$ is the center of the actuator-mirror contact surface, and point $B_{i,j}$ is the center of the actuator-base contact surface. $\left(x_i,y_j,0\right)$ and $\left(x_i,y_j,h\right)$ are the coordinates of point $A_{i,j}$ and point $B_{i,j}$, respectively, where $x_i=ia,y_j=ja$,and h is the height of the actuator in the manufacturing environment. The dash line $\overline{A_{i,j}{B}_{i,j}}$ represents the original position of the ${\left(i, j\right)}^{th}$ actuator at $T_m$. When $T_e$ is not equal to $T_m$, due to the thermal expansion and contraction characterization, point $A_{i,j}$ moves to point $A_{i,j}^{\text{'}}$ and point $B_{i,j}$ moves to point $B_{i,j}^{\text{'}}$. The solid line $\overline{A_{i,j}^{\text{'}}{B}_{i,j}^{\text{'}}}$ represents the position of the ${\left(i, j\right)}^{th}$ actuator at $T_e$.

 

Fig. 2 (a) The XYZ coordinates and the total displacements of the actuator. (b) The XOY and $X^{\text{'}}{O}^{\text{'}}{Y}^{\text{'}}$ coordinates and the shear displacement of the mirror. (c) The $Y^{\text{'}}{O}^{\text{'}}{Z}^{\text{'}}$ section and the normal displacement of the mirror.

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By applying the notation of linear thermal expansion coefficient, the coordinates differences between point $A_{i,j}$ and point $A_{i,j}^{\text{'}}$ are written as Eqs. (1) and (2), while Eqs. (3) and (4) express the coordinates differences between point $B_{i,j}$ and point $B_{i,j}^{\text{'}}$ .

Here ${\delta }_{iA}$ is the temperature induced deformation value along the X-direction of point $A_{i,j}$. ${\delta }_{jA}$ is the temperature induced deformation value along the Y-direction of point $A_{i,j}$. ${\delta }_{iB}$ and ${\delta }_{jB}$ is the corresponding value of point $B_{i,j}$. ${\alpha }_G$ and ${\alpha }_B$ are the linear thermal expansion coefficients of the mirror and steel base, respectively. $\Delta T_e$ represents the difference between $T_e$ and $T_m$, which is expressed as Eq. (5).

Considering the practical operation environment, the temperature of the DM in the working environment $T_e$ is hardly equal to the temperature in the manufacturing environment $T_m$. When the temperature $T_e$ of the DM is not equal to the original manufacturing temperature $T_m$, $\Delta T_e\ne 0, {\delta }_{iA}\ne {\delta }_{iB}$ and ${\delta }_{jA}\ne {\delta }_{jB}$, which indicate that solid line $\overline{A_{i,j}^{\text{'}}{B}_{i,j}^{\text{'}}}$ is not normal to the mirror. Consequently, the actuator is tilted and stretched. The tensile stress appears in the actuator to resist the tensile deformation, which is influenced by two factors: one is the reaction force when the actuator is stretched, another is the heat expansion and contraction. The contraction along the ideal tilted axis $\overline{A_{i,j}^{\text{'}}{B}_{i,j}^{\text{'}}}$ induced by the tensile stress is supposed as the total reverse displacements $\Delta h_{ij}$. As shown in Figs. 2(b) and 2(c), the second coordinate system $X^{\text{'}}{O}^{\text{'}}{Y}^{\text{'}}$ is defined by selecting point $A_{i,j}$ as the origin $O^{\text{'}}$ and selecting the direction of the vector from point $A_{i,j}$ to point $A_{i,j}^{\text{'}}$ as the forward direction of the $X^{\text{'}}$ axis. Figure 2(c) shows the cross section $Y^{\text{'}}{O}^{\text{'}}{Z}^{\text{'}}$ and the normal displacements $\Delta h_{ijz}$ (blue line) caused by $\Delta h_{ij}$. The shear displacements $\Delta h_{ijx}$ and $\Delta h_{ijy}$ are illustrated by the blue line in Fig. 2(b). The normal displacements $\Delta h_{ijz}$, the shear displacements $\Delta h_{ijx}$ and $\Delta h_{ijy}$ could be expressed in Eq. (6).

Where the angle ${\alpha }_{i,j}$ is associated with the row number i and the column number j [Fig. 2(b), Eq. (7)]. ${\alpha }_{i,j}$ is the incline angle between the Z-direction and the direction of the tilted axis $\overline{A_{i,j}^{\text{'}}{B}_{i,j}^{\text{'}}}$ [Fig. 2(c), Eq. (8)].

Specially, ${\alpha }_{i,j}$ is equal to $\pi /2$ when $j=0$. ${\delta }_{Ai,j}$ is the total displacement from point $A_{i,j}$ to point $A_{i,j}^{\text{'}}$ on the $XOY$ plane [Eq. (9)], and all the subscript A could be replaced by B in Eq. (9) to represent the total displacement from point $B_{i,j}$ to point $B_{i,j}^{\text{'}}$.

For a DM, the surface shape of the mirror is driven and controlled by the vertical movements of the PZT actuators to generate the conjugate surface of the distorted wavefront. Therefore, the normal displacement $\Delta h_{ijz}$ [Eq. (6)] appearing on the mirror surface could be corrected by the pushing and pulling of the actuators vertically, while the shear displacements $\Delta h_{ijx}$ and $\Delta h_{ijy}$ of the actuators could not be compensated and would result in the dynamic HFD in the residue [8]. Therefore, in a DM, the TID in the correction residual could be eliminated if $\Delta h_{ijx}$ and $\Delta h_{ijy}$ are always equal to zero even if the $\Delta T_e$ is not equal to zero [Eq. (6)].

In order to keep $\Delta h_{ijx}$ and$\Delta h_{ijy}$ always equal to zero, the angle ${\theta }_{i,j}$ in Eq. (6) should be set to zero. By introducing Eqs. (1)–(4) and (9), Eq. (8) could be rewritten as Eq. (10).

Where the structure parameters $\alpha$ is not equal to zero. $\Delta T_{eB}$ is the difference between the DM base’s temperature in the working environment and $T_m$, while $\Delta T_{eG}$ is the difference between the DM mirror’s temperature in the working environment and $T_m$. Obviously, in the working environment with steady temperature field, $\Delta T_{eB}$ and $\Delta T_{eG}$ are both equal to$\Delta T_e$. As the materials of the steel base and the mirror in a DM could not be the same, the thermal expansion coefficient ${\alpha }_G$ is not equal to ${\alpha }_B$. From Eq. (10), in the working environment with the temperature not equal to the original manufacture temperature (most probably happening in the real practice), the angle ${\theta }_{i,j}$ could still be kept to zero if the relative parameters are set to satisfy Eq. (11).

A TCM is designed to regulate the temperature field of the steel base to achieve the target $\Delta T_{eB}$, as shown in Fig. 3, which could be installed to the rear surface of the steel base for a manufactured DM. The steel base temperature could be regulated by the heat conduction with the TCM. The mirror and the steel base are indirectly connected by low thermal conductivity PZT-actuators, and the area of the surface contacting with air is much larger than the area of mirror actuators contacting surface. According to the theory of heat conduction, the mirror temperature mainly depends on the convective heat transfer with air, which almost equals to the temperature of the DM in the working environment, even the temperature of the steel base is regulated by the auxiliary TCM. Therefore, Eq. (11) could be rewritten as Eq. (12).

 

Fig. 3 (a)The DM with the TCM. (b) Distribution of four TECs.

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To improve the heat conductivity, the heat-conducting silicon grease is applied on the contacting layer between the steel base and the TCM. The designed TCM consists of four-square array distributed TECs (single TEC size is 40mm × 40mm × 4mm, and the combined size is 80mm × 80mm × 4mm, 12V/5A, TEC1204) [Fig. 3(b)], a water-cooling block, and a temperature control circuit. The stable temperature of the steel base depends on $T_e$ and the TCM temperature $T_{tcm}$. Thus, the stable $\Delta T_{eB}$ could be expressed as Eq. (13) based on the theory of heat conduction.

Apparently, for the stable working environment with the constant value of $T_e$, the parameter $\Delta T_{eB}$ could always be controlled solely and directly by adjusting the temperature of the TCM. Based on this method, the angle ${\theta }_{i,j}$ in Eq. (6) could be kept zero and the TID in the residual could be eliminated. Thus, the application field of a DM would be extended to an unrestricted working environment with a wider ambient temperature range.

3. Simulation analysis of temperature induced distortion

To evaluate the TCM’s compensation capability on the TID of the DM, a finite element model is built in COMSOL Multiphysics software [23,24]. In the simulation model, the DM consists of a continuous plate BK7 mirror (84mm $\times$ 84mm $\times$ 1mm full size), a stainless-steel base (84mm $\times$ 84mm $\times$ 1mm full size) and 49 square distributed discrete PZT posts (5mm-diameter, 36mm-length). Primary parameters of the materials as constants in small temperature variation range are listed in Table 1.

Table 1. Material parameters in the finite element simulation

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The whole model is built using the thermal stress interface from the structural mechanics module. In the simulation, the degrees of freedom for the four corners of the steel base are set to zero, and the Z-direction of the steel base’s rear surface is also set to zero. The initial surface shape of the mirror is set to be an ideal plane without any distortion. The heat transfer is by means of the natural convective heat flux. For the heating boundary condition, the convective heat transfer coefficient of all outer surfaces is set 10 $\text{ W}/({\text{m}}^2\text{K}$), while the coefficient of the base TCM contacting surface is set 25 $\text{ W}/({\text{m}}^2\text{K}$). The original manufacturing environment temperature $T_m$ of the DM is set 20 $℃$.

3.1 TID characteristics of DM

In the simulation, the working temperature of the DM is set $\Delta T_e$deviation from the original manufacturing environment temperature $T_m$. Figures 4(a) and 4(b) show the temperature gradient map and the displacement map of the whole DM without any distortion compensation. From Fig. 4(a), when the temperature of the steal base is not controlled, $\Delta T_{eB}$ and $\Delta T_{eG}$ are both equal to 5$℃$ ($\Delta T_e$). The thermal expansion coefficients difference between the mirror and the steal base will result in the expansion mismatch and consequently lead to the tilt of the actuators in the DM [Fig. 4(b)].

 

Fig. 4 The temperature gradient maps and the displacement maps of the DM at $\Delta T_e=$5$℃$. The temperature gradient map (a) and the displacement map (b) without any distortion compensation. The temperature gradient map (c) and the displacement map (d) with the TCM working at the optimal compensation temperature.

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Accordingly, the whole surface shape (WSS) of the DM is distorted by the TID. Figures. 5(a1) and 5(b1) show the distorted WSSs (i.e. the TID) of the DM at $\Delta {\text{T}}_{\text{e}}=$ −5$℃$ and $\Delta {\text{T}}_{\text{e}}=$5$℃$, respectively. The PV values of the two distorted WSSs are both 0.56 $\text{μm}$. According to the color temperature diagrams, the square distributed local distortion array could be identified clearly. The distribution of the local distortion is well matched with the actuators’ array, which could be identified by print-through of the actuators. The power spectral density (PSD) curves of the WSSs of the DM, illustrating the frequency characterization, are calculated [25,26] and shown in Figs. 5(a2) and 5(b2). The red lines represent the standard lines of the PSD curves [27], while the vertical dashed lines indicate the boundary between the low-frequency distortion and the high-frequency distortion [28]. As shown in Figs. 5(a2) and 5(b2), the PSD curves of the distorted WSSs locate above the standard line from low-frequency to high-frequency, which demonstrates the surface shape of the DM with TIDs are not qualified. The 3D maps of the distorted WSSs after high pass filtering (SSHF) the first 40 orders Zernike aberration of Figs. 5(a1) and 5(b1) are shown in Figs. 5(a3) and 5(b3) respectively, while the 2D maps are displayed in Figs. 5(a4) and 5(b4). The black circles marked in Fig. 5 represents the actuators’ position in the DM. As shown in Figs. 5(a4) and 5(b4), the distribution of the bulges consistent well with that of the actuators. A convexity and concavity appear on both sides of each actuator except for the central actuator in Figs. 5(a4) and 5(b4), which shows the typical characteristic of the dynamic HFD caused by the tilt of the actuators [8]. The dynamic HFD becomes more obvious with the distance from the actuator to the mirror center increasing, which could be explained by that the tilted angle ${\text{θ}}_{\text{i},\text{j}}$ [Eq. (8)] is positively related to the distance. The PV values of the aberration shown in Figs. 5(a4) and 5(b4) are both 0.18 $\text{μm}$. As the self-compensation residues illustrated in Figs. 5(a5) and 5(b5), the TID of the DM couldn’t be well corrected. Consequently, as the compensation result, the HFD will remain on the surface of the DM. The PV values of the fitting residues are both 0.22$\text{μm}$, which are almost equal to the PV values of the HFD shown in Figs. 5(a4) and 5(b4). By comparing the patterns in Figs. 5(a) and 5(b), the TID is opposite and symmetry when the temperature rises or falls of the same degrees.

 

Fig. 5 The TIDs of the DM without temperature compensation. (a1) is the distorted WSS at $\Delta T_e=$-5$℃$. (b1) is the distorted WSS at$\Delta T_e=$5$℃$. (a2) and (b2) are the PSD curves of the distorted WSSs. (a3) and (b3) are the 3D maps of the distorted SSHFs. (a4) and (b4) are the 2D maps of (a3) and (b3), respectively. (a5) and (b5) are the 2D maps of the fitting residues of the distorted WSSs in (a1) and (b1) respectively, while (a6) and (b6) are the 3D maps of the fitting residues.

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Figure 6(a) shows the changing laws of the PV values and the RMS values of the WSS and the SSHF when $\Delta T_e$ varies from −5 $℃$ to 5 $℃$. The PV and RMS curves shown in Fig. 6(a) are origin-symmetric. Besides, the PV/RMS values of the WSS/SSHF increase linearly with the absolute value of $\Delta T_e$. Specifically, the PV value of the WSS increases 0.12$\text{μm}$ when the absolute value of $\Delta T_e$ increases 1$℃$, and the PV value of the SSHF increases 0.04$\text{μm}$. The relationship between the PV values of the TID and the expansion coefficient difference ($\Delta \text{α}$) of the mirror and the steel base is investigated at $\Delta T_e=$ 5 $℃$ [Fig. 6(c)]. From the black line shown in Fig. 6(c), the PV value of the WSS increases linearly with the fitting slope 0.06 $\text{μm}/({10}^{-6}/℃)$.

 

Fig. 6 The surface shape distortion of the DM with (a) different temperature difference and (c)different thermal expansion coefficient difference at $\Delta T_e=$ 5$℃$. (b) and (d) are the self-compensation residues of (a) and (c).

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The TID of the DM depends on the thermal expansion coefficient difference and the temperature variation based on the curves in Fig. 6. As the thermal expansion coefficients of two different materials could hardly achieve the same value, the TID would always exist on the surface shape of a DM if there is temperature variation. As shown in Figs. 6(b) and 6(d), the PV values of the self-compensation residues are almost equal to the PV values of the SSHF, which are linearly increasing with the temperature and thermal expansion coefficient difference.

3.2 Depression of TID by using TCM on DM

To eliminate the TID, a TCM analyzed in section 2 is designed and installed on the steel base of the manufactured DM [Fig. 3]. In the simulation, the working temperature of the DM is also set $\Delta T_e$ deviation from the original manufacturing environment temperature $T_m$. Meanwhile, the TCM is set to the temperature $T_{tcm}$ and the temperature of the steel base is affected, while the temperature of the mirror still equals to the working environment temperature. Figures 4(c) and 4(d) show the temperature gradient map and the displacement map of the whole DM with the TCM working at the optimal compensation temperature. As shown in Fig. 4(c), when the temperature of the steal base is regulated by the TCM working at the optimal compensation temperature, $\Delta T_{eB}$ is equal to 1.8$℃$, while $\Delta T_{eG}$ is still equal to 5 $℃$ ($\Delta T_e$). As the expansion mismatch is compensated, the tilt of the actuators is corrected [Fig. 4(d)]. Therefore, the TIDs shown in Figs. 5(a1) and 5(b1) turn into the patterns shown in Figs. 7(a1) and 7(b1) when $\Delta T_e$ equals to −5$℃$ and $5℃$, respectively.

 

Fig. 7 The TIDs with the TCM working at the optimal compensation temperature. (a1) is the pre-compensated WSS at $\Delta T_e=$ −5$℃$. (b1) is the pre-compensated WSS at$\Delta T_e=$5$℃$. (a2) and (b2) are the PSD curves of the compensated WSSs in (a1) and (b1). (a3) and (b3) are the 3D maps of the pre-compensated SSHFs. (a4) and (b4) are the 2D maps of (a3) and (b3), respectively. (a5) and (b5) are the 2D maps of the fitting residues of the WSS in (a1) and (b1) respectively, while (a6) and (b6) are the 3D maps.

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The pre-compensated WSSs and the final fitting residuals are depicted in Figs. 7(a) and 7(b) respectively, when the temperature of the steel base is regulated to achieve the target $\Delta T_{eB}$ [Eq. (12)] at $\Delta T_e=$ −5$℃$ and $\Delta T_e=$ 5$℃$. Note that in the pre-compensation process, only the TCM working to compensate the TID, while in the self-compensation process, the DM is controlled to realize the final correction. The PV values of the pre-compensated WSSs are both 0.10$\text{μm}$, as shown in Figs. 7(a1) and 7(b1). Different from Fig. 4, the convexity [Fig. 5(a2)] and concavity [Fig. 5(b2)] on both sides of each actuator no longer exist. All the convex hulls [Fig. 7(a1)] and the concave hulls [Fig. 7(b1)] on the surface shape move to the position of each actuator. Figures 7(a3) and 7(b3) show the 3D maps of the pre-compensated SSHFs after filtering the first 40 orders Zernike aberration of Figs. 7(a1) and 7(b1), while the 2D maps are shown in Figs. 7(a4) and 7(b4). The PV values of Figs. 7(a3) –7(a4) and 7(b3) –7(b4) are 0.08$\text{μ}$m. Figures 7(a2) and 7(b2) display the PSD curves of the pre-compensated WSSs of the DM with the TCM working at the optimal compensation temperature. Although the PSD curve still locates above the standard line from low-frequency to high-frequency, the gap between the PSD curve and the standard line becomes smaller, comparing to Figs. 5(a2) and 5(b2).

Thus, by using the TCM, the distorted WSSs of the DM shown in Figs. 5(a1) and 5(b1) are improved to the pre-compensated WSSs shown in Figs. 7(a1) and 7(b1), which do not contain dynamic HFD caused by the actuators tilt. This indicates that incline angle ${\theta }_{i,j}$ reaches to zero [Eq. (10)] and the dynamic HFD caused by the shear components $\Delta h_{ijx}$ and $\Delta h_{ijy}$ [Eq. (6)] are eliminated. Therefore, only the normal component $\Delta h_{ijz}$ [Eq. (6)], which could be corrected by the DM itself, remains on the surface shape. Based on the new surface shapes shown in Figs. 7(a1) and 7(b1), the DM could achieve better self-compensation results [Figs. 7(a5) and 7(b5)]. As shown in Figs. 7(a5) and 7(b5), the PV values of the fitting residuals of the DM self-compensation could achieve as good as 5nm, which indicates that the TID of the DM could be well depressed by using the TCM as pre-compensation method and the DM as the distortion-compensation approach.

Figures 8(a) and 8(b) show the relationship between the PV value of the WSS and the set compensation temperature $T_{TCM}$ of the TCM at different environment temperature. Correspondingly, the varying tendency of the PV value of the SSHF is illustrated in Figs. 8(c) and 8(d). As shown in Fig. 8, for different environment temperatures, there always exists an optimal compensation temperature $T_{TCM}$, where the PV value of the WSS/SSHF could achieve the minimum. For instance, as the red curve shown in Fig. 8(a), the optimal compensation temperature of the TCM for the 15$℃$ environment temperature is 23$℃$.

 

Fig. 8 The PV values of the TIDs with the TCM working at different temperatures. (a)–(d) show the influence of the set compensation temperature of the TCM at different $T_e$. (e) shows the influence of the $\Delta T_e$ on the optimal $\Delta T_{TCM}$, the $\Delta T_{eB}$ and the PV of the WSS.

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As shown in Fig. 8(e), summarized from Figs. 8(a) and 8(b), the optimal $\Delta T_{TCM}$ (blue solid circles) and $\Delta T_{eB}$ (red solid circles) exist for each $\Delta T_e$. The optimal $\Delta T_{TCM}$ is the difference between the optimal TCM temperature $T_{TCM}$ and the manufacturing temperature $T_m$. For instance, the optimal $\Delta T_{TCM}$ and $\Delta T_{eB}$ are 3$℃$ and −2$℃$ when $\Delta T_e$ is equal to −5$℃$. The optimal $\Delta T_{eB}$ is linear with the $\Delta T_e$, and the fitting slope is 0.36, which is almost the value of the ${\alpha }_G/{\alpha }_B$ by introducing the material parameters listed in Table 1. The PV value (green hollow diamonds) of the WSS at the optimal $\Delta T_{eB}$ is also linear with $\Delta T_e$ (0.02$\text{μm}/℃$ fitting slope). As Fig. 8(e) shown, different from Figs. 6(b) and 6(d), the PV values of the self-compensation residue at the optimal $\Delta T_{eB}$ are nearly equal to zero and stable. This indicate that the residual TID could be compensated by the DM itself even the remained normal component increases with $\Delta T_e$ when the TCM working at the optimal temperature.

4. Experiments and results

The interferometric technique is the widely used to measure the optical surface shape. However, the accuracy of the interferometer is heavily influenced by the stability of the temperature. The deflectometry system (DS) [29,30] is chosen to measure the evolution process of the surface shape of the DM. As shown in Figs. 9(a) and 9(b), a DS consisting of a LCD display (1024 × 768 pixels, UM-900 of Lilliput Co., Ltd), a CCD camera (1384 × 1036 pixels, GS3-U3-14S5M-C of Point Grey Research Inc.) is built. The lab-manufactured 49-actuator DM (manufacture temperature $T_m=$ 23$℃$) [Fig. 9(c)] with an 84mm × 84mm continuous mirror plate (>98%HR@1064nm and >50%R@632nm) and square distributed stacked PZT actuators (P885.91, Physik Instrumente GmbH, 5mm × 5mm × 36mm, 38μm stroke) array was rigidly connected to the steel base. The auxiliary TCM consisting of the TECs and a water-cooling block is screwed to the steel base as shown in Figs. 9(c) and 9(d). The material parameters of the steel base and the mirror are listed in Table 1.

 

Fig. 9 (a) is the experiment setup; (b) is the layout of the experiment setup containing the control circuits; (c) is the lab-manufactured DM with a TCM; (d) is the bottom of the water-cooling block.

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Before measurement, the DS should be calibrated to acquire the relationship between the camera coordinate and the world coordinate [30]. After calibration, the reference phase of the measured DM is acquired, setting the ambient temperature equal to the mounting temperature ($T_m$). Two sets of grating patterns with sinusoidal intensity profile along the X and Y directions are displayed on the LCD and imaged on the CCD camera by the reflection of the surface of the DM [Fig. 9(a)]. To obtain high measurement accuracy, the LCD produces four phase-shift fringe images and the CCD captures four times repeatedly. A phase-shift algorithm is applied to computer the phase distribution on the DM. The measurement result is used as the reference phase, which reflects the original surface shape of the DM. When the ambient temperature is set to the target working temperature $T_e$, the surface shape of the DM will change accordingly, which results in a new phase distribution. Then, the four phase-shift fringe images and algorithm are applied to computer the new phase distribution. Base on the reference phase, the phase difference of each pixel is calculated and the variation of the surface shape of the DM could be achieved using a reconstruct algorithm [27].

In the experiment, the TIDs of the DM without the TCM working are measured first. When the ambient temperature is set to 17.3 $℃$ or 28.7 $℃$ ($\Delta T_e=$-5.7$℃$, 5.7$℃$), the distorted WSSs of the DM are shown in Figs. 10(a1) and 10(b1). Figures 10(a2) and 10(b2) display the PSD curves of the distorted WSSs of the DM without the TCM working, at $\Delta T_e=$-5.7$℃$ and 5.7$℃$. As shown in Figs. 10(a2) and 10(b2), the PSD curves of the distorted WSSs locate above the standard line from low-frequency to high-frequency, which demonstrates the surface shape of the DM with TIDs are not qualified.

 

Fig. 10 The TIDs of the DM without the TCM working. (a1) is the distorted WSS at $\Delta T_e=$-5.7$℃$, PV value = 2.17μm. (b1) is the distorted WSS at $\Delta T_e=$5.7$℃$, PV value = 1.97μm. (a2) and (b2) are the PSD curves of the WSSs. (a3) is the SSHF after filtering the first 40-orders Zernike aberration of (a1), PV value = 0.71μm, with the actuators’ position marked by black circles. (b3) is the SSHF after filtering the first 40-orders Zernike aberration of (b1), PV value = 0.69μm. with the actuators’ position marked by black circles. (a4) and (b4) are the influence of $\Delta T_e$ on the PV and RMS of the distorted WSSs.

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The convex hulls [Fig. 10(a)] and the concave hulls [Fig. 10(b)] appear with an approximate square matrix distribution on the surface shape. Figures 10(a3) and 10(b3) show the SSHF of Figs. 10(a1) and 10(b1) with 7 × 7 square matrix black circles representing the actuators’ distribution, respectively. The convex and the concave hulls locate at both sides of the actuators [Figs. 10(a3) and 10(b3)]. Different from the simulation, in the experiment, due to the inhomogeneity of the assembly stress, not all the hulls could be observed on the surface shape of the DM. Figures 10(a4) and 10(b4) show that the PV and RMS values of the distorted WSSs and SSHFs increase linearly with the absolute value of $\Delta T_e$.

Figures 11(a1) –11(a2) and Figs. 11(b1) –11(b2) show the WSSs and the SSHFs of the compensation residue by the DM at $\Delta T_e=$-5.7 $℃$ and 5.7 $℃$, respectively, without the TCM working. As shown in Fig. 11(a1), after compensation, the HFD still remain on the surface of the DM. Consequently, the convex and concave hulls appear more distinct in the compensation residue [Figs. 11(a1) and 11(b1)] than in the initial distorted WSSs [Figs. 10(a1) and 10(b1)]. At $\Delta T_e=$ −5.7 $℃$, the PV values of the WSSs before and after compensation are 2.17$\text{μm}$ [Fig. 10(a1)] and 1.46$\text{μm}$ [Fig. 11(a1)], while the PV values of the SSHFs are 0.71$\text{μm}$ [Fig. 10(a3)] and 1.02$\text{μm}$ [Fig. 11(a2)]. At $\Delta T_e=$ 5.7 $℃$, the PV values of the WSSs before and after compensation are 1.97$\text{μm}$ [Fig. 10(b1)] and 1.41$\text{μm}$ [Fig. 11(b1)], while the PV values of the SSHFs are 0.69$\text{μm}$ [Fig. 10(b3)] and 0.75$\text{μm}$ [Fig. 11(b2)]. Figures 11(a3) and 11(b3) show the PSD curves of the initial distorted WSSs [Figs. 10(a1) and 10(b1)] in black and the compensation residues without TCM working [Figs. 11(a1) and 11(b1)] in green. As shown in Figs. 11(a3) and 11(b3), the PSD curves of the compensation residues are almost the same with that of the initial distorted WSSs, which both locate above the standard line.

 

Fig. 11 The compensation residues of the TIDs without the TCM working. (a1) is the compensation residue at $\Delta T_e=$ −5.7 $℃$, PV = 1.46μm, RMS = 0.27 $\text{μm}$. (b1) is the compensation residue at $\Delta T_e=$5.7$℃$, PV = 1.41μm, RMS = 0.29$\text{μm}$. (a2) (PV = 1.02μm, RMS = 0.09$\text{μm}$) and (b2) (PV = 0.75μm, RMS = 0.08$\text{μm}$) are the SSHFs of the compensation residues after filtering the first 40-orders Zernike aberration. (a3) and (b3) are the PSD curves of the TIDs and the compensation residues without TCM working at $\Delta T_e=$ −5.7 $℃$ and $\Delta T_e=$5.7 $℃$, respectively.

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Experiment results above indicate that the TID (i.e. the distorted WSS) is the dynamic HFD, which could not be effectively compensated and will remain on the surface of the DM. Restricted by the phenomenon of the unwanted and uncorrectable TID, the correction capability of the DM is seriously limited in the practical working environment for the temperature variation.

A hybrid close-loop control algorithm (HCLA) is proposed to compensate the TID of the manufactured DM using an auxiliary TCM. As shown in Fig. 12, the HCLA consists of two major compensation steps, including the TCM pre-compensation and the self-compensation of the DM. In the TCM pre-compensation step, the steel base’s temperature is affected directly by the TCM, which is driven by the computer controlled TECs. Along with the temperature variation of the steel base, the TID is compensated accordingly and the HFD of the DM disappears from the surface shape of the DM. When the HFD corresponding to the actuators couldn’t be observed in the WSS of the DM, the self-compensation of the DM is triggered. In the self-compensation step, the normal displacement on the surface shape is compensated by the pushing and pulling activities of the actuators of the DM. Here, in our experiment, the PV value of the residue WSS (judgement value) is set as the ending criterion for the close-loop algorithm. If the judgement value reaches 0.5$\text{μm}$, the hybrid close-loop algorithm is accomplished and a well-compensated surface shape is finally achieved. If the judgement value is still higher than 0.5$\text{μm}$, the HFD caused by the shear displacements is considered remaining on the surface and the hybrid close-loop algorithm returns to the first step.

 

Fig. 12 The hybrid close-loop control algorithm.

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In the experiment, $\Delta T_{eB}$ of the steel base is controlled by the TCM and measured in real time. Figures 13(a1) and 13(b1) depict the pre-compensation results of the TIDs after the TCM compensation step while $\Delta T_{eB}$ is measured −3.5 $℃$ at $\Delta T_e=$ −5.7 $℃$ and 3.2 $℃$ at $\Delta T_e=$5.7 $℃$. The WSSs of the TCM-compensation results are shown in Figs. 13(a1) and 13(b1) when $\Delta T_e=$ −5.7 $℃$ and 5.7 $℃$, respectively. The convex and the concave hulls in Fig. 10 almost disappear from the surface shape of the DM with the PV values of the WSSs decreasing from 2.17$\text{μm}$ and 1.97$\text{μm}$ [Figs. 10(a1) and 10(b1)] to 1.61$\text{μm}$ and 1.48$\text{μm}$ [Figs. 13(a1) and 13(b1)].

 

Fig. 13 (a1) and (b1) are the WSSs of the pre-compensation results of the TIDs after the first step (only the TCM working) at $\Delta T_e=$-5.7$℃$ (PV = 1.61$\text{μm}$, RMS = 0.32$\text{μm}$) and $\Delta T_e=$ 5.7$℃$ (PV = 1.48$\text{μm}$, RMS = 0.30$\text{μm}$). (a2) and (b2) are the self-compensation results when the hybrid close-loop algorithm is accomplished at $\Delta T_e=$ −5.7$℃$ (PV = 0.44$\text{μm}$, RMS = 0.09$\text{μm}$) and $\Delta T_e=$ 5.7$℃$ (PV = 0.45$\text{μm}$, RMS = 0.09$\text{μm}$). (a3) and (b3) are the SSHF of the self-compensation residues after filtering the first 40-orders Zernike aberration at $\Delta T_e=$ −5.7$℃$ (PV = 0.22μm, RMS = 0.03$\text{μm}$) and $\Delta T_e=$ 5.7$℃$ (PV = 0.22μm, RMS = 0.03$\text{μm}$). (a4) and (b4) are the PSDs of the initial distorted WSSs and the compensation residues after the hybrid close-loop algorithm is accomplished at $\Delta T_e=$ −5.7$℃$ and $\Delta T_e=$ 5.7$℃$.

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Figures 13(a2) –13(a3) and 13(b2) –13(b3) depict the self-compensation residues of the TID when the hybrid close-loop algorithm is accomplished. The WSSs of the residue at $\Delta T_e=$ −5.7 $℃$ and 5.7 $℃$ are shown in Figs. 13(a2) and 13(b2). As seen from Figs. 13(a2) and 13(b2), no convex and concave hulls in square array distribution corresponding to the actuators could be observed. At $\Delta T_e=$ −5.7 $℃$, the PV value of the WSS after self-compensation is 0.44$\text{μm}$ [Fig. 13(a2)], only 20% of the initial distorted WSS [Fig. 10(a1)], while the PV value of the SSHF after self-compensation [Fig. 13(a3)] is 0.22$\text{μm}$, only 31% of the initial distorted SSHF [Fig. 10(a3)]. Correspondingly, at $\Delta T_e=$ 5.7 $℃$, the PV value of the WSS after self-compensation [Fig. 13(b2)] is 0.45$\text{μm}$, which is 23% of the initial distorted WSS [Fig. 10(b1)], while the PV value of the SSHF after self-compensation [Fig. 13(b3)] is 0.21$\text{μm}$, which is 30% of the initial distorted SSHF [Fig. 10(b3)]. Figures 13(a4) and 13(b4) show the PSD curves of the initial distorted WSSs [Figs. 10(a1) and 10(b1)] in black and the self-compensation residues [Figs. 13(a2) and 13(b2)] in green when the hybrid close-loop algorithm is accomplished. As shown in Figs. 13(a4) and 13(b4), the PSD curves of the self-compensation residues shift away from the black curves and get very close to the red standard line, which demonstrates that the TIDs could be effectively compensated when the hybrid close-loop control is achieved.

Thus, by using the TCM, the initial TID of the DM shown in Figs. 10(a1) and 10(b1) are improved to the surface shapes shown in Figs. 13(a1) and 13(b1), which do not contain dynamic HFD caused by temperature variation. Based on the surface shapes shown in Figs. 13(a1) and 13(b1), a HCLA is carried out and a well-compensated WSS is finally achieved. As shown in Figs. 13(a2) and 13(b2), the PV values of the compensation residues of the DM could achieve as good as 23% of the initial TID. The application field of a DM is not restricted by the working temperature and could be extended to the working environment with a wider ambient temperature range.

In the experiment, some parameters are not as ideal as that set in the simulation, especially the influence function, the target temperature of the TCM and the residual assembly stress between the mirror and the 49-actuators. Thus, limited by the experimental condition, when the hybrid close-loop algorithm is accomplished in the experiment, the self-compensation residues are not as good as that in the simulation.

5. Conclusion

In conclusion, the mechanism and the compensation method of the TID of the DM is investigated theoretically and experimentally. The actuators’ tilt caused by the thermal expansion coefficient difference between the mirror and the steel base is the essential reason of the TID. Due to the tilt actuators’ shear components, the TID couldn’t be effectively corrected by the DM itself. To eliminate the actuators’ tilt, an auxiliary temperature module is proposed to regulate the steel base’s temperature. The compensation principle and the feasibility of the TCM are investigated. A finite element model is built to investigate the influence of the temperature and the thermal expansion coefficient differences on the TID and the TCM’s compensation capability. Simulation results show that the TID, positively related to the temperature and the thermal expansion coefficient difference, could be compensated effectively by the TCM. In experiment, a HCLA is adopted to control the compensation process of the TCM and the DM. Experiment results show that the TID is well depressed by using the TCM and the DM itself. Finally, a well-compensated DM surface shape is achieved in our experiment.