Surface Analysis and Optimization of Large Aperture Mirror Supported by Elastic Expansion Sleeves

In order to meet the stability of the support structure of arrayed large aperture mirror and improve the surface accuracy of the mirror, the elastic expansion sleeve is proposed as the back-support structure of the large aperture mirror. The feasibility of the support method is verified by theoretical analysis and experiment. The PV value of additional wavefront distortion is about 75.3nm<λ/5(λ=632.8nm) when the torque of single mandrel screw is 40N·cm, which meets the technical specifications. The change trend of the support-induced surface is compared between the finite element simulation and the measured results under different mandrel screw torque conditions. When the support torque is increased in equal increments, there is a linear positive correlation between the PV value of support-induced surface and the support torque. Based on the change trend of the support-induced surface, the optimization design scheme of the initial surface is proposed. And the simulation results show that the optimized induced distortion is smaller than that of the ideal surface (the compensation ratio is 7.68%), which reduces the additional wavefront distortion under the actual mounting support.


Introduction
In high-power laser systems, large aperture mirrors play a crucial role [1,2].The surface accuracy and structural stability of the mirror have a direct impact on beam positioning [3,4].Currently, the clamping and support structures used for large aperture mirrors mainly include peripheral clamping, side clamping, and back support [5,6].Scholars both domestically and internationally have conducted in-depth research on these structures and proposed a series of optimization methods.
The large aperture mirror studied in this paper is a component of SG-II main device and is used in the Mega-Joule Level Inertial Confinement Fusion (ICF) facility.It adopts a back support structure with elastic expansion sleeves [7,8], which not only accommodates an array layout of mirror elements but also reduces the radial installation space.
In this paper, the feasibility of the support method is verified through an analysis of additional surface distortion.The reliability of the finite element analysis model is validated by comparing the supportinduced surfaces obtained from finite element analysis and experimental measurements under different mandrel screw torques.The relationship curve between mandrel screw torque and support-induced surface PV value is derived.An optimization design scheme for the initial surface of the mirror is proposed, which compensates to some extent for the support-induced surface and reduces additional

Analysis of Additional Wavefront Distortion in Elastic Expansion
The support structure of the mirror in this study is composed of three elastic expansion sleeves at the back.The elastic expansion sleeve consists of a canister and a shaft, which are used in conjunction with a cushion.One end of the elastic expansion sleeve is connected to the mirror counterbore, while the other end is fixed to the cushion.The outer ring of the cushion has screw holes for the canister, and the inner ring has screw holes for the shaft.The dimensions of the mirror are 610mm × 440mm × 85mm, and the material used is H-K9L.Both elastic expansion sleeve and the cushion are made of 4J45 alloy steel material, with similar thermal expansion coefficients (figure 1).Compared to traditional canisters with a single gap slot, the outer surface of the elastic expansion sleeves in this study features multiple gap slots, which results in more uniform force distribution among the connecting components and smaller motion transmission errors.
In this experiment, a large-diameter interferometer with a diameter of 600mm was used to measure the surface of the mirror supported by the three-point canister at the back.The experiment was conducted at a room temperature of 25°C, with the mirror placed vertically on the interferometer measurement platform.Before each measurement, the mirror was left undisturbed for one day.During the experiment, the single mandrel screw is loaded with torque  = 75.3nm, is smaller than 0 PV (figure 3), meeting the technical specifications for the additional wavefront distortion of the mirror.The elastic expansion sleeve structure proposed in this study can meet the operational requirements under practical conditions.

Comparison of Support-induced Surface between Simulation and Experiment
In this paper, the overall surface ( ) ( , ) n Q x y of the mirror is primarily influenced by three factors: the selfweight-induced surface ( In the equation, ( + constitutes a bare mirror surface.The additional wavefront distortion 2 PV can be express as: In finite element analysis [9], boundary conditions are set to analyze the additional surface () In finite element simulation, starting with a mandrel screw torque of 40 N cm  , the variation trend of the support-induced surface was analyzed in mandrel screw torque conditions.A finite element model was created in ANSYS finite element software, and the material parameter table is shown in table 1.Based on the constraint relationship observed in the mirror measurement experiment, external constraints were added to establish a model of a large aperture mirror supported by the elastic expansion sleeve.The coordinate data of each node after the mirror's Z-direction deformation due to mounting is derived.This data includes rigid displacement terms such as translation and tilt.The Zernike polynomials are used to fit and separate the rigid displacement terms [10].
The surface data obtained from the experiment was decoupled and separated into the induced-surface of the support using numerical calculations in MetroPro software (as shown in figure 4).The calculation formula for this separation process is given by equation (3).

The Relationship between Mandrel Screw Torques and Support-induced Distortion
To further analyze the effect of the mandrel screw torque ( T ) on the support-induced distortion, measurements and finite element simulations of the surface deformation were conducted based on different mandrel screw torque values within the range of 40 with an increment of 5 The comparison of the support-induced distortion between the experimental measurements and finite element simulations is shown in figure 6.From figure 6, it can be observed that in both experimental and simulation results, the PV value of the support-induced distortion is linearly positively correlated with the mandrel screw torque T .When ) between the experiment and simulation, it is observed that the predicted PV value under the simulation model changes faster than the measured model.When the mandrel screw torque changes by 1

N cm
 , the difference in PV value between the simulated support-induced distortion and the measured support-induced distortion is 0.00422λ.This error is approximately 1/50 of the technical specification for the additional wavefront distortion of the mirror (λ/5).The analysis results indicate that the finite element model has a high accuracy and can be used for the analysis and prediction of the support-induced distortion, providing greater safety.
Analyzing figure 8, it is observed that the compensation value PV The compensation ratio  is influenced by the design parameters of the mirror's initial surface (in this study, the central sagitta and curvature of the mirror).Under the same design of initial surface parameters, the compensation value PV  of the support-induced surface is linearly positively correlated with the mandrel screw torque T , while the compensation ratio  remains constant.However, for different designs of the mirror's initial surface parameters, the compensation ratio  will vary.

Conclusion
In this study, a model of large-aperture mirror supported by elastic expansion sleeves was established.The measured additional face shape distortion under met the design criteria.Simulated support-induced surface distortions and measured support-induced surface distortions were calculated for different mandrel screw torque conditions.The mandrel screw torque showed a linear positive correlation with the support-induced surface distortion PV value.An optimized design proposal with an initial concave-convex surface is proposed.Simulation results demonstrated that under the same mounting conditions, the concave-convex mirror surface generated smaller overall surface distortion (PV compensation ratio ) compared to the ideal surface, thus reducing the additional wavefront distortion of the mirror supported by elastic expansion sleeves.

Figure 1 .
Figure 1.Back support structure of mirror.
deformation maps of the mirror before and after loading are shown in figure 2. (a) Without mounting.(b) With mounting.

Figure 2 .
Figure 2. Surface deformation contours of mirror.Experimental design criteria indicate that the additional wavefront distortion of the mirror should be within 0 PV = λ/5 (λ = 632.8nm).The distortion of the bare mirror, 1 PV = 298.1nm.under the condition of mandrel screw torque 40 T N cm = , four experiments were conducted, and the average value of the experimental results yielded an average mounted surface of PV = 373.4nm.The additional wavefront distortion of the mirror,2

Figure 3 .
Figure 3. Experimental measurement results of mirror distortion.
of the mirror, the initial surface ( of the mirror, and the supportinduced surface () due to the elastic expansion sleeve.

yy
y due to the self-weight of the mirror and the support-induced surface ( of the mirror accurately in finite element simulations.The measured surface results in actual experiments include the bare mirror surface of the mirror due to self-weight remains constant, making the support-induced surface ( ) 3 ( , ) n Q x y the only comparative indicator between finite element simulations and experiments.

Figure 4 .
Figure 4. Numerical decoupling calculation process of support-induced surface.In the equation: () ( , ) n m z x y represents the induced-surface caused by the elastic expansion sleeve.

y
represents the surface figure without mounting force.Equation (3) is equivalent to equation (1).The induced-surface () the elastic expansion sleeve corresponds to the support-induced surface () 3 ( , ) n Q x y .The surface figure with mounting force () same mounting conditions, the measured and simulated support-induced surface results are shown in figure 5.The overall deformation of the mirror is basically consistent in both cases, with a trend of outward bulging and inward concavity.The maximum deformation occurs at the periphery of the mirror, while the deformation in the central light-transmitting region is smaller.In both surface deformations, the distortions caused by the three-point support at the back are clearly visible.The measured results are consistent with the finite element simulation results, indicating good reliability of the finite element model.(a) Experiment.(b) Simulation.

Figure 5 .
Figure 5. Support-induced surface results of experiment and simulation.

Figure 6 .
Figure 6.Variation of PV values with different torques for each mandrel screw.

,
the predicted PV value from the simulation is greater than the measured PV value.The slope of PV value change in the measured support-induced distortion ( a K ) and the calculated PV value change in the finite element model ( b K ) for the mirror are: represents the measured change in PV value of the support-induced distortion, and b P  represents the simulated change in PV value of the support-induced distortion.Comparing the slope of PV value change (

y
of the support-induced surface distortion is linearly positively correlated with the mandrel screw torque.If disregarding the additional surface ( caused by the mirror's own weight, the compensation ratio  for the overall surface distortion can be determined.

Table 1 .
Simulation model material parameters.