- Email: [email protected]
tilt mirror
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Dynamic surface parameter analysis for frequency response of tip/ tilt mirror
T
Guangzhou Xu* Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Tip/tilt mirror Dynamic surface parameter Discrete error Opto-mechanical integrated simulation
The tip/tilt mirror performs controllable high-speed swing to stabilize the imaging beam and compensate for the external micro-vibration for optical telescope. The calculation of surface parameter under dynamic driven frequency is the design foundation of tip/tilt mirror. To resolve the dynamic surface parameter calculation problem, the frequency response analysis of tip/tilt mirror is carried out; the discrete error and elimination method of optical surface for tip/tilt mirror is proposed as well to improve the surface calculation precision of tip/tilt mirror; the optomechanical simulation technology is selected to process the dynamic surface change data of tip/ tilt mirror and it has been proved the effectiveness to obtain the dynamic surface parameter of tip/tilt mirror. The calculation results demonstrate the nonlinear change rule between the surface change parameter and the driven frequency; the surface change is not obvious under the lower micro-vibration driven frequency, but is obvious under the higher driven frequency; the microvibration frequency and environmental adaption of tip/tilt mirror are determined based on the dynamic surface parameter simulation results. The research of dynamic surface parameter for frequency response of tip/tilt mirror provides a new technical reference for the dynamic surface simulation and structural design of tip/tilt mirror.
1. Introduction For the high-resolution spatial optical telescope, the resolution of the telescope can be easily influenced by the external microvibration caused by the momentum wheel and micro-gravity. Generally, the satellite can provide the attitude control accuracy with about 10″, but as for the high-resolution space optical telescopes, the satellite attitude control accuracy cannot satisfy the highprecision observation requirements of the telescope. To decrease the influence of image quality of the external micro-vibration for the optical telescope, the related image stabilization technology is produced. The optical image stabilization technology is an effective method to compensate for the optical axis disturbance in real time. The typical optical image stabilization unit is composed of the tip/ tilt mirror unit, the image shift detection unit and the image shift compensation controller unit [1] and as shown in Fig. 1. The tip/tilt mirror unit is the executive part to compensate for optical axis disturbance. It is generally composed of the tip/tilt mirror, the driving platform and the flexible mounting structure and as shown in Fig. 2. The tip/tilt mirror acts between the incident beam and it performs the controllable high-speed swing to stabilize the imaging beam and compensate for the influence of external micro-vibration on the image quality for the entire optical system. As the important optical assembly, the tip/tilt mirror swings under the control of driven platform and is widely used in the satellite optical sensing detector. For example, with the help of tip/tilt mirror unit of optical image stabilization, the Solar-B telescope
⁎
Corresponding author. E-mail address: [email protected].
https://doi.org/10.1016/j.ijleo.2019.163964 Received 8 October 2019; Accepted 2 December 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
G. Xu
Fig. 1. Optical image stabilization unit.
Fig. 2. The tip/tilt mirror unit.
obtains the solar magnetic field image with the resolution of 0.3″ and the JWST telescope will gain the high-resolution image with 0.1″ [2]. Currently, the aperture of requirement for tip/tilt mirror is getting larger from Φ70 mm to Φ160 mm and even more larger; the tip/tilt mirror needs to adapt to the external micro-vibration excitation especially the high-frequency excitation; that is the mirror needs to keep good performance under the external micro-vibration excited frequency. The research of dynamic surface change under the driven excitation frequency of tip/tilt platform has become the important technical problem for the design of structure and external micro-vibration adaption for tip/tilt mirror. Besides, the relationship between the dynamic surface change and the driven frequency is an attractive research problem as well. The simulation of dynamic surface parameter under different driven frequency of tip/tilt platform mainly involves the following contents: the tip/tilt mirror structure, the frequency response analysis of tip/tilt mirror and the calculation of dynamic surface parameter.
2. Structure of tip/tilt unit As discussed above, the tip/tilt mirror unit assembly is made up of tip-tilt mirror, tip/tilt platform and flexible support structure. Based on the current technical requirement, the effective aperture of tip/tilt mirror is Φ160 mm with SiC material. The flexible support is designed to decrease the stress influence caused by the mechanical connection and thermal deformation between the mirror and the platform. The material of flexible support is 4J32 Invar alloy. The tip/tilt mirror assembly is made up of tip/tilt mirror and flexible support. The swing part of the platform is driven by 4 piezoelectric actuators. The flexure hinge is the necessary structural form to realize good kinematics performance of tip/tilt platform. The tip/tilt mirror assembly is connected to the tip/tilt platform by 4 bolts and the mirror is glued to the flexible support. The tip/tilt unit assembly and the tip/tilt assembly is shown in Figs. 3 and 4 .
Fig. 3. Tip/tilt mirror unit assembly. 2
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
G. Xu
Fig. 4. Tip/tilt mirror assembly.
3. Optical surface parameter simulation of tip/tilt mirror The optical surface parameter simulation of tip/tilt mirror mainly refers to the dynamic surface parameter calculation under the different driven excitation frequency. To resolve this problem, the following procedure should be taken care. First, the finite element frequency response analysis of tip/tilt mirror should be carried out. Second, the dynamic surface parameter of tip/tilt mirror need to be performed. When the surface parameter calculation of tip/tilt mirror is executed, the finite element discrete error should be recognized and eliminated. In addition, surface parameter calculation under different excitation frequency involves the surface calculation of the rigid body displacement and surface distortion. As we know, the opto-mechanical integrated simulation technology is the important technical method to solve the image quality evaluation problem of optical payload under the external mechanical or thermal loads. Therefore, opto-mechanical integrated simulation technology is selected and researched to obtain the surface change parameter in different driven excitation frequency of tip/tilt platform. 3.1. Frequency response analysis of tip/tilt mirror Frequency response analysis is the effective method to calculate the steady-state response of structure. External loads is the function of driven frequency in the analysis of frequency response. The tip/tilt mirror excutes the controllable high-speed swing with the driven signal just as Φ = A × sin(2π × f × t), Where A is the amplitude value and f is the driven frequency. Frequency response analysis of tip/tilt mirror is to achieve the optical surface displacement change under the sine kinematics excitation. As we know, before conducting the frequency response of tip/tilt mirror, the first thing is to build the finite element model of tip/tilt mirror. Considering the move characteristics and simulation goal, the finite element model of tip/tilt mirror is simplified. The finite element model of tip/tilt platform is omitted and only the movement of the platform is taken care; that is the finite element model of tip/tilt mirror and its flexible support structure is built; the tip/tilt platform is simplified to the pivot point and the rigid connection element:RBE2 (for Nastran). The sine load is applied to the pivot point to simulate the kinematics excitation of the tip/tilt platform. The finite element model of tip/tilt mirror is shown in Fig. 5. As above discussion, the excitation is just as:Φ = A × sin(2π × f × t). Based on the design requirement, A = 0.2 mrad, and the range of frequency f changes from 20 Hz to 300 Hz. After executing the frequency response analysis by means of finite element method, the displacement distribution of tip/tilt mirror under certain excitation frequency of tip/tilt platform is shown in Fig. 6. The mirror surface produces the obvious displacement change under the certain driven frequency with swinging of tip/tilt mirror. The surface displacement of tip/tilt mirror is composed of rigid body displacement and the distortion displacement. We can distinguish the above two kinds of displacements and obtain the optical surface change parameter with the help of the subsequent optomechanical integrated simulation method. 3.2. Dynamic surface parameter calculation of tip/tilt mirror Generally, the PV and RMS value can be selected to describe the change of optical surface. To improve the calculation precision of
Fig. 5. Finite element model of tip/tilt mirror. 3
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
G. Xu
Fig. 6. Displacement of tip/tilt mirror under certain frequency.
the optical surface parameters of the tip/tilt mirror, the finite element discrete error of the tip/tilt mirror and its elimination method are studied as well. In addition, the opto-mechanical simulation technology is adopted to further analyze the dynamic surface parameter change detailed. the detailed changes of the optical surface of the tip/tilt mirror under different driven frequencies of can be comprehensively evaluated based on opto-mechanical integrated simulation technology. The mainly calculation procedure of dynamic surface parameter for tip/tilt mirror is as follows. First, the optical surface of finite element discrete error for tip/tilt mirror is eliminated to improve the surface parameter calculation precision. Second, the surface change is fitted with the help of opto-mechanical integrated simulation technology and the Zernike fitting coefficient under different driven frequency is obtained as well. Third, the fitting coefficient of tip/tilt mirror surface is transferred to the software Zemax and then the surface change analysis is carried out with the help of Zemax data procedure function. 3.2.1. Discrete error and elimination of optical surface When the frequency response of tip/tilt mirror assembly is carried out, the tip/tilt mirror assembly should be discretized first, the discretization procedure will produce the discrete error [3]. As for the general structural analysis, the discrete error can be omitted, but for the optical surface analysis, because the surface change magnitude is nanometer scale. The Discrete error of tip/tilt mirror should be considered and analyzed. Discrete error of optical surface can be defined as the discrete nodes of optical surface deviates original geometrical model, and then produce the position error of the nodes. Therefore, the discrete error of tip/tilt mirror should be eliminated firstly to achieve the high-precision optical surface parameter. The technical method to eliminate the discrete error is to use the projection node of discrete node to the theoretical plane instead of the finite element discrete node; that is to take ”▵” instead of ”□” (Figs. 7 and 8). Two methods can be taken to finish the above node replacement process. The first one is to search the discrete nodes automatically with related algorithm, and then project them to the theoretical optical plane. Another one is with the help of the finite element preprocessing tool, discrete nodes is selected manually, then project the selected node to the original optical plane. The first approach is more conveniently to be realized based on the developed program. 3.2.2. Opto-mechanical integrated simulation technology Opto-mechanical integrated simulation involves several essential algorithms. The node displacement conversion algorithm of optical surface and fitting algorithm of node displacement change in sagittal direction is the foundation to realize the opto-mechanical integrated simulation. Optical surface can be defined by the sagittal equation. The change of optical surface should be expressed in the sagittal direction as well [4]. There is a thing that needs to be cared is sagittal displacement is not equal to the change of finite element node displacement of optical surface. The reason is node displacement contains three coordinate components and it also includes the displacement change of the node position along the radial direction. The node displacement of optical surface calculated by the finite element analysis needs to transfer the node displacement to the node sagittal direction. The displacement conversion method for node displacement to the sagittal direction is shown in Fig. 4. The node P0 on the initial optical surface moves to the point P1 due to the external loads. The displacement in the optical axis direction is ΔZ and the displacement is ΔR in radial direction. It is necessary to calculate the parameter TP1 with converting the node
Fig. 7. Discrete error elimination. 4
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
G. Xu
Fig. 8. Node displacement conversion algorithm.
displacement to the sagittal direction, and use TP1 to describe the node displacement change of the optical surface.
⎧ TP1 = ΔZ + ΔZsag ΔZ = Z (R P0 ) + Z (R P0 + ΔR) ⎨ ⎩ sag
(1)
In the formula (1), Z = Z(R) is the sagittal equation of optical surface. The above displacement conversion algorithm based on sagittal equation is also selected in the subsequent surface fitting algorithm based on Zernike polynomials. When the opto-mechanical integrated simulation is performed, it is necessary to transfer the mirror surface change to the optical data format which can be recognized by the optical program, for example Zemax, and then evaluate the influence of external loads on optical parameters such as wavefront and image quality. Hence, the polynomial fitting of optical surface change is the first work to do to realize the opto-mechanical integration simulation. Fringe Zernike polynomials are selected as basis function to realize surface change fitting and the definition of Fringe Zernike polynomials can refer the Zamax help files. Gram-Schmidt method is used to construct fitting vectors as well. The constructed vectors are orthogonal and linearly independent. The optical surface change are expressed by the linear combination of Fringe Zernike polynomials. The solution of contradictory equation is solved by the generalized inverse method. The generalized inverse method is equivalent to the least square method in fitting the wavefront aberration function, but the derivation process and program implementation are more concise. The detailed derivation process to realize the Zernike fitting of optical surface change is as follows [5]. Let n is the node number of finite element model for optical surface and m is the item number of Zernike polynomial selected; Fn×1 is a vector formed by converting the displacement of finite element node to the direction of optical axis. Un×m is the matrix that is composed of column vector Uj with the jth basis function (j = 1, 2, .., m); Cm×1 is the linear combination coefficient obtained by fitting optical surface change. Because the basis function set Uj is linearly independent on the point set xi (i = 1, 2, .. ., n), it is equivalent to the vector set Φj is linearly independent.
⎛ Uj (x1) ⎜ Uj (x2) Φj = ⎜ · · ⎜ · ⎜U (x ) ⎝ j n
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
(2) m
Therefore, Uj can form a subspace of R . (3)
Un × m Cm × 1 = Fn × 1 Gram-Schmidt method is used to orthogonalize Un×m, and orthogonalization matrix Vn×m is obtained.
(4)
Un × m = Vn × m Rm × m Rm×m is the middle transition matrix from matrix Vnm to matrix Unm.
Vn × m Rm × m Cm × 1 = Fn × 1
(5)
Rm × m Cm × 1 = Qm × 1
(6)
Vn × m Qm × 1 = Fn × 1
(7)
Contradictory equation has the unique the least norm least squares solution.
Qm × 1 = Vn+× mFn × 1
(8)
Vn+× m = (V T V )−1V T
(9)
After getting of solution of equation (9), the Qm×1 is calculated by formula (8). At last, the Zernike polynomial linear combination coefficient Cm×1 is obtained. 5
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
G. Xu
Table 1 The fitting coefficient of three typical driven frequency. No./Freq.
31.2 Hz
165.6 Hz
277.6 Hz
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
−1.5093E-04 7.7038E-07 1.6217E-01 −7.4762E-10 −3.5472E-10 −1.0471E-08 8.7280E-09 1.3314E-06 −6.2362E-09 3.9559E-09 2.5711E-08 −1.0931E-08 −1.1954E-08 2.2771E-08 −5.4975E-07 −5.8347E-09 6.7097E-09 −1.4709E-08 −4.1988E-08 −1.4942E-08 3.9682E-08 7.4964E-09 −3.8144E-08 1.2172E-07 2.9577E-09 −7.4454E-10 3.7824E-08 −2.8623E-08 −6.7969E-09 1.6173E-08 −9.2318E-10 −2.9564E-08 1.9152E-08 −4.0424E-08 −3.6938E-09 6.7204E-09 −5.3175E-09
−1.7631E-04 2.2979E-05 1.6868E-01 −1.1673E-07 1.2338E-07 −5.7756E-09 1.8070E-08 4.0262E-05 8.7576E-08 −6.5032E-09 −2.4501E-08 −4.6525E-08 1.5022E-08 1.8056E-09 −1.5745E-05 −1.5181E-08 1.7619E-08 2.6872E-09 6.9456E-08 −1.1975E-08 −4.4725E-08 1.3036E-08 1.9430E-08 2.7254E-06 −4.4943E-08 −2.5067E-08 4.0870E-08 −1.7464E-08 3.4619E-08 4.5439E-08 −2.8875E-08 1.9043E-08 −2.3713E-08 −4.1560E-08 −2.2909E-07 3.7552E-08 3.4597E-08
−2.3194E-04 7.3910E-05 1.8224E-01 −3.8388E-07 4.1345E-07 −9.3151E-09 2.0582E-08 1.2204E-04 2.0279E-07 −5.9227E-08 −9.6199E-08 −2.1840E-07 −1.7984E-08 3.7998E-09 −4.7609E-05 −1.0826E-07 9.0529E-08 1.2747E-08 1.0217E-07 −1.8749E-08 −4.5587E-08 9.2654E-10 −1.0794E-08 8.2979E-06 −2.9411E-08 −3.1246E-08 1.0947E-07 −9.5173E-08 3.8385E-08 1.0227E-07 2.9848E-08 −9.1913E-09 −2.6806E-08 −4.4936E-08 −7.4882E-07 2.6949E-08 9.4727E-08
4. The dynamic surface parameter calculation of tip/tilt mirror The dynamic surface parameter analysis of tip/tilt mirror refers to the surface parameter distribution under the different excitation frequency. The frequency range of tip/tilt platform is changed from 20 Hz to 300 Hz, surface parameter is changed under the excitation of different driven frequency of tip/tilt platform. The surface change value under different driven frequency of tip/tilt mirror is calculated based on the above discussed surface calculation method. The detailed procedure is as follows. First, the fitting coefficient of tip/tilt mirror surface is gotten by means of opto-mechanical method. Second, the fitting coefficient is transferred to optical program Zemax. Third, the surface change data processing and graphical display can be carried out with the help of Zemax. The Fringe Zernike fitting coefficient of three typical driven frequency is as Table 1 shown.
Fig. 9. Surface contour with typical driven frequency. 6
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
G. Xu
Table 2 Surface RMS of tip/tilt mirror under different driven frequency. No.
Freq. (Hz)
Surface RMS (λ)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
20 31.2 42.4 53.6 64.8 76 87.2 98.4 109.6 120.8 132 143.2 154.4 165.6 176.8 188 199.2 210.4 221.6 232.8 244 255.2 266.4 277.6 288.8 300
0.00035 0.0008 0.0015 0.0024 0.0035 0.0048 0.00635 0.0081 0.0101 0.0123 0.01475 0.01745 0.0204 0.0236 0.02705 0.03075 0.03475 0.03905 0.04365 0.04855 0.05375 0.0593 0.06525 0.0715 0.07815 0.08525
The surface change contour of tip/tilt mirror under three typical driven frequency is shown in Fig. 9. The surface contour shows that the surface change distribution of tip/tilt mirror under the different driven frequency is similar, but the surface parameter value is different. The detailed calculation results of tip/tilt surface RMS parameter under different driven frequency is shown in Table 2. The surface RMS variation curve under the different driven frequency is shown in Fig. 10. According to Table 2 and Fig. 10, the following valuable conclusions can be drawn. The surface change shows the nonlinear characteristics with the increase of the driven frequency. When the driven frequency is relatively less, the surface change is not obvious and the surface change is obvious under the higher driven frequency. As for the current tip/tilt mirror with aperture Φ160 mm, the surface RMS change is less than 0.01λ when the driven frequency is less than 100Hz; the surface RMS change is less than 0.02λ with the driven frequency being less than 150Hz; when the driven frequency is less than 180 Hz, the surface RMS change is less than 0.03λ. The surface RMS change increases rapidly when the driven frequency is higher than 200 Hz and the surface RMS is only 0.085λ at the driven frequency with 300Hz. Current tip/tilt mirror can only works on the driven frequency less than 150 Hz with surface change 0.02λ based on the design requirement. That is the higher driven frequency which will lead to the higher surface wavefront change and then make influence to the optical system image quality. 5. Conclusions To analyze the dynamic surface parameter of frequency response for tip/tilt mirror, the dynamic surface parameter calculation method is researched and is mainly composed of finite element discrete error and its eliminating technique, the Zernike polynomials
Fig. 10. The surface RMS under different driven frequency. 7
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
G. Xu
fitting method and data processing procedure combining with Zemax. The Dynamic surface parameter of frequency response for tip/ tilt mirror is calculated with the proposed method under the different driven frequency of tip/tilt platform. The calculation result shows the nonlinear change rule between the surface change parameter and the driven frequency. The surface change is not obvious under the lower micro-vibration driven frequency, but the surface change is obvious on the higher driven frequency. The analysis conclusions reveal the surface change rule under the different driven frequency and are important for the design of tip/tilt mirror. In other words, when the tip/tilt mirror is designed, the first thing that should be considered is if the designed tip/tilt mirror can satisfy the micro-vibration frequency environment and the dynamic surface parameter calculation is need to be carried out as well. The research of dynamic surface parameter for frequency response of tip/tilt mirror provides a new technical reference and has the significant engineering and scientific value for the simulation and design of tip/tilt mirror. Acknowledgement This study was supported by CAS ”Light of West China” Program (No.XAB2017A10). In addition, many thanks to the listed reference authors and some anonymous researchers, their study inspire us and help us a lot to do this work. References [1] L.H. Yang, L.Z. Jun, X.P. Mei, An overview on image stabilization method of space-born remote sensing systems, Spacecraft Recov. Rem. Sens. 31 (2010) 52–57. [2] K. Ichimoto, S. Tsuneta, Y. Suematsu, T. Shimizu, M. Otsubo, Y. Kato, M. Noguchi, The solar optical telescope onboard the solar-b, Proc. SPIE 5847 (2004) 1142–1150. [3] G.Z. Xu, P. Ruan, J.F. Yang, F. Li, X.T. Yan, S. Wang, High-precision algorithm of surface parameter of parabolic primary mirror for space solar telescope, Optik 127 (2016) 10687–10696. [4] R.C. Juergens, P.A. Coronato, Improved method for transfer of fea result s to optical codes, Proc. SPIE 5174 (2003) 105–115. [5] K.B. Doyle, V.L. Genberg, G.J. Michels, Integrated Optomechanical Analysis, SPIE Press, 2002.
8
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Dynamic surface parameter analysis for frequency response of tip/ tilt mirror
T
Guangzhou Xu* Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Tip/tilt mirror Dynamic surface parameter Discrete error Opto-mechanical integrated simulation
The tip/tilt mirror performs controllable high-speed swing to stabilize the imaging beam and compensate for the external micro-vibration for optical telescope. The calculation of surface parameter under dynamic driven frequency is the design foundation of tip/tilt mirror. To resolve the dynamic surface parameter calculation problem, the frequency response analysis of tip/tilt mirror is carried out; the discrete error and elimination method of optical surface for tip/tilt mirror is proposed as well to improve the surface calculation precision of tip/tilt mirror; the optomechanical simulation technology is selected to process the dynamic surface change data of tip/ tilt mirror and it has been proved the effectiveness to obtain the dynamic surface parameter of tip/tilt mirror. The calculation results demonstrate the nonlinear change rule between the surface change parameter and the driven frequency; the surface change is not obvious under the lower micro-vibration driven frequency, but is obvious under the higher driven frequency; the microvibration frequency and environmental adaption of tip/tilt mirror are determined based on the dynamic surface parameter simulation results. The research of dynamic surface parameter for frequency response of tip/tilt mirror provides a new technical reference for the dynamic surface simulation and structural design of tip/tilt mirror.
1. Introduction For the high-resolution spatial optical telescope, the resolution of the telescope can be easily influenced by the external microvibration caused by the momentum wheel and micro-gravity. Generally, the satellite can provide the attitude control accuracy with about 10″, but as for the high-resolution space optical telescopes, the satellite attitude control accuracy cannot satisfy the highprecision observation requirements of the telescope. To decrease the influence of image quality of the external micro-vibration for the optical telescope, the related image stabilization technology is produced. The optical image stabilization technology is an effective method to compensate for the optical axis disturbance in real time. The typical optical image stabilization unit is composed of the tip/ tilt mirror unit, the image shift detection unit and the image shift compensation controller unit [1] and as shown in Fig. 1. The tip/tilt mirror unit is the executive part to compensate for optical axis disturbance. It is generally composed of the tip/tilt mirror, the driving platform and the flexible mounting structure and as shown in Fig. 2. The tip/tilt mirror acts between the incident beam and it performs the controllable high-speed swing to stabilize the imaging beam and compensate for the influence of external micro-vibration on the image quality for the entire optical system. As the important optical assembly, the tip/tilt mirror swings under the control of driven platform and is widely used in the satellite optical sensing detector. For example, with the help of tip/tilt mirror unit of optical image stabilization, the Solar-B telescope
⁎
Corresponding author. E-mail address: [email protected].
https://doi.org/10.1016/j.ijleo.2019.163964 Received 8 October 2019; Accepted 2 December 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
G. Xu
Fig. 1. Optical image stabilization unit.
Fig. 2. The tip/tilt mirror unit.
obtains the solar magnetic field image with the resolution of 0.3″ and the JWST telescope will gain the high-resolution image with 0.1″ [2]. Currently, the aperture of requirement for tip/tilt mirror is getting larger from Φ70 mm to Φ160 mm and even more larger; the tip/tilt mirror needs to adapt to the external micro-vibration excitation especially the high-frequency excitation; that is the mirror needs to keep good performance under the external micro-vibration excited frequency. The research of dynamic surface change under the driven excitation frequency of tip/tilt platform has become the important technical problem for the design of structure and external micro-vibration adaption for tip/tilt mirror. Besides, the relationship between the dynamic surface change and the driven frequency is an attractive research problem as well. The simulation of dynamic surface parameter under different driven frequency of tip/tilt platform mainly involves the following contents: the tip/tilt mirror structure, the frequency response analysis of tip/tilt mirror and the calculation of dynamic surface parameter.
2. Structure of tip/tilt unit As discussed above, the tip/tilt mirror unit assembly is made up of tip-tilt mirror, tip/tilt platform and flexible support structure. Based on the current technical requirement, the effective aperture of tip/tilt mirror is Φ160 mm with SiC material. The flexible support is designed to decrease the stress influence caused by the mechanical connection and thermal deformation between the mirror and the platform. The material of flexible support is 4J32 Invar alloy. The tip/tilt mirror assembly is made up of tip/tilt mirror and flexible support. The swing part of the platform is driven by 4 piezoelectric actuators. The flexure hinge is the necessary structural form to realize good kinematics performance of tip/tilt platform. The tip/tilt mirror assembly is connected to the tip/tilt platform by 4 bolts and the mirror is glued to the flexible support. The tip/tilt unit assembly and the tip/tilt assembly is shown in Figs. 3 and 4 .
Fig. 3. Tip/tilt mirror unit assembly. 2
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
G. Xu
Fig. 4. Tip/tilt mirror assembly.
3. Optical surface parameter simulation of tip/tilt mirror The optical surface parameter simulation of tip/tilt mirror mainly refers to the dynamic surface parameter calculation under the different driven excitation frequency. To resolve this problem, the following procedure should be taken care. First, the finite element frequency response analysis of tip/tilt mirror should be carried out. Second, the dynamic surface parameter of tip/tilt mirror need to be performed. When the surface parameter calculation of tip/tilt mirror is executed, the finite element discrete error should be recognized and eliminated. In addition, surface parameter calculation under different excitation frequency involves the surface calculation of the rigid body displacement and surface distortion. As we know, the opto-mechanical integrated simulation technology is the important technical method to solve the image quality evaluation problem of optical payload under the external mechanical or thermal loads. Therefore, opto-mechanical integrated simulation technology is selected and researched to obtain the surface change parameter in different driven excitation frequency of tip/tilt platform. 3.1. Frequency response analysis of tip/tilt mirror Frequency response analysis is the effective method to calculate the steady-state response of structure. External loads is the function of driven frequency in the analysis of frequency response. The tip/tilt mirror excutes the controllable high-speed swing with the driven signal just as Φ = A × sin(2π × f × t), Where A is the amplitude value and f is the driven frequency. Frequency response analysis of tip/tilt mirror is to achieve the optical surface displacement change under the sine kinematics excitation. As we know, before conducting the frequency response of tip/tilt mirror, the first thing is to build the finite element model of tip/tilt mirror. Considering the move characteristics and simulation goal, the finite element model of tip/tilt mirror is simplified. The finite element model of tip/tilt platform is omitted and only the movement of the platform is taken care; that is the finite element model of tip/tilt mirror and its flexible support structure is built; the tip/tilt platform is simplified to the pivot point and the rigid connection element:RBE2 (for Nastran). The sine load is applied to the pivot point to simulate the kinematics excitation of the tip/tilt platform. The finite element model of tip/tilt mirror is shown in Fig. 5. As above discussion, the excitation is just as:Φ = A × sin(2π × f × t). Based on the design requirement, A = 0.2 mrad, and the range of frequency f changes from 20 Hz to 300 Hz. After executing the frequency response analysis by means of finite element method, the displacement distribution of tip/tilt mirror under certain excitation frequency of tip/tilt platform is shown in Fig. 6. The mirror surface produces the obvious displacement change under the certain driven frequency with swinging of tip/tilt mirror. The surface displacement of tip/tilt mirror is composed of rigid body displacement and the distortion displacement. We can distinguish the above two kinds of displacements and obtain the optical surface change parameter with the help of the subsequent optomechanical integrated simulation method. 3.2. Dynamic surface parameter calculation of tip/tilt mirror Generally, the PV and RMS value can be selected to describe the change of optical surface. To improve the calculation precision of
Fig. 5. Finite element model of tip/tilt mirror. 3
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
G. Xu
Fig. 6. Displacement of tip/tilt mirror under certain frequency.
the optical surface parameters of the tip/tilt mirror, the finite element discrete error of the tip/tilt mirror and its elimination method are studied as well. In addition, the opto-mechanical simulation technology is adopted to further analyze the dynamic surface parameter change detailed. the detailed changes of the optical surface of the tip/tilt mirror under different driven frequencies of can be comprehensively evaluated based on opto-mechanical integrated simulation technology. The mainly calculation procedure of dynamic surface parameter for tip/tilt mirror is as follows. First, the optical surface of finite element discrete error for tip/tilt mirror is eliminated to improve the surface parameter calculation precision. Second, the surface change is fitted with the help of opto-mechanical integrated simulation technology and the Zernike fitting coefficient under different driven frequency is obtained as well. Third, the fitting coefficient of tip/tilt mirror surface is transferred to the software Zemax and then the surface change analysis is carried out with the help of Zemax data procedure function. 3.2.1. Discrete error and elimination of optical surface When the frequency response of tip/tilt mirror assembly is carried out, the tip/tilt mirror assembly should be discretized first, the discretization procedure will produce the discrete error [3]. As for the general structural analysis, the discrete error can be omitted, but for the optical surface analysis, because the surface change magnitude is nanometer scale. The Discrete error of tip/tilt mirror should be considered and analyzed. Discrete error of optical surface can be defined as the discrete nodes of optical surface deviates original geometrical model, and then produce the position error of the nodes. Therefore, the discrete error of tip/tilt mirror should be eliminated firstly to achieve the high-precision optical surface parameter. The technical method to eliminate the discrete error is to use the projection node of discrete node to the theoretical plane instead of the finite element discrete node; that is to take ”▵” instead of ”□” (Figs. 7 and 8). Two methods can be taken to finish the above node replacement process. The first one is to search the discrete nodes automatically with related algorithm, and then project them to the theoretical optical plane. Another one is with the help of the finite element preprocessing tool, discrete nodes is selected manually, then project the selected node to the original optical plane. The first approach is more conveniently to be realized based on the developed program. 3.2.2. Opto-mechanical integrated simulation technology Opto-mechanical integrated simulation involves several essential algorithms. The node displacement conversion algorithm of optical surface and fitting algorithm of node displacement change in sagittal direction is the foundation to realize the opto-mechanical integrated simulation. Optical surface can be defined by the sagittal equation. The change of optical surface should be expressed in the sagittal direction as well [4]. There is a thing that needs to be cared is sagittal displacement is not equal to the change of finite element node displacement of optical surface. The reason is node displacement contains three coordinate components and it also includes the displacement change of the node position along the radial direction. The node displacement of optical surface calculated by the finite element analysis needs to transfer the node displacement to the node sagittal direction. The displacement conversion method for node displacement to the sagittal direction is shown in Fig. 4. The node P0 on the initial optical surface moves to the point P1 due to the external loads. The displacement in the optical axis direction is ΔZ and the displacement is ΔR in radial direction. It is necessary to calculate the parameter TP1 with converting the node
Fig. 7. Discrete error elimination. 4
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
G. Xu
Fig. 8. Node displacement conversion algorithm.
displacement to the sagittal direction, and use TP1 to describe the node displacement change of the optical surface.
⎧ TP1 = ΔZ + ΔZsag ΔZ = Z (R P0 ) + Z (R P0 + ΔR) ⎨ ⎩ sag
(1)
In the formula (1), Z = Z(R) is the sagittal equation of optical surface. The above displacement conversion algorithm based on sagittal equation is also selected in the subsequent surface fitting algorithm based on Zernike polynomials. When the opto-mechanical integrated simulation is performed, it is necessary to transfer the mirror surface change to the optical data format which can be recognized by the optical program, for example Zemax, and then evaluate the influence of external loads on optical parameters such as wavefront and image quality. Hence, the polynomial fitting of optical surface change is the first work to do to realize the opto-mechanical integration simulation. Fringe Zernike polynomials are selected as basis function to realize surface change fitting and the definition of Fringe Zernike polynomials can refer the Zamax help files. Gram-Schmidt method is used to construct fitting vectors as well. The constructed vectors are orthogonal and linearly independent. The optical surface change are expressed by the linear combination of Fringe Zernike polynomials. The solution of contradictory equation is solved by the generalized inverse method. The generalized inverse method is equivalent to the least square method in fitting the wavefront aberration function, but the derivation process and program implementation are more concise. The detailed derivation process to realize the Zernike fitting of optical surface change is as follows [5]. Let n is the node number of finite element model for optical surface and m is the item number of Zernike polynomial selected; Fn×1 is a vector formed by converting the displacement of finite element node to the direction of optical axis. Un×m is the matrix that is composed of column vector Uj with the jth basis function (j = 1, 2, .., m); Cm×1 is the linear combination coefficient obtained by fitting optical surface change. Because the basis function set Uj is linearly independent on the point set xi (i = 1, 2, .. ., n), it is equivalent to the vector set Φj is linearly independent.
⎛ Uj (x1) ⎜ Uj (x2) Φj = ⎜ · · ⎜ · ⎜U (x ) ⎝ j n
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
(2) m
Therefore, Uj can form a subspace of R . (3)
Un × m Cm × 1 = Fn × 1 Gram-Schmidt method is used to orthogonalize Un×m, and orthogonalization matrix Vn×m is obtained.
(4)
Un × m = Vn × m Rm × m Rm×m is the middle transition matrix from matrix Vnm to matrix Unm.
Vn × m Rm × m Cm × 1 = Fn × 1
(5)
Rm × m Cm × 1 = Qm × 1
(6)
Vn × m Qm × 1 = Fn × 1
(7)
Contradictory equation has the unique the least norm least squares solution.
Qm × 1 = Vn+× mFn × 1
(8)
Vn+× m = (V T V )−1V T
(9)
After getting of solution of equation (9), the Qm×1 is calculated by formula (8). At last, the Zernike polynomial linear combination coefficient Cm×1 is obtained. 5
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
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Table 1 The fitting coefficient of three typical driven frequency. No./Freq.
31.2 Hz
165.6 Hz
277.6 Hz
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
−1.5093E-04 7.7038E-07 1.6217E-01 −7.4762E-10 −3.5472E-10 −1.0471E-08 8.7280E-09 1.3314E-06 −6.2362E-09 3.9559E-09 2.5711E-08 −1.0931E-08 −1.1954E-08 2.2771E-08 −5.4975E-07 −5.8347E-09 6.7097E-09 −1.4709E-08 −4.1988E-08 −1.4942E-08 3.9682E-08 7.4964E-09 −3.8144E-08 1.2172E-07 2.9577E-09 −7.4454E-10 3.7824E-08 −2.8623E-08 −6.7969E-09 1.6173E-08 −9.2318E-10 −2.9564E-08 1.9152E-08 −4.0424E-08 −3.6938E-09 6.7204E-09 −5.3175E-09
−1.7631E-04 2.2979E-05 1.6868E-01 −1.1673E-07 1.2338E-07 −5.7756E-09 1.8070E-08 4.0262E-05 8.7576E-08 −6.5032E-09 −2.4501E-08 −4.6525E-08 1.5022E-08 1.8056E-09 −1.5745E-05 −1.5181E-08 1.7619E-08 2.6872E-09 6.9456E-08 −1.1975E-08 −4.4725E-08 1.3036E-08 1.9430E-08 2.7254E-06 −4.4943E-08 −2.5067E-08 4.0870E-08 −1.7464E-08 3.4619E-08 4.5439E-08 −2.8875E-08 1.9043E-08 −2.3713E-08 −4.1560E-08 −2.2909E-07 3.7552E-08 3.4597E-08
−2.3194E-04 7.3910E-05 1.8224E-01 −3.8388E-07 4.1345E-07 −9.3151E-09 2.0582E-08 1.2204E-04 2.0279E-07 −5.9227E-08 −9.6199E-08 −2.1840E-07 −1.7984E-08 3.7998E-09 −4.7609E-05 −1.0826E-07 9.0529E-08 1.2747E-08 1.0217E-07 −1.8749E-08 −4.5587E-08 9.2654E-10 −1.0794E-08 8.2979E-06 −2.9411E-08 −3.1246E-08 1.0947E-07 −9.5173E-08 3.8385E-08 1.0227E-07 2.9848E-08 −9.1913E-09 −2.6806E-08 −4.4936E-08 −7.4882E-07 2.6949E-08 9.4727E-08
4. The dynamic surface parameter calculation of tip/tilt mirror The dynamic surface parameter analysis of tip/tilt mirror refers to the surface parameter distribution under the different excitation frequency. The frequency range of tip/tilt platform is changed from 20 Hz to 300 Hz, surface parameter is changed under the excitation of different driven frequency of tip/tilt platform. The surface change value under different driven frequency of tip/tilt mirror is calculated based on the above discussed surface calculation method. The detailed procedure is as follows. First, the fitting coefficient of tip/tilt mirror surface is gotten by means of opto-mechanical method. Second, the fitting coefficient is transferred to optical program Zemax. Third, the surface change data processing and graphical display can be carried out with the help of Zemax. The Fringe Zernike fitting coefficient of three typical driven frequency is as Table 1 shown.
Fig. 9. Surface contour with typical driven frequency. 6
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
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Table 2 Surface RMS of tip/tilt mirror under different driven frequency. No.
Freq. (Hz)
Surface RMS (λ)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
20 31.2 42.4 53.6 64.8 76 87.2 98.4 109.6 120.8 132 143.2 154.4 165.6 176.8 188 199.2 210.4 221.6 232.8 244 255.2 266.4 277.6 288.8 300
0.00035 0.0008 0.0015 0.0024 0.0035 0.0048 0.00635 0.0081 0.0101 0.0123 0.01475 0.01745 0.0204 0.0236 0.02705 0.03075 0.03475 0.03905 0.04365 0.04855 0.05375 0.0593 0.06525 0.0715 0.07815 0.08525
The surface change contour of tip/tilt mirror under three typical driven frequency is shown in Fig. 9. The surface contour shows that the surface change distribution of tip/tilt mirror under the different driven frequency is similar, but the surface parameter value is different. The detailed calculation results of tip/tilt surface RMS parameter under different driven frequency is shown in Table 2. The surface RMS variation curve under the different driven frequency is shown in Fig. 10. According to Table 2 and Fig. 10, the following valuable conclusions can be drawn. The surface change shows the nonlinear characteristics with the increase of the driven frequency. When the driven frequency is relatively less, the surface change is not obvious and the surface change is obvious under the higher driven frequency. As for the current tip/tilt mirror with aperture Φ160 mm, the surface RMS change is less than 0.01λ when the driven frequency is less than 100Hz; the surface RMS change is less than 0.02λ with the driven frequency being less than 150Hz; when the driven frequency is less than 180 Hz, the surface RMS change is less than 0.03λ. The surface RMS change increases rapidly when the driven frequency is higher than 200 Hz and the surface RMS is only 0.085λ at the driven frequency with 300Hz. Current tip/tilt mirror can only works on the driven frequency less than 150 Hz with surface change 0.02λ based on the design requirement. That is the higher driven frequency which will lead to the higher surface wavefront change and then make influence to the optical system image quality. 5. Conclusions To analyze the dynamic surface parameter of frequency response for tip/tilt mirror, the dynamic surface parameter calculation method is researched and is mainly composed of finite element discrete error and its eliminating technique, the Zernike polynomials
Fig. 10. The surface RMS under different driven frequency. 7
Optik - International Journal for Light and Electron Optics 203 (2020) 163964
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fitting method and data processing procedure combining with Zemax. The Dynamic surface parameter of frequency response for tip/ tilt mirror is calculated with the proposed method under the different driven frequency of tip/tilt platform. The calculation result shows the nonlinear change rule between the surface change parameter and the driven frequency. The surface change is not obvious under the lower micro-vibration driven frequency, but the surface change is obvious on the higher driven frequency. The analysis conclusions reveal the surface change rule under the different driven frequency and are important for the design of tip/tilt mirror. In other words, when the tip/tilt mirror is designed, the first thing that should be considered is if the designed tip/tilt mirror can satisfy the micro-vibration frequency environment and the dynamic surface parameter calculation is need to be carried out as well. The research of dynamic surface parameter for frequency response of tip/tilt mirror provides a new technical reference and has the significant engineering and scientific value for the simulation and design of tip/tilt mirror. Acknowledgement This study was supported by CAS ”Light of West China” Program (No.XAB2017A10). In addition, many thanks to the listed reference authors and some anonymous researchers, their study inspire us and help us a lot to do this work. References [1] L.H. Yang, L.Z. Jun, X.P. Mei, An overview on image stabilization method of space-born remote sensing systems, Spacecraft Recov. Rem. Sens. 31 (2010) 52–57. [2] K. Ichimoto, S. Tsuneta, Y. Suematsu, T. Shimizu, M. Otsubo, Y. Kato, M. Noguchi, The solar optical telescope onboard the solar-b, Proc. SPIE 5847 (2004) 1142–1150. [3] G.Z. Xu, P. Ruan, J.F. Yang, F. Li, X.T. Yan, S. Wang, High-precision algorithm of surface parameter of parabolic primary mirror for space solar telescope, Optik 127 (2016) 10687–10696. [4] R.C. Juergens, P.A. Coronato, Improved method for transfer of fea result s to optical codes, Proc. SPIE 5174 (2003) 105–115. [5] K.B. Doyle, V.L. Genberg, G.J. Michels, Integrated Optomechanical Analysis, SPIE Press, 2002.
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