Optical Pupil Shift Correction Method for Large Ground-Based Optical Telescopes

Featured Application

The primary and secondary mirror misalignment errors and the bending and sinking compensation method proposed in this paper can be applied to the mounting and observation of large telescopes.

Abstract

Cassegrain telescopes with larger apertures suffer significant optical pupil shifting errors caused by gravity and temperature gradients. This study constructs an optical model of a large Cassegrain telescope and a method to calculate and compensate its shifting error. First, the optical structure is simplified by incorporating only the most relevant telescope components, and the system optics is modeled using ray tracing. A computer-aided mounting-based method for calculating the misalignment error of primary and secondary mirrors is proposed, with the spatial position change of secondary mirrors as the input. Next, a compensation method based on the coma-free point theory of the Cassegrain system is proposed, with the main mirror’s optical axis as the reference. Finally, using a 4 m aperture telescope as an example, the gravity- and thermal-deformation-induced shifting error is simulated. Based on this simulation, the secondary mirror position is adjusted using a secondary mirror-adjustment mechanism Hexapod platform to compensate for the misalignment error. Both gravity and temperature are major causes of shift in the pupil position, with a maximum shift of 18 mm. After compensation, this shift is controlled within 1 mm. The modeling method and the simulation process mentioned in this research can also be used in the other relevant fields.

1. Introduction

The increasing knowledge of the observable universe is driving the demand for telescopes with large numerical apertures to observe farther objects in space. The Cassegrain-type telescope has a large aperture and is used in radio astronomy, Earth stations, and satellite communication. A recent example of such an instrument is the James Webb Telescope. The increasing aperture of Cassegrain telescopes, and consequently their truss weight and back intercept, has amplified the degradation of imaging quality caused by the misalignment error due to structural and temperature changes, especially for the adaptive optics high-resolution imaging system. The compensation method of the error has received special focus and caused difficulties in the research of ground-based large-aperture telescopes [1].

The gravitational bending and sinking of the secondary mirror, as well as changes in the relative position of the primary and secondary mirrors due to temperature changes, are the main causes of the misalignment error of an optical system. After the optical system is folded into the Couder imaging system, it causes the system optical axis, and consequently the system optical pupil position, to shift [2,3,4].

For the adaptive optics system, the Hartmann measurement instrument and deformation mirror are placed at the optical pupil position of the primary mirror. The optical pupil shift causes the pupil exit surface of the primary mirror to draw a circle on the surface of the deformation mirror. As the aperture of the primary mirror increases, and the back intercept distance of the system becomes longer and the circular motion begins to severely influence the corrective effect of the imaging system [5,6].

In terms of offset error correction, the European Southern Observatory first used a piece of correction technology based on active optics on the New Technology Telescope (NTT). However, the active support technology for the primary mirror does not solve in principle the optical pupil shift problem. Lucimara et al. analyzed the sensitivity of the third-order coma of the two-reflection system to off-centering [7,8,9].

The alignment of the optical pupil and field of view in an off-axis three-mirror system by computer-assisted misalignment was demonstrated by Cook et al. [10,11], while R. Upton et al. undertook the active alignment of the Advanced Technology Solar Telescope by means of optical analysis [12]. For the alignment of off-axis optical systems, G. Ju and X. Zhang et al. realized the alignment of off-axis two reflectors using aberration fields and nodal aberration theory [13,14,15]. All of the above methods are used during the period of preliminary alignment of the telescope; however, the optical pupil shift correction proposed in this paper is generated by the bending and sinking of the ground-based large-aperture telescope and the temperature change, so the method proposed here is more advanced and novel in terms of the compensation in the observation process.

In addition, Kim used the inverse optimization method to adjust a Cassegrain telescope, adjusting the wavefront aberration from 0.283λ to 0.194λ. Fuerschbach used the nodal aberration theory to study the aberration type corresponding to the freeform surface located at any position in the system. However, these methods did not consider the optical pupil shift, and consequently could not correct the misalignment error [16,17].

The adjustment mechanism of the secondary mirror can compensate the system aberration produced by the deviation in the position of the mirror. However, from the system’s perspective, this compensation–correction method cannot correct the change of the optical axis of the main system [18].

Therefore, to compensate the main system’s misalignment error and the optical axis change, we proposed a method based on the coma-free theory of the Cassegrain main system and the position-aberration compensation of the secondary mirror. Thus, the changes in the system aberration and optical axis shift were compensated in real time, and the correction effect of the adaptive optical system was improved. The overall optical model of the telescope was used to calculate the degrees of bending and sinking variation at different elevation angles and calibrate the optical axis, which helped eliminate the effect of optical pupil shift on the adaptive optical imaging system. This method not only improves the calibration accuracy but also reduces the number of calibration iterations needed to improve the system efficiency [19,20].

A ground-based optical telescope with an aperture of up to 4 m was used to study the Cassegrain main system-based optical axis calibration method and optical pupil shift compensation technology. First, the components of the telescope were described in detail, and a simplified overall optical model of the telescope is constructed based on ray tracing. Then, the causes of the system pupil shift, the pupil calibration method, and the compensation process were introduced, based on the card-type master-system coma-free theory, a Hartmann-and-calibration-camera dual detector pupil-calibration compensation method. Finally, the number of effective sub-aperture points of the Hartmann detector in the calibration process of the adaptive optical imaging system was used to verify the effectiveness of the model and the compensation method.

2. Structural Optics Hybrid Modeling of the Telescope

The diagram of the 4 m class ground-based telescope system described in this article is illustrated in Figure 1, the optical system of which is in the form of a Cassegrain. The dimensions of the system are 12.5 m × 5.8 m (height × diameter) and a total mass of approximately 90 tons. The optical structure of the telescope mainly includes a tracking frame, primary mirror, secondary mirror, Couder system, adaptive optical imaging system, and multiple imaging terminals. To simplify the analysis process, we only considered the first five parts.

The tracking frame is the carrier of the telescope’s optical system, which precisely tracks targets in select areas in the sky through the high-precision rotation of the elevation and azimuth axis systems. The interconnection between the tracking frame and each optical element is depicted in Figure 1.

The complexity of the 4 m class telescope system makes it difficult to model every component in detail during optical analysis, despite eliminating other imaging terminals. Therefore, the model is simplified as follows.

• The overall gravity- and temperature-induced deformations of the telescope’s tracking frame are uniformly affected by the deviation in the position of the secondary mirror from the primary mirror. This assumption is used as the basis for discussion on the optical-axis deviation of the primary system.
• The telescope azimuth axis and elevation-bearing-load margin are extremely high, and it is regarded as an ideal body in the process of tracking, i.e., the influences of bearing friction and clearance on the optical system are dismissed.
• The positional accuracy of the main mirror of the whiffletree hydraulic support system has an extremely high resolution. The main mirror’s optical axis does not change. The hydraulic correction process has a short response time. In the process of following the elevation movement, the hydraulic correction process is regarded as the ideal movement, dismissing the main mirror’s position offset error and correction delay.

With these assumptions, we applied the ray-tracing method to establish the overall optical model of the telescope (Figure 2).

According to the working condition of a telescope, the positional accuracy of the secondary mirror plays a decisive role in changing its optical axis. The position of the secondary mirror relative to the primary mirror is expressed by the finite element analysis of the position of the nodes on the surface of the secondary mirror (Figure 3).

Therefore, in the process of the ray-tracing of a telescope, we consider the change in the secondary mirror’s nodal position relative to the primary mirror.

The primary and secondary mirror nodal displacements can be expressed as:

$$\left\{\begin{array}{c}{q}_f\\ q_s\end{array}\right\}=\left[\begin{array}{c}{\Phi }_f\\ {\Phi }_s\end{array}\right]\left\{x\right\},$$

(1)

where $\left\{q_f\right\}$ and $\left\{q_s\right\}$ are the nodal displacement vectors of the primary and secondary mirrors, and $\left[{\Phi }_f\right]$ and $\left[{\Phi }_s\right]$ are the selection matrices associated with the primary and secondary mirrors.

The element at the mirror nodal position is 1, and the rest of the elements are 0. In this study, because the main mirror is equipped with a hydraulic active support to the maintenance of the absolute position of its constant, the expression for the nodal displacement can be simplified as

$$\left\{q_s\right\}=\left[{\Phi }_s\right]\left\{x\right\},$$

(2)

By using a combination of the displacement and current position of the mirror node, we can express the current position of the secondary mirror as

$$\left\{x_s\right\}=\left\{x_s^0\right\}+\left\{q_s\right\}=\left[{\Phi }_s\right]\left(\left\{x^0\right\}+\left\{x\right\}\right),$$

(3)

Based on the current position of the secondary mirror surface relative to each node on the primary mirror, a subsequent optical performance analysis was performed. Based on the coma-free theory, the secondary mirror was adjusted by controlling the Hexapod platform to actively compensate for aberrations and deviations in the optical axis.

3. Calculation of Optical Pupil Shift and Active Compensation

3.1. Causes of Optical Pupil Shift

We consider the classic primary and secondary mirror system with a Cassegrain structure as an example. The misalignment error of the optical system is mainly due to the gravitational bending and sinking of the primary and secondary mirrors, as well as temperature-induced changes in their relative positions, resulting in a shift in the system’s optical axis. An unstable optical axis of the system can cause a change in the position of the focal point of the primary and secondary mirrors, causing a shift in the pupil surface of the primary mirror [21,22,23,24].

For the adaptive optical imaging system, the Hartmann measurement and deformation mirror systems were placed at the optical pupil position of the primary mirror. The pupil shift caused the mapping surface of the primary mirror to circumscribe the deformation mirror. As the aperture of the primary mirror increased, the influence of drawing the circle becomes more evident, which severely impedes the correction of the adaptive optical high-resolution imaging system and, consequently, the quality of image. From the perspective of the telescope’s structure, the main causes of the shift may include [24]:

• The position deviation of the primary mirror is caused by the deformation of its support assembly: The use of flexible rods in the primary mirror-support structure to reduce the deviation of the optical material and the temperature change of the structural material, which could lead to a decrease in the stiffness of the structure, cause the primary mirror under the action of gravity to deviate its position and angle relative to the primary mirror cell. This deviation varies with the telescope elevation angle. In this study, the active optical technology ensured an absolute position deviation of 0.02 mm for the primary mirror, which had a negligible effect on the optical axis offset [25,26,27,28] (Figure 4).
• The deviation of the secondary mirror is due to the gravitational deformation of the truss: As illustrated in Figure 1, the span of the truss is very large in height (approximately 5.5 m). Despite the use of large-diameter bars in the truss structure, the position and angular deviation of the secondary mirror occurs with the change of elevation angle, owing to the high structural load from, for example, the ring beam and four wing beams [29,30] (Figure 5).
• Thermal deformation causes the misalignment error: The change in the ambient temperature will result in the thermal deformation of the structure. Therefore, the primary and secondary mirrors will deviate from the theoretical position, which is the major cause of the interval deviation of the primary and secondary mirrors. With the increase in the telescope aperture, the influence of deviation of the primary mirror optical axis relative to the perpendicular four-way plane is aggravated. Deviations in the primary and secondary mirror intervals will also trigger a deviation in the relative position of the primary and secondary mirrors [31] (Figure 5).

3.2. Process of Optical Pupil Shift Compensation

During the tracking process of the telescope, the deviation in the positions of the primary and secondary mirror induced by the structural deformation affected the wave aberration of the primary system. The change in the optical axis of the primary system led to the optical pupil shift, which ultimately affected the efficacy of correction of the adaptive optical imaging system. The aberration and optical-axis deviation vary with the telescope elevation angle. Based on the coma-free theory of the Cassegrain system, these defects can be compensated by adjusting the position and attitude of the secondary mirror by manipulating the Hexapod platform. The specific compensation process is illustrated in Figure 6.

First, the current position of the primary and secondary mirror nodes is calculated by Equation (3). Then, based on the current position of the secondary mirror, the coma-free rotation point of the primary and secondary mirrors is calculated using the coma-free rotation point theory. This rotation point is used as the reference, and the calculation of control inputs to the Hexapod platform based on target imaging off-target amount. Thus, the shift is compensated. Finally, the compensated shift is analyzed and evaluated using the adaptive optical correction effect.

3.3. Nonlinear Least-Squares Fitting to Calculate Secondary Mirror Position

The determination of the current position of the secondary mirror is crucial to the compensation process. In the telescope system model, the position of the secondary mirror is represented as several discrete nodes on its surface. Least-squares fitting was employed to obtain the best-fit quadratic surface formed by the discrete nodes on the mirror surface, the origin of which is the current position of the secondary mirror. The process of fitting is as follows.

In the geometric model, we assumed $n$ number of $a$ vectors, each of which corresponds to $m$ measurements of $Y$. The expression relation is expressed as

$$Y=F\left(a\right)+q,$$

(4)

where $F$ is the continuous differentiable function of vectors $a$ and $q$ is the error vector.

For given secondary mirror surface measurements $Y$, the nonlinear least-squares fitting expression for $a$ is

$$\min q^{T}q={\left[Y-F(a)\right]}^T\left[Y-F(a)\right],$$

(5)

Newton’s method was used to solve these nonlinear estimates in the following iterative format.

$${\left.\frac{\partial F}{\partial a}\right|}_{a_k}=Y-F(a_k),$$

(6)

$$a_{k+1}=a_k+\lambda \Delta a,$$

(7)

where the expression on the left-hand side is the Jacobi matrix.

$$J_{mn}=\frac{\partial F_m}{\partial a_n},$$

(8)

To solve this problem, the pointing vector $F$ at the nearest point of the surface sampling points toward the ideal surface, and its partial derivative matrix value, which is the input condition for the nonlinear fit, was calculated.

A schematic of the quadratic surface in three-dimensional space is illustrated in Equation (8). The best-fit spherical center position is $Y_c$ and the best-fit radius is $R$, which can be expressed by the following equation:

$${‖Y-Y_c‖}^2-Q=R^2,$$

(9)

where $Q$ is the second-order surface fitting error. In this study, the second-order surface coefficient of the secondary mirror was −1.054 and the curvature was 1030 mm. The primary and secondary mirror spacing were up to 5600 mm; on this basis, the secondary surface fitting error is negligible; therefore, Equation (9) can be simplified as

$${‖Y-Y_c‖}^2=R^2,$$

(10)

For each sampling point $Y_i$, the quadratic surface with its corresponding point $Y_i{}^{\prime }$ is expressed as

$$Y_i{}^{\prime }=Y_i+R\frac{Y_i-Y_c}{‖Y_i-Y_c‖},i=1,2,\cdots ,m,$$

(11)

The deviation of the sampling point $Y_i$ from its counterpart $Y_i{}^{\prime }$ is expressed as

$$Y_i{}^{″ }=Y_i-Y_i{}^{\prime }=\left[‖Y_i-Y_c‖-R\right]\frac{Y_i-Y_c}{‖Y_i-Y_c‖},$$

(12)

The expression for the Jacobi matrix by Newton’s method from the expression of $Y_i{}^{\prime }$ is

$$J=\frac{\partial }{\partial }\left[Y_c+R\frac{Y_i-Y_c}{‖Y_i-Y_c‖}\right],$$

(13)

From these expressions and initial data, the aspheric-fitted linear equation system expression for the secondary mirror is expressed as

$$\left[\begin{array}{cccc}{J}_{X_1{}^{\prime },R}& J_{X_1{}^{\prime },X_C}& J_{X_1{}^{\prime },Y_C}& J_{X_1{}^{\prime },Z_C}\\ J_{Y_1{}^{\prime },R}& J_{Y_1{}^{\prime },X_C}& J_{Y_1{}^{\prime },Y_C}& J_{Y_1{}^{\prime },Z_C}\\ J_{Z_1{}^{\prime },R}& J_{Z_1{}^{\prime },X_C}& J_{Z_1{}^{\prime },Y_C}& J_{Z_1{}^{\prime },Z_C}\\ ⋮& ⋮& ⋮& ⋮\\ J_{X_N{}^{\prime },R}& J_{X_N{}^{\prime },X_C}& J_{X_N{}^{\prime },Y_C}& J_{X_N{}^{\prime },Z_C}\\ J_{Y_N{}^{\prime },R}& J_{Y_N{}^{\prime },X_C}& J_{Y_N{}^{\prime },Y_C}& J_{Y_N{}^{\prime },Z_C}\\ J_{Z_N{}^{\prime },R}& J_{Z_N{}^{\prime },X_C}& J_{Z_N{}^{\prime },Y_C}& J_{Z_N{}^{\prime },Z_C}\end{array}\right]\left[\begin{array}{c}\Delta R\\ \Delta X_C\\ \Delta Y_C\\ \Delta Z_C\end{array}\right]=\left[\begin{array}{c}{X}_1{}^{″}\\ Y_1{}^{″}\\ Z_1{}^{″}\\ ⋮\\ X_N{}^{″}\\ Y_N{}^{″}\\ Z_N{}^{″}\end{array}\right],$$

(14)

The initial value of the corresponding vector for the measured data is obtained as

$$Y_C=\frac{1}{n}{\displaystyle \sum _{i=1}^n{Y}_i},$$

(15)

$$R=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{\left|Y_i-\overline{Y_i}\right|}^2}},$$

(16)

3.4. Effect of Secondary Mirror Position Error

The effect of wavefront aberration caused by the change in the secondary mirror’s position according to the aberration theory is comprehensively analyzed in this section. The change in position of the secondary mirror affects mainly the wavefront of astigmatism and coma for the main system. If the central vertex of the secondary mirror is considered the coordinate origin, the optical axis direction is the Z-axis, and the vertical optical-axis direction is established along the x and y-axes. The deviation of the secondary mirror can be defined as node displacement $d_z$ in the z direction, node displacement $d_x$ and $d_y$ in the x and y directions, and node rotations $t_x$ and $t_y$ around the x- and y-axes in a total of five variations. Figure 7 illustrates the diagram of $d_y$, $t_x$, and $d_z$ deviations.

According to the third-order vector wave aberration theory, for the telescope system with a card structure, the wavefront of coma and astigmatism when the secondary mirror is in the ideal position are expressed as

$$W_{coma}=({\displaystyle \sum _j{W}_{131j}H\cdot \rho })(\rho \cdot \rho ),$$

(17)

$$W_{ast}=\frac{1}{2}({\displaystyle \sum _j{W}_{222j}{H}^2}){\rho }^2,$$

(18)

where $j$ is the number of surfaces of the optical elements in the optical system and $W_{131j}$ and $W_{222j}$ are the third-order coma and astigmatism coefficients of the wave aberration of the jth surface system. The vector schematic is shown in Figure 8.

As the node displacement $dx$ and $dy$ or node rotation $Tx$ and $Ty$ of the $j\mathrm{th}$ plane would offset the axis of symmetry of the aberration of this plane by ${\sigma }_j$, the equations of coma aberration and image dispersion would become

$${W^{\prime }}_{coma}=[({\displaystyle \sum _j{W}_{131j}H}-{\displaystyle \sum _j{W}_{131j}}{\sigma }_j)\cdot \rho ](\rho \cdot \rho ),$$

(19)

and

$${W^{\prime }}_{ast}=\frac{1}{2}[{\displaystyle \sum _j{W}_{222j}{H}^2}-2H(W_{222j}{\sigma }_j)+{\displaystyle \sum _j{W}_{222j}{\sigma }_j}]\cdot {\rho }^2,$$

(20)

respectively.

For the coma, an additional term $\left({\displaystyle \sum _j{W}_{131j}}{\sigma }_j\right)\cdot \rho \cdot (\rho \cdot \rho )$ is added to Equation (19). For astigmatism, $H(W_{222j}{\sigma }_j)\cdot {\rho }^2$ and ${W^{\prime }}_{ast}=\frac{1}{2}[{\displaystyle \sum _j{W}_{222j}{\sigma }_j}]\cdot {\rho }^2$ are added to Equation (20). If the deviation is small, the on-axis field-of-view (FOV) image dispersion increment term will be negligible relative to the coma increment term. For the telescope system with a cassette structure, when the displacement occurred along the optical axis direction, the aberration was defocused and spherical, which can be corrected by moving the secondary mirror along the axial direction. When the vertical optical axis direction has shift and tilt, a constant coma is produced in the FOV, which can be corrected by translating or tilting the secondary mirror along the vertical optical axis. The telescope system error mainly comprises the relative position errors of the primary and secondary mirrors. For the on-axis FOV, if the aberration is corrected, the central axis of the primary and secondary mirrors will intersect at a coma-free point (Figure 9).

The distance between the coma-free point and the secondary mirror center vertex $Z_{cfp}$ satisfies [32]:

$$Z_{cfp}=\frac{2L(m_2^2-1)}{{(m_2+1)}^2[(m_2-1)-(m_2+1)b_{s2}]},$$

(21)

where $L$ is the distance between the central vertex of the secondary mirror and the center of the focal plane in the ideal case and $m_2$ is the secondary mirror magnification, i.e., the ratio of the focal length of the system to the focal length of the primary mirror. In addition, $m_2=f^{\prime }/f_1{}^{\prime }$, where $f^{\prime }$ is the focal length of the telescope system, $f_1{}^{\prime }$ is the focal length of the primary mirror, and $b_{s2}$ is the aspheric coefficient of the secondary mirror. If rotated about this point, the coma difference resulting from the node displacement of the secondary mirror $d_y$ and the node rotation by $t_x$ can always cancel each other out in the on-axis FOV (Figure 10).

Therefore, after correction of the on-axis FOV, secondary mirror position error can be corrected by rotating the secondary mirror around the coma-free point.

4. Simulation Analysis and Experimental Verification

The 4 m telescope illustrated in Figure 11 was used as an example to validate the Hartmann-and-calibration dual detector optical-pupil calibration-compensation method based on the coma-free theory of the Cassegrain system. The optical axis deflection and optical pupil shift introduced through gravitational deformation and temperature change were analyzed. The shift error was actively compensated by secondary mirror positioning, attitude adjustment, and telescope pointing adjustment. Finally, the proposed model and compensation method were validated based on the number of effective sub-aperture points of the Hartmann detector in the actual calibration process of the adaptive optical imaging system.

4.1. Compensation of Optical-Axis Deviation under the Influence of Gravity

The best-fit aspheric surface of the secondary mirror was calculated from the deformation results of several discrete nodes on the secondary mirror. The position and angular deviations of this mirror with respect to the optical axis of the primary mirror were obtained. The optical aberration was compensated by adjusting the position and angle of the Hexapod platform. Before and after the compensation, the variation in the angular deviation of the secondary mirror relative to the main mirror’s optical axis with respect to the elevation angle is depicted in Figure 12. The variation in the position deviation with respect to the elevation angle is displayed in Figure 13.

The deviations in position and angle were used as inputs for analyzing the optical-axis deviation of the main system in the simplified optical model. The variations in the optical-axis deviation of the main system with the elevation angle before and after compensation are depicted in Figure 14.

Figure 12 leads to the conclusion that the optical axis still deviates significantly after compensation. Based on the coma-free theory of the primary and secondary mirrors of the Cassegrain system, the optical axis was compensated by rotating the secondary mirror around the coma-free point, with respect to the target offset of the calibration camera. The optical-axis deviation after compensation is illustrated in Figure 15.

Subsequently, since the Hartmann detector is the conjugate surface of the primary mirror, the effectiveness of the compensation method was verified by the pupil shift of effective sub-aperture on the Hartmann detector during the actual calibration of the adaptive optical imaging system. The pupil shift of effective detection sub-apertures of the Hartmann detector before and after compensation is shown in Figure 16.

The following conclusions can be drawn from the calculated results of the secondary mirror deviations shown in Figure 12 and Figure 13 and the optical-axis deviations of the main system in Figure 14 and Figure 15.

The maximum deviations in the position and angle of both the primary and secondary mirrors before compensation occurred when the optical axis was orientated in the horizontal direction. The maximum angular deviation was close to 38″, and the maximum position deviation 0.5 mm. After the Hexapod platform compensation, the maximum angular deviation decreased to 7.1″, and the maximum position deviation to 0.0175 mm.

Compared with a decrease in angular deviation, the compensated position deviation decreased by a smaller margin, i.e., from 0.5 to 0.0175. This is because the secondary mirror’s secondary surface fitting error was omitted in the nonlinear least-squares fitting process for simplicity. A nonlinear spherical fit was performed on the current position of the hyper-mirror node. The geometric line passing through the fitted central position and node of the secondary surface was used as the optical axis. This axis deviated from the actual one by a certain margin, which led to an initial deviation in the primary and secondary mirror positions.

Based on the coma-free theory of the Cassegrain system, we found that the optical-axis compensation of the secondary mirror rotating about the coma-free system did not change the image quality of the main system. It only changed the optical axis of the main system, which is essentially the process of aligning the optical axis of the secondary mirror with the main mirror. The maximum deviation of the optical axis before compensation was 123″. After compensation, the maximum deviation decreased to 3.5″. As indicated in the Hartmann detector imaging diagram, as the target azimuth and elevation angle are varied, the uncompensated beam impinges on the pupil surface shifted by 18 mm after stabilization, and the compensated pupil shift plummeted to 1 mm (Figure 16). Thus, the optical system performance is significantly improved.

4.2. Compensation of Optical-Axis Deviation Induced by Temperature Variations

The thermal deformation analysis of the telescope system at different ambient temperatures (−20–50 °C) is illustrated in Figure 11. The variation in the angular deviation of the secondary mirror with temperature before and after compensation displays in Figure 17, obtained using the same analysis method as in Section 3.2. Figure 18 illustrates the variation in the position deviation with temperature.

The same compensation method was used for the optical-axis deviation caused by temperature change. The variation in the optical-axis deviation of the telescope main system with temperature before and after the compensation is shown in Figure 19.

Figure 19 leads to the conclusion that the optical-axis deviation after compensation reduces to within 5.5″. Additionally, the secondary mirror did not require to be rotated about the coma-free point for compensation.

The following conclusions were drawn from the calculations of the secondary mirror deviations before and after compensation amid temperature variations (Figure 17 and Figure 18) and those of the optical-axis deviations of the main system (Figure 19).

Temperature variations induce relatively small angular and positional deviations in the telescope. In fact, the deviations of the secondary mirror with the compensation control of the Hexapod platform are close to zero.

As the ambient temperature increases, the variation of the control input of the Hexapod platform is strictly linear. In application, the ambient temperature compensation can be considered linear.

The maximum position and angular deviations of the primary and secondary mirrors before compensation were –30 °C each. The maximum angular deviation was approximately 24″, and the maximum position deviation was approximately 3.5 mm. After Hexapod platform compensation, the maximum angular deviation dwindled to 0.0058″ and the maximum position deviation was reduced to 0.015 mm. The maximum deviations of the optical axis before and after compensation were 21.3″ and 5.16″, respectively. The optical pupil shift caused by the deviation of the optical axis after compensation was within the acceptable range. Thus, the performance of the optical system was augmented.

5. Conclusions

We investigated the degradation of adaptive optical image correction in a ground-based large optical telescope due to the optical pupil shift caused by gravitational deformation and temperature gradient. Methods to estimate the optical axis shift and actively compensate the secondary mirror adjustment platform were also investigated. A simplified optical model of the telescope was established based on ray tracing. The causes of the optical pupil shift and compensation process were meticulously studied. Based on the coma-free theory of the Cassegrain system, a method to compensate for the calibration of the optical pupil was proposed. The experimental results proved that both gravitational deformation and temperature gradients caused the disorientation of the optical axis of the entire system, and the pupil shift of the conjugate surface of the primary mirror reached 18 mm. The secondary mirror bend compensation and rotation compensation about the coma-free point remarkably reduced the pupil shift to less than 1 mm. Finally, the effectiveness of the proposed model and compensation method were verified by the pupil shift of effective sub-aperture on the Hartmann detector in the actual calibration process of an adaptive optical imaging system.

The improvement of the image resolution of the adaptive telescope is related to the number of wavefront detection sub-apertures. Due to the optical pupil shift caused by the number of invalid and effective sub-apertures having limited differences, the method outlined in this paper will not only essentially improve the image resolution of the telescope but fundamentally solve the environmental reasons for the instability of correction. Thus, the theory presented in this paper has specific limitations for adaptive telescopes.

6. Patents

Name of Patent: A method of optical pupil shift correction method for large aperture adaptive telescope. Chinese Patent Number. ZL 2022 1 0809076.3.