# Correction of wavefront rotation between interferometer and shack-hartmann sensor using bending modes

## Abstract

In some active optics system, the influence function, which is the surface deformation ability of an actuator, is measured by directly detecting the mirror surface using an interferometer and the surface correction of the primary mirror is done using the Shack-Hartmann (S-H) sensor. However, if the wavefronts of the interferometer and the S-H sensor have an obvious rotation, a large error will be introduced to the correction force calculations. In this paper, bending modes are used to detect this wavefront rotation. Bending modes are a series of orthonormal stiffness-increasing modes calculated using the influence functions. Two methods, optimum search and top-line (a line passes through the surface center and the top of surface deformation peaks) detection, are developed and tested in simulation using the experimental data of a 620-mm active optics system. The detection errors of the two methods are 0.286° and 0.085° in the simulation, respectively. The rotation detection method is then tested on this 620-mm system. The simulation and experimental results show that top-line detection is a suitable method for the detection of wavefront rotation.

## Keywords

Wavefront rotation;active optics Bending mode## Abbreviations

- DCT
discrete cosine transform

- LMS
least mean square

- RMSE
root mean square error

- S-H
Shack-Hartmann

- SORT
Starfire Optical Range Telescope

- SVD
singular value decomposition

## Introduction

Measurements of influence functions, which help finding the surface deformation ability of an actuator, consumes large amount of time. For example, in the active optics system of a 4 m SiC mirror in our lab, there are 54 actuators; and it therefore needs at least 540 min for a loop of influence function measurement (about 5 min for each force application and surface detection and at least twice of these for one actuator). Such measurement requires a time-consuming stable detection condition, so the mirror surface is planned to be directly detected by a Zygo interferometer in a closed detection tower during the influence function measurement to obtain a high precision result. After this measurement, the mirror surface is detected by the S-H sensor of the active optics system. Then the problem left is that the wavefronts detected by the interferometer and the S-H sensor may have a rotation angle. If the angle is larger than 1.0°, it will obviously hamper the correction ability of the active optics system. Hence, wavefront rotations should be detected and adjusted before active corrections. On the other hand, it is also difficult to include additional components in the system just for detecting such rotations. Therefore, detections of wavefront rotations without additional components are necessary for surface corrections.

Bending modes are first used on the active optics system of Starfire Optical Range Telescope (SORT), and then successfully applied on other active optics systems [1, 2, 3, 4, 5].They are a series of orthonormal modes created via singular value decomposition (SVD) on the influence matrix of the actuators. Mode stiffness is arranged in an increasing order, and mode surfaces can be properly rebuilt on the mirror of the active optics system. Therefore, we apply the first order bending mode of the system to detect wavefront rotations and the rotation detections are tested on a 620-mm active optics experimental system in our lab.

The primary mirror surface is detected by an S-H wavefront sensor that is shown in Fig. 2. It contains a camera, a light source, a beamsplitter, a lenslet and two collimation lenses. This sensor sends a spherical lightwave to the primary mirror and collimates the reflected light to the lenslet to acquire a dot image and to calculate the wavefront. The lenslet has a pitch of 300 μm and a focal length of 7 mm, which results in a wide detection range to deal with the large astigmatism of the primary mirror aberration in the passive support state. The system has 272 sample points that are fitted to 32 Zernike polynomials, excluding two tilts and defocus. Zernike polynomials are useful in expressing wavefront data [6, 7].

## Bending mode correction strategy

### Influence function

where *f*_{i} is the active force added on the *i*th actuator, and *f*_{i} ⋅ *x*_{i} and *f*_{i} ⋅ *y*_{i} are *x-axis* and *y-axis* moment of the *i*th actuator, where *x*_{i} and *y*_{i} are the coordinates of the actuator. This means that at least three more actuators are needed to balance the net force and net moment.

*F*

_{i}, which is

*f*N on the

*i*th actuator and 0 N on other actuators, the balance force vectors are.

*X*and

*Y*are the vectors of coordinates

*x*and

*y*, and

*F*

_{N},

*F*

_{X}and

*F*

_{Y}are the balance force vectors for net force and net moment of the

*x-axis*and

*y-axis*. If

*X*and

*Y*satisfy.

then *F*_{N}, *F*_{X}, and *F*_{Y} are independent and the balanced force of *F*_{i} is *F*_{i} − *F*_{N} − *F*_{X} − *F*_{Y}. Actuators set at the regular position commonly satisfy Eq. 3, which is the case for the 620-mm active optics system.

The influence force of each actuator is obtained by adding a unit force on this actuator and balancing the force on all actuators by using the former method. The rank of the influence force matrix is three less than the number of actuators because of the balance distribution. Therefore, the rank of the influence matrix is also three less than the number of actuators.

### Bending mode

*F*added on the active support causes a surface aberration

*W*:

where *A* is the influence matrix of the actuators, *F*_{A} is the influence force matrix, and *a* is calculated by a least squares fitting such as *a* = (*A*^{T}*A*)^{−1}*A*^{T} ∗ *W*. However, the correction may require many iteration steps to acquire an acceptable surface, and the correcting force may be too large to add onto the actuators. To obtain a suitable correcting force, a better fitting base than the influence function is needed.

*A*. The mode matrix

*B*is obtained by performing SVD on

*A*, such as.

*B*is a combination of the influence matrix

*A*, so the mode surface can be properly rebuilt on the primary mirror by using its mode force. The mode force

*F*

_{B}is defined as

*B*is orthonormal, the mode coefficients

*b*and the correcting force

*F*are.

The columns of *F*_{B}, which are mode force vectors related to bending modes, are arranged in an increasing order [2, 8]. This means that lower order mode deformation occurs more easily, and fitting a surface aberration by low order modes could effectively optimize the correcting force amplitude with an acceptable fitting precision.

Further discuss about the orthonormality of bending mode in Zernike base is shown in Appendix.

## Wavefront rotation detection

A high precision influence matrix measurement is usually performed using a detection system directly aimed at the primary mirror using a high-quality light source. This high-precision influence matrix is then used in the active optics system. Then, the S-H wavefront sensor of the active optics system detects the primary surface deformation. However, if the wavefront has an obvious rotation with respect to the influence matrix, the correction ability will be largely deteriorated.

Since the active optics system detects primary mirror deformations, the detection of a specific deformation could aid in determining the wavefront rotation. This detection requires no additional measurement components. A unique mirror deformation such as the influence of one actuator could perform this function, but the low-order bending mode is much more ‘soft’ that a small mode force can make a large deformation for detection. Moreover, the low-order mode surface has a low spatial frequency that is easily detected and introduces less detection error. Therefore, using the first order mode could result in high-precision detection.

As mentioned in Section 2.2, the mode surface could be precisely rebuilt on the primary mirror by adding its mode force on the actuators. Then, the detection is performed by adding a mode force on the support, detecting the wavefront of the rebuilt mode, and comparing the wavefront surface with the mode surface to determine the rotation angle.

For a regular structure support system, the first order bending mode is like an astigmatism, which has two symmetric valleys and peaks. Therefore, the rotation *α* detected using this mode can be either of two symmetric rotations: *α* and 180 ° + *α*. An additional detection is needed to ensure that the true result is obtained, using an asymmetric surface, such as the influence function of one actuator or an asymmetric low order mode.

### Zernike coefficient rotation

The detected wavefront of the S-H wavefront sensor is described in terms of Zernike coefficients. The Zernike coefficients are easy to be rotated since all the Zernike polynomials are circularly symmetric or orthogonal in pairs. Rotation of orthogonal polynomial pairs is performed by multiplying a rotation matrix with their coefficients. The angle of the rotation matrix is equal to the phase angle of the Zernike polynomial pair, which is equal to the rotation angle multiplied by the rotation order of the Zernike polynomial. For example, a rotation of *α* corresponds to a phase angle 3*α* for a trefoil. Rotating the Zernike coefficients of the surface directly could introduce less fitting error than rotating the surface image rebuilt by the Zernike coefficients.

### Optimum search detection

The simulation of wavefront rotation detection is performed using the experimental data from Section 4.1. The calculated modes #1 and #5 surfaces and their rebuilt surfaces are used. The rebuilt mode surfaces are rotated by a certain angle *θ* to simulate the rotated wavefront surfaces of the rebuilt mode. Mode #1 is used for the main detection and mode #5 for the additional detection.

To detect the rotation angle of the two surfaces, an optimum search could be a suitable approach. The rotated wavefront surface is the search target. The calculated mode surface is rotated to fit the target, and the rotation angle *α* is the parameter to be optimized. This is a simple two-way search. The optimum search begins with an initial rotation angle *α*_{0}, an initial searching rate *φ*_{0} and a rate scale factor *ρ*. In the searching step *i*, the calculated mode surface rotated by *α*_{i} is compared with those rotated by *α*_{i} + *φ*_{i} and *α*_{i} − *φ*_{i}. If the mode surface rotated by *α*_{i} + *φ*_{i} or *α*_{i} − *φ*_{i} fits the target better, then *a*_{i + 1} = *a*_{i} + *φ*_{i} or *a*_{i + 1} = *α*_{i} − *φ*_{i}, and *φ*_{i + 1} = *φ*_{i}. If it does not, then *a*_{i + 1} = *a*_{i} and *φ*_{i + 1} = *ρ* ⋅ *φ*_{i}. The search is terminated after a certain number of searching steps or if the search rate *φ*_{i} is less than a given value.

Comparison of each search step is performed using an evaluation function. The function evaluates the fitting of the rotated mode surface and target surface. The root mean square error (RMSE) and cross-correlation coefficient of the two surfaces and RMSE of the discrete cosine transform (DCT) of the two surfaces are tested as evaluation functions. The DCT image is transformed in 4 × 36 blocks divided in polar coordinates.

*θ*= 30 ° ,

*α*

_{0}= 20 ° ,

*φ*

_{0}= 22.5 ° , and

*ρ*= 0.8. The target surface is the rebuilt mode #1 surface rotated by

*θ*= 30°. The first attempt is direct RMSE, which is to determine the RMSE of the rotated mode surface and target surface. The searching rate reduces to 0.12413 ' ' in 100 steps, the searching result is

*α*= 29.714°, and the detection error is 0.286°. Then, the DCT RMSE and cross-correlation coefficient are tested under the same conditions. However, the result of cross-correlation coefficient is

*α*= 29.691°, which is close to that of direct RMSE, and the result of DCT RMSE is equal to that of direct RMSE. The detection error is mainly caused by the difference between the calculated mode surface and rebuilt mode surface. The searching value

*α*

_{i}of the three evaluation functions is shown in Fig. 4, and they are almost the same. Direct RMSE, which requires less computation, is chosen as the evaluation function for the optimum search.

### Top-line detection

The precision of the optimum search is limited by the difference between the calculated mode and rebuilt mode. To obtain better detection for further correction, we use another approach based on the mode surface features. Mode #1 has two obvious aligned peaks. Then, a top-line that passes through the surface center and the tops of two peaks is selected for detection.

To identify the top-line, a series of rings, which are concentric with the surface, are set on the surface. The top-point of the two peaks on each ring is selected. The top-line angle is detected by a least mean square (LMS) fitting of these top-points. The angle between the top-lines of the source surface (calculated mode #1) and target surface (rotated rebuilt mode #1) is the detected rotation angle. The detection mainly depends on the features of the mode #1 surface, which are obvious and introduce little detection error. This mode #1 top-line detection also involves the issue of 180° symmetry solution, and requires an additional detection to acquire the correct result.

The top-line detection is tested with the same simulation data used in Section 3.2. The target surface is still the rebuilt mode #1 surface rotated by *θ*_{0} = 30°. The detection uses 44 rings to find 88 top-points. In each ring, the maximum point is found as the top-point of one peak, and a search in the opposite direction of this top-point is processed to detect the top-point of the other peak. Then, a top-line is detected by an LMS of these 88 top-points. The searching precision of the top-point is 1.40625 ' '. The top-line angle of the calculated mode #1 surface is 92.028°, and that of the rotated rebuilt mode #1 surface is 121.943°. The detected rotation angle is 29.915°, and the detection error is 0.085°, which is much better than the values obtained in the optimum search. The calculated mode #1 and the rotated rebuilt mode #1 with their top-lines are shown in Fig. 5.

## Results of the experiments on 620-mm active optics system

### Influence detection and mode rebuilding

The bending mode and mode force of this active support system are calculated using the detected influence functions. A series of mode forces *F*_{B} is set on the active support to acquire the rebuilt mode surface. As a result, the first 6 modes can be well rebuilt on the system, and the rebuilt mode #1 is 92.1% similar to the calculated mode #1 in RMS. High order modes could be rebuilt as well, but they contain some low-order modes. Therefore, to obtain a good correction result, several correction iterations from high-order modes to low-order modes are performed.

### Active correction using bending mode

*λ*. (

*λ*= 632.5 nm).

### Wavefront rotation detection

To obtain a rotated detection system, the camera of the S-H wavefront sensor is rotated by approximately 10°. Then, the detected wavefront surface is also rotated by approximately 10°. To achieve a good surface correction, the wavefront surface should be rotated back before fitting with bending modes.

*N*is detected, which has a large astigmatism and an obvious rotation. First, a direct surface correction without rotation correction is performed on the rotated system. The result is not good; the mirror surface is 0.647

*λ*in RMS and still has a large astigmatism. Then, another correction process is started where the detected surface is rotated by −11.950° before fitting with bending modes. The first correction is done in three iterations using 15 modes, 12 modes, and 2 modes. The correction result is 0.108

*λ*in RMS, similar to the result of the non-rotated detection system. The surfaces of the normal state and the two correction surfaces are shown in Fig. 11, and the Zernike coefficients of the normal state surface and final correction surface with rotation correction is shown in Fig. 12.

## Conclusions

A procedure for active correction and rotation detection using bending modes is described in this paper. Bending modes are a series of orthonormal modes arranged in increasing order of stiffness. These modes can be well rebuilt on an active optics system. Considering these characteristics, two rotation detection methods using bending modes were developed. These detection methods require no additional components and their detection precision is high. These two methods, namely optimum search and top-line detection, were firstly tested via simulation using experimental data from a 620-mm active optics system. The detection errors were 0.286° and 0.085°, respectively. Then, the rotation detection was tested on the 620-mm active optics system. Two active corrections were performed in non-rotation state and rotation state with top-line rotation detection. The similarity in such correction results means that the rotation detection is effective. In our future work, we plan to apply these rotation detection methods to the active optics system of a 4-m SiC primary mirror.

## Notes

### Acknowledgements

Not applicable.

### Authors' contributions

YZ, TC and HL conceived the presented idea. YZ developed the methodology. YZ and HL collected and analyzed the data. YZ discussed the result and wrote the paper. TC supervised the entire work. All authors read and approved the final manuscript.

### Funding

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

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