## Abstract

Abstract: A new method is proposed for testing a rotationally symmetric aspheric surface with several annular subapertures based on a Hartmann–Shack sensor. In consideration of the limited sampling of Hartmann–Shack subapertures in the matching annular subaperture, a new algorithm for whole-aperture wavefront reconstruction from annular subaperture Hartmann–Shack gradient data is established. The algorithm separates the tip, tilt, and defocus misalignments for each annular subaperture by introducing annular Zernike polynomials. The performance of the algorithm is evaluated for different annular subaperture configurations, and the sensitivity of the algorithm to the detector error of the wavefront gradient is analyzed. The algorithm is verified by the experimental results.

© 2010 OSA

## 1. Introduction

Aspheric elements can be used in variety of optical systems, especially large telescope systems, to improve the imaging quality with fewer optical elements. The common methods for measuring aspheric surfaces, such as the null measuring method with compensator, have a lot of drawbacks such as high cost and introduction of errors of the auxiliary optics. A subaperture measuring method has been developed to measure aspheric surfaces with low cost, which can overcome measurement dynamic ranges and sampling limitations of measuring instruments. The subaperture measuring method was first introduced in a subaperture interferometric method in the 1980s [1]. Many researchers have explored different subaperture shapes for measuring different types of mirrors and different whole-aperture wavefront reconstruction algorithms from subaperture wavefront data [2–15].

An annular subaperture interferometric method was first proposed by Y. M. Liu and G. N. Lawrence for measuring rotationally symmetric aspheric surfaces [5]. Because of the advantages of low cost and flexibility, many researchers have explored the method, especially the reconstruction algorithm [5–11]. The reconstruction algorithm based on circle Zernike polynomials was proposed and demonstrated by Y. M. Liu *et al.* [5], and this method can calculate full-aperture Zernike coefficients from the subaperture Zernike coefficients obtained by commercial interferogram reduction software. Following the idea of Y. M. Liu *et al.* [5], X. Hou *et al.* [6,7] presented a full-aperture wavefront reconstruction method from annular subaperture interferometric data by introducing annular Zernike polynomials. Another reconstruction method with successive overlapping phase maps was presented in 1993 by Melozzi *et al.* [8]. By starting from the inner phase map and removing the misalignment errors from its adjacent annular map, the whole-aperture wavefront is reconstructed by repeating the process. Following the work by Otsubo *et al.,* Granados-Agustín *et al.* [11] improved the method by simultaneously fitting misalignment errors among multiple overlapping subapertures [12].

Subaperture interferometric methods may be restricted in measuring aspheric surfaces because the interferometer is sensitive to the measuring environment and has a smaller measurement dynamic range. The subaperture measuring method based on the Hartmann–Shack sensor is an alternative method for testing optical elements effectively. Compared with an interferometer, the Hartmann–Shack sensor is a wavefront sensing instrument that measures the wavefront gradient with a larger measurement dynamic range. The subaperture measuring method can measure the optical element in a bad measuring environment. D. R. Neal *et al.* [16] presented a rectangle subaperture stitching method using a Hartmann–Shack sensor for measuring flat surfaces, and the method was successfully applied to the measurement of nanotopographic features on silicon wafers with high speed.

In this paper, a new method of annular subaperture measuring method based on the Hartmann–Shack sensor is proposed for testing rotationally symmetric aspheric surfaces. Compared with the annular subaperture interferometric method, the new method can test aspheric surfaces with fewer annular subapertures in bad measurement environments. The principle of the annular subaperture measuring method based on the Hartmann–Shack sensor is to use a series of spherical wavefronts with different curvature radii to match different annular subapertures of an aspheric surface until the whole-aperture aspheric surface is measured. In the matching subaperture, the departure between the spherical wavefront and the tested aspheric surface is in the measurement dynamic range of the Hartmann–Shack sensor.

There are several ways to reconstruct the whole-aperture wavefront. One way is where the whole-aperture wavefront is joined together from the subaperture wavefront data, which should be reconstructed from the annular subaperture Hartmann–Shack gradient data in advance. In this way, algorithms such as the one developed by Y. M. Liu *et al.* [5] can be applied to reconstruct the whole-aperture wavefront. This kind of algorithm, which has been verified in the subaperture interferometric method, will encounter many difficulties in the subaperture measuring method based on the Hartmann–Shack sensor. For example, the matching annular subaperture may be too narrow to correctly reconstruct the wavefront because of low sampling of Hartmann–Shack subapertures in the annular subaperture. Even if all the annular subaperture wavefronts can be reconstructed, the error of the annular subaperture wavefront reconstruction will be introduced into the whole-aperture wavefront. The procedure of the reconstruction algorithm is complicated. Another way is to reconstruct the whole-aperture wavefront after the stitching gradient map is obtained. For example, the method developed by T. D. Raymond *et al.* [17] first removes the relative tilts from the subaperture gradient data to get the stitching gradient data, and then it reconstructs the whole-aperture wavefront with a reconstruction algorithm. In this method, an overlapped region is required between the adjacent measurement subapertures.

In light of the drawbacks of the above algorithms, a new algorithm is established to reconstruct the whole-aperture wavefront directly from the annular subaperture gradient data. The overlapped region between the adjacent annular subapertures is not necessary. Even when the matching annular subaperture is too narrow, the new algorithm can still reconstruct the whole-aperture wavefront from the valid Hartmann–Shack gradient data of all the annular subapertures. Errors in annular subaperture wavefront reconstruction will not be introduced in the whole-aperture wavefront by avoiding annular subaperture wavefront reconstruction. By introducing annular Zernike polynomials, the algorithm can separate annular subaperture misalignments from the whole-aperture wavefront and avoid the coupling error of the different modal. The procedure of the reconstruction algorithm is simple.

In this paper, our purpose is to establish a new reconstruction algorithm from annular subaperture Hartmann–Shack gradient data. The performance of the algorithm is evaluated by numerical simulation and experiment of measuring a spherical surface and a parabolic surface. In Section 2, the principle of the algorithm is described. In Section 3, the reconstruction algorithm is verified to perform well in different annular subaperture configurations, and the algorithm is not sensitive to detector error of the wavefront gradient. In Section 4, in order to verify the algorithm precision a spherical surface is measured by two annular subapertures at different defocusing positions. The result is compared with the result, which is directly measured at the vertex curvature with a spherical wavefront by the Hartmann–Shack sensor. By comparing the two results, the precision of the reconstruction algorithm can be evaluated. The reconstruction algorithm is applied to measure a parabolic surface. In Section 5, the theory and the experiment are analyzed to reach the conclusion.

## 2. The whole-aperture wavefront reconstruction algorithm

As shown in Fig. 1 , the whole-aperture surface with a central obstruction of ${\epsilon}_{0}$ is divided into three annular subapertures. The whole-aperture polar coordinate $(R,T)$is defined in a global normalized coordinate system as shown in Eq. (1),

where ${\epsilon}_{0}$is the central obstruction of the whole-aperture surface. The inner annular subaperture region ${\epsilon}_{0}\le R\le {R}_{out1},0\le T\le 2\pi $ will be defined in a local normalized coordinate system as shown in Eq. (2),where $({r}_{1},{t}_{1})$ is the subaperture polar coordinate in a local normalized coordinate system, and ${R}_{out1}$ is the outer boundary of the inner annular subaperture. By the same rule, the coordinates of other annular subapertures can be defined. The relation of the global normalized coordinate system and the local normalized coordinate system can be shown as Eq. (3),where*K*is the number of the annular subapertures, $({r}_{j},{t}_{j})$ is the $j\_th$ annular subaperture local normalized coordinate, $(R,T)$ is the global normalized coordinate, and ${\epsilon}_{j}$ is the central obstruction ratio of the $j\_th$ annular subaperture.

The annular subaperture misalignments of piston, tip, tilt, and defocus can be represented by the first four annular Zernike polynomials as Eq. (4),

where $w{m}_{j}$ is the $j\_th$ annular subaperture misalignment, ${z}_{i,j}$ is the expression of the $i\_th$ annular Zernike polynomials for the $j\_th$ annular subaperture, and ${a}_{i,j}$ is the $i\_th$ annular Zernike coefficient for the $j\_th$ annular subaperture misalignment error. The annular Zernike polynomials are orthogonal and normalized in annular normalized subapertures, which can be found in [18] and [19].The whole-aperture wavefront *W* including annular subaperture misalignments can be represented as a linear combination of annular Zernike polynomials as shown in Eq. (5),

*L*is the number of the annular Zernike modes,

*M*is the number of the annular subapertures, ${A}_{i}$ is the $i\_th$ annular Zernike coefficient for the global wavefront, and ${Z}_{i}$ is the expression of the $i\_th$ annular Zernike polynomials.

Operating the two sides of Eq. (5), Eq. (5) can be expressed as Eq. (6),

*σ*is the sum area of the Hartmann–Shack subaperture in a global normalized coordinate system, and $X=R\mathrm{cos}(T),Y=R\mathrm{sin}(T)$.

For the $k\_th$ Hartmann–Shack subaperture in the $j\_th$ matching annular subaperture, the gradient ${G}_{k,j}$ can be expressed as shown in Eq. (7),

Equation (7) can be expressed in matrix form as shown in Eq. (8),

*S*is the sum of the valid Hartmann–Shack subapertures in all matching annular subapertures as shown in Eq. (9), and $AA$ and

*G*can be shown as Eq. (10) and Eq. (11).

The T matrix, which can solve all annular subaperture misalignment errors, can be expressed in Eq. (12),

and ${z}_{x,p}^{j}(k),{z}_{y,p}^{j}(k)$ can be solved by Eq. (14),

The C matrix, which can reconstruct the whole-aperture wavefront, can be expressed in Eq. (15),

and the value of ${Z}_{X,P}(k),{Z}_{Y,P}(k)$ can be solved by Eq. (16),

By solving Eq. (8), $AA$, which includes the whole-aperture annular Zernike coefficients, can be solved by the criterion of least square.

Because the Hartmann–Shack sensor only measures the gradient of the wavefront, the above algorithm is insensitive to the piston due to annular subaperture misalignment, air turbulence, and the piston error on the tested surface. So the precision of the method is not affected by the piston misalignment, and this method is fit for measuring the surface with continuous gradient.

## 3. Simulation analysis

The performance of the whole-aperture wavefront reconstruction algorithm is analyzed in this section. The analysis focuses on the performance of the algorithm with different annular subaperture configurations and the sensitivity of the algorithm to detecting errors of the wavefront gradient.

The annular subaperture configuration depends on Hartmann–Shack subaperture array, the aperture, and curvature radius of the tested aspheric surfaces. Various annular subaperture configurations may be encountered in practical measurements. Especially, the border annular subaperture may be so narrow that there is only one subaperture sampling frequency. To measure the performance of the algorithm in different annular subaperture configurations, three configurations of the two annular subapertures for a $40\times 40$ subaperture array are defined for the whole-aperture with an obstruction ratio of 0.2, as shown in Fig. 2 .

To measure the sensitivity of the algorithm to detecting errors of the wavefront gradient, some zero-mean and Gaussian distribution random noises are added to the wavefront gradient data of every annular subaperture. The noise-to-signal ratio of the gradient data can be defined as Eq. (17),

where*τ*is the noise-signal ratio,

*δ*is the standard deviation of the added noise gradient data, and

*S*is the standard deviation of the signal gradient data.

The simulation procedure is described as follows:

- 1. The original whole-aperture wavefront with a central obstruction of 0.2 is generated with a series of Zernike polynomials.
- 2. The whole-aperture wavefront is divided into three concentric annular subapertures, as shown in Fig. 2.
- 3. Each annular aperture is measured with different misalignments of piston, tip, tilt, and defocus, and then the gradient data of all the annular subapertures is added with noise of different levels.
- 4. The whole-aperture wavefront is reconstructed from the gradient data by the algorithm established in Section 2. Whole-aperture annular Zernike coefficients can be obtained.
- 5. The whole-aperture wavefront is generated with whole-aperture annular Zernike coefficients.
- 6. The reconstructed whole-aperture wavefront is compared with the original whole-aperture wavefront, and the residual wavefront between them is calculated.

The simulation result for Configuration 1 is shown in Fig. 3 . The original wavefront is shown in Fig. 3(a). The isometric of three annular subaperture wavefronts with different subaperture misalignments is shown in Fig. 3(b). The valid Hartmann–Shack subaperture spot arrays of the three annular subapertures are shown in Figs. 3(c), 3(d), and 3(e), respectively. The gradient data is added with Gaussian noise $\tau =0.1$. Using the reconstruction algorithm in Section 2, the whole-aperture wavefront is reconstructed as shown in Fig. 3(f). The residual wavefront between the original wavefront and the reconstructed wavefront is shown in Fig. 3(g).

By averaging 100 simulation results for per noise level, the root-square-mean values and peak-valley values of the residual wavefront for different cases of annular subaperture configurations and different noise levels are shown in Table 1 .

From the results in Table 1 and Fig. 3, the reconstruction algorithm performs well for different annular subaperture configurations. The precision of the reconstruction algorithm is very high when there is no added noise for the three configurations of the annular subaperture. Although the precision of the reconstruction algorithm decreases with the increase of noise level, the precision of the algorithm remains at a relative higher level. We can come to the conclusion that the reconstruction algorithm is relative insensitive to the detector errors of the wavefront gradient.

## 4. Experiment verification

The major components of the measuring system are shown in Fig. 4 . A 0.635 μm light source is collimated by a relay-imaging telescope. A converging lens produces a series of spherical wavefronts to match the tested surface. The tested surface is mounted with a 5-axis adjustor capable of controlling the tip, tilt, and defocus during measurement operation. The reflected light from the tested surface passes through an aperture that limits the matching annular subaperture to avoid crosstalk of subaperture spots. Then the reflected light is imaged, using a 5:1 relay telescope, onto the lenslet array. Each lenslet in the array has a 0.13 mm square aperture and a focal length of 4 mm. Light incident on each lenslet of the array produces a focal spot on a digital charged-coupled device (CCD) camera.

During the measurement operation, the far-field focal spot of the reflected light from the tested surface is measured by CCD 2 to help the alignment of the tested surface. In order to determine the matching annular subaperture, the tested surface is marked by a so-called characterized fiduciary and then is imaged onto CCD 3 by the relay imaging telescope. The fill factor of the Hartmann–Shack subaperture at the boundary of the matching annular subaperture is calculated, and then the fill factor of the Hartmann–Shack subaperture can be used to determine the valid Hartmann–Shack subapertures in the matching annular subaperture.

#### 4.1 Results of measuring spherical surface

In this section, a spherical surface with poor optical quality is chosen in the experiment to verify the algorithm. The spherical surface is measured with two defocus annular subapertures. The images of the valid Hartmann–Shack subaperture spot array in the two annular subapertures are shown in Figs. 5(a) and 5(b). The valid Hartmann–Shack subaperture spots in the matching annular subaperture are circled. The whole-aperture wavefront of the spherical surface is reconstructed from the two annular subaperture gradients, shown in Fig. 6 . The $\text{PV}$value and the$\text{RMS}$value are $\text{0 .736\lambda}$and$\text{0 .151\lambda}$, respectively.

For comparison, the spherical surface is directly tested at its vertex curvature center by a spherical wavefront. The whole-aperture valid subaperture spot array is shown in Fig. 5(c), and the whole-aperture wavefront is shown in Fig. 7 . The $\text{PV}$value and$\text{RMS}$value are $\text{0 .712\lambda}$and $\text{0 .148\lambda}$, respectively. The residual wavefront between the two above wavefronts is shown in Fig. 8 , and the $\text{PV}$value and $\text{RMS}$value are $\text{0 .073\lambda}$and $\text{0 .011\lambda}$, respectively.

Because we take the whole-aperture wavefront directly measured by the Hartmann–Shack sensor as a comparison, the residual wavefront shown in Fig. 8 can evaluate the precision of the reconstruction algorithm. The experimental result shows that the precision of the reconstruction algorithm is high and gets agreement with the simulation result in Section 3.

In order to test the precision of the algorithm for different defocuses, every annular subaperture is tested at a different defocusing position, which the optical system permits. The PV value and RMS value of the residual wavefront in different cases is shown in Table 2 . The annular subaperture misalignments are separated by the algorithm from the Hartmann–Shack gradient data. The precision of the algorithm is relatively unaffected by a change in defocus. The results shown in Table 2 can also get us to the conclusion that the magnitude of the higher-order aberration introduced through the defocus method remains at a lower level.

#### 4.2 Results of testing parabolic surface

In this section, the reconstruction algorithm is applied to measure a parabolic surface. The experimental setup for testing the parabolic surface is shown in Fig. 9(a) . Images of the Hartmann–Shack subaperture spot arrays of the two matching annular subapertures are shown in Figs. 9(b) and 9(c).

The whole-aperture wavefront is reconstructed from annular subaperture Hartmann–Shack gradient data as shown in Fig. 10 . The $\text{PV}$value and $\text{RMS}$value are $\text{5 .541\lambda}$ and$\text{1 .319\lambda}$, respectively. For comparison, the parabolic surface is tested by an infrared interferometer as shown in Fig. 11 , and the $\text{PV}$value and $\text{RMS}$value are $\text{5 .698\lambda}$ and $\text{1 .310\lambda}$, respectively.

The wavefront maps obtained by the two methods are in good agreement. The residual wavefront between the two above wavefronts is shown in Fig. 12 , and the$\text{PV}$value and$\text{RMS}$value are $\text{0 .248\lambda}$ and $\text{0 .028\lambda}$, respectively.

The Hartmann–Shack subaperture spots are broadened because of the strong curvature of the parabolic surface, so the detector error of the wavefront gradient may be larger than the detector error of the wavefront gradient in testing spherical surfaces. The detector error of the wavefront gradient influences the precision of the experimental result. The experimental result of measuring a parabolic surface still proves that the reconstruction algorithm is feasible and efficient in measuring an aspheric surface with acceptable precision.

## 5. Conclusion

An annular subaperture measuring method based on the Hartmann–Shack sensor is proposed for measuring rotationally symmetric aspheric surfaces, for which accurately combining the subaperture gradient data corrupted by misalignments and noise into a complete surface figure is the key problem. In this paper, the whole-aperture wavefront algorithm from subaperture wavefront gradient data is established. By numerical simulations, the algorithm performs well in different subaperture configurations and is insensitive to the detector error of the wavefront gradient. The precision of the algorithm is verified by measuring a spherical surface with the annular subaperture measuring method. The precision of the algorithm is high in measuring a spherical surface. The reconstruction algorithm is applied to test a parabolic surface with acceptable precision.

Though the experiments are carried out for two annular subapertures, the algorithm can be used for reconstructing a whole-aperture wavefront with more annular subapertures based on the larger micro-lens array of the Hartmann–Shack sensor.

## Acknowledgements

The author is grateful to the Advanced Optical Center of Institute of Optics and Electronics of the Chinese Academy of Sciences for the help in the experiment. The author also gives thanks to Professor Wenhan Jiang and Professor Changhui Rao for their advice regarding the paper.

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