Absolute calibration of Hartmann-Shack wavefront sensor by spherical wavefronts

Optics Communications 283 (2010) 910–916

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Absolute calibration of Hartmann-Shack wavefront sensor by spherical wavefronts Jinsheng Yang a,b,c,*, Ling Wei a,b,c, Hongli Chen a,b,c, Xuejun Rao a,b, Changhui Rao a,b a

The Laboratory on Adaptive Optics, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China The Key Laboratory on Adaptive Optics, Chinese Academy of Sciences, Chengdu 610209, China c Graduate School of Chinese Academy of Sciences, Beijing 100080, China b

a r t i c l e

i n f o

Article history: Received 27 July 2009 Received in revised form 3 November 2009 Accepted 4 November 2009

a b s t r a c t A method for absolute calibration of Hartmann-Shack wavefront sensor (HSWFS), in which the wavefront differences of several spherical wavefronts are used to determine parameters of HSWFS, is proposed in this paper. The calibration method is introduced and the experiment results and error analysis are presented. Across a pupil with diameter of 2.6 mm, a lenslet array of 20  20 sub-apertures with square configuration, and focal length 4 mm, is used to sample the incident wave. The results indicate the uncertainty of the Hartmann-Shack wavefront sensor calibrated by the proposed method, is improved to less than k/60 PV value and k/500 RMS value (k = 635 nm) with modal reconstruction method. Furthermore, the factors affected the results are analyzed. The error analysis suggested that the influences of the factors on the accuracy of reconstruction can be controlled to an accept level. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction The Hartmann-Shack wavefront sensor (HSWFS), which is often used in adaptive optics for astronomy and laser beam diagnosis, is a well-known apparatus for measuring wavefront [2,6]. In the past several years, high accuracy measurements of wavefront are urgently needed in some fields, such as thermal and gravitational effects analyzing, mechanical deformities and optical aberrations of optical systems measurement. In most situations, the sensor has to be calibrated before it works to eliminate the errors induced by manufacture and assembling of the sensor [1,5]. Traditionally, the HSWFS is calibrated by a plane beam, but the aberrations of the plane beam cannot be eliminated from the system. It is fatal to high accuracy measurement. New method for accurately calibrating HSWFS becomes an issue has to be resolved. A HSWFS is composed of a lenslet array and a photodetector (usually a CCD camera) fixed on the focal plane of the lenslet array. The measured wavefront is spatially sampled using a lenslet array which creates a spot pattern on the CCD camera. The displacement of the spots created by the measured wavefront from the spots of a reference wavefront (usually a plane beam) allows for the measurement of the average wavefront slopes across each microlens aperture. The original wavefront shape is usually obtained by fitting them to a polynomial basis [3,7]. Recently, Alexander Chernyshov and Uwe Sterr calibrated a HSWFS by measuring several spherical wavefronts [4]. The * Corresponding author. Address: The Laboratory on Adaptive Optics, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China. E-mail address: [email protected] (J. Yang). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.11.022

reported results show that the calibrated sensor reaches a high accuracy. The method of fitting acquired data to a polynomial is used to determine the parameters of HSWFS. In the process of the data fitting, the differences between the real curvatures of spherical wavefront and the curvatures which are measured by the HSWFS are used as known. However, because the real curvatures are unknown before the determination of the parameters of HSWFS, the differences are also unknown quantities. In this paper, based on the principle of spherical wavefront calibration, we proposed a method for calibrating HSWFS. An apparatus for the calibration experiment is also setup. The parameters of the HSWFS, the distance between the lenslet array and the CCD camera chip f, distance between centers of lenslet can be determined exactly. A normal HSWFS was calibrated through the method above. The results show that the method can reduce the uncertainty of the HSWFS to a level of k/500 RMS value in measuring spherical wavefront. Furthermore, we analyze the factors which affect the accuracy of the sensor. 2. The method of the parameters determination of HSWFS The distorted wavefront U(x, y) is spatially sampled by lenslet array, the displacements of the light spots center from optical axes of the respective microlenses in x, y direction r(x), r(y). The values of r(x) and r(y) can be determined by Eq. (1) [4]:

rðxÞ ¼ f

@U ; @x

rðyÞ ¼ f

@U ; @y

the the are the

ð1Þ

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where U(x, y) denotes the phase of wavefront on the lenslet array surface plane, f is the distance between the lenslet array and the CCD camera chip (usually the focal length of the lenslet array). The phase of a spherical wavefront radiant from a point source could be described as:

Uðx; yÞ ¼

1 2 ðx þ y2 Þ; 2Z

ð2Þ

where Z is the distance between the point source and the surface of the lenslet array (Fig. 1), (x, y) is the coordinates of the point on the lenslet array surface. From the Eqs. (1) and (2), the displacements r(x), r(y) at the sub-aperture number (i, j), i and j are the numbers of the line and column.

rðxÞ ¼ f

  @U f ¼ xi;j ; @x i;j Z

rðyÞ ¼ f

  @U f ¼ y : @y i;j Z i;j

ð3Þ

¼ fT

1 ðxi;j  xi1;j Þ Z

1 þ T; Z

Q y ¼ ½iT þ rðyi;j Þ  ½ðj  1ÞT þ rðyi;j1 Þ ¼ T þ f 1 ¼ fT þ T; Z

ð4Þ 1 ðy  yi;j1 Þ Z i;j ð5Þ

where T is the distance between the adjacent microlenses optical axes. Eqs. (4) and (5) show that the distances of the adjacent light spots of spots pattern are independent to the coordinates. The distance of spots center on the CCD chip is described as the product of the number of the pixels J and the size of the pixel S, then

1 Q ¼ Q x ¼ Q y ¼ JS ¼ fT þ T; Z

ð6Þ

fT fT Z¼ ¼ : Q  T JS  T

ð7Þ

Eq. (7) shows the relationship of the radius of curvature of spherical wavefront Z and the parameters f, S, and T of the HSWFS. Because of the errors induced by manufacture and assemblage, the exact values f0, T0, and S0 are not known. Suppose that the differences between the actual values and the values designed are df, dT, and dS. Then

T ¼ T 0 þ dT f ¼ f0 þ df S ¼ S0 þ dS dT=T  1 df =f  1 dS=S  1:

1 Z

q¼ ¼

JS  T : fT

ð9Þ

Then, the difference between the real curvature q0 and the curvature qmeas calculated from the spot pattern is

dq ¼ qmeas  q0 ¼ q0

ð8Þ

S0 ¼

Q T0  J J

ð10Þ

ð11Þ

Eq. (10) becomes

  df 1 dT dS   : f0 f0 T 0 S0

ð12Þ

Because the distance between the spot light source and the lenslet array cannot be measured exactly. A reference point is selected on the box of the HSWFS. The distance from the spot light source to the reference point is Zref. The relationship among Z0, Zref, and dZ is shown in Fig. 2.

q0 ¼

1  qref þ q2ref dZ; Z ref  dZ

ð13Þ

where qref = 1/Zref . In this study, one method is used for identification of the HSWFS parameters. The process is below. Considering the parameter Z cannot be measured accurately, it is necessary to find another way to calculate the parameters. But the distance between two positions of the light source can be measured accurately. According to Eq. (7), the distance between two positions can described as

DZ ¼ Z 1  Z 2 ¼

fT fT  ; J1 S  T J2 S  T

ð14Þ

where DZ is the distance between two positions, J1, J2 are the number of pixels of the adjacent spots distance in different positions. So, the distance between the lenslet array and the CCD camera chip f, can be obtained from equation

f ¼

ðJ 1 S  TÞðJ2 S  TÞ : ðJ2  J 1 ÞS

ð15Þ

After the identification of parameter f, Eq. (12) becomes

dq ¼ qmeas  q0 ¼ 

  JS0 dT dS :  T 0 f0 T 0 S0

ð16Þ

And Eq. (16) becomes

dq ¼ qmeas  q0 ¼ 

Fig. 1. Light spot pattern in the plane of the CCD chip for a spherical wave-front.

  df JS dT dS  0  f0 T 0 f0 T 0 S0

In the case of Z  f, we have Q  T0

dq ¼ qmeas  q0 ¼ q0

Therefore, the distances between the adjacent light spots in two directions, which created by the spherical wavefront, can be described as

Q x ¼ ½iT þ rðxi;j Þ  ½ði  1ÞT þ rðxi1;j Þ ¼ T þ f

For the purpose of calculation, the radius of curvature Z is changed to the form of curvature q. Eq. (7) becomes

  1 dT dS :  f0 T 0 S0

Fig. 2. Setup of the experiment.

ð17Þ

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Table 1 Parameters of HSWFS. CCD camera

Parameter

Microlens array

Parameter

Manufacturer Type

Basler A312f

0.13 mm  0.13 mm 20  20

Number of pixels Pixel size

782  582

Distance T Number of sub-aperture Focal length

8.3 lm  8.3 lm

Diameter

£2.6 mm

4 mm

Considering Eqs. (13) and (17), the relationship between qmeas and qref is

  1 dT dS :  f0 T 0 S0

ð18Þ

This expression describes the relationship between the measured curvature and the reference curvature. From the Eqs. (15) and (18), the parameters of f0, T0, S0, and dZ can be determined. The process of parameters’ determination is shown as follows. Firstly, according to the Eq. (15) parameter f0 is determined. Because f is the parameter that influences the results most, it is reasonable to assume that the real values S0, and T0 are equal to the design values. With different values of DZ, a series of f are obtained and the mean value of the obtained value is the real value of f0. The rest parameters are identified by fitting to the data according to Eq. (15) with a series of measured distances. In Eq. (18), the distance T and pixel size S are not independent, so it is not possible to determine them both. In this study, considering the manufacture, the parameter S is assumed to the real value. 3. Experimental setup The setup for the calibration of HSWFS is shown in Fig. 2. The radiation of a 0.635 lm semiconductor laser (LASER 2) is coupled in a single mode fiber. The spherical wavefront radiated from the end of the fiber is used as the calibrating wave. The maximum output power of the fiber is 10 mW. This is enough for the calibration. The He–Ne laser (LASER 1) is used to align the components. During the testing, it is found that the tilt of HSWFS, the shift of HSWFS, the location accuracy of the spot light source, and other factors affect the reconstruction accuracy of the sensor. How the factors affect the results and their influences on the results are analyzed later. In the study reported, the parameters of the HSWFS used are shown in Table 1. 4. Results Fig. 3 shows the image acquired from HSWFS, where the spots’ centroids are used to calculate the parameters and the local slope of measured wavefront. For each measurement point, 100 frames of the CCD were averaged to reduce the effect of air turbulence and of electronic readout noise. Twenty-two images of spot pattern in different positions are obtained and 21 exact DZ are acquired at the same time. According to Eq. (15), we can acquire 21 values of f. The real value of the distance between the lenslet array and the CCD chip is determined by averaging the values of f0 calculated, f0 = 4.246 ± 0.030 mm (Fig. 4). Other parameters can be determined by fitting to the data according to Eq. (16). The result of fitting curve is shown in Fig. 5. According to the fitting curve, the parameters are obtained, dZ = 10.56 mm, dT = 0.003 lm. To obtain the aberrations of the HSWFS itself, the geometric parameters of the lenslet array are used as a reference to reconstructive the spherical wavefront. The residual errors between

Fig. 3. Image acquired form CCD camera.

4.25

Focal length (mm)

qmeas ¼ q2ref dZ þ qref 

4.2

4.15

4.1

4.05

4

5

10

15

20

Times Fig. 4. The multi-measurement results of the focal length of lenslet array (- average focal length).

the measured wavefront and the real wavefront are the aberrations caused by the sensor. The errors amount to about 0.535k PV and 0.112k RMS for five different spherical wavefronts. Fig. 6 shows typical aberrations of the HSWFS we used. The results indicate that astigmatism is the main aberration. It is necessary for HSWFS to choose a reference to eliminate aberrations of the optics. If we assume that the additional aberrations of the sensor are independent of the curvature of the spherical wavefront. Then, it is reasonable to choose a spherical wavefront as a reference. So, the data that the HSWFS measured through the sensor is the difference between the incident wavefront and the referent spherical wave. In order to reconstruct the wavefront incident to the pupil of the sensor, the defocus aberration of reference spherical wavefront has to be subtracted from the reconstructed data of the HSWFS. Then, the wavefront which has been corrected expressed as follows:

Urec ¼ UHS  Uref ;

ð19Þ

where UHS is the wavefront reconstructed by HSWFS, Uref is the defocus aberration of the referent wave, Urec is the wavefront corrected. The spherical wavefronts with different radii are reconstructed by the sensor, which has been calibrated with a spherical

J. Yang et al. / Optics Communications 283 (2010) 910–916

5.1. Error of centroid

-3

x 10

In the process of wavefront reconstruction, centroid error limited the accuracy of the HSWFS. Because of the pixelization error, photon noise, and readout noise of CCD camera, the error of centroid is crucial to influence the wavefront reconstruction. In the study, we analyzed how the centroid error influences the accuracy of the HSWFS. The method to calculate the errors of centroids are below. We measured the distances of adjacent spots in spot patterns created by different spherical wavefront in y direction. The differences of the distances in corresponding regions are caused by the curvature of the spherical wavefront in theory. In the case of knowing the exact parameters of the sensor, it is easy to be acquired that the real differences caused by the curvature of the spherical wavefronts. Then, the distance errors between the real and calculated differences are caused by centroid error. The errors of distance are show in Fig. 9a. Because of the independence of the centroid error, the relationship of the distance error and the centroid error meets Eq. (20)

1.4

1

ρ

meas

(mm-1)

1.2

0.8

0.6 0.6

913

0.8

1

ρref

1.2

1.4

(mm-1) x 10-3

rc ¼ Fig. 5. The relationship between qmeans and qref and the quadratic fitting curve.

rQ 2

ð20Þ

;

where rQ means the error of distances and rc means the centroid error. The wavefront error rerror caused by the error of centroid rc meets Eq. (21) [8]. -1

0.2

r rerror ¼ pffiffiffiffiffiffic

12F #

Y (mm)

-0.5

0.1

0

0

0.5

-0.1

1

-0.2

-1

-0.5

0

0.5

1

X (mm) Fig. 6. Errors of the HSWFS calibrated.

wavefront of Z = 1.75 m used as a reference. A series of spherical wave with radius of curvature from 0.5 m to 1.7 m are reconstructed by the sensor. The situation of a typical wave with radius of 0.85 m is shown in Fig. 7. The results shows that the difference between the reconstructed wave and the real one is 0.0117k(PV), 0.0013k(RMS). Twenty-one spherical waves with different radii are measured by the sensor. The reconstructive errors are shown in Fig. 8. Fig. 8a shows the PV values of residual errors, and Fig. 8b shows the RMS values of residual errors. The results indicate that the maximum PV error is 0.0163k and RMS error is 0.0019k. This means that in all the measured cases the residual errors are less than k/60 PV and k/500 RMS over the measured diameter of 2.6 mm. 5. Factors affected the results Many factors affected the accuracy of reconstruction, including errors of HSWFS, adjustment process of calibration and measurement. It is necessary to analyze the factors influenced the accuracy when the sensor is calibrated. Main errors of them are analyzed below.

;

ð21Þ

where F# means the f-number of the microlens. Fig. 9b shows that the error rc is less than 6  103 pixel, according to the Eq. (20), the wavefront error induced by centroid error rerror is below to 6.925  104k. 5.2. Error of parameters determination The uncertainty of the HSWFS’s parameters induced the relative error to reconstruction process. It means that the reconstructed wave is in proportion to, but not equal to the real wave. In experiment, it is found that the parameter f affects results most, so we analyzed the parameter f only. Eq. (22) describes the error caused by uncertainty of parameter f.

  wf wf df wf df ¼ Uf  Uf 0 ¼ ¼  f0 þ df f0 f0 þ df f0   df ¼ Uf 0 ; f0 þ df

ð22Þ

where, df is the error of the wavefront reconstructed, df is the uncertainty of value f, Uf is the wavefront reconstructed, Uf0 is the real value of incident wavefront, and f0 is the real value of parameter f. Considering the values we have acquired, df < 0.0162 mm, f0 = 4.246 mm, then, df0 < 0.0038Uf0. A typical spherical wavefront with Z = 0.85 m is used to calculate and the errors is about 9.5  104k RMS value. 5.3. Error of the tilt of the HSWFS During the measurement, it was found that the tilt of the HSWFS affected the determination of parameters like f, and T. To calibrate the parameters exactly, some components are added in the system to eliminate the tilt error, such as the laser LASER1 and the pinhole PH (Fig. 2.). The change of the radius of spherical wavefront dZ induced by the tilt angle h is shown below.

@Z ¼ Z 0 sin h @h

ð23Þ

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J. Yang et al. / Optics Communications 283 (2010) 910–916

1

-1

1

-1

0.8

Y (mm)

-0.5 0.6

0.8 -0.5 0.6

0.4

0

0.4

0

0.2

0.2 0.5

0.5 0

1

0

-0.2 -1

-0.5

0

0.5

1

-0.2

1

-1

-0.5

X (mm)

0

0.5

1

X (mm)

(a)

(b)

(c) Fig. 7. A typical spherical wave-front with radius of curvature 0.85 m reconstructed by the sensor. (a) shows the real spherical wave-front, (b) shows the reconstructed wavefront and (c) shows the difference between the two wave-fronts with 0.0117kPV and 0.0013kRMS.

2

RMS value of residual errors (λ)

PV value of residual errors (λ)

0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004

0.6

0.8

1

1.2

1.4

1.6

Radius of curvature of measured spherical wavefrong (m)

(a)

x 10

-3

1.8 1.6 1.4 1.2 1 0.8 0.6

0.6

0.8

1

1.2

1.4

1.6

Radius of curvature of measured spherical wavefrong (m)

(b)

Fig. 8. Residual errors of wave-front measured in different radii by H-S wave-front sensor: (a) shows the PV values of residual errors and (b) shows the RMS values of residual errors.

where, Z0 is the radius of curvature of spherical wavefront. If we assume value of Z0 is equal to 1 m, then, oZ/oh  102m/degree. The setup which we have established can reduce the uncertainty of the angular position of the HSWFS to the level of less than ±0.03°, which affected the calibration in a very low level. The error induced by the tilt is about 3  106k.

5.4. Error of the shift of sensor The shift error can be described as the distance between the climax of the spherical wavefront on lenslet array and the center of circle of the region used to reconstruct the wavefront (Fig. 10). In theory, the shift error induces the tilt and piston errors to the

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J. Yang et al. / Optics Communications 283 (2010) 910–916 -3

-3

x 10

6

RMS vilues of centriod (pixel)

RMS value of ditance (pixel)

10 9 8 7 6 5 4 3

0.6

0.8

1

1.2

1.4

1.6

1.8

x 10

5.5 5 4.5 4 3.5 3 2.5 2

0.6

0.8

Radius of curvature (m)

1

1.2

1.4

1.6

1.8

Radius of curvature (m)

(a)

(b)

Fig. 9. (a) shows the distance errors in different radii and (b) shows the centroid errors in different radii.

where D is the size of pupil, d0 is the size of pinhole; R0 is the curvature of spherical wavefront diffracted from the pinhole and k is the wavelength. In our study, the reference is a spherical wavefront with 1.75 m curvature of radius. So R0 = 1.75 m, D = 2.6 mm, k = 0.635 lm. The core diameter of the fiber d0 is about 10 lm and DU0 is about 1k. DU = 2.9  103k, and the RMS value is about dpinhole = DU/ 6 = 4.9  104k.

h Light Source Z

Lenslet Array

5.6. Error of the location

Fig. 10. The sketch map of the shift error.

wavefront (Eq. (24)), which does not affected the calibration and the measurement of the sensor if the tilts is eliminated from the reconstructed wave. However, we find that the shift error will produce the astigmatism aberration. The reason that induces the phenomena is the fitting error of Zernike polynomials. The results of our emulation show that the value of the fitting error is about one percent of the tilt error.

UðrÞ ¼

ðr þ hÞ2 1 2 2 ðr þ 2rh þ h Þ; ¼ 2Z 2Z

ð24Þ

where, r is the radius of the measurement pupil, h is the shift error of the sensor, and Z is the radius of curvature of the spherical wavefront. Then, the PV value of fitting error is

dshift 

1 1 rh Utilt ¼ ; 100 100 Z

ð25Þ

where, dshift is the fitting error caused by the shift of HSWFS. In our study, the shift error can be controlled to the level of the size of sub-aperture. Then, r = 1.3 mm, h < 0.13 mm, Z > 0.5 m, so dshift < 5  103k. Assume that the errors obey normal distribution. The RMS values rshift is about dshift/6 (8.3  104k). 5.5. Error caused by the pinhole Appling this method to calibrate HSWFS, the quality of the spherical reference wavefront also affects the accuracy of reconstruction. Even wavefront diffracted from a small pinhole can be influenced by aberrations across the initial aperture. The relationship of initial aberrations DU and the aberrations DU0 of wavefront diffracted from the pinhole meets the equation below [4].

DU <

Dd0 DU0 ; 8R0 k

ð26Þ

During calibrating and measurement, we have to locate the spot light source exactly to meet the requirement. How the error of location affects the wavefront error is shown in Eq. (27).

Du ¼

dZr 2 2Z 2

;

ð27Þ

where, r is the radius of the measurement pupil, Z is the radius of curvature of the spherical wavefront, dZ is the location error of the light source, and DU is the PV value of the wavefront error caused by dZ. In our study, Z > 0.5 m, r = 1.3 mm, dZ < 0.5 mm, then, DU < 3  103k. Assume that the errors obey normal distribution. The RMS values rloc is about DU/6 (5  104k). Assuming that the factors are independent each other, the total error is

rtotal ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2error þ d2f þ r2tilt þ r2shift þ r2loc þ d2pinhloe :

ð28Þ

According to the data above, the total error rtotal is less than 1.6  103k. 6. Conclusions In this study, we have developed a method for absolute calibration of HSWFS, in which the spherical wavefront is used as a reference wave. A normal HSWFS was calibrated through the method we have developed. The uncertainty of the spherical wavefront reconstructed by the calibrated HSWFS reaches k/500 RMS across a diameter of 2.6 mm for the sensor with a focal length of 4 mm. The error analysis suggested that the influences of the factors on the accuracy of reconstruction can be controlled to an accept level. The experiment results and the analysis of factors indicated that the method is feasible to calibrate the HSWFS with high accuracy.

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Acknowledgements This work was performed under the auspices of the High Technology Project of China. The authors are grateful to Prof. Wenhan Jiang for his assistance in reviewing the article. Thanks to Dr. Jing Lu for her help in English correction. References [1] Xinyang Li, Wenhan Jiang, Acta Opt. Sin. 22 (2002) 1236 (in Chinese).

[2] Wenhan Jiang, Hao Xian, Zeping Yang, Lingtao Jiang, Xuejun Rao, Bing Xu, Chinese J. Quantum Electron. 15 (1998) 228. [3] Amos. Talmi, Erez N. Ribak, J. Opt. Soc. Am. A 21 (2004) 632. [4] Alexander Chernyshov, Uwe Sterr, Fritz Riehle, Jürgen Helmcke, Johannes Pfund, Appl. Opt. 44 (2005) 6419. [5] John E. Greivenkamp, Daniel G. Smith, Eric Goodwin, Proc. SPIE 5252 (2004) 372. [6] T.L. Bruno, A. Wirth, A.J. Jankevics, Proc. SPIE 1920 (1993) 328. [7] Naftali Zon, Orr Srour, Erez N. Ribak, Opt. Exp. 14 (2006) 635. [8] Zhou Renzhong, Adaptive Optics, National Defence Industry Press, Beijing, 1996 (in Chinese).