The influence of the sampling dots on th
This article appeared in a journal published by Elsevier. The attached
copy is furnished to the author for internal non-commercial research
and education use, including for instruction at the authors institution
and sharing with colleagues.
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licensing copies, or posting to personal, institutional or third party
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In most cases authors are permitted to post their version of the
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Author's personal copy
Optik 123 (2012) 792–795
Contents lists available at ScienceDirect
Optik
journal homepage: www.elsevier.de/ijleo
The influence of the sampling dots on the analysis of the wave front aberration
by using the covariance matrix method
Xuelian Yu a,b , Yong Yao a,∗ , Yunxu Sun a , Jiajun Tian a , Chao Liu a
a
b
Department of Electronic and Information Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, PR China
Department of Optics Information Science and Technology, Harbin University of Science and Technology, Harbin 150080, PR China
a r t i c l e
i n f o
Article history:
Received 22 February 2011
Accepted 15 June 2011
Keywords:
Fringe analysis
Digital image processing
Wave front aberration
Covariance matrix method
Zernike term
a b s t r a c t
The covariance matrix method is a simple method for solving the Zernike polynomial with the higher
fitting precision. In this paper, it was used to analyze the several optical wave fronts of the fine polished
aluminum disk surface captured by a Twyman-Green interferometer system. We had found that the PV
(peak-to-valley) and rms (root-mean-square) values of the wave front aberration changes with changing
the Zernike term and the expressions for the several optical wave fronts with the different sampling
dots were wrong. By analyzing the relations among the condition number of the coefficients matrix, the
Zernike term, and the number of the sampling dots, it was indicated that the number of the sampling
dots had only reduced the fluctuation the PV and the rms value while the Zernike term increases, but
did not change the case that the expressions for the wave front aberration were wrong when the Zernike
term is larger than 14, especially when the number of the sampling dots is less. Such an analysis will be
valuable in solving the Zernike polynomial for the wave front aberration analysis by using the covariance
matrix method in optical testing.
Crown Copyright © 2011 Published by Elsevier GmbH. All rights reserved.
1. Introduction
Zernike polynomials are widely used to describe wave front
aberrations for the interferogram analysis since the mid-1970s
[1–4], because of their unique properties over a circular pupil and
relation to the classical Seidel aberrations that provide a useful
mathematical expression of the aberration content in a wave front
using similar terms [5–7]. The classical Gram–Schmidt method
and the least-squares matrix inversion method have been applied
to determine the Zernike coefficients since 1980 [8]. Typically
37-term Zernike coefficients are provided to express wave front
aberrations and the theoretical interpretation of the Zernike coefficients stability is also given [9], and experimental interpretation
has been done on it [10].
In the paper, the several optical wave fronts with the different
sampling dots are analyzed by using the covariance matrix method
[11] to solve the Zernike polynomials. This paper is organized as
follows. The principle of the covariance matrix method is given in
Section 2. A thorough processing procedure of experimental data
from the circle interference fringe of the fine polished aluminum
disks surface captured by the Twyman–Green interferometer is
described in Section 3, in which the crucial reconstruction algo-
∗ Corresponding author.
E-mail addresses: yxl-1216@sohu.com, yaoyong@hit.edu.cn (Y. Yao).
rithm is based on Zernike polynomials and the covariance matrix
method and experimental results and related discussion are provided in this section. Finally, the conclusion is presented in Section
4.
2. The covariance matrix method
Zernike polynomials can be written in polar coordinates as products of a radial polynomial function and angular functions. These
polynomials are defined here by
Zn (, ) = Rn ()n ()
(1)
where the indices n is a mode term number. The aberrations and
properties corresponding to the first nine mode terms are listed
in Table 1. The ordering of the Zernike terms chosen is the Fringe
ordering [12].
The interference wave front maybe written as follows:
wi (x, y) = q0 + q1 z1i (x, y) + · · · + qj zji (x, y) + · · · + qn zni (x, y)
i = 1, 2, . . . , m
(2)
where wi (x,y) is the wave front of the interference wave surface,
zii (x,y) is the Zernike polynomial of the j-th order, qn is the coefficient, m is total the number of Zernike polynomials. Used in the
fitting, i is the i-th data point of m, (x,y) is the right angle coordinate
of the i-th data point.
0030-4026/$ – see front matter. Crown Copyright © 2011 Published by Elsevier GmbH. All rights reserved.
doi:10.1016/j.ijleo.2011.06.039
Author's personal copy
X. Yu et al. / Optik 123 (2012) 792–795
793
Table 1
The aberrations and properties corresponding to the first nine mode terms.
Term
Zernike polynomial
Meaning
0
1
2
3
4
5
6
7
8
1
cos
sin
22 − 1
2 cos 2
2 cos 2
(−2 + 32 )cos
(−2 + 32 )sin
1 − 62 + 64
Piston
x-Tilt
y-Tilt
Focus
Astigmatism @0◦ and focus
Astigmatism@45◦ and focus
Coma and x-tilt
Coma and y-tilt
Spherical and focus
Let aij = z(xi , yi ), then
⎧
⎪
⎨ a11 q1 + a12 q2 + · · · + a1n qn = w1
a21 q1 + a22 q2 + · · · + a2n qn = w2
(3)
⎪
⎩ · · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·
am1 q1 + am2 q2 + · · · + amn qn = wm
where m > n, and the equations taken from the fitting procedures
are often ill-conditioned. To solve this problem, in the following
section, the covariance matrix method is presented to determine
the Zernike coefficients.
According to Eq. (2), the mean value of the sampling dots
wi (xi ,yj ) is expressed as
m
w̄ =
1
wi
m
(4)
i=1
Therefore, Eq. (2) may be written in the following form:
w̄ =
n
(5)
qj z̄j
j=0
Fig. 1. The optical wave fronts. (a) The circle interference fringe. (b) The patched
image after thinning.
where z̄i is the mean value of the zji (xi , yi ) of the all sampling dots
m
z̄j =
1
Zji
m
(j = 1, 2, . . . , n)
(6)
i=1
Eq. (2) subtracts Eq. (4), and assuming that
Vki = wi − w̄
(i = 1, 2, . . . , m; k = n + 1)
Vji = zji − z̄j (j = 1, 2, . . . , n; i = 1, 2, . . . , m)
(7)
⎡
(8)
A=⎢
Then
Vki = q1 V1i + q2 V2i + · · · + qj Vji + · · · + qn Vni
(j = 1, 2, . . . n;
i = 1, 2, . . . , m; k = n + 1)
(9)
Aef is defined as the covariance of ze and zf , Aef may be written
as
Aef
m
m
m
i=1
i=1
i=1
(10)
According to the method mentioned above, the covariance
matrix Aef can be written as
⎡
⎢
⎢
⎢
⎢
⎣
A11
A21
..
.
An1
Ak1
A12
A22
..
.
An2
Ak2
···
···
..
.
···
···
A1n
A2n
..
.
Ann
Akn
A1k
A2k
..
.
Ank
Akk
⎤
⎥
⎥
⎥
⎥
⎦
⎢
⎣
A12
A22
..
.
An2
A11
A21
..
.
An1
A1n
A2n
..
.
Ann
···
···
..
.
···
⎤
⎥
⎥
⎦
⎡
⎤
A1k
⎢ A2k ⎥
⎥
B = ⎢.
⎣ .. ⎦
Ank
(12)
By solving the following linear equations, q(q1 , q2 , . . ., qn ) can
be obtained
(13)
Aq = B
Therefore, q0 is obtained by Eq. (5)
1
1
1
Vei Vfi =
(zei − z̄e )(zfi − z̄f ) =
zei zfi − z̄e z̄e
=
m
m
m
(e, f = 1, 2, . . . , n + 1)
where k = n + 1, and the covariance matrix Aef Eq. (11) is expressed
by
q0 = w̄ −
n
(14)
qj z̄j
j=1
System stability and the capability of resistance to interference
can be evaluated by the condition number of the coefficients matrix
A as follows:
(11)
cond(A) =
A
A−1
(15)
where
A
and
A−1
are the vector norms of the coefficients
matrix A and its inverse matrix, respectively. In fact, the condition
number is the measurement of the ill-condition of the matrix.
Author's personal copy
794
X. Yu et al. / Optik 123 (2012) 792–795
Fig. 2. The several optical wave fronts with different sampling dots. (a) 1626; (b) 2958; (c) 3717; (d) 4536; (e) 8327.
Table 2
The relation between the condition numbers of the coefficients matrix and the Zernike term for several wave optical fronts with different sampling dots.
Term
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Sampling dots
1626
2958
3717
4536
8327
9.5138
12.3842
12.4358
12.6570
17.6414
21.0075
21.1219
21.1457
21.5705
21.6538
30.7620
30.9717
48.4236
50.0666
1.2322e +003
1.2355e + 003
1.2844e + 003
1.2937e + 003
1.2982e + 003
1.3137e + 003
1.3251e + 003
1.3469e + 003
1.6905e + 003
2.0995e + 003
2.2348e + 003
10.2311
13.1451
13.4834
13.7405
19.5510
21.3857
21.4407
21.7963
25.2473
25.7738
29.8678
30.5599
39.8543
41.0476
48.8052
49.0863
49.1205
49.6311
50.2438
60.5319
62.0512
76.6824
79.8453
116.6395
119.3460
10.4614
13.8794
14.3187
15.1180
22.6623
26.1135
26.1445
26.6488
33.5285
34.9630
43.4199
45.3274
60.4320
63.9891
82.5783
82.9344
83.2817
83.8998
85.1362
110.6442
114.3302
150.2953
156.3228
245.4941
258.8740
9.9191
14.7010
14.7424
17.1754
25.2256
25.6053
26.1492
26.3069
34.0692
35.8958
46.3328
46.5726
72.4584
79.4444
79.7448
79.9127
80.4978
82.0042
82.2459
111.3772
115.5311
153.3583
154.5748
286.7746
308.8180
9.4362
12.3927
12.7156
13.3658
19.7490
21.3288
21.3696
21.6300
25.7984
26.6742
31.0206
31.7148
42.2423
43.9266
47.8431
48.1825
48.7717
48.9238
49.5579
67.8412
69.0140
82.0988
83.7976
105.5363
109.0962
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X. Yu et al. / Optik 123 (2012) 792–795
795
number increases as the Zernike term increases and is less affected
by the number of sample points when the Zernike term is lower.
Note that, as the Zernike term increases, the condition number is
very large when the number of the sampling dots is small, as shown
in Table 2 and the PV and the rms of the wave front aberration is
also very big simultaneously, as shown in Fig. 3. Fig. 3 also shows
that the number of the sampling dots only reduces the fluctuation
the PV value and the rms value when the Zernike term is higher,
but dot not change the fact that the Zernike coefficients is unstable because the condition number of the coefficients matrix is still
bigger.
4. Conclusion
In conclusion, the influence of the sampling dots on the wave
front aberrations had been analyzed and discussed. By researching on the PV and the rms values of the several optical wave front
aberrations with the different sampling dots, and the condition
number of the coefficients matrix, the results show that the number of the sampling dots only reduces the fluctuation the PV and
the rms value while the Zernike term is higher, but will not change
the cases that the Zernike coefficients is unstable because the condition number of the coefficients matrix is still bigger. This also
further shows that the Zernike term is only a decisive factor for
keeping true expression for the wave front aberration when using
the covariance matrix method to solve the Zernike polynomial for
the wave front aberration analysis.
Fig. 3. The relative curves of the wave front aberration with the different Zernike
term from 11 to 35 for the several optical wave fronts with different sampling dots
in Fig. 2. (a) The PV of the wave front aberration. (b) The rms of the wave front
aberration.
3. Experiment results analysis and discussion
In two-beam interferometry the fringe pattern intensity I(x,y)
as a function of the spatial coordinates is given by
I(x, y) = I1 (x, y) + I2 (x, y) + 2
W (x, y)
I1 (x, y)I2 (x, y) cos 2
(16)
where I1 (x,y) and I2 (x,y) are the intensity distributions of the single
beams which are determined separately by appropriate measurements, and W(x,y) is the wave front or their deviation caused by
the distance between the given reference surface and the sample
surface. If the given reference surface is perfect, the case of the
sample surface can be reflected by fringe pattern.
The optical wave front was given in Fig. 1 and processed by
the FCM. Fig. 1(a) shows the circle interference fringe of the equal
inclination of the fine polished aluminum disk surface captured
by a Twyman-Green interferometer system, and the laser wavelength is 532 nm. The deduction of Zernike coefficients is usually
also influenced by the finite number of sampling dots on interferogram and their inherited measurement errors, the uniform
sampling technique [13] was adopted in the sampling process in
the paper. Fig. 1(b) shows the patched image after thinning by
the automatic processing fringe technique [14]. The several optical wave fronts with the different sampling dots were shown
in Fig. 2.
For the several optical wave fronts with different sampling dots,
Fig. 3 shows the relative curves of the wave front aberration with
the different Zernike term from 11 to 35, and Table 2 give the relations between the condition number of the coefficients matrix and
the Zernike term. For the several optical wave fronts, the condition
Acknowledgments
The authors acknowledge the support from the National
Science Foundation of China (Grant no. 60977034) and the
cooperation project in industry, education and research of
Guangdong province and Ministry of Education of China
(Grant no. 2010B090400306).
References
[1] E.J. Fernández, L. Vabre, Adaptive optics with a magnetic deformable
mirror: applications in the human eye, Opt. Express 14 (2006)
8900–8917.
[2] P.M. Prieto, E.J. Fernández, S. Manzanera, P. Artal, Adaptive optics with a programmable phase modulator: applications in the human eye, Opt. Express 12
(2004) 4059–4071.
[3] E. Dalimier, C. Dainty, Comparative analysis of deformable mirrors for ocular
adaptive optics, Opt. Express 13 (2005) 4275–4285.
[4] P. Kurczynski, H.M. Dyson, B. Sadoulet, Large amplitude wavefront generation and correction with membrane mirrors, Opt. Express 4 (2006)
509–517.
[5] V.N. Mahajan, Zernike circle polynomials and optical aberrations of systems
with circular pupils, Appl. Opt. 33 (1994) 8121.
[6] W. Swantner, W.W. Chow, Gram–Schmidt orthonormalization of Zernike polynomials for general aperture shapes, Appl. Opt. 33 (1994) 1832–1837.
[7] D. Sinclair, Seidel vs. Zernike, http://www.sinopt.com/learning1/desnotes/seiz
ern.html.
[8] D. Malacara, J.M. Carpio, J.J. Sánchez, Wavefront fitting with discrete orthogonal
polynomials in a unit radius circle, Opt. Eng. 29 (1990) 672–675.
[9] J.Y. Wang, D.E. Silva, Wave-front interpolation with Zernike polynomials, Appl.
Opt. 19 (9) (1980) 1510–1518.
[10] X.L. Yu, Y. Yao, Y.X. Sun, J.J. Tian, Stability of Zernike coefficients solved by the
covariance matrix method in the analysis of the wave front aberration, Optik
(2011), doi:10.1016/j.ijleo.2010.10.028.
[11] Y.E. Liu, A simple method for Zernike polynomial fitting in fringe analysis, Acta
Optica Sinica 5 (1985) 361–373 (in Chinese).
[12] J.C. Wyant, K. Creath, Basic wavefront aberration theory for optical metrology,
in: Handbook of Applied Optics and Optical Engineering, Academic, 1992, pp.
27–39.
[13] J.Y. Wang, D.E. Silva, Wave-front interpolation with Zernike polynomials, Appl.
Opt. 19 (1980) 1510–1518.
[14] X.L. Yu, Y. Yao, W.J. Shi, Y.X. Sun, D.Y. Chen, Study on an automatic processing
technique of the circle interference fringe for fine interferometry, Optik 121
(2010) 826–830.
copy is furnished to the author for internal non-commercial research
and education use, including for instruction at the authors institution
and sharing with colleagues.
Other uses, including reproduction and distribution, or selling or
licensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of the
article (e.g. in Word or Tex form) to their personal website or
institutional repository. Authors requiring further information
regarding Elsevier’s archiving and manuscript policies are
encouraged to visit:
http://www.elsevier.com/copyright
Author's personal copy
Optik 123 (2012) 792–795
Contents lists available at ScienceDirect
Optik
journal homepage: www.elsevier.de/ijleo
The influence of the sampling dots on the analysis of the wave front aberration
by using the covariance matrix method
Xuelian Yu a,b , Yong Yao a,∗ , Yunxu Sun a , Jiajun Tian a , Chao Liu a
a
b
Department of Electronic and Information Engineering, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, PR China
Department of Optics Information Science and Technology, Harbin University of Science and Technology, Harbin 150080, PR China
a r t i c l e
i n f o
Article history:
Received 22 February 2011
Accepted 15 June 2011
Keywords:
Fringe analysis
Digital image processing
Wave front aberration
Covariance matrix method
Zernike term
a b s t r a c t
The covariance matrix method is a simple method for solving the Zernike polynomial with the higher
fitting precision. In this paper, it was used to analyze the several optical wave fronts of the fine polished
aluminum disk surface captured by a Twyman-Green interferometer system. We had found that the PV
(peak-to-valley) and rms (root-mean-square) values of the wave front aberration changes with changing
the Zernike term and the expressions for the several optical wave fronts with the different sampling
dots were wrong. By analyzing the relations among the condition number of the coefficients matrix, the
Zernike term, and the number of the sampling dots, it was indicated that the number of the sampling
dots had only reduced the fluctuation the PV and the rms value while the Zernike term increases, but
did not change the case that the expressions for the wave front aberration were wrong when the Zernike
term is larger than 14, especially when the number of the sampling dots is less. Such an analysis will be
valuable in solving the Zernike polynomial for the wave front aberration analysis by using the covariance
matrix method in optical testing.
Crown Copyright © 2011 Published by Elsevier GmbH. All rights reserved.
1. Introduction
Zernike polynomials are widely used to describe wave front
aberrations for the interferogram analysis since the mid-1970s
[1–4], because of their unique properties over a circular pupil and
relation to the classical Seidel aberrations that provide a useful
mathematical expression of the aberration content in a wave front
using similar terms [5–7]. The classical Gram–Schmidt method
and the least-squares matrix inversion method have been applied
to determine the Zernike coefficients since 1980 [8]. Typically
37-term Zernike coefficients are provided to express wave front
aberrations and the theoretical interpretation of the Zernike coefficients stability is also given [9], and experimental interpretation
has been done on it [10].
In the paper, the several optical wave fronts with the different
sampling dots are analyzed by using the covariance matrix method
[11] to solve the Zernike polynomials. This paper is organized as
follows. The principle of the covariance matrix method is given in
Section 2. A thorough processing procedure of experimental data
from the circle interference fringe of the fine polished aluminum
disks surface captured by the Twyman–Green interferometer is
described in Section 3, in which the crucial reconstruction algo-
∗ Corresponding author.
E-mail addresses: yxl-1216@sohu.com, yaoyong@hit.edu.cn (Y. Yao).
rithm is based on Zernike polynomials and the covariance matrix
method and experimental results and related discussion are provided in this section. Finally, the conclusion is presented in Section
4.
2. The covariance matrix method
Zernike polynomials can be written in polar coordinates as products of a radial polynomial function and angular functions. These
polynomials are defined here by
Zn (, ) = Rn ()n ()
(1)
where the indices n is a mode term number. The aberrations and
properties corresponding to the first nine mode terms are listed
in Table 1. The ordering of the Zernike terms chosen is the Fringe
ordering [12].
The interference wave front maybe written as follows:
wi (x, y) = q0 + q1 z1i (x, y) + · · · + qj zji (x, y) + · · · + qn zni (x, y)
i = 1, 2, . . . , m
(2)
where wi (x,y) is the wave front of the interference wave surface,
zii (x,y) is the Zernike polynomial of the j-th order, qn is the coefficient, m is total the number of Zernike polynomials. Used in the
fitting, i is the i-th data point of m, (x,y) is the right angle coordinate
of the i-th data point.
0030-4026/$ – see front matter. Crown Copyright © 2011 Published by Elsevier GmbH. All rights reserved.
doi:10.1016/j.ijleo.2011.06.039
Author's personal copy
X. Yu et al. / Optik 123 (2012) 792–795
793
Table 1
The aberrations and properties corresponding to the first nine mode terms.
Term
Zernike polynomial
Meaning
0
1
2
3
4
5
6
7
8
1
cos
sin
22 − 1
2 cos 2
2 cos 2
(−2 + 32 )cos
(−2 + 32 )sin
1 − 62 + 64
Piston
x-Tilt
y-Tilt
Focus
Astigmatism @0◦ and focus
Astigmatism@45◦ and focus
Coma and x-tilt
Coma and y-tilt
Spherical and focus
Let aij = z(xi , yi ), then
⎧
⎪
⎨ a11 q1 + a12 q2 + · · · + a1n qn = w1
a21 q1 + a22 q2 + · · · + a2n qn = w2
(3)
⎪
⎩ · · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·· · ·
am1 q1 + am2 q2 + · · · + amn qn = wm
where m > n, and the equations taken from the fitting procedures
are often ill-conditioned. To solve this problem, in the following
section, the covariance matrix method is presented to determine
the Zernike coefficients.
According to Eq. (2), the mean value of the sampling dots
wi (xi ,yj ) is expressed as
m
w̄ =
1
wi
m
(4)
i=1
Therefore, Eq. (2) may be written in the following form:
w̄ =
n
(5)
qj z̄j
j=0
Fig. 1. The optical wave fronts. (a) The circle interference fringe. (b) The patched
image after thinning.
where z̄i is the mean value of the zji (xi , yi ) of the all sampling dots
m
z̄j =
1
Zji
m
(j = 1, 2, . . . , n)
(6)
i=1
Eq. (2) subtracts Eq. (4), and assuming that
Vki = wi − w̄
(i = 1, 2, . . . , m; k = n + 1)
Vji = zji − z̄j (j = 1, 2, . . . , n; i = 1, 2, . . . , m)
(7)
⎡
(8)
A=⎢
Then
Vki = q1 V1i + q2 V2i + · · · + qj Vji + · · · + qn Vni
(j = 1, 2, . . . n;
i = 1, 2, . . . , m; k = n + 1)
(9)
Aef is defined as the covariance of ze and zf , Aef may be written
as
Aef
m
m
m
i=1
i=1
i=1
(10)
According to the method mentioned above, the covariance
matrix Aef can be written as
⎡
⎢
⎢
⎢
⎢
⎣
A11
A21
..
.
An1
Ak1
A12
A22
..
.
An2
Ak2
···
···
..
.
···
···
A1n
A2n
..
.
Ann
Akn
A1k
A2k
..
.
Ank
Akk
⎤
⎥
⎥
⎥
⎥
⎦
⎢
⎣
A12
A22
..
.
An2
A11
A21
..
.
An1
A1n
A2n
..
.
Ann
···
···
..
.
···
⎤
⎥
⎥
⎦
⎡
⎤
A1k
⎢ A2k ⎥
⎥
B = ⎢.
⎣ .. ⎦
Ank
(12)
By solving the following linear equations, q(q1 , q2 , . . ., qn ) can
be obtained
(13)
Aq = B
Therefore, q0 is obtained by Eq. (5)
1
1
1
Vei Vfi =
(zei − z̄e )(zfi − z̄f ) =
zei zfi − z̄e z̄e
=
m
m
m
(e, f = 1, 2, . . . , n + 1)
where k = n + 1, and the covariance matrix Aef Eq. (11) is expressed
by
q0 = w̄ −
n
(14)
qj z̄j
j=1
System stability and the capability of resistance to interference
can be evaluated by the condition number of the coefficients matrix
A as follows:
(11)
cond(A) =
A
A−1
(15)
where
A
and
A−1
are the vector norms of the coefficients
matrix A and its inverse matrix, respectively. In fact, the condition
number is the measurement of the ill-condition of the matrix.
Author's personal copy
794
X. Yu et al. / Optik 123 (2012) 792–795
Fig. 2. The several optical wave fronts with different sampling dots. (a) 1626; (b) 2958; (c) 3717; (d) 4536; (e) 8327.
Table 2
The relation between the condition numbers of the coefficients matrix and the Zernike term for several wave optical fronts with different sampling dots.
Term
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Sampling dots
1626
2958
3717
4536
8327
9.5138
12.3842
12.4358
12.6570
17.6414
21.0075
21.1219
21.1457
21.5705
21.6538
30.7620
30.9717
48.4236
50.0666
1.2322e +003
1.2355e + 003
1.2844e + 003
1.2937e + 003
1.2982e + 003
1.3137e + 003
1.3251e + 003
1.3469e + 003
1.6905e + 003
2.0995e + 003
2.2348e + 003
10.2311
13.1451
13.4834
13.7405
19.5510
21.3857
21.4407
21.7963
25.2473
25.7738
29.8678
30.5599
39.8543
41.0476
48.8052
49.0863
49.1205
49.6311
50.2438
60.5319
62.0512
76.6824
79.8453
116.6395
119.3460
10.4614
13.8794
14.3187
15.1180
22.6623
26.1135
26.1445
26.6488
33.5285
34.9630
43.4199
45.3274
60.4320
63.9891
82.5783
82.9344
83.2817
83.8998
85.1362
110.6442
114.3302
150.2953
156.3228
245.4941
258.8740
9.9191
14.7010
14.7424
17.1754
25.2256
25.6053
26.1492
26.3069
34.0692
35.8958
46.3328
46.5726
72.4584
79.4444
79.7448
79.9127
80.4978
82.0042
82.2459
111.3772
115.5311
153.3583
154.5748
286.7746
308.8180
9.4362
12.3927
12.7156
13.3658
19.7490
21.3288
21.3696
21.6300
25.7984
26.6742
31.0206
31.7148
42.2423
43.9266
47.8431
48.1825
48.7717
48.9238
49.5579
67.8412
69.0140
82.0988
83.7976
105.5363
109.0962
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X. Yu et al. / Optik 123 (2012) 792–795
795
number increases as the Zernike term increases and is less affected
by the number of sample points when the Zernike term is lower.
Note that, as the Zernike term increases, the condition number is
very large when the number of the sampling dots is small, as shown
in Table 2 and the PV and the rms of the wave front aberration is
also very big simultaneously, as shown in Fig. 3. Fig. 3 also shows
that the number of the sampling dots only reduces the fluctuation
the PV value and the rms value when the Zernike term is higher,
but dot not change the fact that the Zernike coefficients is unstable because the condition number of the coefficients matrix is still
bigger.
4. Conclusion
In conclusion, the influence of the sampling dots on the wave
front aberrations had been analyzed and discussed. By researching on the PV and the rms values of the several optical wave front
aberrations with the different sampling dots, and the condition
number of the coefficients matrix, the results show that the number of the sampling dots only reduces the fluctuation the PV and
the rms value while the Zernike term is higher, but will not change
the cases that the Zernike coefficients is unstable because the condition number of the coefficients matrix is still bigger. This also
further shows that the Zernike term is only a decisive factor for
keeping true expression for the wave front aberration when using
the covariance matrix method to solve the Zernike polynomial for
the wave front aberration analysis.
Fig. 3. The relative curves of the wave front aberration with the different Zernike
term from 11 to 35 for the several optical wave fronts with different sampling dots
in Fig. 2. (a) The PV of the wave front aberration. (b) The rms of the wave front
aberration.
3. Experiment results analysis and discussion
In two-beam interferometry the fringe pattern intensity I(x,y)
as a function of the spatial coordinates is given by
I(x, y) = I1 (x, y) + I2 (x, y) + 2
W (x, y)
I1 (x, y)I2 (x, y) cos 2
(16)
where I1 (x,y) and I2 (x,y) are the intensity distributions of the single
beams which are determined separately by appropriate measurements, and W(x,y) is the wave front or their deviation caused by
the distance between the given reference surface and the sample
surface. If the given reference surface is perfect, the case of the
sample surface can be reflected by fringe pattern.
The optical wave front was given in Fig. 1 and processed by
the FCM. Fig. 1(a) shows the circle interference fringe of the equal
inclination of the fine polished aluminum disk surface captured
by a Twyman-Green interferometer system, and the laser wavelength is 532 nm. The deduction of Zernike coefficients is usually
also influenced by the finite number of sampling dots on interferogram and their inherited measurement errors, the uniform
sampling technique [13] was adopted in the sampling process in
the paper. Fig. 1(b) shows the patched image after thinning by
the automatic processing fringe technique [14]. The several optical wave fronts with the different sampling dots were shown
in Fig. 2.
For the several optical wave fronts with different sampling dots,
Fig. 3 shows the relative curves of the wave front aberration with
the different Zernike term from 11 to 35, and Table 2 give the relations between the condition number of the coefficients matrix and
the Zernike term. For the several optical wave fronts, the condition
Acknowledgments
The authors acknowledge the support from the National
Science Foundation of China (Grant no. 60977034) and the
cooperation project in industry, education and research of
Guangdong province and Ministry of Education of China
(Grant no. 2010B090400306).
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