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A new method for improving the accuracy of wavefront fitting with Zernike polynomials
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Physics Procedia 19 (2011) 134–138
ICOPE EN 2011
A new meethod forr improv ving the accuracy y of wav vefront ffitting with Zeernike poolynomials W Wang Zhaoomina*, Ch hee Oichoooa, Qu Weijjuana, Xu Qiangsheng Q ga a
Ngee Ann Pollytechnic, 535 Cllementi Road, 599 9489,Singapore
Absttract Zern nike polynomialls have been wiidely used to fitt wavefront andd, by their repreesentation, calcu ulate the deviatiion due to waveefront aberrrations in an ooptical system. In order to ob btain the coeffi cients of the Zernike Z polynom mials, a set off discrete ortho ogonal polyn nomials needs tto be constructeed using the Grram-Schmidt m method on a unittary circle, and the coefficientts are then calcu ulated by fitting the waveffront data and orthogonal o poly ynomials by thee least squares method. Through simulation, it is found therre is a largee deviation betw ween the given coefficients an nd the calculatedd results becau use of limitation ns of the least ssquares method d. This paper proposes a moodified methodd to improve thee accuracy of w wavefront fitting g.
© 2011 Published by ElsevierV. B.V. Selection and/or peer-review under responsibility of the Organising Committee of © 20 010 Published bby Elsevier B.V the ICOPEN 2011 conference Keyw words: Wavefrontt fitting; Zernike polynomials; p Leaast squares methood; Zernike coeffiicients
1. In ntroduction Zernike Z polynoomials have been b successfu ully used for ttheir represen ntation of aberrrations in maany different fields f of op ptics [1, 2]. To know exactt aberrations, Zernike Z coeffi ficients must be b known. Theey can be obtaained by fittin ng the wavefront data w with a set of poolynomials by y the least squuares method.. Where the polynomials p orrthogonal oveer the discrrete unitary ciircle are consttructed using the Gram-Schhmidt method d on basis of Zernike Z polynnomials [3]. Due D to the limitation l of lleast squares method, theree will be residdual coefficieents in the waavefront remov oval of aberrattions. Heree, we describee a modified method m to perfform more acccurate Zernikee coefficients with w respect to the wavefro ont. 2. Wavefront W rep presented by Zernike poly ynomials The T wavefront can be expannded in terms of o orthonormaal Zernike pollynomials ܼ ሺɏǡ Ʌሻ in the foorm
ܹሺɏǡ Ʌሻ ൌ σ ܽ ܼ ሺɏǡ Ʌሻሻ
(1) (
* Corresponding aauthor. Fax: +65-664689491. E-mail E address: [email protected].
1875-3892 © 2011 Published by Elsevier B.V. Selection and/or peer-review under responsibility of the Organising Committee of the ICOPEN 2011 conference doi:10.1016/j.phpro.2011.06.137
Wang Zhaomin et al. / Physics Procedia 19 (2011) 134–138
135
Where ܽ is the Zernike coefficient. The polynomials can be written as
ܼ௩ ሺߩǡ ߠሻ ൌ ඥʹሺ݊ ͳሻܴ ሺߩሻܿߠ݉ݏǡ݉ ് Ͳ ܼௗௗ ሺߩǡ ߠሻ ൌ ඥʹሺ݊ ܼ ሺߩǡ ߠሻ ൌ ξ݊
ͳሻܴ ሺߩሻߠ݉݊݅ݏǡ݉
ͳܴ ሺߩሻǡ݉
(2)
്Ͳ
(3)
ൌͲ
(4)
Where index n represents the order of the polynomial, m is called the azimuthal frequency. ܴ ሺߩሻ is the radial polynomials and ሺିଵሻೞ ሺି௦ሻǨ ሺିሻȀଶ ܴ ሺߩሻ ൌ σ௦ୀ ߩିଶ௦ (5) శೞ షೞ ௦Ǩቀ
మ
ି௦ቁǨቀ
మ
ି௦ቁǨ
The index j is a polynomial-ordering number and has relationship with n and m in the form
ൌ ሾሺʹ݆ െ ͳሻǤହ ͲǤͷሿ௧ െ ͳ
(6)
ʹሼሾʹ݆ ͳ െ ݊ሺ݊ ͳሻሿȀͶሽ௧ ݊݁ݒ݁ݏ݄݅݊݊݁ݓ ൌቊ ʹሼሾʹሺ݆ ͳሻ െ ݊ሺ݊ ͳሻሿȀͶሽ௧ െ ͳ݀݀ݏ݄݅݊݊݁ݓ The Zernike polynomials and the relationship among j, n, and m are given in Table 1[4].
(7)
Table 1 Zernike polynomials j
n
m
Zernike polynomials
Aberration name
1
0
0
1
Piston
2
1
1
ɏ ߠ
x tilt
3
1
1
ɏߠ
y tilt
4
2
0
ξ͵ሺʹɏଶ െ ͳሻ
5
2
2
ξɏଶ ʹߠ
Primary astigmatism at Ͷͷι
6
2
2
ξɏଶ ʹߠ
Primary astigmatism at Ͳι
7
3
1
ξͺሺ͵ɏଷ െ ʹɏሻߠ
Primary y coma
8
3
1
ξͺሺ͵ɏଷ െ ʹɏሻ ߠ
Primary x coma
ଷ
9
3
3
ξͺɏ ͵ߠ
10
3
3
ξͺɏଷ ͵ߠ
11
4
0
ξͷሺɏସ െ ɏଶ ͳሻ
Primary spherical
12
4
2
ξͳͲሺͶɏସ െ ͵ɏଶ ሻ ʹ ߠ
Secondary astigmatism at Ͳι
ସ
ଶ
13
4
2
ξͳͲሺͶɏ െ ͵ɏ ሻʹ ߠ
14
4
4
ξͳͲɏସ Ͷߠ
15
4
4
ξͳͲɏସ Ͷߠ
Secondary astigmatism at Ͷͷι
3. Generation of orthogonal polynomials by Gram-Schmidt orthogonalization Zernike polynomials are orthogonal on continuous unitary circle but not on discrete unitary circle. Thus, GramSchmidt orthogonalization is carried out to construct a set of polynomials which are orthogonal on the discrete unitary circle
Wang Zhaomin et al. / Physics Procedia 19 (2011) 134–138
136
ሺܸଵ
ܸଶ
ǥ ܸିଵ
ܸ ሻ ൌ ሺܼଵ
ܼଶ
ǥ
ͳ ܿଶǡଵ ۇ ܼ ሻ ڭ ۈ ܿିଵǡଵ ܿ ۉǡଵ
ܼିଵ
Ͳ ͳ ڭ ܿିଵǡଶ ܿǡଶ
ǥ ǥ ڰ ǥ ǥ
Ͳ Ͳ ڭ ͳ ܿǡିଵ
Ͳ Ͳ ۊ ۋڭ Ͳ ͳی
(9)
In a general way ିଵ
ܸ ൌ ܼ σୀଵ ܿǡ ܼ
(10)
On the discrete unitary circle, the polynomials ܸ ሺɏǡ Ʌሻ satisfy the following condition σே ୀଵ ܸ ሺɏ ǡ Ʌ ሻ ܸ ሺɏ ǡ Ʌ ሻ ൌ Ͳ
(11)
Then the ܿǡ can be written ିଵ
ଶ ே ܿǡ ൌ σୀଵ ൛ܿିǡ ൣሺσே ୀଵ ܼ ሺɏ ǡ Ʌ ሻܸି ሺɏ ǡ Ʌ ሻሻൗሺσୀଵ ܸି ሺɏ ǡ Ʌ ሻሻ൧ൟ
(12)
Where ݅ ൌ ͳǡʹǡ͵ǡ ǥ ǡ ݆ െ ͳ ൌ ͳ.We get the wavefront expressed by orthogonal polynomials ܸ ܹሺɏǡ Ʌሻ ൌ σ ܾ ܸ ሺɏǡ Ʌሻ
(13)
Here, we define the mean value of the wavefront as
ഥ ൌ ଵ σே ഥ ܹ ୀଵ ܹሺɏ ǡ Ʌ ሻ ൌ σ ܾ ܸఫ
(14)
ଵ
ഥఫ ൌ σே ܸ ሺɏ ǡ Ʌ ሻ, L is the maximum value of the index j. Where ܸ ே ୀଵ Then we reconstruct the Eq.13 by subtracting the mean value of the wavefront
ഥ ൌ σ ܾ ሺܸ ሺɏǡ Ʌሻ െ ܸ ഥఫ ሻ ܹሺɏǡ Ʌሻ െ ܹ
(15)
For the first term of the orthogonal polynomials is piston terms, which mean value is a constant. So we reset the തതതଶ ሻǡ ǥ ǡ ൫ܸ െ ܸ ഥఫ ൯ , it is easy to verify polynomials’ orthogonality on discrete orthogonal polynomials as ሺܸଶ െ ܸ unitary circle. The Eq.15 can be written as
ഥ ൌ σୀଶ ܾ ሺܸ ሺɏǡ Ʌሻ െ ܸ ഥఫ ሻ ܹሺɏǡ Ʌሻ െ ܹ
(16)
4. Least squares fitting We define the measured wavefront data as ܹሺɏ ǡ Ʌ ሻ , the deviation of fitting is Ɂଶ . The wavefront and orthogonal polynomials are fitted with least squares method ഥ ഥ Ɂଶ ൌ σே ୀଵ൫ܹሺɏ ǡ Ʌ ሻ െ ܹ െ σୀଶ ܾ ሺܸ ሺɏ ǡ Ʌ ሻ െ ܸఫ ሺɏ ǡ Ʌ ሻሻ൯
ଶ
(17)
In order to make the Ɂଶ as small as possible, we take differential on the both side of the Eq.17 பஔమ பೕ
మ
ൌ
ಽ തതത ഥ ப σಿ ೖసభቀௐሺೖ ǡೖ ሻିௐିσೕసమ ೕ ሺೕ ሺೖ ǡೖ ሻିണ ሺೖ ǡೖ ሻሻቁ
பೕ
ൌͲ
(18)
Thus, coefficient ܾ can be obtained by solving the equation. ܾ ൌ
തതത ഥ σಿ ೖసభሺௐሺೖ ǡೖ ሻିௐሻሺೕ ሺೖ ǡೖ ሻିണ ሺೖ ǡೖ ሻሻ
(19)
మ തതത σಿ ೖసభሺೕ ሺೖ ǡೖ ሻିണ ሺೖ ǡೖ ሻሻ
Where ݆ ൌ ʹǡ͵ǡ ǥ ܮ. ഥ െ σୀଶ ܾ ሺܸ െ ܸ തതതത and σୀଶ ܾ ܸ ഥఫ ሻ, ܹǡ ഥఫ , so the coefficient of the The first piston term ܸଵ is composed of ܹ െ ܹ piston term can be written as
ഥ െ σୀଶ ܾ ܸ ഥఫ ܾଵ ൌ ܹ
ಽ ഥ ഥ σಿ ೖసభሺௐሺೖ ǡೖ ሻିௐିσೕసమ ೕ ሺೖ ǡೖ ሻሺೕ ሺೖ ǡೖ ሻିണ ሻሻభ ሺೖ ǡೖ ሻ మ σಿ ೖసభ ሺభ ሺೖ ǡೖ ሻሻ
(20)
Wang Zhaomin et al. / Physics Procedia 19 (2011) 134–138
137
5. Zernike coefficcients Since we know w the coefficiients ܾ andܿ e coefficients ܽ can be got by substitutting the Eq.10 0 and , the Zernike Eq.1 13 into the Eq..1 ିଵ
ܹ ൌ ܾଵ ܼଵ σୀଶ ܾ ሺܼ σୀଵ ܿ ܼ ሻ
(2 21)
Comparing C witth Eq.1, we caan get the coeffficients ܽ
ܽ ൌ ܾ σୀାଵ ܾ ܿ
(2 22)
Wheere ݆ ൌ ͳǡʹǡ͵ǡ ǥ ǡ ሺ ܮെ ͳሻ and a ܽ ൌ ܾ . 6. Calculation We W do a simulaation experim ment on the mo odified metho d with respect to least squaares method. E Exit pupil diam meter is 2 mm, the wavvelength is 6333nm, and thee simulated w wavefront with h fixed first 15 terms Zernnike coefficien nts in nt. Tablle 2. Fig.1 shoows the simulaated wavefron
Fig.1. Simulaated wavefront
Through T calcuulation, we geet the fitting results in Taable 2. The calculations c work w without iteration. Fo or the limittation of the lleast squares method, m theree are obviouslly deviations between b fittin ng results giveen by least squ uares meth hod and the ssimulated coeffficients. Whiile the modifiied method giives better ressults either inn coefficients or in stand dard deviationn.
Wang Zhaomin et al. / Physics Procedia 19 (2011) 134–138
138
Table 2 Simulation and the fitting results Index j
Coefficients by simulation
Least squares method
modified method
1
0.1
0.0908
0.0994
2
-0.1
-0.1514
-0.1133
3
0.1
0.1494
0.1000
4
-0.1
-0.0705
-0.1041
5
0.1
-0.0083
0.1000
6
-0.1
-0.0719
-0.0976
7
-0.1
-0.1000
-0.1000
8
0.1
0.1000
0.1014
9
-0.1
-0.1013
-0.1000
10
-0.1
-0.0965
-0.0998
11
-0.1
-0.1000
-0.0998
12
-0.1
-0.1000
-0.1000
13
0.1
0.0986
0.1000
14
-0.1
-0.0990
-0.1000
15
0.1
0.1000
0.1000
Standard deviation
0.1125
0.0124
7. Conclusions The proposed method changes form of wavefront fitting by dividing it into two parts. By introducing a mean value of the wavefront, the first part is finished by fitting the wavefront subtracting of mean value with the orthogonal polynomials subtracting of each polynomial’s mean value. The other part just contains piston term. Coefficient of piston term can be obtained easily by fitting the wavefront removal of σ݆ܮൌʹ ܾ݆ ܸ݆ . To both parts, the orthogonal polynomials coefficients are obtained by least squares method. It can achieve well-fitting results without iterations.
Acknowledgement This study is supported by Translational Research & Development grant NRF2009NRF-TRD001-008 from the Singapore National Research Foundation and Innovation Fund grant MOE2008-IF-1-009 from the Singapore Ministry of Education.
References [1] Born M, Wolf E. Principles of Optics. Oxford: New York; 1999. [2] Wyant J. C, K. Creath. BasicWavefront Aberration Theory for Optical Metrology. Appl. Opt. Optical Eng. XI, 1–53; 1992. [3] Malacara D, J. M. Carpio-Valde´z, J. Javier Sa´nchez-Mondrago´n. Wavefront Fitting With Discrete Orthogonal Polynomials in a Unit Radius Circle. Opt. Eng., 29, 672–675;1990. [4] Malacara D. Optical Shop Testing. John Wiley & Sons, Inc., Hoboken, New Jersey; 2007.
Physics Procedia 19 (2011) 134–138
ICOPE EN 2011
A new meethod forr improv ving the accuracy y of wav vefront ffitting with Zeernike poolynomials W Wang Zhaoomina*, Ch hee Oichoooa, Qu Weijjuana, Xu Qiangsheng Q ga a
Ngee Ann Pollytechnic, 535 Cllementi Road, 599 9489,Singapore
Absttract Zern nike polynomialls have been wiidely used to fitt wavefront andd, by their repreesentation, calcu ulate the deviatiion due to waveefront aberrrations in an ooptical system. In order to ob btain the coeffi cients of the Zernike Z polynom mials, a set off discrete ortho ogonal polyn nomials needs tto be constructeed using the Grram-Schmidt m method on a unittary circle, and the coefficientts are then calcu ulated by fitting the waveffront data and orthogonal o poly ynomials by thee least squares method. Through simulation, it is found therre is a largee deviation betw ween the given coefficients an nd the calculatedd results becau use of limitation ns of the least ssquares method d. This paper proposes a moodified methodd to improve thee accuracy of w wavefront fitting g.
© 2011 Published by ElsevierV. B.V. Selection and/or peer-review under responsibility of the Organising Committee of © 20 010 Published bby Elsevier B.V the ICOPEN 2011 conference Keyw words: Wavefrontt fitting; Zernike polynomials; p Leaast squares methood; Zernike coeffiicients
1. In ntroduction Zernike Z polynoomials have been b successfu ully used for ttheir represen ntation of aberrrations in maany different fields f of op ptics [1, 2]. To know exactt aberrations, Zernike Z coeffi ficients must be b known. Theey can be obtaained by fittin ng the wavefront data w with a set of poolynomials by y the least squuares method.. Where the polynomials p orrthogonal oveer the discrrete unitary ciircle are consttructed using the Gram-Schhmidt method d on basis of Zernike Z polynnomials [3]. Due D to the limitation l of lleast squares method, theree will be residdual coefficieents in the waavefront remov oval of aberrattions. Heree, we describee a modified method m to perfform more acccurate Zernikee coefficients with w respect to the wavefro ont. 2. Wavefront W rep presented by Zernike poly ynomials The T wavefront can be expannded in terms of o orthonormaal Zernike pollynomials ܼ ሺɏǡ Ʌሻ in the foorm
ܹሺɏǡ Ʌሻ ൌ σ ܽ ܼ ሺɏǡ Ʌሻሻ
(1) (
* Corresponding aauthor. Fax: +65-664689491. E-mail E address: [email protected].
1875-3892 © 2011 Published by Elsevier B.V. Selection and/or peer-review under responsibility of the Organising Committee of the ICOPEN 2011 conference doi:10.1016/j.phpro.2011.06.137
Wang Zhaomin et al. / Physics Procedia 19 (2011) 134–138
135
Where ܽ is the Zernike coefficient. The polynomials can be written as
ܼ௩ ሺߩǡ ߠሻ ൌ ඥʹሺ݊ ͳሻܴ ሺߩሻܿߠ݉ݏǡ݉ ് Ͳ ܼௗௗ ሺߩǡ ߠሻ ൌ ඥʹሺ݊ ܼ ሺߩǡ ߠሻ ൌ ξ݊
ͳሻܴ ሺߩሻߠ݉݊݅ݏǡ݉
ͳܴ ሺߩሻǡ݉
(2)
്Ͳ
(3)
ൌͲ
(4)
Where index n represents the order of the polynomial, m is called the azimuthal frequency. ܴ ሺߩሻ is the radial polynomials and ሺିଵሻೞ ሺି௦ሻǨ ሺିሻȀଶ ܴ ሺߩሻ ൌ σ௦ୀ ߩିଶ௦ (5) శೞ షೞ ௦Ǩቀ
మ
ି௦ቁǨቀ
మ
ି௦ቁǨ
The index j is a polynomial-ordering number and has relationship with n and m in the form
ൌ ሾሺʹ݆ െ ͳሻǤହ ͲǤͷሿ௧ െ ͳ
(6)
ʹሼሾʹ݆ ͳ െ ݊ሺ݊ ͳሻሿȀͶሽ௧ ݊݁ݒ݁ݏ݄݅݊݊݁ݓ ൌቊ ʹሼሾʹሺ݆ ͳሻ െ ݊ሺ݊ ͳሻሿȀͶሽ௧ െ ͳ݀݀ݏ݄݅݊݊݁ݓ The Zernike polynomials and the relationship among j, n, and m are given in Table 1[4].
(7)
Table 1 Zernike polynomials j
n
m
Zernike polynomials
Aberration name
1
0
0
1
Piston
2
1
1
ɏ ߠ
x tilt
3
1
1
ɏߠ
y tilt
4
2
0
ξ͵ሺʹɏଶ െ ͳሻ
5
2
2
ξɏଶ ʹߠ
Primary astigmatism at Ͷͷι
6
2
2
ξɏଶ ʹߠ
Primary astigmatism at Ͳι
7
3
1
ξͺሺ͵ɏଷ െ ʹɏሻߠ
Primary y coma
8
3
1
ξͺሺ͵ɏଷ െ ʹɏሻ ߠ
Primary x coma
ଷ
9
3
3
ξͺɏ ͵ߠ
10
3
3
ξͺɏଷ ͵ߠ
11
4
0
ξͷሺɏସ െ ɏଶ ͳሻ
Primary spherical
12
4
2
ξͳͲሺͶɏସ െ ͵ɏଶ ሻ ʹ ߠ
Secondary astigmatism at Ͳι
ସ
ଶ
13
4
2
ξͳͲሺͶɏ െ ͵ɏ ሻʹ ߠ
14
4
4
ξͳͲɏସ Ͷߠ
15
4
4
ξͳͲɏସ Ͷߠ
Secondary astigmatism at Ͷͷι
3. Generation of orthogonal polynomials by Gram-Schmidt orthogonalization Zernike polynomials are orthogonal on continuous unitary circle but not on discrete unitary circle. Thus, GramSchmidt orthogonalization is carried out to construct a set of polynomials which are orthogonal on the discrete unitary circle
Wang Zhaomin et al. / Physics Procedia 19 (2011) 134–138
136
ሺܸଵ
ܸଶ
ǥ ܸିଵ
ܸ ሻ ൌ ሺܼଵ
ܼଶ
ǥ
ͳ ܿଶǡଵ ۇ ܼ ሻ ڭ ۈ ܿିଵǡଵ ܿ ۉǡଵ
ܼିଵ
Ͳ ͳ ڭ ܿିଵǡଶ ܿǡଶ
ǥ ǥ ڰ ǥ ǥ
Ͳ Ͳ ڭ ͳ ܿǡିଵ
Ͳ Ͳ ۊ ۋڭ Ͳ ͳی
(9)
In a general way ିଵ
ܸ ൌ ܼ σୀଵ ܿǡ ܼ
(10)
On the discrete unitary circle, the polynomials ܸ ሺɏǡ Ʌሻ satisfy the following condition σே ୀଵ ܸ ሺɏ ǡ Ʌ ሻ ܸ ሺɏ ǡ Ʌ ሻ ൌ Ͳ
(11)
Then the ܿǡ can be written ିଵ
ଶ ே ܿǡ ൌ σୀଵ ൛ܿିǡ ൣሺσே ୀଵ ܼ ሺɏ ǡ Ʌ ሻܸି ሺɏ ǡ Ʌ ሻሻൗሺσୀଵ ܸି ሺɏ ǡ Ʌ ሻሻ൧ൟ
(12)
Where ݅ ൌ ͳǡʹǡ͵ǡ ǥ ǡ ݆ െ ͳ ൌ ͳ.We get the wavefront expressed by orthogonal polynomials ܸ ܹሺɏǡ Ʌሻ ൌ σ ܾ ܸ ሺɏǡ Ʌሻ
(13)
Here, we define the mean value of the wavefront as
ഥ ൌ ଵ σே ഥ ܹ ୀଵ ܹሺɏ ǡ Ʌ ሻ ൌ σ ܾ ܸఫ
(14)
ଵ
ഥఫ ൌ σே ܸ ሺɏ ǡ Ʌ ሻ, L is the maximum value of the index j. Where ܸ ே ୀଵ Then we reconstruct the Eq.13 by subtracting the mean value of the wavefront
ഥ ൌ σ ܾ ሺܸ ሺɏǡ Ʌሻ െ ܸ ഥఫ ሻ ܹሺɏǡ Ʌሻ െ ܹ
(15)
For the first term of the orthogonal polynomials is piston terms, which mean value is a constant. So we reset the തതതଶ ሻǡ ǥ ǡ ൫ܸ െ ܸ ഥఫ ൯ , it is easy to verify polynomials’ orthogonality on discrete orthogonal polynomials as ሺܸଶ െ ܸ unitary circle. The Eq.15 can be written as
ഥ ൌ σୀଶ ܾ ሺܸ ሺɏǡ Ʌሻ െ ܸ ഥఫ ሻ ܹሺɏǡ Ʌሻ െ ܹ
(16)
4. Least squares fitting We define the measured wavefront data as ܹሺɏ ǡ Ʌ ሻ , the deviation of fitting is Ɂଶ . The wavefront and orthogonal polynomials are fitted with least squares method ഥ ഥ Ɂଶ ൌ σே ୀଵ൫ܹሺɏ ǡ Ʌ ሻ െ ܹ െ σୀଶ ܾ ሺܸ ሺɏ ǡ Ʌ ሻ െ ܸఫ ሺɏ ǡ Ʌ ሻሻ൯
ଶ
(17)
In order to make the Ɂଶ as small as possible, we take differential on the both side of the Eq.17 பஔమ பೕ
మ
ൌ
ಽ തതത ഥ ப σಿ ೖసభቀௐሺೖ ǡೖ ሻିௐିσೕసమ ೕ ሺೕ ሺೖ ǡೖ ሻିണ ሺೖ ǡೖ ሻሻቁ
பೕ
ൌͲ
(18)
Thus, coefficient ܾ can be obtained by solving the equation. ܾ ൌ
തതത ഥ σಿ ೖసభሺௐሺೖ ǡೖ ሻିௐሻሺೕ ሺೖ ǡೖ ሻିണ ሺೖ ǡೖ ሻሻ
(19)
మ തതത σಿ ೖసభሺೕ ሺೖ ǡೖ ሻିണ ሺೖ ǡೖ ሻሻ
Where ݆ ൌ ʹǡ͵ǡ ǥ ܮ. ഥ െ σୀଶ ܾ ሺܸ െ ܸ തതതത and σୀଶ ܾ ܸ ഥఫ ሻ, ܹǡ ഥఫ , so the coefficient of the The first piston term ܸଵ is composed of ܹ െ ܹ piston term can be written as
ഥ െ σୀଶ ܾ ܸ ഥఫ ܾଵ ൌ ܹ
ಽ ഥ ഥ σಿ ೖసభሺௐሺೖ ǡೖ ሻିௐିσೕసమ ೕ ሺೖ ǡೖ ሻሺೕ ሺೖ ǡೖ ሻିണ ሻሻభ ሺೖ ǡೖ ሻ మ σಿ ೖసభ ሺభ ሺೖ ǡೖ ሻሻ
(20)
Wang Zhaomin et al. / Physics Procedia 19 (2011) 134–138
137
5. Zernike coefficcients Since we know w the coefficiients ܾ andܿ e coefficients ܽ can be got by substitutting the Eq.10 0 and , the Zernike Eq.1 13 into the Eq..1 ିଵ
ܹ ൌ ܾଵ ܼଵ σୀଶ ܾ ሺܼ σୀଵ ܿ ܼ ሻ
(2 21)
Comparing C witth Eq.1, we caan get the coeffficients ܽ
ܽ ൌ ܾ σୀାଵ ܾ ܿ
(2 22)
Wheere ݆ ൌ ͳǡʹǡ͵ǡ ǥ ǡ ሺ ܮെ ͳሻ and a ܽ ൌ ܾ . 6. Calculation We W do a simulaation experim ment on the mo odified metho d with respect to least squaares method. E Exit pupil diam meter is 2 mm, the wavvelength is 6333nm, and thee simulated w wavefront with h fixed first 15 terms Zernnike coefficien nts in nt. Tablle 2. Fig.1 shoows the simulaated wavefron
Fig.1. Simulaated wavefront
Through T calcuulation, we geet the fitting results in Taable 2. The calculations c work w without iteration. Fo or the limittation of the lleast squares method, m theree are obviouslly deviations between b fittin ng results giveen by least squ uares meth hod and the ssimulated coeffficients. Whiile the modifiied method giives better ressults either inn coefficients or in stand dard deviationn.
Wang Zhaomin et al. / Physics Procedia 19 (2011) 134–138
138
Table 2 Simulation and the fitting results Index j
Coefficients by simulation
Least squares method
modified method
1
0.1
0.0908
0.0994
2
-0.1
-0.1514
-0.1133
3
0.1
0.1494
0.1000
4
-0.1
-0.0705
-0.1041
5
0.1
-0.0083
0.1000
6
-0.1
-0.0719
-0.0976
7
-0.1
-0.1000
-0.1000
8
0.1
0.1000
0.1014
9
-0.1
-0.1013
-0.1000
10
-0.1
-0.0965
-0.0998
11
-0.1
-0.1000
-0.0998
12
-0.1
-0.1000
-0.1000
13
0.1
0.0986
0.1000
14
-0.1
-0.0990
-0.1000
15
0.1
0.1000
0.1000
Standard deviation
0.1125
0.0124
7. Conclusions The proposed method changes form of wavefront fitting by dividing it into two parts. By introducing a mean value of the wavefront, the first part is finished by fitting the wavefront subtracting of mean value with the orthogonal polynomials subtracting of each polynomial’s mean value. The other part just contains piston term. Coefficient of piston term can be obtained easily by fitting the wavefront removal of σ݆ܮൌʹ ܾ݆ ܸ݆ . To both parts, the orthogonal polynomials coefficients are obtained by least squares method. It can achieve well-fitting results without iterations.
Acknowledgement This study is supported by Translational Research & Development grant NRF2009NRF-TRD001-008 from the Singapore National Research Foundation and Innovation Fund grant MOE2008-IF-1-009 from the Singapore Ministry of Education.
References [1] Born M, Wolf E. Principles of Optics. Oxford: New York; 1999. [2] Wyant J. C, K. Creath. BasicWavefront Aberration Theory for Optical Metrology. Appl. Opt. Optical Eng. XI, 1–53; 1992. [3] Malacara D, J. M. Carpio-Valde´z, J. Javier Sa´nchez-Mondrago´n. Wavefront Fitting With Discrete Orthogonal Polynomials in a Unit Radius Circle. Opt. Eng., 29, 672–675;1990. [4] Malacara D. Optical Shop Testing. John Wiley & Sons, Inc., Hoboken, New Jersey; 2007.