Orthonormal curvature polynomials over a unit circle: basis set derived from curvatures of Zernike polynomials Chunyu Zhao* and James H. Burge College of Optical Sciences, the University of Arizona, 1630 E. University Blvd, Tucson, Arizona 85721, USA * [email protected]
Abstract: Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials. ©2013 Optical Society of America OCIS codes: (220.4840) Testing; (080.1010) Aberrations (global).
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976). M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1980), P464–468. P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007). C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part I: basis set derived from gradients of Zernike polynomials,” Opt. Express 15, 18014–18024 (2007). C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part II: completing the basis set,” Opt. Express 16, 6586–6591 (2008). W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980). G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt. 35, 6162–6172 (1996). R. G. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. 31, 6902– 6907 (1992). A. Gavrielides, “Vector polynomials orthogonal to the gradient of Zernike polynomials,” Opt. Lett. 7(11), 526– 528 (1982). R. E. Parks and D. S. Anderson, “Surface Profile Determination Using a Two-ball Spherometer,” Optical Fabrication and Testing Workshop Technical Notebook, OSA, Tucson, AZ meeting, Nov. 5–7, 1979. P. E. Glenn, “Lambda-over-one-thousand metrology results for steep aspheres using a curvature profiling technique,” Proc. SPIE 1531, 61–64 (1992). P. E. Glenn, “Robust, sub-angstrom-level midspatial-frequency profilometry,” Proc. SPIE 1333, 175 (1990). F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990). M. Schulz, “Topography measurement by a reliable large-area curvature sensor,” Optik (Stuttg.) 112, 86–90 (2001). B. C. Kim, T. Saiag, Q. Wang, J. Soons, R. S. Polvani, and U. Griesmann, “The Geometry Measuring Machine(GEMM) Project at NIST,” in Proceedings of ASPE 2004 Winter Topical Meeting on Free-Form Optics: Design, Fabrication, Metrology, Assembly, 108 (2004). B. C. Kim, M. C. Kwon, B. U. Choo, and I. J. Yoon, “3-D Shape easurement Using Curvature Data,” Proc. SPIE 7389, 73892H (2009).
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31430
17. C. Elstner, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingärtner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113, 154–158 (2002). 18. W. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34(4), 1232– 1237 (1995). 19. S. Ríos, E. Acosta, and S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996). 20. T. M. Apostol, Linear Algebra: A First Course, with Applications to Differential Equations (John Wiley & Sons, 1997), pp. 111–114. 21. R. Upton and B. Ellerbroek, “Gram–Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape,” Opt. Lett. 29, 2840–2842 (2004). 22. D. W. Kim, B. C. Kim, C. Zhao, C. J. Oh, and J. H. Burge, “Algorithms for surface reconstruction from curvature data for freeform aspherics,” Proc. SPIE 8838, 88380B (2013). 23. J. E. Sheedy and R. F. Hardy, “The optics of occupational progressive lenses,” Optometry 76(8), 432–441 (2005).
1. Introduction Zernike polynomials [1,2], which are scalar functions of the two-dimensional pupil coordinates (x, y), are often used in optical testing to represent wavefront or surface figure. These functions are fit to data collected from instruments such as coordinate measurement machines or interferometers. Other instruments measure slope, which is the derivative of the scalar wavefront or surface. Slopes can be measured with shearing interferometry, ShackHartmann sensors, or a scanning pentaprism test [3].The measured slope data, which generally consists of two dimensional vectors, can be fit to an orthonormal set of vector polynomials [4,5], allowing the corresponding wavefront or surface Zernike coefficients to be calculated directly from the vector polynomial coefficients. Many other methods for processing slope data also exist [6–9]. Curvature is an intrinsic property of a continuous surface, and can be measured directly without sensitivity to rigid body motions of displacement or tilt. So high-accuracy curvature measurements can be made in the presence of vibration, alignment errors, or mechanical instability without the errors that limit surface or slope measurements. A variety of methods have been developed for measuring curvature. Parks used a spherometer to measure the curvature along two orthogonal directions of an off-axis parabola surface, then fitted the data to the 2nd derivatives of Zernike polynomials to get the surface Zernike coefficients [10]. Paul Glenn developed a scanning system to measure the curvature by measuring the slope at two close points along the scan line [11,12]. Roddier’s curvature sensor determines the Laplacian of a wavefront by measuring the light intensity distributions at two defocused image positions [13]. Schulz used a laser interferometer as the curvature sensor and scanned it to construct the surface map [14]. Kim adopted Schulz’s approach [15], but later replaced the laser interferometer with a white light interferometer [16] to eliminate the measurement sensitivity to the focus alignment of the sensor. The curvature data can be integrated [6,17] twice to get the wavefront or surface map, similar to integration of slope data. A modal estimate of wavefront or surface from the curvature data provides an alternative; Parks’ method for fitting data to the second derivative of Zernike polynomials is an example of modal estimate methods, so are the approaches taken by Han [18] and Rios et al [19] for data taken from a Roddier type of curvature censor. We should point out that the interferometer type of sensor measures all three elements of the curvature (or the second derivatives) of a wavefront or surface, but other sensors only measure part of the curvature information, as discussed in Section 2. In this paper, we develop a basis set of polynomials that are optimal for modal decomposition of the general curvature measurement data where information of all three second derivatives is included. We follow the development of vector polynomials that create an orthogonal basis for fitting slope data, which is then used directly to calculate surface or wavefront. By further differentiating the Zernike polynomials, using Gram-Schmidt orthogonalization, and applying recursion relations, we develop a closed form definition of orthonormal curvature functions and provide the methodology of applying them for data #195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31431
reduction. The resulting basis functions are defined to be orthonormal over a unit circle, and they are calculated as three element vectors corresponding to the three forms of the second derivative. Each element consists of a linear combination of Zernike polynomials, which allows a straightforward computation from curvature coefficients to Zernike surface or wavefront coefficients. In Section 2, we introduce the curvature definitions with respect to a local normal, and define a curvature operator. In Section 3, the Noll’s notations for Zernike polynomials are presented, and closed form relations for the curvatures of Zernike polynomials are derived. We use these to develop the orthonormal set of curvature polynomials using the GramSchmidt method and present the result in Section 4. The mapping from the orthonormal curvature polynomials to curvatures of scalar functions represented by standard Zernike polynomials is discussed in Section 5. 2. Curvature and curvature operator For a curve defined by z = z ( x ) in x-z plane, the curvature is calculated as j = 226
(1)
Clearly curvature calculation involves second derivative as well as the first derivative, i.e. local slope. In optical testing, often times the curvature measurement is performed normal to the wavefront or surface where slope is negligible. Or, the measurands are often deviations from the nominal, e.g. the surface or wavefront error from a perfect sphere, or aspheric departure from the best fit sphere. In these cases, the slope term is much smaller than 1. So the curvature can be approximated with d2z (2) dx 2 To further validate this approximation, we draw analogy to the surface measurement using an interferometer. As illustrated in Fig. 1, when an interferometer measures a spherical surface, it does not measure the absolute shape represented by a sag function z = z ( x ) , but c ( x) ≅
the deviation from a sphere along the surface normal direction, also known as surface error, which is then plotted as 2-D map. (Since a sphere cannot be mapped to a plane perfectly, nonlinear mapping relations are necessary to create a 2-D map of the shape errors in the sphere, which is not the subject of this paper.) When a laser or white light interferometer [14– 16] is used as a curvature sensor, it is also aligned normal to the surface under test, and it measures the local curvature of the surface at the measurement point. Combined with a scanning mechanism, the interferometer curvature sensor can measure a full surface in high resolution. Like the surface measurement interferometry, the goal is not to measure the absolute surface topography, but the surface error, i.e. deviation from the ideal. If the ideal surface curvatures are subtracted from the measured curvatures, the curvatures of the surface error are obtained which is further processed to yield the surface error. For precision optics, the surface error has slopes on the order of micro radians, or 10−6, which translates to an error of a few parts in 1012 when using Eq. (2) to approximate Eq. (1). For a not-so-high-quality optic with the slope of surface error on the order of milli-radians, the approximation causes only a few parts in 106 of error. The approximation errors are local when the curvature data is fitted modally such that the corresponding surface or wavefront error is on the same order. So, we conclude this approximation is valid for practical metrology applications.
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31432
Fig. 1. Illustration of an interferometer measuring a spherical surface. What is measured is the surface deviation from a sphere along the normal direction, not the absolute sag.
For a 2-dimensional map, z = z ( x ) , again curvature can be approximated as the second d2z . So we define dxdy curvature of a 2-D function as a 3-element vector where each of the three elements depends on one or more of the three second derivatives:
derivatives, but there are three of them: (d 2 z ) / (dx 2 ) , (d 2 z ) / (dy 2 ) and
1 d 2z d 2z 2 + 2 dy 2 dx c1 2 d z = c2 c ( x, y ) = dxdy c3 2 2 1 d z − d z 2 dx 2 dy 2
(3)
Note that the curvature is not a physical vector that exists in a 3-D space, but we define it as a vector just for notational and computational convenience. The first element c1 is half of the Laplacian of the 2-D scalar function. Based on this definition of curvatures, we define a curvature operator as 1 d2 d2 2+ 2 dy 2 dx 2 d CURV = dxdy 2 2 1 d − d 2 2 dx dy 2
(4)
When it acts on a scalar 2-D function, it yields a 3-element curvature vector. Note that the curvature vector consists of three elements. If the sensor is a laser or white light interferometer, the three elements correspond to the measured power and two astigmatisms.
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31433
Whereas, a Roddier type of wavefront sensor only measures the Laplacian of the wavefront, i.e. c1 in the curvature vector. 3. Zernike polynomials and their curvatures In this paper we adopt Noll’s notation and numbering scheme for Zernike polynomials [1] which defines the polynomials as Z even
j
Z odd
j
= n + 1Rnm (r ) 2 cos(mθ ) m≠0 = n + 1Rnm (r ) 2 sin(mθ ) Z j = n + 1Rn0 (r ),
(5)
m=0
where Rnm (r ) =
( n − m )/ 2
s =0
(−1) s (n − s )! r n−2s s ![(n + m) / 2 − s ]![(n − m) / 2 − s ]!
(6)
j: the general index of Zernike polynomials n: the maximum power of the radial coordinate r m: the multiplication factor of the angular coordinate θ n and m have the following relations: m ≤ n and (n - m) is even For a given index j, there is a unique corresponding pair of (n, m), and the parity of j determines the angle dependence of the polynomial. While for a given pair of (n, m), j is ambiguous when m≠0. In some relationships given in the subsequent text, n and m are usually known, but the corresponding j (therefore the sine or cosine angle dependence of the polynomial) depends on other factors. For this reason, we choose to use j(n, m) for the general index of a Zernike polynomial or the polynomials derived in this paper to show that n and m are known and the actual j will be determined by other conditions. As the first step toward deriving an orthonormal set of curvature polynomials, we derive the curvatures of each of the Zernike polynomials Zj, and denote the function as ZCj such that 1 ∂2 1 ∂2 + 2 2 ∂y 2 2 ∂x ∂2 ZC j = CURV ( Z j ) = ∂x∂y 1 ∂2 1 ∂2 2 ∂x 2 − 2 ∂y 2
Z j ( x, y )
(7)
We first take the second derivatives of each Zernike polynomial. Since the derivatives of each Zernike polynomial are combinations of lower order Zernike polynomials, the second derivatives – derivatives of derivatives – are combinations of lower order Zernike polynomials as well. We apply the recursion relationships from Noll to represent the second derivatives as linear combinations of lower order Zernike polynomials, and then we combine them according to the curvature definition in (3) to form the 3-element vector polynomials ZC. The first non-trivial 20 terms are presented in Table 1. These functions provide a complete basis to represent curvatures, but they require further manipulation to create an orthonormal set.
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31434
Table 1. Curvatures of Zernike polynomials: ZCj = CURV(Zj) j
4
5
ZCj
Z1 48 0 0 0 24 Z1 0
j
14
15
6
0 24 0 Z 1
16
7
2Z 3 72 Z 2 −Z 3
17
8
2Z 2 72 Z 3 Z 2
18
9
0 72 Z 2 Z 3
19
10
0 72 − Z 3 Z 2
20
11
12
13
3Z1 + 4Z 4 240 2Z5 2Z 6 8Z 6 120 0 3Z1 + 4 Z 4
8Z 5 120 3Z1 + 4 Z 4 0
ZCj
0 240 − Z 5 Z 6 0 240 Z 6 Z 5
3072Z 2 + 2400Z 8 768Z 3 + 600 Z 7 + 600 Z 9 768Z 2 + 600 Z8 + 600 Z10 3072Z 3 + 2400Z 7 768Z 2 + 600 Z8 − 600 Z10 − 768Z 3 − 600Z 7 + 600Z 9 2400Z10 − 768Z 3 − 600 Z 7 768Z 2 + 600Z 8 2400Z10 − 768Z 3 − 600 Z 7 768Z 2 + 600Z 8 0 600 − Z 9 Z 10
21
0 600 Z10 Z 9
22
4032Z1 + 8400 Z 4 + 5040 Z11 4200Z 5 + 2520 Z13 4200 Z 6 + 2520Z12
23
8400 Z 5 + 5040 Z13 2016Z1 + 4200 Z 4 + 2520 Z11 − 1260 Z14 1260 Z15
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31435
4. An orthonormal set of curvature polynomials We use linear combinations of the ZC terms to create an orthogonal set. We define the inner product of two 3-element vector polynomials A and B in a unit circle as A ⋅B =
1 ∬( A1 B1 + A2 B2 + A3 B3 ) dxdy π
(8)
where the integration is over a unit circle. The inner product is taken of the ZC functions, and some results are shown in Table 2 (the table is symmetric about the diagonal, but only non-zero elements under the diagonal are shown). These ZC polynomials are not orthogonal, as the matrix of inner products listed in Table 2 is not diagonal. Table 2. List of the inner products of the first 13 Zernike curvatures excluding the trivial terms Inner Product ZCj
4
4
48
ZCi 5
5
6
7
8
9
10
11
12
13
14
15
24
6
24
7
432
8
432
9
144
10
144
11
2640
96 15
12
1800
48 15
13
1800
48 15
480
14
480
15
4.1 Orthogonalization of the curvature functions Using the Gram-Schmidt orthogonalization method (the general description for the method can be found in [20], and an optical application can be found in [21]), we construct a new set of polynomials with curvature of Zernike polynomials ZC as basis. The curvatures of Z1 through Z3 are all zero; therefore they are not used in the construction of the new set which we call C. We choose to index the first C polynomial as 4 to maintain its correspondence with Zernike polynomials. The first 33 C polynomials are listed in Table 3. Table 3. List of first 33 orthonormal curvature polynomials Cj as functions of curvatures of Zernike polynomials. n
Cj
C4 = 2
C5 = C6 =
1 48 1 24 1 24
n
C7 =
ZC4 ZC5 ZC6
Cj
3
C8 =
1 3 ⋅144 1
ZC7
ZC8 3 ⋅144 1 C9 = ZC9 144
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31436
C10 =
C11 = C12 = 4
C13 =
4 ⋅ 480 1 3 ⋅ 480 1
( ZC
− 15ZC4
)
( ZC
− 15ZC6
)
11
12
(
ZC13 − 15ZC5 3 ⋅ 480 1 C14 = ZC14 480 1 C15 = ZC15 480
C22 =
C23 =
C24 = 6
1
)
144
ZC10
32 ZC8 ZC16 − 3 4 ⋅1200 1 32 C17 = ZC7 ZC17 − 3 4 ⋅1200
C16 =
C18 = 5
35 21 ZC11 + Z ZC22 − 4 4 4 ⋅ 2520 35 21 ZC13 + Z ZC23 − 4 4 4 ⋅ 2520
(ZC18 −
32 ZC10 ) 3
1
192 98 ZC16 + ZC30 − 25 25 4 ⋅ 4704
C31 =
192 98 ZC19 + ZC31 − 25 25 4 ⋅ 4704
C32 =
192 98 ZC18 + ZC32 − 25 25 4 ⋅ 4704
1
7
1 3 ⋅1200
C30 =
1
35 21 ZC12 + Z ZC24 − 4 4 4 ⋅ 2520
1
32 ZC9 ZC19 − 3 3 ⋅1200 1 C20 = ZC20 1200 1 C21 = ZC21 1200 1 192 98 C29 = ZC17 + ZC29 − 25 25 4 ⋅ 4704 C19 =
1
35 ZC15 ZC25 − 4 3 ⋅ 2520 1 35 C26 = ZC14 ZCC26 − 4 3 ⋅ 2520 1 C27 = ZC27 2520 1 C28 = ZC 28 2520 C25 =
1
1
1
1
1
192 ZC21 ZC33 − 25 3 ⋅ 4704 1 192 C34 = ZC20 ZC34 − 25 3 ⋅ 4704 1 C35 = ZC35 4704 1 C36 = ZC36 4704 From these numerically calculated low order terms, we observe that C33 =
1
• Each term in the orthonormal series is combination of at most three Zernike curvature polynomials • The jth term where n = m is the corresponding jth term of Zernike curvature polynomials scaled by 1/ 2(n 4 − n 2 ).
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31437
• The jth term where n = m + 2 is combination of two Zernike curvature polynomials: – The first term is the corresponding jth term scaled by
1 3 ⋅ 2 ( n4 − n2 )
– The 2nd term is the j′-th term where n′ = n-2, and m′ = m with same angular dependence, scaled by −
4 ( n 2 − 1)
( n − 2)
2
1 3 ⋅ 2 ( n4 − n2 )
• The j-th term where n ≥ m + 4 is combination of three Zernike curvature polynomials: – The first term is the corresponding j-th term scaled by
1 4 ⋅ 2 ( n4 − n2 )
– The 2nd term is the j′-th term where n′ = n-2, and m′ = m with same angular dependence, scaled by −
4 ( n 2 − 1)
( n − 2)
2
1 4 ⋅ 2 ( n4 − n2 )
– The 3rd term is the j′′-th term where n′′ = n-4, and m′′ = m with same angular dependence, scaled by
n 2 ( n + 1)
( n − 2 ) ( n − 3) 2
1 4 ⋅ 2 ( n4 − n2 )
The general form of the C polynomials is then C j ( n, m ) =
4 ( n 2 − 1) ZC ZC j' ( n − 2, m ) + − j ( n, m ) 2 ( n − 2) k ⋅ 2 ( n 4 − n 2 ) (9) 1
ZC j "( n − 4, m ) 2 − − n 2 n 3 ( ) ( ) n 2 ( n + 1)
Where the following rules apply for terms with n > = 3: – 2nd term exists only when n > = (m + 2) – 3rd term exists only when n > = (m + 4) – All three terms have the same θ dependence, i.e., they are all cosθ, or sinθ, or have no θ dependence (symmetric terms) – When n = m, k = 1 – When n = (m + 2), k = 3 – When n ≥ (m + 4), k = 4 4.2 Orthonormal polynomials written in terms of Zernike polynomials Since C polynomials are linear combinations of ZC polynomials, each element of any C polynomials must also be linear combinations of Zernike polynomials. The first three C polynomials, C4, C5 and C6, are unit vectors along three axes representing power, 45° and 90° astigmatisms, respectively. We numerically calculated the Zernike polynomial coefficients for each element of each term of the C polynomials and develop the recursion relations listed in Table 4 for terms with n higher than 2. Some low order C polynomials are listed in Table 5
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31438
as explicit expressions of Zernike polynomials combinations. The key points of the recursion relations are summarized here: 1. C4, C5 and C6 form the unit vectors for orthonormal curvature polynomials. 2. The elements of each C term with radial index of n, depend on only the Zernike polynomials with index n-2, understandably so. 3. The first element of a C polynomial (half the Laplacian of the corresponding scalar function) is 0 when n = m, and is a single scaled Zernike polynomial term otherwise 4. The 2nd or 3rd element of a C polynomial is one Zernike polynomial (or 0) when n-m is either 0 or 2, or when n-m>2 but m = 0; it is a combination of two Zernike polynomials when n-m>2 but m ≠ 0. Table 4. Summary of the recursion relations between the C polynomials and the Zernike polynomials. Cj(1) (j(n,m))
Term Cj(1) = 0
n=m n>m
n-m =2 nm>2
Cj(2) (j(n,m))
nm>2
Cj(3)
m' = m
Term
Zj' (j'(n',m'), n' = n-2) j and j' parity Coefficient
m = 1, j = 7 or 8 m = 2, j = 12 m = 2, j = 13
m' = m =1
opposite
opposite
Cj(2) = 0 m' = m2 = 0 (j' = 4)
m>2
m' = m2
m = 0
m' = m +2=2
m = 1
m' = m =1
m = 2
m' = m2=0
m>2
m' = m2
Condition
Term
Zj” (j”(n”,m”), n” = n-2) j and j” parity Coefficient Condition
NA
2/3
same same (m≠0)
m' = m2
n=m
n-m =2
m' = m
Zj' (j'(n',m'), n' = n-2) j and j' parity Coefficient
1/ 2
1/ 2 − 1/ 2
Condition
Term
Zj” (j”(n”.m”), n” = n-2) j and j” parity Coefficient Condition
j odd j even
1/ 6 NA
1/ 3 opposite
1/ 6 − 1/ 6
(j' odd)
1/2
opposite
1/ 8
opposite
j odd j even
1/2
j odd
0
j even
1/ 8
j odd
− 1/ 8
Zj' (j'(n',m'), n' = n-2)
j even
m” = m +2=3
opposite
m” = m +2=4
opposite
m” = m +2
opposite
1/ 8
j even
− 1/ 8
j odd
1/ 8
j even
− 1/ 8
j odd
1/ 8
j even
− 1/ 8
j odd
Zj” (j”(n”.m”), n” = n-2)
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31439
(j(n,m)) n = m>2
n-m =2
m = 1
m' = m =1
m = 2, j = 12 m = 2, j = 13
m' = m2 = 0 (j’ = 4)
m>2 m = 0
nm>2
j and j' parity
Term m' = m2
m' = m =1
m = 2
m' = m2=0
m>2
m' = m2
Condition
Term
j and j” parity
Coefficient
Condition
1/ 2
same
1/ 6
same
− 1/ 6
j even j odd
1/ 3
Cj(3) = 0 m' = m2 m' = m +2=2
m = 1
Coefficient
1/ 6
same (j' even)
NA
j odd
1/2
1/ 8
same
j even
− 1/ 8
j odd
1/2
j even
0
j odd
same
1/ 8
m” = m +2=4
same
1/ 8
same
1/ 8
m” = m +2
1/ 8
same
m” = m +2=3
Table 5. List of the first 15 terms of the non-trivial C polynomials written in terms of the combinations of Zernike polynomials. C4
Z1 0 0 C9
1 / 2Z2 1 / 2 Z3 0
C14
0 − 1 / 2 Z5 1 / 2 Z6
C5
C6
C7
C8
0 Z1 0
0 0 Z 1
2 / 3 Z1 1 / 6Z2 − 1 / 6 Z3
2 / 3Z 2 1 / 6 Z3 1 / 6Z2
C10
C11
C12
C13
1 / 2Z4 1 / 4 Z5 1 / 4 Z6
2 / 3Z6 0 1 / 3Z 4
2 / 3Z5 1 / 3Z 4 0
C17
C18
− 1 / 2 Z3 1 / 2Z2 0
C15
0 1 / 2 Z6 1 / 2 Z5
C16
1 / 2 Z8 1 / 8( Z 7 + Z 9 ) 1 / 8( Z8 + Z10 )
1 / 2 Z7 1 / 8( Z 8 − Z10 ) 1 / 8(− Z 7 + Z 9 )
2 / 3 Z10 − 1 / 6Z7 1 / 6 Z8
4.3 An example verification of the recursion relations We give an example to verify the recursion relations presented in Sections 4.1 and 4.2. When j = 226 , then n = 20, m = 16 , and Cj has cosine angle dependence. According to the recursion relations presented in Sec. 4.1, k = 4 , and C226 =
1596 8400 ZC188 + ZC152 ZC226 − 324 5508 4 ⋅ 319200
1
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31440
The relevant ZC terms are numerically calculated and shown in Table 6. Then the C226 term is calculated from the ZC terms to yield a Zernike representation which is also presented in Table 6. Table 6. List of ZC terms for calculating C226 and the calculated C226 expressed in Zernike polynomials.
2062032 Z152 + 638400 Z188 − 918540 Z119 − 515508Z151 − 159600 Z185 + 159600Z189 918540 Z120 + 515508Z150 + 159600 Z186 + 159600Z190 418608Z152 − 329460 Z119 − 104652Z151 329460 Z120 + 104652 Z150 0 − 65280Z119 65280Z120
Z C226
Z C188
Z C152
226
1/ 2 Z188 C − 1/ 8 Z185 + 1/ 8 Z189 1/ 8 Z186 + 1/ 8 Z190 From orthogonality of Zernike polynomials, it is obvious that
• the elements of C226 are orthogonal to each other; • the magnitude of C226 is 1; and • C226 is orthogonal to all C terms presented in Table 5. Careful examination of the Zernike representation of C226 confirms the recursion relations presented in Section 4.2. 5. Relating the curvature polynomials coefficients to Zernike coefficients
The set of C polynomials fully spans the space of 3-element vector distributions K(x,y) over the unit circle where a scalar function Φ(x,y) exists such that K(x,y) = CURV(Φ(x,y)). It is useful to represent the vector data using the vector polynomials C and relate to a scalar functions φ that are defined as C(x,y) = CURV(φ(x,y)). According to (9), the scalar functions can be written as 4 ( n 2 − 1) n 2 ( n + 1) Z (10) Z − + Z j ( n, m ) j' ( n − 2, m ) j "( n − 4, m ) 2 2 4 2 − − − n 2 n 2 n 3 ( ) ( ) ( ) k ⋅ 2(n − n ) Vector data K(x,y) is decomposed into a linear combination of the orthonormal C polynomials as
φ j ( n,m ) =
1
K ( x, y ) = α j C j ( x, y )
(11)
Using the definitions of the scalar functions Φ and φj (K(x,y) = CURV(Φ(x,y)), and Cj(x,y) = CURV(φj(x,y))), we have Φ ( x, y ) = α jφ j ( x, y )
(12)
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31441
where the coefficients were found from the vector decomposition in Eq. (11). Since φj is linear combinations of Zernike polynomials, the scalar function Φ can in turn be represented as linear combinations of standard Zernike polynomials: Φ ( x, y ) = α jφ j ( x, y ) = γ j Z j ( x, y )
(13)
Equation (10) indicates that the Zernike polynomial Z j ( n , m ) has contributions from three φ terms – φ j ( n , m ) , φ j '( n + 2, m ) and φ j "( n + 4, m ) . And the coefficients of these standard Zernike polynomials can be found by
γ j ( n, m) =
α j ( n, m )
k1 ⋅ 2 ( n − n 4
2
)
−
2α j ′( n + 2, m ) k 2 ⋅ 2n 2 ( n + 2 )
2
+
α j ′′( n + 4, m ) k3 ⋅ 2 ( n + 1)( n + 2 ) ( n + 3) 2
(14)
where j, j’ and j” have the same parity, and • k3 = 4; • when n = m, k1 = 1 and k2 = 3; • when n = m + 2, k1 = 3 and k2 = 4; and • when n ≥ m + 4, k1 = k2 = 4. This procedure is useful for applications such as processing data from a wavefront curvature sensor. The curvature measurement data can be fit to the C polynomials to give a set of coefficients αj. These are converted directly to a standard Zernike polynomial representation of the wavefront, with coefficients γj. Kim et al outlines the procedure for processing curvature data using this approach and presented an example in [22]. 6. Discussion
Besides the obvious application in processing the curvature measurement data, the basis of curvature functions provides an efficient way to represent optical power variation across an optic, such as a progressive addition lenses for eyeglasses. In this case, the performance of the lens is fully described using the variation of power and astigmatism across the lens. In fact, the common representation of these lenses shows two functions over a circular domain, one is power and the other is magnitude of the astigmatism [23]. While this provides information about the performance, a complete representation requires both astigmatism components, or a total of 3 maps. An interesting application of the curvature polynomials comes from the fact that they form a complete basis for functions defined by second derivatives of continuous functions. The design of the lens power and astigmatism can be effectively performed using this curvature basis, knowing that the result can be integrated and used to determine a continuous shape that can be manufactured onto the lens surface. This mode of design can be very efficient and provide better insight than the current technique of optimizing the surfaces based on a ray trace simulation of performance. 7. Summary
We define a curvature operator that creates a three-element curvature vector from the second derivatives and show how it applies to Zernike polynomials. We then derive a set of curvature polynomials C that is orthonormal, and completely spans the space defined by a continuous circular surface. Each term in the set of C polynomials consists of linear combinations of at most three terms of curvatures of Zernike polynomials. The coefficients of these terms from fitting the measured curvature data can be converted to the coefficients for the corresponding Zernike polynomials representing the subject under test analytically, which makes the
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31442
polynomials an ideal choice for modal estimate of a wavefront or surface whose curvature were measured. Each element of any term of these polynomials comes from linear combinations of at most two Zernike polynomials; a recursion relation is given to generate analytical expressions of the polynomials in terms of lower order Zernike polynomials.
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31443
Abstract: Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials. ©2013 Optical Society of America OCIS codes: (220.4840) Testing; (080.1010) Aberrations (global).
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976). M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1980), P464–468. P. C. V. Mallik, C. Zhao, and J. H. Burge, “Measurement of a 2-meter flat using a pentaprism scanning system,” Opt. Eng. 46, 023602 (2007). C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part I: basis set derived from gradients of Zernike polynomials,” Opt. Express 15, 18014–18024 (2007). C. Zhao and J. H. Burge, “Orthonormal vector polynomials in a unit circle, Part II: completing the basis set,” Opt. Express 16, 6586–6591 (2008). W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980). G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt. 35, 6162–6172 (1996). R. G. Lane and M. Tallon, “Wave-front reconstruction using a Shack-Hartmann sensor,” Appl. Opt. 31, 6902– 6907 (1992). A. Gavrielides, “Vector polynomials orthogonal to the gradient of Zernike polynomials,” Opt. Lett. 7(11), 526– 528 (1982). R. E. Parks and D. S. Anderson, “Surface Profile Determination Using a Two-ball Spherometer,” Optical Fabrication and Testing Workshop Technical Notebook, OSA, Tucson, AZ meeting, Nov. 5–7, 1979. P. E. Glenn, “Lambda-over-one-thousand metrology results for steep aspheres using a curvature profiling technique,” Proc. SPIE 1531, 61–64 (1992). P. E. Glenn, “Robust, sub-angstrom-level midspatial-frequency profilometry,” Proc. SPIE 1333, 175 (1990). F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990). M. Schulz, “Topography measurement by a reliable large-area curvature sensor,” Optik (Stuttg.) 112, 86–90 (2001). B. C. Kim, T. Saiag, Q. Wang, J. Soons, R. S. Polvani, and U. Griesmann, “The Geometry Measuring Machine(GEMM) Project at NIST,” in Proceedings of ASPE 2004 Winter Topical Meeting on Free-Form Optics: Design, Fabrication, Metrology, Assembly, 108 (2004). B. C. Kim, M. C. Kwon, B. U. Choo, and I. J. Yoon, “3-D Shape easurement Using Curvature Data,” Proc. SPIE 7389, 73892H (2009).
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31430
17. C. Elstner, J. Gerhardt, P. Thomsen-Schmidt, M. Schulz, and I. Weingärtner, “Reconstructing surface profiles from curvature measurements,” Optik (Stuttg.) 113, 154–158 (2002). 18. W. Han, “New method for estimating wavefront from curvature signal by curve fitting,” Opt. Eng. 34(4), 1232– 1237 (1995). 19. S. Ríos, E. Acosta, and S. Bara, “Modal phase estimation from wavefront curvature sensing,” Opt. Commun. 123, 453–456 (1996). 20. T. M. Apostol, Linear Algebra: A First Course, with Applications to Differential Equations (John Wiley & Sons, 1997), pp. 111–114. 21. R. Upton and B. Ellerbroek, “Gram–Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape,” Opt. Lett. 29, 2840–2842 (2004). 22. D. W. Kim, B. C. Kim, C. Zhao, C. J. Oh, and J. H. Burge, “Algorithms for surface reconstruction from curvature data for freeform aspherics,” Proc. SPIE 8838, 88380B (2013). 23. J. E. Sheedy and R. F. Hardy, “The optics of occupational progressive lenses,” Optometry 76(8), 432–441 (2005).
1. Introduction Zernike polynomials [1,2], which are scalar functions of the two-dimensional pupil coordinates (x, y), are often used in optical testing to represent wavefront or surface figure. These functions are fit to data collected from instruments such as coordinate measurement machines or interferometers. Other instruments measure slope, which is the derivative of the scalar wavefront or surface. Slopes can be measured with shearing interferometry, ShackHartmann sensors, or a scanning pentaprism test [3].The measured slope data, which generally consists of two dimensional vectors, can be fit to an orthonormal set of vector polynomials [4,5], allowing the corresponding wavefront or surface Zernike coefficients to be calculated directly from the vector polynomial coefficients. Many other methods for processing slope data also exist [6–9]. Curvature is an intrinsic property of a continuous surface, and can be measured directly without sensitivity to rigid body motions of displacement or tilt. So high-accuracy curvature measurements can be made in the presence of vibration, alignment errors, or mechanical instability without the errors that limit surface or slope measurements. A variety of methods have been developed for measuring curvature. Parks used a spherometer to measure the curvature along two orthogonal directions of an off-axis parabola surface, then fitted the data to the 2nd derivatives of Zernike polynomials to get the surface Zernike coefficients [10]. Paul Glenn developed a scanning system to measure the curvature by measuring the slope at two close points along the scan line [11,12]. Roddier’s curvature sensor determines the Laplacian of a wavefront by measuring the light intensity distributions at two defocused image positions [13]. Schulz used a laser interferometer as the curvature sensor and scanned it to construct the surface map [14]. Kim adopted Schulz’s approach [15], but later replaced the laser interferometer with a white light interferometer [16] to eliminate the measurement sensitivity to the focus alignment of the sensor. The curvature data can be integrated [6,17] twice to get the wavefront or surface map, similar to integration of slope data. A modal estimate of wavefront or surface from the curvature data provides an alternative; Parks’ method for fitting data to the second derivative of Zernike polynomials is an example of modal estimate methods, so are the approaches taken by Han [18] and Rios et al [19] for data taken from a Roddier type of curvature censor. We should point out that the interferometer type of sensor measures all three elements of the curvature (or the second derivatives) of a wavefront or surface, but other sensors only measure part of the curvature information, as discussed in Section 2. In this paper, we develop a basis set of polynomials that are optimal for modal decomposition of the general curvature measurement data where information of all three second derivatives is included. We follow the development of vector polynomials that create an orthogonal basis for fitting slope data, which is then used directly to calculate surface or wavefront. By further differentiating the Zernike polynomials, using Gram-Schmidt orthogonalization, and applying recursion relations, we develop a closed form definition of orthonormal curvature functions and provide the methodology of applying them for data #195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31431
reduction. The resulting basis functions are defined to be orthonormal over a unit circle, and they are calculated as three element vectors corresponding to the three forms of the second derivative. Each element consists of a linear combination of Zernike polynomials, which allows a straightforward computation from curvature coefficients to Zernike surface or wavefront coefficients. In Section 2, we introduce the curvature definitions with respect to a local normal, and define a curvature operator. In Section 3, the Noll’s notations for Zernike polynomials are presented, and closed form relations for the curvatures of Zernike polynomials are derived. We use these to develop the orthonormal set of curvature polynomials using the GramSchmidt method and present the result in Section 4. The mapping from the orthonormal curvature polynomials to curvatures of scalar functions represented by standard Zernike polynomials is discussed in Section 5. 2. Curvature and curvature operator For a curve defined by z = z ( x ) in x-z plane, the curvature is calculated as j = 226
(1)
Clearly curvature calculation involves second derivative as well as the first derivative, i.e. local slope. In optical testing, often times the curvature measurement is performed normal to the wavefront or surface where slope is negligible. Or, the measurands are often deviations from the nominal, e.g. the surface or wavefront error from a perfect sphere, or aspheric departure from the best fit sphere. In these cases, the slope term is much smaller than 1. So the curvature can be approximated with d2z (2) dx 2 To further validate this approximation, we draw analogy to the surface measurement using an interferometer. As illustrated in Fig. 1, when an interferometer measures a spherical surface, it does not measure the absolute shape represented by a sag function z = z ( x ) , but c ( x) ≅
the deviation from a sphere along the surface normal direction, also known as surface error, which is then plotted as 2-D map. (Since a sphere cannot be mapped to a plane perfectly, nonlinear mapping relations are necessary to create a 2-D map of the shape errors in the sphere, which is not the subject of this paper.) When a laser or white light interferometer [14– 16] is used as a curvature sensor, it is also aligned normal to the surface under test, and it measures the local curvature of the surface at the measurement point. Combined with a scanning mechanism, the interferometer curvature sensor can measure a full surface in high resolution. Like the surface measurement interferometry, the goal is not to measure the absolute surface topography, but the surface error, i.e. deviation from the ideal. If the ideal surface curvatures are subtracted from the measured curvatures, the curvatures of the surface error are obtained which is further processed to yield the surface error. For precision optics, the surface error has slopes on the order of micro radians, or 10−6, which translates to an error of a few parts in 1012 when using Eq. (2) to approximate Eq. (1). For a not-so-high-quality optic with the slope of surface error on the order of milli-radians, the approximation causes only a few parts in 106 of error. The approximation errors are local when the curvature data is fitted modally such that the corresponding surface or wavefront error is on the same order. So, we conclude this approximation is valid for practical metrology applications.
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31432
Fig. 1. Illustration of an interferometer measuring a spherical surface. What is measured is the surface deviation from a sphere along the normal direction, not the absolute sag.
For a 2-dimensional map, z = z ( x ) , again curvature can be approximated as the second d2z . So we define dxdy curvature of a 2-D function as a 3-element vector where each of the three elements depends on one or more of the three second derivatives:
derivatives, but there are three of them: (d 2 z ) / (dx 2 ) , (d 2 z ) / (dy 2 ) and
1 d 2z d 2z 2 + 2 dy 2 dx c1 2 d z = c2 c ( x, y ) = dxdy c3 2 2 1 d z − d z 2 dx 2 dy 2
(3)
Note that the curvature is not a physical vector that exists in a 3-D space, but we define it as a vector just for notational and computational convenience. The first element c1 is half of the Laplacian of the 2-D scalar function. Based on this definition of curvatures, we define a curvature operator as 1 d2 d2 2+ 2 dy 2 dx 2 d CURV = dxdy 2 2 1 d − d 2 2 dx dy 2
(4)
When it acts on a scalar 2-D function, it yields a 3-element curvature vector. Note that the curvature vector consists of three elements. If the sensor is a laser or white light interferometer, the three elements correspond to the measured power and two astigmatisms.
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31433
Whereas, a Roddier type of wavefront sensor only measures the Laplacian of the wavefront, i.e. c1 in the curvature vector. 3. Zernike polynomials and their curvatures In this paper we adopt Noll’s notation and numbering scheme for Zernike polynomials [1] which defines the polynomials as Z even
j
Z odd
j
= n + 1Rnm (r ) 2 cos(mθ ) m≠0 = n + 1Rnm (r ) 2 sin(mθ ) Z j = n + 1Rn0 (r ),
(5)
m=0
where Rnm (r ) =
( n − m )/ 2
s =0
(−1) s (n − s )! r n−2s s ![(n + m) / 2 − s ]![(n − m) / 2 − s ]!
(6)
j: the general index of Zernike polynomials n: the maximum power of the radial coordinate r m: the multiplication factor of the angular coordinate θ n and m have the following relations: m ≤ n and (n - m) is even For a given index j, there is a unique corresponding pair of (n, m), and the parity of j determines the angle dependence of the polynomial. While for a given pair of (n, m), j is ambiguous when m≠0. In some relationships given in the subsequent text, n and m are usually known, but the corresponding j (therefore the sine or cosine angle dependence of the polynomial) depends on other factors. For this reason, we choose to use j(n, m) for the general index of a Zernike polynomial or the polynomials derived in this paper to show that n and m are known and the actual j will be determined by other conditions. As the first step toward deriving an orthonormal set of curvature polynomials, we derive the curvatures of each of the Zernike polynomials Zj, and denote the function as ZCj such that 1 ∂2 1 ∂2 + 2 2 ∂y 2 2 ∂x ∂2 ZC j = CURV ( Z j ) = ∂x∂y 1 ∂2 1 ∂2 2 ∂x 2 − 2 ∂y 2
Z j ( x, y )
(7)
We first take the second derivatives of each Zernike polynomial. Since the derivatives of each Zernike polynomial are combinations of lower order Zernike polynomials, the second derivatives – derivatives of derivatives – are combinations of lower order Zernike polynomials as well. We apply the recursion relationships from Noll to represent the second derivatives as linear combinations of lower order Zernike polynomials, and then we combine them according to the curvature definition in (3) to form the 3-element vector polynomials ZC. The first non-trivial 20 terms are presented in Table 1. These functions provide a complete basis to represent curvatures, but they require further manipulation to create an orthonormal set.
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31434
Table 1. Curvatures of Zernike polynomials: ZCj = CURV(Zj) j
4
5
ZCj
Z1 48 0 0 0 24 Z1 0
j
14
15
6
0 24 0 Z 1
16
7
2Z 3 72 Z 2 −Z 3
17
8
2Z 2 72 Z 3 Z 2
18
9
0 72 Z 2 Z 3
19
10
0 72 − Z 3 Z 2
20
11
12
13
3Z1 + 4Z 4 240 2Z5 2Z 6 8Z 6 120 0 3Z1 + 4 Z 4
8Z 5 120 3Z1 + 4 Z 4 0
ZCj
0 240 − Z 5 Z 6 0 240 Z 6 Z 5
3072Z 2 + 2400Z 8 768Z 3 + 600 Z 7 + 600 Z 9 768Z 2 + 600 Z8 + 600 Z10 3072Z 3 + 2400Z 7 768Z 2 + 600 Z8 − 600 Z10 − 768Z 3 − 600Z 7 + 600Z 9 2400Z10 − 768Z 3 − 600 Z 7 768Z 2 + 600Z 8 2400Z10 − 768Z 3 − 600 Z 7 768Z 2 + 600Z 8 0 600 − Z 9 Z 10
21
0 600 Z10 Z 9
22
4032Z1 + 8400 Z 4 + 5040 Z11 4200Z 5 + 2520 Z13 4200 Z 6 + 2520Z12
23
8400 Z 5 + 5040 Z13 2016Z1 + 4200 Z 4 + 2520 Z11 − 1260 Z14 1260 Z15
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31435
4. An orthonormal set of curvature polynomials We use linear combinations of the ZC terms to create an orthogonal set. We define the inner product of two 3-element vector polynomials A and B in a unit circle as A ⋅B =
1 ∬( A1 B1 + A2 B2 + A3 B3 ) dxdy π
(8)
where the integration is over a unit circle. The inner product is taken of the ZC functions, and some results are shown in Table 2 (the table is symmetric about the diagonal, but only non-zero elements under the diagonal are shown). These ZC polynomials are not orthogonal, as the matrix of inner products listed in Table 2 is not diagonal. Table 2. List of the inner products of the first 13 Zernike curvatures excluding the trivial terms Inner Product ZCj
4
4
48
ZCi 5
5
6
7
8
9
10
11
12
13
14
15
24
6
24
7
432
8
432
9
144
10
144
11
2640
96 15
12
1800
48 15
13
1800
48 15
480
14
480
15
4.1 Orthogonalization of the curvature functions Using the Gram-Schmidt orthogonalization method (the general description for the method can be found in [20], and an optical application can be found in [21]), we construct a new set of polynomials with curvature of Zernike polynomials ZC as basis. The curvatures of Z1 through Z3 are all zero; therefore they are not used in the construction of the new set which we call C. We choose to index the first C polynomial as 4 to maintain its correspondence with Zernike polynomials. The first 33 C polynomials are listed in Table 3. Table 3. List of first 33 orthonormal curvature polynomials Cj as functions of curvatures of Zernike polynomials. n
Cj
C4 = 2
C5 = C6 =
1 48 1 24 1 24
n
C7 =
ZC4 ZC5 ZC6
Cj
3
C8 =
1 3 ⋅144 1
ZC7
ZC8 3 ⋅144 1 C9 = ZC9 144
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31436
C10 =
C11 = C12 = 4
C13 =
4 ⋅ 480 1 3 ⋅ 480 1
( ZC
− 15ZC4
)
( ZC
− 15ZC6
)
11
12
(
ZC13 − 15ZC5 3 ⋅ 480 1 C14 = ZC14 480 1 C15 = ZC15 480
C22 =
C23 =
C24 = 6
1
)
144
ZC10
32 ZC8 ZC16 − 3 4 ⋅1200 1 32 C17 = ZC7 ZC17 − 3 4 ⋅1200
C16 =
C18 = 5
35 21 ZC11 + Z ZC22 − 4 4 4 ⋅ 2520 35 21 ZC13 + Z ZC23 − 4 4 4 ⋅ 2520
(ZC18 −
32 ZC10 ) 3
1
192 98 ZC16 + ZC30 − 25 25 4 ⋅ 4704
C31 =
192 98 ZC19 + ZC31 − 25 25 4 ⋅ 4704
C32 =
192 98 ZC18 + ZC32 − 25 25 4 ⋅ 4704
1
7
1 3 ⋅1200
C30 =
1
35 21 ZC12 + Z ZC24 − 4 4 4 ⋅ 2520
1
32 ZC9 ZC19 − 3 3 ⋅1200 1 C20 = ZC20 1200 1 C21 = ZC21 1200 1 192 98 C29 = ZC17 + ZC29 − 25 25 4 ⋅ 4704 C19 =
1
35 ZC15 ZC25 − 4 3 ⋅ 2520 1 35 C26 = ZC14 ZCC26 − 4 3 ⋅ 2520 1 C27 = ZC27 2520 1 C28 = ZC 28 2520 C25 =
1
1
1
1
1
192 ZC21 ZC33 − 25 3 ⋅ 4704 1 192 C34 = ZC20 ZC34 − 25 3 ⋅ 4704 1 C35 = ZC35 4704 1 C36 = ZC36 4704 From these numerically calculated low order terms, we observe that C33 =
1
• Each term in the orthonormal series is combination of at most three Zernike curvature polynomials • The jth term where n = m is the corresponding jth term of Zernike curvature polynomials scaled by 1/ 2(n 4 − n 2 ).
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31437
• The jth term where n = m + 2 is combination of two Zernike curvature polynomials: – The first term is the corresponding jth term scaled by
1 3 ⋅ 2 ( n4 − n2 )
– The 2nd term is the j′-th term where n′ = n-2, and m′ = m with same angular dependence, scaled by −
4 ( n 2 − 1)
( n − 2)
2
1 3 ⋅ 2 ( n4 − n2 )
• The j-th term where n ≥ m + 4 is combination of three Zernike curvature polynomials: – The first term is the corresponding j-th term scaled by
1 4 ⋅ 2 ( n4 − n2 )
– The 2nd term is the j′-th term where n′ = n-2, and m′ = m with same angular dependence, scaled by −
4 ( n 2 − 1)
( n − 2)
2
1 4 ⋅ 2 ( n4 − n2 )
– The 3rd term is the j′′-th term where n′′ = n-4, and m′′ = m with same angular dependence, scaled by
n 2 ( n + 1)
( n − 2 ) ( n − 3) 2
1 4 ⋅ 2 ( n4 − n2 )
The general form of the C polynomials is then C j ( n, m ) =
4 ( n 2 − 1) ZC ZC j' ( n − 2, m ) + − j ( n, m ) 2 ( n − 2) k ⋅ 2 ( n 4 − n 2 ) (9) 1
ZC j "( n − 4, m ) 2 − − n 2 n 3 ( ) ( ) n 2 ( n + 1)
Where the following rules apply for terms with n > = 3: – 2nd term exists only when n > = (m + 2) – 3rd term exists only when n > = (m + 4) – All three terms have the same θ dependence, i.e., they are all cosθ, or sinθ, or have no θ dependence (symmetric terms) – When n = m, k = 1 – When n = (m + 2), k = 3 – When n ≥ (m + 4), k = 4 4.2 Orthonormal polynomials written in terms of Zernike polynomials Since C polynomials are linear combinations of ZC polynomials, each element of any C polynomials must also be linear combinations of Zernike polynomials. The first three C polynomials, C4, C5 and C6, are unit vectors along three axes representing power, 45° and 90° astigmatisms, respectively. We numerically calculated the Zernike polynomial coefficients for each element of each term of the C polynomials and develop the recursion relations listed in Table 4 for terms with n higher than 2. Some low order C polynomials are listed in Table 5
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31438
as explicit expressions of Zernike polynomials combinations. The key points of the recursion relations are summarized here: 1. C4, C5 and C6 form the unit vectors for orthonormal curvature polynomials. 2. The elements of each C term with radial index of n, depend on only the Zernike polynomials with index n-2, understandably so. 3. The first element of a C polynomial (half the Laplacian of the corresponding scalar function) is 0 when n = m, and is a single scaled Zernike polynomial term otherwise 4. The 2nd or 3rd element of a C polynomial is one Zernike polynomial (or 0) when n-m is either 0 or 2, or when n-m>2 but m = 0; it is a combination of two Zernike polynomials when n-m>2 but m ≠ 0. Table 4. Summary of the recursion relations between the C polynomials and the Zernike polynomials. Cj(1) (j(n,m))
Term Cj(1) = 0
n=m n>m
n-m =2 nm>2
Cj(2) (j(n,m))
nm>2
Cj(3)
m' = m
Term
Zj' (j'(n',m'), n' = n-2) j and j' parity Coefficient
m = 1, j = 7 or 8 m = 2, j = 12 m = 2, j = 13
m' = m =1
opposite
opposite
Cj(2) = 0 m' = m2 = 0 (j' = 4)
m>2
m' = m2
m = 0
m' = m +2=2
m = 1
m' = m =1
m = 2
m' = m2=0
m>2
m' = m2
Condition
Term
Zj” (j”(n”,m”), n” = n-2) j and j” parity Coefficient Condition
NA
2/3
same same (m≠0)
m' = m2
n=m
n-m =2
m' = m
Zj' (j'(n',m'), n' = n-2) j and j' parity Coefficient
1/ 2
1/ 2 − 1/ 2
Condition
Term
Zj” (j”(n”.m”), n” = n-2) j and j” parity Coefficient Condition
j odd j even
1/ 6 NA
1/ 3 opposite
1/ 6 − 1/ 6
(j' odd)
1/2
opposite
1/ 8
opposite
j odd j even
1/2
j odd
0
j even
1/ 8
j odd
− 1/ 8
Zj' (j'(n',m'), n' = n-2)
j even
m” = m +2=3
opposite
m” = m +2=4
opposite
m” = m +2
opposite
1/ 8
j even
− 1/ 8
j odd
1/ 8
j even
− 1/ 8
j odd
1/ 8
j even
− 1/ 8
j odd
Zj” (j”(n”.m”), n” = n-2)
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31439
(j(n,m)) n = m>2
n-m =2
m = 1
m' = m =1
m = 2, j = 12 m = 2, j = 13
m' = m2 = 0 (j’ = 4)
m>2 m = 0
nm>2
j and j' parity
Term m' = m2
m' = m =1
m = 2
m' = m2=0
m>2
m' = m2
Condition
Term
j and j” parity
Coefficient
Condition
1/ 2
same
1/ 6
same
− 1/ 6
j even j odd
1/ 3
Cj(3) = 0 m' = m2 m' = m +2=2
m = 1
Coefficient
1/ 6
same (j' even)
NA
j odd
1/2
1/ 8
same
j even
− 1/ 8
j odd
1/2
j even
0
j odd
same
1/ 8
m” = m +2=4
same
1/ 8
same
1/ 8
m” = m +2
1/ 8
same
m” = m +2=3
Table 5. List of the first 15 terms of the non-trivial C polynomials written in terms of the combinations of Zernike polynomials. C4
Z1 0 0 C9
1 / 2Z2 1 / 2 Z3 0
C14
0 − 1 / 2 Z5 1 / 2 Z6
C5
C6
C7
C8
0 Z1 0
0 0 Z 1
2 / 3 Z1 1 / 6Z2 − 1 / 6 Z3
2 / 3Z 2 1 / 6 Z3 1 / 6Z2
C10
C11
C12
C13
1 / 2Z4 1 / 4 Z5 1 / 4 Z6
2 / 3Z6 0 1 / 3Z 4
2 / 3Z5 1 / 3Z 4 0
C17
C18
− 1 / 2 Z3 1 / 2Z2 0
C15
0 1 / 2 Z6 1 / 2 Z5
C16
1 / 2 Z8 1 / 8( Z 7 + Z 9 ) 1 / 8( Z8 + Z10 )
1 / 2 Z7 1 / 8( Z 8 − Z10 ) 1 / 8(− Z 7 + Z 9 )
2 / 3 Z10 − 1 / 6Z7 1 / 6 Z8
4.3 An example verification of the recursion relations We give an example to verify the recursion relations presented in Sections 4.1 and 4.2. When j = 226 , then n = 20, m = 16 , and Cj has cosine angle dependence. According to the recursion relations presented in Sec. 4.1, k = 4 , and C226 =
1596 8400 ZC188 + ZC152 ZC226 − 324 5508 4 ⋅ 319200
1
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31440
The relevant ZC terms are numerically calculated and shown in Table 6. Then the C226 term is calculated from the ZC terms to yield a Zernike representation which is also presented in Table 6. Table 6. List of ZC terms for calculating C226 and the calculated C226 expressed in Zernike polynomials.
2062032 Z152 + 638400 Z188 − 918540 Z119 − 515508Z151 − 159600 Z185 + 159600Z189 918540 Z120 + 515508Z150 + 159600 Z186 + 159600Z190 418608Z152 − 329460 Z119 − 104652Z151 329460 Z120 + 104652 Z150 0 − 65280Z119 65280Z120
Z C226
Z C188
Z C152
226
1/ 2 Z188 C − 1/ 8 Z185 + 1/ 8 Z189 1/ 8 Z186 + 1/ 8 Z190 From orthogonality of Zernike polynomials, it is obvious that
• the elements of C226 are orthogonal to each other; • the magnitude of C226 is 1; and • C226 is orthogonal to all C terms presented in Table 5. Careful examination of the Zernike representation of C226 confirms the recursion relations presented in Section 4.2. 5. Relating the curvature polynomials coefficients to Zernike coefficients
The set of C polynomials fully spans the space of 3-element vector distributions K(x,y) over the unit circle where a scalar function Φ(x,y) exists such that K(x,y) = CURV(Φ(x,y)). It is useful to represent the vector data using the vector polynomials C and relate to a scalar functions φ that are defined as C(x,y) = CURV(φ(x,y)). According to (9), the scalar functions can be written as 4 ( n 2 − 1) n 2 ( n + 1) Z (10) Z − + Z j ( n, m ) j' ( n − 2, m ) j "( n − 4, m ) 2 2 4 2 − − − n 2 n 2 n 3 ( ) ( ) ( ) k ⋅ 2(n − n ) Vector data K(x,y) is decomposed into a linear combination of the orthonormal C polynomials as
φ j ( n,m ) =
1
K ( x, y ) = α j C j ( x, y )
(11)
Using the definitions of the scalar functions Φ and φj (K(x,y) = CURV(Φ(x,y)), and Cj(x,y) = CURV(φj(x,y))), we have Φ ( x, y ) = α jφ j ( x, y )
(12)
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31441
where the coefficients were found from the vector decomposition in Eq. (11). Since φj is linear combinations of Zernike polynomials, the scalar function Φ can in turn be represented as linear combinations of standard Zernike polynomials: Φ ( x, y ) = α jφ j ( x, y ) = γ j Z j ( x, y )
(13)
Equation (10) indicates that the Zernike polynomial Z j ( n , m ) has contributions from three φ terms – φ j ( n , m ) , φ j '( n + 2, m ) and φ j "( n + 4, m ) . And the coefficients of these standard Zernike polynomials can be found by
γ j ( n, m) =
α j ( n, m )
k1 ⋅ 2 ( n − n 4
2
)
−
2α j ′( n + 2, m ) k 2 ⋅ 2n 2 ( n + 2 )
2
+
α j ′′( n + 4, m ) k3 ⋅ 2 ( n + 1)( n + 2 ) ( n + 3) 2
(14)
where j, j’ and j” have the same parity, and • k3 = 4; • when n = m, k1 = 1 and k2 = 3; • when n = m + 2, k1 = 3 and k2 = 4; and • when n ≥ m + 4, k1 = k2 = 4. This procedure is useful for applications such as processing data from a wavefront curvature sensor. The curvature measurement data can be fit to the C polynomials to give a set of coefficients αj. These are converted directly to a standard Zernike polynomial representation of the wavefront, with coefficients γj. Kim et al outlines the procedure for processing curvature data using this approach and presented an example in [22]. 6. Discussion
Besides the obvious application in processing the curvature measurement data, the basis of curvature functions provides an efficient way to represent optical power variation across an optic, such as a progressive addition lenses for eyeglasses. In this case, the performance of the lens is fully described using the variation of power and astigmatism across the lens. In fact, the common representation of these lenses shows two functions over a circular domain, one is power and the other is magnitude of the astigmatism [23]. While this provides information about the performance, a complete representation requires both astigmatism components, or a total of 3 maps. An interesting application of the curvature polynomials comes from the fact that they form a complete basis for functions defined by second derivatives of continuous functions. The design of the lens power and astigmatism can be effectively performed using this curvature basis, knowing that the result can be integrated and used to determine a continuous shape that can be manufactured onto the lens surface. This mode of design can be very efficient and provide better insight than the current technique of optimizing the surfaces based on a ray trace simulation of performance. 7. Summary
We define a curvature operator that creates a three-element curvature vector from the second derivatives and show how it applies to Zernike polynomials. We then derive a set of curvature polynomials C that is orthonormal, and completely spans the space defined by a continuous circular surface. Each term in the set of C polynomials consists of linear combinations of at most three terms of curvatures of Zernike polynomials. The coefficients of these terms from fitting the measured curvature data can be converted to the coefficients for the corresponding Zernike polynomials representing the subject under test analytically, which makes the
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31442
polynomials an ideal choice for modal estimate of a wavefront or surface whose curvature were measured. Each element of any term of these polynomials comes from linear combinations of at most two Zernike polynomials; a recursion relation is given to generate analytical expressions of the polynomials in terms of lower order Zernike polynomials.
#195627 - $15.00 USD Received 12 Aug 2013; revised 2 Dec 2013; accepted 3 Dec 2013; published 12 Dec 2013 (C) 2013 OSA 16 December 2013 | Vol. 21, No. 25 | DOI:10.1364/OE.21.031430 | OPTICS EXPRESS 31443
