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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 9 — Apr. 28, 2008
  • pp: 6586–6591
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Orthonormal vector polynomials in a unit circle, Part II : completing the basis set

Chunyu Zhao and James H. Burge  »View Author Affiliations


Optics Express, Vol. 16, Issue 9, pp. 6586-6591 (2008)
http://dx.doi.org/10.1364/OE.16.006586


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Abstract

Zernike polynomials provide a well known, orthogonal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. A related set of orthogonal functions is given here which represent vector quantities, such as mapping distortion or wavefront gradient. Previously, we have developed a basis of functions generated from gradients of Zernike polynomials. Here, we complete the basis by adding a complementary set of functions with zero divergence – those which are defined locally as a rotation or curl.

© 2008 Optical Society of America

1. Introduction

2. The S⃗ polynomials

Sj=ϕj=îϕjx+ĵϕjy.
(1)

The scalar functions ϕj are linear combination of Zernike polynomials. Following Noll’s notation and numbering scheme,3

3. R. J. Noll, “Zernike polynomials and atmospheric turbulence”, J. Opt. Soc. Am. 66, 1976, 207–211. [CrossRef]

ϕj=12n(n+1)Zj,foralljwithn=m,
(2a)

and

ϕj=14n(n+1)(Zjn+1n1Zj(n=n2,m=m)),foralljwith nm.
(2b)

Each S⃗ polynomial is then the linear combinations of gradient of Zernike polynomials following (1) and (2). Since gradient of Zernike polynomials can also be represented by Zernike polynomials,3

3. R. J. Noll, “Zernike polynomials and atmospheric turbulence”, J. Opt. Soc. Am. 66, 1976, 207–211. [CrossRef]

S⃗ polynomials can be written as linear combinations of Zernike polynomials as well. The first 14 non-trivial S⃗ polynomials are listed in Table 1.

Table 1. List of the first 14 non-trivial S⃗ polynomials as linear combinations of Zernike polynomials.

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If A⃗ and B⃗ are two vector polynomials defined over a unit circle, we define their inner product as

(A,B)=1π(AB)dxdy,
(3)

where integration is over unit circle.

S⃗ polynomials are orthonormal, which means

(Si,Sj)=1π((ϕi)(ϕj))dxdy=δij.
(4)

3. Derivation of a complementary set of vector polynomials

Any vector field can be written as4

4. H. F. Davis and A. D. Snider, Introduction to Vector Analysis, (Wm. C. Brown Publisher, 1986).

v=ϕ+×P,
(5)

where ϕ is a scalar and P⃗ is a vector. The divergence of ν⃗ is then

∇•ν⃗=∇2 ϕ+∇•(∇×P⃗)=∇2 ϕ,

and the curl of ν⃗ is

∇×ν⃗=∇×(∇×P⃗)=∇(∇•P⃗)-∇2 P⃗.

T=×P=[îĵk̂xyzPxPyPz].
(6)

This set has to be mutually orthogonal as well.

Like the S⃗ polynomials, T⃗ polynomials are vectors defined in x-y plane only. A convenient choice of P⃗ is vectors along z axis only, i.e. Px=Py=0. We can use a scalar ψ instead to represent P⃗:

P=ψk,̂
(7)

where ψ is a function of x and y: ψ=ψ (x, y). It follows that

Ti=×(ψik̂)=îψiyĵψix.
(8)

The inner product of two T⃗ polynomials is then

(Ti,Tj)=((ψiy)(ψjy)+(ψix)(ψjx))dxdy
=((ψi)(ψj))dxdy.
(9)

We choose a basis of functions {ψi} that we use to generate the T⃗ polynomials to be the same basis as we used to generate the S⃗ polynomials, {ϕi}. By Eq. (4), we know that their choice will create T⃗ polynomials that are mutually orthogonal.

From Eqs. (1) and (8), and with ψi=ϕi, we know that S⃗i(x,y) and T⃗i(x,y) have same magnitude and are orthogonal to each other at any point in a unit circle, therefore (S⃗i,T⃗i)=0. But the sets S⃗ and T⃗ are not fully independent. For all the j with m=n, we can show that

∇×T⃗j=-2 ϕj,

and

2ϕj2Zj(1rr(rr)+1r22θ2)[rn(cosnθsinnθ)]=0,
(10)

which means T⃗j has 0 curl and is therefore not linearly independent of S⃗ polynomials. For example, when j=9 or 10, m=n=3: T9=12(îZ6ĵZ5)=S10 and T10=12(îZ5ĵZ6)=S9 .

For any other pair of i and j, (S⃗i,T⃗j)=0.

Fig. 1. Relations between the S⃗ and T⃗ polynomials. The Laplacian vector fields are the overlap between S⃗ and T⃗. The dashed circles and associated solid arrows illustrate the local behaviors of the vectors in different sets after subtracting the local constant vector.

The S⃗ and T⃗ polynomials can be thought of as vector fields in a unit circle. In vector calculus, S⃗ is known as irrotational vector fields which have zero curl everywhere, and T⃗ is known as solenoidal vector fields which have zero divergence everywhere. The two types vector fields have some overlap where both divergence and curls are everywhere zero, which is known as Laplacian vector field. The overall relationship between S⃗ and T⃗ vector fields is illustrated in Figure 1. The overlapped area contains terms derived from corresponding scalar ϕ polynomials whose Laplacian is 0. If ϕ represents wavefront, these terms correspond to a wavefront that has zero net curvature at any point in the pupil.

It is useful to compare the different types of functions defined here. The S⃗ functions are generated from gradients, thus have no curl. Since S⃗ functions are 2-d vectors defined in a plane, mathematically, we can express the curl as line integral along a closed path in the plane:

Sdl=0.
(11)

The T⃗ functions have no divergence. Again they are 2-d vectors defined in a plane. Mathematically, we express divergence of a 2-d vector as a line integral over a closed path:

Tn̂dl=0,
(12)

where is the unit normal vector pointing out of the closed path.

The intersection, which includes both S⃗ and T⃗, is of the form that fits both Eqs. (11) and (12), having both zero divergence and zero curl. Graphical depictions of the local behavior of the functions are included in Fig. 1: dashed circles represent infinitesimal region and solid arrows represent local vectors (after a constant vector is subtracted.)

Table 2 lists expressions for the first 15 T⃗ polynomials. The plots of first 12 non-trivial T⃗ polynomials are shown in Table 3. The complete set of orthogonal vector polynomials that fully spans the circular domain can be written as the combined set of S⃗ polynomials and independent T⃗ polynomials since the Laplacian type functions are included in both sets. Care must be taken to ensure that the common functions are not counted twice.

Table 2. Analytical expressions of the first 15 T⃗ polynomials.

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Table 3. Plots of the first 12 non-trivial T⃗ polynomials.

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4. Summary

If interested, you can request the MATLAB codes for calculating the S⃗ and T⃗ polynomials from Dr. Chunyu Zhao, czhao@optics.arizona.edu.

References and links

1.

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). [CrossRef] [PubMed]

2.

C. Zhao, et al, “Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique,” Proc. SPIE 6293, 62930k, (2006). [CrossRef]

3.

R. J. Noll, “Zernike polynomials and atmospheric turbulence”, J. Opt. Soc. Am. 66, 1976, 207–211. [CrossRef]

4.

H. F. Davis and A. D. Snider, Introduction to Vector Analysis, (Wm. C. Brown Publisher, 1986).

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(080.1010) Geometric optics : Aberrations (global)
(220.4840) Optical design and fabrication : Testing

ToC Category:
Optical Design and Fabrication

History
Original Manuscript: February 19, 2008
Revised Manuscript: April 18, 2008
Manuscript Accepted: April 19, 2008
Published: April 24, 2008

Virtual Issues
Vol. 3, Iss. 5 Virtual Journal for Biomedical Optics

Citation
Chunyu Zhao and James H. Burge, "Orthonormal vector polynomials in a unit circle, Part II : completing the basis set," Opt. Express 16, 6586-6591 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6586


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References

  1. 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). [CrossRef] [PubMed]
  2. C. Zhao,  et al, "Figure measurement of a large optical flat with a Fizeau interferometer and stitching technique," Proc. SPIE 6293, 62930k (2006). [CrossRef]
  3. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211(1976). [CrossRef]
  4. H. F. Davis and A. D. Snider, Introduction to Vector Analysis, (Wm. C. Brown Publisher, 1986).

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