Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group

Generalization of Zernike polynomials for regular portions of circles and ellipses

Open Access Open Access

Abstract

Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit circle. Here, we present a generalization of this Zernike basis for a variety of important optical apertures. On the contrary to ad hoc solutions, most of them based on the Gram-Schmidt orthonormalization method, here we apply the diffeomorphism (mapping that has a differentiable inverse mapping) that transforms the unit circle into an angular sector of an elliptical annulus. In this way, other apertures, such as ellipses, rings, angular sectors, etc. are also included as particular cases. This generalization, based on in-plane warping of the basis functions, provides a unique solution and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for most common, elliptical and annular apertures are provided.

© 2014 Optical Society of America

1. Introduction

The problem of finding complete orthonormal systems to represent functions defined on finite supports with a given geometry appears in many areas of Physics and Engineering. In particular, Zernike circle polynomials (ZPs) [1] are widely used to represent optical path differences (phase differences or wave aberrations) in wavefronts, or even the sag of optical surfaces (such as the human cornea [2, 3]) as they are well adapted to the circular shape of a majority of conventional optical systems. There are an infinite number of possible systems, but ZPs (or lineal combinations of them [4]) show important advantages and interesting properties. Among these properties, they permit to establish a link with the traditional Seidel theory of aberrations [5], which is based on a third order Taylor series expansion, and with further extensions of the Seidel theory to 5th order, etc. On the one hand, the monomials of the Taylor series have a clear physical meaning as they represent different types of aberrations (defocus, astigmatism, coma, etc.) but are not orthogonal, which limits both theoretical and practical developments. On the other hand, higher order ZPs contain lower order terms, as a necessary balance to get zero average [6]. As a result of this cross-talk between higher and lower orders, the link of the ZPs to the Seidel theory is not evident. Nevertheless, the theoretical and practical advantages of orthonormal polynomials, make that ZPs became the standard way to describe the phase of wavefronts [7] (or the wave aberration or optical path differences) in many fields ranging from atmospheric optics [8], optical design and testing [9] or visual optics (the ANSI Z80.28 standard for reporting aberrations in the human eye is based on ZPs). Even though the circle is the most common optical aperture, there are other geometries, such as the annular pupils in large telescopes [10] or rectangular in anamorphic systems, etc. Furthermore, even in the case of circular apertures, the effective pupil becomes elliptical for off-axis field angles [11]. The eccentricity of the ellipse increases with field angle, and can reach high values for wide angle lenses. The importance of this problem motivated the development of a series of ad hoc solutions, most of them based on the Gram-Schmidt (G-S) method to obtain orthonormal basis on different types of apertures [12, 13] such as ellipses [11], rectangles [14], annuli [15], circular sectors [16], etc. In some particular cases, such as rectangles, Legendre [14] or Chebyshev polynomials [17] were also proposed, but their integration within Seidel or Zernike theoretical frameworks is difficult. An extensive catalogue of polynomial basis functions on different types of apertures can be found in [18]. The G-S method has several advantages but also important drawbacks. The main advantage is that it is a quite general linear method, and thus, the resulting basis functions are linear combinations of the initial ZPs. The main drawbacks are that there is not a unique solution (the final basis may depend on the ordering of the initial system, or on the particular implementation [19], refining algorithms [20, 21], etc.) and that one has to find an ad hoc solution for every type of aperture [12–17]. These, in turn, hinder the physical interpretation of the associated expansion coefficients (especially for the higher orders due to a cumulative effect associated to the G-S method). In addition, the G-S method is especially well-suited for numerical implementation, which means even further optimization for specific parameters of the apertures (eccentricity or orientation of the ellipse, radius of the central obscuration, etc.). Somewhat more general analytical expressions can be obtained too using a non-recursive method [22], but the computational cost may dramatically increase with the order of the polynomial, which may become an effective limitation [11].

In this context, our goal was to develop a general framework able to provide a common formulation under a unique criterion and providing a unique general solution for most of the usual optical apertures. Our approach is based on finding the mapping that transforms the unit circle into the desired aperture geometry. That is to find the diffeomorphism (i.e. a mapping that has a differentiable inverse mapping) [23] that transforms the unit circle into the connected set within the plane that represents the optical aperture. This mapping means warping (and rotating within the plane) the input basis functions so that they fit into the new aperture geometry. On the contrary to that ad hoc solutions, that warping permits not only uniqueness, but also a high level of invariance of the mathematical properties and physical meaning of the basis functions (tilt, defocus, astigmatism, coma, etc.) and hence a natural generalization of the aberration theory. One of the simplest possible mappings consists of the affine transformation (composed of scaling along x and y, and rotation) which maps the circle into the ellipse. Here, the resulting basis functions are linear combinations of the initial ones (i.e. polynomials), and the associated metrics are proportional (i.e. Euclidean). Other mappings, in particular those transforming the circle into (circular or elliptical) annuli, are non-linear and thus, in general, the warped basis functions are not polynomials and the associated metric may not be Euclidean.

In the present work, we will consider the angular sector of an elliptical annulus with arbitrary orientation as our most general case of mapping of the circle, since other geometries such as circles, ellipses, annuli, sectors, etc., correspond to particular values of the parameters of the general sector. Square, rectangular, hexagonal, etc., geometries are not considered in the present study.

2. Theoretical basis

The generalization of unit circle polynomials (or, in general, any complete and orthogonal set of basis functions in the circle) to deformations or partitions of the disc can be achieved by applying a diffeomorphism of the unit circle (or unit disc) D into a connected set M within the plane (see Fig. 1):

 figure: Fig. 1

Fig. 1 Mapping of the unit circle D onto a connected set M through diffeomorphism φ(u,v).

Download Full Size | PDF

R=A1

The diffeomorphism is an especially useful transformation in this context, since it is a bijection and its inverse exists and is differentiable as well:

u:=(u,v)=φ1(x,y):=(u(x,y),v(x,y)),
and the Jacobian of the inverse transformation φ1will beJ(x,y):=(ux). Now, let us choose a complete set of basis functions Zj (for example ZPs) orthonormal on D with metricdudv. These functions are orthonormal also under the change of variables(u,v)(x,y)=φ(u,v); and taking into account thatdudv=|J(x,y)|dxdy, then we have:
δi,j=1πDZi(u,v)Zj(u,v)dudv=1πMZi(φ1(x,y))Zj(φ1(x,y))|J(x,y)|dxdy
where π is the area of the unit circle D. This means that the new functions resulting from this change of variables
ds:=dxdy=rdrdθ
are orthonormal on M with metric |J(x,y)|dxdy. We can obtain a further generalization by considering the product of these functions with a continuous function Q(x,y) so that:

K¯j(x,y):=Q(x,y)Zj(φ1(x,y)).

The resulting functions are orthonormal on Mwith metric Q2(x,y)|J(x,y)|dxdy. In particular, we can take the trivial case Q(x,y)=1, so that K¯j(x,y)=Kj(x,y) are orthonormal functions on with metric |J(x,y)|dxdy. Another interesting particular case is when we take Q(x,y)=|J(x,y)|, in shuch a way that the resulting functions K¯j(x,y) are orthonormal on but now with the Euclidean metric dxdy.

It is straightforward to show that the set {K¯j(x,y)} forms a complete system in L2(M) with metric Q2(x,y)|J(x,y)|dxdy. For any function f¯(x,y) on L2(M), we can define f(u,v) on L2(D) (with Euclidean metric dudv) as follows:

f(u,v):=Q1(φ(u,v))f¯(φ(u,v))=jcjZj(u,v),
that is the expansion of f on the basis set Zj, where the coefficients are given by the projections of f(u,v) on the basis functions:

cj:=1πDf(u,v)Zj(u,v)dudv.

If we now apply the change of coordinates to the definition of the new basis functions, and solve for Zj we have:

Zj(u,v)=Q1(φ(u,v))K¯j(φ(u,v));
and then we find that
f¯(x,y):=jcjK¯j(x,y)
where the coeffients are also obtained through projection onto the set of basis functions on M:

cj:=1πMf¯(x,y)K¯j(x,y)Q2(x,y)|J(x,y)|dxdy.

Note that the metric will be dxdyonly when Q(x,y)=|J(x,y)|. The above formulation assumes Cartesian coordinates, but equivalent results are obtained in polar coordinates.

We want to remark that the choice of Q(x,y)has important consequences. In practrice, there are two alternative ways to obtain the expansion coeffients cj, either computing the inner products (projections) with the basis functions, as in Eq. (10), or through least squares fit, which is especially useful for sampled functions [24]. In the last case, the effect of the non Euclidean metric (Q(x,y)=1) is that we will have a weighted least squares problem. On the one hand, using Q(x,y)=|J(x,y)| implies a deformation of the basis functions which might potentially alter their physical meaning. In fact some properties such as zero mean or the possibility of computing the variance of the function as the squared sum of the coefficients can be lost in that case. We know that the mean of functionsKj(x,y), μj=1πMKj(x,y)|J(x,y)|dxdy=0, j0as a consequence of their orthogonality; but the mean of K¯j(x,y), μ¯j=1πMK¯j(x,y)dxdy=1πMKj(x,y)|J(x,y)|dxdymay be different from zero depending on the Jacobian. The varianceσ2 of any function f(x,y) isσ2=1πMf(x,y)2|J(x,y)|dxdy(1πMf(x,y)|J(x,y)|dxdy)2. If we replace f(x,y)by its expansion jcjKj(x,y) we have σ2=jcj2as far as Kj(x,y)have zero mean (μj=0,j0), since the left integral becomes the sum ofcj2δj,j, whereas the right integrals are equal to zero, except for j = 0, for which we have(c0)2. Thus we have σ2=jcj2c02=j>0cj2. If we now consider the expansion on the system K¯j(x,y) then we arrive to σ2=jc¯j2(jc¯jμ¯j)2j>0c¯j2. As we discuss further below, despite the complexity added by the non-Euclidean metric to the computation of inner products or to the weighted least squares fit, it seems more convenient to setQ(x,y)=1. Nevertheless, when J(x,y)=Constant then Q(x,y)=|J(x,y)|=Constant is a simple re-normalization factor.

3. Standard portions of circles and ellipses

In this section we particularize the above general formulation to standard partitions of circles and ellipses (annuli, angular sectors, etc.). To this end we consider the angular sector G of an elliptical annulus with arbitrary orientation as the most general mapping considered here. As shown in Fig. 2, the geometry of this general sector G is described by 6 parameters, whereas other shapes (annuli, ellipses, etc.) are obtained as particular cases for specific values of these parameters. The mapping from the unit circle (radius R = 1) into the ellipse can be obtained through a linear affine transformation that is the composition of scaling x and y (to obtain semi axes A and B with BA) and rotation by angle α (formed by the major axis A with the x axis). The (concentric) elliptical annulus requires another parameter 0h<1, that is the proportionality constant between its inner and outer elliptical boundaries, h=a/A=b/B (a=hA and b=hBare the inner semiaxes). Finally the angular sector will be the area inside the angular interval[θ1,θ2]. Thus, our general mapping will be determined by six parameters: A, B, α, h, θ1 andθ2. This also includes the mapping of the unit circle into another circle with arbitrary radiusR=A1.

 figure: Fig. 2

Fig. 2 General angular sector [θ1,θ2] of an elliptical (semiaxes A, B) annulus (h = a/A = b/B) with orientation α.

Download Full Size | PDF

We can obtain a more compact expression for the mapping φ by considering it as the composition of two mappings φ=φ2φ1. In the second mapping φ2we apply a rotation by angle α so that:

(x,y)=φ2(X,Y)=(XcosαYsinα,Xsinα+Ycosα)
and
(X,Y)=φ21(x,y)=(xcosα+ysinα,xsinα+ycosα)
that is a change to the scaled variables(X,Y):=(AX,BY):=(Arcosθ,Brsinθ), that is r:=(X/A)2+(Y/B)2, θ:=arctan((AY)/(BX)).

Now we can apply the mapping φ1 that transforms the unit circle into the angular ring sector:

(X,Y)=φ1(u,v)=(A[h+(1h)ρ]cos[θ1+θ2θ12πϕ],B[h+(1h)ρ]sin[θ1+θ2θ12πϕ])
and
(u,v)=φ11(X,Y)=(r(X,Y)h1hcos[2π(θ(X,Y)θ1)θ2θ1],r(X,Y)h1hsin[2π(θ(X,Y)θ1)θ2θ1])
where ρ:=u2+v2, ϕ:=arctan(v/u); r(X,Y) and θ(X,Y)were defined above. Then, the complete inverse mapping φ1=φ11φ21is:

(u,v)=φ1(x,y)=(r(x,y)h1hcos[2π(θ(x,y)θ1)θ2θ1],r(x,y)h1hsin[2π(θ(x,y)θ1)θ2θ1]),

withr(x,y)=(xcosα+ysinαA)2+(xsinα+ycosαB)2;θ(x,y)=arctan(A(xsinα+ycosα)B(xcosα+ysinα)); θ1=arctan[ABtan(θ1α)]; and θ2=arctan[ABtan(θ2α)]. As a result of the composition of two transformations, we have an intermediate change of variables(x,y)(X,Y)(u,v)or in polar coordinates: (r,θ)(r,θ)(ρ,ϕ).

We can see that(x,y)=(rcosθ,rsinθ), with r=x2+y2,θ=arctan(y/x), are now confined within the sector G depicted in Fig. 2. The angular interval is [θ1:=α+θ¯1,θ2:=α+θ¯2] where θ¯1,θ¯2are angles with the X axis. On the one hand, we have θ¯=arctan((B/A)tanθ)so that θ¯1=arctan((B/A)tanθ1) and θ¯2=arctan((B/A)tanθ2) with θ¯1<θ¯2 and θ¯1,θ¯2[0,2π]. On the other hand 0ρ1 and0ϕ2π. Then we have:

h2(XA)2+(YB)2=[h+(1h)ρ]2=r21
and

θ1θ=arctan(AYBX)=θ1+θ2θ12πϕθ2.

In Eq. (14a), we can see thatr=h+(1h)ρ. Sinceρ[0,1], then r[h,1]and Eq. (14b) means that θ[θ1,θ2]and hence (x,y)=(rcosθ,rsinθ) take values within the sectorG:={(x,y),hr1,θ1θθ2}. The Jacobian of this transform is

J(x,y)=2π(rh)ABr(1h)2(θ2θ1).

It is noteworthy that the Jacobian is not constant only for annuli and annular sectors (see Table 1).

Tables Icon

Table 1. Range or constant values for the parameters corresponding to the different mappings of the unit circle. All parameters are positive.

3.1. General system

Now, we can write the new basis functions on G. We will consider Q(x,y)=1:

Gj(x,y):=Kj(x,y)=Kj(XcosαYsinα,Xsinα+Ycosα)=Kj(ArcosθcosαBrsinθsinα,Arcosθsinα+Brsinθcosα):=Zj(rh1hcos[2π(θθ1)θ2θ1],rh1hsin[2π(θθ1)θ2θ1]).

Functions Gj form a complete orthonormal system on G, the general angular sector of the elliptical ring, with metric (differential of surface area) ds:=|J(x,y)|dxdy=|J(rcosθ,rsinθ)|rdrdθ. It is straightforward to show that functions Gj: (1) are orthonormal (i.e. their inner products are Kronecker's deltas); (2) any square-integrable function, defined on G can be expressed as a linear combination of functions Gj; and (3) when the general sector of an elliptical annulus tends to the unit circle, that is when A1, B1,α0, h0, θ10and θ22π, then Gj(x,y)Zj(x,y). Table 1 lists the ranges of the parameters as well as the Jacobian (Eq. (15) for the different particular geometries (annulus, sectors, ellipses, etc.)

3.2. Zernike polynomials

In what follows we will consider that Zj=Znm are ZPs (here j is a combination of the two indexes: the order of the polynomial n, and the angular frequency m. Different authors use different ordering, and hence different j. Probably the most accepted ordering is the one proposed by Noll [8]). Their expression in polar coordinatesρ:=u2+v2, ϕ:=arctan(v/u) is:

Znm(ρ,ϕ):={NnmRn|m|(ρ)cos(mϕ),m0,NnmRn|m|(ρ)sin(mϕ),m<0,
where
Nnm=2(n+1)(1+δm0)andRn|m|(ρ)=s=0(n|m|)/2(1)s(ns)!s!(n|m|2s)!(n+|m|2s)!ρn2s
If we apply the mapping described above, then we obtain
Znm(rh1h,2π(θθ1)θ2θ1)={NnmRn|m|(rh1h)cos(m2π(θθ1)θ2θ1),m0,NnmRn|m|(rh1h)sin(m2π(θθ1)θ2θ1),m<0,
with r=r(cos(θα)/A)2+(sin(θα)/B)2; θ=arctan(ABtan(θα)); θ1=arctan[ABtan(θ1α)]; and θ2=arctan[ABtan(θ2α)].

Figure 3 shows one example for the particular case of Z33 (trefoil).

 figure: Fig. 3

Fig. 3 Basis functions obtained for the particular case of the Zernike circle polynomial Z33(trefoil) corresponding to the different cases listed in Table 1. The fringes in the interferograms correspond to 1 wavelength of optical path difference.

Download Full Size | PDF

4. Particular cases: ellipses and annuli

In this Section we analyze in detail two cases mostly relevant in optics: ellipses and annuli, departing from the system of Zernike circle polynomials.

4.1. Elliptical apertures

As we said above, the mapping from the unit circle into an ellipse is a linear transformation involving three parameters A, B (it is common to use the eccentricity e=1B2/A2as a measure of the elongation. This parameter will be used for the explicit expressions of the polynomials listed in Table 2), and α, and thus: (x,y)=φ(u,v)=(AucosαBvsinα,Ausinα+Bvcosα), and its inverse mapping: (u,v)=φ1(x,y)=((xcosα+ysinα)/A,(xsinα+ycosα)/B). The Jacobian J(x,y)=1/(AB) is constant (i.e. Euclidean metric), so it is possible to use Q=|J(x,y)|=1/AB as a simple re-normalization factor. The set of basis functions on the ellipse EA,B,α:={(x,y),(xcosα+ysinα)2/A2+(xsinα+ycosα)2/B21} will be:

Tables Icon

Table 2. Expressions of the orthogonal elliptical polynomials up to order n = 4 where A is the major semi axis; e=1B2/A2 is the eccentricity; and α is the orientation of the ellipse.

E¯nm(x,y):=1ABEnm(x,y)=1ABZnm((xcosα+ysinα)/A,(xsinα+ycosα)/B)

For the particular case of ZPsZnm(ρ,ϕ), we only need to simplify Eq. (18) by setting h=0;θ'1=0; and θ'2=2π to obtain the system E¯nm(r,θ):

E¯nm(r,θ)={1ABNnmRn|m|(r)cos(mθ),m0,1ABNnmRn|m|(r)sin(mθ),m<0,
where Nnmand Rn|m|were given in Eq. (17b) and the variables (r,θ) were defined right after Eq. (18). Since this is a particular case of G, these functions (1) are orthonormal; (2) any square-integrable function, defined on E can be expressed as a linear combination of functions Enm; and (3) when the ellipse transforms to the unit circle, that is when A1, B1andα0, then Enm(x,y)Znm(x,y). The specific orthogonal elliptical polynomials are listed in Table 2 up to order 4. Several representative examples, corresponding to various Zernike wavefront aberrations: tilt, defocus, astigmatism, coma, trefoil and spherical aberration are represented in Fig. 4 for the particular case of α = 137.5° and e = 0.74. This would correspond to the effective pupil for an off-axis (skew) wavefront passing through a circular aperture at field angle 32.4° and azimuth 54.3° (Only positive values of m are shown since m < 0 are rotated versions of the same aberration modes).

 figure: Fig. 4

Fig. 4 Orthogonal elliptical polynomials corresponding to Zernike wavefront aberrations: tilt E11, defocus E20, astigmatismE22, coma E31, trefoilE33and spherical aberrationE40. The fringes in the interferograms correspond to 1 wavelength of optical path difference.

Download Full Size | PDF

B. Annular apertures

The mapping of the unit circle into the annulusφ:DO, with a and A (with a = hA) being the radii of the inner and outer circular boundaries respectively, can be expressed as:

(x,y)=φ(u,v)=(A[h+(1h)ρ]cosϕ,A[h+(1h)ρ]sinϕ)
and its inverse:

(u,v)=φ1(x,y)=(rhAA(1h)cosθ,rhAA(1h)sinθ).

With this mapping r=A[h+(1h)ρ]; since ρ[0,1], then r[hA,A]. This means that the Jacobian J(x,y)=rhArA2(1h)2 is positive, but it is not constant, and hence the metric is not Euclidean (unless we set Q=|J(x,y)|=rhArA(1h)). The functions:

Onm(r,θ):=Znm(rhAA(1h)cosθ,rhAA(1h)sinθ)
or
O¯nm(r,θ):=rhArA(1h)Znm(rhAA(1h)cosθ,rhAA(1h)sinθ)
form complete orthonormal systems on the annulus O:={(x,y),a=hArA,0θ2π} with metric ds:=rhArA2(1h)2dxdy=rhAA2(1h)2drdθ for system {Onm} or ds:=dxdy=rdrdθ for system {O¯nm}. The general expression in polar coordinates for {Onm}results:

Onm(r,θ)={2(n+1)(1+δm0)s=0(n|m|)/2(1)s(ns)!s!(n+|m|2s)!(n|m|2s)!(rhAA(1h))n2scos(mθ),m0,2(n+1)(1+δm0)s=0(n|m|)/2(1)s(ns)!s!(n+|m|2s)!(n|m|2s)!(rhAA(1h))n2ssin(mθ),m<0.

Again, this system is complete, orthonormal and reduces to Znmwhen h0 andA1. The specific annular functions (polynomial quotients) are listed in Table 3 up to order 4, and representative examples are shown in Fig. 5 for h = 0.33 which is close to that of the central obscuration of the Hubble telescope.

Tables Icon

Table 3. Expressions of the orthogonal annular basis functions (polynomial quotients) up to order n = 4. A and a = hA are the outer and inner radii.

 figure: Fig. 5

Fig. 5 Orthogonal annular polynomials corresponding to equivalent Zernike wavefront aberrations.

Download Full Size | PDF

5. Discussion and conclusions

We presented a general method to obtain complete orthonormal systems in any connected set M. This consists of applying a diffeomorphism to a given (complete and orthonormal) system on the unit circle to transform the circle basis functions into the new system on the desired geometry. This type of mapping is a bijection, so that it is invertible. In addition, its inverse is differentiable, so it has an associated Jacobian. All these properties are essential for establishing a rigorous and robust theoretical framework. We then particularized the method to circles, ellipses and their standard portions: annuli and angular sectors, etc. The general mapping of the unit circle into an angular sector of an elliptical ring can be obtained by the composition of four mappings: for the annuli: r(rh)/(1h) where h and 1 are the radii of the inner and outer (unit disk) boundaries of the annulus. Similarly for the angular sectors: θ2π(θθ1)/(θ2θ1). The scaling: xAx and yBy transforms the unit circle into a horizontal ellipse. Finally the in-plane rotation by angle α permits to have an arbitrary orientation. The eight geometries, listed in Table 1, are obtained by composing one, two, three or four of these transformations. The associated Jacobians are constant except for the radial mappingr(rh)/(1h), since the Jacobian to pass from Cartesian to polarcoordinates is not constant. Thus, the simplicity of this framework, based on simple changes of variables is another interesting aspect (even though the composition of up to four of these changes may yield to long expressions).

These systems are especially important in the wave theory of aberrations and optical image formation [5, 7]. As we said in the Introduction, there is not a complete agreement on a unified theory yet. Nevertheless, the good mathematical properties of ZPs are making that they are becoming a standard among the scientific community, but even more in technologies embedded in many industrial tools (optical design), devices (surface metrology, optical testing) and even in clinical apparatus in ophthalmology (corneal topographers, ocular aberrometers, etc.) In addition to form a complete orthonormal system they have compact expressions both in polar and Cartesian coordinates, and they also show an interesting list of additional properties. The main drawbacks come from the fact that ZPs represent a highly convenient but arbitrary choice, and from some difficulties for unification with the Seidel (third order and further extensions to 5th and higher orders) theory of aberrations. The wave aberration theory based on orthogonal systems, such as Zernike polynomials, is superior to the Seidel theory for nearly diffraction-limited optical system because the image quality, as measured by the Strehl ratio, Sr, (peak of the point spread function normalized to that of the Airy disk) [5] can be predicted from the wavefront variance, and hence from the expansion coefficients as: Sr1k2σ2=1k2j>0cj2(where k=2π/λ is the wave number). However this approximation is valid only when the aberrations are small, i.e. when σ is not much greater than λ/14 (Maréchal criterion for diffraction-limited optical quality) [5]. Thus, the wavefront varianceσ2, or the root mean square (RMS) wavefront errorσ, are good metrics for optical image only whenσ<<λ. Thus the quadratic sum of the wavefront coefficients σ2, or the RMS σ, are good metrics for the wavefront quality, but in aberrated systems they are not good metrics for predicting image quality.

Nevertheless, the need for a unified theoretical framework for describing wavefronts (optical aberrations) and optical surfaces, in terms of complete orthogonal systems, is patent. Even in the context of ZPs one can find that different authors use a variety of units (wavelengths, micrometers, etc.), normalizations (use of Nnm or not), ordering (since they have two indexes n, m there are many different possibilities to define a single index j which were used by different authors), or even opposite sign conventions. Among all these possibilities the ordering proposed by Noll [8] seems widely accepted, but the Optical Society of America adopted a different ordering for reporting aberrations in the human eye (ANSI standard Z80.28). This disparity of criteria existing even for the ZPs may explode when extending the Zernike theory to other optical apertures, such ellipses, annuli, sectors, etc. In fact there is a large list of publications with different polynomials which are ad hoc solutions for every type of optical apertures [11–18]. Most of these polynomials are obtained through G-S orthonormalization. A relevant improvement over the classical G-S method can be obtained by a non-recursive algorithm [19] which shows better numerical stability. It consists of inverting the Gram matrix, but considering only up to a given order n, and using the Cholesky decomposition. Additional numerical stability can be obtained through re-orthogonalization [20] or by an iterative process [21] up to machine precision accuracy. This non-recursive G-S algorithm provided probably the best results published in the literature [18]. The main limitation is that the symbolic implementation of these methods requires heavy loads of computer memory [11]. We want to remark that the systems of orthogonal polynomials obtained with these improved G-S methods are perfectly valid for ad hoc applications. However, uniqueness is not guaranteed as the result depends on the initial ordering (index j), or on the particular algorithm. In addition, the physical meaning of each polynomial could change after orthogonalization, especially for higher order polynomials, due to the cumulative effect inherent to the G-S method. As a result, the basis functions obtained for different apertures may have different properties and physical meaning, which seems far from the goal of a unified theory.

We believe that the mapping method proposed here, implemented as a change of variables, overcomes most of these difficulties and drawbacks, and provides a common framework, especially well-suited for a unified theory of aberrations. In addition to the theoretical relevance, the unified formulation may be the starting point for developing standards for academic, industrial or even clinical (ophthalmic optics) use.

The two particular cases of Section 4 are especially relevant. In the first case, the importance is clear if we realize even the circular aperture (most common in optics) becomes elliptical off-axis. The size (A) eccentricity (e) and orientation (α) of the ellipse changes continuously with field angle and azimuth. These changes can be large in wide angle optical systems, and especially important in vision (both biological and artificial), where the field of view can be of the order of 180° or even more. Thus, the need for having a unified method to describe wavefronts corresponding to different field angles seems clear. The proposed change of variables (mapping) is a simple affine transformation from the circle to the ellipse, to adapt the modes (polynomials) to the scales (A and B) and orientation of the effective aperture, which changes as we move from the optical axis to a peripheral field angle. This affine transformation (scaling and rotation) is general for virtually any type of aperture (annuli, polygons, etc.) while it keeps a reasonable level of invariance, and hence a similar physical meaning of the expansion coefficients. This mapping should be applied for representing off-axis wavefronts, especially when the field of view is significantly wide.

The case of annular apertures is relevant not only for being a typical aperture in telescopes, but also due to important differences associated to its particular topology. Other apertures, such as circles, ellipses, or even angular sectors are simply connected sets, whereas the central obscuration of annuli makes that they are connected (but not simply) sets. This has important consequences. The deformation (mapping) of the circle necessary to arrive at the annulus (intuitively this would be a sort of stretching of an infinitesimal central hole to form the finite obscuration) is not uniform, as it depends on the distance to the center (ρ). This lack of uniformity makes that the original polynomials become polynomial quotients (see Table 3) and that the Jacobian is not a simple constant normalization factor. Thus, in this case, we have a non-Euclidean metric and hence the original plane element of area becomes curved now. That curvature becomes patent if we compare the upper left panels of Figs. 4 and 5 which represent E11 (ellipse) and O11 (annulus) respectively. InE11 the mapping is linear and hence we can see tilted (rotated) but straight fringes, whereas O11 shows vertical but curved fringes. As we said in Section 2, it is possible to choose Q(x,y)=|J(x,y)|but this does not eliminate the curvature of the fringes. This choice does not seem compatible with a unified framework as it can potentially change the properties of the basis functions and the physical meaning of the associated coefficients.

Acknowledgments

This research was supported by the Spanish Ministry of Economía y Competitividad and the European Union, grant FIS2011-22496, and by the Government of Aragón, research group E99.

References and links

1. F. Zernike, “Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastme-thode,” Physica (Utrecht) 1(7-12), 689–704 (1934). [CrossRef]  

2. J. Schwiegerling, J. E. Greivenkamp, and J. M. Miller, “Representation of videokeratoscopic height data with Zernike polynomials,” J. Opt. Soc. Am. A 12(10), 2105–2113 (1995). [CrossRef]   [PubMed]  

3. D. R. Iskander, M. J. Collins, and B. Davis, “Optimal modeling of corneal surfaces with Zernike polynomials,” IEEE Trans. Biomed. Eng. 48(1), 87–95 (2001). [CrossRef]   [PubMed]  

4. W. Lukosz, “Der Einfluss der Aberrationen aud die optische Ubertragungsfunktion bei kleinen Orts-Frequenzen,” Opt. Acta (Lond.) 10(1), 1–19 (1963). [CrossRef]  

5. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge UK, 1999).

6. V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Appl. Opt. 33(34), 8125–8127 (1994). [CrossRef]   [PubMed]  

7. B. R. A. Nijboer, The diffraction theory of aberrations, Ph.D. thesis, University of Groningen, 1942.

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

9. D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, Hoboken, NY, USA, 2007).

10. S. R. Restaino, S. W. Teare, M. DiVittorio, C. Gilbreath, and D. Mozurkewich, “Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials,” Opt. Eng. 42(9), 2491–2495 (2003). [CrossRef]  

11. J. A. Díaz and R. Navarro, “Orthonormal polynomials for elliptical wavefronts with an arbitrary orientation,” Appl. Opt. 53(10), 2051–2057 (2014). [CrossRef]   [PubMed]  

12. W. Swantner and W. W. Chow, “Gram-Schmidt orthonormalization of Zernike polynomials for general aperture shapes,” Appl. Opt. 33(10), 1832–1837 (1994). [CrossRef]   [PubMed]  

13. R. Upton and B. Ellerbroek, “Gram-Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape,” Opt. Lett. 29(24), 2840–2842 (2004). [CrossRef]   [PubMed]  

14. V. N. Mahajan, “Orthonormal aberration polynomials for anamorphic optical imaging systems with rectangular pupils,” Appl. Opt. 49(36), 6924–6929 (2010). [CrossRef]   [PubMed]  

15. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71(1), 75 (1981). [CrossRef]  

16. J. A. Díaz and V. N. Mahajan, “Orthonormal aberration polynomials for optical systems with circular and annular sector pupils,” Appl. Opt. 52(6), 1136–1147 (2013). [CrossRef]   [PubMed]  

17. F. Liu, B. M. Robinson, P. J. Reardon, and J. M. Geary, “Analyzing optics test data on rectangular apertures using 2-D Chebyshev polynomials,” Opt. Eng. 50(4), 043609 (2011). [CrossRef]  

18. V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis (SPIE, 2013).

19. G. M. Dai and V. N. Mahajan, “Nonrecursive determination of orthonormal polynomials with matrix formulation,” Opt. Lett. 32(1), 74–76 (2007). [CrossRef]   [PubMed]  

20. L. Giraud and J. Langou, “Robust selective Gram-Schmidt reorthogonalization,” SIAM J. Sci. Comput. 25, 417–441 (2003). [CrossRef]  

21. W. Hoffmann, “Iterative algorithms for Gram–Schmidt orthogonalization,” Computing 41(4), 335 (1989). [CrossRef]  

22. V. N. Mahajan and G. M. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24(9), 2994–3016 (2007). [CrossRef]   [PubMed]  

23. B. O'Neill, Elementary Differential Geometry, Rev. 2nd ed. (Academic Press, New York, NY, 2006).

24. J. L. Rayces, “Least-squares fitting of orthogonal polynomials to the wave-aberration function,” Appl. Opt. 31(13), 2223–2228 (1992). [CrossRef]   [PubMed]  

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1 Mapping of the unit circle D onto a connected set M through diffeomorphism φ(u,v).
Fig. 2
Fig. 2 General angular sector [ θ 1 , θ 2 ] of an elliptical (semiaxes A, B) annulus (h = a/A = b/B) with orientation α.
Fig. 3
Fig. 3 Basis functions obtained for the particular case of the Zernike circle polynomial Z 3 3 (trefoil) corresponding to the different cases listed in Table 1. The fringes in the interferograms correspond to 1 wavelength of optical path difference.
Fig. 4
Fig. 4 Orthogonal elliptical polynomials corresponding to Zernike wavefront aberrations: tilt E 1 1 , defocus E 2 0 , astigmatism E 2 2 , coma E 3 1 , trefoil E 3 3 and spherical aberration E 4 0 . The fringes in the interferograms correspond to 1 wavelength of optical path difference.
Fig. 5
Fig. 5 Orthogonal annular polynomials corresponding to equivalent Zernike wavefront aberrations.

Tables (3)

Tables Icon

Table 1 Range or constant values for the parameters corresponding to the different mappings of the unit circle. All parameters are positive.

Tables Icon

Table 2 Expressions of the orthogonal elliptical polynomials up to order n = 4 where A is the major semi axis; e= 1 B 2 / A 2 is the eccentricity; and α is the orientation of the ellipse.

Tables Icon

Table 3 Expressions of the orthogonal annular basis functions (polynomial quotients) up to order n = 4. A and a = hA are the outer and inner radii.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

R=A1
u :=(u,v)= φ 1 (x,y):=(u(x,y),v(x,y)),
δ i,j = 1 π D Z i (u,v) Z j (u,v)dudv= 1 π M Z i ( φ 1 (x,y)) Z j ( φ 1 (x,y))| J(x,y) |dxdy
ds:=dxdy=rdrdθ
K ¯ j (x,y):=Q(x,y) Z j ( φ 1 (x,y)).
f(u,v):= Q 1 (φ(u,v)) f ¯ (φ(u,v))= j c j Z j (u,v),
c j := 1 π D f (u,v) Z j (u,v)dudv.
Z j (u,v)= Q 1 (φ(u,v)) K ¯ j (φ(u,v));
f ¯ (x,y):= j c j K ¯ j (x,y)
c j := 1 π M f ¯ (x,y) K ¯ j (x,y) Q 2 (x,y)| J(x,y) |dxdy.
(x,y)= φ 2 (X,Y)=(XcosαYsinα,Xsinα+Ycosα)
(X,Y)= φ 2 1 (x,y)=(xcosα+ysinα,xsinα+ycosα)
(X,Y)= φ 1 ( u,v )=(A[h+(1h)ρ]cos[ θ 1 + θ 2 θ 1 2π ϕ],B[h+(1h)ρ]sin[ θ 1 + θ 2 θ 1 2π ϕ])
(u,v)= φ 1 1 (X,Y)=( r (X,Y)h 1h cos[ 2π( θ (X,Y) θ 1 ) θ 2 θ 1 ], r (X,Y)h 1h sin[ 2π( θ (X,Y) θ 1 ) θ 2 θ 1 ] )
(u,v)= φ 1 (x,y)=( r ( x,y )h 1h cos[ 2π( θ ( x,y ) θ 1 ) θ 2 θ 1 ], r ( x,y )h 1h sin[ 2π( θ ( x,y ) θ 1 ) θ 2 θ 1 ] ),
h 2 ( X A ) 2 + ( Y B ) 2 = [h+(1h)ρ] 2 = r 2 1
θ 1 θ =arctan( AY BX )= θ 1 + θ 2 θ 1 2π ϕ θ 2 .
J(x,y)= 2π( r h) AB r (1h) 2 ( θ 2 θ 1 ) .
G j (x,y):= K j (x,y)= K j ( XcosαYsinα,Xsinα+Ycosα ) = K j ( A r cos θ cosαB r sin θ sinα,A r cos θ sinα+B r sin θ cosα ) := Z j ( r h 1h cos[ 2π( θ θ 1 ) θ 2 θ 1 ], r h 1h sin[ 2π( θ θ 1 ) θ 2 θ 1 ] ).
Z n m (ρ,ϕ):={ N n m R n |m| (ρ)cos(mϕ), m0, N n m R n |m| (ρ)sin(mϕ), m<0,
N n m = 2(n+1) ( 1+ δ m0 ) and R n | m | (ρ)= s=0 ( n| m | ) /2 (1) s (ns)! s!( n|m| 2 s )!( n+|m| 2 s )! ρ n2s
Z n m ( r h 1h , 2π( θ θ 1 ) θ 2 θ 1 )={ N n m R n |m| ( r h 1h )cos(m 2π( θ θ 1 ) θ 2 θ 1 ), m0, N n m R n |m| ( r h 1h )sin(m 2π( θ θ 1 ) θ 2 θ 1 ), m<0,
E ¯ n m (x,y):= 1 AB E n m (x,y)= 1 AB Z n m ((xcosα+ysinα)/A,(xsinα+ycosα)/B)
E ¯ n m ( r , θ )={ 1 AB N n m R n |m| ( r )cos(m θ ), m0, 1 AB N n m R n |m| ( r )sin(m θ ), m<0,
(x,y)=φ(u,v)=(A[h+(1h)ρ]cosϕ,A[h+(1h)ρ]sinϕ)
(u,v)= φ 1 (x,y)=( rhA A( 1h ) cosθ, rhA A( 1h ) sinθ).
O n m (r,θ):= Z n m ( rhA A( 1h ) cosθ, rhA A( 1h ) sinθ)
O ¯ n m (r,θ):= rhA r A( 1h ) Z n m ( rhA A( 1h ) cosθ, rhA A( 1h ) sinθ)
O n m (r,θ)={ 2(n+1) ( 1+ δ m0 ) s=0 ( n| m | ) /2 (1) s (ns)! s!( n+|m| 2 s )!( n|m| 2 s )! ( rhA A( 1h ) ) n2s cos(mθ), m0, 2(n+1) ( 1+ δ m0 ) s=0 ( n| m | ) /2 (1) s (ns)! s!( n+|m| 2 s )!( n|m| 2 s )! ( rhA A( 1h ) ) n2s sin(mθ), m<0.
Select as filters


Select Topics Cancel
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.