An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems

Robert W. Gray; Christina Dunn; Kevin P. Thompson; Jannick P. Rolland

doi:10.1364/OE.20.016436

1. Introduction

As optical design and analysis simulation environments advance, they are not only developing new, faster, more effective algorithms, they are also expanding to interact and interface more seamlessly with related fields such as interferometric optical testing of individual surfaces and optical alignment and interferometric acceptance testing of assembled optical systems. As these fields coalesce into a common environment, there are situations where methods and features developed in isolation in one field are enabling when placed in a related field. A case in point is to combine the circular aperture Fringe Zernike polynomial of optical testing [1,2] with the field dependent wave aberration function of H.H. Hopkins [3], which is a basis in optical design.

In the 1970s, the field of interferometric optical testing emerged as a means for measuring the deviation of an as-fabricated optical surface from a desired nominal shape, often a sphere or conic [4]. It was found early on that Zernike polynomials provided an excellent metric basis for describing and understanding errors in the shape of an optical surface. In testing an optical surface, the most common geometry involves a test where the interferometer output is focused using a transmission sphere to a point that is then collocated at the center of curvature of the optical surface. The resulting Zernike polynomial coefficients that are the output of the modern digital interferometer characterize the departure of the surface under test from a sphere as coefficients of a polynomial orthonormalized over a circular bounding aperture, often limited to a particular order in aperture dependence. A key point to be made is that in the initial application of Zernike polynomials to the testing of individual optical surfaces there was no motivation to consider field dependence, only aperture dependence was being sought.

In the same time period, the field of computer-aided optical design was emerging on a commercial basis, in the form of programs such as CODE V^®. In optical design, both the aperture dependence and the field dependence of the aberrations are important. While the computer-aided formulation of optical design emerged in the 1970s, the theoretical basis was created much earlier. For aberration theorists working in optical design, a common starting point is a form of the wave aberration expansion written out by H.H. Hopkins [3] in the 1950s. This formulation is an important basis for lens design, but not commonly seen, or used, in optical testing. Recently, Sasian [5] has reformulated the work of Hopkins and provided this work as an accessible computational tool for optical design and analysis. It is significant to note that the equations that result in the wave aberration coefficients rely only on the parameters of a paraxial raytrace (marginal and chief ray height and angle by surface) and on aspheric deformation coefficients of a rotationally symmetric optical system. By contrast, the coefficients of a Zernike polynomial representation are computed based on fitting by a least squares technique with a dense sample of the wavefront departure over the aperture both in measurement (interferometry) and in simulation (by real raytracing).

While the wave aberration function of Hopkins guides the theoretical studies and expectations for optical design forms, real raytrace methods are used to present the performance of an optical system during the optical design process. The supporting graphical display of important information gathered by real raytrace over aperture and field of an optical system design includes transverse ray aberration plots and astigmatic and distortion field curves. These are only useful for understanding the limiting aberrations as a function of field of view to those trained in optical design. These methods are highly efficient, a feature required in the 1970s, but no longer with the increase in raytrace speeds of over 8 orders of magnitude in the last two decades. An emerging aberration display that directly isolates the aberration types and displays them versus field of view, first shown in 1985 [6], is to compute and display real ray based Zernike coefficient pair magnitudes and orientations over a somewhat dense grid at the image plane (a 51 × 51 patch in the field). Termed full field displays, this approach has the advantageous feature that it allows studying not only a summary measure of image quality, RMS wavefront error or MTF, but, more useful for the optical designer, the dominant aberration type within different regions in the field of view. While this approach is excessively redundant for a rotationally symmetric form, there is no significant time penalty, typically requiring less than 5 seconds compared to the 30 minute compute time when it was introduced. Significantly, this method is directly extensible to alignment sensitivity studies (i.e. optical systems without rotational symmetry, including freeform surfaces), without restriction.

Zernike polynomials were early on rapidly adopted by both the design and the testing communities and the supporting fields of optical engineering and optical alignment because they are orthogonal and complete over a unit radius circular pupil, they represent balanced aberrations, the H.H. Hopkins’ aberration expansion of optical design could be related to the Zernike polynomials, there is an established methodology for finding the expansion coefficients, and as previously mentioned, they can be used to describe any surface shape error. As a result, both communities could communicate effectively the shape of an optical component and in many cases extract important information from the component terms. Unlike a summary measure of image quality, like RMS wavefront error, a Zernike polynomial database for an optical system at a particular point in its field of view or an optical component surface gives information on the state figuring (Z₉ spherical aberration), the state of mounting (Z₅, Z₆ astigmatism), and the state of alignment (Z₇, Z₈ coma). Here we have used the coefficient numbering associated with the Fringe form of the Zernike polynomial [1]. There are many equivalent expressions for Zernike polynomials that differ in the ordering of the coefficients (for example the Born and Wolf, “standard” version is ordered based on mathematical properties rather than optical aberration forms) and in the normalization of the coefficients (the Fringe form is 0-P normalized, others are RMS normalized). For example, lens designers prefer the Fringe ordering of the Zernike polynomials because this ordering places 3rd order spherical aberration before rather than after 5th order elliptical coma [2].

One significant advantage to an optical analysis in simulation based on real raytracing using Zernike coefficients is that it immediately conveys the existence of higher order aberration terms, or not. In fact, in using these displays one realizes that an important piece of information is missing from the current highly efficient suite of experienced designer displays; there is no display to directly show the balance of field linear third order coma with field cubed third order aperture coma even though there is such a display for astigmatism and distortion. This is primarily due to the existence of the Coddington close skew raytrace, which allows quantifying the astigmatism by tracing only two real rays, thereby fitting within the requirements for efficiency from the early days of optical design, where ray trace speed was 5 minutes per ray surface versus modern ray trace speeds that exceed 10,000,000 ray surfaces per second. No equally efficient method was developed for coma, even though it is clearly useful. With the full field display, the balance of lower order field coma with higher order in field coma is directly illustrated by the display of the Fringe Zernike pair Z₇, Z₈. With a judicious choice of scaled and oriented symbol for each Zernike term, or pair, a circle for spherical aberration, a combination of an oriented open triangle and circle (ice cream cone) for coma, and an undirected line without end arrows for astigmatism, these displays take on many of the useful characteristics of a 2D full field spot diagram. John Rogers has recently explored a variation on the term by term full field display where multiple aberration types are simultaneously displayed with the dominant term also dominating the information content in its local space in the field of view [7].

Other researchers have used Zernike polynomials for describing field aberrations resulting from the misalignment of rotationally symmetric optical systems. Over ten years ago, McLeod [8] used Zernike polynomials to describe field astigmatism aberration resulting from misalignment between the primary and secondary mirrors of a Ritchey-Chretien telescope. Rakich [9] describe using the plate method [10] for simplifying the third-order analysis of optical systems and uses Zernike polynomials to resolve individual plate contributions to the system aberration into the coefficients for coma and astigmatism in Zernike terms. Noethe and Guisard [11] present measurements of the field astigmatism for the VLT, which are the coefficients of their Zernike polynomials Z₄ and Z₅ (our Z₅ and Z₆). Matsuyama and Ujike [12] have developed “functions that are orthogonal to each other and expressed by a simple combination of Zernike function(s) of pupil coordinates and Zernike function(s) of field coordinates.” Within the field of astronomical telescopes, prior work stopped at third order, mainly because the practical limitations of signal to noise ratio generally limit the usefulness of higher order aberrations in the astronomical alignment context. The new generation of relatively wide-field atmospherically corrected systems under development will require expanding to higher order. We derive here the explicit functional form for the wavefront aberration expansion in Zernike polynomials of the pupil coordinates having expansion coefficients as functions of field and the wavefront aberration function’s W-coefficients through 6th order plus 8th order spherical in wavefront aberration.

As an overview, in Section 2 the H.H. Hopkins formulation of the wave aberration function used by aberration theorists working in the field of optical design is introduced and extended first to accommodate field points located anywhere in the field of view and second to change the reference sphere to result in a measurement metric related to the minimum RMS wavefront error. This is followed by the introduction of an expression for the Fringe Zernike polynomial and the corresponding coefficients. Section 3 presents three examples that illustrate the effectiveness of this approach where one case shows how one can readily determine whether aberrations above sixth order in wavefront are beginning to dominate towards the edges of the aperture and/or field of view. In this case it is also shown how to use this information to estimate the coefficients for the next order(s) of aberration within the type being displayed. Section 4 provides a summary and conclusions.

2. Adding field dependence to the aperture dependent Zernike polynomial description of the wave aberration function

The scalar form of the wave aberration function for a rotationally symmetric optical system as a generalization of the fourth order form presented in Hopkins [3] is

W = \sum_{j} \sum_{p}^{\infty} \sum_{n}^{\infty} \sum_{m}^{\infty} {(W_{k l m})}_{j} H^{k} ρ^{l} \cos^{m} (θ - ϕ),

where

k = 2 p + m

and

l = 2 n + m

are a result of the conditions for rotational and mirror invariance, which the aberration function must satisfy for rotationally symmetric systems. In Eq. (1), the sum over j is a sum over all surfaces,

W_{k l m}

are the expansion coefficients, H is the normalized radius to the field point, ρ is the normalized radius to the pupil point, θ is the angle between the field point vector

\vec{H}

and the x-axis, and ϕ is the angle between the pupil point vector

\vec{ρ}

and the x-axis. Here we have adopted the extension proposed by Shack [13] of allowing the field of view to be studied in not one, as has been done historically, but rather in two dimensions, as illustrated in Fig. 1 . The resulting expression through sixth order in wavefront is then, as an extension to Eq. (77) in Hopkins [3],

\begin{array}{l} W = Δ W_{20} ρ^{2} + Δ W_{11} H ρ \cos (θ - ϕ) + W_{040} ρ^{4} + W_{131} H ρ^{3} \cos (θ - ϕ) \\ + W_{220 S} H^{2} ρ^{2} + W_{222} H^{2} ρ^{2} \cos^{2} (θ - ϕ) + W_{311} H^{3} ρ \cos (θ - ϕ) \\ + W_{060} ρ^{6} + W_{151} H ρ^{5} \cos (θ - ϕ) + W_{240 S} H^{2} ρ^{4} + W_{242} H^{2} ρ^{4} \cos^{2} (θ - ϕ) \\ + W_{331 S} H^{3} ρ^{3} \cos (θ - ϕ) + W_{333} H^{3} ρ^{3} \cos^{3} (θ - ϕ) + W_{420 S} H^{4} ρ^{2} + W_{422} H^{4} ρ^{2} \cos^{2} (θ - ϕ) \\ + W_{511} H^{5} ρ \cos (θ - ϕ) + W_{080} ρ^{8} . \end{array}

A terminology commonly used in optical design to identify this term set is provided in Table 1 of the Appendix.

Fig. 1 The aperture and field vectors placed in the coordinate system to be used for the wave aberration contribution in terms of Zernike polynomials.

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To further position the comparison to the corresponding Zernike polynomial form of the wavefront expansion, the terms in cosine raised to a power, cos^m(θ – ϕ), used in the expansion in Eq. (2), can be rewritten using standard trigonometric identities into expressions involving cos[m(θ – ϕ)]. This conversion results in the following modifications of Eq. (2)

\begin{array}{l} W = Δ W_{20} ρ^{2} + Δ W_{11} H ρ \cos (θ - ϕ) + W_{040} ρ^{4} + W_{131} H ρ^{3} \cos (θ - ϕ) \\ + W_{220 M} H^{2} ρ^{2} + \frac{1}{2} W_{222} H^{2} ρ^{2} \cos (2 θ - 2 ϕ) + W_{311} H^{3} ρ \cos (θ - ϕ) \\ + W_{060} ρ^{6} + W_{151} H ρ^{5} \cos (θ - ϕ) + W_{240 M} H^{2} ρ^{4} + \frac{1}{2} W_{242} H^{2} ρ^{4} \cos (2 θ - 2 ϕ) \\ + W_{331 M} H^{3} ρ^{3} \cos (θ - ϕ) + \frac{1}{4} W_{333} H^{3} ρ^{3} co s (3 θ - 3 ϕ) + W_{420 M} H^{4} ρ^{2} + \frac{1}{2} W_{422} H^{4} ρ^{2} \cos (2 θ - 2 ϕ) \\ + W_{511} H^{5} ρ \cos (θ - ϕ) + W_{080} ρ^{8}, \end{array}

where in Eq. (3) we have made use of the relations Eq. (4) through Eq. (7) to give the wave aberration expansion in terms of the paraxial based wave aberration coefficients of H.H. Hopkins in a form that can be used throughout a two dimensional field of view.

W_{220 M} = W_{220 S} + \frac{1}{2} W_{222} .

W_{331 M} = W_{331 S} + \frac{3}{4} W_{333} .

W_{240 M} = W_{240 S} + \frac{1}{2} W_{242} .

W_{420 M} = W_{420 S} + \frac{1}{2} W_{422} .

Turning now to the wavefront aberration function commonly used in commercial interferometry, a Zernike polynomial set for a circular aperture is given by

Z_{n}^{\pm m} (ρ, ϕ) = Z_{j} = R_{n}^{m} (ρ) {\begin{matrix} \cos (m ϕ) f o r + m \\ \sin (m ϕ) f o r - m \end{matrix},

where m is a positive (or zero) integer, and

R_{n}^{m} (ρ)

is the radial factor given by

R_{n}^{m} (ρ) = \sum_{s = 0}^{(n - m) / 2} \frac{{(- 1)}^{s} (n - s)!}{s! (\frac{n + m}{2} - s)! (\frac{n - m}{2} - s)!} ρ^{n - 2 s} .

The norm of a Zernike polynomial is then given by

N_{n m} = {| Z_{n}^{\pm m} (ρ, ϕ) |}^{2} = \int_{0}^{2 π} \int_{0}^{1} Z_{n}^{\pm m} (ρ, ϕ) Z_{n}^{\pm m} (ρ, ϕ) ρ d ρ d ϕ = \frac{π (1 + δ_{0 m})}{2 (n + 1)} .

Table 2 in the Appendix provides a list of the Zernike polynomials to be used in the wavefront aberration function expansion in both standard and Fringe notation.

Since the Zernike polynomials are orthogonal and complete over the unit radius circle, we can expand the wavefront aberration function W in terms of the Zernike polynomials. The coefficients of this expansion are calculated as

A_{n}^{\pm m} (H, θ) = \frac{1}{N_{n m}^{}} \int_{0}^{2 π} \int_{0}^{1} W (H, θ, ρ, ϕ) Z_{n}^{\pm m} (ρ, ϕ) ρ d ρ d ϕ .

where N_{n m} is the norm for the Zernike polynomial

Z_{n}^{\pm m} (ρ, ϕ)

.

Using Eqs. (3,8,9,10) in Eq. (11) we can now write the wavefront aberration function as

\begin{array}{l} W = (\frac{1}{2} Δ W_{20} + \frac{1}{3} W_{040} + \frac{1}{4} W_{060} + \frac{1}{5} W_{080} + \frac{1}{2} W_{220 M} H^{2} + \frac{1}{3} W_{240 M} H^{2} + \frac{1}{2} W_{420 M} H^{4}) [Z_{0}^{0} (ρ, ϕ)] \\ + (Δ W_{11} + \frac{2}{3} W_{131} + \frac{1}{2} W_{151} + W_{311} H^{2} + \frac{2}{3} W_{331 M} H^{2} + W_{511} H^{4}) H \cos (θ) [Z_{1}^{1} (ρ, ϕ)] \\ + (Δ W_{11} + \frac{2}{3} W_{131} + \frac{1}{2} W_{151} + W_{311} H^{2} + \frac{2}{3} W_{331 M} H^{2} + W_{511} H^{4}) H \sin (θ) [Z_{1}^{- 1} (ρ, ϕ)] \\ + (\frac{1}{2} Δ W_{20} + \frac{1}{2} W_{040} + \frac{9}{20} W_{060} + \frac{2}{5} W_{080} + \frac{1}{2} W_{220 M} H^{2} + \frac{1}{2} W_{240 M} H^{2} + \frac{1}{2} W_{420 M} H^{4}) [Z_{2}^{0} (ρ, ϕ)] \\ + (\frac{1}{2} W_{222} + \frac{3}{8} W_{242} + \frac{1}{2} W_{422} H^{2}) H^{2} \cos (2 θ) [Z_{2}^{2} (ρ, ϕ)] + (\frac{1}{2} W_{222} + \frac{3}{8} W_{242} + \frac{1}{2} W_{422} H^{2}) H^{2} \sin (2 θ) [Z_{2}^{- 2} (ρ, ϕ)] \\ + (\frac{1}{3} W_{131} + \frac{2}{5} W_{151} + \frac{1}{3} W_{331 M} H^{2}) H \cos (θ) [Z_{3}^{1} (ρ, ϕ)] + (\frac{1}{3} W_{131} + \frac{2}{5} W_{151} + \frac{1}{3} W_{331 M} H^{2}) H \sin (θ) [Z_{3}^{- 1} (ρ, ϕ)] \\ + (\frac{1}{6} W_{040} + \frac{1}{4} W_{060} + \frac{2}{7} W_{080} + \frac{1}{6} W_{240 M} H^{2}) [Z_{4}^{0} (ρ, ϕ)] \\ + (\frac{1}{4} W_{333}) H^{3} \cos (3 θ) [Z_{3}^{3} (ρ, ϕ)] + (\frac{1}{4} W_{333}) H^{3} \sin (3 θ) [Z_{3}^{- 3} (ρ, ϕ)] \\ + (\frac{1}{8} W_{242}) H^{2} \cos (2 θ) [Z_{4}^{2} (ρ, ϕ)] + (\frac{1}{8} W_{242}) H^{2} \sin (2 θ) [Z_{4}^{- 2} (ρ, ϕ)] \\ + (\frac{1}{10} W_{151}) H \cos (θ) [Z_{5}^{1} (ρ, ϕ)] + (\frac{1}{10} W_{151}) H \sin (θ) [Z_{5}^{- 1} (ρ, ϕ)] \\ + (\frac{1}{20} W_{060} + \frac{1}{10} W_{080}) [Z_{6}^{0} (ρ, ϕ)] + (\frac{1}{70} W_{080}) [Z_{8}^{0} (ρ, ϕ)] . \end{array}

This result, by inspection, provides the field (H, θ) dependence of the Zernike coefficients, which are summarized in Table 3 of the Appendix.

With the realization that $\sin (m θ) = 0$ when m = 0 we see that the result can be written in the form

W = \dots + g (H^{2}) H^{m} \cos (m θ) Z_{n}^{+ m} + g (H^{2}) H^{m} \sin (m θ) Z_{n}^{- m} + \dots,

where

g (H^{2}) = \sum_{q = 0}^{} C_{q} H^{2 q},

and the

C_{q}

include the W-coefficients.

Because the Zernike polynomials occur in orthogonal pairs, we can associate an orientation angle a with each pair defined with the equation

α = \arg (g (H^{2}) H^{m} \cos (m θ) + i g (H^{2}) H^{m} \sin (m θ)),

where arg(.) is the argument of the complex number and i is the square root of minus one.

Equation (12) opens an important pathway that now connects the field dependent wave aberration of optical design with the field independent formulation of Zernike polynomials that have been used in optical testing. This bridge now brings essential new insights into optical alignment and optical testing and becomes particularly effective when working to develop an understanding into the emerging class of misalignment-induced aberration fields.

3. Examples

Three telescope models are presented as qualitative examples. The first telescope is a telescope reported by James G. Baker [14], shown in Fig. 2(a) . Full field displays based on real raytrace using optical design software CODE V^® were produced for Fringe Zernikes Z₅/Z₆, Z₇/Z₈, and Z₉. A Matlab^® function was written to compute the Fringe Zernike coefficients using Eq. (12). The W-expansion coefficients used for the calculations were produced using the CODE V^® FORDER macro [15]. The Matlab^® function produced plots for the field dependent coefficients of Z₅/Z₆, Z₇/Z₈, Z₉, which are provided in Fig. 3 adjacent to the corresponding real ray based full field display.

Fig. 2 Optical layouts of (a) Baker [14] (Media 1) and (b) James Webb [16] (Media 2) telescope models. Arrows indicate the location of the aperture stops.

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Fig. 3 Qualitative comparison of a real ray based computation of the Zernike coefficients from a grid of field points to the analytic wave aberration function prediction of the field dependence for a Baker telescope model. (a) The left plots are real ray based fitting of the exit pupil wavefront to the Fringe Zernike polynomials computed on a grid of points (normalized field range ± 1°), while (b) the right plots are based on analytic calculations using the Zernike coefficients in Eq. (12) (wavefront expanded through 6th order) with a normalized field range ± 1°. Top row: Z₅ + Z₆ (Astigmatism). Middle row: Z₇ + Z₈ (Coma). Bottom row: Z₉ (Spherical).

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The second model is an early version of the James Webb telescope [16], shown in Fig. 2(b). The qualitative comparison between the full field plots produced by real raytrace and fitting to the Fringe Zernike polynomial are compared to the comparable computation based on Eq. (12) are presented in Fig. 4 . This optical system has its field view strongly limited by 5th order astigmatism, a field quartic aberration.

Fig. 4 Qualitative comparison of a real ray based computation of the Zernike coefficients from a grid of field points to the analytic wave aberration function prediction of the field dependence for a James Webb telescope model. (a) The left plots are real ray based fitting of the exit pupil wavefront to the Fringe Zernike polynomials computed on a grid of points (normalized field range ± 0.25°), while (b) the center plots are based on analytic calculations using Eq. (12) (wavefront expanded through 6th order) with a normalized field range ± 0.25°. The right plot (c) shows the result for astigmatism expanded through 8th order (Eq. (16)) providing a far better qualitative match to the real ray trace results. Top row: Z₅ + Z₆ (Astigmatism). Middle row: Z₇ + Z₈ (Coma). Bottom row: Z₉ (Spherical).

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The third model is a recent proprietary 3-mirror telescope design that is included here to demonstrate the comatic field balance discussed earlier. Again the coefficients that are displayed are astigmatism (Z₅/Z₆), coma (Z₇/Z₈), and spherical aberration Z₉. The comparison between the full field plots produced by a field point by field point fitting and accumulation of the wavefront data to the Fringe Zernike polynomial using real raytrace data in CODE V^® and the Matlab^® function that includes the analytic field dependence are presented in Fig. 5 .

Fig. 5 Qualitative comparison of a real ray based computation of the Zernike coefficients from a grid of field points to the analytic wave aberration function prediction of the field dependence for a proprietary telescope model. (a) The left plots are real ray based fitting of the exit pupil wavefront to the Fringe Zernike polynomials computed on a grid of points (normalized field range ± 15°), while (b) the center plots are based on analytic calculations using the Zernike coefficients in Eq. (12) (wavefront expanded through 6th order) with a normalized field range ± 15°. The right plots (c) show the results for the wavefront expanded through 8th order (Eq. (16)). Top row: Z₅ + Z₆ (Astigmatism). Middle row: Z₇ + Z₈ (Coma). Bottom row: Z₉ (Spherical).

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It is important to realize that except in the case of large 2-mirror astronomical telescopes, where the limiting aberrations are strictly 3rd order, what are commonly referred to as the “low order” Zernike terms, of Z₅/Z₆, Z₇/Z₈, Z₉, are in fact not low order when field dependence is included. These terms contain both the low and the higher order dependence with field.

For all the (Z₅ + Z₆) (astigmatism) plots, the orientation angle calculated from Eq. (15) is further divided by 2. This is a result of the way that CODE V^® displays the (Z₅ + Z₆) plots and is not an omission of Eq. (15). Further, for the James Webb and the proprietary telescope models, it was found that the astigmatism plots could not be matched without expanding the wavefront aberration function through 8th order, which provided additional field dependent terms. And finally for the proprietary telescope model, it was found that the coma (Z₇ + Z₈) and spherical (Z₉) plots could not be well matched without expanding the wavefront aberration function through 8th order, which provided higher order field dependent (H) terms than what is provided in Eq. (12). For these aberrations, selecting θ = 0, the form of the Zernike expansion of the wavefront aberration function W includes

\begin{array}{l} W = ... + (w_{5, 2} H^{2} + w_{5, 4} H^{4} + w_{5, 6} H^{6}) [Z_{5}^{} (ρ, ϕ)] + (w_{7, 1} H^{} + w_{7, 3} H^{3} + w_{7, 5} H^{5}) [Z_{7}^{} (ρ, ϕ)] + \\ + (w_{9, 0} + w_{9, 4} H^{2} + w_{9, 4} H^{4}) [Z_{9}^{} (ρ, ϕ)] + ... . \end{array}

The w_i,j coefficients were determined by a least squares fit to the Zernike coefficients that CODE V^® produced over 11 field points along the θ = 0 axis. In using Eq. (16) for performing the least squares fit of the data, it is worth noting that the least squares fit is not a fit to the Zernike polynomials. It is a fit of the data to the expansion coefficient functions in field parameter H. Therefore, in performing this fit we do not need to be concerned with maintaining the orthogonality property of the Zernike polynomials over a discrete set of points.

The remaining plots show qualitative agreement between the CODE V^® and Matlab^® implementation of the corresponding coefficient equations of Eq. (12). The need to add higher order field (H) terms to the Zernike expansion coefficient functions indicates that these systems are not limited by 6th order wavefront aberrations.

While not particularly powerful in the rotationally symmetric form, even here it can be used to more readily visualize important aberration balancing points in the field of view. As seen in Fig. 3, for a well corrected lens with performance that is limited by a combination of 3rd and 5th order aberrations, coma and spherical are balanced at the edge of the format, a point not readily seen with the current set of optical design evaluation tools, for predominantly historical reasons. In fact, and in many ways this is the point, in this design, executed by James G. Baker, not only is the spherical aberration balanced using an offsetting mixture of 3rd and 5th order field constant spherical aberration, the design also balances the field quadratic oblique spherical aberration (often called secondary astigmatism by the mathematicians but not by the lens design community), which is seen clearly now in the coefficients of the Z₉ term. In Fig. 3, the right set of full field displays is in fact computed analytically from the wave aberration coefficients while the left set are the same calculations using simply a real ray based ray trace and an aperture only formulation of the Zernike polynomial fit of the exit pupil wavefront on a grid of field points.

4. Conclusion

This paper reports on a methodology that gives explicitly the field dependence to the coefficients of the Fringe Zernike polynomial expansion of the wavefront aberration function, typically used as a means of representing an as-fabricated optical surface. CODE V^® models of three optical systems (astronomical telescopes) were used to qualitatively compare the field dependence of the Zernike expansion coefficients of the wavefront aberration function using data produced by real raytracing and by the equations presented. When the equations resulted in a poor match to the real raytracing, we have shown that extending the equations to the next higher aberration order in field parameter resulted in good qualitative agreement.

With this bridge established between optical design and optical testing, a rich new environment for studying and understanding the misalignment induced aberrations of optical systems is emerging that is proving to be valuable to extracting key insights into methods for effectively aligning high performance optical systems. Therefore, future research and papers will extend the equations develop to rotationally symmetric optical systems that have one or more decentered and/or tilted optical surfaces and to the development of alignment procedures for such systems. Additionally, since many astronomical telescope designs have central obscurations, this work will be extended to annular Zernike polynomials.

APPENDIX

Table 1. The wave aberration coefficients (types) of a rotationally symmetric optical system and their common names in lens design [3].

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Table 2. Subset of the Zernike polynomials.

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Table 3. Field dependence of the Zernike coefficients in terms of wave aberration coefficients with field dependence.

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Acknowledgments

R. W. Gray was supported by a NASA Graduate Student Researchers Program (GSRP) fellowship and the II-VI Foundation. This research also benefitted from support by the NYSTAR Foundation (C050070) and the National Science Foundation (EECS-1002179). We thank Synopsys, Inc. for the educational license for CODE V^®.

References and links

1. The Fringe Zernike polynomial was developed by John Loomis at the University of Arizona, Optical Sciences Center in the 1970s, and is described on page C-8 of the CODE V® Version 10.4 Reference Manual (Synopsys, Inc.) (2012)

2. ISO 10110–5:1996(E), “Optics and optical instruments – Preparation of drawings for optical elements and systems – Part 5: Surface form tolerances.

3. H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon Press), p. 48 (1950).

4. M. P. Rimmer, C. M. King, and D. G. Fox, “Computer program for the analysis of interferometric test data,” Appl. Opt. 11(12), 2790–2796 (1972). [CrossRef] [PubMed]

5. J. Sasián, “Theory of sixth-order wave aberrations,” Appl. Opt. 49(16), D69–D95 (2010). [CrossRef] [PubMed]

6. K. P. Thompson, “Beyond optical Design: interaction between the lens designer and the real world,” The International Optics Design Conference, Proc. SPIE 554, 426 (1985).

7. J. R. Rogers, personal communication, (2012).

8. B. A. McLeod, “Collimation of fast wide-field telescopes,” PASP 108, 217–219 (1996). [CrossRef]

9. A. Rakich, “Calculation of third-order misalignment aberrations with the Optical Plate diagram,” SPIE-OSA 7652 (2010).

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11. L. Noethe and S. Guisard, “Final alignment of the VLT,” Proc. SPIE 4003 (2000).

12. T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev. 11(4), 199–207 (2004). [CrossRef]

13. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope on its aberration field,” Proc. SPIE 251, 146–155 (1980).

14. J. G. Baker, I. King, G. H. Conant, Jr., W. R. Angell, Jr., and E. Upton, “Technical report no. 2 – The utilization of automatic calculating machinery in the field of optical design,” May 31, 1952.

15. M. R. Rimmer, Optical Aberration Coefficients, Master Thesis, University of Rochester, 1963.

16. J. C. Mather, ed., Astronomical Telescopes and Instrumentation Glasgow, (SPIE), 5487, SPIE, Bellingham, WA (2004).

H ^k	ρ ^l	cos ^m(φ)	Field Dependence	Coeff.	Transverse Aberration Name
	2	0	Constant	ΔW₂₀	Focus, Reference Sphere
	1	1	Linear	ΔW₁₁	Tilt, Reference Sphere
4th Order Wave Aberration Types
0	4	0	Constant	W₀₄₀	3rd order spherical
1	3	1	Linear	W₁₃₁	3rd order coma
2	2	0	Quadratic	W_220SW_220M	3rd order sagittal focal surface 3rd order medial focal surface
2	2	2	Quadratic	W₂₂₂	3rd order astigmatism
3	1	1	Cubic	W₃₁₁	3rd order distortion (does not degrade imagery)
6th Order Wave Aberration Types
0	6	0	Constant	W₀₆₀	5th order spherical
1	5	1	Linear	W₁₅₁	Field linear 5th order coma
2	4	0	Quadratic	W_240S W_240M	5th order sagittal focal surface due to oblique spherical 5th order medial focal surface due to oblique spherical
2	4	2	Quadratic	W₂₄₂	5th order oblique spherical
3	3	1	Cubic	W_331S W_331M	5th order field-cubed 3rd order coma 5th order medial field-cubed 3rd order coma
3	3	3	Cubic	W₃₃₃	5th order elliptical coma (Trefoil in testing)
4	2	0	Quartic	W_420S W_420M	5th order sagittal astigmatic focal surface 5th order medial astigmatic focal surface
4	2	2	Quartic	W₄₂₂	5th order astigmatism
5	1	1	Quintic	W₅₁₁	5th order distortion (does not degrade imagery)
8th Order Wave Aberration Type
0	8	0	Constant	W₀₈₀	7th order spherical aberration

Fringe  $Z_{j} (ρ, ϕ)$	Norm  $N_{n m}^{}$	Zernike Polynomial	Standard  $Z_{n}^{\pm m} (ρ, ϕ)$
$Z_{1} (ρ, ϕ)$	$π$	$1$	$Z_{0}^{0} (ρ, ϕ)$
$Z_{2} (ρ, ϕ)$	$\frac{π}{4}$	$ρ \cos (ϕ)$	$Z_{1}^{1} (ρ, ϕ)$
$Z_{3} (ρ, ϕ)$	$\frac{π}{4}$	$ρ \sin (ϕ)$	$Z_{1}^{- 1} (ρ, ϕ)$
$Z_{4} (ρ, ϕ)$	$\frac{π}{3}$	$2 ρ^{2} - 1$	$Z_{2}^{0} (ρ, ϕ)$
$Z_{5} (ρ, ϕ)$	$\frac{π}{6}$	$ρ^{2} \cos (2 ϕ)$	$Z_{2}^{2} (ρ, ϕ)$
$Z_{6} (ρ, ϕ)$	$\frac{π}{6}$	$ρ^{2} \sin (2 ϕ)$	$Z_{2}^{- 2} (ρ, ϕ)$
$Z_{7} (ρ, ϕ)$	$\frac{π}{8}$	$(3 ρ^{3} - 2 ρ) \cos (ϕ)$	$Z_{3}^{1} (ρ, ϕ)$
$Z_{8} (ρ, ϕ)$	$\frac{π}{8}$	$(3 ρ^{3} - 2 ρ) \sin (ϕ)$	$Z_{3}^{- 1} (ρ, ϕ)$
$Z_{9} (ρ, ϕ)$	$\frac{π}{5}$	$6 ρ^{4} - 6 ρ^{2} + 1$	$Z_{4}^{0} (ρ, ϕ)$
$Z_{10} (ρ, ϕ)$	$\frac{π}{8}$	$ρ^{3} \cos (3 ϕ)$	$Z_{3}^{3} (ρ, ϕ)$
$Z_{11} (ρ, ϕ)$	$\frac{π}{8}$	$ρ^{3} \sin (3 ϕ)$	$Z_{3}^{- 3} (ρ, ϕ)$
$Z_{12} (ρ, ϕ)$	$\frac{π}{10}$	$(4 ρ^{4} - 3 ρ^{2}) \cos (2 ϕ)$	$Z_{4}^{2} (ρ, ϕ)$
$Z_{13} (ρ, ϕ)$	$\frac{π}{10}$	$(4 ρ^{4} - 3 ρ^{2}) \sin (2 ϕ)$	$Z_{4}^{- 2} (ρ, ϕ)$
$Z_{14} (ρ, ϕ)$	$\frac{π}{12}$	$(10 ρ^{5} - 12 ρ^{3} + 3 ρ) \cos (ϕ)$	$Z_{5}^{1} (ρ, ϕ)$
$Z_{15} (ρ, ϕ)$	$\frac{π}{12}$	$(10 ρ^{5} - 12 ρ^{3} + 3 ρ) \sin (ϕ)$	$Z_{5}^{- 1} (ρ, ϕ)$
$Z_{16} (ρ, ϕ)$	$\frac{π}{7}$	$20 ρ^{6} - 30 ρ^{4} + 12 ρ^{2} - 1$	$Z_{6}^{0} (ρ, ϕ)$
$Z_{25} (ρ, ϕ)$	$\frac{π}{9}$	$70 ρ^{8} - 140 ρ^{6} + 90 ρ^{4} - 20 ρ^{2} + 1$	$Z_{8}^{0} (ρ, ϕ)$

Fringe ^Zernike	$A_{n}^{m}$	Expansion Coefficient Function	Zernike Aberr. Name
Reference Sphere Parameters
Z₁(ρ,ϕ)	$A_{0}^{0}$	$\begin{array}{l} \frac{1}{2} Δ W_{20} + \frac{1}{3} W_{040} + \frac{1}{4} W_{060} + \frac{1}{5} W_{080} + \frac{1}{2} W_{220 M} H^{2} \\ + \frac{1}{3} W_{240 M} H^{2} + \frac{1}{2} W_{420 M} H^{4} \end{array}$	Piston
Z₂(ρ,ϕ) Z₃(ρ,ϕ)	$A_{1}^{+ 1}$   $A_{1}^{- 1}$	$(Δ W_{11} + \frac{2}{3} W_{131} + \frac{1}{2} W_{151} + W_{311} H^{2} + \frac{2}{3} W_{331 M} H^{2} + W_{511} H^{4}) H \cos (θ)$ $(Δ W_{11} + \frac{2}{3} W_{131} + \frac{1}{2} W_{151} + W_{311} H^{2} + \frac{2}{3} W_{331 M} H^{2} + W_{511} H^{4}) H \sin (θ)$	Tilt (Centroid)
Z₄(ρ,ϕ)	$A_{2}^{0}$	$\begin{array}{l} \frac{1}{2} Δ W_{20} + \frac{1}{2} W_{040} + \frac{9}{20} W_{060} + \frac{2}{5} W_{080} + \frac{1}{2} W_{220 M} H^{2} \\ + \frac{1}{2} W_{240 M} H^{2} + \frac{1}{2} W_{420 M} H^{4} \end{array}$	Focus (Field Curvature)
4th Order Wave Aberration Types
Z₅(ρ,ϕ) Z₆(ρ,ϕ)	$A_{2}^{+ 2}$   $A_{2}^{- 2}$	$(\frac{1}{2} W_{222} + \frac{3}{8} W_{242} + \frac{1}{2} W_{422} H^{2}) H^{2} \cos (2 θ)$   $(\frac{1}{2} W_{222} + \frac{3}{8} W_{242} + \frac{1}{2} W_{422} H^{2}) H^{2} \sin (2 θ)$	Astigmatism
Z₇(ρ,ϕ) Z₈(ρ,ϕ)	$A_{3}^{+ 1}$   $A_{3}^{- 1}$	$(\frac{1}{3} W_{131} + \frac{2}{5} W_{151} + \frac{1}{3} W_{331 M} H^{2}) H \cos (θ)$   $(\frac{1}{3} W_{131} + \frac{2}{5} W_{151} + \frac{1}{3} W_{331 M} H^{2}) H \sin (θ)$	Coma
Z₉(ρ,ϕ)	$A_{4}^{0}$	$\frac{1}{6} W_{040} + \frac{1}{4} W_{060} + \frac{2}{7} W_{080} + \frac{1}{6} W_{240 M} H^{2}$	Spherical
6th Order Wave Aberration Types
Z₁₀(ρ,ϕ) Z₁₁(ρ,ϕ)	$A_{3}^{+ 3}$   $A_{3}^{- 3}$	$(\frac{1}{4} W_{333}) H^{3} \cos (3 θ)$   $(\frac{1}{4} W_{333}) H^{3} \sin (3 θ)$	Elliptical Coma (Trefoil)
Z₁₂(ρ,ϕ) Z₁₃(ρ,ϕ)	$A_{4}^{+ 2}$   $A_{4}^{- 2}$	$(\frac{1}{8} W_{242}) H^{2} \cos (2 θ)$   $(\frac{1}{8} W_{242}) H^{2} \sin (2 θ)$	Oblique Spherical
Z₁₄(ρ,ϕ) Z₁₅(ρ,ϕ)	$A_{5}^{+ 1}$   $A_{5}^{- 1}$	$(\frac{1}{10} W_{151}) H \cos (θ)$   $(\frac{1}{10} W_{151}) H \sin (θ)$	5th Coma
Z₁₆(ρ,ϕ)	$A_{6}^{0}$	$\frac{1}{20} W_{060} + \frac{1}{10} W_{080}$	5th Spherical
8th Order Wave Aberration Type
Z₂₅(ρ,ϕ)	$A_{8}^{0}$	$\frac{1}{70} W_{080}$	7th Spherical

H ^k	ρ ^l	cos ^m(φ)	Field Dependence	Coeff.	Transverse Aberration Name
	2	0	Constant	ΔW₂₀	Focus, Reference Sphere
	1	1	Linear	ΔW₁₁	Tilt, Reference Sphere
4th Order Wave Aberration Types
0	4	0	Constant	W₀₄₀	3rd order spherical
1	3	1	Linear	W₁₃₁	3rd order coma
2	2	0	Quadratic	W_220SW_220M	3rd order sagittal focal surface 3rd order medial focal surface
2	2	2	Quadratic	W₂₂₂	3rd order astigmatism
3	1	1	Cubic	W₃₁₁	3rd order distortion (does not degrade imagery)
6th Order Wave Aberration Types
0	6	0	Constant	W₀₆₀	5th order spherical
1	5	1	Linear	W₁₅₁	Field linear 5th order coma
2	4	0	Quadratic	W_240S W_240M	5th order sagittal focal surface due to oblique spherical 5th order medial focal surface due to oblique spherical
2	4	2	Quadratic	W₂₄₂	5th order oblique spherical
3	3	1	Cubic	W_331S W_331M	5th order field-cubed 3rd order coma 5th order medial field-cubed 3rd order coma
3	3	3	Cubic	W₃₃₃	5th order elliptical coma (Trefoil in testing)
4	2	0	Quartic	W_420S W_420M	5th order sagittal astigmatic focal surface 5th order medial astigmatic focal surface
4	2	2	Quartic	W₄₂₂	5th order astigmatism
5	1	1	Quintic	W₅₁₁	5th order distortion (does not degrade imagery)
8th Order Wave Aberration Type
0	8	0	Constant	W₀₈₀	7th order spherical aberration

Fringe  $Z_{j} (ρ, ϕ)$	Norm  $N_{n m}^{}$	Zernike Polynomial	Standard  $Z_{n}^{\pm m} (ρ, ϕ)$
$Z_{1} (ρ, ϕ)$	$π$	$1$	$Z_{0}^{0} (ρ, ϕ)$
$Z_{2} (ρ, ϕ)$	$\frac{π}{4}$	$ρ \cos (ϕ)$	$Z_{1}^{1} (ρ, ϕ)$
$Z_{3} (ρ, ϕ)$	$\frac{π}{4}$	$ρ \sin (ϕ)$	$Z_{1}^{- 1} (ρ, ϕ)$
$Z_{4} (ρ, ϕ)$	$\frac{π}{3}$	$2 ρ^{2} - 1$	$Z_{2}^{0} (ρ, ϕ)$
$Z_{5} (ρ, ϕ)$	$\frac{π}{6}$	$ρ^{2} \cos (2 ϕ)$	$Z_{2}^{2} (ρ, ϕ)$
$Z_{6} (ρ, ϕ)$	$\frac{π}{6}$	$ρ^{2} \sin (2 ϕ)$	$Z_{2}^{- 2} (ρ, ϕ)$
$Z_{7} (ρ, ϕ)$	$\frac{π}{8}$	$(3 ρ^{3} - 2 ρ) \cos (ϕ)$	$Z_{3}^{1} (ρ, ϕ)$
$Z_{8} (ρ, ϕ)$	$\frac{π}{8}$	$(3 ρ^{3} - 2 ρ) \sin (ϕ)$	$Z_{3}^{- 1} (ρ, ϕ)$
$Z_{9} (ρ, ϕ)$	$\frac{π}{5}$	$6 ρ^{4} - 6 ρ^{2} + 1$	$Z_{4}^{0} (ρ, ϕ)$
$Z_{10} (ρ, ϕ)$	$\frac{π}{8}$	$ρ^{3} \cos (3 ϕ)$	$Z_{3}^{3} (ρ, ϕ)$
$Z_{11} (ρ, ϕ)$	$\frac{π}{8}$	$ρ^{3} \sin (3 ϕ)$	$Z_{3}^{- 3} (ρ, ϕ)$
$Z_{12} (ρ, ϕ)$	$\frac{π}{10}$	$(4 ρ^{4} - 3 ρ^{2}) \cos (2 ϕ)$	$Z_{4}^{2} (ρ, ϕ)$
$Z_{13} (ρ, ϕ)$	$\frac{π}{10}$	$(4 ρ^{4} - 3 ρ^{2}) \sin (2 ϕ)$	$Z_{4}^{- 2} (ρ, ϕ)$
$Z_{14} (ρ, ϕ)$	$\frac{π}{12}$	$(10 ρ^{5} - 12 ρ^{3} + 3 ρ) \cos (ϕ)$	$Z_{5}^{1} (ρ, ϕ)$
$Z_{15} (ρ, ϕ)$	$\frac{π}{12}$	$(10 ρ^{5} - 12 ρ^{3} + 3 ρ) \sin (ϕ)$	$Z_{5}^{- 1} (ρ, ϕ)$
$Z_{16} (ρ, ϕ)$	$\frac{π}{7}$	$20 ρ^{6} - 30 ρ^{4} + 12 ρ^{2} - 1$	$Z_{6}^{0} (ρ, ϕ)$
$Z_{25} (ρ, ϕ)$	$\frac{π}{9}$	$70 ρ^{8} - 140 ρ^{6} + 90 ρ^{4} - 20 ρ^{2} + 1$	$Z_{8}^{0} (ρ, ϕ)$

An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems

Abstract

1. Introduction

2. Adding field dependence to the aperture dependent Zernike polynomial description of the wave aberration function

3. Examples

4. Conclusion

APPENDIX

Acknowledgments

References and links

Supplementary Material (2)

Cited By

Figures (5)

Tables (3)

Equations (16)

Optics Express