Standards for Reporting the Optical Aberrations of Eyes

S652 Journal of Refractive Surgery Volume 18 September/October 2002

ABSTRACTIn response to a perceived need in the vision

community, an OSA taskforce was formed at the1999 topical meeting on vision science and its appli-cations (VSIA-99) and charged with developing con-sensus recommendations on definitions, conven-tions, and standards for reporting of optical aber-rations of human eyes. Progress reports were pre-sented at the 1999 OSA annual meeting and at VSIA-2000 by the chairs of three taskforce subcommitteeson (1) reference axes, (2) describing functions, and(3) model eyes. [J Refract Surg 2002;18:S652-S660]

The recent resurgence of activity in visualoptics research and related clinical disciplines(eg, refractive surgery, ophthalmic lens

design, ametropia diagnosis) demands that thevision community establish common metrics, termi-nology, and other reporting standards for the speci-fication of optical imperfections of eyes. Currentlythere exists a plethora of methods for analyzing andrepresenting the aberration structure of the eye butno agreement exists within the vision community ona common, universal method for reporting results.In theory, the various methods currently in use bydifferent groups of investigators all describe thesame underlying phenomena and therefore it shouldbe possible to reliably convert results from one rep-resentational scheme to another. However, the prac-tical implementation of these conversion methods iscomputationally challenging, is subject to error, andreliable computer software is not widely available.All of these problems suggest the need for opera-

tional standards for reporting aberration data andto specify test procedures for evaluating the accura-cy of data collection and data analysis methods.

Following a call for participation1, approximately20 people met at VSIA-99 to discuss the proposal toform a taskforce that would recommend standardsfor reporting optical aberrations of eyes. The groupagreed to form three working parties that wouldtake responsibility for developing consensus recom-mendations on definitions, conventions and stan-dards for the following three topics: (1) referenceaxes, (2) describing functions, and (3) model eyes. Itwas decided that the strategy for Phase I of this pro-ject would be to concentrate on articulating defini-tions, conventions, and standards for those issueswhich are not empirical in nature. For example, sev-eral schemes for enumerating the Zernike polyno-mials have been proposed in the literature.Selecting one to be the standard is a matter ofchoice, not empirical investigation, and thereforewas included in the charge to the taskforce. On theother hand, issues such as the maximum number ofZernike orders needed to describe ocular aberra-tions adequately is an empirical question which wasavoided for the present, although the taskforce maychoose to formulate recommendations on suchissues at a later time. Phase I concluded at theVSIA-2000 meeting.

REFERENCE AXIS SELECTION

SummaryIt is the committee’s recommendation that the

ophthalmic community use the line-of-sight as thereference axis for the purposes of calculating andmeasuring the optical aberrations of the eye. Therationale is that the line-of-sight in the normal eyeis the path of the chief ray from the fixation point tothe retinal fovea. Therefore, aberrations measuredwith respect to this axis will have the pupil centeras the origin of a Cartesian reference frame.

Standards for Reporting the Optical Aberrationsof EyesLarry N. Thibos, PhD; Raymond A. Applegate, OD, PhD; James T. Schwiegerling, PhD;Robert Webb, PhD; VSIA Standards Taskforce Members

From the School of Optometry, Indiana University, Bloomington, IN(Thibos); College of Optometry, University of Houston, TX (Applegate);Department of Ophthalmology, University of Arizona, Tucson, AZ(Schweigerling); Schepens Research Institute, Boston, MA (Webb).

This is an expanded version of a report originally published as: ThibosLN, Applegate RA, Schwiegerling JT, Webb R. Standards for Reporting theOptical Aberrations of Eyes. In: Lakshminarayanan V, ed (Optical Societyof America, Washington, D.C., 2000). Vision Science and Its Applications.TOPS-35:232-244.

Secondary lines-of-sight may be similarly construct-ed for object points in the peripheral visual field.Because the exit pupil is not readily accessible inthe living eye whereas the entrance pupil is, thecommittee recommends that calculations for speci-fying the optical aberration of the eye be referencedto the plane of the entrance pupil.

BackgroundOptical aberration measurements of the eye from

various laboratories or within the same laboratoryare not comparable unless they are calculated withrespect to the same reference axis and expressed inthe same manner. This requirement is complicatedby the fact that, unlike a camera, the eye is a decen-tered optical system with non-rotationally symmet-ric components (Fig 1). The principle elements of theeye’s optical system are the cornea, pupil, and thecrystalline lens. Each can be decentered and tiltedwith respect to other components, thus rendering anoptical system that is typically dominated by comaat the foveola.

The optics discipline has a long tradition of spec-ifying the aberration of optical systems with respectto the center of the exit pupil. In a centered opticalsystem (eg, a camera, or telescope) using the centerof the exit pupil as a reference for measurement ofon-axis aberration is the same as measuring theoptical aberrations with respect to the chief rayfrom an axial object point. However, because theexit pupil is not readily accessible in the living eye,it is more practical to reference aberrations to theentrance pupil. This is the natural choice for objec-tive aberrometers which analyze light reflected fromthe eye.

Like a camera, the eye is an imaging device

designed to form an in-focus inverted image on ascreen. In the case of the eye, the imaging screen isthe retina. However, unlike film, the “grain” of theretina is not uniform over its extent. Instead, thegrain is finest at the foveola and falls off quickly asthe distance from the foveola increases. Conse-quently, when viewing fine detail, we rotate our eyesuch that the object of regard falls on the foveola(Fig 2). Thus, aberrations at the foveola have thegreatest impact on an individual’s ability to see finedetails.

Two traditional axes of the eye are centered onthe foveola, the visual axis and the line-of-sight, butonly the latter passes through the pupil center. Inobject space, the visual axis is typically defined asthe line connecting the fixation object point to theeye’s first nodal point. In image space, the visualaxis is the parallel line connecting the second nodalpoint to the center of the foveola (Fig. 3, left). In con-trast, the line-of-sight is defined as the (broken) linepassing through the center of the eye’s entrance andexit pupils connecting the object of regard to thefoveola (Fig. 3, right). The line-of-sight is equivalentto the path of the foveal chief ray and therefore isthe axis which conforms to optical standards. Thevisual axis and the line of sight are not the sameand in some eyes the difference can have a largeimpact on retinal image quality.2 For a review of theaxes of the eye see.3 (To avoid confusion, we notethat Bennett and Rabbetts4 re-define the visual axisto match the traditional definition of the line ofsight. The Bennett and Rabbetts definition is

Journal of Refractive Surgery Volume 18 September/October 2002 S653

Standards for Reporting Optical Aberrations of Eyes/Thibos et al

Figure 1. The cornea, pupil, and crystalline lens are decenteredand tilted with respect to each other, rendering the eye a decenteredoptical system that is different between individuals and eyes withinthe same individual.

Figure 2. An anatomical view of the macular region as viewed fromthe front and in cross section (below). a: foveola, b: fovea,c: parafoveal area, d: perifoveal area. (From Histology of the HumanEye by Hogan MJ, Alvarado JA, and Weddell JE. W.B. SaundersCompany Publishers, 1971, page 491.)

counter to the majority of the literature and is notused here.)

When measuring the optical properties of the eyefor objects which fall on the peripheral retina out-side the central fovea, a secondary line-of-sight maybe constructed as the broken line from object pointto center of the entrance pupil and from the centerof the exit pupil to the retinal location of the image.This axis represents the path of the chief ray fromthe object of interest and therefore is the appropri-ate reference for describing aberrations of theperipheral visual field.

METHODS FOR ALIGNING THE EYE DURING MEASUREMENT

SummaryThe committee recommends that instruments

designed to measure the optical properties of the eyeand its aberrations be aligned co-axially with theeye’s line-of-sight.

BackgroundThere are numerous ways to align the line of

sight to the optical axis of the measuring instru-ment. Here we present simple examples of an objec-tive method and a subjective method to achieveproper alignment.

Objective MethodIn the objective alignment method schematically

diagramed in Fig. 4, the experimenter aligns thesubject’s eye (which is fixating a small distant targeton the optical axis of the measurement system) tothe measurement system. Alignment is achieved bycentering the subject’s pupil (by adjusting a bite bar)on an alignment ring (eg, an adjustable diametercircle) which is co-axial with the optical axis of themeasurement system. This strategy forces the opti-cal axis of the measurement device to pass throughthe center of the entrance pupil. Since the fixationtarget is on the optical axis of the measurementdevice, once the entrance pupil is centered withrespect to the alignment ring, the line-of-sight is

co-axial with the optical axis of the measurementsystem.

Subjective MethodIn the subjective alignment method schematical-

ly diagramed in Figure 5, the subject adjusts theposition of their own pupil (using a bite bar) untiltwo alignment fixation points at different opticaldistances along and co-axial to the optical axis of themeasurement device are superimposed (similar toaligning the sights on rifle to a target). Note thatone or both of the alignment targets will be defo-cused on the retina. Thus the subject’s task is toalign the centers of the blur circles. Assuming thechief ray defines the centers of the blur circles foreach fixation point, this strategy forces the line ofsight to be co-axial with the optical axis of the mea-surement system. In a system with significantamounts of asymmetric aberration (eg, coma), thechief ray may not define the center of the blur circle.In practice, it can be useful to use the subjectivestrategy for preliminary alignment and the objec-tive method for final alignment.



Figure 3. Left panel illustrates the visual axis and right panel illus-trates the line of sight.

Figure 4. Schematic of a generic objective alignment systemdesigned to place the line of sight on the optical axis of the mea-surement system. BS: beam splitter, FP: on axis fixation point.

Figure 5. Schematic of a generic subjective alignment systemdesigned to place the line of sight on the optical axis of the mea-surement system. BS: beam splitter, FP: fixation point source.

Conversion Between Reference AxesIf optical aberration measurements are made

with respect to some other reference axis, the datamust be converted to the standard reference axis(see the tools developed by Susana Marcos at ourtemporary website: http//color.eri.harvard/stan-dardization). However, since such conversionsinvolve measurement and/or estimation errors fortwo reference axes (the alignment error of the mea-surement and the error in estimating the new refer-ence axis), it is preferable to have the measurementaxis be the same as the line-of-sight.

DESCRIPTION OF ZERNIKE POLYNOMIALSThe Zernike polynomials are a set of functions

that are orthogonal over the unit circle. They areuseful for describing the shape of an aberratedwavefront in the pupil of an optical system. Severaldifferent normalization and numbering schemes forthese polynomials are in common use. Below wedescribe the different schemes and make recom-mendations towards developing a standard for pre-senting Zernike data as it relates to aberration the-ory of the eye.

Double Indexing SchemeThe Zernike polynomials are usually defined in

polar coordinates (�, �), where � is the radial coordi-nate ranging from 0 to 1 and � is the azimuthal com-ponent ranging from 0 to 2�. Each of the Zernikepolynomials consists of three components: a normal-ization factor, a radial-dependent component and anazimuthal-dependent component. The radial compo-nent is a polynomial, whereas the azimuthal compo-nent is sinusoidal. A double indexing scheme is use-ful for unambiguously describing these functions,with the index n describing the highest power(order) of the radial polynomial and the index mdescribing the azimuthal frequency of the sinusoidalcomponent. By this scheme the Zernike polynomialsare defined as

(1)

where Nmn is the normalization factor described

in more detail below and R|m|n(�) is given by

(2)

This definition uniquely describes the Zernikepolynomials except for the normalization constant.

The normalization is given by(3)

where �m0 is the Kronecker delta function (ie, �m0 =1 for m = 0, and �m0 = 0 for m does not = 0). Note thatthe value of n is a positive integer or zero. For agiven n, m can only take on values -n, -n + 2, -n +4,...n.

When describing individual Zernike terms, thetwo index scheme should always be used. Below are-some examples.

Good“The values of Z-1

3 (�, �) and Z24 (�, �) are 0.041 and

-0.121, respectively”“Comparing the astigmatism terms, Z -2

2 (�, �) andZ2

2(�, �) ...”Bad'The values of Z7 (�, �) and Z12 (�, �) are 0.041 and

-0.121, respectively”“Comparing the astigmatism terms, Z5 (�, �) and

Z6 (�, �) ...”

Single Indexing SchemeOccasionally, a single indexing scheme is useful

for describing Zernike expansion coefficients. Sincethe polynomials actually depend on two parameters,n and m, ordering of a single indexing scheme isarbitrary. To avoid confusion, a standard singleindexing scheme should be used, and this schemeshould only be used for bar plots of expansion coeffi-cients (Fig 6). To obtain the single index, j, it is con-venient to lay out the polynomials in a pyramid withrow number n and column number m as shown inTable 1. The single index, j, starts at the top of thepyramid and steps down from left to right. To con-vert between j and the values of n and m, the fol-lowing relationships can be used:

j = n(n + 2) + m (mode number) (4)2

n = roundup -3 + �9 + 8j (radial order) (5)2

m = 2j - n(n + 2) (angular frequency) (6)



Coordinate SystemTypically, a right-handed coordinate system is

used in scientific applications as shown in Figure 7.For the eye, the coordinate origin is at the center ofthe eye’s entrance pupil, the +x axis is horizontalpointing to the right, the +y axis is vertical pointingup, and the +z Cartesian axis points out of the eyeand coincides with the foveal line-of-sight in objectspace, as defined by a chief ray emitted by a fixationspot. Also shown are conventional definitions of thepolar coordinates

r = �x2 + y2 and � = arctan(y/x)

This definition gives x = rcos� and y = rsin�. Wenote that Malacara5 uses a polar coordinate system

in which x = rsin� and y = rcos�. In other words, � ismeasured clockwise from the +y axis (Fig 1b),instead of counterclockwise from the +x axis(Fig 1a). Malacara’s definition stems from early(pre-computer) aberration theory and is not recom-mended. In ophthalmic optics, angle � is called the“meridian” and the same coordinate system appliesto both eyes.

Because of the inaccessibility of the eye’s imagespace, the aberration function of eyes are usuallydefined and measured in object space. For example,objective measures of ocular aberrations use lightreflected out of the eye from a point source on theretina. Light reflected out of an aberration-free eyewill form a plane-wave propagating in the positivez-direction and therefore the (x,y) plane serves as a



Table 1—Zernike Pyramid(Row number is polynomial order n, column number is sinusoidal frequency m, table entry is the

single-index j.)

Figure 6. Example of a bar plot using the sin-gle index scheme for Zernike coefficients.

Figure 7. Conventional right-handed coordi-nate system for the eye in Cartesian andpolar forms.



Table 2Listing of Zernike Polynomials up to the 7th Order (36 Terms)

natural reference surface. In this case the wavefrontaberration function W(x,y) equals the z-coordinateof the reflected wavefront and may be interpreted asthe shape of the reflected wavefront. By these con-ventions, W>0 means the wavefront is phase-advanced relative to the chief ray. An examplewould be the wavefront reflected from a myopic eye,converging to the eye’s far-point. A closely relatedquantity is the optical path-length difference (OPD)between a ray passing through the pupil at (x,y) andthe chief ray point passing through the origin. In thecase of a myopic eye, the path length is shorter formarginal rays than for the chief ray, so OPD<0.Thus, by the recommended sign conventions,OPD(x,y) = -W(x,y).

Bilateral symmetry in the aberration structure ofeyes would make W(x,y) for the left eye the same asW(-x,y) for the right eye. If W is expressed as aZernike series, then bilateral symmetry wouldcause the Zernike coefficients for the two eyes to beof opposite sign for all those modes with odd

symmetry about the y-axis (eg, mode Z-22). Thus, to

facilitate direct comparison of the two eyes, a vectorR of Zernike coefficients for the right eye can beconverted to a symmetric vector L for the left eye bythe linear transformation L=M❋R, where M is adiagonal matrix with elements +1 (no sign change)or -1 (with sign change). For example, matrix M forZernike vectors representing the first 4 orders(15 modes) would have the diagonal elements [+1,+1, -1, -1, +1, +1, +1, +1, -1, -1, -1, -1, +1, +1, +1].

A STANDARD ABERRATOR FOR CALIBRATIONThe original goal was to design a device that

could be passed around or mass-produced to cali-brate aberrometers at various laboratories. We firstthought of this as an aberrated model eye, but thatlater seemed too elaborate. One problem is that thesubjective aberrometers needed a sensory retina intheir model eye, while the objective ones needed areflective retina of perhaps known reflectivity. Wedecided instead to design an aberrator that could beused with any current or future aberrometers, withwhatever was the appropriate model eye.

The first effort was with a pair of lenses thatnearly cancelled spherical power, but when dis-placed sideways would give a known aberration.That scheme worked, but was very sensitive to tilt,and required careful control of displacement. Thesecond design was a trefoil phase plate (OPD = Z3

3 =�r3sin3�) loaned by Ed Dowski of CDM Optics, Inc.This 3rd order aberration is similar to coma, butwith three lobes instead of one, hence the commonname “trefoil.” Simulation of the aberration functionfor this plate in ZEMAX is shown in Figures 8 and9. Figure 8 is a graph of the Zernike coefficientsshowing a small amount of defocus and 3rd orderspherical aberration, but primarily C3

3 . Figure 9shows the wavefront, only one-half micron (onewave) peak to peak, but that value depends on �,above.



Figure 8. Zernike coefficients of trefoil phaseplate from ZEMAX model (note different num-bering convention from that recommendedabove for eyes).

Figure 9. Wavefront map for trefoil phase plate from the ZEMAXmodel.

We mounted the actual plate and found that ithad even more useful qualities: As the phase plate istranslated across the pupil, it adds some C2

2,horizontal astigmatism. When the plate is perfectlycentered, that coefficient is zero. Further, the slopeof C2

2(�x) measures the actual pupil.(7)

so(8)

and similarly(9)

This means that �Z33 = 3Z2

2 �x and then, since W =Cm

n Zmn , we get a new term proportional to �x.

Plotting the coefficient C22 against �x, we need to

normalize to the pupil size. That could be useful asa check on whether the aberrator is really at thepupil, or whether some smoothing has changed thereal pupil size, as measured. Figures 10-13 confirmthis behavior and the expected variation with rota-tion (3�).

Although the phase plate aberrator works inde-pendently of position in a collimated beam, someaberrometers may want to use a converging ordiverging beam. Then it should be placed in a pupilconjugate plane. We have not yet built the mount forthe phase plate, and would appreciate suggestions-for that. Probably we need a simple barrel mountthat fits into standard lens holders—say 30 mm out-side diameter. We expect to use a standard pupil,but the phase plate(s) should have 10 mm clearaperture before restriction. The workshop seemed tofeel that a standard pupil should be chosen. Shouldthat be 7.5 mm?

We have tested the Z33 aberrator, but it may be a

good idea to have a few others. We borrowed thisone, and it is somewhat fragile. Bill Plummer ofPolaroid thinks he could generate this and otherplates in plastic for “a few thousand dollars” for eachdesign. Please send suggestions as to whether otherdesigns are advisable, and as to whether we willwant to stack them or use them independently([email protected]). That has someimplications for the mount design, but not severeones. We suggest two Z3

3 plates like this one, andperhaps a Z0

6 , fifth order spherical.

At this time, then, our intent is to have one ormore standard aberrators that can be inserted intoany aberrometer. When centered, and with a stan-dard pupil, all aberrometers should report the sameZernike coefficients. We do not intend to includepositioners in the mount, assuming that will be dif-ferent for each aberrometer.

Another parameter of the design is the value of �.That comes from the actual physical thickness andthe index of refraction. Suggestions are welcomehere, but we assume we want coefficients that arerobust compared to a diopter or so of defocus.

The index will be whatever it will be. We willreport it, but again any chromaticity will depend onhow it is used. We suggest that we report theexpected coefficients at a few standard wavelengthsand leave interpolation to users.



Figure 10. Wavefront map from the aberrator, using the SRRaberrometer.

Figure 11. The phase plate of Figure 10 has been movedhorizontally 4 mm.

PLANS FOR PHASE II

Reference Axes Sub-committee—Develop a shareware library of software tools

needed to convert data from one ocular referenceaxis to another (eg, convert a wavefront aberrationfor the corneal surface measured by topographyalong the instrument’s optical axis into a wavefrontaberration specified in the eye’s exit pupil planealong the eye’s fixation axis).

—Generate test datasets for evaluating softwaretools

Describing Functions Subcommittee—Develop a shareware library of software tools

for generating, manipulating, evaluating, etc. therecommended describing functions for wavefrontaberrations and pupil apodizing functions.

—Develop additional software tools for convertingresults between describing functions (eg, convertingTaylor polynomials to Zernike polynomials, or con-verting single-index Zernikes to double-indexZernikes, etc.)

—Generate test datasets for evaluating softwaretools

Model Eyes Subcommittee—Build a physical model eye that can be used to

calibrate experimental apparatus for measuring theaberrations of eyes

—Circulate the physical model to all interestedparties for evaluation, with results to be presentedfor discussion at a future VSIA meeting

REFERENCES1. Thibos LN, Applegate RA, Howland HC, Williams DR, Artal

P, Navarro R, Campbell MC, Greivenkamp JE,Schwiegerling JT, Burns SA, Atchison DA, Smith G, SarverEJ. A VSIA-sponsored effort to develop methods and stan-dards for the comparison of the wavefront aberration struc-ture of the eye between devices and laboratories. In: VisionScience and Its Applications. Washington, D.C: OpticalSociety of America; 1999:236-239.

2. Thibos LN, Bradley A, Still DL, Zhang X, Howarth PA.Theory and measurement of ocular chromatic aberration.Vision Res 1990;30:33-49.

3. Bradley A, Thibos LN. (Presentation 5) athttp://www.opt.indiana.edu/lthibos/ABLNTOSA95.

4. Bennett AG, Rabbetts RB. Clinical Visual Optics, 2nd ed.Philadelphia, PA: Butterworth; 1989.

5. Malacara D. Optical Shop Testing, 2nd ed. New York, NY:John Wiley & Sons, Inc; 1992.



Figure 12. Zernike coefficients are stable against horizontaldisplacement, except for C 3

3 .Figure 13. Zernike coefficients C3

3 and C-33 as a function of rotation

of the phase plate about the optic axis.

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Standards for Reporting the Optical Aberrations of Eyes

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