Representation of Wavefront Aberrations

1Slide

The goal of this tutorial is to provide a brief introductionto how the optical imperfections of a human eye arerepresented by wavefront aberration maps and howthese maps may be interpreted in a clinical context.

4th Wavefront Congress - San Francisco - February 2003

Representation of

Wavefront Aberrations

Larry N. Thibos

School of Optometry, Indiana University,

Bloomington, IN 47405

[email protected]

http://research.opt.indiana.edu/Library/wavefronts/index.htm

2Slide

My lecture is organized around the following 5 questions.

•First, What is an aberration map? Because the aberrationmap is such a fundamental description of the eye’ opticalproperties, I’m going to describe it from three different, butcomplementary, viewpoints: first as misdirected rays oflight, second as unequal optical distances between object andimage, and third as misshapen optical wavefronts.

•Second, how are aberration maps displayed? The answerto that question depends on whether the aberrations aredescribed in terms of rays or wavefronts.

•Third, how are aberrations classified? Several methods areavailable for classifying aberrations. I will describe for youthe most popular method, called Zernike analysis.

lFourth, how is the magnitude of an aberration specified? Iwill describe three simple measures of the aberration mapthat are useful for quantifying the magnitude of optical errorin a person’s eye. Other measures based on the quality of theretinal images are more sophisticated conceptually, but maybe more important for predicting the visual impact of ocularaberrations.

lLastly, how may we interpret the spatial derivatives of theaberration map?

Lecture outline

• What is an aberration map?– Ray errors– Optical path length errors– Wavefront errors

• How are aberration maps displayed?– Ray deviations– Optical path differences– Wavefront shape

• How are aberrations classified?– Zernike expansion

• How is the magnitude of an aberration specified?– Wavefront variance– Equivalent defocus– Retinal image quality

• How are the derivatives of the aberration map interpreted?

3Slide

To illustrate the concepts associated with aberration mapsI will use a series of clinical examples of normal andabnormal eyes. Our goal in analyzing these eyes usingaberrometry is to describe the nature of the opticalproblem and how it may be diagnosed by inspection of theaberration map and its associated aberration coefficients.

Clinical Examples

• Prism (“first Zernike order”)– Video-keratoscopic errors

• Sphero-cylindrical (“second Zernike order”)– myopia, astigmatism

• Keratoconus (“third Zernike order”)– Vertical coma

• LASIK (“fourth Zernike order”)– Spherical aberration

• Dry eye (“irregular higher order”)

4Slide

To begin with the question “What is an aberration map” itis helpful to consider an emmetropic eye which is free ofaberrations. Such an eye is optically perfect and thereforemay be used as a “Gold Standard” for judging the opticalimperfections of real eyes. The key property of a perfecteye is that it focuses a distant point source of light into aperfect image on the retina.

We can account for this perfect retinal image threedifferent ways.

• Firstly, in a perfect eye, all of the rays emerging from apoint source of light at the eye’s far point P that passthrough the eye’s pupil will intersect at a common point P-prime on the retina.

• Secondly, the perfect eye has the property that theoptical distance from the object to the image is the samefor every point of entry in the eye’s pupil.

• Thirdly, the wavefront of light focused by the eye has aperfectly spherical shape.

The gold standard of optical quality depicted here is for thecase of an emmetropic eye with relaxed accommodation.

Optically perfect eye is the “Gold Standard”

Key points:

• All rays from P intersect at common point P on the retina.

• The optical distance from object P to image P is the same forall rays.

• Wavefront converging on retina is spherical.

perfect retinal image

Rays from distant pointsource, P. P

5Slide

In general, we would like to have a gold standard that alsoworks for an object point other than infinity so that we candescribe the aberrations of an accommodating eye or anametropic eye.

Fortunately, the same three conditions for perfectiondevised for an emmetropic eye with a far-point at infinityapply also in this more general case of a near-sighted orfar-sighted eye.

The last two ways of describing a perfect eye are based onthe concept of “optical distance”, which is an importantconcept in optics that is very useful for interpreting theaberration map of eyes. The meaning of the term “opticaldistance” is explained in the next slide.

General case: accommodating or ametropic eye

Key points:

• All rays from P intersect at common point P on the retina.

• The optical distance from object P to image P is the samefor all rays.

• Wavefront converging on retina is spherical.

perfectretinal

image ofobject pointObject point

raysP

P

6Slide

Definition: Optical path length

Optical Path Length specifies distances in wavelengths ( ).

Optical distance = physical distance X refractive index

Equal optical length => equal phase

Water Air

In everyday life we measure distances in physical units ofmeters. However, in optics it is common to measure distanceswith a ruler that is calibrated in wavelengths of light. Such aruler effectively measures the number of times light mustoscillate in traveling from the object to the image. If all raysoscillate the same number of times, then the light will arrive atthe retina with the same phase and therefore will interfereconstructively to produce a perfect image.

For example, light has a shorter wavelength when itpropagates through the watery medium of the eye compared toair. Consequently, rays traveling only a short physical distancein water undergo the same number of oscillations, andtherefore travel the same optical distance, as a longer path inair.

A simple formula for computing optical distance is to multiplythe physical distance times refractive index.

This concept of optical path length is very useful forunderstanding the imaging property of lenses as shown in thenext slide.

Spheres are perfect wavefronts

Point sources producespherical wavefronts.

Spherical wavefrontscollapse onto point images.

Lens

A lens forms an image by refracting rays. If the optical distancetaken by every ray that passes through the lens is the same, then allthe rays will arrive at the image plane with the same phase to form aperfect image. Thus the perfect eye is one which provides the same opticaldistance from object to image for all the rays passing through the eye’spupil.

This concept of optical distance is also useful for understandingwavefronts of light. A wavefront is the locus of points which are thesame optical distance from their source point. When lightpropagates in a homogeneous medium, equal optical distances areequal physical distances. Consequently, the wavefronts produced bya point source are perfect spheres because all points on a sphere arethe same distance from the center of the sphere.

By the same line of reasoning, a converging spherical wavefront willcollapse down to a perfect point image. Thus we may conclude thata perfect eye converts spherical expanding wavefronts into sphericalcollapsing wavefronts. If we require that an eye be well focused fordistant objects, then the sphrerical wavefronts arriving at the eye willbe plane waves. Nevertheless, the perfect eye will focus these planewaves into spherical wavefronts centered on the retinalphotoreceptors.

Notice in this drawing that the rays of light emanating from a pointsource are always perpendicular to the wavefront surface. Thisperpendicular relationship makes rays and wavefrontsinterchangeable and complementary concepts. If you know wherethe rays are, you can draw the wavefront, and visa-versa.

8Slide

Now that we know what a perfect eye is like, we candefine an aberrated eye 3 ways that correspond to our 3ways of defining optical perfection.

Firstly, the rays do NOT focus at a common retinallocation.

Secondly, the optical path distance from an object pointto the retinal image is NOT the same for all rays passingthrough the pupil. The difference in optical distancebetween any ray and the center ray is called the OpticalPath Difference, or OPD.

And thirdly, the wavefronts inside the eye are NOTspherical, they are distorted.

Aberrated eye

Key points:

• Rays do NOT intersect at the same retinal location.

• The optical distance from object to retina is NOT the samefor all rays. OPD = Optical Path Difference

• Wavefront is NOT spherical.

Pflawed retinal image

Rays from distant pointsource, P. P

What is an aberration map?

An aberration map is a visualization of howthe eye’s aberrations vary across the pupil.

3 Formats:• Map #1: ray deviations• Map #2: optical path length differences• Map #3: wavefront shape

2 Viewpoints:• Light propagating towards the retina• Reflected light propagating away from the eye

With this background in optical theory, we can now answer the firstquestion posed: What is an aberration map?

By definition, an aberration map is a graphical display or“visualization” of how aberrations vary across the eye’s pupil.

Since we have defined aberrations 3 different ways, there are 3different maps we can draw.

Firstly, we can show how each ray deviates from a perfect ray.

Second, we can show how the optical distance from object to imagevaries across the pupil.

And thirdly, we can show how the shape of the wavefront differs froma sphere.

Although these three descriptions are equivalent, they representdifferent ways of thinking about aberrations that are all valuable intheir own way for interpreting clinical cases of optical dysfunction.Thus, when we interpret an aberration map of an eye we may do sothree different ways, in terms of where rays strike the retina, in termsof the optical distance from the object to the retina through differentpoints of entry in the pupil, and in terms of the shape of the wavefrontof light produced by the eye.

Not only do we have 3 formats for displaying the aberration map, buteach of these 3 formats can be used from two different viewpoints.The first viewpoint is of light propagating from the external world intothe eye towards the retina. The second viewpoint is of light reflectedby the fundus back out of the eye, propagating away from theindividual.

These two viewpoints are interchangeable conceptually, but they arevery different from a practical point of view. Fortunately, by havingtwo different viewpoints available for investigating the samephenomenon, we are able to overcome a serious practical problem.

The problem is that the eye is a closed system so it is difficult for aclinician to inspect the light rays and wavefronts once they enter theeye to see if they are perfect. For this reason, objective methods havebeen devised for measuring the optical aberrations of eyes using lightthat is reflected out of an eye due to a laser beam that is focused ontothe retina at some point P -prime.

Optically, it doesn’t matter whether we think of point P as the objectand P-prime as the image, or visa-versa. The rays in this diagram donot depend on the direction of propagation of the light. Similarly, theoptical distance between P and P’ is independent of the direction oflight propagation. This means we can avoid the practical problem ofhaving a closed system if we characterize the eye’s aberrations interms of the light reflected out of the eye from a small spot of lightplaced on the retina with a narrow laser beam.

If the eye is perfect, then a point source located on the retina at pointP’ will emerge from the eye as a perfect spherical wavefront centeredon the far-point P outside of the eye. So, in principle, all we need todo to characterize the optical quality of a real eye is to capture awavefront of light emerging from the eye and compare it to a sphere.Any deviations from a sphere are called “aberrations” and aquantitative description of these deviations across the pupil is called a“wavefront aberration function” or simply an “aberration map”.

To say the same thing in terms of optical path lengths, all we need tomeasure is the difference in optical path length between object pointand image point through every possible pupil location.

To say the same thing a third time, now in the language of ray optics,all we need to measure is the direction of individual rays of light asthey emerge from the eye’s pupil. By comparing the actual rays to theperfect rays we can construct a ray aberration map.

The perfect eye

objectpoint

Image

point

rays

reflected

wavefront

P

P

LASER

11Slide

In practice, an instrument called an “aberrometer” isused to measures the shape of the wavefront of lightreflected back out of the eye. If that wavefront is asphere centered on the eye’s far point, then the eye isoptically perfect except for a focusing error associatedwith myopia or hyperopia. To eliminate this focusingerror as well and make the eye truly perfect, the far pointmust be at infinity. Since a spherical wavefront withinfinite radius of curvature is a plane wave, the perfectemmetropic eye is characterized by the plane wave itreflects from a point source focused on the retina.

A plane wave is a very convenient reference wavefrontthat makes it is easy to specify the aberrations of an eye.Aberrations are given by the difference between theactual wavefront and a reference wave in the x-y plane ofthe pupil.

P

P

z

y

x

Wavefront

Deviations of the wavefront from a perfect sphere indicatethe presence of optical aberrations.

Aberrometers measure the wavefront reflected from point P

Reference sphere

Wavefront

Error

12Slide

In summary, the wavefront aberration map of an eye indicatesthe distance between the reflected wavefront and the idealplane wavefront located in the plane of the eye’s pupil. Thisdistance between the reflected wavefront and the pupil planerepresents an error of optical path length which varies frompoint-to-point across the pupil. The aberration map cantherefore be quantified by a mathematical function W thatdepends on the x- and y-coordinates of points inside the eye’spupil. Thus we see that the aberration map can be understoodand displayed using all 3 formats of ray optics, optical patherrors, and wavefront shape errors regarless of whether thelight is propagating into the eye or reflected out of the eye.

P

z

y

x

Wavefront

The surface Z(x,y) is also the error W(x,y) between the reflectedwavefront and the ideal reference plane wave in the pupil plane.

Specification of the wavefront error of an eye

Z(x,y)Ideal reference

plane-wave

Pupilcircle

ErrorW(x,y)

13Slide

Coordinate System for Aberration Maps

y

x

rEvery point inpupil plane has alocation inrectangular (x,y)coordinates andin polar (r, )coordinates.

The aberrationfunction W(x,y)or W(r, ) may bedefined witheither coordinatesystem.

This raises my second question, “How are aberrationmaps displayed?” To display aberration maps we firsthave to agree on a coordinate system for the eye’s pupilplane. The natural coordinate system for use inophthalmic optics is the standard mathematicalcoordinate system shown here. Every point in the pupilplane can be located uniquely by its x-y coordinates ofthe rectangular coordinate frame, or in terms of the polarcoordinates r and theta.

The aberration function W can therefore be describedeither as a function of the x-y rectangular coordinates ofthe eye, or as a function of the r-theta polar coordinatesof the eye.

Typically we use the same coordinate system for eithereye. However, to take account of the bi-lateral symmetryof eyes, it is often best to flip the coordinate system aboutthe y-axis so that the positive x-axis points nasally ratherthan to the right.

How are aberration maps displayed?

Ray Errors Optical DistanceErrors

Wavefront Error

Aberration type: negative vertical coma

Having agreed on a coordinate system, we are able to displayaberration maps 3 ways using the three formats just described.

Here we show aberration maps for a particular type of aberrationcalled “coma”, plotted these 3 different ways.

On the left we have a graphical depiction of the ray errors producedat various locations. The red circle denotes the pupil margin and thearrows shows how each ray is deviated as it emerges from the pupilplane.

This pattern of ray aberrations indicates that rays near the upper andlower margins of the pupil are deflected upwards relative to theperfect ray, while the rays near the middle of the pupil are deflecteddownwards relative to the perfect ray.

In the center figure we see that the optical distance from object toimage is greater over most of the lower half of the pupil compared tothe upper half.

The figure at the right shows the shape of the wavefront aberrationfunction. This diagram shows that the phase of the light is retardedover most of the lower half of the pupil and advanced in the upperhalf.

Notice that the wavefront map on the right agrees with the opticaldistance map in the middle, but has opposite sign. This is because along optical path causes phase retardation, and a short optical pathcauses phase advancement.

Notice also that the ray map on the left agrees with the wavefrontmap on the right. To connect these two maps, recall that each of thevectors in the map at the left indicates the local slope of the surface onthe right. Consequently, these vectors are showing the local directionof propagation of the wavefront, which is always perpendicular to thesurface of the wavefront.

Thus we see that although the three maps look quite different, they alldescribe the same optical aberration from three different perspectives.

15Slide

Clinically, the preceeding example of vertical coma iscommonly associated with the corneal disease Keratoconus. Ifthe protruding cone of keratoconus is in the inferior cornea,then the optical path of rays in the inferior half of the pupil willbe longer because rays will spend relatively more time in waterthan in air. Because light travels slower in water, thewavefront will be retarded in the vicinity of the ectasia.

This example shows how to conceive of optical path errors andwavefront errors based on a simple model of the physical causeof an aberration.

Wavefront Aberration in Keratoconus

Advanced phase <= Short optical path

Retarded phase <= Long optical path

Reference

Ectasia

16Slide

Clinical Example: LASIK-induced spherical aberration

P P′

n=1

P P′

n>1

Before surgery

After surgery

Here is another example showing how to interpret theocular aberrations of a clinical condition, in this caseLASIK refractive surgery.

The top of this slide shows an optical diagram for amyopic eye that is otherwise free of optical aberrations.For such an eye, light reflected from a point of light atretinal point P-prime would form a perfect image at theeye’s far-point, P. Consequently, the wavefrontconverging on P will be a perfect sphere.

After LASIK surgery, the cornea has become flatter at theapex and so the rays emerging from the center of thepupil will be parallel and focused at infinity, as expectedfor an emmetropic eye. However, outside this centralregion the cornea is still highly curved and so the eye isstill myopic for marginal rays. This combination ofemmetropia for paraxial rays and myopia for marginalrays is a classical case of positive spherical aberration.

The aberration maps for an eye with spherical aberrationare shown on the next slide.

17Slide

Aberration maps for LASIK

Ray Errors Optical DistanceErrors

Wavefront Error

Aberration type: Seidel spherical aberration

0

0.5

1

Looking first at the ray aberration map on the left, we see thatthe central region of the pupil is largely free of aberrations.However, the ray aberrations grow rapidly near the pupilmargin where corneal curvature is greater.

The map of optical path length for this example of sphericalaberration reveals that paraxial rays follow a longer path to adistant far point than do the marginal rays, which implies thatthe phase of the central rays will lag behind the phase of themarginal rays. This explains why the wavefront error map onthe right shows a concave wavefront with the central arealagging behind the marginal portion of the wavefront.

Please keep in mind that this example, and the previous one, areidealized cases created for tutorial purposes. The aberrationstructure of real human eyes is usually complicated by thepresence of a variety of different kinds of aberrations. In orderto describe the complicated aberration structure of real eyes, itis very useful to have a method available for systematicallyclassifying aberrations. With such a method we maydecompose any aberration map, no matter how complicated,into a combination of basic, elementary forms.

This leads to our third question, How are aberrations classified?

18Slide

How are aberrations classified?

Mathematical descriptions of wavefront aberrations:

• Seidel aberrations

• Taylor series expansion of aberration map

• Zernike expansion of aberration map

Classical aberration theory was developed by Seidel forsymmetrical optical systems. Unfortunatley, the lack ofsymmetry in human eyes requires a more generaltreatment, such as Taylor series expansion of thewavefront map, or Zernike expansion of the aberrationmap.

Each method has its advantages and disadvantages. Fortoday’s lecture I’ve selected the Zernike method forillustrating the concept of systematic classification ofaberrations into fundamental forms.

19Slide

Some of these fundamental forms are familiar toeveryone in the field of ophthalmic optics.

For example, the wavefront error for a myopic orhyperopic eye has a spherical shape that is definedmathematically in terms of a simple equation thatcontains the key expression x-squared plus y-squared.

There are some other constants in the equation whicharen’t essential, but which provides some convenientfeatures. For example, the minus-1 in the equation forcesthe average error to be zero. All of the fundamentalaberrations in Zernike’s expansion except the first oneshare this property of having zero mean error.

Wavefront error for defocus

20Slide

The equations Zernike used to represent wavefronterrors often look much simpler if written not in terms ofthe X-Y coordinates of a rectangular coordinate system,but instead in terms of the polar coordinates r and theta.

An eye with astigmatism, for example, will reflect asaddle-shaped wavefront which has a rather simplealgebraic equation, when written in polar form.

Now that we see how to visualize the eye’s wavefrontaberration function and how to describe itmathematically with an equation, the next step is tocombine simple wavefronts like those I”ve been showingto make more complex wavefronts that can describe theaberration structure of real eyes.

Wavefront error for astigmatism

r = radius

= meridian

To do this we need a catalog of basic shapes that we can addtogether. Since the basic building blocks of Zernike will be thebasis for describing the aberration structure of eyes, they areknown as “basis functions”.

Each Zernike basis function is the product of two other functions,one of which depends only on radius and the other which dependsonly on meridian.

For example, the astigmatism wavefront is described by theformula:

square-root of 6 times r-squared times cosine of 2 theta

Notice that the term involving the radius variable “r” is apolynomial. The largest exponent in this polynomial is 2, so it iscalled a “second-order” polynomial. The trigonometric terminvolving the meridian variable theta is a co-sinusoidal harmonic,in this case with a frequency of 2. The normalization constantsquare-root of 6 is not absolutely necessary but its inclusionprovides a mathematically convenient property that the variance ofthe waveform will be 1.

This basic pattern of a normalizing constant times a polynomialtimes a harmonic occurs for all of the Zernike functions. Anotherexample is a basis function called “coma” which is the product of a3rd order polynomial and a sinusoidal harmonic of frequency 1.By convention, the superscript is given a minus sign when thetrigonometrical harmonic is a sine function, and is given a plussign when the harmonic is a cosine function.

Anatomy of Zernike basis functions

Z2+2 = 6 r2 ⋅c o s ( 2)

Z3−1 = 8 (3r 3 − 2r) ⋅sin( )Coma:

Astigmatism:

polynomial harmonicnormalization constant

order frequency

The first 21 Zernike basis functions, or “modes” as they areoften called, are shown in this catalog.

The Zernike catalog is best viewed as a pyramid, rather like theperiodic table of the elements. Each row in the pyramidcorresponds to a given order of the polynomial component ofthe function and each column corresponds to a differentmeridional frequency. Again, by convention, co-sinusoidalharmonics are assigned positive frequencies and sinusoidalharmonics are assigned negative frequencies.

Some of these aberrations have names like “sphericalaberration” and “coma” but most are simply identified by theirfrequency and order. Although it is possible to number thesemodes in sequence starting from the top of the pyramid, this isnot the most intuitive naming scheme. Instead, the OpticalSociety of America has recommended a double-script notationwhich designates each basis function according to its order andfrequency. The order is used as a subscript and the frequency isused as a superscript to unambiguously and convenientlyidentify each mode.

Traditionally, ophthalmology and optometry have beenconcerned solely with Zernike aberrations of the first andsecond orders. The first order terms are called vertical andhorizontal prism, and the second order terms are calledastigmatism and defocus. Only these 5 aberrations of the firstand second order can be corrected by conventional spectaclelenses or contact lenses. The current resurgence of interest invisual optics research and refractive surgery is focusedprimarily on the measurement and correction of the higherorder aberrations in the 3rd, 4th,and 5th orders.

Meridional frequency

Rad

ial ord

er

0

1

2

3

4

5

0 1 2 3 4 5-1-2-3-4-5

Zfn

f = frequency

n = order

Periodic tableof Zernikefunctions

23Slide

OSA Standards for Reporting Ocular Aberrations*

• Double -script notation

– Subscript = radial order

– Superscript = meridional frequency

• Meridian measured counter-clockwisefrom horizontal

• Normalizing constants produce unitvariance

* Thibos, Applegate, Schwiegerling, & Webb (2000). Standards for reportingthe optical aberrations of eyes. In Lakshminarayanan, V. (Ed.), Trends inOptics and Photonics. Washington, DC: Optical Society of America.Reprinted in Customized Corneal Ablation (2001) by MacRae, Krueger, andApplegate. Slack Inc.

It is an unfortunate fact of history that a variety of waysfor defining the Zernike polynomials exists in the opticsliterature. To avoid potential confusion, a taskforce wasformed in the year 2000 under the auspices of the OpticalSociety of America to recommend standards forreporting optical aberrations of the eye. The fullrecommendations of the taskforce are published in theseries “Trends in Optics and Photonics” and wasreprinted in the recent book by MacRae, Krueger, andApplegate.

The main recommendations are:

(1) the use of a double-script notation for identifyingeach Zernike polynomial, with a subscript indicatingradial order and a sperbscript indicating meridionalfrequency

(2) the meridian angle be measured counter-clockwisefrom the horizonatal, which is the standard ophthalmicconvention for specifying the axis of astigmatism

(3) and the use of normalization constants which makeall of the Zernike polynomials have unit variance.

24Slide

To summarize Zernike analysis, we describe the aberrationstructure of an eye mathematically as the weighted sum ofZernike basis functions. Such a description is called a “Zernikeexpansion” of the wavefront aberration.

The weight which must be applied to each basis function, ormode, when computing the sum is called an aberrationcoefficient.

Each aberration coefficient is just a number, with physicalunits typically specified in microns, or sometimes they arereported in the units of the wavelength of light.

The aberration coefficients of a Zernike expansion areanalogous to the Fourier coefficients of a Fourier expansion,which are in turn analogous to the energy spectrum of a lightsource. Thus it is common language to speak of chromaticspectrum of light, or the “Fourier spectrum” of a waveform,and in the same way we may speak of the “Zernike spectrum”of the eye’s optical aberrations.

Zernike expansion = weighted sum of modes

W(r, ) = a00 ⋅ Z0

0

+a1−1 ⋅ Z1

−1 + a1+1 ⋅ Z1

+1

+a2−2 ⋅ Z2

−2 + a20 ⋅ Z2

0 +K

= anf ⋅ Zn

f

frequency∑

order∑

a nf = aberration coefficient (weight)

Znf = Zernike basis function (mode)

25Slide

One convenient feature of the Zernike expansion I’ve justdescribed is that every mode except the zero-order modehas zero mean, and they all are scaled so as to have unitvariance. This puts all of the modes on a common basisso their relative magnitudes can be compared easily.Furthermore, the variance of a weighted Zernike mode isjust the square of the aberration coefficient.

An attractive feature of the set of Zernike functions isthat they are mutually orthogonal, which means they areindependent of each other mathematically. The practicaladvantage of orthogonality is that we can determine theamount of defocus, or astigmatism, or any other Zernikemode occurring in an aberration function without havingto worry about the presence of other modes.

Furthermore, the orthogonality of the Zernike basisfunctions makes it easy to calculate the total variance in awavefront as the sum of the variances in the individualcomponents.

Statistics of Zernike modes

Mean Znf( ) = 0 for all n > 0, all f

Variance Znf( ) =1, for all n, f

Variance a ⋅ Znf( ) = a2, a = aberration coefficient

Total variance = sum of individual variances = a2

modes∑

Normal eye, weak higher-order aberrations

Phase = - OPD

z=d24o8i.od.sh.ex01.sr1.zc;

6 mm pupil

I can imagine that some of you are thinking that this lecturehas had more than enough optical theory for such an earlyhour of the morning. However, the reward for yourperseverance is that we now have all of the tools we need tobegin interpreting aberration maps of real human eyes.

Over the course of the next several slides I will show you someexamples of aberration maps from normal, healthy eyes. Itwould be a mistake to think that normal eyes are not aberrated.In fact, all eyes are aberrated to some degree. Whatdistinguishes the normal from abnormal eye is the magnitudeof these aberrations. Thus it is important to gain experiencewith normal eyes first so that we will be able to recognize eyesthat are outside the normal range.

In order to focus our attention on the higher-order aberrations,I have assumed that the lower-order aberrations of prism,defocus, and astigmatism can be corrected perfectly withspectacle lenses. Thus, all we are seeing in this aberration mapis the effect of the higher order aberrations.

The first example is of an eye with very little aberration in thehigher order modes. The maximum difference between peakand valley measurements is less than 1 micron for this eye overa 6mm pupil. When I look at this map, I am struck by theprominent combination of a single peak and a single valley.This is characteristic of the aberration coma and since the axisconnecting peak to valley is nearly horizontal, this eye appearsto be dominated by horizontal coma.

27Slide

Normal eye, weak higher-order aberrations

Zernike Spectrum Order Distribution

Meridional Frequency

Rad

ial O

rder

0 0.1 0.2

RMS Error ( m)µ-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

7

H. coma

A quantitative Zernike expansion of this aberration mapprovides a large number of aberration coefficients toassess. A convenient way to visualize this 2-dimensionalZernike spectrum graphically is to build a pyramid ofboxes in the shape of the periodic table of Zernike basisfunctions as shown on the left. Each box is colored using agrey scale of intensity to indicate the value of thecorresponding Zernike coefficient. White signifies a largepositive value, black signifies a large negative value, andgrey signifies zero.

The most prominent mode in the spectrum of this eyeoccurs for the third order, with meridional frequency of +1.This is indeed the mode called horizontal coma, just as wesuspected.

The Zernike spectrum is sometimes simplified bycombining all of the information contained in a given rowof the pyramid. The correct way to combine aberrationcoefficients on the same row of the pyramid is to sum theirsquared values. The square root of this sum is called RMSerror, which stands for “root mean squared”. For this eyewe clearly see an exponential decline in RMS error withZernike order that is a common feature of normal eyes.

28Slide

Normal eye, moderate higher-order aberrations

z=d24oa5.od.sh.ex02.sr1.zc;

Phase = - OPD

6 mm pupil

This next example is of a normal eye with a moderatelevel of higher-order aberrations. The pattern in theaberration map for this eye is very different from theprevious eye. Here we see 4 prominent peaks in theaberration function, which suggest the presence of alarge amount of a 4th order aberration called“quadrafoil”.

29Slide

Normal eye, moderate higher-order aberrations

Zernike Spectrum Order Distribution

0 0.1 0.2

RMS Error ( µ m)Meridional Frequency

Rad

ial O

rder

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

QuadrafoilTrefoilV. coma

This visual inspection is confirmed by a quantitativeanalysis that reveals a strong negative mode in the 4throw, 4th column. The spectrum also shows twoprominent modes in the third order. The one withfrequency -1 is vertical coma and the one with frequency-3 is called “trefoil”. The distribution by order showsthat this eye has equal amounts of third and fourth orderaberrations, which is rather unusual for normal eyes.

30Slide

Normal eye, strong higher-order aberrations

Phase = - OPD

z=d24o9l.os.sh.ex03.sr1.zc;

6 mm pupil

My third example of normal eyes has a relatively largeamount of higher order aberration. This pattern is moredifficult to assess visually. The prominent pairing ofpeak and valley suggests the presence of vertical coma,but there evidently are other modes as well thatcomplicate the aberration map.

31Slide

Normal eye, strong higher-order aberrations

Zernike Spectrum Order Distribution

RMS Error (µm)Meridional Frequency

Rad

ial O

rder

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

7

0 0.4 0.8

These observations are supported by the Zernikespectrum, which shows several stong components, oneof which is vertical coma.

Notice that the magnitude of these aberrations is still lessthan 1 micro meter of RMS error for any single Zernikeorder.

32Slide

Irregular aberrations: tear film disruption

Phase = - OPD

6 mm pupil

a1_AB_40sec.m

A clinical example of a highly irregular aberration mapis shown here for a case of a dry eye patient with adisrupted tear film. The red and yellow areas of thephase map on the left indicate areas where optical pathlength is relatively short and the phase is relativelyadvanced. These are regions where the corneal tear filmhas been replaced by air as it becomes thinner andeventually breaks up.

33Slide

Irregular aberrations: tear film disruption

Zernike Spectrum Order Distribution

Meridional Frequency

Rad

ial O

rder

-10 -8 -6 -4 -2 0 2 4 6 8 10

0123456789

10

0 0.2 0.4

RMS Error (µ m)

Because of the complicated shape of the aberration map,the Zernike spectrum for this dry eye example containsmany significant terms with much higher radial orderand meridional frequency than usually occurs in thenormal eye.

34Slide

One last example shows the aberration map before andafter LASIK surgery to correct 3 diopters of myopia.Before surgery this eye had excellent optical quality asindicated by the nearly flat aberration map and smallZernike coefficients. After surgery the eye was nearlyemmetropic but the optical quality was not as good.Now the aberration map is not as flat and the Zernikeanalysis shows significant increase in the magnitude ofspherical aberration.

Clinical Applications: LASIK refractive surgery

Before surgery After surgery

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

0

0

-1

-1

1

1.2

3 4 5 6 7 8 9 10

0RM

S e

rror

(µm

)

Zernike order

1.0

3 4 5 6 7 8 9 100

RM

S e

rror

(µm

)

Zernike order

1.0

3 4 5 6 7 8 9 10

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

0

-1-2

-2

-3

-3

-1

-1

35Slide

How is the magnitude of an aberration specified?

4 common metrics:

§Wavefront variance

§RMS error

§Equivalent defocus

§Retinal image quality

These clinical examples give a brief glimpse of the greatvariety of aberration maps encountered in normalhuman eyes. Although the experienced clinician canmake a qualitative assessment of the eye’s optical systemby inspecting the aberration map visually, somesimplification is needed to answer questions like:“Which eye is more aberrated?”

This issue of judging the degree of aberration of an eyeleads me to my fourth question, “How is the magnitudeof an aberration specified?”

Of the four common metrics listed here, the first threeare closely related to each other but the fourth representsa completely different approach that requires a littlemore optical theory to understand fully.

36Slide

X

Y

WFE = Z(x,y)

RMS Error = Wavefront Variance

Wavefront Variance =1

# pointsZ (x, y) − Z ( )2

x, y∑

The first, and most popular metric of the strength of anaberration is wavefront variance. The variance of awavefront is a measure of how much it differs from aplane wave. It is easily computed by summing thesquared heights of every point on the curve. However, ifyou know the aberration coefficients for an eye you don’tneed to carry out this calculation because the wavefrontvariance is just the sum of squared Zernike coefficients.

The quantity called “RMS error” is another name for thesquare root of wavefront variance. It is a popular metricbecause the physical units are in microns.

One of the awkward features of wavefront variance isthat it changes when pupil diameter changes. To getaround this problem, we can use an alternative metriccalled “equivalent defocus” which factors out pupildiameter.

37Slide

“Equivalent defocus” is defined as the amount ofdefocus required to produce the same wavefrontvariance as found in one or more higher-orderaberrations.

A simple formula allows us to compute equivalentdefocus in diopters if we know the pupil size and thetotal wavefront variance in other Zernike modes.

One of the appealing features of equivalent defocus as anaberration metric is that it factors out pupil size bynormalizing RMS error by pupil area. Empirically wehave found that equivalent defocus is largelyindependent of pupil diameter in normal eyes. Thissimplifies the quantification of a person’s higher-orderaberrations because it becomes possible to summarize apatient’s aberrations by saying, for example, that thepatient has 1 diopter of equivalent defocus withouthaving to specify pupil size, in the same way that we cansay a myopic eye is defocused by 1 diopter, withouthaving to specify pupil size.

Clinical interpretation of wavefront variance

Equivalent defocus is:the amount of defocus required to produce

the same RMS wavefront error producedby one or more higher-order aberrations.

Equivalent Defocus (D) = 4 3RMS Error ( m)

Pupil Area (mm2 )

38Slide

Retinal image quality: step 1

Aberration map Auto-correlation Optical TransferFunction

c20 = c3

−1 = 0.25 m

A completely different approach to specifying themagnitude of aberrations is to assess their effect onretinal image quality. Optical theory tells us that if weknow the aberration map, then we can compute retinalimage quality by a three step process.

Step 1 is to perform a mathematical operation called“autocorrelation” on the aberration map which yields theoptical transfer function for the eye.

39Slide

Retinal image quality: step 2

Point-spreadFunctionFourier Transform

Optical TransferFunction

3.75 arcmin

Step 2 is to perform a different mathematical operationcalled a Fourier transform on the optical transferfunction to produce the eye’s point-spread function,which is a simulation of the retinal image formed by apoint source of light.

40Slide

Retinal image quality: step 3

Point-spreadFunction Convolution Retinal Image

3.75 arcmin10 ′

Step 3 is to reproduce the point-spread function for everypoint source in the object. This corresponds to yetanother mathematical operation called convolution,which yields a simulation of the retinal image for anyobject of interest, in this case an eye chart.

As you might imagine, there are many ways to assess thequality of the retinal image for a complex object in thereal world. Currently there is active debate in the visionscience community about which metric of image qualityis the best predictor of visual performance.

41Slide

The final question I wish to address is how to interpretthe spatial derivatives of the aberration map.

Recall that rays are always perpendicular to wavefrontsso if a wavefront is tilted, then the direction of the raywill change relative to a perfect reference ray drawnparallel to the chief ray emerging from the center of thepupil. In other words, the amount of horizontaldisplacement, delta X, and the amount of verticaldisplacement, delta Y, of a ray is determined by the localslope of the wavefront.

Since the slope of the wavefront is computedmathematically by taking the spatial derivativehorizontally, and vertically, we can conclude that the rayaberrations are directly proportional to the firstderivatives of the wavefront aberration function. Indeed,this is how many aberrometers, including the Shack-Hartmann wavefront sensor works: it measures rayaberrations, which are then interpreted as wavefrontslopes, and these slopes are then integrated to obtain thewavefront.

How are derivatives of aberration maps interpreted?

objectpoint

Chief rayP

P

LASER

Reference ray

∆∆y

x

Wavefront slope determinesray aberrations

Wavefront Slope and Curvature: Defocus (z20)

(RMS=0.5 m, pupil=6mm, S= 0.385 D)

2nd

deriv.

1st derivative

collapse

tails

Let us now apply this line of thinking to some concrete examples,such as defocus, shown here. The conventional aberration map fordefocus is shown in the upper left figure.

The first derivative of the aberration map gives the slope of thewavefront at every point in the pupil. We can visualize the slopeinformation with the ray aberration map shown in the lower leftfigure. Each arrow in this field of arrows shows the direction ofpropagation of the light ray at the corresponding point in the pupil.In this example all of the arrows are pointing towards the center,which indicates that the rays of light are all converging to a commonfocal point.

One of the nice feataures of the ray aberration map is that it can beused to derive a geometrical-optics approximation to the point-spread function called a “spot diagram”. We do this by collapsing allof the arrows so their tails coincide. The tips of the arrows, shown byred spots in the figure at the bottom right, indicate where each ray oflight reflected out of the eye will intersect a distant screen. Or, tochange our viewpoint, the spots show where the rays from a distantpoint source will intersect the retina in this myopic eye.

To go one step further in this kind of analysis, the second spatialderivative may be used to determine the curvature of the wavefrontaberration map. Curvature, and its close cousin vergence, are afamiliar concepst to clinicains and are calibrated in the familiar unitsof diopters. The figure in the upper right corner shows the averageof the horizontal and vertical curvatures, or vergences, in diopters,for this same, defocused eye. Notice that this map shows only onecolor, which means the curvature is the same everywhere in thepupil. This makes sense because a defocused eye produces aspherical wavefront and a spherical wavefront has the samecurvature everywhere.

43Slide

Wavefront Slope and Curvature: Astigmatism (z22)

(RMS=0.5 m, pupil=6mm, Jo= 0.27D)

Another example of the first and second derivatives of awavefront aberration function is shown here for Zernikeastigmatism. In this case the ray aberration map showsthat some rays are converging towards the optical axisand some are diverging. However, the spot diagram isstill circular, as we would expect for an astigmaticsystem with zero spherical equivalent.

Note that the mean curvature is zero everywhere in thepupil, which indicates that this wavefront is similar to aplane wave in that it will come to focus only at infinity.This result occurs because the wavefront has a saddleshape everywhere, with equal but opposite curvature inthe horizontal and vertical directions. The average ofthese curvatures is zero everywhere in the pupil becauseequal but opposite curvatures cancel out.

44Slide

Wavefront Slope and Curvature: Coma (z31)

(RMS=0.5 m, pupil=6mm, Meq= 0.385 D)

My last example of spatial derivativesof the aberrationmap is for coma. Note the complicated ray aberrationmap, which leads to a complex spot diagram which is thegeometrical approximation to the exact point-spreadfunction that can be computed by the Fourier opticsmethods described earlier.

Currently we don’t have enough experience looking atthese spatial derivative maps to know whether theyprovide valuable insight into clinical problems.However I am optimistic that they will prove their worthin the end. The easy transition from the map ofwavefront slopes to a spot diagram of the eye’s pointspread function is certainly a useful feature that is easyto understand. Also, curvature maps have a longtradition in ophthalmic optics for understanding theshape of the cornea and the optical properties of eyes so Isuspect that these maps will also provide valuableinsight, in the fullness of time.

45Slide

This concludes my tutorial.

Details of the OSA standards for reporting opticalaberrations of eyes is available on the web at the addressshown here.

For further reading I can recommend recent featureissues the Journal of Refractive Surgery which contain anumber of tutorial papers as well as current researchreports in the general area of visual optics.

I can also recommend the new monograph by MacRae,Krueger, and Applegate as an excellent introduction tothe general topic of wavefront aberrations of the eye.

Lastly, a copy of my lecture slides and speaker notes isavailable on the WEB at the address shown above.

Reference materials

OSA Standards for Reporting OpticalAberrations of eyes http://research.opt.indiana.edu/Library/OSAStandard.pdf

Feature issues (2000, 2001, 2002)J. Refractive Surgery (Sept. issue)

MonographCustomized Corneal Ablation: The Quest for Super Vision ,

MacRae, Krueger, & Applegate (eds.) Slack, Inc. (2001)

Lecture slideshttp://research.opt.indiana.edu/Library/wavefronts.htm

46Slide

The end

Visual Optics Group at Indiana University

Larry Thibos, PhD

Arthur Bradley, PhD

Donald Miller, PhD

Carolyn Begley, OD

Nikole Himebaugh, OD

Xin Hong, PhD

Xu Cheng, MD

Fan Zhou, BS

Pete Kollbaum, OD

Kevin Haggerty, BS

Vision Research at

http://www.opt.indiana.edu

National Institutes ofHealth (R01-EY05109)

Borish Center forOphthalmic Research

Support

47Slide

Supplementary slides follow

48Slide

How are aberrations measured?

• Subjective–Vernier alignment–Spatially resolved refractometer–Subjective aberroscope

• Objective–Objective aberroscope–Laser ray tracing–Shack-Hartmann wavefront sensor

A variety of techniques have been invented over theyears for measuring the eye’s aberrations. In the nexttalk, Ron Kruger will discuss these various technologiesfor measuring aberrations. However, it is worthmentioning one of these techniques now to demonstratehow any given method can be described 3 ways: interms of ray aberrations, optical path length differences,and wavefront shape. Consider, for example, the Shack-Hartmann wavefront sensor.

49Slide

The principle of the Shack-Hartmann aberrometer is easilyunderstood in terms of ray optics as an elaboration of the 17thcentury Scheiner Disk which Thomas Young later used to makehis famous optometer.

Imagine that we are able to place a point source of light on thepatient’s fundus. Light reflected back out of the eye from thissource will tell us everything we need to know about the eye’saberrations if we can determine the direction of each ray of lightas it emerges from the eye’s pupil. Scheiner’s disk allows us toisolate individual rays to determine their direction ofpropagation. If the pair of rays isolated by Scheiner’s disk donot intersect at the eye’s far point, then the eye is opticallyaberrated and we may quantify the magnitude of this aberrationin terms of the horizontal and vertical deviation of a ray fromthe perfect reference ray.

Scheiner Optometer (1619)

Scheiner’s disk

∆y

∆x

Reference ray

Ametropic eye

Retinalpointsource

50Slide

In principle we can track the direction of many rayssimultaneously by adding more holes to Scheiner’s disk to makewhat is known as a Hartmann screen. By capturing theseisolated rays with photographic film or the CCD sensor inside avideo camera, we can determine the direction of propagation ofeach ray as it passes through a specific point in the pupil. Theend result is a map of the ray aberrations of the eye.

To convert Scheiner’s subjective method into an objectiveoptometer, we reverse the direction of light propagation. Inother words, we put a spot of light on the retina which thenbecomes a point source which reflects light back out of the eye.

Next, drill some more holes in Scheiner’s disk so it becomes aHartmann screen. That way each aperture in the Hartmannscreen isolates a narrow pencil of rays emerging from the eyethrough a specific part of the pupil. The rays then intersect avideo sensor which tells us the horizontal and verticaldisplacement of the ray from the non-aberrated, referenceposition.

Thus we have a Hartmann aberrometer for objectivelymeasuring the ray aberrations of the eye. Now fill the individualapertures of the Hartmann screen with tiny lenses and you havea Hartmann-Shack aberrometer, or as I would prefer to say, aScheiner-Hartmann-Shack aberrometer.

Hartmann screen

Point source

d

CCD sensor

Objective Scheiner/Hartmann Aberrometer

51Slide

The modern version of this idea suggested by Dr. RolandShack and Ben Platt in 1965 replaces the Hartmannscreen with an array of tiny lenses which are moreefficient at capturing light and focusing it onto the CCDsensor.

Having described the principle of operation of theScheiner-Hartmann-Shack aberrometer in terms of rayaberrations, I will now repeat my description using thelanguage of wave optics so that you will appreciate whythis instrument is often called a wavefront sensor ratherthan a ray sensor.

Hartmann-Shack wavefront sensor

Retinal point

source

Lenslet array

CCD sensor

Relay lenses

Scheiner-Hartmann-Shack Aberrometer

52Slide

We can see how the wavefront sensor works by considering aside view of the lenslet array as shown on the right-hand sideof this figure.

For a perfect, emmetropic eye the reflected wavefront will be aplane wave perpendicular to a parallel bundle of rays. Thiswavefront is sub-divided by the lenslet array into smallerwavefronts that are focused by the lenses in the array into aperfect lattice of point images. When viewed from the front,asshown on the left, each image falls on the optical axis of thecorresponding lenslet. The overall result is a regular grid ofevenly spaced spots of light.

Principle of Wavefront Sensor

Micro-lensletarray

Video sensor

Sub-divide the wavefrontwith micro-lenslets.

Local slope determines spotposition on video sensor.

Perfect wavefront

Frontview

53Slide

If the eye is aberrated, then the reflected wavefront will bedistorted and the individual rays will not be parallel to eachother. Because rays are perpendicular to wavefronts, thedirection of propagation of each ray is determined by the localslope of the wavefront over each lenslet. It is wavefront slopewhich determines where the spot is focused.. Thus anaberrated wavefront produces a disordered collection of spotimages.

By analyzing the displacement of each spot from itscorresponding lenslet axis, we can deduce the slope of theaberrated wavefront when it entered the corresponding lenslet.Mathematical integration of this slope information yields theshape of the aberrated wavefront.

Principle of Wavefront Sensor

Micro-lensletarray

Video sensor

Displacement of spots fromreference grid indicates localslope of aberrated wavefront.

Aberrated wavefront

Frontview

54Slide

Slope of Wavefront: Defocus

This is a more detailed display of wavefront slopes.