Representation of Wavefront Aberrations

Transcription

1 1 4th Wavefront Congress - San Francisco - February 2003 Representation of Wavefront Aberrations Larry N. Thibos School of Optometry, Indiana University, Bloomington, IN thibos@indiana.edu The goal of this tutorial is to provide a brief introduction to how the optical imperfections of a human eye are represented by wavefront aberration maps and how these maps may be interpreted in a clinical context.

2 2 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? My lecture is organized around the following 5 questions. First, What is an aberration map? Because the aberration map is such a fundamental description of the eye optical properties, I m going to describe it from three different, but complementary, viewpoints: first as misdirected rays of light, second as unequal optical distances between object and image, and third as misshapen optical wavefronts. Second, how are aberration maps displayed? The answer to that question depends on whether the aberrations are described in terms of rays or wavefronts. Third, how are aberrations classified? Several methods are available for classifying aberrations. I will describe for you the most popular method, called Zernike analysis. Fourth, how is the magnitude of an aberration specified? I will describe three simple measures of the aberration map that are useful for quantifying the magnitude of optical error in a person s eye. Other measures based on the quality of the retinal images are more sophisticated conceptually, but may be more important for predicting the visual impact of ocular aberrations. Lastly, how may we interpret the spatial derivatives of the aberration map?

3 3 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 ) To illustrate the concepts associated with aberration maps I will use a series of clinical examples of normal and abnormal eyes. Our goal in analyzing these eyes using aberrometry is to describe the nature of the optical problem and how it may be diagnosed by inspection of the aberration map and its associated aberration coefficients.

4 4 Optically perfect eye is the Gold Standard Rays from distant point source, P. P perfect retinal image 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 for all rays. Wavefront converging on retina is spherical. To begin with the question What is an aberration map it is helpful to consider an emmetropic eye which is free of aberrations. Such an eye is optically perfect and therefore may be used as a Gold Standard for judging the optical imperfections of real eyes. The key property of a perfect eye is that it focuses a distant point source of light into a perfect image on the retina. We can account for this perfect retinal image three different ways. Firstly, in a perfect eye, all of the rays emerging from a point source of light at the eye s far point P that pass through the eye s pupil will intersect at a common point P- prime on the retina. Secondly, the perfect eye has the property that the optical distance from the object to the image is the same for every point of entry in the eye s pupil. Thirdly, the wavefront of light focused by the eye has a perfectly spherical shape. The gold standard of optical quality depicted here is for the case of an emmetropic eye with relaxed accommodation.

5 5 General case: accommodating or ametropic eye P Object point Key points: rays All rays from P intersect at common point P on the retina. The optical distance from object P to image P is the same for all rays. Wavefront converging on retina is spherical. P perfect retinal image of object point In general, we would like to have a gold standard that also works for an object point other than infinity so that we can describe the aberrations of an accommodating eye or an ametropic eye. Fortunately, the same three conditions for perfection devised for an emmetropic eye with a far-point at infinity apply also in this more general case of a near-sighted or far-sighted eye. The last two ways of describing a perfect eye are based on the concept of optical distance, which is an important concept in optics that is very useful for interpreting the aberration map of eyes. The meaning of the term optical distance is explained in the next slide.

6 6 Definition: Optical path length Optical Path Length specifies distances in wavelengths ( ). Equal optical length => equal phase Water Air Optical distance = physical distance X refractive index In everyday life we measure distances in physical units of meters. However, in optics it is common to measure distances with a ruler that is calibrated in wavelengths of light. Such a ruler effectively measures the number of times light must oscillate in traveling from the object to the image. If all rays oscillate the same number of times, then the light will arrive at the retina with the same phase and therefore will interfere constructively to produce a perfect image. For example, light has a shorter wavelength when it propagates through the watery medium of the eye compared to air. Consequently, rays traveling only a short physical distance in water undergo the same number of oscillations, and therefore travel the same optical distance, as a longer path in air. A simple formula for computing optical distance is to multiply the physical distance times refractive index. This concept of optical path length is very useful for understanding the imaging property of lenses as shown in the next slide.

7 Spheres are perfect wavefronts Point sources produce spherical wavefronts. Spherical wavefronts collapse onto point images. Lens A lens forms an image by refracting rays. If the optical distance taken by every ray that passes through the lens is the same, then all the rays will arrive at the image plane with the same phase to form a perfect image. Thus the perfect eye is one which provides the same optical distance from object to image for all the rays passing through the eye s pupil. This concept of optical distance is also useful for understanding wavefronts of light. A wavefront is the locus of points which are the same optical distance from their source point. When light propagates in a homogeneous medium, equal optical distances are equal physical distances. Consequently, the wavefronts produced by a point source are perfect spheres because all points on a sphere are the same distance from the center of the sphere. By the same line of reasoning, a converging spherical wavefront will collapse down to a perfect point image. Thus we may conclude that a perfect eye converts spherical expanding wavefronts into spherical collapsing wavefronts. If we require that an eye be well focused for distant objects, then the sphrerical wavefronts arriving at the eye will be plane waves. Nevertheless, the perfect eye will focus these plane waves into spherical wavefronts centered on the retinal photoreceptors. Notice in this drawing that the rays of light emanating from a point source are always perpendicular to the wavefront surface. This perpendicular relationship makes rays and wavefronts interchangeable and complementary concepts. If you know where the rays are, you can draw the wavefront, and visa-versa.

8 8 Aberrated eye Rays from distant point source, P. P flawed retinal image Key points: Rays do NOT intersect at the same retinal location. The optical distance from object to retina is NOT the same for all rays. OPD = Optical Path Difference Wavefront is NOT spherical. Now that we know what a perfect eye is like, we can define an aberrated eye 3 ways that correspond to our 3 ways of defining optical perfection. Firstly, the rays do NOT focus at a common retinal location. Secondly, the optical path distance from an object point to the retinal image is NOT the same for all rays passing through the pupil. The difference in optical distance between any ray and the center ray is called the Optical Path Difference, or OPD. And thirdly, the wavefronts inside the eye are NOT spherical, they are distorted.

9 What is an aberration map? An aberration map is a visualization of how the 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 first question 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 3 different 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 image varies across the pupil. And thirdly, we can show how the shape of the wavefront differs from a sphere. Although these three descriptions are equivalent, they represent different ways of thinking about aberrations that are all valuable in their own way for interpreting clinical cases of optical dysfunction. Thus, when we interpret an aberration map of an eye we may do so three different ways, in terms of where rays strike the retina, in terms of the optical distance from the object to the retina through different points of entry in the pupil, and in terms of the shape of the wavefront of light produced by the eye. Not only do we have 3 formats for displaying the aberration map, but each of these 3 formats can be used from two different viewpoints. The first viewpoint is of light propagating from the external world into the eye towards the retina. The second viewpoint is of light reflected by the fundus back out of the eye, propagating away from the individual. These two viewpoints are interchangeable conceptually, but they are very different from a practical point of view. Fortunately, by having two different viewpoints available for investigating the same phenomenon, we are able to overcome a serious practical problem.

10 The perfect eye P Image point rays reflected wavefront P object point LASER The problem is that the eye is a closed system so it is difficult for a clinician to inspect the light rays and wavefronts once they enter the eye to see if they are perfect. For this reason, objective methods have been devised for measuring the optical aberrations of eyes using light that is reflected out of an eye due to a laser beam that is focused onto the retina at some point P -prime. Optically, it doesn t matter whether we think of point P as the object and P-prime as the image, or visa-versa. The rays in this diagram do not depend on the direction of propagation of the light. Similarly, the optical distance between P and P is independent of the direction of light propagation. This means we can avoid the practical problem of having a closed system if we characterize the eye s aberrations in terms of the light reflected out of the eye from a small spot of light placed on the retina with a narrow laser beam. If the eye is perfect, then a point source located on the retina at point P will emerge from the eye as a perfect spherical wavefront centered on the far-point P outside of the eye. So, in principle, all we need to do to characterize the optical quality of a real eye is to capture a wavefront of light emerging from the eye and compare it to a sphere. Any deviations from a sphere are called aberrations and a quantitative 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 to measure is the difference in optical path length between object point and 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 as they emerge from the eye s pupil. By comparing the actual rays to the perfect rays we can construct a ray aberration map.

11 11 Aberrometers measure the wavefront reflected from point P Reference sphere Wavefront y Error Wavefront z P P x Deviations of the wavefront from a perfect sphere indicate the presence of optical aberrations. In practice, an instrument called an aberrometer is used to measures the shape of the wavefront of light reflected back out of the eye. If that wavefront is a sphere centered on the eye s far point, then the eye is optically perfect except for a focusing error associated with myopia or hyperopia. To eliminate this focusing error as well and make the eye truly perfect, the far point must be at infinity. Since a spherical wavefront with infinite radius of curvature is a plane wave, the perfect emmetropic eye is characterized by the plane wave it reflects from a point source focused on the retina. A plane wave is a very convenient reference wavefront that makes it is easy to specify the aberrations of an eye. Aberrations are given by the difference between the actual wavefront and a reference wave in the x-y plane of the pupil.

12 12 Specification of the wavefront error of an eye Wavefront Z(x,y) y Ideal reference plane-wave z Error W(x,y) Pupil circle P x The surface Z(x,y) is also the error W(x,y) between the reflected wavefront and the ideal reference plane wave in the pupil plane. In summary, the wavefront aberration map of an eye indicates the distance between the reflected wavefront and the ideal plane wavefront located in the plane of the eye s pupil. This distance between the reflected wavefront and the pupil plane represents an error of optical path length which varies from point-to-point across the pupil. The aberration map can therefore be quantified by a mathematical function W that depends on the x- and y-coordinates of points inside the eye s pupil. Thus we see that the aberration map can be understood and displayed using all 3 formats of ray optics, optical path errors, and wavefront shape errors regarless of whether the light is propagating into the eye or reflected out of the eye.

13 13 Coordinate System for Aberration Maps Every point in pupil plane has a location in rectangular (x,y) coordinates and in polar (r, ) coordinates. y r x The aberration function W(x,y) or W(r, ) may be defined with either coordinate system. This raises my second question, How are aberration maps displayed? To display aberration maps we first have to agree on a coordinate system for the eye s pupil plane. The natural coordinate system for use in ophthalmic optics is the standard mathematical coordinate system shown here. Every point in the pupil plane can be located uniquely by its x-y coordinates of the rectangular coordinate frame, or in terms of the polar coordinates r and theta. The aberration function W can therefore be described either as a function of the x-y rectangular coordinates of the eye, or as a function of the r-theta polar coordinates of the eye. Typically we use the same coordinate system for either eye. However, to take account of the bi-lateral symmetry of eyes, it is often best to flip the coordinate system about the y-axis so that the positive x-axis points nasally rather than to the right.

14 How are aberration maps displayed? Ray Errors Optical Distance Errors Wavefront Error Aberration type: negative vertical coma Having agreed on a coordinate system, we are able to display aberration maps 3 ways using the three formats just described. Here we show aberration maps for a particular type of aberration called coma, plotted these 3 different ways. On the left we have a graphical depiction of the ray errors produced at various locations. The red circle denotes the pupil margin and the arrows shows how each ray is deviated as it emerges from the pupil plane. This pattern of ray aberrations indicates that rays near the upper and lower margins of the pupil are deflected upwards relative to the perfect ray, while the rays near the middle of the pupil are deflected downwards relative to the perfect ray. In the center figure we see that the optical distance from object to image is greater over most of the lower half of the pupil compared to the upper half. The figure at the right shows the shape of the wavefront aberration function. This diagram shows that the phase of the light is retarded over most of the lower half of the pupil and advanced in the upper half. Notice that the wavefront map on the right agrees with the optical distance map in the middle, but has opposite sign. This is because a long optical path causes phase retardation, and a short optical path causes phase advancement. Notice also that the ray map on the left agrees with the wavefront map on the right. To connect these two maps, recall that each of the vectors in the map at the left indicates the local slope of the surface on the right. Consequently, these vectors are showing the local direction of propagation of the wavefront, which is always perpendicular to the surface of the wavefront. Thus we see that although the three maps look quite different, they all describe the same optical aberration from three different perspectives.

15 15 Wavefront Aberration in Keratoconus Reference Advanced phase <= Short optical path Ectasia Retarded phase <= Long optical path Clinically, the preceeding example of vertical coma is commonly associated with the corneal disease Keratoconus. If the protruding cone of keratoconus is in the inferior cornea, then the optical path of rays in the inferior half of the pupil will be longer because rays will spend relatively more time in water than in air. Because light travels slower in water, the wavefront will be retarded in the vicinity of the ectasia. This example shows how to conceive of optical path errors and wavefront errors based on a simple model of the physical cause of an aberration.

16 16 Clinical Example: LASIK-induced spherical aberration n=1 n>1 P P Before surgery P P After surgery Here is another example showing how to interpret the ocular aberrations of a clinical condition, in this case LASIK refractive surgery. The top of this slide shows an optical diagram for a myopic eye that is otherwise free of optical aberrations. For such an eye, light reflected from a point of light at retinal point P-prime would form a perfect image at the eye s far-point, P. Consequently, the wavefront converging on P will be a perfect sphere. After LASIK surgery, the cornea has become flatter at the apex and so the rays emerging from the center of the pupil will be parallel and focused at infinity, as expected for an emmetropic eye. However, outside this central region the cornea is still highly curved and so the eye is still myopic for marginal rays. This combination of emmetropia for paraxial rays and myopia for marginal rays is a classical case of positive spherical aberration. The aberration maps for an eye with spherical aberration are shown on the next slide.

17 17 Aberration maps for LASIK Ray Errors Optical Distance Errors 1 Wavefront Error Aberration type: Seidel spherical aberration Looking first at the ray aberration map on the left, we see that the central region of the pupil is largely free of aberrations. However, the ray aberrations grow rapidly near the pupil margin where corneal curvature is greater. The map of optical path length for this example of spherical aberration reveals that paraxial rays follow a longer path to a distant far point than do the marginal rays, which implies that the phase of the central rays will lag behind the phase of the marginal rays. This explains why the wavefront error map on the right shows a concave wavefront with the central area lagging behind the marginal portion of the wavefront. Please keep in mind that this example, and the previous one, are idealized cases created for tutorial purposes. The aberration structure of real human eyes is usually complicated by the presence of a variety of different kinds of aberrations. In order to describe the complicated aberration structure of real eyes, it is very useful to have a method available for systematically classifying aberrations. With such a method we may decompose any aberration map, no matter how complicated, into a combination of basic, elementary forms. This leads to our third question, How are aberrations classified?

18 18 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 for symmetrical optical systems. Unfortunatley, the lack of symmetry in human eyes requires a more general treatment, such as Taylor series expansion of the wavefront map, or Zernike expansion of the aberration map. Each method has its advantages and disadvantages. For today s lecture I ve selected the Zernike method for illustrating the concept of systematic classification of aberrations into fundamental forms.

19 19 Wavefront error for defocus Some of these fundamental forms are familiar to everyone in the field of ophthalmic optics. For example, the wavefront error for a myopic or hyperopic eye has a spherical shape that is defined mathematically in terms of a simple equation that contains the key expression x-squared plus y-squared. There are some other constants in the equation which aren t essential, but which provides some convenient features. For example, the minus-1 in the equation forces the average error to be zero. All of the fundamental aberrations in Zernike s expansion except the first one share this property of having zero mean error.

20 20 Wavefront error for astigmatism r = radius = meridian The equations Zernike used to represent wavefront errors often look much simpler if written not in terms of the 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 a saddle-shaped wavefront which has a rather simple algebraic equation, when written in polar form. Now that we see how to visualize the eye s wavefront aberration function and how to describe it mathematically with an equation, the next step is to combine simple wavefronts like those I ve been showing to make more complex wavefronts that can describe the aberration structure of real eyes.

21 Anatomy of Zernike basis functions order frequency Astigmatism: Z 2 +2 = 6 r 2 cos(2) normalization constant polynomial harmonic Coma: Z 3 1 = 8 (3r 3 2r) sin( ) To do this we need a catalog of basic shapes that we can add together. Since the basic building blocks of Zernike will be the basis for describing the aberration structure of eyes, they are known 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 depends only on meridian. For example, the astigmatism wavefront is described by the formula: square-root of 6 times r-squared times cosine of 2 theta Notice that the term involving the radius variable r is a polynomial. The largest exponent in this polynomial is 2, so it is called a second-order polynomial. The trigonometric term involving the meridian variable theta is a co-sinusoidal harmonic, in this case with a frequency of 2. The normalization constant square-root of 6 is not absolutely necessary but its inclusion provides a mathematically convenient property that the variance of the waveform will be 1. This basic pattern of a normalizing constant times a polynomial times a harmonic occurs for all of the Zernike functions. Another example is a basis function called coma which is the product of a 3rd order polynomial and a sinusoidal harmonic of frequency 1. By convention, the superscript is given a minus sign when the trigonometrical harmonic is a sine function, and is given a plus sign when the harmonic is a cosine function.

22 Periodic table of Zernike functions Z f n f = frequency n = order Radial order Meridional frequency The first 21 Zernike basis functions, or modes as they are often called, are shown in this catalog. The Zernike catalog is best viewed as a pyramid, rather like the periodic table of the elements. Each row in the pyramid corresponds to a given order of the polynomial component of the function and each column corresponds to a different meridional frequency. Again, by convention, co-sinusoidal harmonics are assigned positive frequencies and sinusoidal harmonics are assigned negative frequencies. Some of these aberrations have names like spherical aberration and coma but most are simply identified by their frequency and order. Although it is possible to number these modes in sequence starting from the top of the pyramid, this is not the most intuitive naming scheme. Instead, the Optical Society of America has recommended a double-script notation which designates each basis function according to its order and frequency. The order is used as a subscript and the frequency is used as a superscript to unambiguously and conveniently identify each mode. Traditionally, ophthalmology and optometry have been concerned solely with Zernike aberrations of the first and second orders. The first order terms are called vertical and horizontal prism, and the second order terms are called astigmatism and defocus. Only these 5 aberrations of the first and second order can be corrected by conventional spectacle lenses or contact lenses. The current resurgence of interest in visual optics research and refractive surgery is focused primarily on the measurement and correction of the higher order aberrations in the 3rd, 4th,and 5th orders.

23 23 OSA Standards for Reporting Ocular Aberrations* Double -script notation Subscript = radial order Superscript = meridional frequency Meridian measured counter-clockwise from horizontal Normalizing constants produce unit variance * Thibos, Applegate, Schwiegerling, & Webb (2000). Standards for reporting the optical aberrations of eyes. In Lakshminarayanan, V. (Ed.), Trends in Optics and Photonics. Washington, DC: Optical Society of America. Reprinted in Customized Corneal Ablation (2001) by MacRae, Krueger, and Applegate. Slack Inc. It is an unfortunate fact of history that a variety of ways for defining the Zernike polynomials exists in the optics literature. To avoid potential confusion, a taskforce was formed in the year 2000 under the auspices of the Optical Society of America to recommend standards for reporting optical aberrations of the eye. The full recommendations of the taskforce are published in the series Trends in Optics and Photonics and was reprinted in the recent book by MacRae, Krueger, and Applegate. The main recommendations are: (1) the use of a double-script notation for identifying each Zernike polynomial, with a subscript indicating radial order and a sperbscript indicating meridional frequency (2) the meridian angle be measured counter-clockwise from the horizonatal, which is the standard ophthalmic convention for specifying the axis of astigmatism (3) and the use of normalization constants which make all of the Zernike polynomials have unit variance.

24 24 Zernike expansion = weighted sum of modes 0 W(r, ) = a 00 Z a 1 Z 1 + a 1 Z 1 +a 2 2 Z a 0 2 Z 0 2 +K = a n f order frequency Z n f a n f Z n f = aberration coefficient (weight) = Zernike basis function (mode) To summarize Zernike analysis, we describe the aberration structure of an eye mathematically as the weighted sum of Zernike basis functions. Such a description is called a Zernike expansion of the wavefront aberration. The weight which must be applied to each basis function, or mode, when computing the sum is called an aberration coefficient. Each aberration coefficient is just a number, with physical units typically specified in microns, or sometimes they are reported in the units of the wavelength of light. The aberration coefficients of a Zernike expansion are analogous to the Fourier coefficients of a Fourier expansion, which are in turn analogous to the energy spectrum of a light source. Thus it is common language to speak of chromatic spectrum 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.

25 25 Statistics of Zernike modes f Mean( Z n ) = 0 for all n > 0, all f f Variance( Z n ) =1, for all n, f f Variance( a Z n ) = a 2, a = aberration coefficient Total variance = sum of individual variances = a 2 modes One convenient feature of the Zernike expansion I ve just described is that every mode except the zero-order mode has zero mean, and they all are scaled so as to have unit variance. This puts all of the modes on a common basis so their relative magnitudes can be compared easily. Furthermore, the variance of a weighted Zernike mode is just the square of the aberration coefficient. An attractive feature of the set of Zernike functions is that they are mutually orthogonal, which means they are independent of each other mathematically. The practical advantage of orthogonality is that we can determine the amount of defocus, or astigmatism, or any other Zernike mode occurring in an aberration function without having to worry about the presence of other modes. Furthermore, the orthogonality of the Zernike basis functions makes it easy to calculate the total variance in a wavefront as the sum of the variances in the individual components.

26 Normal eye, weak higher-order aberrations Phase = - OPD 6 mm pupil z=d24o8i.od.sh.ex01.sr1.zc; I can imagine that some of you are thinking that this lecture has had more than enough optical theory for such an early hour of the morning. However, the reward for your perseverance is that we now have all of the tools we need to begin interpreting aberration maps of real human eyes. Over the course of the next several slides I will show you some examples of aberration maps from normal, healthy eyes. It would be a mistake to think that normal eyes are not aberrated. In fact, all eyes are aberrated to some degree. What distinguishes the normal from abnormal eye is the magnitude of these aberrations. Thus it is important to gain experience with normal eyes first so that we will be able to recognize eyes that 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 with spectacle lenses. Thus, all we are seeing in this aberration map is the effect of the higher order aberrations. The first example is of an eye with very little aberration in the higher order modes. The maximum difference between peak and valley measurements is less than 1 micron for this eye over a 6mm pupil. When I look at this map, I am struck by the prominent combination of a single peak and a single valley. This is characteristic of the aberration coma and since the axis connecting peak to valley is nearly horizontal, this eye appears to be dominated by horizontal coma.

27 27 Normal eye, weak higher-order aberrations Radial Order Zernike Spectrum 0 H. coma Meridional Frequency Order Distribution RMS Error ( µ m) A quantitative Zernike expansion of this aberration map provides a large number of aberration coefficients to assess. A convenient way to visualize this 2-dimensional Zernike spectrum graphically is to build a pyramid of boxes in the shape of the periodic table of Zernike basis functions as shown on the left. Each box is colored using a grey scale of intensity to indicate the value of the corresponding Zernike coefficient. White signifies a large positive value, black signifies a large negative value, and grey signifies zero. The most prominent mode in the spectrum of this eye occurs for the third order, with meridional frequency of +1. This is indeed the mode called horizontal coma, just as we suspected. The Zernike spectrum is sometimes simplified by combining all of the information contained in a given row of the pyramid. The correct way to combine aberration coefficients on the same row of the pyramid is to sum their squared values. The square root of this sum is called RMS error, which stands for root mean squared. For this eye we clearly see an exponential decline in RMS error with Zernike order that is a common feature of normal eyes.

28 28 Normal eye, moderate higher-order aberrations Phase = - OPD 6 mm pupil z=d24oa5.od.sh.ex02.sr1.zc; This next example is of a normal eye with a moderate level of higher-order aberrations. The pattern in the aberration map for this eye is very different from the previous eye. Here we see 4 prominent peaks in the aberration function, which suggest the presence of a large amount of a 4th order aberration called quadrafoil.

29 29 Normal eye, moderate higher-order aberrations Zernike Spectrum Order Distribution Radial Order Trefoil V. coma Quadrafoil Meridional Frequency RMS Error ( µ m) This visual inspection is confirmed by a quantitative analysis that reveals a strong negative mode in the 4th row, 4th column. The spectrum also shows two prominent modes in the third order. The one with frequency -1 is vertical coma and the one with frequency -3 is called trefoil. The distribution by order shows that this eye has equal amounts of third and fourth order aberrations, which is rather unusual for normal eyes.

30 30 Normal eye, strong higher-order aberrations Phase = - OPD 6 mm pupil z=d24o9l.os.sh.ex03.sr1.zc; My third example of normal eyes has a relatively large amount of higher order aberration. This pattern is more difficult to assess visually. The prominent pairing of peak and valley suggests the presence of vertical coma, but there evidently are other modes as well that complicate the aberration map.

31 31 Normal eye, strong higher-order aberrations Radial Order Zernike Spectrum Meridional Frequency Order Distribution RMS Error (µm) These observations are supported by the Zernike spectrum, which shows several stong components, one of which is vertical coma. Notice that the magnitude of these aberrations is still less than 1 micro meter of RMS error for any single Zernike order.

32 32 Irregular aberrations: tear film disruption Phase = - OPD 6 mm pupil a1_ab_40sec.m A clinical example of a highly irregular aberration map is shown here for a case of a dry eye patient with a disrupted tear film. The red and yellow areas of the phase map on the left indicate areas where optical path length is relatively short and the phase is relatively advanced. These are regions where the corneal tear film has been replaced by air as it becomes thinner and eventually breaks up.

33 33 Irregular aberrations: tear film disruption Radial Order Zernike Spectrum Meridional Frequency Order Distribution RMS Error (µ m) Because of the complicated shape of the aberration map, the Zernike spectrum for this dry eye example contains many significant terms with much higher radial order and meridional frequency than usually occurs in the normal eye.

34 Slide 34 Clinical Applications: LASIK refractive surgery 3 Before surgery 3 After surgery RMS error (µm) Zernike order RMS error (µm) Zernike order One last example shows the aberration map before and after LASIK surgery to correct 3 diopters of myopia. Before surgery this eye had excellent optical quality as indicated by the nearly flat aberration map and small Zernike coefficients. After surgery the eye was nearly emmetropic but the optical quality was not as good. Now the aberration map is not as flat and the Zernike analysis shows significant increase in the magnitude of spherical aberration.

35 35 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 great variety of aberration maps encountered in normal human eyes. Although the experienced clinician can make a qualitative assessment of the eye s optical system by inspecting the aberration map visually, some simplification is needed to answer questions like: Which eye is more aberrated? This issue of judging the degree of aberration of an eye leads me to my fourth question, How is the magnitude of an aberration specified? Of the four common metrics listed here, the first three are closely related to each other but the fourth represents a completely different approach that requires a little more optical theory to understand fully.

36 36 Wavefront Variance = 1 # points x, y ( Z(x, y) Z ) 2 WFE = Z(x,y) Y X RMS Error = Wavefront Variance The first, and most popular metric of the strength of an aberration is wavefront variance. The variance of a wavefront is a measure of how much it differs from a plane wave. It is easily computed by summing the squared heights of every point on the curve. However, if you know the aberration coefficients for an eye you don t need to carry out this calculation because the wavefront variance is just the sum of squared Zernike coefficients. The quantity called RMS error is another name for the square root of wavefront variance. It is a popular metric because the physical units are in microns. One of the awkward features of wavefront variance is that it changes when pupil diameter changes. To get around this problem, we can use an alternative metric called equivalent defocus which factors out pupil diameter.

37 37 Clinical interpretation of wavefront variance Equivalent defocus is: the amount of defocus required to produce the same RMS wavefront error produced by one or more higher-order aberrations. Equivalent Defocus (D) = 4 3 RMS Error ( m) Pupil Area (mm 2 ) Equivalent defocus is defined as the amount of defocus required to produce the same wavefront variance as found in one or more higher-order aberrations. A simple formula allows us to compute equivalent defocus in diopters if we know the pupil size and the total wavefront variance in other Zernike modes. One of the appealing features of equivalent defocus as an aberration metric is that it factors out pupil size by normalizing RMS error by pupil area. Empirically we have found that equivalent defocus is largely independent of pupil diameter in normal eyes. This simplifies the quantification of a person s higher-order aberrations because it becomes possible to summarize a patient s aberrations by saying, for example, that the patient has 1 diopter of equivalent defocus without having to specify pupil size, in the same way that we can say a myopic eye is defocused by 1 diopter, without having to specify pupil size.

38 38 Retinal image quality: step 1 Aberration map Auto-correlation Optical Transfer Function c 2 0 = c 3 1 = 0.25 m A completely different approach to specifying the magnitude of aberrations is to assess their effect on retinal image quality. Optical theory tells us that if we know the aberration map, then we can compute retinal image quality by a three step process. Step 1 is to perform a mathematical operation called autocorrelation on the aberration map which yields the optical transfer function for the eye.

39 39 Retinal image quality: step 2 Optical Transfer Function Fourier Transform Point-spread Function 3.75 arcmin Step 2 is to perform a different mathematical operation called a Fourier transform on the optical transfer function to produce the eye s point-spread function, which is a simulation of the retinal image formed by a point source of light.

40 40 Retinal image quality: step 3 Point-spread Function Convolution Retinal Image 3.75 arcmin 10

41 41 How are derivatives of aberration maps interpreted? Reference ray y x P Chief ray P object point Wavefront slope determines ray aberrations LASER The final question I wish to address is how to interpret the spatial derivatives of the aberration map. Recall that rays are always perpendicular to wavefronts so if a wavefront is tilted, then the direction of the ray will change relative to a perfect reference ray drawn parallel to the chief ray emerging from the center of the pupil. In other words, the amount of horizontal displacement, delta X, and the amount of vertical displacement, delta Y, of a ray is determined by the local slope of the wavefront. Since the slope of the wavefront is computed mathematically by taking the spatial derivative horizontally, and vertically, we can conclude that the ray aberrations are directly proportional to the first derivatives of the wavefront aberration function. Indeed, this is how many aberrometers, including the Shack- Hartmann wavefront sensor works: it measures ray aberrations, which are then interpreted as wavefront slopes, and these slopes are then integrated to obtain the wavefront.

42 Wavefront Slope and Curvature: Defocus (z 20 ) 2 nd deriv. 1 st derivative collapse tails (RMS=0.5 m, pupil=6mm, S= D) Let us now apply this line of thinking to some concrete examples, such as defocus, shown here. The conventional aberration map for defocus is shown in the upper left figure. The first derivative of the aberration map gives the slope of the wavefront at every point in the pupil. We can visualize the slope information with the ray aberration map shown in the lower left figure. Each arrow in this field of arrows shows the direction of propagation 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 common focal point. One of the nice feataures of the ray aberration map is that it can be used to derive a geometrical-optics approximation to the pointspread function called a spot diagram. We do this by collapsing all of the arrows so their tails coincide. The tips of the arrows, shown by red spots in the figure at the bottom right, indicate where each ray of light reflected out of the eye will intersect a distant screen. Or, to change our viewpoint, the spots show where the rays from a distant point source will intersect the retina in this myopic eye. To go one step further in this kind of analysis, the second spatial derivative may be used to determine the curvature of the wavefront aberration map. Curvature, and its close cousin vergence, are a familiar concepst to clinicains and are calibrated in the familiar units of diopters. The figure in the upper right corner shows the average of the horizontal and vertical curvatures, or vergences, in diopters, for this same, defocused eye. Notice that this map shows only one color, which means the curvature is the same everywhere in the pupil. This makes sense because a defocused eye produces a spherical wavefront and a spherical wavefront has the same curvature everywhere.

43 43 Wavefront Slope and Curvature: Astigmatism (z 22 ) (RMS=0.5 m, pupil=6mm, Jo= 0.27D) Another example of the first and second derivatives of a wavefront aberration function is shown here for Zernike astigmatism. In this case the ray aberration map shows that some rays are converging towards the optical axis and some are diverging. However, the spot diagram is still circular, as we would expect for an astigmatic system with zero spherical equivalent. Note that the mean curvature is zero everywhere in the pupil, which indicates that this wavefront is similar to a plane wave in that it will come to focus only at infinity. This result occurs because the wavefront has a saddle shape everywhere, with equal but opposite curvature in the horizontal and vertical directions. The average of these curvatures is zero everywhere in the pupil because equal but opposite curvatures cancel out.

44 44 Wavefront Slope and Curvature: Coma (z 31 ) (RMS=0.5 m, pupil=6mm, Meq= D) My last example of spatial derivativesof the aberration map is for coma. Note the complicated ray aberration map, which leads to a complex spot diagram which is the geometrical approximation to the exact point-spread function that can be computed by the Fourier optics methods described earlier. Currently we don t have enough experience looking at these spatial derivative maps to know whether they provide valuable insight into clinical problems. However I am optimistic that they will prove their worth in the end. The easy transition from the map of wavefront slopes to a spot diagram of the eye s point spread function is certainly a useful feature that is easy to understand. Also, curvature maps have a long tradition in ophthalmic optics for understanding the shape of the cornea and the optical properties of eyes so I suspect that these maps will also provide valuable insight, in the fullness of time.

45 45 Reference materials OSA Standards for Reporting Optical Aberrations of eyes Feature issues (2000, 2001, 2002) J. Refractive Surgery (Sept. issue) Monograph Customized Corneal Ablation: The Quest for Super Vision, MacRae, Krueger, & Applegate (eds.) Slack, Inc. (2001) Lecture slides This concludes my tutorial. Details of the OSA standards for reporting optical aberrations of eyes is available on the web at the address shown here. For further reading I can recommend recent feature issues the Journal of Refractive Surgery which contain a number of tutorial papers as well as current research reports in the general area of visual optics. I can also recommend the new monograph by MacRae, Krueger, and Applegate as an excellent introduction to the general topic of wavefront aberrations of the eye. Lastly, a copy of my lecture slides and speaker notes is available on the WEB at the address shown above.

46 46 Visual Optics Group at Indiana University Support 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 National Institutes of Health (R01-EY05109) Borish Center for Ophthalmic Research Vision Research at The end

47 47 Supplementary slides follow

48 48 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 the years for measuring the eye s aberrations. In the next talk, Ron Kruger will discuss these various technologies for measuring aberrations. However, it is worth mentioning one of these techniques now to demonstrate how any given method can be described 3 ways: in terms of ray aberrations, optical path length differences, and wavefront shape. Consider, for example, the Shack- Hartmann wavefront sensor.

49 49 Scheiner Optometer (1619) Ametropic eye y x Reference ray Scheiner s disk Retinal point source The principle of the Shack-Hartmann aberrometer is easily understood in terms of ray optics as an elaboration of the 17th century Scheiner Disk which Thomas Young later used to make his famous optometer. Imagine that we are able to place a point source of light on the patient s fundus. Light reflected back out of the eye from this source will tell us everything we need to know about the eye s aberrations if we can determine the direction of each ray of light as it emerges from the eye s pupil. Scheiner s disk allows us to isolate individual rays to determine their direction of propagation. If the pair of rays isolated by Scheiner s disk do not intersect at the eye s far point, then the eye is optically aberrated and we may quantify the magnitude of this aberration in terms of the horizontal and vertical deviation of a ray from the perfect reference ray.

50 50 Objective Scheiner/Hartmann Aberrometer d Point source CCD sensor Hartmann screen In principle we can track the direction of many rays simultaneously by adding more holes to Scheiner s disk to make what is known as a Hartmann screen. By capturing these isolated rays with photographic film or the CCD sensor inside a video camera, we can determine the direction of propagation of each ray as it passes through a specific point in the pupil. The end result is a map of the ray aberrations of the eye. To convert Scheiner s subjective method into an objective optometer, we reverse the direction of light propagation. In other words, we put a spot of light on the retina which then becomes a point source which reflects light back out of the eye. Next, drill some more holes in Scheiner s disk so it becomes a Hartmann screen. That way each aperture in the Hartmann screen isolates a narrow pencil of rays emerging from the eye through a specific part of the pupil. The rays then intersect a video sensor which tells us the horizontal and vertical displacement of the ray from the non-aberrated, reference position. Thus we have a Hartmann aberrometer for objectively measuring the ray aberrations of the eye. Now fill the individual apertures of the Hartmann screen with tiny lenses and you have a Hartmann-Shack aberrometer, or as I would prefer to say, a Scheiner-Hartmann-Shack aberrometer.

51 51 Scheiner-Hartmann-Shack Aberrometer Hartmann-Shack wavefront sensor Retinal point source CCD sensor Lenslet array Relay lenses The modern version of this idea suggested by Dr. Roland Shack and Ben Platt in 1965 replaces the Hartmann screen with an array of tiny lenses which are more efficient at capturing light and focusing it onto the CCD sensor. Having described the principle of operation of the Scheiner-Hartmann-Shack aberrometer in terms of ray aberrations, I will now repeat my description using the language of wave optics so that you will appreciate why this instrument is often called a wavefront sensor rather than a ray sensor.

52 52 Principle of Wavefront Sensor Sub-divide the wavefront with micro-lenslets. Local slope determines spot position on video sensor. Micro-lenslet array Front view Video sensor Perfect wavefront We can see how the wavefront sensor works by considering a side view of the lenslet array as shown on the right-hand side of this figure. For a perfect, emmetropic eye the reflected wavefront will be a plane wave perpendicular to a parallel bundle of rays. This wavefront is sub-divided by the lenslet array into smaller wavefronts that are focused by the lenses in the array into a perfect lattice of point images. When viewed from the front,as shown on the left, each image falls on the optical axis of the corresponding lenslet. The overall result is a regular grid of evenly spaced spots of light.

53 53 Principle of Wavefront Sensor Displacement of spots from reference grid indicates local slope of aberrated wavefront. Micro-lenslet array Front view Video sensor Aberrated wavefront If the eye is aberrated, then the reflected wavefront will be distorted and the individual rays will not be parallel to each other. Because rays are perpendicular to wavefronts, the direction of propagation of each ray is determined by the local slope of the wavefront over each lenslet. It is wavefront slope which determines where the spot is focused.. Thus an aberrated wavefront produces a disordered collection of spot images. By analyzing the displacement of each spot from its corresponding lenslet axis, we can deduce the slope of the aberrated wavefront when it entered the corresponding lenslet. Mathematical integration of this slope information yields the shape of the aberrated wavefront.

54 54 Slope of Wavefront: Defocus This is a more detailed display of wavefront slopes.