Thibos, Larry N.*
The Prentice Medal is named after Charles F. Prentice, a founding father of professional Optometry. In the second edition of his treatise on ophthalmic lenses, published in 1907, Prentice said:1
“The prism diopter stands unchallenged in its unique ability to harmonize all of the refracting elements in the optometrical lens-case by establishing a complete and inseparable relationship between prisms and lenses.”
What is this “complete and inseparable relationship” that plays such a fundamental role in optometric optics and makes the Prentice rule even more important today than it was in Prentice’s time a century ago?
As every student of optometry knows, the Prentice rule describes how lenses refract rays of light. Fig. 1 shows a bundle of parallel rays from a distant point of light, like a star in the sky, entering a lens parallel to the optical axis of the lens. For an ideal spherical lens, all of these light rays come to focus at a single point at a distance d from the lens. The symbol F in the Prentice equation is the dioptric power of the lens, which is equal to the inverse of distance d. The letter c in the Prentice equation refers to the displacement of an incoming ray from the optical axis. If we isolate any ray for inspection, we see from the geometry of right triangles that the angle Δ with which the refracted ray strikes the optical axis is equal to c/d, but because 1/d is the lens power, the angle Δ is the product of ray displacement and lens power. That is the Prentice rule: Δ = c F.
To connect the Prentice rule with wavefronts, we draw a continuous surface that is perpendicular to every ray of light. Now draw a tangent line to this wavefront surface at the foot of our isolated ray so we can see how the slope of the tangent line (which is also the slope of the wavefront) makes the same angle Δ with a reference plane perpendicular to the optical axis. That means the Prentice rule also tells us the slope of the wavefront at every point on the wavefront. Moreover, if we rearrange the Prentice equation to form a ratio of wavefront slope Δ to the displacement c, the result is 1/d, which is the vergence of the emerging ray and also the wavefront vergence at each ray location on the wavefront.
This way of interpreting the Prentice equation is an important change in viewpoint. We are no longer talking about the lens itself, but the wavefront produced by that lens. This focuses our attention on the wavefront of light rather than the lens that shaped the wavefront. Moreover, the diagram in Fig. 1 defines a perfect wavefront, the gold standard for wavefront analysis. Rearranging the Prentice equation showed that, if the ratio of Δ/c is constant, then all of the rays have the same vergence, which means that they all cross the optical axis at exactly the same focus point. That is what makes a wavefront perfect.
Of course, no lens is perfect, and that goes for eyes as well. Fig. 2 shows an aberrated lens forming an aberrated wavefront from a distant point source. The diagram is misleading (yet commonly found in textbooks) because many of the rays are not even in the plane of the diagram—it’s only our imagination that is in the plane of the diagram. The rays coming out of the plane of the diagram are called “skew rays.” Because skew rays don’t intersect the optical axis, the meaning of angle Δ as an angle of intersection between the ray and optical axis ceases to be useful. Nevertheless, we can always draw a tangent plane perpendicular to the ray (even if the ray is skew) so we can measure the angle Δ between a ray and a plane. That angle tells us the slope of the wavefront at a specific point on the wavefront. If we can measure wavefront slope at many ray locations, then we can use calculus to integrate those wavefront slope measurements to reveal the wavefront’s shape.
This generalization of the Prentice rule to become a statement about wavefront slope is the foundation of modern wavefront aberrometry. The rule applies even if the wavefront is not a perfect sphere, so the rays do not have the same vergence and do not cross the axis at the same place. As illustrated in Fig. 3, the Prentice rule works also for skew rays associated with nonprincipal meridia of an astigmatic wavefront. For such rays, wavefront slope has two components, the usual meridional component associated with wavefront radial vergence and the tangential component responsible for skewness that prevents the ray from intersecting the optical axis. Regardless of the shape of the wavefront, the meridional slope is sufficient to reconstruct wavefront shape by mathematical integration.
What does this wavefront theory of geometrical optics have to do with the eye? Instead of light arriving at a lens from a star, imagine a point source of light reflected from the retina and emerging from the eye’s optical system as a wavefront (Fig. 4). As previously shown, if we can measure the slope of the emerging wavefront at many points on the wavefront, then we have all of the information needed to reconstruct the shape of that wavefront, thereby obtaining a comprehensive description of the eye’s optical aberrations. Although it is possible to accomplish the same result by measuring the intersection point of light rays with an image plane near the focus point, that approach is more difficult because the rays overlap and get confused near the focus point, so they must be isolated and measured sequentially. By measuring wavefront slope near the eye, where individual rays are well separated, it becomes possible to make many measurements simultaneously.
If Prentice were alive today, I’m sure he would immediately understand the principle of modern wavefront analysis because this 21st century methodology is just a clinical application of his 20th century rule relating wavefront slope to wavefront shape. However, Prentice wasn’t the first person to think this way. The basic idea for isolating rays of light to measure wavefront slopes to determine refractive errors is at least 400 years old. The earliest reference I know is the Scheiner treatise2 on physiological optics published in 1619. Scheiner was a contemporary of Galileo and Kepler, and he devised a simple device now called Scheiner disk shown in Fig. 5 to isolate a pair of rays to determine the eye’s refractive error. Some 200 years later, Thomas Young used the Scheiner principle to build an optometer that he used to measure the astigmatism of the human eye for the first time.
Workers in other areas of optics are perhaps more familiar with Hartmann’s rediscovery of the Scheiner disk in the early 20th century as a way to measure the quality of rays and lenses. In fact, Hartmann and Prentice were contemporaries so we might speculate that if they had known of each other’s work, we would be celebrating the 100th birthday of wavefront aberrometry today. The schematic diagram in Fig. 6 shows how the Scheiner-Hartmann screen isolates many individual rays simultaneously as they emerge from an eye. The intersection of each isolated ray with a light detector tells us the direction of the ray and thus the slope of the wavefront in the plane of the Hartmann screen.
The modern version of the Scheiner disk principle suggested by Dr. Roland Shack and Ben Platt replaces the Hartmann screen with an array of tiny lenses, which are more efficient at capturing light and focusing it onto a video sensor. As shown in Fig. 7, a relay telescope images the eye’s entrance pupil onto the lenslet array so that the wavefront measurements include corneal refraction but is conceptually equivalent to placing the lenslets in the pupil plane. This schematic diagram of a modern wavefront aberrometer shows a laser beam being used to place a small spot of light on the retina. The spot would be placed on the fovea to measure ocular aberrations along the principal line of sight but could be instead placed anywhere on the peripheral retina to measure aberrations along secondary lines of sight associated with peripheral vision. When light from this retinal “beacon” reflects back out of the eye, it is captured by a pair of relay lenses that focus the eye’s pupil plane onto an array of lenses that subdivide the light into many small beams. Depending on how each little beam is deflected as propagates to form a spot image on the sensor, we can figure out the slope of the wavefront over each lenslet. Although this device is commonly known as a “Shack-Hartmann” aberrometer, the actual historical lineage is: Scheiner, Young, Hartmann, Prentice, Shack, Liang (who patented the idea!).
The invention of technology for measuring wavefront shape opens up a whole new world of ocular wavefront aberrometry: the systematic classification of refractive errors according to wavefront shape. Basic wavefront shapes can be classified by extending the 20th century practice of decomposing refractive errors into focus errors (associated with myopia and hyperopia) and astigmatic errors, as illustrated in the top half of Fig. 8. In the 21st century, thanks to wavefront technology, many more basic shapes are needed to describe eyes. Coma and trefoil are two examples of many basic shapes described mathematically by Zernike polynomials.
Frits Zernike (1888–1966) is a Nobel laureate famous for his invention of phase microscopy. In the process, he invented a set of eponymous mathematical functions and then kindly donated them to vision science for describing the optical properties of eyes. The first 21 Zernike basis functions, or “modes” as they are often called, are shown in Fig. 9. This pyramid of shapes is logically organized into rows and columns, rather like the periodic table of the elements. Each row of the pyramid corresponds to a given order of the polynomial component of the function, and each column corresponds to a different meridional frequency of the sinusoidal component. The basic idea of Zernike wavefront analysis is to mathematically decompose a wavefront into a weighted sum of basic shapes. The weights, or amount of each component, are called Zernike coefficients Cnm, where n is the pyramid row number and m is the column number. The six modes in the top three rows are the basis of 20th century optometry but are now recognized as just the tip of the Zernike pyramid. An unlimited number of modes below the tip represent the expanding scope of optometry and ophthalmology into the 21st century. From this perspective, the second-order power vector components M, J0, and J45 are history. They belong to the 20th century before we grasped the larger landscape revealed by Zernike analysis.
The preceding diagrams give ample evidence of the three-dimensional nature of optical wavefront analysis. Unfortunately, most optics textbooks contain only two-dimensional drawings that attempt to explain three-dimentional concepts. Imagine instead a textbook constructed like a child’s storybook that reveals Cinderella’s castle when the book opens! An example of unfolding optical origami useful for envisioning the geometry of rays and wavefronts is illustrated in Fig. 10 (see electronic appendix online for a working model, available at http://links.lww.com/OPX/A136). Follow the instructions to fold the airplane to reveal the optical axis, the chief ray connecting pupil center to an off-axis object point, the meridional plane, the sagittal plane, and the AIRPLANE!
Looking back in time, I would say 1997 was the year our optical mind-set began to change because of wavefront aberrometry. That was the year Optometry and Vision Science published a pair of feature issues highlighting exploratory work being done using the new wavefront concept for describing the optical aberrations of the eyes and their correction. To introduce one of those issues, I wrote an editorial entitled “The New Visual Optics,” which said, in part3:
“Contemporary visual optics research is changing our mindset, our way of thinking about the optical system of the eye, and in the process is re-defining the field of visual optics. All of the eye’s optical imperfections will one day be represented comprehensively by a two-dimensional map in the plane of the pupil. This pupil map will look much like the corneal topographic maps currently used to describe the shape of the corneal surface. Interpretation of the two maps will be quite different, however, because the pupil map describes how wavefronts entering the eye from each point in the visual world become distorted from the perfect spherical shape needed to form the ideal retinal image.”
The future arrived much sooner than anyone anticipated. Just 3 years after that editorial, the modern era of clinical wavefront aberrometry began with the birth of the 21st century and the introduction of the first commercial aberrometer for performing wavefront refraction based on the Hartmann-Shack wavefront sensor. This introduction of aberrometry technology into the clinic also introduced a change in attitude: higher order aberrations were no longer a nuisance to be avoided. Instead, aberrations will be taken into account when prescribing the best possible correcting lens based on a key principle: the best prescription is the one that optimizes retinal image quality.
But how can a clinician know if image quality is optimized when the only person in the world who can see the retinal image is the patient? The clinician can’t look inside the eye to see if the image is well focused, so how can he or she know that image quality has been maximized by a particular combination of correcting lenses? The answer is that we can calculate the retinal image even though we cannot observe it. Optical theory tells us that the wavefront aberration map is sufficient to enable an accurate calculation of retinal images, including the effects of diffraction and interference, using the principles of physical optics. As illustrated in Fig. 11, the first step uses the measured wavefront aberration map of an eye to compute the retinal image of a single point of light. Because any object in the world is a collection of point sources, each of which is blurred the same way, we can add these partially overlapping images of all the object points using a mathematical calculation called “convolution.” The result is a rendering of the retinal image of any object, for example, an eye chart. In this way, clinicians are empowered with a tool capable of providing insight into the patient’s problem when faced with the task of deciding which correcting lens is better, number 1 or number 2?
Given an ability to compute images from wavefront measurements, we can begin to address the simplest and most fundamental of optometric questions: where is the far point of an aberrated eye? According to the wavefront refraction principle previously given, the far point is the axial location for the object that optimizes the quality of the retinal image when accommodation is relaxed. This definition leaves unstated the criterion by which retinal image quality should be judged. Finding a useful metric of image quality is challenging because, even for a simple point of light, rays passing through different parts of the pupil come to focus at different axial distances. An example is shown in Fig. 12 for an eye with positive spherical aberration. If the visual object is located at the distance where paraxial rays come into focus, the retinal image is clear and sharp but has relatively low contrast. Alternatively, if the object is located at the distance where marginal rays focus, the image will appear darker with more contrast but is not as legible or sharp. These conflicting criteria of “sharper” versus “darker” are well known to clinicians, but only by taking into account the effects of spherical aberration does an optical explanation emerge and an understanding develop of how this particular aberration affects vision.
An important insight gained by wavefront analysis is that the optometrist’s persistent question, “which is better?” does not have a single correct answer because many criteria, in addition to “sharper” or “darker,” can be used for judging image quality. For example, the patient may judge perceived quality based on some internal notion of personal esthetics, a sense of beauty that makes one corrective lens preferable to another. Or quality may be based on the fidelity of the image—whether it meets some expectation or platonic notion of what the object ought to look like. Both of these criteria are internal, subjective, and difficult to standardize for reliable and consistent use in a scientific or clinical setting.
An objective approach to the quantification of image quality defines functional optical quality in terms of measureable visual performance. This approach leads to the formulation of a fundamental hypothesis that links the world of the psyche (mind) to the physical world of observable events. That psychophysical linking hypothesis is better images yield better visual performance. Conversely, better performance indicates better images. Thus, for the image illustrated in Fig. 13, if visual performance on a specific task (such as letter identification) is better when viewing through the lens instead of around it, then the lens is providing superior image quality regardless of the esthetic appeal or fidelity of the image to the original object.
An objective paradigm for deciding which correcting lens is better begins by using a measured wavefront map to calculate the retinal image of a single point of light (called the point-spread function [PSF]). The image of a point is typically too complicated for direct evaluation, as illustrated in Fig. 14, but it can be reduced to a single number using an image quality calculator. This metric value is then used to decide which correcting lens is optimum. Some metrics of image quality are more easily specified in the spatial frequency domain, in which case the calculator would be based on the eye’s optical transfer function (OTF), which can also be computed from the wavefront map.
The efficacy of image quality calculations can be improved by taking into account the early stages of neural processing of the retinal image. For example, Fig. 15 illustrates convolution of the optical PSF (which describes how light from a point source spreads laterally) with a neural PSF (which describes how the neural response to a point stimulus on the retina spreads laterally) yields a visual PSF (which describes the neural response to a single point of light in the object). The ratio of the maximum value of this visual PSF for the patient’s eye to the maximum value achieved for an optically perfect eye is a metric of image quality called the visual Strehl ratio that is useful for predicting visual acuity. This metric may also be computed in the spatial frequency domain by multiplying the OTF by the neural contrast sensitivity function of the observer to achieve a visual transfer function. The volume under this visual transfer function, compared with that obtained for an optically perfect eye, is an equivalent measure of the visual Strehl ratio.
Fig. 16 summarizes the computational process of wavefront refraction, which is the process of finding the optimum spherocylindrical correcting lens that maximizes the quality of the retinal image.4 Conceptually, the idea is equivalent to having a tiny OD inside the eye to assess retinal image quality using the image quality calculator. A virtual lens is then adjusted by mathematically adding a spherical wavefront to correct defocus (and/or an astigmatic wavefront to correct astigmatism) to the eye’s wavefront to maximize the quality of the computed retinal image.
Only a small step of imagination is needed to broaden the concept of wavefront refraction to objectively determine the refractive state of an aberrated accommodating eye. As indicated in Fig. 17 (see animated version online), all ocular aberrations change when the crystalline lens accommodates, but the primary change is for Zernike defocus (C20). The same computational procedure used to determine the optimum correcting lens of the relaxed eye applies also to the accommodating eye by optimizing an appropriate metric of image quality.
A hypothetical example of wavefront measurement of refractive state of an accommodating eye is illustrated in Fig. 18. The typical human eye has positive spherical aberration when accommodation is relaxed, in which case the optimum focus plane is closer to the eye for marginal rays than for paraxial rays. When eyes accommodate to a near target, spherical aberration typically changes sign to become negative. Consequently, the relative locations of focal planes for paraxial and marginal rays become reversed, as illustrated in the bottom panel of the figure. That expands the range of accommodation if the goal is to produce a sharper image but compresses the range of accommodation if the goal is to produce an image with maximum contrast. That insight would not have been achieved without taking aberrations into account when examining retinal image quality.
Fig. 19 summarizes the wavefront method for computing the refractive state of an accommodating eye. The patient views a stimulus that evokes accommodation while the clinician simultaneously measures the eye’s wavefront aberration and measures image quality. The question is was the target in the optimum location? Or would retinal image quality have been greater if the target were slightly closer or farther away? If image quality can be improved by defocusing the measured wavefront, that indicates the presence of an accommodative error that can be quantified as the difference between the optimum target vergence and the actual target vergence.
Given this wavefront approach to measuring refractive state, a new perspective emerges on many current issues in accommodation research. For example, is retinal image quality maximum for all states of accommodation? Or are we all perpetual laggards who don’t accommodate enough to maximize the quality of our retinal images? How do changes in higher order aberrations and pupil size during accommodation affect refractive state? The notion that the pupil plays a role in determining refractive state is a radical idea that arises only when aberrations are taken into account. How do age-related changes in accommodation affect retinal image quality? That issue is becoming more relevant with the aging population and our need to better understand the visual consequences of presbyopia. Do presbyopia therapies based on aberration concepts work? If not, why not?
Wavefront aberrometry is everywhere in optometry, ophthalmology, and visual optics; it is truly the “wave” of the future, and the future has arrived. In this lecture, I’ve commented only on objective refraction of the relaxed eye and the measurement of refractive state during accommodation. In addition, wavefront technology is being used to study tear film optics and the ocular surface. Wavefront sensors are the key component of adaptive optics systems for high-resolution imaging of the fundus. Wavefront aberrometers help assess treatment strategies using contact lenses or refractive surgery for correcting, or exploiting, higher order aberrations. Wavefront ideas may even help unlock the mysteries of myopia development by revealing the optical conditions that control ocular growth.
These are exciting times for visual optics research and its clinical applications. The American Academy of Optometry plays a critical role in translating advances in knowledge gained by research into patient care. By giving voice to optometric research, this Academy makes its greatest contribution, by way of the practitioner, to the public’s well-being.
Larry N. Thibos
School of Optometry Indiana University
800 E. Atwater Ave.
Bloomington, IN 47405-3680