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Representation of Wavefront Aberrations Slide 1 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 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. Slide 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. lFourth, 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. lLastly, how may we interpret the spatial derivatives of the aberration map? Slide 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. Slide 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. Slide 5 General case: accommodating or ametropic eye P rays perfect P retinal image of Object point object point 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. 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. Slide 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. 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. Slide 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. 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.
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