Utilizing computer analysis of a simplified eye model, along with a description of the eye's retina structure in the region of highest resolution, allows us to better understand the basics of how the eye functions and the limits of visual resolution. Armed with this understanding, the optical engineer will be better equipped to predict the performance of a typical observer and of visual instruments when they are used in conjunction with the eye. This article provides data and examples intended to encourage that understanding.
The eye model
A number of established eye models have served well over the years to describe the makeup and function of the eye. Much of the data presented here has been taken from one of those established models.1 The eye, as an optical device, is quite basic and easy to understand. It may be thought of as a hollow sphere, about one inch in diameter, filled with a waterlike fluid. The transparent surface through which light enters the eye is called the cornea. The convex outer surface of the cornea provides the majority of optical power found in the eye. A few millimeters behind the cornea there is a relatively thick, double convex lens, the eyelens. Just in front of the eyelens, there is an opaque disk with an opening at its center. That opening will change diameter (involuntarily) to control the amount of light entering the eye. The disk, which also gives the eye its color, is referred to as the iris. The eyelens is able to change its shape, and thus its lens power. In so doing, it is able to focus the incoming light precisely on the image surface at the back of the eye, the retina. While the retina covers nearly half the inside surface of the eye, the most useful portion in terms of our ability to resolve detail is a 0.3-mm-diam portion in that area referred to as the fovea. It is this 0.3-mm-diam area that will be considered here, when discussing the limits of visual resolution.
Figure 1 shows details of the model eye developed for purposes of this article. As the parameters of the eye will vary between models, it is equally true that they will vary from person to person, depending on such factors as; age, size, gender, ethnicity, health, etc. The values shown in Fig. 1 are believed to represent valid data for a typical case involving an average, middle-aged person. The primary purpose for generating this model has been to permit legitimate computer analysis of the eye using modern optical design and analysis software. Figure 1 also contains the spectral response curve for the eye. The area under this curve has been divided into three equal areas and an area-centered wavelength has been selected to represent each of these areas during computer analysis.
Note that, for this model of the eye, the exterior surface of the cornea has been assumed to be aspheric, with a conic constant of -0.5. This indicates that the surface is midway between being spherical and paraboloidal. In the case of the actual eye, there are several other aspheric surfaces, as well as some gradient index (of refraction) materials within the eyelens. The model has been simplified by assigning all aspheric contributions to the outer surface of the cornea, while noting the amount of residual spherical aberration in the model and making it essentially equal to that found in the typical eye. Assuming a nominal iris/pupil diameter of 3.5 mm, this model has approximately 1 wave of spherical aberration and 3/4-wave of chromatic aberration for those wavelengths selected in Fig. 1.
The ability of the young, healthy eye to focus on objects over a range of distances is referred to as accommodation. This change is accomplished by a change in the shape of the eyelens. When viewing objects at infinity, the controlling muscles are relaxed and the power of the eyelens is at its minimum. When viewing objects closer to the eye, the radius on the front surface is made steeper and the power of the eyelens is increased. While several other more subtle changes take place to the thickness and the second radius of the eyelens, it is valid for purposes of our computer analysis to make changes to just the front radius. The standard point of maximum accommodation is at an object distance of 254 mm (10 in.). This distance is often referred to as the near point of a typical eye. For purposes of this discussion on visual resolution, the eye model is configured such that the radius of the eyelens front surface is reduced to 6.2 mm, making the lens nearly equiconvex. This change reduces the EFL of the eye from 17.4 to 16.7 mm. When set up for analysis with an object at 254 mm and the eye model per Fig. 1 (near focus), the magnification factor from the object surface to the fovea is 0.068.
The Modulation Transfer function (MTF) for the optics of the model eye with a pupil diameter of 3.5 mm, viewing an object at 254 mm, has been computed for the image at the fovea with the results shown in Fig. 2. Figure 2 contains a second curve, referred to as the aerial image modulation (AIM) curve of the fovea. This curve has been derived from Fig. 5.4 of Reference 1. While many other complex factors contribute, the AIM curve is in large part a result of the structure of the retina in the critical region of the fovea. Note that, as is the case for photographic film, where the AIM curve assumes optimum focus, exposure, and film processing, the AIM curve in Fig. 2 assumes all other aspects of the visual system are functioning properly. In Fig. 2, the AIM and the MTF curves intersect at a frequency of 110 cycle/mm. This indicates that, for the typical eye under the conditions described, the maximum resolution will be 110 cycle/mm (at the retina). At higher frequencies, the visual system requires a higher modulation of the image than is produced by the optics of the eye.
The retina contains two types of light sensing elements: rods and cones. While the rods are more sensitive to light, they do not deliver the resolving capability of the cones. In the critical region of the fovea there are only cones, at a density of approximately 200,000 cones per square millimeter.2 For the diameter of 0.3 mm, this means there are approximately 15,000 cones within the critical region of the fovea. The light sensitive area of each cone is about 1.5 m in diameter, while these cones are spaced approximately 2.5 m from center to center.1
One of the more intriguing aspects of this example is the scale of things at that point where the image is formed within the eye. Figure 3 shows an illustration of the conditions that exist when the eye is being used to examine a standard, high-contrast 3-bar chart located at the near point of the eye. At the magnification of 0.068x, a target of 7.5 cycle/mm will be imaged onto the fovea with a frequency of 110 cycle/mm. Analysis indicates that this target will be (just barely) resolved by the typical viewer, assuming all other factors are nominal. It will be found that this target frequency of 7.5 cycle/mm, at a distance of 254-mm, corresponds quite closely to the often quoted visual resolution limit of 2 arc min per resolved line pair. In the enlarged view in Fig. 3, a portion of the fovea roughly equal to the diameter of a human hair (0.002 in.) is shown, with the 110-cycle/mm image superimposed. A thoughtful review of this illustration leaves one with new respect and appreciation for the precision and performance of the human visual system.
This article contains data on a model of the human eye suitable for computer analysis using modern optical design software. Details of the model are presented along with a more general description of the actual eye and its function. Computer analysis reveals the limit of visual resolution in terms of the typical 3-bar target. An illustration of this case of maximum performance highlights the precision and remarkable capabilities of the eye and the human visual system. Data contained here can be used to determine the compatibility of any optical design intended for visual applications with the performance limits of the eye.
1. Modern Optical Engineering, Chap. 5, McGraw-Hill, 1990
2. MIL-HDBK-141, Chap. 4
Bruce H. Walker
Bruce H. Walker is the author of Optical Engineering Fundamentals, an optics textbook published by McGraw-Hill, and numerous articles dealing with the subjects of optical engineerng and lens design. He is an independent consultant.