Chapter 9 – Optics of the Normal Eye
PETER C. MAGNANTE
The optics of the eye can best be understood in terms of the optical characteristics of its components, the cornea, pupil, crystalline lens, and retina, and how they function in combination.
The quality and characteristics of the different optical components and the combination are described in the following terms.
• Chromatic aberration
• Spherical aberration
• High-order aberrations
• Light scattering
• Retinal factors
• Focal length
• Depth of focus
Optical function is best described by a series of tests.
• Visual acuity
• Contrast sensitivity
• Modulation transfer function
• Wave-front testing
• Vernier testing
• Detection of fast-moving objects (flicker)
• Dark adaptation
Eye clinicians define the abnormal eye by comparison with the normal. Thus the theoretical limits of the best quality, or threshold, image for the normal, emmetropic eye must be known. In this chapter the optical variables that determine the thresholds of image quality for distant objects for the human eye are discussed.
INDIVIDUAL OPTICAL ELEMENTS OF THE EYE
The cornea’s anterior surface is approximately spherical with a radius of curvature that is typically 8?mm. This surface is responsible for about two thirds of the eye’s refractive power. The corneal stroma must be transparent for high-quality image formation on the retina, yet the normal human cornea scatters 10% of the incident light. By comparison, the corneal stroma of the eagle is almost as transparent as glass.  This factor (along with the larger pupil size and finer cone diameter) is why the resolution of the eagle eye is better than 120 cycles per degree, which is equivalent to a Snellen acuity of 20/5 (6/1.5).
The aspherical shape of the cornea’s anterior surface affects the quality of the retinal image. Astigmatism is caused by this surface having different radii of curvature along different meridians. A survey of normal eyes shows that almost every human eye has a baseline corneal astigmatism of at least 0.25–0.50D.  Spherical aberration is caused by the corneal surface’s radius of curvature changing (generally increasing) with distance from the center of the pupil to the pupillary margin. The amount of spherical aberration contributed by the cornea varies with pupillary aperture and individual corneal shape. For a pupil 4?mm in diameter, spherical aberration varies from +0.21D to +1.62D, depending on the specific corneal form.
The iris, which gives the eye its color, expands or contracts to control the amount of light admitted to the eye. The pupil formed by the iris can range in diameter from 8?mm in very dim light down to about 1.5?mm under very bright conditions. There is a strong association between visual acuity and pupillary diameter. For example, visual acuity has been shown to improve steadily as background illumination increases up to a value of 3400cd/m2 . Also, as the eye focuses on objects close at hand, the pupil gets smaller.
Retinal image quality, as determined by optical aberrations such as spherical aberration, tends to improve with decreasing pupil diameter, because optical aberrations decrease with decreasing pupil size. On the other hand, retinal image quality, as determined by diffraction, tends to improve with increasing pupil diameter. For most eyes the best retinal images are obtained when the pupil diameter is about 2.4?mm, which is the diameter at which the effects of aberration and diffraction are balanced optimally. Thus the optimal pupillary size seems to be determined by several influences. In fact, Campbell and Gregory have shown that pupil size tends to be adjusted automatically to give optimal visual acuity over a wide range of luminance.
Crystalline Lens Factors
The crystalline lens, which has about one third of the eye’s refractive power, enables the eye to change focus. When the eye views nearby objects, the ciliary muscle changes the shape of the crystalline lens making it more bulbous and, consequently, optically more powerful. The lens of a young adult can focus over a range greater than 10D. Presbyopia, which begins at about 40 years of age, is the inability of the eye to focus (accommodate) due to hardening of the crystalline lens with age. When the eye can no longer accommodate at the reading distance, positive spectacle lenses of about 2–3D are prescribed to correct the difficulty.
The normal 20-year-old crystalline lens scatters about 20% of the incident light. The amount of scatter is more than double this in the normal 60-year-old lens. Such scatter significantly diminishes contrast sensitivity. Also, the normal 20-year-old lens absorbs about 30% of incident blue light. At age 60, this absorption increases to about 60% of the incident blue light.  The
Figure 9-1 Spherical aberrations produced by lenses of the same shape. A, A glass lens. B, A fish lens. The variation in index of refraction is responsible for the elimination of spherical aberration in the fish lens. (From Fernald RD, Wright SE. Nature. 1983;301:618–20.)
increase of blue light absorption with age results in subtly decreased color discrimination, as well as decreased chromatic aberration.
The variation in index of refraction of the crystalline lens (higher index in the nucleus, lower index in the cortex) is responsible for neutralization of a good part of the spherical aberration caused by the human cornea. Figure 9-1 shows how this variation of index of refraction in the spherical fish lens almost eliminates its spherical aberration when compared with a spherical glass lens.
Because the index of refraction of the ocular components of the eye varies with wavelength, colored objects located at the same distance from the eye are imaged at different distances with respect to the retina. This phenomenon is called axial chromatic aberration. In the human eye the magnitude of chromatic aberration is approximately 3D. However significant colored fringes around objects generally are not seen because of the preferential spectral sensitivity of human photoreceptors. Studies have shown that humans are many times more sensitive to yellow-green light with a central wavelength at 560?nm than to red or blue light.
The variation of refractive power with pupil diameter, which causes light rays to focus at different distances from the retinal plane, is called spherical aberration. The eye’s spherical aberration, in addition to depending on pupil diameter, depends on individual corneal contour, accommodative state, and the age of the lens. For a normal photopic eye, spherical aberration may vary from approximately 0.25D to almost 2D.
A comet-like tail or directional flare appearing in the retinal image, when a point source is viewed, is a manifestation of another aberration called coma. Because the eye is a somewhat nonaxial imaging device, and because the cornea and lens are not perfectly centered with respect to the pupil, coma generally is present in all human eyes.
HIGHER ORDER ABERRATIONS.
The eye’s aberrations of even higher order than the so-called primary aberrations, which include astigmatism, spherical aberration, and coma, are now being measured with wave front sensors, and the data are being used to control photorefractive surgical lasers with the hope of achieving aberration-corrected vision.
Another significant optical factor that degrades vision is intraocular light scatter. The mechanism of light scatter is different from the aberrations discussed above, each of which deviates the direction of light rays coming from points in object space to predictable and definite directions in image space. With light scattering, incoming light rays are deflected from their initial (i.e., prescattered) direction into random (postscattered) directions, which generally lie somewhere within a cone angle of approximately a degree or so. Therefore a dioptric value cannot be placed on the blur caused by light scatter. A glaring light worsens the effect of light scatter on vision. Thus a young, healthy tennis player may not see the ball when it is nearly in line with the Sun. Light scattering is the mechanism associated with most cataracts and causes significant degradation of vision due to image blur, loss of contrast sensitivity, and veiling glare.
An image may be considered as made up of an array of point-like regions. When a picture on the video screen is viewed with a magnifying glass, these small regions, called pixels, are seen clearly. Similarly, the elements that form a photographic image are the silver halide grains in the film’s emulsion. Likewise, the pixel elements comprising a retinal image are the cone and rod photoreceptors. It is the finite size of these photoreceptors that ultimately determines the eye’s ability to resolve fine details.
The finest details in a retinal image can be resolved only within the foveal macular area. This area is an elliptical zone of about 1.5?mm in maximal width, having an angular size of approximately 0.3 degrees about the eye’s visual axis. It contains over 2000 tightly packed light-sensitive cones. The cones themselves have diameters of 1–2?µm (a dimension comparable to 3–4 wavelengths of green light) and are separated by about 0.5?µm. Cone size is an important factor in determining the ultimate resolution of the human eye ( Fig. 9-2 ). In much of the fovea no nerve fiber layer, ganglion cell layer, inner plexiform layer, or inner nuclear layer is present, and in the very center of the fovea no outer nuclear layer is present. Only the outer plexiform and cones exist.
Another important aspect of the cone receptors is their orientation. Each cone functions as a “light pipe” or a fiber optic, which is directed to the second nodal point of the eye ( Fig. 9-3 ). This orientation optimally receives the light that forms an image and partially prevents this light from being scattered to neighboring cones.
Another retinal factor that helps to improve vision is the configuration of the foveal pit, which is a small concavity in the retina. This recessed shape acts as an antiglare device in which the walls of the depression prevent stray light, within the internal globe of the eye, from striking the cones at the center of the depression. Finally, the yellow macular pigment may be considered to act as a blue filter that limits chromatic aberration and also absorbs scattered light, which is predominantly of shorter wavelength (i.e., the blue end of the spectrum).
Resolution and Focal Length Factors
A derivation of the theoretical diffraction-limited resolution of a normal emmetropic human eye must consider the eye’s optimal pupil diameter, its focal length, which is associated with its axial length, and the anatomical size of the photoreceptors. A point object imaged by a diffraction-limited optical system has an angular
Figure 9-2 Retinal mosaic (rhesus monkey) in an area adjacent to the fovea. The large circles are rods and the clusters of small circles are cones. This section gives a perspective of the different receptor sizes. (From Wassle H, Reiman HJ. The mosaic of nerve cells in mammalian retina. Proc R Soc Lond B. 1978;200:441–61.)
diameter in radians (diameter at one half the peak intensity of the Airy disc) given by equation (9-1) .
In equation (9-1) , let pupil diameter be 2.4?mm which, for a normal eye, is the largest pupil diameter for which spherical aberration is insignificant, and let the wavelength be 0.00056?mm (yellow-green light) to find the diffraction-limited angular diameter = 0.00028 radians (or, equivalently, 0.98 minutes of arc). Note that this angular diameter matches the angular resolution of an eye with 20/20 Snellen acuity, because the black-on-white bands of the letter E on the 20/20 line of the Snellen chart are spaced 1 minute of arc apart.
The spatial diameter in millimeters of the diffraction-limited Airy disc on the retina is found by multiplying the angular diameter, given by equation (9-1) , by the effective focal length of the eye.
Using the angular diameter found from equation (9-1) and a value of 17?mm for the eye’s effective focal length (i.e., second nodal point to retina distance) in equation (9-2) results in the diffraction-limited spatial diameter = 0.0048?mm (i.e., 4.8?µm).
It is interesting to use our results to make a comparison with Kirschfield’s estimate that about five receptors are needed to scan the Airy disc in order to obtain the maximal visual information available. If we assume that the foveal cones are approximately 1.5?µm in diameter and are separated by about a 0.5?µm of space, then the distance between neighboring cones is 2.0?m. We estimate the number of receptors covered by the Airy disc by calculating in equation (9-3) the ratio of the area of the Airy disc to the area occupied by a single cone.
Using equation (9-3) we find that approximately six receptors are covered by the Airy disc in close accord with Kirschfield’s estimate of five. Thus given an eye with maximal sensitivity to yellow light and an optimal pupil size of 2.4?mm, we find that the human eye’s 17?mm effective focal length and, correspondingly, its 24?mm axial length are properly sized to achieve optimal resolution
Figure 9-3 Orientation of the photoreceptors. They all point toward the second nodal point of the eye.
for the cone sizes present. The higher resolution of the eagle’s eye compared with the human’s eye probably results from a larger pupil size-to-focal–length ratio, cones of smaller diameter, and a clearer cornea and lens.
Depth of Focus
An optical system with a fairly large depth of focus enables a fixed-focus camera to give sharp pictures of both a mountain in the distance and a subject 6?ft (1.8?m) away, an insect or a small animal to see objects clearly from 30?ft (9?m) to 4 inches (10?cm) away with no accommodation mechanism, and a presbyopic patient to read a newspaper through a pinhole with no reading correction. Depth of focus of an imaging system is defined to be the distance range (usually in millimeters) from the best-focused image distance where the resolution does not change or, equivalently, the blur caused by defocusing goes unnoticed. Depth of focus also can be expressed in diopters when the dioptric equivalent is the additional power needed for an optical system to change its focal length by an amount equal to the depth of focus. Depth of field, which is related to but different from depth of focus, is defined as the distance range that an object may move (toward or away from a fixed-focus optical system) and still be considered in focus.
In Figure 9-4 , the eye is represented in a simplified form with only one refractive element. We define the following symbols.
The object which can move from infinity to a near point
Focal length of the model eye
Distance from the retina where the object at the near point comes to focus
Photoreceptor size determining the eye’s ultimate resolution
Refractive index of the model eye
n/f, which is the dioptric power of the eye when an object is viewed at infinity
n/(f + x), which is the dioptric power of the eye when an object is viewed at the near point
The depth of focus may be expressed by equation (9-4) .
Equation (9-5) is obtained by the method of similar triangles shown in Figure 9-4 .
Figure 9-4 A model eye with a single refraction surface. For definitions and explanation, see text.
Substitution of equation (9-5) in equation (9-4) results in equation (9-6) .
Equation (9-6) shows that the depth of focus (D1 – D2 ), in diopters, is proportional to the product of the index of refraction (n) and the limiting photoreceptor or grain size (c), and inversely proportional to the product of the focal length of the system (f) and the pupil size (p). Therefore a small pupillary aperture brings objects at a wide range of distances from the lens into focus. Associated with the smaller pupillary size may be a larger blur spot due to diffraction. However if the blur spot is no larger than the size of the receptor, the blur will be unnoticed. Examples based on equation (9-6) are shown in Box 9-1 .
The examples of Box 9-1 use the reduced eye, where f = 22.2?mm and n = 1.33. In the more accurate human schematic eye of Gullstrand, f = 17?mm and n > 1.33, because the different indices of refraction of the cornea, aqueous, lens, and vitreous are taken into account. For comparison, a clinically orientated study of humans showed that pupils of diameter 1–2?mm produced a mean depth of focus >4D.    The image-enhancing mechanism in the retina and brain may have helped to increase the subjects’ perceived depth of focus in the clinical studies cited when compared with our calculated results found in Box 9-1 .
The example in Box 9-2 shows that, even without a mammalian accommodation system, the fly theoretically can see objects from infinity to within <1?mm. Of course, this theoretical range does not account for the degradation effect of diffraction on fine details. However, it may be assumed that the fly is most interested in large objects.
Depth of focus and visual acuity are quite different among the species. For example, although a human and a falcon have a comparable depth of focus, the falcon has a much better acuity than that of humans. On the other hand the bat's eye, compared with those of the human and falcon, has a relatively large depth of focus of about 10D and relatively poor visual acuity. As noted in equation (9-6) , a larger retinal grain size implies larger depth of focus. However, it also implies reduced resolution.
Figure 9-5 shows an out-of-focus eye viewing a distant point source of light and how reducing pupil size with a pinhole (or a stenopeic slit) diminishes the size of the blurred retinal image. The diagram illustrates how a small aperture placed in front of the pupil of an out-of-focus eye (e.g., myopic or hyperopic) can improve visual acuity. The pinhole not only helps to reduce focus error and increase depth of focus, it also helps to correct both regular
Determination of Depth of Focus. The Reduced Human Eye Model with One Refracting Surface
p = 3?mm
f = 22.2?mm (for reduced eye model with one refracting surface)
c = three cones (assume each cone is 1.5?µm in diameter and spacing between cones is 0.5?µm); the total cluster of three cones = 4.5?µm + 2 spaces = 5.5?µm
n = 1.333
Calculation of depth of focus:
This figure falls within the experimental literature, which shows a range of depth of foci in human subjects from +0.04D to +0.47D.
Example 1 is calculated for a resolution system of 1 minute of arc or the equivalent of 20/20 (6/6) visual acuity, and a high level of contrast. If the system's limit is 20/40 (6/12), the angle of resolution may be doubled to 2 minutes, which covers six cones or 11?µm. Assume the eye has a 2?mm pupil.
Calculation of depth of focus:
The system in Example 2 [i.e., 20/40 (6/12) resolution], but using a 1?mm pinhole.
Calculation to Show the Depth of Focus for the Fly's Eye
In the compound eye of the fly the organization is based on ommatidia with separate groups of receptors (about eight) under each lens.
p = 26?µm
f = 50?µm
c = 2?µm
n = 1.365
Calculation of depth of focus:
and irregular astigmatism, as well as other higher order optical aberrations. Disadvantages of using a small-aperture system are increased diffraction blur and decreased retinal illuminance. For example, the diffraction blur caused by a 0.5?mm pinhole will reduce noticeably the visual acuity of someone who tests normally as having 20/20 vision. Also, if a small aperture is used with a camera in order to take pictures indoors that have a large depth of field, bright lights will be needed to illuminate the scene.
On the other hand, a small-aperture lens can efficiently deliver good illuminance provided the lens has a correspondingly short focal length. The key parameter that governs the efficiency of illumination of an imaging system, as well as its theoretical optical resolution, is the size of the cone angle of the light from a point object which comes through the aperture and converges onto a point in the image plane. The term numerical aperture, or NA, which is used commonly to describe microscope objectives, is a measure of this cone angle. An alternate parameter (equivalent to
Figure 9-5 The principle of the pinhole. A narrower pupillary aperture decreases the angle of the cone of light that produces the blur circle. Ultimately the blur circle, albeit dimmer, is the size of the limiting cluster of photoreceptors (i.e., pixel-size or grain-size equivalent). A stenopeic slit may be considered as a line of pinholes.
Figure 9-6 Spacing of gap in the letter C. This letter is used to determine the minimal separable spacing or resolution of the eye at the retina. The gap in the letter C subtending 1 minute, when imaged on the retina, has a dimension (Y) representing the resolution of the eye where tan (1 minute) = Y/0.67 (with Y in inches); therefore Y = 0.00019 inches (or 4.8?m). The overall size of the letter is X = 0.349 inches, and the gap dimension is 0.070 inches (see text).
the cone angle), which is used commonly to describe photographic lenses, is the “speed” or “f-number” of the lens. The f-number is the ratio of the lens's focal length to its entrance aperture diameter. For example, an f/8 lens has a focal length that is 8 times greater than its aperture diameter. Note that the f-number may be the same for a hawk's eye as for the facet of an insect's eye.
VISUAL ACUITY TESTING
The idea that the minimal separation between two point sources of light was a measure of vision dates back to Hooke in 1679, when he noted, “tis hardly possible for any animal eye well to distinguish an angle much smaller than that of a minute: and where two objects are not farther distant than a minute, if they are bright objects, they coalesce and appear as one.” In the early nineteenth century, Purkinje and Young used letters of various sizes for “judging the extent of the power of distinguishing objects too near or too remote for perfect vision.” Finally, in 1863, Professor Hermann Snellen of Utrecht developed his classic test letters. He quantitated the lines by comparison of the visual acuity of a patient with that of his assistant who had perfect vision. Thus 20/200 (6/60) vision meant that the patient could see at 20?ft (6?m) what Snellen's assistant could see at 200?ft (60?m). 
The essence of correct identification of the letters on the Snellen chart is to see the clear spaces between the black elements of the letter. Thus in Figure 9-6 , the angular spacing between the bars of the C is 1 minute for the 20/20 (6/6) letter. The entire letter has an angular height of 5 minutes. To calculate the height, x, of a 20/20 (6/6) letter use equation (9-11) .
From equation (9-11) , x = 0.0291 feet (0.349 inches). In like manner, the 20/200 (6/60) letter is 10 times taller, or 3.49 inches (8.87?cm) high.
The Snellen acuity test traditionally is done at a distance of 20?ft (6?m). At this distance very little accommodation is required by the patient. For hospital patients, testing must often be carried
Figure 9-7 Visual acuity charts. Standard Snellen and Bailey–Lovie charts.
out in a smaller room. If the doctor stands at the foot of the bed and the patient sits, propped up at the head of the bed, the distance between them is about 5?ft (1.5?m). Thus the classic Snellen chart, with its conventional notations, may be used if the chart is reduced to one fourth its original size. Admittedly, a test at 5?ft (1.5?m) requires the emmetropic patient to accommodate 0.67D.
Over the years it has become apparent that projection of the Snellen chart onto a screen in a darkened examination room does not give an accurate replication of “everyday” visual function. For example, the high contrast black-on-white letters do not represent the contrast of most objects seen in everyday life. The dark examination room, which is devoid of glare sources, also is not representative of most daytime visual tasks.
As the projector bulb ages or collects dirt and as the projection lens becomes dusty, the contrast of the letters projected on the chart decreases. Thus a change in readings between patient visits may not always arise from a significant change in the visual status of the patient. At present, British standards require 480–600 lux to illuminate distant wall charts and 1200 lux to illuminate projected charts.
As the letters become smaller on the Snellen chart, the number of letters per line increases. Thus one error per line means a different score for each line. It, therefore, is necessary to establish criteria by which it can be agreed that a patient has seen the line. Some clinicians credit a patient if more than one half the letters are identified correctly. Others require identification of all the letters before credit is given. Also remember that no orderly progression of size change exists from line to line. Thus a two-line change on the Snellen chart going from the 20/200 (6/60) line to the 20/80 (6/24) represents an improvement of visual acuity by a factor of 2.5, whereas a two-line change going from the 20/30 (6/9) line to the 20/20 (6/6) line represents an improvement by only a factor of 1.5.
Another problem is that the identification of different letters of the same size has been shown to vary in difficulty. Thus A and L are easier to identify than E. The Bailey–Lovie chart ( Fig. 9-7 ), designed by two Australian optometrists and modified by Ferris et al. in 1982, uses 10 letters of similar difficulty with 5 different letters per line and has uniform size change between neighboring lines. Another approach is to use the Landolt ring test in which circles, each with an open gap, of decreasing size are used in successive lines, with the orientation of the gaps in the circles randomly changing.
The 20/20 (6/6) Snellen line represents the ability to see 1 minute of arc, which is close to the theoretical diffraction limit, but the occasional patient can see the 20/15 (6/4.5) or 20/10 (6/3) line. Four explanations suggest themselves. First, some individuals may have cone outer segment diameters of less than 1.5?µm, which would give a finer-grain mosaic having cone separations of less than 1 minute of arc. Second, longer eyes provide slightly magnified retinal images, thereby tending to yield better acuities. Third, some eyes may have less aberration than others, which would allow them to function optimally with larger pupils having, consequently, better diffraction-limited performance. Finally, the neural image enhancement mechanisms may be slightly more efficient in certain favored individuals.
CONTRAST SENSITIVITY TESTING
Visual acuity testing is relatively inexpensive, takes little time to perform, and describes visual function with one notation, such as 20/40 (6/12). Best of all, for over 150 years it has provided an end point for the correction of a patient's refractive error. Yet contrast sensitivity testing, a time-consuming test born in the laboratory of the visual physiologist and described by a graph rather than a simple notation, recently has become a popular clinical test. It describes a number of subtle levels of vision not accounted for by the visual acuity test; thus it more accurately quantifies the loss of vision in cataracts, corneal edema, neuro-ophthalmic diseases, and certain retinal diseases. Although these advantages have been known for a long time, the recent enhanced popularity has arisen because of patients with cataracts. As lifespan increases, more patients who have cataracts request medical help. Very often, their complaints of objects that appear faded or objects that are more difficult to see in bright light are not described accurately by their Snellen acuity scores. Contrast sensitivity tests and glare sensitivity tests do quantitate these complaints.
Contrast sensitivity testing is similar to Snellen visual acuity testing in that it tests using several differently sized letters or grid patterns. However it is different from visual acuity testing, because the letters (or grid patterns) are displayed in six or more shades of gray instead of the standard black letters of the Snellen chart. Thus contrast sensitivity testing reports show a contrast threshold (i.e., lightest shade of gray just perceived) for each of several letter (or grid pattern) sizes.
The components of a conventional newspaper photo consist of various regions associated with the scene where each region is filled in with a definite density of black dots depicting that region's contrast or level of gray. Such newspaper photos may have over 100 half-tone levels (i.e., densities of black dots) to represent the different contrast levels in the scene. A video engineer will describe the ability of an electronic display device to faithfully depict contrast levels by citing the display's gray-scale resolution. For example, a video monitor may be said to have 8-bit gray-scale resolution, which means that 28 = 256 different levels of gray (ranging from white to black) can fill in the various regions that make up a picture on the monitor's screen.
Whereas a black letter on a white background is a scene of high contrast, a child crossing the road at dusk and a car looming in a fog are scenes of low contrast. The contrast of a target on a background is defined by equation (9-12) .
As an example, suppose a photometer measures the luminance of a target at 100 units of light and the luminance of the background at 50 units of light. Substitution into equation (9-12) gives equation (9-13) .
Suppose the contrast of a target of a certain size is 0.33, which also may represent a particular older patient's threshold, which means that this patient cannot detect similar sized targets of lower contrast. The older patient's contrast sensitivity (CS) is the reciprocal of the contrast, namely CS = 3.0. On the other hand, a young, healthy subject viewing a target of the same size may have a contrast threshold of 0.01 with a corresponding CS = 100. Occasionally subjects (for certain size targets) have even better contrast thresholds. A subject could have a contrast threshold of 0.003, which converts into a CS of 333. In the visual psychology literature, CS often is described in logarithmic terms. For example, associated with CS = 10 is log(CS) = 1, with CS = 100 is log(CS) = 2, and with CS = 1000 is log(CS) = 3, and so on.
Both the visual scientist and the optical engineer use a series of alternating black and white bars as targets. The optical engineer describes the fineness of a target by the number of line pairs per millimeter (a line pair consists of a dark bar with a white space next to it). The higher the number of line pairs per millimeter, the finer is the target. For example, about 82 line pairs per millimeter imaged on the retina of an eye with a focal length of 21?mm is equivalent to a periodic black-white target in object space, where the white space between two black spaces subtends approximately a minute of arc (like the letter E of the Snellen chart viewed at 20 feet). Equivalently, with a Snellen chart viewed at 20 feet, 109 line pairs per millimeter on the retina is equivalent to the 20/15 (6/4.5) letters.
The visual scientist generally describes a periodic bar pattern in terms of its spatial frequency as perceived at the test distance—the units are cycles per degree (cpd). A cycle is a black bar and a white space. To convert Snellen units into cpd at the 20?ft (6?m) testing distance, the Snellen denominator is divided into 600 (180). For example, 20/20 (6/6) converts into 30?cpd. Likewise, 20/200 (6/60) converts into 3?cpd.
So far, targets have been described as high-contrast dark bars of different spatial frequency against a white background. These also are known as square waves or Foucault gratings. However, in optics very few images can be described as perfect square waves with perfectly sharp edges. Diffraction tends to make most edges slightly fuzzy, as do spherical aberration and oblique astigmatism, particularly in the case of the optics of the eye. If the light intensity is plotted across a strongly blurred image of a Foucault grating, a sine wave pattern results. Sine wave patterns have great appeal, because they can be considered the essential elements from which any pattern can be constructed. The mathematician can break down any alternating pattern, be it an electrocardiogram or a trumpet's sound wave, into a unique sum of sine waves. This mathematical decomposition of patterns into sinusoidal components is known as a Fourier transformation. Joseph Fourier, a French mathematician, initially developed this waveform language to describe sound waves and vibrations. Fourier's theorem describes the way that