4
The Optical Building Blocks of Eyes
Almost 2 km below the turbulent surface waters of the open ocean, the softskin smooth-head, a small black fish, slowly swims through the inky darkness. Its tiny eyes pierce the blackness in search of the rare and fitful sparks of living light produced by other animals, the only light this fish will ever see. These tiny punctate flashes of light, brilliant against the shroud of darkness behind, may signal the arrival of a seldomly encountered mate or announce the presence of an equally rare meal. Localizing the point, and acting accordingly, will be a matter of urgency—it may be days or even weeks before such an opportunity arises again. But was that faint glimmer really a light? Did the fish’s eyes collect enough photons to be sure the flash was real? And where exactly was it? Are the fish’s eyes acute enough to be certain of where the flash was located? The answers to these questions ultimately depend on the optical structure of the eyes themselves. Only if the eye is sufficiently sensitive to light and has visual sampling stations adequately dense will the flash be localized clearly. And obviously this visual world—and the eye morphology best suited to it—is very different from that of a coral-reef fish experiencing a colorful and brilliantly lit extended scene. How eyes evolve to best provide their owners with reliable information thus depends on where and how the eyes are used. Even the most rudimentary of eyes—with appropriate levels of sensitivity and resolution—can be highly successful. How successful eyes are constructed is the topic of this chapter.
Irrespective of their optical specializations—a topic we return to in chapter 5—all eyes have one thing in common: they collect and absorb light arriving from different places in the environment, thus giving animals information about the relative distribution of light and dark in the surrounding world, including the contrasts and positions of objects. This information is used to support a variety of visual tasks, such as identifying and avoiding predators, detecting and pursuing prey or conspecifics, and orientating and navigating within the habitat. Although some animals use their eyes to perform more or less all of these tasks, others do not. As nicely summarized by Land and Nilsson (2006), all visual systems evolved within one of two main categories, being either “general purpose” or “special purpose.” The eyes of vertebrates and the more advanced invertebrates (such as arthropods and cephalopods) are of the first type: these eyes have evolved to perform many different visual tasks and are accompanied by the advanced brains necessary to analyze all of them. Animals with “special purpose” visual systems—for example, jellyfish, worms, bivalves, and gastropods—have visual systems optimized for one primary purpose, such as to detect the shadow of an approaching predator. Their eyes, brains, and nervous hardware are often coadapted to perform this special task with maximum neuronal efficiency.
But no matter whether general purpose or special purpose, all eyes must fulfill certain basic building requirements in order to provide their owners with reliable visual information. To see what these are, it is necessary to first define what an eye actually is or, possibly more importantly, what it is not.
What Is an Eye?
The simplest type of visual organ—found in many smaller invertebrates and larvae (notably of worms and insects)—is an aggregation of one or more photoreceptors on the body surface, shielded on one side by a pigment cell containing screening pigment granules. Such “eye spots” are unable to detect the direction from which light is incident (i.e., they do not possess spatial vision) and are therefore little more than simple detectors of light intensity. Because spatial vision, no matter how crude, is considered to be the hallmark of a “true eye,” eye spots are not considered true eyes. But for those invertebrates that possess them, eye spots are able to detect the presence or absence of light and to compare its intensity sequentially in different directions, thus allowing animals to avoid light, to move toward it, or to detect sudden changes in its brightness.
In a “true eye,” photoreceptors need to be exposed to light in one or more directions and shaded from light in others, thus creating spatial vision. The easiest way to achieve this is to push the pigment-backed layer of light-sensitive photo receptors into the epidermis, creating a photoreceptor-lined invagination or “pit.” Such “pigment-pit eyes” (figure 4.1) are common in many invertebrate lineages. Because the photoreceptors each occupy different positions in the pigment-lined pit, they are each able to receive light from different directions in space but are shaded in many other directions, thus creating crude spatial vision. In fact, the deeper and narrower the pit, the narrower the region of space from which photoreceptors receive light and the better the spatial resolution. As a result, pit eyes are considered to be true eyes.
Figure 4.1 Pigment-pit eye of the arc clam Anadara notabilis. (A) A semithin longitudinal section through two pigment pit eyes located on the edge of the clam’s mantle. Scale bar: 40 μm. (From Nilsson, 1994) (B) A schematic diagram of the pigment cup eye in A showing its everse invaginated retina of rhabdomeric photoreceptors (exaggerated in size for clarity; small stacked circles represent the microvilli). Each photoreceptor receives light from a different broad region of space (blue shading), thus endowing the eye with crude spatial resolution. (Redrawn with permission from Nilsson, 1994)
This brief discussion of eye spots and pigment-pit eyes highlights two design principles that are common to all true eyes: (1) each photoreceptor has a unique physical position in the retina and thus receives light from a restricted region of the outside world, and (2) light from that region reaches the photoreceptor through an aperture or “pupil” that restricts the entrance of light from other regions (i.e., shades the photoreceptor from unwanted light). This pupil not only affects the resolution but also sets the sensitivity because a larger pupil supplies more light to the retina. Even greater resolution and sensitivity are achieved from another watershed advance during the evolution of vision: the appearance of refractive lenses (Nilsson and Pelger, 1994). Such lenses—typical of compound eyes and camera eyes—focus an image of the outside world onto the underlying retina and have the potential to dramatically improve visual performance.
Good spatial resolution and high sensitivity to light, arguably the two most highly desirable properties of any eye, generally trade off against each other: in an eye of given size, it is difficult to maximize one without compromising the other. Which of the two is maximized ultimately depends on what the animal has evolved to see and, in particular, what light level the animal is usually active in. What is very useful for our purposes is that this trade-off also reveals the basic design principles for constructing a successful eye.
Building a Good Eye: A Trade-off between Resolution and Sensitivity
What makes a successful eye? To answer this question, consider the simplified schematic camera eye shown in figure 4.2A. In analogy to its namesake, a camera eye has a retina of visual cells that senses an image formed by an overlying cornea and lens. These eyes are found in all vertebrates, including ourselves, as well as in many invertebrates, and we discuss this eye type in more detail in chapter 5. Even though we have used camera eyes as an example, the reasoning developed below applies to all types of eye.
In most camera eyes the photoreceptors are packed into an orderly matrix around the back of the eye that collectively receives a focused image of the outside world. Each photoreceptor is responsible for collecting and absorbing light from a single small region in this image (and thus from a single small region of the outside world). In this sense, to borrow the digital camera term, each photoreceptor defines a single “pixel” in the visual image; two neighboring photoreceptors therefore define two neighboring “pixels.” And as is well known from digital cameras, the density of pixels sets the finest spatial detail that can be reconstructed: smaller and more densely packed pixels have the potential to reconstruct—or “sample”—finer details. However, this is true only if each pixel can collect enough light. If, in the quest for higher resolution, the pixel becomes too small—that is, if the region of space from which the pixel collects photons becomes too small—then it will collect too little light to reliably sample the image. This of course becomes more and more problematic as light levels fall: at any given light level the minimum visual pixel size (and the maximum resolution) will be limited by the sensitivity of the eye.
Figure 4.2 A schematic camera eye viewing a visual scene (the trunk and bark of a eucalyptus tree). (A) Array of photoreceptors in the retina, whose photoreceptive segments have a length l and a diameter d, receives a focused cone of light of angular half-width θ through a circular pupil of diameter A. Angular separation of two neighboring photoreceptors in the retina is the interreceptor angle Δφ. Each photoreceptor has a narrow receptive field (of angular width Δρ) and thus receives light from only a small part (or “pixel”) of the visual scene. The focal length f of the eye is defined as the distance between the optical nodal point N (in this case located at the center of the lens) and the focal plane at the distal tips of the photoreceptors. (B) Electro physiologically measured receptive fields of single photoreceptors in the superposition compound eyes of two species of hawkmoths, the day-active Macroglossum stellatarum and the nocturnal Deilephila elpenor. The half-width of the receptive field Δρ, also known as the “acceptance angle,” is wider in the nocturnal species (3.0°) than in the diurnal species (1.3°). The receptive field is roughly Gaussian in shape, although in nocturnal species such as Deilephila, where the cone of incident light can be especially broad, optical cross talk (see text) may induce significant off-axis flanks (as evidenced here), thereby reducing spatial resolution. (B adapted from Warrant et al., 2003)
Thus, inherent in the design of any eye is an unavoidable trade-off-between resolution and sensitivity. In an eye that strives for higher resolution the pixels must become smaller and more densely packed. In doing this the photon sample collected by each, and the reliability of the light intensity measurement made by each, necessarily declines. For an eye adapted for vision in bright light, when photon samples are large, this is not likely to be a problem. But for a nocturnal eye the situation is probably quite different—larger and more coarsely packed pixels, each catching as many of the available photons as possible, are instead likely to be the preferred design.
This trade-off between resolution and sensitivity is readily revealed by the few anatomical parameters shown in figure 4.2A. Let us begin by considering sensitivity.
Sensitivity
Why does a small photon catch limit the ability of photoreceptors to discriminate the contrasts of fine spatial details? The answer to this question is beautifully explained in the classic studies of Selig Hecht, Simon Schlaer, and Maurice Pirenne from 70 years ago (Hecht et al., 1942; Pirenne, 1948). They were the first to recognize, on the basis of psychophysical experiments, that individual photorecepters (rods) in the human retina are capable of detecting single photons, an ability probably shared by all animals. However, despite this apparently remarkable performance, they also recognized that this was insufficient on its own to discriminate objects in dim light. In fact, our threshold for detecting the mere presence of a light source requires at least 5–14 photons (Pirenne, 1948). It turns out that the problem for object discrimination in dim light (at least in part) lies in the random and unpredictable nature of photon arrival at the retina. Pirenne explained this point in his well-known illustration of a grid of 400 photoreceptors receiving photons from the image of a perfectly dark circular object on a brighter background (figure 4.3; Pirenne, 1948). At the dimmest light level (1×: figure 4.3, upper left panel), the sporadic and random arrival of photons from the background results in their absorption by only six photoreceptors—in this situation the object remains indistinguishable. Even at a 10-fold higher background light level (10×: figure 4.3, upper right panel), the object is still disguised, hidden in the random “noise” of photon absorptions. After a further 10 times increase in light intensity (100×: figure 4.3, lower left panel), the dark object becomes noticeable, but not without a degree of uncertainty. It is not until light levels are increased by yet a further 10 times that the object can be distinguished clearly (1000×: figure 4.3, lower right panel).
Figure 4.3 Spatial resolution and light level. A matrix of 400 photoreceptors (small circles) image a black disk (large circle at center) at four different light levels that differ in intensity by factors of 10. Photoreceptors excited by light are shown as white; inactive photoreceptors are shown as black. (Adapted and redrawn from Pirenne, 1948)
Pirenne’s famous illustration highlights one of the main problems for vision in dim light—noise arising from the random arrival of photons, sometimes referred to as photon “shot noise.” The effect of this noise is even more devastating for realistic visual scenes that contain a great spectrum of contrasts from large to small than it is for a simple high-contrast black-or-white binary scene like the one used by Pirenne. The reason for this, as predicted by Pirenne’s demonstration, is that the smaller the contrast of an object (or an internal detail of an object) the earlier it will be erased by shot noise as light levels fall. Thus, at any given light level, the smallest detectable contrast will be set by the level of photon shot noise.
At about the same time as Pirenne, two scientists—the American physicist Albert Rose (1942) and the Dutch biologist Hugo de Vries (1943)—independently quantified the effects of photon shot noise on the discrimination of visual contrast. They recognized that because the arrival and absorption of photons are stochastic (and governed by Poisson statistics), quantum fluctuations will limit the finest contrast that can be discriminated. If we say that a photoreceptor absorbs N photons during one visual integration time, the uncertainty or shot noise associated with this sample will be √N photons. In other words, the photoreceptor will absorb N ± √N photons (Rose, 1942; de Vries, 1943). The visual signal-to-noise ratio, defined as N/√N (which equals √N), improves with increasing photon catch, implying that photon shot noise and contrast discrimination are worse at lower light levels. This is the famous “Rose–de Vries” or “square root” law of visual detection at low light levels: the visual signal-to-noise ratio, and thus contrast discrimination, improves as the square-root of photon catch.
To illustrate this, imagine two neighboring photoreceptors (which we will call P1 and P2) that sample photons from a contrast border in a scene. Let P1 sample photons from the brighter side of the border and P2 from the darker side. Let’s also say that the contrast is such that the brighter side of the border reflects twice as many photons as the darker side. Consider first a very low level of illumination. Imagine that during one visual integration time P1 samples six photons from the brighter side of the contrast border and P2 samples three photons from the darker side. Due to shot noise and the square-root law, these two samples will have an uncertainty associated with them: 6 ± 2.4 photons and 3 ± 1.7 photons, respectively. In other words, the noise levels would be about half the magnitude of the signals, and in this case the samples would not differ significantly. Thus, at the level of the photoreceptors the contrast border would be drowned by noise and remain invisible. But in much brighter light it is a different story. Imagine instead that P1 samples 6 million photons, and P2 3 million. Now the samples are 6,000,000 ± 2,449 photons and 3,000,000 ± 1,732 photons. These noise levels are a minuscule fraction of the signal levels, and there is no question that at this light intensity the contrast border would easily be seen. Indeed, even borders of significantly lower contrast would likely be seen.
Thus, the finest contrast that can be discriminated by the visual system depends on the light level. Moreover, photon shot noise also sets the upper limit to the visual signal-to-noise ratio (which in our simple example above is around 2 at the lower light level and about 2000 at the higher light level). In reality, however, the signal-to-noise ratio is always lower than this. This is because there are two further sources of noise that degrade vision in dim light even more.
Shot noise is an unavoidable consequence of the properties of light and is thus an external or “extrinsic” source of noise. In addition to this there are also two internal, or “intrinsic,” sources of noise. The first of these, referred to as “transducer noise,” is inherent in the photoreceptor’s response to single photons. Even though Hecht, Schlaer, and Pirenne recognized quite early that visual systems must be capable of responding to single photons, it was not until nearly two decades later that the American physiologist Stephen Yeandle actually recorded such responses from a photoreceptor in the horseshoe crab Limulus (Yeandle, 1958). These small but distinct electrical responses, which he termed “bumps,” are now commonly recorded from photoreceptors in both vertebrates and invertebrates (figure 4.4A). Six years later, together with Michealangelo Fourtes, Yeandle established that there is a 1:1 relationship between transduced photons and bumps (Fuortes and Yeandle, 1964): a single bump results from the absorption and transduction of no more than a single photon, and a single transduced photon leads to no more than a single bump. What also became apparent was that despite being the response to an invariant stimulus—a single photon—bumps were, in contrast, highly variable, changing in their latency, duration, and amplitude (see figure 4.4A). This inability of the photoreceptors to produce an identical electrical response to each absorbed photon introduces visual noise. This source of noise, originating in the biochemical processes leading to signal amplification, degrades the reliability of vision (Lillywhite, 1977, 1981; Lillywhite and Laughlin, 1979; Laughlin and Lillywhite, 1982).
Figure 4.4 Quantum events in photoreceptors. (A) Single-photon responses recorded intracellularly in a photoreceptor of the nocturnal bee Megalopta genalis (measured in response to continuous dim light). (From Frederiksen et al., 2008) (B) Dark noise recorded in a rod photoreceptor of the nocturnal toad Bufo marinus in complete darkness, showing the two main components of the noise: (1) discrete “dark events” (red dots) indistinguishable from responses to real photons and (2) a continuous low-amplitude response fluctuation (e.g., within the green box). This suction electrode recording shows rod outer segment membrane currents at 22°C. (Adapted from Baylor et al., 1980) Note the much slower response speed in toads compared to bees (A).
The second source of intrinsic noise, referred to as “dark noise,” arises because the biochemical pathways responsible for transduction are occasionally activated—even in perfect darkness (Barlow, 1956). Two components of dark noise have been identified in recordings from photoreceptors (figure 4.4B; Baylor et al., 1980): (1) a continuous low-amplitude fluctuation in measured electrical activity (sometimes called membrane or channel noise) and (2) discrete “dark events,” electrical responses that are indistinguishable from those produced by real photons. The continuous component arises from spontaneous thermal activation of rhodopsin molecules or of intermediate components in the phototransduction chain (such as phosphodiesterase) (Rieke and Baylor, 1996). The amplitude of this membrane noise is negligible in insects (Lillywhite and Laughlin, 1979) but can be quite significant in vertebrate photoreceptors, particularly cones. “Dark events” also arise due to spontaneous thermal activations of rhodopsin molecules. In those animals in which dark events have been measured, they are rare (e.g., insects, crustaceans, toads, and primates) (Lillywhite and Laughlin, 1979; Baylor et al., 1980, 1984; Dubs et al., 1981; Doujak, 1985; Henderson et al., 2000; Katz and Minke, 2012). At their most frequent they occur around once per minute at 20°C, although in most species they occur a lot less frequently. At very low light levels both components of dark noise can significantly contaminate visual signals (Rieke, 2008) and even set the ultimate limit to visual sensitivity (as found in the nocturnal toad Bufo bufo) (Aho et al., 1988, 1993a). And because dark noise is the result of thermal activation of the molecular processes of transduction, it should come as no surprise that it is more pronounced at higher retinal temperatures. Thus, in cold-blooded animals living in cold environments such as the deep sea (where water temperatures are around 4°C), dark noise is probably quite low. Nevertheless, dark noise may still be sufficient to limit the visibility of bioluminescent flashes.
These two sources of intrinsic noise—transducer noise and dark noise—further degrade visual reliability in dim light because they raise the contrast threshold for visual discrimination above that resulting from shot noise alone (Laughlin and Lillywhite, 1982). Taken together, intrinsic and extrinsic noise pose a serious threat to the reliability of vision in dim light.
How then can visual reliability—and thus contrast discrimination—be improved in dim light? One way is to have an optical system that captures more light (e.g., a wider lens and larger pupil). Another way is to have retinal circuitry optimized to reduce noise and maximize the visual signal-to-noise ratio. The photoreceptors themselves can be optimized for signaling in dim light as in nocturnal insects (e.g., Laughlin, 1996; Frederiksen et al., 2008), or the underlying circuitry can filter out noise by using a strategy of nonlinear thresholding as has so beautifully been demonstrated at rod-rod bipolar cell synapses in the retina of nocturnal mice (Okawa and Sampath, 2007; Okawa et al., 2010; Pahlberg and Sampath, 2011). Finally, an eye can simply give up the attempt to see fine spatial detail altogether and have larger “pixels,” instead improving the discrimination of coarser details. All of these solutions improve the reliability of vision at low light levels and are commonplace in animals that have evolved to see well in dim light (as we discuss in detail in chapter 11). How eyes maximize their sensitivity to light is the topic to which we turn next.
The Optical Sensitivity of Eyes to Extended Scenes and Point Sources
The features of eyes that maximize their sensitivity have been nicely summarized by Michael Land in his famous sensitivity equation (Land, 1981). This equation accounts for the most common type of visual scenes experienced by animals: “extended” scenes. Unlike scenes dominated by point sources, such as those experienced by deep-sea animals that search the inky depths for tiny points of bioluminescence, extended scenes are characterized by objects visible as a panorama throughout three-dimensional space.
Good sensitivity to an extended scene results from a pupil of large area (πA2/4) and photoreceptors each viewing a large solid angle (or “pixel”) of visual space (πd2/4 f2 steradians) and absorbing a substantial fraction of the incident light (kl/[2.3 + kl] for broad-spectrum terrestrial daylight). The optical sensitivity S of the eye to an extended broad-spectrum scene (in units of square micrometers times steradians) is then simply given by the product of these three factors (Kirschfeld, 1974; Land, 1981; Warrant and Nilsson, 1998):
In mesopelagic deep-sea habitats, where daylight is essentially monochromatic (blue light of around 480 nm wavelength), the following expression is more accurate (Land, 1981; Warrant and Nilsson, 1998):
In both equations A is the diameter of the pupil, f the focal length of the eye, and d, l, and k the diameter, length, and absorption coefficient of the photoreceptors, respectively (figure 4.2A). Wider pupils, shorter focal lengths, or larger photoreceptors (i.e., wider “visual pixels”) all increase S. These are common features of eyes adapted for vision in dim light (see chapter 11).
The absorption coefficient k is a constant that describes the efficiency with which a photoreceptor absorbs light—a photoreceptor with a larger value of k absorbs a greater fraction of the incident light per unit length than a photoreceptor with a lower value. If the photoreceptor length has units of micrometers, then k has units of inverse micrometers. Thus, a photoreceptor with k = 0.01 μm–1 absorbs a fraction of 0.01 (or 1%) of the propagating incident light for every micrometer of photoreceptor through which the light passes. Photoreceptor absorption coefficients vary over a roughly 10-fold range (Warrant and Nilsson, 1998) and tend to be smaller in invertebrates (ca. 0.005–0.01 μm–1) than in vertebrates (ca. 0.01–0.05 μm–1). The reason for this is most likely the fact that in vertebrate photoreceptors the ratio of photoreceptive membrane to cytoplasm is greater than it is in the photoreceptors of invertebrates. The total fraction of light absorbed by the photoreceptor (the final bracketed term in equations 4.1a and 4.1b) can now easily be calculated. If, for example, the photoreceptor is 100 μm long, then this means that for white light (equation 4.1a), the total fraction of light absorbed is 100 ⋅ 0.01/(2.3 + 100 ⋅ 0.01) = 0.303. That is, around 30% of the incident light is absorbed by this photoreceptor.
To put all this into perspective, imagine an eye of optical sensitivity S μm2 sr that views an extended scene of average radiance R photons μm–2 s–1 sr–1. For a nocturnal terrestrial scene, R will be roughly 8 orders of magnitude lower than for the same scene illuminated by bright sunshine. The total number of photons N absorbed per second from the scene by a photoreceptor in the eye is then simply
Thus, eyes of greater optical sensitivity absorb more photons from an extended scene of given intensity. In other words, the highly sensitive superposition compound eyes of a nocturnal hawkmoth (see chapter 5) will absorb several hundred times as many photons every second as the apposition compound eyes of a day-active honeybee that views exactly the same scene. This example highlights the great variation in optical sensitivities found in the animal kingdom, which vary over many orders of magnitude and are fairly tightly correlated with the light levels in which animals are active (table 4.1).
Even very small eyes, like those of nocturnal moths, can have high optical sensitivities. This is because the optical sensitivity is also dependent on the eye’s “F-number,” a parameter that describes the eye’s light-gathering capacity and does not depend on eye size. The F-number is simply the ratio of the focal length to the pupil diameter, f/A. F-numbers are quite familiar to photographers because camera lenses of lower F-number produce brighter images. The same is true in eyes, as an inspection of equation 4.1 can testify: a higher optical sensitivity results from a lower F-number (i.e., from higher A or lower f). And because f and A tend to be correlated (i.e., larger eyes tend to have longer focal lengths and larger pupils), even small eyes can have very high optical sensitivities. The tiny camera eyes of the nocturnal net-casting spider Dinopis subrufus—a formidable nocturnal hunter (Austin and Blest, 1979)—have an optical sensitivity that is over 100 times greater than that of our own much larger eyes (calculated for dark-adapted foveal vision).
Finally, as we mentioned above, many deep-sea animals have eyes adapted to see pinpoints of bioluminescence against a profoundly dark background. The image of a point source on the retina is, by definition, also a point of light. For a visual detector to collect all the light from this point, the “pixel” it represents need not be any larger than the image itself. Receptors viewing large solid angles of space do not improve the detection of a point source. In fact, one would predict that an eye adapted to reliably detect a point source of light against a dark background should only need a large pupil and long photoreceptors of high absorption coefficient. Thus, in the deep sea, the optical sensitivity (now with units of square micrometers) of an eye to monochromatic bioluminescent point sources is simply:
where all symbols have the same meanings as before (figure 4.2A). If accurate localization of the point source is also necessary, then one would predict the eye to have good spatial resolution as well. As we see in chapter 6, these conditions are fulfilled in the eyes of many deep-sea animals that rely on the detection of bioluminescence for survival and reproduction.
For a human observer, with large eyes and large dark-adapted pupils (ca. 8 mm diameter), the night sky is ablaze with stars. This, however, is not the case for the net-casting spider Dinopis. Despite its superior optical sensitivity for viewing dimly lit extended scenes, the spider is now at a distinct disadvantage when viewing stars—with a pupil area 36 times smaller than that of humans, only a small fraction of the stars visible to us would be visible to the spider!
Table 4.1 The optical sensitivities S and photoreceptor acceptance angles Δρ of a selection of dark-adapted eyes (in order of decreasing S)
Notes: This table is expanded from an original given by Land (1981). S has units μm2 sr. Δρ is calculated using equation 4.4 and has units of degrees. A = diameter of aperture (μm); d and l = diameter and length of the photreceptor, respectively (μm); f = focal length (μm). See chapter 5 for a full description of eye types. NSup = nocturnal superposition eye; MSup = mesopelagic superposition eye; DSup = diurnal superposition eye; DCam = diurnal camera eye; NCam = nocturnal camera eye; LMCam = lower mesopelagic camera eye; ECam = epipelagic camera eye; LCam = littoral camera eye; DApp = diurnal apposition eye; NApp = nocturnal apposition eye; LMApp = lower mesopelagic apposition eye; Pig = pigment cup eye; Lmirr = littoral concave mirror eye.
a Focal length is calculated from Matthiessen’s ratio: f = 1.25A.b Rhabdom length quoted as double the actual length due to the presence of a tapetum.c S was calculated with k = 0.0067 μm–1, using equation 4.1b for monochromatic light. All other values were calculated with equation 4.1a for broad-spectrum light.d Values taken from frontal eye.e Posterior medial (PM) eye;f Anterior lateral (AL) eye. Sources: Invertebrates (Warrant, 2006) and vertebrates (Warrant and Nilsson, 1998).
Resolution
As we have just seen, optical sensitivity is independent of eye size, which might lead the reader to wonder why animals would ever bother having bigger eyes. It turns out that small eyes are seriously limited in another property critical for vision—resolution. Small eyes have small apertures and short focal lengths, and their retinas have limited space for photoreceptors. All of these limitations have serious implications for resolution.
A higher physical packing density of photoreceptors, which is specified by their smaller angular separation, will increase the eye’s spatial resolution. This angular separation is defined by the interreceptor angle Δφ (figure 4.2A), an angle that depends on the physical separation of photoreceptors in the retina (given by the photoreceptors’ center-to-center spacing rcc) and the eye’s focal length f:
In many retinas the photoreceptors are so tightly packed that they touch. The retina is then said to be “contiguous,” meaning that rcc is the same as the photoreceptor’s diameter d. The interreceptor angle is then simply given by:
Spatial resolution can thus be increased by reducing the diameter of the photoreceptors or by increasing the eye’s focal length. A downside of reducing d is that it decreases the photoreceptor’s photon capture area—and thus the eye’s optical sensitivity (equation 4.1)—and this may not be desirable in dim light. In fact, as discussed in chapter 11, photoreceptors are often pooled neurally to effectively enlarge the retinal “pixels” so that they can collectively capture more light in dim conditions. The rod photoreceptors of our own retina are typically pooled in the hundreds for precisely this reason.
The downside of increasing the eye’s focal length is that this is only possible by having a larger eye, and this may be impractical for many reasons. The limited body size of the animal itself, the constraints associated with its mode of locomotion, or the extra energy costs associated with running a larger eye are just some of the problems that may prevent the evolution of a larger eye. As we describe in chapter 6, some remarkable animals have nevertheless managed to pull this off, either by simply having much larger eyes than other closely related animals of the same size or by cramming a small piece of a much larger eye onto their otherwise small heads.
In the absence of all other effects (such as the quality of the optical image focused on the retina; see below) the interreceptor angle Δφ would set the finest spatial detail that could be seen. In reality, however, the finest spatial detail is determined by the size of the photoreceptor’s “receptive field,” that is, the size of the region of visual space from which the photoreceptor is capable of receiving photons. The diameter of this roughly circular receptive field is sometimes called the photoreceptor’s “acceptance angle” Δρ (figure 4.2A), and in its simplest geometric form this is given by:
Values of Δρ in different eyes are given in table 4.1. Notice how eyes having higher optical sensitivity frequently have broader acceptance angles (and lower spatial resolution), revealing the unavoidable trade-off-between resolution and sensitivity.
This description of the receptive field as a circular region of space is somewhat simplistic: in reality the receptive field of a photoreceptor is equivalent to its sensitivity to light arriving from different angles of incidence φ (figure 4.2B). For axial light (φ = 0°), sensitivity is maximal (100%), but for every other direction it is less, falling steadily as the angle of incidence is increased. A typical receptive field profile is roughly Gaussian in shape: in three dimensions it resembles a bell.
Notice that equations 4.4 and 4.3b are the same, that is, in a contiguous retina Δρ = Δφ. Thus, to achieve maximal spatial resolution Δρ could be minimized in exactly the same way as Δφ, with the same downsides. However, in real retinas Δρ is not commonly smaller than Δφ. This is because eyes typically possess one or more optical limitations (e.g., aberrations) that blur the image formed in the retina. This blurring broadens Δρ and can coarsen spatial resolution to a value below that predicted by the photoreceptor matrix. This is readily seen in the two-dimensional receptive field of the nocturnal elephant hawk moth Deilephila elpenor (figure 4.2B), measured using intracellular electrophysiology. This has a half-width (i.e., acceptance angle Δρ) of 3.0° (Warrant et al., 2003), which is considerably larger than the geometric value of 0.9° obtained using equation 4.4 (table 4.1). Moreover, the experimental curve is somewhat more flared than a Gaussian of the same half-width—the substantial flanks visible at higher values of φ are the result of optical cross talk (see below), an optical flaw that can seriously degrade spatial resolution.
Optical Limitations on Image Quality
The first limitation is unavoidable and a property of all lenses, namely diffraction (figure 4.5). Diffraction results from an important physical characteristic of light itself—its wave nature. No matter how optically perfect an imaging system otherwise is, there is no escaping diffraction. To see its effects on image quality, imagine a lens that focuses the image of a distant star (figure 4.5A). Even though stars are infinitesimally small points of light, their images on the retina (or on the image sensor of a telescope) are far from infinitesimal and are always blurred due to diffraction. Light from the star reaches the lens as a series of straight parallel wavefronts, and as these strike the outer surface of the lens they are refracted. Those regions of the wavefront near the thinner edge of the lens pass through it rather quickly, but those regions entering near the thick center of the lens take longer, and by the time they exit they are out of phase with other regions of the wavefront. Instead of being straight, each exiting wavefront is curved—those regions of the wavefront arriving in phase at the image plane of the lens reinforce to create brighter image areas, whereas those arriving out of phase annihilate to create dimmer image areas. As a result, the image of the star turns out to be a complicated circular pattern consisting of a central bright peak surrounded by light and dark rings (figure 4.5B,C). This pattern is known as the Airy diffraction pattern, named for the British astronomer George Biddell Airy, who first described it in 1835. Its central bright peak is known as the Airy disk. Because in the worst cases the Airy disk may cover several photoreceptors, it leads to uncertainty regarding the exact position of the star and thus degrades spatial resolution.
The size of the Airy disk depends on the wavelength of the stimulating light (λ) and the diameter of the lens aperture (pupil) through which this light has passed (A). The half-width ΔρA of the Airy disk’s intensity profile (figure 4.5A) is given by
and its diameter (i.e., the diameter of the entire central peak measured to the center of the first dark ring) is 2.44ΔρA. Thus, in eyes (or telescopes) with large lenses (i.e., large A), the effects of diffraction can be quite minor. But in eyes with very small lenses the width of the Airy disk could be large, with rather dramatic consequences for resolution. A good example can be found among very tiny insects, whose compound eyes may have ommatidial lens diameters of 10 μm or less. If we consider that such an eye is viewing a point source of monochromatic green light that has a wavelength of 500 nm (or 0.5 μm), then the diameter of the Airy disk would be 2.44 × 0.5/10 radians, or about 7° across. In a compound eye, the interreceptor angle is defined by the angular separation of the ommatidia (see chapter 6), and this could easily be a lot less than 7°. Thus, even though the packing of photoreceptors might predict a certain spatial resolution, it might not be realized because of diffraction.
Figure 4.5 Diffraction. (A) Monochromatic light (from a point source at infinity) of wavelength λ passes through the aperture of a lens (defined by a pupil of diameter A). The propagating wavefronts are diffracted, and the resulting image on the retina is a symmetric pattern of light and dark rings called the Airy diffraction pattern, whose central peak (the Airy disk) has a half-width of λ/A. (B) The Airy diffraction pattern seen graphically in three dimensions. (C) The Airy diffraction pattern seen on the image plane from above, showing its characteristic light and dark rings and its conspicuous central peak. (B and C courtesy of Yakir Gagnon)
But even in eyes with large apertures where the Airy disk is as small as, or smaller than, the receptor spacing (such as in large vertebrate eyes), there are other optical defects that may blur the image sufficiently that resolution is nonetheless lower than predicted by the retinal grain. Many lenses (and eyes) suffer from aberrations, notably spherical and chromatic aberration (figure 4.6), and curiously the effects of these imperfections worsen as lenses become larger relative to focal length—quite the opposite to diffraction.
The first of these aberrations—spherical aberration—arises because of the simple inability of lenses to focus incoming parallel light rays (e.g., from a star) to a single point (figure 4.6A,B). In fact, in the case of positive spherical aberration, rays entering the periphery of the lens will be focused to a position closer to the rear of the lens than rays entering near the center of the lens. The actual position of best focus is that for which the lateral spread of rays is least. In such an eye, our image of the star—already blurred by diffraction—would now be blurred even more. To make matters worse, the position of least lateral spread may not even coincide with the retinal surface. Spherical aberration can be corrected by making the lens surface aspheric, but this solution has limited success because generally only those rays entering the eye parallel, or nearly parallel, to its optical axis are corrected: rays entering at other angles are less well focused than they would be in a spherical lens. Nevertheless, this solution has arisen many times, and our own cornea is aspheric for this reason. Correction of spherical aberration can also be achieved by an appropriate gradient of refractive index within the interior of the lens (with refractive index falling from the lens center to its edge; see chapter 5), and this is frequently the situation found in camera eyes (e.g., spiders: Blest and Land, 1977; cephalopod mollusks: Sweeney et al., 2007; fish: Matthiessen, 1882). Even our own lens corrects spherical aberration in this manner.
Chromatic aberration (figure 4.6C,D) arises because the transparent material of the lens is invariably dispersive; that is, its refractive index is higher for shorter wave-lengths of light. Thus, light of shorter wavelength is refracted more strongly than light of longer wavelength. This means that if parallel rays of white light are incident on the lens, the shorter wavelengths (e.g., ultraviolet light) will be brought to focus closer to the rear surface of the lens than the longer wavelengths (e.g., red light). As with spherical aberration, this variation in the position of best focus for different wavelengths leads to a reduction in the quality of the image through blurring. Again, the position of best focus is the position where the least lateral spreading of rays occurs. One way of correcting for chromatic aberration is to replace the single lens element with two or more separate elements, each with a different refractive index and shape, the exact combination carefully chosen to bring rays of different wavelengths to a common focal point. However, no known examples of this exist in nature. A remarkable solution that has evolved, notably in fish, are lenses constructed of several concentric shells of different refractive indices. Each shell of such a “multifocal lens” is responsible for bringing a specific band of wavelengths to a crisp focus on the retina (Kröger et al., 1999). Another possible correction actually exploits the aberration by placing photoreceptors of a certain spectral sensitivity at the position corresponding to the best focal plane of the corresponding wavelengths. This strategy results in a layered (or “tiered”) retina, with photoreceptors sensitive to shorter wavelengths lying in layers positioned distally to other layers containing photoreceptors sensitive to longer wavelengths. This strategy is apparent in the eyes of some jumping spiders (Blest et al., 1981), where it may play a role in accurate depth perception (Nagata et al., 2012).
Figure 4.6 Optical aberrations. (A) Spherical aberration occurs because rays of white light (from a point source at infinity) that enter the periphery of a lens are focused to a position closer to the lens than those entering at the center. The position of best focus coincides with the position of least lateral spreading of the rays—the “circle of least confusion” (colc: arrows). The focal plane (fp) for central rays lies further from the lens. The blurring effects of spherical aberration are evident in an image of a globular cluster of stars taken using a high-powered optical telescope (B, lower image). In the absence of this aberration (B, upper image), individual stars are much more clearly resolved (which is particularly obvious at the center of the cluster). (C) Chromatic aberration occurs because rays of white light (from a point source at infinity) that enter the lens suffer optical dispersion. This arises because the refractive index of the lens material is not the same for each of the constituent wavelengths. Consequently, rays of shorter wave-length are focused to a position closer to the lens than rays of longer wavelength. Chromatic aberration can be seen as rainbow-colored flare in the image of a weather vane (D). (Photograph courtesy of Tony Karp, www.timuseum.com)
Finally, in some curious eyes, the optical image is as crisp and diffraction-limited as one could wish, but the retina is in totally the wrong place to receive a focused image (figure 4.7)! Even though this is not common, it has independently evolved in several lineages, and in these cases it is invariably adaptive. The typical arrangement is to have the retina located too close to the backside of the lens so that the image plane is located a long way behind the retinal surface. If the eye again views a star, this means that the image of the star, although reasonably sharp some distance behind the retina, will be a wide blurry spot on the retinal surface. Such “underfocused” eyes are thereby poorly resolved. Good examples are insect ocelli (figure 4.7A,B). These camera eyes, which are usually found as a triangular formation of three ocelli between the two compound eyes, are often used together as a kind of optical gyroscope to stabilize flight (see Mizunami, 1994, for this and other functions of ocelli). Most (but by no means all) ocelli are heavily underfocused and are thereby capable only of detecting broad changes in light intensity. But this they do quickly and with great sensitivity, a perfect adaptation for monitoring body pitch and roll during flight, when the relative quantities of dark ground and bright sky that occupy the visual fields of the three ocelli are constantly changing (Wilson, 1978; Stange, 1981). The camera eyes of the box jellyfish Tripedalia cystophora (figure 4.7C,D) are another excellent example. These jellyfish—which inhabit mangrove swamps in Central America—are remarkable in that they possess 24 eyes of two different types distributed over four sensory clubs—or “rhopalia”—that hang from stalks close to the margin of the bell. Each rhopalium carries a small upper and a large lower camera eye as well as two pairs of simpler pigment-pit eyes (Nilsson et al., 2005). The upper camera eyes are particularly interesting. As their name suggests, these eyes have an upwardly directed visual field, and these are ideal for viewing the overlying mangrove canopy through the water surface. However, despite an exquisite gradient of refractive index that endows the lens with razor-sharp aberration-free optics, the focal plane of the lens is well below the retina, and the resulting image perceived by the eye is blurry. This blurry image means that only the coarsest details of the scene are registered, in this case the broad dark edge of the mangrove canopy seen against the bright sky. This is the only dorsally located object of interest for jellyfish, and the upper eyes—which act as spatial low-pass filters—ensure that this coarse detail is seen with maximum reliability. Jellyfish rarely stray from the canopy edge—out in the open waters of the lagoon they would likely starve—so holding station close to the mangroves with this reliable visual cue is vitally important (Garm et al., 2011). The two large and downward-pointing lower camera eyes have a similarly underfocused design, and serve a similar purpose: to reliably detect and avoid the broad dark mangrove roots.
Thus, diffraction, aberrations, and image location together determine the quality of the optical image that is focused on the retina and can significantly widen the photoreceptor’s acceptance angle Δρ. This, however, is only half the story, because in the end the quality of the visual image—that is, the quality of the image actually perceived—depends not only on the optical properties of the lenses but also on those of the retina. Even if the eye’s optics can focus the crispest diffraction-limited image imaginable, the photoreceptors themselves may fail to preserve spatial details present in the image. The reason for this lies in the inability of photoreceptors to capture all of the light that is intended for them because either (1) the angle of incident light rays is too great for the photoreceptor to capture by total internal reflection or (2) the photoreceptor is so thin that it functions as a waveguide and propagates light as one or more waveguide modes, much of the energy of which remains outside the photoreceptor. In both cases this leaked light can be absorbed by neighboring photoreceptors, which then (incorrectly) interpret the light as having arrived from their own receptive fields. Such “optical cross talk” between photoreceptors broadens their acceptance angle Δρ (i.e., widens their receptive fields) and degrades spatial resolution.
Figure 4.7 Underfocused eyes. (A) The large median (MO) and lateral (LO) ocelli of the nocturnal bee Megalopta genalis are located on the dorsal head surface between the two compound eyes (CE). (B) The lens (l) of the median ocellus is astigmatic and has two focal planes (a and b) well below the retina (r) where images (in this case of a striped grating) are sharply focused (insets). (C) A single rhopalium of the box jellyfish Tripedalia cystophora showing the upper (ULE) and lower (LLE) lens eyes and the two pairs of pigment pit eyes (PE). (D) The spherical lens of the upper lens eye has a parabolic gradient of refractive index that brings incoming rays to a sharp focus below the retina. (A,B from Warrant et al., 2006; C,D from Nilsson et al., 2005, with kind permission. Scale bars in A, B, and C: 100 μm; B [insets] and D: 50 μm)
Optical Cross Talk between Photoreceptors
Whether or not a ray is totally internally reflected depends on how steeply the ray strikes the photoreceptor wall (figure 4.8A): if it strikes too steeply the ray will pass out of the photoreceptor. For a given refractive index difference between the photoreceptor and its surround, there is a minimum critical angle of incidence (θc) for rays striking the wall such that they undergo total internal reflection: rays incident at the wall with angles less than θc will pass out of the photoreceptor. This critical angle also sets the maximum angle of incidence (θmax) with which rays can strike the distal tips of the photoreceptors and remain trapped by total internal reflection. θmax for meridional rays (rays passing through the axis of the photoreceptor) depends on the refractive indices of the photoreceptor ni and the external cellular medium no (Warrant and McIntyre, 1993):
Thus, the greater the internal refractive index relative to the external refractive index the greater is θmax—in other words, the wider the focused cone of light rays that can be captured by the photoreceptor. How broad is this cone for real photoreceptors? In insects, typical values for ni are 1.36–1.40 and for no around 1.34 (Kirschfeld and Snyder, 1975; Nilsson et al., 1988), and what is immediately obvious is that the difference between them is only slight: Δn ≈ 0.02–0.06. When these values are used in equation 4.6, one obtains θmax ≈ 10°. This means that for total light trapping, the maximum allowable angular width of a cone of light rays perfectly (and idealistically) coincident at the center of the photoreceptor would be twice this angle (see figure 4.2A), that is, 20°. In many eyes, especially those with high F-numbers adapted for vision in bright light, this condition is easily met, and all light incident on the photoreceptor is captured. However, in many eyes adapted for vision in dim light (with low F-numbers), the cone of incident light can be up to 80° wide, which means that a tremendous quantity of light escapes from the photoreceptor, significantly broadening the photoreceptor’s receptive field and degrading spatial resolution. In some eyes—such as the superposition compound eyes of some dung beetles—this spread of light turns out to be the dominant determinant of spatial resolution, even though the image focused by the optics is actually quite good (Warrant and McIntyre, 1990).
Not surprisingly, many solutions exist for overcoming this problem (figure 4.8). The most common solutions involve some type of shielding of the photoreceptors, such that the escape of incident rays is prevented (Land, 1984; Warrant and McIntyre, 1991). Some eyes have no shielding whatsoever (as in figure 4.8A), implying that the loss of spatial resolution caused by light spread in the retina is acceptable for normal visual behavior. A better solution is a shield of light-absorbing pigment granules (figure 4.8B). Although this solution eliminates light spread, it does so by absorbing light, so in eyes adapted for dim light it is usually not adopted. Another interesting solution, not involving shielding, occurs when photoreceptors have a noncylindrical shape, a feature of some moth and crustacean eyes (figure 4.8C). A barrel-shaped photo receptor, while still having only a slightly higher refractive index than its surroundings, can trap a much wider cone of incident rays. The final and most complete solution is to encase the photoreceptor (or groups of photoreceptors, as in some fish) in a reflective material that traps every ray incident on the photoreceptor, regardless of its incident angle (figure 4.8D). These reflective structures—which are associated with a tapetum—are either reflective pigment granules (as in fish and crustaceans) or air-filled tracheoles (as in insects). Unlike screening pigments, a reflective shield also bounces the light back into the photoreceptor for further absorption by the visual pigment, enhancing both spatial resolution and sensitivity!
Figure 4.8 Solutions for optical cross talk in the retina. (A) A photoreceptive segment of refractive index ni that is surrounded by a cellular matrix of refractive index no will totally internally reflect incident light rays that have an angle of incidence less than or equal to θmax (ray 1). Angles of incidence greater than this (ray 2) result in the ray passing out of the segment. θc is the minimum (critical) angle with which a ray can strike the segment wall and still be internally reflected. (B–D) Three common solutions to cross talk in the retina. (B) A sheath of light-absorbing screening pigment. (C) A noncylindrical photoreceptive segment. (D) A sheath of reflective tapetal structures (reflective pigment granules or tracheoles). (Adapted from Warrant and McIntyre, 1993)
A second serious problem for resolution arises when the outer segments of vertebrate rods and cones and the rhabdoms of invertebrates are very thin (less than about 2 μm in diameter). Photoreceptors as thin as 1 μm are commonly found in both vertebrates and invertebrates, but in some extreme cases they may be even thinner: the rhabdomere R7 in the male killer fly Coenosia attenuata has a diameter of only 0.7 μm (Gonzalez-Bellido et al., 2011). Such photoreceptor diameters approach the wavelength range of visible light (0.3–0.7 μm). In fact, the diameter of the killer fly R7 rhabdomere is similar to the wavelength of red light! When light propagates through a photoreceptor that has a diameter similar to the wavelength of the light, the observed optical phenomena can no longer be described by conventional geometric ray optics (as we used above). Rather, the phenomena will obey the principles of waveguide optics, and the photoreceptor will behave as a waveguide (Snyder, 1975). Waves of light propagating in a waveguide interfere, leading to the production of waveguide “modes,” stable patterns of light within the waveguide. Modes can be grouped into several orders—first-(fundamental), second-, third-, fourth-, … order modes (e.g., Snyder and Love, 1983)—and the number that propagate depends on the diameter of the waveguide: fewer (and lower-order) modes are propagated in thinner waveguides. However, the fundamental mode is always propagated, regardless of the waveguide’s diameter. As the waveguide diameter is increased, more and more modes will be propagated until, eventually, the number of propagating modes becomes so large that the laws of geometric optics again become applicable. Waveguide modes have been observed in both vertebrate and invertebrate photoreceptors (e.g., Enoch and Tobey, 1981; Nilsson et al., 1988), and as we see in chapter 5, they play a critical role in the function of afocal apposition compound eyes.
An important property of waveguide modes is that not all of their energy is propagated inside the waveguide: a proportion of the mode energy propagates outside the waveguide, and this proportion is greater the higher the order of the mode (Snyder, 1975, 1979; Nilsson et al., 1988; Hateren, 1989; Land and Osorio, 1990). This property has a considerable influence on spatial resolution: the more light that is propagated outside the target photoreceptor, the more light that can be absorbed by neighboring photoreceptors, and the worse the resolution.
Putting It All Together: Resolution, Sensitivity, and the Discrimination of Visual Contrast
A nocturnal spider straining to see the fleeting movements of its favorite prey in the dark has eyes in which the trade-off between resolution and sensitivity has been tipped in favor of sensitivity. The opposite is true of a soaring eagle, whose acute vision—which allows it to spot small rodents on the ground far below—is only possible in bright daylight. How has this unavoidable trade-off affected what these animals can actually see?
The answer to this question really comes down to a matter of how fine a contrast they can see, that is to say, to their “contrast sensitivity.” We have seen above that the density of photoreceptors in an eye (Δφ)—and more importantly, the size of the receptive field of each (Δρ)—sets the finest spatial detail that can be reconstructed. But as we have also seen, this is only true if the eye has sufficient sensitivity: even though a greater density of smaller “visual pixels”—to again borrow the digital camera term—has the potential for higher spatial resolution, the pixels risk being too insensitive to achieve it. This becomes more and more true as light levels fall.
Not surprisingly, the eyes of animals active in dim light—such as nocturnal spiders—tend to maximize sensitivity by having larger and less densely packed photo receptors (i.e., each with a wider Δρ). An unavoidable consequence of this is that vision becomes coarser. Reliable contrast discrimination is then confined to a smaller range of coarser image details, with all finer spatial details drowned by visual noise. The opposite is true of animals active in bright daylight, such as eagles: the eyes of these animals tend to maximize spatial resolution by having smaller and more densely packed visual cells, and the abundance of light allows them to reliably discriminate a broad range of spatial details, from coarse to fine. Thus, it is not only the density of the visual pixels but also the light intensity and the optical sensitivity of the eye (and thus the relative magnitude of the noise) that sets the range of spatial details that can be seen.
The ability of imaging devices to faithfully record the spatial details of a scene—whether they are eyes, cameras, or telescopes—has long interested vision scientists and optical engineers alike. But what exactly do we mean when we talk about “spatial details”? One of the most common ways to define spatial detail is in terms of a pattern of black-and-white stripes known as a “grating” (figure 4.9A): coarser details are described by coarser gratings with wider stripes, and finer details are described by finer gratings with narrower stripes. How coarse or fine a grating is depends on its “spatial wavelength” λs, that is, on the distance between the centers of two consecutive black or white stripes. The inverse of this wavelength defines the grating’s “spatial frequency” ν (in grating cycles per unit distance), with finer gratings having higher frequency. Obviously, the ability of an eye or camera to resolve the stripes of a particular grating depends on how far away it is, so in order to remove this ambiguity, spatial frequency is usually defined on the basis of the angle that a single stripe cycle subtends at the lens (i.e., grating cycles per degree): a given physical grating (of set λs) closer to the lens will have a lower spatial frequency than the same grating further way.
Figure 4.9 The modulation transfer function (MTF). (A) Four sinusoidal gratings of increasing spatial frequency (left to right): 0.9, 1.4, 2.0, and 4.0 cycles/°. (B) When a grating of 100% contrast is imaged by a lens, the contrast in the image falls as spatial frequency increases. The frequency at which image contrast becomes zero is the optical cutoff frequency νco.
The finest spatial detail that an imaging device can discriminate is then simply given by the highest spatial frequency that it can resolve. Lens manufacturers often quantify this for a lens by measuring its “modulation transfer function” (or MTF), and this is also useful for understanding the performance of eyes. In such a measurement, the lens images a series of gratings of maximal (100%) contrast at ever increasing spatial frequency. The contrast in the image, relative to that of the object, is then measured for each grating. The resulting plot of image contrast versus spatial frequency—the MTF (figure 4.9B)—shows that gratings of lower spatial frequency are imaged with almost no loss of contrast, but as spatial frequency increases less and less image contrast remains. At and beyond a certain spatial frequency—the optical cutoff frequency νco—all contrast is lost, and the lens is incapable of resolving finer details. For high-quality diffraction-limited lenses where optical defects (e.g., aberrations) are minimal, the optical cutoff frequency can be quite high, which means that the lens is able to resolve very fine spatial detail. The opposite is true for poorer lenses that lack correction for aberrations and other imperfections.
Unlike commercial lenses, the highest spatial frequency that eyes can resolve depends not only on the quality of the optical image but also on the retina. The density of the photoreceptors, the optical cross talk between them, and the extent of visual noise all play their parts to further limit the finest spatial detail that can be seen, especially in eyes adapted for vision in dim light. As an example, the camera eye of the nocturnal net-casting spider Dinopis subrufus has a lens that forms a crisp aberration-free image, allowing it to resolve gratings of spatial frequency up to at least 0.8 cycles/° (Blest and Land, 1977). But this resolution is not preserved in the retina: due to the eye’s low F-number (0.58), the cone of light focused on the retina is a hefty 82° wide, causing significant cross talk. With an interreceptor spacing Δφ of 1.5° this leads to photoreceptors with receptive fields that have acceptance angles Δρ of around 2.3° and significant side flanks (see figure 4.2B; Laughlin et al., 1980). If we were to approximate this receptive field by a Gaussian of half-width Δρ = 2.3°, then the MTF would also be a Gaussian (figure 4.10A; Snyder, 1977). Further, if we define νco as the frequency at which the MTF falls (say) to 1% of maximum, then νco = 0.49 cycles/° (figure 4.10B). This is only a little more than half the spatial frequency passed by the optics! In reality, the receptive field’s side flanks (see figure 4.2B) would probably reduce νco even more.
The situation worsens further if one also accounts for noise. This can be seen if we calculate the number of photons N that Dinopis (with eyes of optical sensitivity S) can sample per second from a uniform extended object with radiance R. As we saw in equation 4.2, N is simply given by the product of S and R. From table 4.1, S for Dinopis is 101 μm2 sr. A typical starlight value of R is around 1 photon μm–2 s–1 sr–1 (in green light, λ = 555 nm: Land, 1981). Thus, when Dinopis views an object illuminated by starlight, each of its photoreceptors absorbs about 100 photons every second. What noise will be associated with such a sample? One source, as we saw above, is photon shot noise, the noise arising from the random and unpredictable nature of photon arrival (where noise is simply given by √N). Thus, in our example of Dinopis, this noise is then √100 or 10 photons s–1. Furthermore, unlike the “signal,” which is the number of absorbed photons N multiplied by the eye’s MTF (Snyder, 1977), this shot noise is independent of spatial frequency. Thus, if one plots the signal and shot noise together (figure 4.10B), the signal steadily declines with spatial frequency due to the declining MTF while the noise level remains constant. Eventually a spatial frequency is reached at which the two curves cross—beyond this frequency the signal becomes lower than the noise, and above this “maximum detectable spatial frequency” (νmax) the contrasts of all finer spatial frequencies are drowned in the noise (Warrant, 1999). In our example of Dinopis, if one assumes a Gaussian MTF, then νmax is 0.35 cycles/° (figure 4.10B), which is a reduction by about 30% compared to the cutoff without noise (0.49 cycles/°). In reality, νmax is likely to be even lower again because other sources of noise—particularly those inherent within the photoreceptors themselves—can be quite substantial, equaling or even exceeding shot noise levels (Laughlin and Lillywhite, 1982). Spatial resolution declines even further as light levels fall. At light levels 10 times dimmer than starlight (i.e., N = 10 photons s–1, √N = 3.2 photons s–1), νmax would fall to just 0.25 cycles/°.
Figure 4.10 The modulation transfer function (MTF) of an eye. (A) The range of spatial frequencies detectable by an eye depends on the size of the photoreceptor’s receptive field (inset). Narrower receptive fields allow a wider range and a higher optical cutoff frequency νco (green curves) than broader receptive fields (red curves). (B) Signal and (photon shot) noise in the eye of the nocturnal spider Dinopis sub rufus, which has a photoreceptor acceptance angle of 2.3° and views a scene illuminated by starlight. The maximum detectable spatial frequency νmax is defined as the frequency at which signal equals noise. νco is here defined as the frequency at which the signal has fallen to 1% of maximum, but other criteria could also be used.
Highly sensitive eyes like those of Dinopis are well adapted for coarse but reliable spatial vision in dim light, and this of course is a prerequisite for being able to hunt prey at night. Much of the reason for this ability is the structure of the eye itself—the camera eyes of Dinopis have the potential to supply substantially more spatial information per unit mass than (say) the compound eyes of their arthropod relatives, the insects (Laughlin, 2001). This naturally highlights the importance of eye design in any discussion of visual performance, so before continuing with that particular topic any further, we next turn our attentions to the major eye types of the animal kingdom.