The human visual systemIn the course of several billion years, Earth’s lifeforms have evolved numerous schemes for usingelectromagnetic radiation in the wavelength range3500-9000 Å (infra-red through ultra-violetlight). Plant chlorophyll, bacterial rhodopsin, pho-tosensitive spots, compound insect eyes, camera-like eyes, and the pits of a pit viper----all exist totransform light into something else.For plants and bacteria, light provides the energythat powers their chemical factories, through theprocess known as photosynthesis1. In higher or-ganisms, eyes and their equivalents transduce lightinto nerve impulses, thereby enabling the posses-sors of a visual sense rapidly to detect changes intheir environment. Such changes can presage danger, indicate asource of food, a mate, resynchronize the circadianclocks----all of which are intimately connectedwith survival. Therefore vision must have beenunder intense evolutionary pressure to improve toan optimum level consonant with the overalleconomy of the organism. That is, Darwin’s con-cept of the origin and improvement of visionthrough natural selection explains why the eyes ofhigher organisms perform so well, without the needof teleological argument.In this chapter we study the mammalian eye----pri-marily the human eye----explaining its functioningin terms of optics and quantum mechanics. Weshall be particularly interested in discussing thevarious optimizations and engineering compro-mises to be found in eyes.1. Simple lensesAs the sketch to the above right shows, the humaneye is constructed along much the same lines as abox camera: light entering through a hole at thefront (the pupil) is focused on a light-sensitivelayer at the back (the retina). The simplest ap-proach to the physics of lenses treats light as raysrather than electromagnetic waves. That is, weimagine that light emanating from a point movesthrough a homogeneous transparent medium in astraight line.The key to understanding lenses is what happenswhen light falls on an interface between differentmedia. To begin with we imagine the interface is aplane surface, as shown on p. 100. Snell’s Lawrelates the angle between the normal to the surface(dashed line) and the incident ray, to the anglebetween the normal and the refracted ray:1 sinθi = n sinθr ,Horizontal section of the right eye, from abovePhysics of the Human Body 99Chapter 11: The human visual system1. You might not be aware that humans also employ photosynthesis: under the influence of photosynthe-sis, made possible by ultraviolet rays from the Sun, a sterol compound from the liver (dehydro-cholesterol) is converted to vitamin D3. This supplies enough vitamin D3 for human needs.where n is the index of refraction of the mediumand 1 is the index of refraction of the vacuum. The keys to understanding lenses (that is, refrac-tion at curved interfaces) are first, to regard smallpatches of the surface as flat; and second, to treatthe two curved surfaces separately.Consider the upper of the two figures below. Weneed to relate the distances u and v to the radiusR of the lens surface (which for simplicity we taketo be spherical) and the index of refraction n. Thetriangle ABC has sides u ⁄ sinθ, u+R and R, respec-tively. From the law of sines we have(u + R) sinθ = R sin(π − θi) ≡ R sinθi .The triangle BCD gives us the relationR sinθr = (v − R) sinϕ .We also see that the shared altitude of the two righttriangles, ABE and BDE, isBE___ ≡ AB___ sinθ = BD___ sinϕ .Hence in the limit of small anglesuv ≈ sinθsinϕoru + Rv − R ≈ n uvleading to the relation1u + nv ≈ n − 1R .Now to combine the effects for two curved surfaceswe simply apply the formula again, but this timekeeping in mind that for the right-hand surface theimage distance and radius of curvature must beconsidered negative: that is, w < 0 and r < 0,whencenv − 1|w| = n − 1−|r| .When we combine the two we get100 Physics of the Human BodySimple lenses1u + 1|w| = (n − 1) 1R + 1|r| =df 1fwhere f is called the focal length of the lens.The physical interpretation of the focal length is itis that point where light rays coming from infinitelyfar away (u = ∞, that is parallel rays) are broughtto a point; or alternatively if a point source of lightis placed at the focal distance on one side of thelens, parallel rays emerge on the other side.2. Problems with lensesA simple lens made of a uniform material, whosesurfaces are elements of a sphere, suffers fromseveral problems in forming images. The easiest tounderstand is chromatic aberration: in general theindex of refraction of a transparent material de-pends on the wavelength of the light. The focallength is different for each color, hence the lensfocuses multicolored light to a rainbow-like ring(the best compromise) rather than to a point. Thisreduces the resolution. High quality camera lensescompensate chromatic aberration by joining layersof different kinds of glass, with opposite variationsof refractive index with wavelength. The opossiteeffects tend to cancel.A second difficulty is spherical aberration----lightrays passing through the lens far from its axisconverge at a different point from axial rays. Tosome extent this can be compensated by restrictingthe entrance aperture of the lens so that rays passthrough close to the axis. (It can also be reducedby varying the shape of the lens from spherical.)A third proble is astigmatism----rays coming frompoints off the axis are not imaged at the samedistance behind the lens as rays originating nearthe axis. This effect can be compensated to someextent by choosing the radii of the two surfacescarefully, by restricting the lens aperture, and alsoby varying the lens shape from spherical.Not all the light incident on an interface betweenmedia of different indices of refraction gets trans-mitted. Some is reflected at each surface. Thebigger the didiscontinuity of refractive index, themore light is reflected. Since the best lens materialshave high refractive index, they tend to reflect themost light. Thus lenses suffer from loss of lightintensity, excluding whatever light is absorbed bythe medium (since no medium but vacuum isperfectly transparent). Manufacturers of highquality lensesapply varioussurface coat-ings to lessenthe interfacediscontinuity.By properly ad-justing the spa-tial variation ofn in the transi-tion region, such coatings can reduce substantiallythe reflectivity of the surface2.How important are