The mammalian eye is designed to collect light and focus it onto the retina. The retina consists of an array of cells, each having the ability to detect light falling on its surface. Most of the refraction (and thus focusing) of incoming light rays takes place at the interface between air and the cornea. The lens does the fine tuning, changing the focal length so the image lands exactly on the retina. The tuning is necessary since the eye must be able to bring into focus light from objects as close as 0.1 m as well as light from an infinitely distant source.
Spatial resolution is the ability of the eye to distinguish waves coming from different directions. For example, if a distant car is facing you at night with its headlights on, you can see two distinct headlights, since the light from the two headlights approaches your eye from two directions (see figure). If the car is far enough away, however, your eye lacks the resolution to distinguish the headlights, and you see only one light source.
Resolution is measured in degrees or radians. For instance, if your eye can just resolve two headlights which are 1.5 m apart on a car which is 1 km away, then the angular separation of the lights is approximately 1.5 m/1000 m = 1.5 x 10–3 radians. The resolution of your eye is 1.5 x 10–3 rad or 0.09 degrees (since 1 rad ≈57˚) or 5 seconds of an arc. To a good approximation, the angular separation of two light sources (or features on any sort) is the ratio of spatial separation Δxto distance from the point of reference L. (See figure.) Thus the better the resolution, the smaller the resolution angle.
Ultimately the spatial resolution of any detector, including the eye, is limited by diffraction, which is the spreading of waves. When waves pass through an aperture, they spread on the other side subtending an angle given by
θdiff = λ/d (1)
where θdiff is measured in radians, λ is the wavelength of the wave involved, and d is the diameter of the hole through which the waves must pass. Of course, diffraction is the physical limit of the resolution. The actual resolution of a detector may be much poorer than equation (1) would indicate if it is poorly designed. The human eye, when functioning properly, is essentially diffraction limited.
For the following problems use c = 3.0 x 108 m/s. Green light has a wavelength of 520 nm in a vacuum. (1 nm = 10–9 m)
A certain eye does not focus correctly, being either near- or far-sighted. The front to back length of the eye is 0.025 m, but the focusing power of the resting eye is 35 diopters. What should be the approximate power of the appropriate corrective lens?