The coronavirus has led to events being canceled, people being isolated, and classes being disrupted. What can I do to help? I plan to post to this blog more often, so students can learn how physics is applied to medicine and biology. I can’t promise daily posts, but I’ll do what I can.
The maximum photopic (bright-light) resolution of the eye is limited by four effects: diffraction of the light passing through the circular aperture of the pupil (5–8 μm), spacing of the receptors (≈ 3 μm), chromatic and spherical aberrations (10–20 μm), and noise in eyeball aim (a few micrometers)… The total standard deviation is (62+32+152+52)1/2 = 17 μm in the image on the retina. Since the diameter of the eyeball is about 2 cm, this corresponds to an angular size… of (17 × 10^6)/(2 × 10^2) = 8.5 × 10^4 rad = 0.048 ° = 2.9 min of arc.
Let’s examine the factors contributing to acuity, one by one.
The Rayleigh criterion specifies the minimum angular separation, θmin, of two objects that can just be resolved. The criterion can be expressed as θmin = 1.22 λ/ D, where λ is the wavelength of light and D is the diameter of the pupil. If we use light from the center of the visible spectrum-say green light with wavelength 550 nm-and a pupil diameter of 2.5 mm, we get θmin = 0.00027 radians, which is 0.015° or 0.93 minutes of arc. If we take the eyeball diameter to be 2 cm, that translates into a minimum separation on the retina of 5.4 μm.
The Spacing of Receptors
According to The First Steps in Seeing, by Robert Rodieck, in the fovea cones have a density of about 0.1 per square micron. That translates roughly into a 3 micron separation between cones. The cone density is down by a factor of ten in other parts of the retina.
Chromatic and Spherical Aberration
Chromatic aberration arises because the index of refraction of the eye, including the lens, depends on wavelength of the light. Therefore, different colors form images at different locations. Spherical aberration arises because a spherical lens is not ideal for forming images; off-axis rays have a different focal point than on-axis rays. The eye and its lens, however, are not truly spherical, so when we speak of spherical aberration in the context of vision, we mean heterogeneities in the imaging system that cause the image to be blurred. Rodieck says
At night the pupil is fully open, and the spread of photons is due mainly to the optical imperfections of the eye; the effects of these imperfections increase rapidly with pupil size. The other factor that contributes to the spread of photons is intrinsic to the nature of how photons go from place to place, and is termed diffraction. This factor is not significant here, but in daylight, when the pupil is small, the spread of photons in the retinal image is due mainly to diffraction.
Noise in Eyeball Aim
Rodieck explains how your gaze is always moving, even when staring at a stationary object.
Gazing at a stationary object also involves smooth eye movements. This is because your head is always in slight motion as the muscle of your body and neck attempt to maintain your posture. Thus when you look as steadily as possible at some small stationary object, such as a pebble on the ground, your slight head movements cause the image of the pebble to move on your retina.
This motion has some noise, which limits our visual acuity.
A Snellen chart is the traditional way to measure visual acuity. You stand 6 meters (20 feet) from the chart and read the letters with one eye. If you plan to print out this chart, you need to make sure it is the correct size; the topmost “E” should be 87.3 mm tall. In that case, the 20/20 row corresponds to letters that subtend 5 minutes of arc.
Another test of visual acuity arises because the second star from the end of the handle of the Big Dipper is actually a double star: Mizar and Alcor. They are separated by 12 minutes of arc. George Bohigian published an article in Survey of Ophthalmology (Volume 53, Pages 536–539, 2008) about “ An Ancient Eye Test-Using the Stars.” He begins
A common vision test in ancient Persia used the double star of the Big Dipper in the constellation Ursa Major or the Big Bear. This vision evaluation test was given to elite warriors in the ancient Persian army and was called “the test” or “the riddle.” The desert Arabs, especially the Bedouins, used the separation of Mizar and Alcor as a test of good vision. The separation of these two stars is known as the Arab Eye Test , and has been used in antiquity to test children’s eyesight. This article explores the origin, history, and practicality of this eye test and how it correlates with the present-day Snellen visual acuity test.
The Arab Eye Test using the double star of Mizar and Alcor remains a practical test of visual acuity and visual function as it was over 1000 years ago. This test is somewhat equivalent to the 20/20 in the Snellen visual acuity nomenclature. This is the first report that correlates the Mizar-Alcor naked eye test with the current Snellen visual acuity test. With the spread of Islam from Spain to Central Asia, the Arabs brought their knowledge of astronomy mixed with the traditions of Greece, India, Babylonia, and Persia to Western civilization.
Throughout our history the stars have been a constant guide to navigation, measure the seasons, to divine the future, and to measure eyesight. The Arab Eye test is an example of how a natural phenomenon has been used for a practical purpose.