One important question about vision has to do with the number of photons required for seeing. The question has to be refined, because an immediate problem arises in deciding where these photons must be absorbed. Do they all have to be absorbed in the same rod (cones are neglected, since we know they are used for high light-intensity vision)? Can a single rod effectively absorb more than one photon in any arbitrarily short time interval? Do rods or groups of rods cooperate in vision? Do they cooperate in experiments to determine the minimum number of photons that may be detected?
Studies of the sensitivity of the eye to light are carried out by exposing the eye to flashes of dim light and measuring the fraction of times (the probability) that the flash is seen. If light of any intensity could be seen, the experiment would be uninteresting. But the results of these experiments are more than normally interesting.
There are two ways of doing the experiment. The most direct way is to shine progressively dimmer light flashes into the eye and to determine the smallest intensity that is normally seen. The actual mechanics of this experiment is of interest. The subject sits in a dark box with his teeth gripping a plaster cast specially made to fit the subject's teeth. This, plus some clamps on the subject's head, permits him to gaze directly ahead at a very dim red light which also helps to fix the subject's eyes. The rods are quite insensitive to red, so this light doesn't disturb the experiment. Dark-adaptation was achieved by having the subject sit in total darkness for 45 minutes. An aperture which can be illuminated with light of arbitrary intensity and wavelength is situated at an angle of about 20 degrees away from the line of sight. The size of the aperture is chosen so that the image formed on the retina covers only about 500 rods. The amount of light needed for vision can be estimated from the fact that stars of the 8th magnitude are visible, corresponding to about 10 -9 erg/sec. In the laboratory setup, even less energy was needed for detection.
The subjects were able to detect light of wavelength 510 mp whose intensity corresponded to about 10~ 10 energy units (ergs) incident on the eye. This corresponds to about 100 photons. This energy has to be corrected for the various reflections and absorptions experienced by photons before they reach the retina and are absorbed. At the cornea about 5% is reflected, leaving 95 photons to continue. Half of these are absorbed in the humors of the eye, leaving about 45 photons. Eighty percent of these are absorbed in the nonsensitive portions of the retinal structure, leaving about 10 photons to be absorbed for this barely detectable flash.
In the actual experimental situation, it was found that light of a particular intensity could always be seen. But when a slightly lower intensity was used, the subject was always able to see the flash in a certain percentage of cases. That is, for an intensity which can always be seen, one cannot be found just slightly lower which is never seen. Therefore, for each subject, that intensity which was seen 60% of the time was arbitrarily taken as the threshold for that particular run. There were variations from individual to individual, and even the same person exhibited a day-to-day variation. In the experiments being discussed, seven subjects were studied, and the range for threshold seeing was 2.1 to 5.7 X 10-10 erg. One particular subject had a range of 4.83 to 5.68 X 10-10 erg.
At a wavelength of 510 mµ, the energy of a quantum is 3.89 X 10-12 erg. Thus, the actual range in photons is from
which is 54 to 158 photons. Since about 10% of the photons get through to the retina, the number of photons needed for threshold sensitivity is 5 to 16.
We have 5 to 16 photons falling on 500 rods. On each rod, therefore, there is an average (taking the 5-photon number for calculating purposes) of 1 photon per 100 rods, or a chance of 0.01 that any one rod received a photon. The chance that 2 photons will fall on the same rod is then approximately 0.01 X 0.01 or 0.0001—one chance in 10,000. Since a total of only 5 photons reaches the rods, it is highly improbable that two photons per rod are needed to produce a response. Thus, one photon must react with one rod, affecting one molecule of the visual pigment, and in turn this reaction initiates a sequence of other chemical reactions leading finally to stimulation of a single nerve fiber. It must then be true that the requirement for 5 photons is exerted at the level of the nerves. The nerve excitation resulting from summing the effects of 5 individual nerve fibers is what is finally sufficient to trigger the optical nerve into transmitting an impulse to the brain.
On quite general grounds, a result greater than one photon is to be expected. If one photon were the threshold, then a single firing of a nerve fiber would result in vision. A single nerve fiber can be excited randomly by heat, fluctuations in chemical reactions in the retina, pressure, etc., so that we should be seeing flashes all the time. Therefore, we would guess that more than one photon would be needed, so as to reduce the random triggering of vision by these agents when it is irrelevant to the seeing processes. Of course, psychologists might maintain the entirely possible position that the triggerings occur but are suppressed by the mind so as to free the eye for purposeful vision.
A number such as 5 photons is interesting because the fluctuations in this number (from the Poisson Distribution) yield a standard deviation of √, which is about 2.3. Two standard deviations (2s) equal about 5. Therefore, in some 5% of the cases where an average of 5 photons was delivered to the retina, the fluctuations should be great enough to result in no photons getting through and therefore the subject would not see the flash that the experimenter thought he was giving.
This leads us to another way of approaching the visual threshold problem. Consider that, for a subject to detect a flash, a threshold of n photons must be equaled or exceeded on the retina in the area being stimulated. When the number of photons arriving is less than n, the subject will see no flash at all; when it is n or greater, the flash will be seen each time. Thus, if this simple model were correct, the frequency of seeing flashes of increasing intensity would rise abruptly from zero to 100% at the threshold value of n. However, there are two sources of variation in such an experiment: biological variability and physical variability. We can't do anything about biological variability, for the term includes factors which we don't understand or cannot control. So we focus attention on physical variability. In this situation it means that when one shines n photons, on the average, on a retina, there will be fluctuations of this number. Indeed, we know that the standard deviation of the number is √. That is, in one third of the cases, the number will differ from n by √ or more. Now, when the average is n, the fraction of times there will be n or more is not unity, as in the simple nonfluctuating example first mentioned in the paragraph, but is something less than unity. For any given intensity of light, the flash will contain a fluctuating number of photons, and only flashes containing at least the threshold number will be seen. As the intensity increases, a steadily increasing proportion of flashes will exceed the threshold number. Finally, intensities will be reached for which, despite the fluctuations, all flashes will exceed the threshold number of photons.
The simple model and the fluctuation model are presented schematically. A flash of a given intensity is presented repeatedly to the subject, and the fraction of times he sees the flash is recorded. Then, in succeeding experiments, using a series of intensities, similar data are recorded. The data are then plotted to show the fraction of times the flash is seen for each incident intensity.The theoretical analysis utilizes the Poisson formula. For any average number of incident photons, we can figure out the fraction of times there will be a minimum of, say, 1 photon hitting the retina. This curve is plotted on the figure below with a 1 beside it. Similarly, we can figure out the fraction of times there will be at least 2 photons hitting the retina for any average intensity. And a similar calculation can be carried out for any threshold number of photons.
The figure above shows these curves for various values of this threshold number. Next we try to fit the calculated curves of the figure above to the actual data obtained. Since the shape of the curve depends strongly on the value of the threshold number, it is possible to distinguish quite well between possible curves. In this way, by trying successively 3, 4, 5, 6, 7, etc. photons for the threshold, it is possible to show that the best number is about 6 photons. This means that at low intensities there were few times that the flash was seen because there wrere few flashes which chanced to contain 6 photons, since the average number of photons was less than 6. When an intensity wTas reached such that an average of 6 photons reached the retinal absorbers, not all the flashes were seen because some of the flashes contained 0, 1, 2, 3, 4, and 5 photons, none of which could be seen. When very high intensities were used, all flashes had at least 6 photons, and so all were seen. In fact, as shown in the latest schema, the data for three different subjects gave 5, 6, and 7 photons as the number which best fitted the data. Note that this experiment could have been done simply by fitting the curves to the data—it is not necessary to measure the absolute intensities, so long as we know the ratio of the intensities.
It is of interest to point out that this experiment could have been done in precisely this way 150 or more years ago. If someone had had the inspiration to fit the results with the already known Poisson Distribution, the quantum nature of light could have been discovered that long ago.