44 Light and the Eye \1 : 4 



The number was reduced by the fraction (about f) which he found to be 

 absorbed in the eye. The final number, then, should be the minimum 

 or number of photons necessary for threshold vision. At least it would 

 be if this number were much larger than one, in which case all pulses 

 could be considered as having equal numbers of photons. Otherwise, 

 the entire data would have to receive a probability-type interpretation. 

 Early estimates based on this method indicated that about 150 photons 

 were necessary at the cornea, and about 30 of these reached the retina 

 for a just visible flash. As this number was redetermined during the 

 1920's and 1930's, it decreased steadily from 30 down to one or two. 

 This small number violates the original basis of the determinations 

 because the number of photons in a light pulse, the number absorbed 

 along the way, and even the fraction absorbed in the retina of those which 

 get there, are subject to probability considerations. In general, one 

 cannot measure these probabilities separately. However, the average 

 number of photons b absorbed by a single receptor of the retina will be 

 proportional to the intensity /, provided the eye does not move ; that is, 



b = kl (6) 



The proportionality constant k will vary with many factors including the 

 size of the test patch, the pupil opening, the wavelength, and the length 

 of the flash. It is clearly desirable to carry out an experiment to measure 

 the threshold number of photons independently of k. The following 

 mathematical manipulations indicate how to design an experiment 

 which satisfies this criterion. 



The number of photons absorbed by a photoreceptor in the retina 

 during a given flash is an integer. It may have any positive value, or it 

 may be zero. However, the average number of photons need not be 

 an integer but will have a definite value b. The probability P that 

 m photons will be absorbed during a flash by the photoreceptor will be 

 given by the Poisson probability distribution, namely: 



P(m) = e —L. (7) 



ml 



Vision will occur if some given integral number n or more photons are 

 absorbed during the exposure. The probability P n that n or more 

 photons will be absorbed in a flash is given by 



P n = § P(m) = 1-2 nm) (8) 



m = n 



Now, one may plot computed values for P n against log b, giving curves 

 such as those shown in Figure 11. Notice that each of these has a 

 different slope. Although the value of b is not known, the value of the 



