THE T W O - Q U A N T A E X P L A N A T I O N OF THE

D E P E N D E N C E OF T H E T H R E S H O L D VALUES

ON T H E VISUAL ANGLE AND T H E TIME

OF OBSERVATION

By M. A. Bouman O H. A. van der Velden

Physisch Laboratorium, Utrecht.

In 194zt, three independent approaches were developed, towards establishing the number of quanta effectively absorbed by the visual purple and necessary for light-perception (van der Velden, 1944. These methods were independent of the loss of light caused by reflection and absorption in the various parts of the eye, so that the absolute values of the energies employed' are of no importance. The methods are:

A. Determination of the relation between the chance of o,bser- vation of a flash and the average number of quanta in the flash for very small visual angles and for short flash-times.

After the war we received' the p-pers by Hecht and his co-work- ers (1942), who had already used this method in 1942. These results indicated that a co-operation of at least 5, 6 or 7 quanta (dependent on the observer) were necessary for the perception of light. Our results (van der Velden, 1944, 1946; Bouman O van der Velden (1947)), on the other hand, suggested a co-operation of two quanta, independent of the observer.

B. Determination of the dependence of the average number of quanta for a fixed chance of observation on the duration of the flash for a small visual angle.

It is obvious, that when more than one quantum is necessary for light-perception, one has to ex,pect a dependence of the average number of quanta m the flash N, necessary for a certain chance of observation, on the duration of the flash t.

Assuming that tl~e retinal condition existing after the absorption of the first quantum will have a finite life-time r it is possible to calculate the number N as a function of t, at least to give the slope

of this curve for large values of t T

In Fig. 1 these curves are given for the cases that 1, 2 or 3 quanta are necessary for the perception of light. The experimental resuIts are in accordance with the two-quanta case, Because of

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the big difference in the slope for the cases of 1, 2 or 3 quanta the one and three quanta cases are impossible. ~r proved to be about 0,02 sec.

C. Determination of the dependence of the average number of quanta for a fixed chance of observation on the visual angle of the flash for a short time of observation.

3 0

2 0

IO

;O%

/ 4

3

o

K = 3 / /

f I

f f

f f

J

K = 2 j "

f f /

J

14=1

©5 ; ~ @ - ' 0.4 I 2 4 10 2 0 4 0

Fig. 1.

The theorehcal number of quanta, necessary for a chance of observation of 6 0 % as a function of t, the duration of the flash, for 1, 2 and 3 quanta. The experimental results are given for a small visual

angle (4 min . ) .

For the same reason as given for B, such a dependence can be expected when more than one quantum is necessary for the percep- tion of light. Assuming that the condition caused by the absorption of a quantum w~ll have an extension corresponding "to an area, with a diameter D of the retina. If the second quantum is absorbed out- side this area no light will be perceived. In Fig. 2 some theoretical curves are given for the dependence of the number of quanta, necessary for a fixed chance of observation, on the diameter d of the flash. Again, the experimental results are completely described by the hypothesis, that two quanta are necessary for the perception of light.

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I/ o ~ 2

/

~ - ,.4' - -

O I O,2 0.4 1 4 10 20 40

Fig. 2.

T h e theoretical number of quanta~ necessa ry for a chance of observa t ion of 60 ~o as a funct ion of the v isual angle d of the light spot, for 1, 2 and 3 quanta . T h e exper imenta l

results are g iven for a n ine of f lash which w a s small compared with ~r.

R E F E R E N C E S .

Bouman, M. A. ~ van de Velden, H. A. (1947), J. opt. Soc. Amer. 37, 908.

Hecht, S. (1942), J. opt. Soc. Amer. 32, 42. van der Velden, H. A. (1944). Physica. II, 179. van der Velden, H. A. (1946). Ophthalmologica, 111, 321.

M. H. P IRENNE, University College, London.

On the basis of the theory of quantum fluctuations, Van der Velden and Bouman conclude from various experiments that the minimum number (n) of quanta which must be absorbed by the rods in order to elicit a visual response is 2.

The experiments of Hecht, Shlaer and Pirenne, on the other hand, have led on a similar basis to values of n ranging from 5 to 8. I should like to refer here to experiments made in the same connexion on visual acuity at very low brightnesses (Pirenne, 1945). This paper is being prepared for publication in full. The experiments

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made so far lead to values of n ranging from 4 to 10; the value

n = 3, and a l ' o r t i o r i n = - 2 and n = 1, seem to be incompatible with the experimental results.

It is probable that all such derivations of n are affected by various errors, but that the sources of error differ in different methods. Critical comparison of the results of various methods is thus likely to show eventually what is the correct estimate of n.

R E F E R E N C E .

Pirenne, M. H. (1945). Proc. Camb. phil. Soc. 42, 78--82.

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