THE QUANTA EXPLANATION

The blue luminescence (3.3) in terms of n isunchanged, but n as a function of J has now amore complicated relationship than (3.2); how-ever, J/n2 remains constant at high but not atlow exciting intensities. When 32>>1 (4.1) is notchanged, but (4.2) becomes

bl(l+u) + (5.3)U2 an c+^/

while (4.8) becomes

J 1 b, I aL1 +. (5.4)

Ii tUi (,u111)l an+-yJ

At high temperatures when y, which is of theform Se-ElkT where E=depth of the electrontraps, is very high, the effect of the traps is of

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

course negligible. At low temperatures the valueof (1J/Il -1) is too large by a factor

[1 +aL/(aen+y)]'-,

and this may explain an increase in the slopecorresponding to b1.

The results at these temperatures are nowbeing investigated more fully. Meanwhile it issuggested that the physical basis of the equationsset up in this paper is the most reasonable onewhich can include both the simpler results ob-served so far at higher temperatures and otherson phosphors with any number of different kindsof luminescence or killer centers.

We wish to thank Mr. J. A. M. van Moll andthe Directors of Philips Lamps, Ltd., for per-mission to publish this work.

VOLUME 38, NUMBER 3 MARCH, 1948

The Quanta Explanation of Vision and the Brightness Impression for VariousTimes of Observation and Visual Angles

M. A. BOUMAN AND H. A. VAN DER VELDENPhysical Institute of the University of Utrecht, The Netherlands

(Received July 30, 1947)

In the previous papers (see references 1-3) it was provedthat the statistical nature of light and certain properties ofthe human apparatus for vision determine the thresholdvalues for various times of observation and visual angles(expressed among others by the laws of Ricco, Talbot, andPiper) and the visual acuity. The behavior of these visualfunctions agreed with the two-quanta case: light is per-ceived when two quanta are effectively absorbed in thevisual purple within an area corresponding to a visualangle of D (0') and within a time (0.02 sec.). Asthe chance is negligible that the two quanta are absorbedin one rod, it proved that a rod reacts on the absorptionof 1 quantum. The two stimuli will cooperate in the nervesystem and give rise to light perception.

From experiments described in the present paper weconclude that after the absorption of a couple of quanta,

INTRODUCTION

ROM the explanation presented by the two-quanta hypothesis for the threshold values.

and visual acuity',2' 3 for various visual angles' H. A. v. d. Velden, Physica 9, 179 (1944).2 H. A. v. d. Velden, Ophthalmologica 111, 321 (1946).3 M. A. Bouman and H. A. v. d. Velden, J. Opt. Soc. Am.

37, 908 (1947).

all quanta absorbed within an area 0 corresponding with avisual angle o and a time T referred to this first couple ofquanta, cause a stimulus which contributes to the bright-ness impression and the total number of stimuli within anarea 0 and a time T determines the brightness impression.

The experiments mentioned consist in the determinationof the dependence of the average numbers of quanta of aflash necessary for a fixed brightness impression on thevisual angle and the time of observation.* The area 0 and the time T are the cause for the existence

of the laws of Talbot and Ricco for the brightness impres-sion. As soon as the time of the flash t or the visual angled exceeds the value T or 0, the average number of quantanecessary for a constant brightness impression is propor-tional to d c.q. t. The area 0 has the nature of a recipientunit and T of an averaging time.

and times of observations, several kinds of co-operation are found to exist between the rodsactivated by the absorption of a quantum: lightis perceived when two quanta are effectivelyabsorbed in the visual purple within an areacorresponding to a visual angle D (-10') andwithin a time (-0.03 sec.). As the chance that

231

M. A. 'BOTJMAN AND H. A. VAN DER VELDEN

d c b a

e f

h2

a2

nfl

FIG. 1.

the two quanta are absorbed within one rod isnegligible, it proves that a rod reacts on theabsorption of one quantum. The two stimuli willcooperate somewhere in the nerve system andgive rise to a light perception. The presence ofactivated rods gives rise to a condition of theapparatus for vision so as to decrease the chancefor light perception for a certain time and over acertain area of the visual field.

As it can be expected that these effects areimportant for the brightness impression, we per-formed experiments suitable for the study ofthese phenomena concerning brightness impres-sion.

We determine the dependence of the averagenumber of quanta necessary for a certain bright-ness impression on the visual angle and time ofobservation for several values of the subjectivebrightness.

1. THE EXPERIMENTS

For the experiments about the dependence onthe time we asked the observer to compare thebrightness impressions of two successive flashesof different duration but equal visual angle. Theobserver had to change the intensity of one ofthe flashes till he was unable to see any differencein brightness impression of the two flashes.

The tungsten ribbon filament lamps I and II(Fig. 1) were imaged by the lenses a, and a2 onthe screen of the disk b, in which two openingswere made, so that the light of I could only passthrough the one, and that of II through theother opening. The images of I and II via themirror e are thrown by the lens d on the artificialpupil c. The size of the openings in the diskc andthe velocity were such as to make the flash-timefor the light of II always 0.1 sec. The flash-time

for the light of I could be regulated between0.05 sec. and 1 sec. The 'time between the twoflashes was always about 0.5 sec. The screen ofthe lens d, completely filled up by the light of Ior II, is the object for observation. The visualangle is determined by the diaphragm f. In frontof the pupil e a Schott VG2 filter g was placedwith a maximum transmission at 5300A. Thehead of the observer was fixed by means of amouth-mounting piece. The dark-adapted righteye was used and the spot of the retina on whichthe flashes were received was situated 7 nasalfrom the fovea, just like in all our previousexperiments.'-3 The fixation of the right eyewas performed by the observation of a red-fixation light with the left eye.*

For several intensities of the test flash of 0.1sec. of lamp II the observer adjusted the currentsof lamp II to every fixated time of the flash ofII for which the brightness impression of the twoflashes seemed to be equal. Every measurementwas repeated several times and the averagecurrent value was determined. The experimentswere performed for various visual angles bychanging the diaphragm f.

In Fig. 2a and 2b we give, for some visualangles, the average number of quanta necessaryfor the same brightness impression as a functionof the flash-time.

The same experimental arrangement wasadapted to the experiments concerning thedependence on the visual angle.

The diaphragm f was removed. The visualangles of the two flashes for which the time wasnow always equal, were determined by the twodiaphragms hi and h2 of the lenses a, and a2. Ofthe two beams issuing from lens d the one dueto flash II has always the visual angle 30', asregulated by the diaphragm h, while the visualangle of the beam due to flash I can be variedfrom 5' to 200' by means of the diaphragm h.The condition of the eye and its fixation were

* The intensities were determined according to the linesdescribed in the previous paper (see reference 3): The lampsI and II were calibrated, so that the average number ofquanta of every wave-length incident on the eve could becomputed with the aid of the measured properties of thefilters, the geometry of the arrangement, and the reflectingand absorbing properties of the lenses and mirror. As wedid not tige monochromatic light, the number of quanta ofevery wave-length was given a special weight accordingthe scoptopic luminosity curve, which was given the valueone in the optimal value.

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THE QUANTA EXPLANATION

quite similar as in the experiments describedconcerning the dependence on time.

For several intensities of the test flash of 30'of lamp II the observer adjusted the currents oflamp I for which the two flashes seemed to beequally bright for every fixated visual angle ofthe flash of I. Again the average of severalmeasurements was determined. The time be-tween the two flashes was again about 0.5 sec.The experiments were performed for severalflash-times. In Fig. 3 we give the curves indi-cating the dependence on visual angle of theaverage number of quanta of a flash for a certainbrightness impression.

2. DISCUSSION

1. The Dependence on Time

When in an area D a very small number ofquanta per second is absorbed by statisticalfluctuations in the time between two succeedingabsorptions, there will be a slight chance thatmore than one quantum is absorbed in a timeT. 1,2,3 When this happens light is perceived bythe cooperation of the nerve stimuli. The dura-tion of this process of an elementary light im-pression begins at the moment the first quantumis absorbed and ends when the light impressiondisappears. The brightness-impression and dura-tion will depend on:

a. The accidental number of absorbed quantaand the accidental distribution in time of theabsorptions. We know that two absorptionswithin r sec. cause light impression, but aboutthe influence of extra absorbed quanta on thebrightness impression we know nothing.

b. The way in which the pulses of the acti-vated rods are conducted by the retina, opticalnerve, brain center to consciousness. Even if thesame rods, as regards time and place, are acti-vated in exactly the same way, it is not sure thatthe way of conducting is also the same, so thatlight impressions are not necessarily identical indurations, extension, and brightness.

For very low intensities of the illuminated areaof the retina the frequency of the elementarylight impression is low. The average time betweenthe beginning of the separate impressions isdetermined by the chance for the absorption oftwo quanta within r.

For increasing intensity this time decreases,and for a sufficient increase it will become smallerthan the duration of an elementary light impres-sion. For these intensity values we may expectsingularities to occur in the behavior of the

5 _ _

102 -E -o -

d1,2 - ___ -- 09

10 2 5 1 sc

2 - -10--

5 - ___---

5

2-

10

2

5 N~~~~trehl 600,, 0

io152 5 lO 2 5 1 sec.

5-

(b)

FIG. 2. Curves representing the average number ofquanta N of a flash necessary for a constant brightnessimpression as a function of the duration t of the flash. Thelowvest curve represents the thresholds N60%, the averagenumber of quanta for which the chance of observation ofthe flash is 60 percent.

233

1. A. BOUMAN AND H. A. VAN DER VELDEN

5 1O 20 50 OO 200'

FIG. 3. Curves representing the average number ofquanta N of a flash necessary for a constant brightnessimpression as a function of the visual angle of the flash.

brightness impression because the cooperation ofthe processes of elementary light impressionsbecomes possible. For the comparison of bright-ness impressions only the established equal or"just different" values are suitable. For the studyof possible singularities one can investigate how acertain brightness impression is obtained forseveral times of observation. It is not sure thatwe need the same intensity in order to obtainthe same brightness impression for different times.

In Fig. 2a and 2b we show this dependence fortwo visual angles and several brightness impres-sions. Each curve represents the average numberof quanta necessary for a fixed brightnessimpression. We give also the threshold curveN6o%. This is the average number of quantanecessary for a chance of observation of 60percent.

For increasing flash-times the number ofquanta per sec. decreases for the threshold N60%.The average time between the two absorptionswhich give rise to the light impression increases,and the chance for extra absorptions within acertain time of the absorption of the two quantadecreases.

In the lowest curve for a constant brightnessimpression of Fig. 2a the chance of observationof the flash is still < 1 for t values < 1 sec. Thefrequency of the elementary light impressionsis small and they will be seen separately. Themoments for the perception of light are com-pletely determined by the chance for the absorp-tion of two quanta within r sec. The shape ofthe curve deviates from the curve N60% whent>0.1 sec. We can conclude that for a certainbrightness impression of the visual process of anelementary light impression the average numberof quanta per sec. may not decrease below adefinite value, as the deviations are such thatfor values of t, for which NS60% is still proportionalto t, the number of quanta for the constantbrightness impression becomes proportional to t.

It can be concluded from the curves for highervalues of the brightness impression where con-tinuous light is perceived, that for a certainbrightness impression the average number ofabsorbed quanta per 0.11 sec. (or ca. 3) mustbe constant.

2. The Dependence on the Visual Angle

For the elementary light impression discussedunder 1. the points a and b will also be importantfor the extension of the impression.

When a flash of very low intensity is givenhaving duration tT, and visual angle which islarge compared with D, there will be a chancefor the absorption of more than one quantum bythe statistical fluctuations in an area D.',','When this occurs, the stimuli will produce anelementary light impression. The average dis-tance of these impressions is determined by thechance just mentioned. With increasing intensitythis chance increases, so that the average dis-tance between the elementary light impressionsdecreases. An increasing part of the illuminatedarea will seem to be filled with light. The possi-bility of cooperation of the elementary impres-sions increases. It is now important to know theaverage number of quanta as a function of thevisual angle for a constant brightness impression.

The data concerning this dependence are givenin Fig. 3. In a way quite analogous to the time-dependence, we can conclude that the averagenumber of quanta per cm2 may not be, smallerthan a definite value, so that the effective

L

234

in

THE QUANTA EXPLANATION

absorption of more quanta, within a certain areaof the place of absorption of the two quantaresponsible for the occurrence of the light im-pression, is important for the brightness impres-sion. It can further be concluded that for aconstant brightness impression the average num-ber of absorbed quanta per area 0 with a visualangle o of about 25' (2D) must be constant.

3. CONCLUSION

In the previous paragraph it was proved thatfor a certain brightness impression the averagenumber of effective absorptions per 0.11 sec. perarea 25' must be constant for all combinationsof time and visual angle of the test flash. Thebehavior of threshold values N60% for variousvisual angles flash-times, and visual acuity mustbe described with the aid of the statisticalfluctuations in time and place of the absorptionsof the quanta in the separate rods: the two-quanta explanation. The nature of the chemicalreactions of the photo-chemical material turnedout to be unimportant for these intensities. Forthe measurements of Figs. 2 and 3, especially forthose brightness impressions for which the con-ditions for the light impression are slightly morethan satisfied, and for which the nature of thebrightness impression determines the shape ofthe curve, the intensities are equal or only alittle higher than those occurring in the thresholdmeasurements and visual acuity. The chemicalproperties of the visual purple will here beunimportant also.

In our opinion, the hypothesis of the laws ofTalbot and Ricco, for the origin of brightnessimpression, being of a physiological or psycho-logical nature, must be considered to be correct.

From the conditions for the occurrence of alight impression, it appeared that two stimuli, oftwo different rods within a time T and a visualangle D, cooperate and result in a light impres-sion, and the effective absorption of one quantumin a rod causes a stimulus.

It is now quite sure that stimuli can cooperatewithin an area D' and time r. From the experi-ments described in the present article we canconclude that the number of stimuli within anarea o-2D and a time T-3r, caused by thesame number of quanta, cooperate to create thebrightness impression. As the first absorbed

quantum does not give rise to a stimulus fromwhich a light impression results, one may doubt,the two-quanta condition once being satisfied,whether every extra quantum causes a stimuluswhich increases the brightness impression. As itappeared from the data of Polyak that allrecipient elements (rods and cones) can influenceeach other by the nerve system, it seems to bevery probable that every extra stimulus in anarea in which already two quanta are absorbedincreases the brightness impression. In a subse-quent paper it will be proved that this assump-tion is confirmed by the behavior of the contrastsensitivity.

The behavior of the threshold values did notenable us to decide whether the two quanta mustbe absorbed within an area D' of the retina orwithin a distance D of each other. At presentwe cannot ascertain from the experiments of thebrightness impression whether the distance o isthe maximum distance within which stimuli cancooperate, or whether there is an area of diametero in the retina within which cooperation ispossible.

By the slight difference in the size of D and oit is, moreover, difficult to insure that the co-operation of all stimuli is already complete when2 quanta, corresponding to the two-quanta-explanation, are absorbed within a distance D,situated within 0. Especially when D' and 0 aresharply bordered areas of the retina, it might bepossible that complete cooperation of all stimuliwithin 0 is only possible when all areas D'belonging to 0 have absorbed two quanta. Asfor larger visual angles the two-quanta shape inthe curves of the brightness impression disap-pears for lower intensities compared with smallervisual angles it seemed probable that one coupleof quanta within D is sufficient for the completecooperation of all stimuli within 0.

4. REMARKS

It appeared that the simultaneous occurrenceof large visual angles and long flash-times in theexperiments of the threshold values causes devia-tions from the two-quanta explanation. Thechance for a light impression is decreased by theabsorption of quanta after 0.1 sec. from thebeginning of the flash and over an area 0.

The behavior of the visual acuity as a function

235

M36 . A. BOUMAN AND H. A. VAN DER VELDEN

of the intensity depended on the shape of thetest object that had to be recognized.3 When theilluminated areas were surrounded completely bydark areas the behavior agreed with the de-pendence predicted by the threshold values forcircular spots. When the illuminated areas ad-joined an illuminated periphery the visual acuitywas proportional to the intensity, in accordancewith the two-quanta explanation. The diminu-tion just mentioned appeared to be less impor-tant. This phenomenon is now clear from thefacts found by the experiments of the presentpaper. The quanta absorbed in the surroundingsadjoining the test-figure facilitate the brightnessimpression of the boundary of the illuminatedareas of the test figure.

The area 0 has the nature of a recipient unitas the observer has always the same brightnessimpression when the total average number ofquanta in this area is the same, regardless of thedistribution of the places of absorption. More-over, it is impossible to recognize a figure ofwhich the details are smaller than the area 0.

The size of the recipient unit found by tenDoescate4 on this spot of the retina is smaller.The visual acuity, of which the upper limit isdetermined by the area 0, was the basis for his

4 J. ten Doescate, Ophthalmologica 112, (1946).

computation. He used the experiments ofWertheim who did not apply the same test figure.Moreover, the area may differ for differentpersons.

The area 0 was till now assumed to be welldefined. The quanta absorbed near the border ofthe area can increases the brightness impressionor size of the light impression. As the curves ofFig. 3 show rather sharp bends, it is probablethat the border of the area is rather well defined.

The time T has the nature of an averagingtime; the observer gets always the same bright-ness impression when the total average numberof quanta within this time is the same, regard-less of the distribution in time of the separateabsorptions.* In our opinion the critical frequency, which is

for the dark-adapted eye and high intensities inthe periphery almost independent of the intensityand is about 10, is determined by the time T.

The time T, within which stimuli of the rodscan cooperate, may not be sharply defined. Thequanta absorbed at the end of a time T canincrease the brightness impression or increasethe time of the light impression. As the curvesof Fig. 2 are rather well-defined it seemed againto be probable that the time T is also ratherwell defined.

236