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Experimental Studies of Light Quantum Fluctuations by the Visual Method
S. I. Vavilov
1. “Classical” and Quantum Fluctuations of Light
In the theory of light it is usually tacitly assumed that a light source is fully characterized by the spectrum, the radiation energy, and the state of polarization. Accordingly, all the physical properties of practically “monochromatic” light should be determined by the frequency of the light oscillations, the power of the luminous flux, and the state of polarization. In reality this is true only for “macro-optics,” the optics of large dimensions of light sources, long observation times, and considerable powers of the luminous flux. On closer study it becomes clear, however, that behind “macro-optics” there lies “micro-optics,” and that the three features listed are insufficient for explaining and describing all the characteristics of a light source and of a “monochromatic” beam. The interference capability of a source depends in a very distinctive way on the nature of the elementary radiator, on its multipolarity, on whether in the experiment we are dealing with a system of dipoles, quadrupoles, magnetic dipoles, etc. In phenomena taking place in rapidly interrupted light, the mean duration of elementary emission and the laws governing the decay of luminescence are of great importance. Finally, when working with light sources of very low energy, statistical oscillations—fluctuations of the luminous flux—begin to manifest themselves.
In principle (and, to a certain extent, in practice), there may exist “monochromatic” radiations with the same frequency, energy, and polarization, but differing from one another in their interference capability, duration of luminescence, and the character of their fluctuations.
Light fluctuations are one of the most delicate aspects of optical phenomena. To observe them it is necessary to experiment with extremely
weak light fluxes. At the same time, theoretically, fluctuations are a necessary property of the natural light flux, both in the classical, purely wave interpretation and from the modern quantum point of view.
An ordinary light source (in the classical treatment) consists of a multitude of radiating moving particles interacting with one another, colliding, receiving new impulses for radiation, or, conversely, ceasing to radiate upon collisions. Taking into account the statistical chaos of these processes, it is easy to understand that, when the number of luminous particles is relatively small and when their ensemble is observed over very short intervals of time, statistical deviations from the mean value in the light flux must be expected. Such fluctuations will be an expression of the disorder of the molecular motions of the luminous medium. It is important, however, to note the following features of these “classical” light fluctuations. First of all, they must not depend on the intensity of the light flux entering the instrument. One may, for example, place between the light source and the instrument any light filter, attenuating the light by an arbitrarily large factor, but the fluctuations must remain the same, since they are determined only by the molecular motion in the light source itself. On the other hand, from the classical point of view it is possible to construct a light source devoid of fluctuations. Let us imagine, for example, molecules dissolved in a very viscous substance and fluorescing under the action of an external factor free of fluctuations. From the classical point of view, fluorescent molecules not subject to the exciting or quenching action of a viscous medium (in agreement with experiment) will emit light continuously and steadily, without fluctuations.
It is obvious that “classical” light fluctuations must depend to a very great extent on the physical state of the light source, and only for the conditions of equilibrium thermal radiation of an absolutely black body can these fluctuations be determined independently of the composition and structure of that body, as a function of its temperature.
The quantum nature of radiation fundamentally changes the character of fluctuation phenomena. Even in the case just considered, when from the classical point of view light fluctuations may be absent, quantum fluctuations, determined by the spontaneous independence of the acts of radiation of individual molecules, must be fully manifested. In the general case, however, these quantum fluctuations must be superposed by statistical oscillations caused by molecular motions and mutual perturbations of the particles. For example, in the forced return of an excited molecule from a metastable state to a labile one with subsequent emission, “classical” fluctuations must appear, reflecting the statistical disorder of the molecular forcing perturbations, and at the same time
there will occur quantum deviations corresponding to the spontaneous character of the return from ordinary excited states to normal ones.
For the energy of radiation in a closed cavity of a heated black body, the mean square of the total fluctuations (classical and quantum) is expressed by Einstein’s two-term formula, showing the additive character of both kinds of fluctuations[^1]:
\[ (\overline{\Delta E_0})^2 = h\nu E_0 + \frac{c^3}{8\pi\nu^2\Delta\nu}\, \frac{E_0^2}{v_0}, \tag{1} \]
where (according to Planck’s formula)
\[ E_0= \frac{8\pi h\nu^3}{c^3}\, \frac{\Delta\nu}{e^{\frac{h\nu}{kT}}-1}\, v_0 . \]
Here \(v_0\) is the volume of the cavity, \(\nu\) is the radiation frequency, and \(\Delta\nu\) is the spectral interval under consideration. Using the written value of \(E_0\), Einstein’s formula is conveniently rewritten in the following form:
\[ (\overline{\Delta E_0})^2 = h\nu E_0 \left( 1+ \frac{1}{e^{\frac{h\nu}{kT}}-1} \right). \tag{2} \]
For visible light[^1], for example, with a wavelength of \(5000\ \text{Å}\) and for a temperature \(T\sim 300^\circ\),
\[ (\overline{\Delta E_0})^2 \approx h\nu E_0, \tag{2'} \]
but, for example, for \(T\sim 3000^\circ\) the second term must be taken into account in the expression for the fluctuations; they will be a measurable function of temperature. Thus fluctuations can, at least in principle, be used, for example, to measure the temperature of very hot stars, provided, of course, that it is possible to maintain the mean intensity of the star’s radiation constant during the experiment.
The principal property of quantum fluctuations of light, radically distinguishing them from “classical” fluctuations, consists in the fact that these fluctuations must be observed in any state of a constant light source, provided only that the resolving power of the light flux is sufficiently large. The source may be self-luminous, scattering extraneous light, incandescent, or luminescent—in all cases quantum fluctuations must occur, observable with sufficient attenuation of the light beam. As has already been said, these quantum fluctuations may under certain conditions be complicated by “classical,” molecular fluctuations, which reflect the nature of the emitter.
2. POSSIBILITY OF VISUAL OBSERVATION OF QUANTUM FLUCTUATIONS
The theoretical investigation of quantum fluctuations has until now been limited to the equilibrium temperature radiation of an absolutely black body. Experimentally, fluctuations of hard light radiations, X-rays, and $\gamma$-rays have already long been accessible to measurement with the aid of sensitive ionization chambers and Geiger–Müller type counters. Modern photocells and photon counters, however, have not been brought to the sensitivity and stability necessary for investigating fluctuations in the visible part of the spectrum. Meanwhile, a systematic study of fluctuation phenomena in this region is of specific interest, since the principal information about other optical properties of matter is especially abundant precisely for visible light.
In the absence of objective instruments, the investigation naturally had to turn again to the aid of the eye, which has already rendered invaluable services in the development of optics. In 1932, Barnes and Czerny[^3] suggested the possibility of observing quantum fluctuations of light with the aid of an eye adapted to darkness. The high sensitivity of the eye is well known, after it has spent a sufficiently long time (about 1 hour) in complete darkness. This sensitivity fluctuates somewhat (sometimes by many times) for different observers and for the same observer at different times. It depends on the place on the retina of the eye on which the image falls, and on the wavelength of the light used. For wavelengths of light of about $525\,m\mu$ the threshold of visual sensation, i.e. the minimum visually perceived energy in observing a light spot of diameter about $6'$, casting an image at a distance of $8^\circ$ from the central depression at the bottom of the eye (fovea centralis), is, according to our measurements described below[^11], about 200 photons per second, with relative fluctuations, for four different observers, by a factor of two. According to the measurements of Hecht and his collaborators[^12], for $\lambda = 510\,m\mu$, for a luminous spot with angular dimensions of $10'$ and when observing with a region of the retina located $20^\circ$ from the fovea, the minimum mean value of the light energy causing a visual sensation corresponds to approximately 100 photons, with fluctuations of the values by about a factor of three. The indicated numbers refer to the energy falling on the eye. There is no doubt that on the way to the retina part of the energy is lost owing to reflection and absorption in the ocular media. Finally, in the retina itself the absorption of light is incomplete. Therefore the actual number of photons producing the minimum visual sensation in the retina is much smaller than the indicated figures and must be measured in a few tens or even in single photons. Under such conditions, according to the laws of statistics, the fluctuations must be very considerable.
The consideration of the inevitability of quantum fluctuations under threshold conditions of observation by the eye was also made the basis of the work of Barnes and Czerny. The concrete experiments carried out by them were, however, wholly inconclusive and manifestly erroneous. The experiments were performed under such conditions that it was quite impossible to avoid numerous and very strong physiological fluctuations, well known to physiologists and psychologists² and much more noticeable and sharp than the expected quantum fluctuations. One of such experiments consisted, for example, in the following: a circular surface 15 cm in diameter with 50 small, randomly arranged green luminous spots on it was observed from a distance of 0.5 m with the naked eye. The whole pattern was visible only for 0.1 sec.; for 0.9 sec. it was screened from the observer by a rotating sector. Thus, the adapted observer, with an extremely mobile eye, for the greater part of the time had no point of support (“fixation point” of physiological optics) and, of course, could not remain in an unchanged position. The involuntary movement of the eyeball during the dark pause between flashes, in the absence of a fixation point, must inevitably have led to sharp fluctuations of the observed brightness, in particular because the rather large field of vision (about 18°) corresponded to different places on the retina with very strongly differing twilight sensitivity.
Another experiment of Barnes and Czerny they themselves describe as follows: “One can use a lamp for a dark room with a green filter, employed for developing panchromatic plates. On the light filter there is first placed a thick layer of white paper, and then black paper with suitable holes. The brightness of the holes can be regulated by the lamp rheostat with a slider or by changing the distance of the head. The fluctuations can be observed only after spending several minutes in the dark.” If, however, one tries to repeat this simple experiment, it is easy to become convinced that the “fluctuations” are quite noticeable not only under conditions of visual threshold, but also at brightnesses hundreds of times exceeding threshold. In other words, quantum fluctuations have nothing to do with it here, and the phenomenon is fully explained by physiological causes.
In addition, it may be asserted that even if all physiological effects were excluded, with the exception of one—namely, the property of the eye to retain a visual impression for fractions of a second—nevertheless quantum fluctuations in the experiment just described could not be observed. Owing to the duration of visual impression, physical fluctuations would, under the conditions described, have to be blurred and averaged, just as they are averaged when observing threshold brightnesses from a large luminous surface. A surface covered with individual luminous spots has no great advantages in this respect over a uniformly
luminous surface of the same dimensions. It is difficult to understand this experiment of Barnes and Czerny, especially since the same authors, in other experiments mentioned above, used the method of short flashes.
From the energy measurements carried out by Barnes and Czerny, it follows that their final experiments were performed at such intensities that quantum fluctuations should, theoretically, have been quite noticeable. The conditions of the experiments, however, were such that physiological fluctuations must also necessarily have occurred, and moreover on a large scale, masking the quantum fluctuations. Thus Barnes and Czerny, while correctly pointing out the possibility of visual observation of quantum fluctuations, did not prove their existence experimentally either qualitatively or, still less, quantitatively.
3. FUNDAMENTALS OF THE VISUAL METHOD FOR MEASURING QUANTUM FLUCTUATIONS
The original consideration of Barnes and Czerny concerning the necessity of noticeable fluctuations in light fluxes corresponding to the visual threshold served as the occasion for a series of our experimental works in 1932–1941, set forth below 4–11*).
From what was said in § 2 regarding the experiments of Barnes and Czerny, it follows that observation of quantum fluctuations in a continuous light flux is hardly possible because of the finite duration of the visual impression and the resulting averaging of the fluctuations. Nor is observation of fluctuations possible for large angular dimensions of the luminous surface. Further, in order to observe quantum fluctuations it is necessary to fix the eye. Conversely, if three conditions are observed:
1) short duration of the flashes,
2) small dimensions of the image on the retina,
3) fixation of the eye,
physical fluctuations at threshold light fluxes must inevitably be observed, provided only that the quantum conceptions of the emission and absorption of light are correct.
It is important to emphasize that visual observation and measurement are greatly facilitated by the fact that there is a sharp threshold of visual sensation. The visual effect in the region of the threshold changes discontinuously; it at once falls to zero at a certain “threshold” value of the light energy. If there were no such threshold and the visual sensation only gradually weakened, then the presence of fluctuations could be judged only by the rapid, difficult-to-measure fluctuations in the brightness of the flashes. Owing to the existence of the threshold,
*) The work was carried out by E. M. Brumberg, K. B. Panshin, Z. M. Sverdlov, T. V. Timofeeva, and the author.
flashes whose energy is less than some limiting value are simply not visible. Thus there is a very sharp qualitative indication of fluctuations: the flashes are either visible or not. Small changes in the brightness of flashes in the threshold region are practically imperceptible \(^{4}\).
Let \(n\) denote the mean number of monochromatic photons of frequency \(\nu\) absorbed in the retina during one brief flash, and let \(z\) denote the actual number of photons absorbed during the given flash. Owing to the presence of a sharp threshold, only those flashes will be perceived for which
\[ z \geqslant n_0 . \tag{3} \]
In what follows it is assumed that \(n_0\) is a constant quantity. If, in fact, \(n_0\) itself fluctuates quite irregularly from flash to flash about some mean value (for physiological reasons), then in the final formulas (as will be seen below from the derivation) \(n_0\) is equivalent to this mean value. Slow changes of \(n_0\) from one group of observations to another are also possible, again for physiological reasons. These latter variations of \(n_0\) must cause deviations of the determined probabilities from the theoretical ones and may cause scatter of the experimental points.
In any optical experiment the properties of light can be determined only by its effects. In experimenting with fluctuations, at the final stage we always in fact deal only with fluctuations of the absorbed energy. Only under the condition of complete absorption, when the number of photons entering the medium and absorbed in it is the same, must the fluctuations in the light flux and in the absorbing medium coincide. With incomplete absorption, the character of the fluctuations (in the classical statistics used here) must depend only on the fluctuations of absorption. It is clear, for example, that if the incident flux is quite regular in its structure and there are no fluctuations in it, but only fluctuations of absorption exist, then the final result will be determined by the fluctuations of absorption. Conversely, in the absence of fluctuations in absorption, the final fluctuations are determined by those in the incident light flux.
In what follows, in accordance with experiment (§ 2), we assume that
\[ n_0 \gg 1 . \tag{4} \]
For the probability of the number \(z\), with mean \(n\), according to the Poisson formula we have
\[ p(z)=\frac{n^z}{e^n z!}. \tag{5} \]
In accordance with what has just been said, \(n\) in this formula denotes the mean number of photons absorbed in the retina.
On the basis of the restriction (4), the probability \(P\) that \(z\) has any value between \(n_0\) and \(\infty\) can be expressed by the integral:
\[ P=\frac{1}{\sqrt{2\pi n}}\int_{n_0}^{\infty} e^{-\frac{(z-n)^2}{2n}}\,dz . \tag{6} \]
Introduce the notation
\[ y=\frac{z-n}{\sqrt{2n}} . \tag{7} \]
Using it, we can rewrite (6) in the following form:
\[ P=\frac{1}{\sqrt{\pi}}\int_{y_0}^{\infty} e^{-y^2}\,dy =\frac{1}{2}-\frac{1}{2\sqrt{\pi}}\int_{0}^{y_0} e^{-y^2}\,dy =\frac{1}{2}-\frac{\varphi}{2}; \tag{8} \]
where
\[ \varphi=\frac{1}{\sqrt{\pi}}\int_{0}^{y_0} e^{-y^2}\,dy \tag{9} \]
denotes the tabulated Gaussian integral, and
\[ y_0=\frac{n_0-n}{\sqrt{2n}} . \tag{10} \]
In processing our measurements, we introduced a simplification of formula (8), based on the fact that the most important and practically interesting interval of variation of probabilities lies approximately between 0.0 and 0.8. In this interval, within the limits of experimental errors, \(\varphi\) differs hardly at all from \(y_0\), as is evident from Table 1.
Table 1
| \(\varphi\) | 0.00 | 0.10 | 0.20 | 0.30 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 |
|---|---|---|---|---|---|---|---|---|---|
| \(y_0\) | 0.00 | 0.09 | 0.18 | 0.28 | 0.38 | 0.48 | 0.60 | 0.74 | 0.91 |
On this basis, the probability \(P\) (8) can be approximately expressed as follows:
\[ P \simeq \frac{1}{2}-\frac{y_0}{2} =\frac{1}{2}-\frac{1}{2}\frac{(n_0-n)}{\sqrt{2n}} . \tag{11} \]
Let
\[ \frac{n}{n_0}=x . \tag{12} \]
Using \(x\), we can rewrite (11) in the following form:
\[ P \simeq \frac{1}{2}-\frac{1}{2}\sqrt{\frac{n_0}{2}}\,\frac{(1-x)}{\sqrt{x}} . \tag{13} \]
The absolute value of \(x\) is conveniently determined graphically from the experimental data by the condition following from the complete formula (8):
\[ \text{for } x=1 \qquad P=\frac{1}{2}. \tag{14} \]
Having found experimentally the values of \(P\) corresponding to relative values of \(x\), we can graphically, by condition (14), find \(P=\frac{1}{2}\), for which \(x=1\). From this the conversion factor is determined, making it possible to pass from relative values of \(x\) to absolute ones.
On the basis of the more exact formula (8), the probability must have the form shown in Fig. 1, where \(P\) is plotted as a function of \(\frac{1-x}{\sqrt{x}}\) (proportional to \(y_0\)). Formula (13), corresponding to a straight line, practically coincides with the middle rectilinear part of the curve. The slope of the rectilinear segment is
\[ k=\frac{1}{2}\sqrt{\frac{n_0}{2}}. \tag{15} \]
Whence
\[ n_0=8k^2. \tag{16} \]
Fig. 1. Probability \(P\) as a function of \(\frac{1-x}{\sqrt{x}}\).
The theory set forth forms the basis of the method for processing measurements of visual quantum fluctuations on the basis of determining \(P\) at varying values of \(x\).
4. EXPERIMENTAL ARRANGEMENTS AND MEASUREMENT PROCEDURE
The conditions of short duration of the observed light flashes, sufficient smallness of the image on the retina, and fixation of the eye, necessary for observing fluctuations, can be readily realized. Fig. 2 gives a photograph from above of the experimental arrangement of our experiments in 1933.1 The light of a four-volt incandescent lamp, powered by an accumulator and placed in lantern \(L_1\), passes through ground glass, a green filter, and a circular aperture 1 mm in diameter into the tube \(R_1R_2\). At both ends of the tube polarizing prisms are located. By rotating one prism relative to the other, the light can be weakened, without changing its spectral composition, to any degree. Between the light source and the tube is placed a disk \(S\) with an aperture, which slowly (1 revolution per second) rotates by means of a synchronous motor \(M\), periodically admitting the light for 0.1 second and blocking it for 0.9 second. The light of the second lamp \(L_2\), passed through a red filter, was reflected by a small mirror \(m\) placed in front of the rotating disk. The image of the red
the fixation spot was located at the same distance from the eye (60 cm) as the green spot from lamp \(L_1\). In contrast to the green spot, the red fixation spot was not intercepted by the rotating disk and was at all times within the field of vision of the eye. The angular distance of the red fixation spot from the green one was equal, in our first experiments, to \(4^\circ\).
Fig. 2. Photograph of the first setup for measuring visual quantum fluctuations.
Between the disk and the tube \(R_1R_2\) there was then placed, in order to attenuate the light by a definite known number of times, a stack of glass plates. Each plate attenuates the light, owing to double reflection, as photometric measurements showed, by 7%. By introducing into the path of the light beam, in the dark, a definite number of glass plates, or, conversely, by reducing their number, it is easy to regulate the luminous flux in quantitative steps near the threshold conditions.
On the second table there is an astronomical chronograph \(CB\), with a coil of telegraph paper tape and electrically controlled pens. These pens are connected with the rotating disk \(S\) in such a way that each revolution of the disk corresponds to a mark on the paper tape. By means of the electric key \(K\) on the other table, the observer can make his own time marks on the moving tape.
The experiment is carried out as follows. The process of adaptation of the observer’s eye lasts a full hour. The observer’s chin rests on a special support, as is usual in experiments in physiological optics. Through the tube \(R_1R_2\) the eye sees both points \(L_1\) and \(L_2\). The brightness of both points is regulated partly by the polarizing prisms \(R_1\) and \(R_2\), and partly by the resistances \(W_1\) and \(W_2\). The eye is fixed throughout on the red point, as a result of which the green point is observed peripherally at a distance of \(4^\circ\) from the fovea.
For the observations a rather prolonged preliminary training is required (5–10 sessions, each lasting at least an hour). The aim of this training is to accustom the eye to fixation, to peripheral vision, and at the same time to the attentiveness required for timely registration of the observed flashes by pressing the key. Our many years’ experience with a dozen observers has shown that the first measurements (before training) are, in all cases, disorderly in character, indicating insufficient “training” of the eye. The state of the observer (fatigue, time of day, illness, etc.) plays a major role. Measurements are greatly hindered by the so-called “intrinsic light of the retina,” i.e. light clouds drifting before the eye in darkness. For experiments it is necessary to choose periods when this phenomenon is absent in the observer or is greatly weakened.
Setting the disk \(S\) into rotation, the observer, while constantly fixing the eye on the red spot, gradually lowers the brightness of the green spot (which under these conditions appears colorless) down to threshold.
The first new qualitative result of this procedure, one of fundamental importance, is that under such conditions there is essentially no sharp threshold of visual sensation. As the brightness of the green spot is lowered, the observer first notices that each passage of the disk aperture corresponds to a flash of diminishing brightness. Then, however, a genuine fluctuation regime begins: the flashes, with further very slight weakening of the light, cease to correspond to every passage of the aperture. For some passages there are no flashes. If the light is decreased still further, the flashes are observed more and more rarely and, finally, become so rare that they can easily be missed. From the qualitative side, this phenomenon confirms with striking clarity the presence of light quantum fluctuations.
For quantitative measurements the observer must mark each flash (under the conditions of the fluctuation regime) by pressing the key \(K\). After some number (sufficient for statistical calculations) of such marks, the observer inserts an additional glass plate into the stack \(G\) and again marks the flashes on the chronograph tape. To obtain such measurements for 4–5 different numbers of glass plates requires 1–2 hours. The marks of flashes on the paper tape and the numbers of all passages of the aperture
of the disk, automatically recorded on the same tape, give everything required for determining the probability of the appearance of flashes. This probability is obviously equal to the ratio of the number of observed flashes to the number of revolutions of the disk. On the other hand, the number of glass plates inserted in the path of the light beam serves as a measure of the relative value \(x\), i.e., of the mean number of photons corresponding to each flash.
In our experiments of subsequent years (1938–1941) the apparatus was somewhat modified[^9], while, however, retaining all the original essential parts. Its scheme is given in Fig. 3.
Fig. 3. Scheme of the second apparatus for measuring visual quantum fluctuations.
The observer’s head, as in the former apparatus, rests on a chin rest. The eye is fixed on the red point \(S\) (attenuated, if desired, by means of a rheostat) through the glass plate \(G\). By changing the inclination of this plate and its distance from the axis of the apparatus, one can vary the angle between the fovea and the tested retinal region. To this retinal region the light from the source reaches through the diaphragm \(O\), covered with milk glass, through an aperture in the disk \(D\) (rotated by means of the synchronous motor \(M\)), through the light filter \(F\), and, finally, through the stack of glass plates \(P\) and the photographic wedge \(K\). The diaphragm \(O\) is illuminated by the incandescent lamp \(L\), fed by an accumulator battery. The light is directed into the eye by the mirror \(m\). For absolute measurements of the energy corresponding to the visual threshold, the mirror \(m\) and the milk-glass plate in front of the diaphragm \(O\) are removed, and the diaphragm is illuminated by the “black body” \(N\).
In this apparatus the angular dimensions of the observed circular light spot were usually equal to \(6'\). Most observations were made at a distance of \(8^\circ\) from the fovea.
For measurements at various wavelengths, instead of the lamp \(L\) there was placed a monochromator with a lamp and with additional filters protecting against scattered light of other wavelengths.
To compare the number of photons \(N\) incident on the pupil of the eye in the case when \(n_0\) photons are absorbed in the retina, it is necessary to know the absolute value of the energy incident on the pupil of the eye during the flash. For this purpose, as already indicated, the black body \(N\) (Fig. 3), moved up to the diaphragm \(O\) (with the ground glass removed), is used. Knowing the temperature of the black body \(\Theta\), the dimensions of the aperture \(O\), the spectral transmission of the filter \(F\), the distance of the diaphragm \(O\) from the pupil, and the diameter of the observer’s pupil under conditions of dark adaptation, one can, on the basis of the laws of black-body radiation, determine the power of the luminous flux incident on the pupil. The temperature of the black body was about \(1100^\circ\) K. The accuracy of the values of the energy \(N\) given below is approximately \(10\%\). The diameter of the pupil used under complete adaptation was determined by photographing the eye in the light of a flash lamp. A scale was pressed against the eye and photographed together with the pupil.
In view of the inconvenience of constant work with the black body, the lamp \(L\), under a definite burning regime and when observed through the light filter \(F\), was compared with the black body. Subsequently it, as a secondary standard, served for relative and absolute measurements. On the basis of (12) and (14), for \(P = \frac{1}{2}\), \(n = n_0\). If, in addition to the fluctuation curve \(F(x)\), the absolute value of the energy for certain \(x\), obtained by the indicated method, is known, then, of course, all values of \(x\) can be converted into absolute ones, and in this way \(N\) will be determined from the \(x\) corresponding to \(P = \frac{1}{2}\). It is important to note that, in this way, in the results reported below \(N\) was determined under the same conditions and from the same measurements as \(n_0\).
Fig. 4. Average number of photons \(\overline{N}\) in a single flash, producing the impression of constant brightness as a function of flash duration (according to measurements by Bouman and van der Velden).
The choice of the duration of an individual flash as one tenth of a second in our experiments was determined by preliminary experiments which showed that, for the same energy of the flash (under conditions close to threshold), for any flash durations \(\ll 0.1\) seconds the brightness remains unchanged; for durations greater than \(0.1\) seconds the brightness decreases and greater energy is required to maintain the former brightness. These preliminary experiments are confirmed by the recent measurements of Bouman and van der Velden\(^{16}\).
In Fig. 4 there is reproduced from this work one of the experimental curves giving the number of photons \(\overline{N}\) of a single flash, cre—
giving the impression of the same brightness as a function of the duration of the flash for a field of view of about \(4'\). It is evident from the figure that the region of constant brightness begins at approximately \(0.1\) sec., in accordance with the figure we have adopted.
In concluding the description of the procedure for fluctuation measurements, one should indicate the method for checking the statistically random character of the fluctuations. Such a check is necessary, since if this condition is not fulfilled, then systematic influences, having nothing in common with quantum fluctuations, undoubtedly intervene in the phenomenon.
Fig. 5. Check of the statistical randomness of fluctuations.
Let us denote by \(w\) the probability of an observed flash for a given passage. The probability \(p_n\) that after a flash there will follow \(n\) passages without flashes and then one passage with a flash will be
\[ p_n=(1-w)^n w \]
or
\[ \log p_n = n \log (1-w)+\log w . \]
This formula can also be used to check the randomness of the fluctuations. Let us denote by \(a_n\) the number of intervals corresponding to \(n\) “dark” passages on the paper tape. Then
\[ p_n=\frac{a_n}{\sum a_n}. \]
Determining from the marks on the paper tape the values of \(p_n\) by this formula and taking logarithms of the numbers obtained, in the case of statistical randomness we should obtain a straight line. As an example we give data relating to one tape obtained from B.’s measurements. In Table II the values of \(n\), \(a_n\), \(p_n\) are compared, and in Fig. 5 the dependence of \(\log p_n\) on \(n\) is represented graphically. From the slope of—
Table II
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | ||
|---|---|---|---|---|---|---|---|---|
| \(n\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | Number of passages 352 |
| \(a_n\) | 64 | 60 | 28 | 11 | 4 | 2 | 1 | \(\sum a_n=170\) |
| \(p_n\) | 0.38 | 0.35 | 0.17 | 0.06 | 0.02 | 0.01 | 0.005 | \(w=\dfrac{170}{352}=0.48\) |
obtained straight line \(w=0.52\), from the intersection of this straight line with the ordinate axis \(w=0.50\), is the directly calculated value \(w=0.48\). Thus, the marks on the chronograph tape are indeed distributed at random. Similar results were also found on other tapes analyzed by the indicated method.
5. RESULTS OF FLUCTUATION MEASUREMENTS FOR THE BLUE-GREEN REGION OF THE SPECTRUM
During the years 1932–1941, hundreds of series of fluctuation measurements were carried out at the State Optical Institute, with ten observers taking part. The measurements were analyzed: 1) with respect to the fulfillment of the linear dependence (13), 2) with respect to the absolute value of \(n_0\), i.e. the number of photons corresponding to the threshold on the grid.
To explain the methods of analyzing the measurements, Table III is given; it presents the results of the numerical processing of one tape, obtained by B. in 1932 with a green filter \((\lambda \sim 530\ \mathrm{m}\mu)\). In this table \(G\) is the number of glass plates introduced into the light beam and weakening the light, \(\alpha\) is the corresponding transmission of light in relative units, with the initial value chosen arbitrarily equal to 1. \(\mathfrak{N}\) is the number of flashes, and \(N\) is the number of passages of the disk.
Table III
| \(G\) | \(\alpha\) | \(x\) | \(\dfrac{1-x}{\sqrt{x}}\) | \(\mathfrak{N}\) | \(N\) | \(P\) |
|---|---|---|---|---|---|---|
| 6 | 1.00 | 1.24 | \(-0.215\) | 37 | 38 | 1.00 |
| 5 | 0.93 | 1.16 | \(-0.148\) | 26 | 32 | 0.81 |
| 4 | 0.86 | 1.08 | \(-0.077\) | 35 | 80 | 0.44 |
| 3 | 0.80 | 1.00 | \(0.000\) | 48 | 111 | 0.43 |
| 2 | 0.75 | 0.96 | \(0.074\) | 44 | 142 | 0.31 |
| 1 | 0.70 | 0.93 | \(0.139\) | 65 | 327 | 0.20 |
To compute \(x\), one may proceed (as indicated in § 4) in the following way. The tabulated values of \(P\) are plotted as a function of the number of plates, and from an interpolation straight line one finds the transmission coefficient corresponding to \(P=\dfrac{1}{2}\). From this there is obtained a conversion factor allowing \(\alpha\) to be transformed into \(x\).
In Fig. 6 are given the graphically processed results of the measurements of 1938[^9] for four observers—B., Sv., S., and E. From Fig. 6 one can obtain an idea of the typical scatter of the experimental points. A very large body of material, collected over many years of work by ten observers, fully confirmed the existence of the approximate linear dependence between \(P\) and \(\dfrac{1-x}{\sqrt{x}}\), as predicted by the theory, in all cases without exception.
Most of the measurements were carried out in the blue-green region of the spectrum, \(500\text{--}550\ \mathrm{m}\mu\), in which, as our experience has shown, within the limits of experimental error there is no dependence of the fluctuations on wavelength, and \(n_0\)
for one and the same observer. For different observers, as is seen from the experimental results presented below, \(n_0\) fluctuates within very wide limits.
Fig. 6. Results of fluctuation measurements by four observers.
Table IV gives the principal summary of the mean results obtained for \(k\) and, respectively, \(n_0\) by different observers during the period from 1933 to 1941. The table indicates the observer’s initials, the year of the experiments, \(\omega\)—the peripheral angle of observation, \(\lambda\)—the mean value of \(k\) and \(n_0\), obtained from a series of observations (a series comprises from 3 to 10 rows of observations). For some observations the mean value \(N\) is also given, determined by the method described in § 4. In the last column the ratio \(\dfrac{N}{n_0}\) is given.
The fluctuations of the values of \(k\) and \(N\) for one and the same observer on different days may be judged from Table V, in which the values of \(N\) and \(k\) are given for two observers on different days of the same year.
Consideration of Tables IV and V leads to the following conclusions:
1) For one observer, with the other conditions of the experiment kept unchanged, \(k\) is practically constant in the wavelength interval 500—550 mµ.
2) The value of \(k\) for different observers and in different experiments fluctuates by more than a factor of two (\(n_0\), proportional to the square of \(k\), correspondingly changes almost by a factor of 6).
3) For one and the same observer on different days \(k\) sometimes fluctuates by a factor of 1.4 (\(n_0\), correspondingly, by a factor of 2). However, in view of
Table IV
| Observer | Year | \(\omega\) | \(\lambda\) | \(\bar{k}\) | \(\bar{n}_0\) | \(N\) | \(\dfrac{N}{\bar{n}_0}\) |
|---|---|---|---|---|---|---|---|
| B.) | 1932 | \(4^\circ\) | 530 | 2.4 | 47 | — | — |
| B.( | 1936 | \(10^\circ\) | 510 | 1.9 | 30 | — | — |
| B.) | 1938 | \(8^\circ\) | 525 | 1.7 | 23 | 168 | 7.1 |
| P.) | 1936 | \(10^\circ\) | 500 | 1.0 | 8.0 | — | — |
| P.( | 1936 | \(10^\circ\) | 510 | 1.0 | 8.0 | — | — |
| P.( | 1936 | \(10^\circ\) | 520 | 1.2 | 11.0 | — | — |
| P.) | 1936 | \(10^\circ\) | 540 | 1.2 | 12.0 | — | — |
| Ch.) | 1938 | \(8^\circ\) | 525 | 1.9 | 29 | 231 | 8.0 |
| Ch.( | 1940 | \(7^\circ\) | 500 | 1.8 | 26 | — | — |
| Ch.) | 1940 | \(7^\circ\) | 550 | 1.8 | 26 | — | — |
| Sv. | 1938 | \(8^\circ\) | 525 | 1.5 | 18 | 324 | 18.0 |
| E. | 1938 | \(8^\circ\) | 525 | 2.0 | 32 | 108 | 3.4 |
| T.) | 1940 | \(7^\circ\) | 500 | 1.9 | 29 | — | — |
| T.) | 1940 | \(7^\circ\) | 550 | 1.7 | 23 | — | — |
| Zh.) | 1941 | \(7^\circ\) | 500 | 1.3 | 13 | — | — |
| Zh.( | 1941 | \(7^\circ\) | 500 | 1.4 | 16 | — | — |
| Zh.) | 1941 | \(7^\circ\) | 550 | 1.7 | 23 | — | — |
| Sm.( | 1940 | \(7^\circ\) | 500 | 2.1 | 35 | — | — |
| Sm.) | 1940 | \(7^\circ\) | 550 | 2.3 | 42 | — | — |
| L.) | 1940 | \(7^\circ\) | 500 | 1.0 | 8.0 | — | — |
| L.) | 1940 | \(7^\circ\) | 550 | 1.0 | 8.0 | — | — |
| Average value | — | — | — | 1.6 | 20 | 208 | 9.0 |
Table V
| Observer Ch., 1938 | \(N\) | \(k\) | Observer Sv., 1938 | \(N\) | \(k\) |
|---|---|---|---|---|---|
| Date | \(N\) | \(k\) | Date | \(N\) | \(k\) |
| 16.XII | 224 | 2.0 | 2.X | 333 | 1.4 |
| 20.XII | 192 | 1.8 | 5.X | 324 | 1.8 |
| 22.XII | 291 | 1.8 | 10.X | 323 | 1.4 |
| 23.XII | 255 | 1.9 | 22.X | 335 | 1.4 |
| 25.XII | 246 | 1.5 | 20.XI | 304 | 1.7 |
| 26.XII | 228 | 1.9 | 23.XI | 318 | 1.4 |
| 28.XII | 242 | 2.3 | 27.XI | 333 | 1.5 |
| 29.XII | 221 | 1.6 | |||
| 30.XII | 178 | 2.2 | |||
| Average | 231 | 1.9 | Average | 324 | 1.5 |
with the small accuracy with which the slope of the straight lines \(k\) can be determined, there are not sufficient grounds for regarding the scatter of the values of \(k\) and, still less, \(n_0\) among different observers and in the same observer on different days as always real.
4) The quantity \(N\), i.e. the number of photons in the threshold region falling on the pupil of the eye, fluctuates on different days for one and the same observer within limits considerably smaller than \(n_0\), as is seen from Table V. At the same time, for different observers \(N\) differs very greatly, by a factor of 3.
5) \(N\) is always considerably greater than \(n_0\); on the average
\[ \frac{N}{n_0}=9.0; \]
for different observers the ratio
\[ \frac{N}{n_0} \]
shows very large variations, more than by a factor of 5.
The very considerable difference between \(n_0\) and \(N\) may at first sight seem to sharply contradict our basic assumption about the quantum character of the observed fluctuations. Such a conclusion, however, is entirely incorrect. On the contrary, data on the optical properties of the ocular media obtained over the last decade necessarily require that
\[ \frac{n_0}{N}\ll 1. \]
By the time our first work was completed, it was possible to speak only in general terms of “energy losses caused by reflection, absorption, and scattering in the eyeball,” as the reason for the sharp divergence between \(N\) and \(n_0\). At the present time there are a number of physiological investigations that make this aspect of the question more concrete.
Not to mention the small loss of light due to reflection at the surface of the eye, it is necessary to take into account the considerable absorption of light in the ocular media up to the retina. According to Roggenbau and Wetthauer\(^{17}\), the absorption of light in the region around 510 m\(\mu\) in all ocular media up to the retina amounts to about 50%. On the other hand, the rods, in which the absorption of light leading to the visual act takes place, do not completely cover the surface of the retina, especially in regions far from the fovea. According to Osterberg\(^{18}\), the total number of rods on the retina is approximately \(1.2\cdot 10^8\). The diameter of the rods is about \(2\mu\). The indicated number of rods is distributed over a surface of about \(10\ \text{cm}^2\), i.e. on the average only about one third of the entire retinal surface is occupied by rods. In the region corresponding to the peripheral distance \(\omega=8^\circ\), on \(1\ \text{mm}^2\) of the retina, according to Osterberg, there are approximately \(1.6\cdot 10^5\) rods; consequently, only 0.5 of the area in this place is covered by rods.
Further, on the very surface of the retina there are nervous and other tissues which to some (apparently small) extent absorb and scatter the incident light. Finally, absorption in the rods themselves, in the visual purple, is not complete even at the maximum (in contrast to what we assumed in the first work). According to
According to Wald,^19 the absorption coefficient of visual purple in the region around \(500\,m\mu\), per unit surface of the frog retina, is \(33.6\%\), for the rabbit \(4.2\%\), and for the rat \(13.0\%\).
Let \(\alpha\) denote the fraction of the incident light transmitted by the ocular media to the retina, \(\beta\) the transmission in the layers of the retina itself that screen the rods, \(\gamma\) the fraction of the retinal area covered by rods in the region of the eye under study, and finally \(\delta\) the fraction of light absorbed in the rods. Then it is obvious that
\[ n_0 \leqslant N\alpha\beta\gamma\delta . \tag{17} \]
The sign \(<\) corresponds here to the possibility that the yield of the physicochemical process determining the visual act is \(<1\).
Using the figures given above for our experimental conditions, one may write:
\[ n_0 \leqslant N \cdot \frac{1}{2}\cdot \frac{1}{2}\,\beta\delta, \]
i.e.
\[ \frac{N}{n_0} \geqslant \frac{4}{\beta\delta}. \]
Comparing this with the experimental mean result \(\dfrac{N}{n_0}=9.0\), we arrive at the conclusion:
\[ \beta\delta \simeq \frac{4}{9}. \]
As has been said, there is no information concerning the magnitude of \(\beta\); in any case it is probable that \(\beta\) differs little from unity. As for \(\delta\), corresponding to the fraction of light absorbed by visual purple in the rods, for the region around \(500\,m\mu\), according to the estimates of physiologists, its value is not less than \(0.2\).^12 Thus the result obtained is in agreement with physiological data. It is impossible, and makes no sense, to speak of quantitative agreement in this question, if only because all quantitative data on the ocular media vary extremely from one eye to another. Nor can statistical methods be applied here, in view of the very small number of well-studied cases.
There is no doubt that all four parameters \(\alpha,\beta,\gamma,\delta\) can vary within very wide limits among different people, whereas their considerable variations over a short interval of time in one and the same person are unlikely. This conclusion is consistent with experiment, as is seen from Tables IV and V.
Thus the experiments described, in addition to confirming the basic assumption of the quantum character of the observed fluctuations, yield results that are very essential for the study of the properties of the eye. Simultaneous determination of \(n_0\) and \(N\) by the method presented makes it possible to measure separately, in the living eye, the sensitivity of the retina and the fraction of light actively manifested in the visual act.
6. FLUCTUATION MEASUREMENTS BY Z. GECHT AND HIS COLLABORATORS
In 1941 a report appeared on visual fluctuation measurements by Z. Gecht, S. Shlaer, and M. Pirenne[^13], carried out in New York. These experiments were performed according to a scheme quite coinciding with ours, differing only in nonessential details and in a considerably smaller scope. The measurements led to quantitative results for \(n_0\) that agree with the results obtained by some of our observers and differ considerably from the results of others.
The arrangement of the American investigators is shown in Fig. 7. The light of an incandescent lamp, supplied by a storage battery, is focused on the slits of a double monochromator \(M_1M_2\) and, finally, on the diaphragm \(P\) near the eye. The observer’s head rests on a chin rest; moreover, as is sometimes done in experiments in physiological optics, the observer’s teeth were clamped on a certain object. The eye is fixed on the spot \(FP\) at an angular peripheral distance of \(20^\circ\). The light is roughly attenuated by the neutral filter \(F\) and finely by the neutral wedge \(W\). The luminous flux corresponding to \(\lambda = 510\ \mathrm{m}\mu\) is opened for \(0.004\) sec. by means of a special device \(S\), which makes it possible, upon pressing a button, to give individual pulses of this duration after any interval of time. The experimenter changes the intensity of the light by moving the wedge \(W\). The observer, by pressing a key, produces a flash and reports whether he saw it or not. Usually about 50 flashes of one and the same intensity were produced. About \(1\frac{1}{2}\) hours are required for a full series of observations. Independently of the fluctuation measurements, measurements of the number of photons falling on the pupil \((N)\) were made with the aid of a thermoelement with a galvanometer. The short duration of the flash, \(0.004\) sec., was chosen “for greater assurance.”
Fig. 7. Diagram of the apparatus for measuring visual quantum fluctuations by Gecht, Shlaer, and Pirenne.
The processing of the fluctuation data, while not differing in principle from ours, was formally somewhat different. The value of the probability \(P\), according to formula (8), was plotted from Fry’s tables[^20] as a function of the logarithm of the mean value of the number of photons per flash. For each \(n_0\) this gives its own curve, and the entire plane \((P, \log n_0)\) can be covered continuously by a family of such curves for a continuously varying parameter \(n_0\) (Fig. 8). The advantage of using lo
...of a logarithmic scale along the abscissa consists in the following: by plotting on the experimental graph the measured values of $P$ as a function of the logarithms of the relative values $n$, one can then, by shifting this graph, drawn for example on tracing paper, over the grid (Fig. 8), find the theoretical curve that best fits the experimental data. The drawback of this method, in our opinion, as compared with that set forth in § 3, is that it is designed for a very accurate determination of $P$. In our method interpolation by means of a straight line smooths out random errors.
Fig. 8. Grid $P(\log n)$ for various values of $n_0$, used by Hecht, Shlaer, and Pirenne for processing fluctuation measurements.
In the published paper results are given for three observers (apparently corresponding to one series of measurements). In Table VI these results are compared with the numbers $N$ of photons incident on the retina at threshold.
Table VI
| Observer | $\omega$ | $\lambda$ | $\bar n_0$ | $N$ | $\dfrac{N}{n_0}$ |
|---|---|---|---|---|---|
| S. H. | $20^\circ$ | 510 | 6 | 115 | 19 |
| S. S. | $20^\circ$ | 510 | 7 | 78 | 11 |
| M. H. P. | $20^\circ$ | 510 | 5 | 100 | 20 |
| Mean value | — | 6 | 96 | 17 |
Comparing Table VI with Table IV, we see that for two of our observers, P. and L., the values $n_0=8$ are close to the values obtained by the American observers. The closeness of the values of $n_0$ for all three American observers is apparently accidental*). As is seen from Table IV, the scatter of the values of $n_0$ with a larger number of experiments and observers becomes very large. The differences in the mean values $N$ in Tables IV and VI are explained by work at different peripheral angles ($8^\circ$ and $20^\circ$). An angle of $20^\circ$ corresponds to a considerably greater sensitivity of the retina than an angle of $8^\circ$.
Thus the American data agree with our measurements**).
*) In Pirenne’s paper15, one of Hecht’s collaborators, it is mentioned, in contrast to the principal communication11, that $n_0$ varied within the limits from 5 to 14 photons.
**) The first communication by Hecht and his collaborators on fluctuation measurements did not contain any mention of our work of 1933–1938.4—8 When publishing our new measurements of 1938–1941 on fluctuations in 1942, we pointed out that our publications of previous years had evidently remained unknown to the American authors, since they are not mentioned in their communication.
In 1943 one of the participants in the American work, Pirenne, published in England2 the results of fluctuation measurements under binocular and monocular observation. It is known that the absolute threshold when observing with two eyes is somewhat (approximately twice) lower than with one eye. The natural supposition is that this occurs because the probability of obtaining a sufficient number of photo—
In these lines, first of all, one is struck by the assertion that our first work continues the work of Barnes and Czerny. From § 2 it is clear that, on the experimental side, this work was erroneous; the only correct element in it was the theoretical indication of the necessity for the existence of quantum fluctuations. There can be no question of any fundamental difference between our experimental work and that of Hecht; all the essential parts of our method and apparatus (as is clear from a comparison of §§ 3, 4, and 5 with § 6) were reproduced by Hecht and his collaborators. Our measurements were not a continuation of the work of Barnes and Czerny, but its complete negation in the experimental part.
The American authors further completely distort the content of our work by asserting that we supposed that the energy falling on the pupil fluctuates. On p. 938 of our first work of 19331 they could read the following: “The only method for obtaining \(n_0\) is provided only by statistical measurements of the kind we have applied. Thus, on the other hand, one may hope to obtain the whole curve of the twilight sensitivity of the eye, and moreover with the advantage that the actual number of photons absorbed by the retina at the threshold of excitation will be found. All energy losses that arise in the eyeball as a result of reflection, absorption, and scattering, and that inevitably affect any energetic methods, drop out automatically in statistical measurements. By this method it is also possible, probably, to settle the interesting question of the true sensitivity of the living retina in the long-wave ultraviolet part of the spectrum. It is known that ultraviolet rays in the region from 400 to 310 mµ are visible at high intensities, and the question remains open whether the small sensitivity of the eye in this region is caused by the actual insensitivity of the retina or only by absorption in the media of the eyeball.” The quoted lines need no further explanation.
Finally, the American investigators incriminate us for expecting differences in the fluctuations for different wavelengths. In the following paragraph it is shown that this is not an expectation, but a fact proved by a number of measurements by many observers. Hecht and his collaborators for some reason confined themselves to the a priori assertion of the independence of \(n_0\) from \(\lambda\), without giving a single measurement, despite the fact that their apparatus included a monochromator that made it easy to pass into any region of the visible spectrum.
new, greater than \(n_n\), increases if one looks with two eyes. This hypothesis, necessarily following from the assumption of quantum fluctuations, was tested by Piéron. The arrangement was approximately the same as that shown in Fig. 8. A round luminous spot of angular size \(10'\) was observed at a peripheral distance of \(20^\circ\). The duration of the flash was 0.004 sec. Both eyes were fixed on one and the same fixation point. In each experiment the probability \(P\) was determined separately for the right and left eyes, \(P_l\) and \(P_r\), and the probability for both eyes \(P_{lr}\).
If \(P_l\) and \(P_r\) correspond to completely independent fluctuation processes, then one may write:
\[ P_{lr}=1-(1-P_r)(1-P_l)=P_r+P_l-P_lP_r . \]
The results of the experiments are given in Table VII. The stated relation of probabilities is well confirmed, i.e. the fluctuation processes in each of the eyes are completely independent. This is a necessary, but still insufficient, condition for the physical character of the fluctuations, since, generally speaking, one can imagine physiological causes of a statistical type that are independent for each eye.
Table VII
| Experiment No. | \(P_l\) | \(P_r\)* | \(P_{lr}\) | Calculated \(P_{lr}\) |
|---|---|---|---|---|
| 1 | 0.40 | 0.46 | 0.75 | 0.68 |
| 2 | 0.04 | 0.10 | 0.13 | 0.136 |
| 3 log brightness 1.1 | 0.44 | 0.32 | 0.56 | 0.62 |
| 3′ log brightness 1.4 | 0.84 | 0.88 | 1.00 | 0.98 |
| 4 | 0.24 | 0.37 | 0.48 | 0.52 |
| 5 | 0.198 | 0.258 | 0.378 | 0.405 |
In 1944 the method of visual quantum fluctuations was once again “discovered” in Utrecht by van der Velden \(^{25*}\). The problem, the method of observation, and the processing of the results coincide in principle with those described in our first work of 1933. The source of light was an incandescent lamp (a band lamp). The principal attenuation of the light was produced by two opal glasses. The size of the luminous spot was limited by diaphragms. The color \((\lambda \sim 530\,\mathrm{m}\mu)\) was determined by the corresponding light filter (Schott UG2). The position of the observer’s head was fixed by clamping with the teeth some fixed object. In front of the right eye there was a diaphragm limiting the pupil. The flashes were produced by means of a photographic “Compur” shutter. The intensity of the flashes was finely regulated by a rheostat in the lamp circuit. The innovation of van der Velden’s apparatus consisted in the placement of the fixation point. It could be observed only with the left eye, while the right eye could see
* Van der Velden’s first publication of 1944 contains no references to Soviet and American works. In subsequent publications only the American studies are discussed.
only the flashes under study. The observer himself opened the shutter at arbitrary intervals of time. The observations were carried out by two persons. The duration of the flashes was about 0.01 sec., the size of the observed spot \(<4''\)*). The angular distance from the fovea was \(\sim 7^\circ\). The statistical measurements were processed on the basis of the Poisson formula by a method not differing from the practical procedure applied by Hecht and his collaborators.
The results of van der Velden differ decisively, on the quantitative side, from the figures obtained by us and by the American investigators. On the basis of measurements by two observers, and subsequently \(^{26}\) also by a third, it follows that \(n_0=2\) in all three cases. According to our observations (and also the American ones), \(n_0\) varies greatly for different eyes (in our observations the smallest value is \(n_0=8\), in the American ones \(n_0=5\)). In one of our papers \({}^7\) of 1936 there are given (cf. Fig. 9) episodic observations by K. B. Panshin (not included in Table IV because of this episodic character) for \(\lambda=500\,m\mu\). An exact calculation by Poisson’s formula gives in this case \(k=1.2\) and \(n_0=1.4\). The peculiarity of these observations consisted in the fact that the peripheral angle, i.e. the distance between the fixation point and the observed spot, was very large, \(\omega=26^\circ\).
Fig. 9. Fluctuation measurements by K. B. Panshin (1936).
Bouman and van der Velden \({}^{26}\) attempted to explain the discrepancy of their measurements by the fact that in the experiments of other investigators one and the same eye was used both for fixation and for fluctuation observations. The fixation spot supposedly “illuminated” the investigated place of the retina. In this connection the following must be noted.
In our observations, from the very beginning the possible influence of the light of the fixation point was taken into account, for which purpose comparative measurements were made with red fixation spots of different brightness. Considerable changes in the latter did not, however, appreciably affect the fluctuation measurements.
If, furthermore, the indicated explanation were correct, one would have to expect that the magnitude of the ordinary visual threshold (for incident energy), determined in the presence of a fixation spot in the visual field, would always be greater than in the ab—
*) De Vries \({}^{27}\), discussing van der Velden’s work, writes of sizes \(12''\), i.e. 20 times smaller.
absence of a fixation spot. This, however, is at variance with the measurements known so far.
In addition, one may cite experiments carried out by us in connection with the dependence of \(n_0\) on \(\lambda^{11}\), discussed in the next paragraph.
With peripheral vision and in the presence of a red fixation point, the eye is naturally accommodated to the latter. Owing to the chromatic aberration of the eye, the light spot under investigation (with dimensions of about \(3'\)) will be defocused to a greater or lesser degree, depending on the difference between the refraction of light of the given \(\lambda\) and the refraction of the light of the fixation point.
If the angular dimensions of the light spot are such that they lie within the limits of applicability of Ricco’s law,*) then the blurring of the image of the spot will not affect the measured value of the threshold, since under these conditions the threshold value does not depend on the dimensions of the image on the retina. On the contrary, outside the range of applicability of Ricco’s law one should naturally expect some dependence of the measured threshold on the color of the fixation point.
On the basis of these considerations we measured the relative values of the dark threshold with fixation points of different colors; the angular dimensions of the light spot under investigation were \(3'\), \(\omega = 7^\circ\). The light was obtained through a glass monochromator; the color of the fixation point was varied by passing the light of an incandescent lamp through three filters with mean transmissions at about 430, 525, and 650 m\(\mu\). The eye was dark-adapted for about an hour; threshold setting was carried out with the aid of a gray wedge. Table VIII gives the mean results of two series of experiments for one observer in the blue-green region of the spectrum. Within the accuracy of the measurements it may be asserted, as is clear from the table, in which the relative threshold values are given, that the value of the latter does not depend to any appreciable extent on the color, and consequently in general on the fixation point.
Table VIII
| \(\lambda\) | \(\lambda_f = 430\) m\(\mu\) | \(\lambda_f = 525\) m\(\mu\) | \(\lambda_f = 650\) m\(\mu\) |
|---|---|---|---|
| 500 m\(\mu\) | 23 | 22 | 25 |
| 550 m\(\mu\) | 23 | 21 | 24 |
Thus the explanation given above by Bouman and van der Velden is doubtful.
Other factors to which the same authors point as an important cause of the discrepancy\(^{26,6}\) are the duration of the flashes and the angular dimensions of the observed spot. The duration of the flashes in the main
*) At very low powers of the falling light, the perceived brightness is directly proportional to the area of the image on the retina (within certain limits of area, the smaller the greater the power of the light). This is Ricco’s law. As is known, at high powers the perceived brightness does not depend on the area of the image.
in the Dutch experiments was about 0.02 sec, whereas in the American measurements a duration of 0.004 sec was used. The size of the spot in the Dutch experiments was \(3'\), and in the American ones \(10'\). As can be understood, however, from Bouman and van der Velden’s own measurements, one example of which is given in Fig. 4, the value of the threshold for flashes whose duration \(\tau \leq 0.1\) sec depends little on \(\tau\). The influence of the angular dimensions of the observed spot, on the other hand, begins to be appreciable only at values \(>10'\), again according to the Dutch authors’ own measurements.
Bouman and van der Velden, beginning with their first paper\(^{25}\), attach universal significance to their result \(n_0 = 2\), assuming that two photons correspond to the minimum possible visual stimulus, associated with some minimum element of the optical image and of resolving power. On the basis of this assumption they make various theoretical estimates of the influence of the duration of the flash and the size of the observed spot on fluctuation measurements, comparing them with their own measurements.
To date, however, there is no explanation for the sharp discrepancy between the principal conclusion of the Dutch investigators concerning the universality of \(n_0 = 2\) and the many other existing measurements, which testify to very large fluctuations in the values of \(n_0\) among individual observers.
A repetition of the experiments of Bouman and van der Velden with a large number of observers is therefore absolutely necessary.
Independently of the interpretation of these experiments, it may be said with certainty that they once again confirm the assumption that visual observation of quantum fluctuations of the light flux is possible, and that our methodology is correct. The variations of \(n_0\) in different setups and under different experimental conditions should naturally be compared with the fact that the threshold itself changes under different conditions. Here this question, as being of a purely biological character, is not discussed. Our measurements presented in § 8 have some bearing on it.
7. VISUAL FLUCTUATIONS IN DIFFERENT PARTS OF THE SPECTRUM
In the first series of our experiments a difference of \(n_0\) with spectrum was found. Table IX gives the spectral measurements of two observers, B. and P., carried out in 1936\(^{6,7}\).
Within the large errors of such measurements, carried out on different days, i.e., under conditions in which, for one and the same observer, \(k\), and still more \(n_0\), may fluctuate very considerably, it may be asserted, in accordance with what was said earlier, that in the blue-green region, approximately from 450 to 550 m\(\mu\), \(k\) is practically the same; it increases at the blue end and begins to fall again in the visible ultraviolet region, reaching a minimum value (according to P.’s measurements) at about 380 m\(\mu\), where \(k\) and \(n_0\) have the same value.
Table IX
Observer B.
| \(\lambda\) (mµ) | 400 | 430 | 450 | 475 | 510 | 525 | 550 | 600 |
|---|---|---|---|---|---|---|---|---|
| \(k\) | 3.2 | 3.5 | 3.1 | 2.4 | 2.1 | 2.4 | 3.0 | 3.8 |
| \(n_0\) | 81 | 93 | 79 | 47 | 32 | 46 | 72 | 115 |
Observer P.
| \(\lambda\) (mµ) | 340 | 360 | 380 | 400 | 420 | 440 | 460 | 480 | 510 | 540 | 580 | 620 | 640 | 660 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(k\) | 2.3 | 1.5 | 1.0 | 1.7 | 3.5 | 2.4 | 1.5 | 1.2 | 1.0 | 1.2 | 2.2 | 3.5 | 5.0 | 6.0 |
| \(n_0\) | 42 | 18 | 8 | 23 | 93 | 46 | 18 | 12 | 8 | 12 | 39 | 97 | 200 | 280 |
as in the region of 510 mµ. The high sensitivity of the retina in this part of the ultraviolet spectrum was later confirmed by observations of the spectral sensitivity of an eye lacking a lens\(^8\).
At the orange-red end of the spectrum, \(k\) and \(n_0\) increase very strongly. In our initial fluctuation studies, not having experimental information on the optical properties of the retina, we assumed that light is absorbed in it practically completely. This led to the hypothesis of proportionality between \(N\) and \(n_0\) and gave us grounds to expect that the spectral dependence of \(n_0\) would be approximately of the same form as the twilight visibility curve.
New physiological data refute these assumptions. Direct measurements of the amount of visual purple in the retinas of various animals show that even at the absorption maximum, near 510 mµ, complete absorption is not reached\(^ {19}\), and consequently in the blue and red regions it is very small. Further, the spectral curve for the absorption coefficient of rhodopsin, at least in its central part, coincides with the twilight visibility curve\(^ {21}\). At the same time, the bleaching of rhodopsin, which apparently constitutes the basic initial link in the process of twilight visual perception, is the same in different parts of the spectrum for one and the same number of absorbed photons\(^ {22}\). All these facts, contrary to our initial assumption, force us to expect that, for the rod apparatus, \(n_0\) should not depend on \(\lambda\).
To reconcile this conclusion with the fluctuation measurements, new, more extensive experimental material was needed, since fluctuation measurements, by their very nature, cannot attain great accuracy.
New experiments^10 were carried out with the apparatus described in § 4. The light source was still an incandescent lamp, whose light was passed through a glass monochromator. Additional light filters were used to protect against stray light. In order to check the constancy of the experimental conditions, measurements were first made with the luminous flux being decreased and then, after the probability of flashes had become very small, in the reverse direction.
Fig. 10. Dependence of retinal sensitivity at threshold on wavelength according to fluctuation measurements.
Large series of spectral fluctuation measurements were performed by five observers. For each \(\lambda\), each observer carried out from 5 to 15 series of observations. The angular dimensions of the observed light spot for all \(\lambda\) were \(3'\); the investigated region of the retina was at an angular distance \(\omega = 7^\circ\) from the fovea. The duration of the flashes was, as before, \(0.1\) sec. Table X gives a summary of the mean values of \(k\) obtained by the five observers for different wavelengths in 1940 and 1941. Under each value of \(k\), the minimum and maximum value of \(k\) found by the observer for the indicated wavelength during the time of the measurements, which sometimes extended over a week and even a month, are marked in parentheses. In the last column are given the mean values of \(\bar n_0\), computed from \(k\) by formula (16). In Fig. 10 the values of \(\dfrac{1}{\bar n_0}\), which may be called the spectral sensitivity, are represented graphically for each observer. From the table and the figure it is seen that: 1) a sharp increase of \(k\), or correspondingly of \(n_0\), toward long wavelengths was found by all observers, in agreement with our previous measurements; 2) an increase of \(k\) and \(n_0\) toward short wavelengths was found by four observers and is absent in one; 3) in the central part of the spectral curve, approximately from 500 to 600 m\(\mu\),
Table X
| \(\lambda\) (mμ) | T. | Sm. | L. | S-ch. | Zh. | Mean \(k\) | Mean \(n_0\) |
|---|---|---|---|---|---|---|---|
| 442 | 2.3 (1.5—3.8) |
2.9 (2.8—3.0) |
0.9 (0.7—1.2) |
2.4 (2.2—2.7) |
2.0 (1.2—3.1) |
2.1 | 35 |
| 500 | 2.0 (1.3—3.2) |
2.1 (2.1—2.3) |
1.0 (0.6—1.1) |
1.8 (1.4—1.9) |
1.7 (0.9—2.6) |
1.7 | 23 |
| 550 | 1.7 (1.2—1.9) |
2.3 (1.7—2.5) |
1.0 (0.7—1.1) |
1.8 (1.1—2.4) |
1.7 (1.3—1.9) |
1.7 | 23 |
| 605 | 1.9 (1.1—3.2) |
2.7 (2.1—3.2) |
0.8 (0.7—1.2) |
1.9 (1.6—2.2) |
1.6 (1.3—1.8) |
1.8 | 25 |
| 635 | 2.3 (1.7—3.4) |
3.2 (2.3—3.9) |
1.3 (1.1—1.6) |
3.4 (2.2—4.7) |
2.3 (1.9—2.5) |
2.5 | 50 |
| 670 | 3.1 (2.0—5.8) |
3.4 (3.1—4.0) |
2.0 (1.3—2.7) |
3.8 (2.3—5.3) |
2.8 (2.3—3.2) |
3.0 | 72 |
\(k\) and \(n_0\) are practically constant. These conclusions are especially clear from Fig. 11, which graphically depicts the mean value \(\bar n_0\) for all five observers as a function of \(\lambda\).
In Fig. 10 the twilight visibility curve is plotted with a dashed line. It is clear that one cannot speak of a quantitative coincidence of this curve and the fluctuation curves.
The explanation of the results set forth, at least in the interval from 500 to 700 mμ, should be sought above all in the interaction of the two receptive apparatuses of the retina, the rod and the cone.
First of all it must be taken into account that the circumstances listed above in the present paragraph: 1) the parallelism of rhodopsin absorption and the twilight-visibility curve and 2) the constancy of the quantum yield of rhodopsin bleaching as a function of \(\lambda\), have been experimentally established only approximately in the interval from 500 to 600 mμ, but here, as the totality of our experiments shows,
(cf. Fig. 11), \(n_0\) is constant and, consequently, the fluctuation results agree with the other data on the eye.
In the long-wavelength part of the spectrum (\(\lambda > 600\,m\mu\)) the sensitivity of the rods, owing to the small absorption by rhodopsin, is very low, and the cones must begin to compete noticeably with them. A qualitative indication of this is the well-known fact that the sensation of color is preserved in threshold vision in the long-wavelength part of the spectrum. According to the old measurements of Charpentier\({}^{23}\), the ratio of the chromatic threshold to the achromatic one (i.e., of the cone threshold to the rod threshold) for the extreme red part of the spectrum is 3.6, for orange—5.5, and for yellow—9.6.
Fig. 11. Mean number of photons \(n_0\) for the threshold on the retina as a function of \(\lambda\).
According to Wentworth’s measurements\({}^{24}\), this ratio for \(\lambda = 672\,m\mu\) and \(\omega = 4^\circ\) is 3.0; for \(5^\circ\)—3.7, for \(10^\circ\)—4.0. Thus there is no doubt that in the long-wavelength part of the spectrum, \(\lambda > 600\,m\mu\), in addition to the rods, the cones also take part in visual perception near the threshold in a very appreciable proportion.
The nature of the interaction of rods and cones under threshold conditions has not been studied; in particular, it is unknown whether one may speak of additivity of this action. Therefore, for a quantitative statistical calculation of this case, the necessary data are not yet available. From the observed fact of an increase of \(n_0\) in the long-wavelength part of the spectrum, one may in any case conclude that excitation of the cones requires the absorption of a significantly larger number of photons than excitation of the rods.
In the short-wavelength part of the spectrum, for \(\lambda < 500\,m\mu\), the parallelism between the absorption of light in rhodopsin and the curve of twilight sensitivity is sharply disturbed, the usual explanation being the assumption of very large inactive absorption both outside the retina and in the rods themselves. Fluctuation measurements show that here, in addition, in the visual process, besides the bleaching of rhodopsin, other, as yet unidentified factors begin to play a role. Given the generally poor state of knowledge of the mechanism of visual perceptions in the short-wavelength part of the spectrum, fluctuation measurements in this region must for the time being be regarded as empirical material, especially since, as is evident from the curves in Fig. 10, cases of constancy of \(n_0\) are also encountered in this region. The sensitivity of the retina in the near ultraviolet region may, furthermore, attain the same magnitude as in the blue-green spectrum (see Table IX).
8. FLUCTUATIONS UNDER CONDITIONS OF AN ARTIFICIALLY ELEVATED THRESHOLD
The possibility of observing visual quantum fluctuations is determined by the extremely low value of the energy corresponding to the dark threshold, attained after prolonged adaptation. If the process of adaptation does not take place in complete darkness (“light adaptation”), a threshold is likewise reached, but it is considerably higher than the dark threshold. This circumstance can be checked by measuring fluctuations. We saw above [formula (17)] that
\[ n_0 \leq N\alpha\beta\gamma\delta, \]
where \(n_0\) is the minimum number of photons that must be absorbed in the rods of the retina in order to produce a visual sensation at threshold (under the given conditions), \(N\) is the minimum number of photons falling on the pupil under the same conditions, \(\alpha\) is the fraction of light transmitted by the ocular media to the retina, \(\beta\) is the fraction of light transmitted by the layers of the retina itself, \(\gamma\) is the fraction of the retinal area covered by rods in the region under investigation, and \(\delta\) is the fraction of light absorbed by the rods.
There is no doubt that, in going from dark to light adaptation, \(N\) increases, acquiring some value \(N'\), which grows as the illumination is intensified; on the other hand, \(\delta\) may at the same time decrease somewhat as a consequence of the greater bleaching of the visual purple in the rods. If bleaching were accompanied only by a decrease in the concentration of rhodopsin, it would have to be assumed that \(n_0\) remains unchanged in the transition to light adaptation. In reality, under light adaptation the chemical conditions in the rods change owing to the accumulation of decomposition products of rhodopsin, as a result of which a change in \(n_0\) itself is possible. This question can be resolved by the fluctuation method.
The experiments\(^{11}\) were carried out in monochromatic light \(\lambda = 500\ \mathrm{m}\mu\); \(\omega = 7^\circ\) from the fovea, the angular dimensions of the observed light spot being about \(3'\). During the course of a day the observer made two comparative series of fluctuation measurements—under complete dark adaptation and under a certain light adaptation, the level of which was varied in different experiments. Light adaptation was achieved by illuminating the room with a very weak light source (an electric lamp powered by an accumulator). Before the observations the eye was exposed to this illumination for about 15 minutes. During the measurements themselves, when the eye was pressed to the tube of the apparatus with the photographic wedge \(K\) (Fig. 3), the adaptive light regime was somewhat disturbed. To reduce the distorting influence of this factor, the observer removed the eye from the tube as often as possible, again exposing it to the action of the established light flux.
In addition to the fluctuation experiments, after and before each series of measurements, the “static” threshold was determined with the stopped
motor \(M\) (Fig. 3); the eye was still fixed on the red point, and the light, visible peripherally \((\omega = 7^\circ)\), was extinguished by a wedge up to threshold. Table XI gives a summary of the measurement results. In the first column is indicated the date of the experiment; in the second—the slope of the fluctuation straight line \(k\) for dark adaptation; in the third—the slope \(k'\) for light adaptation; in the fourth—the ratio of the light threshold to the dark threshold \(\dfrac{N'}{N}\); in the fifth the ratio:
\[ r = \frac{N'}{N}\left(\frac{k}{k'}\right)^2 . \tag{18} \]
A comparison of the values of \(k\) and \(k'\) in Table XI undoubtedly shows that in the transition from dark adaptation to light adaptation \(n_0\)
Table XI
| Observer T. | Observer T. | Observer T. | Observer T. | Observer T. | Observer Zh. | Observer Zh. | Observer Zh. | Observer Zh. | Observer Zh. |
|---|---|---|---|---|---|---|---|---|---|
| Date | \(k\) | \(k'\) | \(\dfrac{N'}{N}\) | \(r\) | Date | \(k\) | \(k'\) | \(\dfrac{N'}{N}\) | \(r\) |
| 1940 | 1941 | ||||||||
| 30.XI | 1.5 | 2.6 | 1.7 | 0.57 | 26.II | 1.9 | 2.7 | 2.0 | 0.99 |
| 1.XII | 1.6 | 2.7 | 2.0 | 0.70 | 28.III | 1.2 | 1.9 | 2.6 | 1.06 |
| 13.XII | 2.5 | 2.7 | 1.5 | 1.27 | 29.III | 1.7 | 2.4 | 2.7 | 0.60 |
| 19.XII | 2.2 | 8.2 | 11.0 | 0.79 | 31.III | 1.3 | 2.9 | 5.0 | 1.41 |
| 30.XII | 2.5 | 2.7 | 1.5 | 1.27 | 1.IV | 0.9 | 2.2 | 5.4 | 0.89 |
| 1941 | 2.IV | 1.8 | 2.6 | 2.0 | 0.96 | ||||
| 4.I | 1.7 | 3.2 | 2.6 | 0.73 | 3.IV | 1.1 | 2.4 | 5.2 | 1.00 |
| 9.I | 1.7 | 3.4 | 2.4 | 0.60 | 8.IV | 1.7 | 2.8 | 2.9 | 1.14 |
| 13.I | 1.7 | 3.1 | 2.1 | 0.63 | 10.IV | 1.1 | 3.0 | 6.4 | 0.78 |
| 16.I | 1.8 | 3.0 | 3.3 | 1.18 | |||||
| Average | 1.9 | — | — | 0.87 | Average | 1.3 | — | — | 0.98 |
increases considerably. On the other hand, \(r\) is approximately equal to unity, as is seen from the last column of the tables. Therefore, on average
\[ \frac{n'_0}{n_0} \approx \frac{N'}{N}. \tag{19} \]
Using formula (17), from this we find:
\[ \alpha\beta\gamma\delta \approx \alpha'\beta'\gamma'\delta', \tag{20} \]
but \(\alpha, \beta, \gamma\), as slowly varying optical constants of the ocular
media cannot change in passing from dark adaptation to light adaptation and, consequently,
\[ \delta \simeq \delta', \]
i.e., the total absorption of rhodopsin in the rods in our experiments, in passing from dark adaptation to light adaptation, practically did not change; that is, bleaching was relatively small. It is natural to expect that on passing to illuminations considerably greater than in our experiments, such a simple state of affairs should be disrupted. However, measurement of fluctuations at high illuminations and a corresponding increase of \(n_0\) is hardly feasible.
The increase of \(k\) in the transition from dark adaptation to light adaptation provides important new evidence of the physical quantum nature of the observed fluctuations. It is difficult to imagine a physiological process that could lead to such a dependence of the fluctuations on the threshold height, and moreover one having the quantitative character (19). One may, on the other hand, note that the increase of \(k\) in passing from dark adaptation to light adaptation is quite explicable from the quantum point of view, but is not necessary. One may imagine, for example, the transition from one state to another as bleaching of rhodopsin and the corresponding lowering of its concentration.
With the other conditions of the medium preserved, \(n_0\) could then remain unchanged at both the dark and the light threshold.
9. SOME APPLICATIONS OF THE METHOD OF VISUAL FLUCTUATIONS TO PHYSICAL INVESTIGATIONS
The diverse experimental material set forth in the preceding sections proves, in its entirety, beyond any doubt the quantum nature of the phenomenon studied.
At the same time, fluctuation measurements, having received their physical explanation, open a new path to the profound investigation of the living eye. We have seen that by this method it was possible for the first time to determine the true sensitivity of the retina at the threshold of visual sensation. The study of fluctuations in various regions of the spectrum reveals new features in the visual apparatus at the orange-red and violet ends of the spectrum. In § 8 measurements under light adaptation are described, revealing the behavior of visual purple under these conditions. One may hope that many features of visual perception will be disclosed in the study of the dependence of fluctuations on the peripheral angle of perception.
At the same time, there can hardly be any doubt as to the great physical significance of the fluctuation method as a method that literally makes tangible to the eye the quantum nature of phenomena. It has already been noted above that, owing to the incomplete absorption by the retina, in fluctuation phenomena the quantum nature of both the light flux and the absorbing medium is manifested simultaneously. If absorption in the retina were...
complete, the phenomenon “in itself” would be determined only by fluctuations of the luminous flux reaching the visual purple. However, owing to the property of Poisson’s law (5), which determines the statistics, the fluctuations depend only on one variable, the mean number \(n\) of absorbed photons, irrespective of what determines this number—the incident luminous flux or the value of the absorption coefficient. Therefore visual quantum fluctuations can be used with complete certainty for studying processes connected with the fundamental features and the nature of light.
In our first experiments\(^{4}\) the fluctuation method was applied to the investigation of coherent rays. It is well known that interference fringes can be photographed at extremely low intensities (exposures lasting many days). From the wave point of view this is self-evident; from the corpuscular point of view it is incomprehensible, since in the interfering beams under these conditions one partner is always absent.
The experiments described below show, however, quite visually that interference at low intensities is no more comprehensible from the wave point of view either.
For the experiments the apparatus shown in Fig. 2 was used, with only one difference: between the rotating disk \(S\) and the tube \(R_1R_2\) there was placed a Fresnel biprism with its refracting edge directed horizontally. In this way two coherent green spots, symmetrically situated with respect to the red fixation point (Fig. 12), were visible in the field of view.
Fig. 12. Arrangement of luminous spots in experiments with coherent rays and double refraction.
When the threshold brightness was reached, both points fluctuated quite distinctly relative to one another, and very rarely were they visible at one and the same time. This phenomenon of objective relative oscillations of coherent rays has catastrophic significance for the wave theory, if one tries to defend it in the present case as well. The experiment admits only one interpretation: each of the rays interferes “with itself.”
The phenomenon was studied quantitatively; the probability \(p\) of the simultaneous appearance of both coherent spots was determined. Obviously, the same value \(n_0\) corresponds to both spots; therefore, regarding the fluctuations in the two spots as independent, by the theorem of multiplication of probabilities we find:
\[ p = P^2 \]
or, using (13),
\[ \sqrt{p} = \frac{1}{2} - \frac{1}{2}\sqrt{\frac{n_0}{2}\frac{(1-x)}{\sqrt{x}}}. \tag{21} \]
In Fig. 13 are given the results of one series of measurements carried out by B. in 1933. Along the ordinate axis \(\sqrt{p}\) is plotted. In accordance with (21) a curve is obtained with slope \(k = 2.3\), which corresponds to \(n_0 = 43\) (cf. Table IV for observer B.). These measurements prove that the fluctuations in both coherent beams occur independently and have a quantum nature.
A similar experiment was also made with a Wollaston polarizing prism, upon which natural light from the source was incident. The two green spots (Fig. 12) were in this case polarized in two mutually perpendicular planes.
As in the preceding experiment, both spots fluctuated entirely independently. From this observation it follows that
Fig. 13. Results of the experiment with coherent beams.
Fig. 14. Results of the experiment with double refraction.
natural light, at sufficiently small intensities, is polarized at each moment in a different way (fluctuations of the state of polarization). It must be emphasized that this fact is somewhat different from the “elementary polarization” possible from the point of view of wave theory, which should exist only for an extremely short time. In our experiment the state of definite polarization of natural light lasts at least \(0.1\) sec. The difference lies in the extreme rarefaction of the light beam under our conditions. The great duration is compensated by the small mean number of photons.
In this case as well, the probability \(p\) of a simultaneous flash of both spots was measured as a function of \(x\). The results are represented graphically in Fig. 14, where along the ordinate axis \(\sqrt{p}\) is again plotted in accordance with formula (21). The observations were made by B. Here again there is a rectilinear dependence; the slope of the straight line is \(k = 2.5\), which corresponds to 50 photons.
It is obvious that, after the established basic facts concerning fluctuations of any light beam of sufficiently small power, similar results should be expected in other analogous experiments as well. One may
one could, for example, observe relative fluctuations of both symmetrical spectra of first order with a diffraction grating. This would mean that at low intensities the diffraction pattern does not arise everywhere at once, but is formed gradually, statistically.
There is hardly any need to multiply experiments of this kind, since they follow logically from one and the same experimentally proven fluctuation principle, which may be formulated as follows:
Every light beam, isolated in any way whatsoever, at sufficiently low power exhibits power fluctuations that occur quite independently and irrespective of oscillations in any other beam.
One may undoubtedly hope that in the near future it will be possible to produce photoelements possessing, in the visible part of the spectrum, no less sensitivity than the eye. With them, of course, it will be possible to repeat all the physical experiments described. The advantage of the visual method lies in the fact that at the same time it provides a new, very subtle means for investigating the eye.
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-
However, from the more detailed article by Hecht and his collaborators, printed in 1942 in the Journal of Physiology, it became clear that our works were known to the authors. At the same time, they briefly mention them in so distorted a form in comparison with their actual content that we are compelled to give here a complete quotation from the corresponding place in the American article: “Barnes and Czerny (1932),” the authors write, “and, following them, Brumberg and Vavilov (1933) stated that in the energy necessary for vision there must exist fluctuations, and both groups of investigators strove to find them. But neither group noted the place where the source of the fluctuations is to be sought, supposing that it must be sought in the energy falling on the cornea. Brumberg and Vavilov even expected differences in the fluctuations for different wavelengths owing to the greater energy necessary for seeing red light, for example, in comparison with blue-green, in accordance with the twilight visibility curve.” ↩