Correlation between photons in two coherent beams of light

By R. HANBURY BROWN
University of Manchester, Jodrell Bank Experimental Station
AND
R. Q. TWISS
Services Electronics Research Laboratory, Baldock

In an earlier paper¹, we have described a new type of interferometer which has been used to measure the angular diameter of radio stars². In this instrument the signals from two aerials A₁ and A₂ (Fig. 1) are detected independently and the correlation between the low-frequency outputs of the detectors is recorded. The relative phases of the two radio signals are therefore lost, and only the correlation in their intensity fluctuations is measured; so that the principle differs radically from that of the familiar Michelson interferometer where the signals are combined before detection and where their relative phase must be preserved.

Fig. 1 A new type of radio interferometer (a), together with its analogue (b) at optical wave-lengths | high-resolution version |

This new system was developed for use with very long base-lines, and experimentally it has proved to be largely free of the effects of ionospheric scintillation². These advantages led us to suggest¹ that the principle might be applied to the measurement of the angular diameter of visual stars. Thus one could replace the two aerials by two mirrors M₁, M₂ (Fig. 1) and the radio-frequency detectors by photoelectric cells C₁, C₂, and measure, as a function of the separation of the mirrors, the correlation between the fluctuations in the currents from the cells when illuminated by a star.

It is, of course, essential to the operation of such a system that the time of arrival of photons at the two photocathodes should be correlated when the light beams incident upon the two mirrors are coherent. However, so far as we know, this fundamental effect has never been directly observed with light, and indeed its very existence has been questioned. Furthermore, it was by no means certain that the correlation would be fully preserved in the process of photoelectric emission. For these reasons a laboratory experiment was carried out as described below.

The apparatus is shown in outline in Fig. 2. A light source was formed by a small rectangular aperture, 0.13 mm × 0.15 mm in cross-section, on which the image of a high-pressure mercury arc was focused. The 4358 A. line was isolated by a system of filters, and the beam was divided by the half-silvered mirror M to illuminate the cathodes of the photomultipliers C₁, C₂. The two cathodes were at a distance of 2.65 m from the source and their areas were limited by identical rectangular apertures O₁, O₂, 9.0 mm × 8.5 mm in cross-section. (It can be shown that for this type of instrument the two cathodes need not be located at precisely equal distances from the source. In the present case their distances were adjusted to be roughly equal to an accuracy of about 1 cm) In order that the degree of coherence of the two light beams might be varied at will, the photomultiplier C₁ was mounted on a horizontal slide which could be traversed normal to the incident light. The two cathode apertures, as viewed from the source, could thus be superimposed or separated by any amount up to about three times their own width. The fluctuations in the output currents from the photomultipliers were amplified over the band 3–27 Mc/s and multiplied together in a linear mixer. The average value of the product, which was recorded on the revolution counter of an integrating motor, gave a measure of the correlation in the fluctuations. To obtain a significant result it was necessary to integrate for periods of the order of one hour, so very great care had to be taken in the design of the electronic equipment to eliminate the effects of drift, of interference and of amplifier noise.

Fig. 2 Simplified diagram of the apparatus | high-resolution version |

Assuming that the probability of emission of a photoelectron is proportional to the square of the amplitude of the incident light, one can use classical electromagnetic wave theory to calculate the correlation between the fluctuations in the current from the two cathodes. On this assumption it can be shown that, with the two cathodes superimposed, the correlation S(0) is given by:

It can also be shown that the associated root-mean-square fluctuations N are given by:

where A is a constant of proportionality depending on the amplifier gain, etc.; T is the time of observation; a(n) is the quantum efficiency of the photocathodes at a frequency n; n₀(n) is the number of quanta incident on a photocathode per second, per cycle bandwidth; b_n is the bandwidth of the amplifiers; m/(m−1) is the familiar excess noise introduced by secondary multiplication; a₁, a₂ are the horizontal and vertical dimensions of the photocathode apertures; q₁, q₂ are the angular dimensions of the source as viewed from the photocathodes; and l₀ is the mean wave-length of the light. The integrals are taken over the complete optical spectrum and the phototubes are assumed to be identical. The factor is determined by the dimensionless parameter h defined by

which is a measure of the degree to which the light is coherent over a photocathode. When , as for a point source, f(h) is effectively unity; however, in the laboratory experiment it proved convenient to make h₁, h₂ of the order of unity in order to increase the light incident on the cathodes and thereby improve the ratio of signal to noise. The corresponding values of f(h₁), f(h₂) were 0.62 and 0.69 respectively.

When the centres of the cathodes, as viewed from the source, are displaced horizontally by a distance d, the theoretical value of the correlation decreases in a manner dependent upon the dimensionless parameters, h₁ and d/a₁. In the simple case where , which would apply to an experiment on a visual star, it can be shown that S(d), the correlation as a function of d, is proportional to the square of the Fourier transform of the intensity distribution across the equivalent line source. However, when h 1, as in the present experiment, the correlation is determined effectively by the apparent overlap of the cathodes and does not depend critically on the actual width of the source. For this reason no attempt was made in the present experiment to measure the apparent angular size of the source.

The initial observations were taken with the photocathodes effectively superimposed (d = 0) and with varying intensities of illumination. In all cases a positive correlation was observed which completely disappeared, as expected, when the separation of the photocathodes was large. In these first experiments the quantum efficiency of the photocathodes was too low to give a satisfactory ratio of signal to noise. However, when an improved type of photomultiplier became available with an appreciably higher quantum efficiency, it was possible to make a quantitative test of the theory.

A set of four runs, each of 90 min. duration, was made with the cathodes superimposed (d = 0), the counter readings being recorded at 5-min. intervals. From these readings an estimate was made of N_e, the root mean square deviation in the final reading S(0) of the counter, and the observed values of S_e(0)/N_e are shown in column 2 of Table 1. The results are given as a ratio in order to eliminate the factor A in equations (1) and (2), which is affected by changes in the gain of the equipment. For each run the factor

was determined from measurements of the spectrum of the incident light and of the d.c. current, gain and output noise of the photomultipliers; the corresponding theoretical values of S(0)/N are shown in the second column of Table 1. In a typical case, the photomultiplier gain was 3 × 10⁵, the output current was 140 mamp., the quantum efficiency a(n₀) was of the order of 15 per cent and n₀(n₀) was of the order of 3×10^–3. After each run a comparison run was taken with the centres of the photocathodes, as viewed from the source, separated by twice their width (d = 2a), in which position the theoretical correlation is virtually zero. The ratio of S_e(d), the counter reading after 90 minutes, to N_e, the root mean square deviation, is shown in the third column of Table 1.

The results shown in Table 1 confirm that correlation is observed when the cathodes are superimposed but not when they are widely separated. However, it may be noted that the correlations observed with d = 0 are consistently lower than those predicted theoretically. The discrepancy may not be significant but, if it is real, it was possibly caused by defects in the optical system. In particular, the image of the arc showed striations due to imperfections in the glass bulb of the lamp; this implies that unwanted differential phase-shifts were being introduced which would tend to reduce the observed correlation.

This experiment shows beyond question that the photons in two coherent beams of light are correlated, and that this correlation is preserved in the process of photoelectric emission. Furthermore, the quantitative results are in fair agreement with those predicted by classical electromagnetic wave theory and the correspondence principle. It follows that the fundamental principle of the interferometer represented in Fig. 1 is sound, and it is proposed to examine in further detail its application to visual astronomy. The basic mathematical theory together with a description of the electronic apparatus used in the laboratory experiment will be given later.

We thank the Director of Jodrell Bank for making available the necessary facilities, the Superintendent of the Services Electronics Research Laboratory for the loan of equipment, and Mr. J. Rodda, of the Ediswan Co., for the use of two experimental phototubes. One of us wishes to thank the Admiralty for permission to submit this communication for publication.

Oct. 5

1. Hanbury Brown, R., and Twins, R. Q., Phil. Mag., 45, 663 (1954).
2. Jennison, R. C., and Das Gupta, M. K., Phil. Mag. (in the press).

Brannen and Ferguson¹ have reported experimental results which they believe to be incompatible with the observation by Hanbury Brown and Twiss² of correlation in the fluctuations of two photoelectric currents evoked by coherent beams of light. Brannen and Ferguson suggest that the existence of such a correlation would call for a revision of quantum theory. It is the purpose of this communication to show that the results of the two investigations are not in conflict, the upper limit set by Brannen and Ferguson being in fact vastly greater than the effect to be expected under the conditions of their experiment. Moreover, the Brown–Twiss effect, far from requiring a revision of quantum mechanics, is an instructive illustration of its elementary principles. There is nothing in the argument below that is not implicit in the discussion of Brown and Twiss, but perhaps I may clarify matters by taking a different approach.

Consider first an experiment which is simpler in concept than either of those that have been performed, but which contains the essence of the problem. Let one beam of light fall on one photomultiplier, and examine the statistical fluctuations in the counting-rate. Let the source be nearly monochromatic and arrange the optics so that, as in the experiments already mentioned, the difference in the length of the two light-paths from a point A in the photocathode to two points B and C in the source remains constant, to within a small fraction of a wave-length, as A is moved over the photocathode surface. (This difference need not be small, nor need the path-lengths themselves remain constant.) Now it will be found, even with the steadiest source possible, that the fluctuations in the counting-rate are slightly greater than one would expect in a random sequence of independent events occurring at the same average rate. There is a tendency for the counts to 'clump'. From the quantum point of view this is not surprising. It is typical of fluctuations in a system of bosons. I shall show presently that this extra fluctuation in the single-channel rate necessarily implies the cross-correlation found by Brown and Twiss. But first I propose to examine its origin and calculate its magnitude.

Let P denote the square of the electric field in the light at the cathode surface in one polarization, averaged over a few cycles. P is substantially constant over the photocathode at any instant, but as time goes on it fluctuates in a manner determined by the spectrum of the disturbance, that is, by the 'line shape'. Supposing that the light contains frequencies around n₀, we describe the line shape by the normalized spectral density g(n–n₀). The width of the distribution g, whether it be set by circumstances in the source itself or by a filter, determines the rate at which P fluctuates. For our purpose, the stochastic behaviour of P can be described by the correlation function which is related in turn to g(n–n₀) by³

For the probability that a photoelectron will be ejected in time dt, we must write aPdt, where a is constant throughout the experiment. It makes no difference whether we think of P as the square of an electric field-strength or as a photon probability density. (In this connexion the experiment of Forrester, Gudmundsen and Johnson⁴ on the photoelectric mixing of incoherent light is interesting.) Assuming one polarization only, and one count for every photoelectron, we look at the number of counts n_T in a fixed interval T, and at the fluctuations in n_T over a sequence of such intervals. From the above relations, the following is readily derived:

and it has been assumed in deriving (2) that T Gtt₀. Now is just the average counting-rate and t₀, a correlation time determined by the light spectrum, is approximately the reciprocal of the spectral band-width Dn in particular, if Dn is the full width at half intensity of a Lorentzian density function, t₀ = (pDn)^–1, while if Dn is the width of a rectangular density function, t₀ = Dn^–1. We see that the fractional increase in mean-square fluctuation over the 'normal' amount is independent of T, and is about equal to the number of counts expected in an interval 1/Dn. This number will ordinarily be very much smaller than one. The result, expressed in this way, does not depend on the counting efficiency.

If one insists on representing photons by wave packets and demands an explanation in those terms of the extra fluctuation, such an explanation can be given. But I shall have to use language which ought, as a rule, to be used warily. Think, then, of a stream of wave packets, each about c/Dn long, in a random sequence. There is a certain probability that two such trains accidentally overlap. When this occurs they interfere and one may find (to speak rather loosely) four photons, or none, or something in between as a result. It is proper to speak of interference in this situation because the conditions of the experiment are just such as will ensure that these photons are in the same quantum state. To such interference one may ascribe the 'abnormal' density fluctuations in any assemblage of bosons.

Were we to carry out a similar experiment with a beam of electrons, we should, of course, find a slight suppression of the normal fluctuations instead of a slight enhancement; the accidentally overlapping wave trains are precisely the configurations excluded by the Pauli principle. Nor would we be entitled in that case to treat the wave function as a classical field.

Turning now to the split-beam experiment, let n₁ be the number of counts of one photomultiplier in an interval T, and let n₂ be the number of counts in the other in the same interval. As regards the fluctuations in n₁ alone, from interval to interval, we face the situation already analysed, except that we shall now assume both polarizations present. The fluctuations in orthogonal polarizations are independent, and we have, instead of (2),

where n₁/T has been written for the average counting-rate in channel 1. A similar relation holds for n₂. Now if we should connect the two photomultiplier outputs together, we would clearly revert to a single-channel experiment with a count n = n₁+n₂. We must then find:

From (4) and (5) it follow that:

This is the positive cross-correlation effect of Brown and Twiss, although they express it in a slightly different way. It is merely another consequence of the 'clumping' of the photons. Note that if we had separated the branches by a polarizing filter, rather than a half-silvered mirror, the factor 1/2 would be lacking in (4), and (5) would have led to which is as it should be.

If we were to split a beam of electrons by a non-polarizing mirror, allowing the beams to fall on separate electron multipliers, the outputs of the latter would show a negative cross-correlation. A split beam of classical particles would, of course, show zero cross-correlation. As usual in fluctuation phenomena, the behaviour of fermions and the behaviour of bosons deviate in opposite directions from that of classical particles. The Brown–Twiss effect is thus, from a particle point of view, a characteristic quantum effect.

It remains to show why Brannen and Ferguson did not find the effect. They looked for an increase in coincidence-rate over the 'normal' accidental rate, the latter being established by inserting a delay in one channel. Their single-channel rate was 5×10⁴ counts per sec., their accidental coincidence rate about 20 per sec., and their resolving time about 10^–8 sec. To analyse their experiment one may conveniently take the duration T of an interval of observation to be equal to the resolving time. One then finds that the coincidence-rate should be enhanced, in consequence of the cross-correlation, by the factor (1+t₀/2T). Unfortunately, Brannen and Ferguson do not specify their optical band-width; but it seems unlikely, judging from their description of their source, that it was much less than 10¹¹ cycles/sec., which corresponds to a spread in wave-length of rather less than 1 A. at 4358 A. Adopting this figure for illustration, we have t₀ = 10^–11 sec., so that the expected fractional change in coincidence-rate is 0.0005. This is much less than the statistical uncertainty in the coincidence-rate in the Brannen and Ferguson experiment, which was about 0.01. Brown and Twiss did not count individual photoelectrons and coincidences, and were able to work with a primary photoelectric current some 10⁴ times greater than that of Brannen and Ferguson. It ought to be possible to detect the correlation effect by the method of Brannen and Ferguson. Setting counting efficiency aside, the observing time required is proportional to the resolving time and inversely proportional to the square of the light flux per unit optical band-width. Without a substantial increase in the latter quantity, counting periods of the order of years would be needed to demonstrate the effect with the apparatus of Brannen and Ferguson. This only adds lustre to the notable achievement of Brown and Twiss.

E. M. PURCELL

Lyman Laboratory of Physics,
Harvard University,
Cambridge, Massachusetts.

1. Brannen, E., and Ferguson, H. I. S., Nature, 178, 481 (1956).
2. Brown, H. R., and Twiss, R. Q., Nature, 177, 27 (1956).
3. Lawson, J. L., and Uhlenbeck, G. E., 'Threshold Signals', p. 61 (McGraw-Hill, New York, 1950).
4. Forrester, A. I., Gudmundsen, R. A., and Johnson, P. O., Phys. Rev., 99, 1691 (1955).

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