Heterodyning of Light

S. I. Borovitskii and G. S. Gorelik

Recently an article appeared by Forrester, Gudmundsen, and Johnson[^1] describing experiments in which they succeeded in obtaining one of the phenomena that may be described as interference of two incoherent light radiations. The American investigators succeeded, with the aid of a photoelectric converter specially developed for this purpose, in carrying out heterodyning, or, as they put it, mixing of two optical spectral lines whose difference frequency belongs to the region of ultrahigh radio frequencies.

The idea of such an experiment, and of certain other related experiments, was expressed in 1947 by one of the authors of the present review[^2]. Independently, a short time later, the idea of the possibility of observing heterodyning of two close optical spectral lines was expressed by Forrester, Parkins, and Gerjuoy[^3].

This idea is elementary*).

Let us consider optical radiation whose spectrum has the form shown in Fig. 1. Along the abscissa is plotted frequency, and along the ordinate, spectral density. We have a doublet whose splitting considerably exceeds the widths of both components. We shall assume that these components are statistically independent (incoherent), for example obtained from different sources or from different atoms of one and the same source. Then the time pattern

*) Nevertheless, it sometimes gives rise to objections based, in our opinion, on insufficient understanding of questions of coherence. As S. Ya. Braude recently informed us, as early as 1945 he had expressed considerations about the possibility of heterodyning two incoherent optical radiations (the work was not published); however, they encountered decisive objections from some physicists. A more detailed clarification of certain questions connected with the possibility of observing interference of incoherent optical radiations had already been given in the pages of this journal[^4] (see also[^5], p. 428).

will have the form shown in Fig. 2, a. Beatings occur with frequency \(F=f_2-f_1\), chaotically modulated (in amplitude and phase) by some stationary random process. The correlation time \(\tau\) of this random process is of the order of \(1/\delta\) (the correlation time is, roughly speaking, the time required, on average, for the value of the amplitude and phase of the beatings to change appreciably).

Fig. 1.

Let our radiation fall on the cathode of a photoelectric converter. Under ordinary experimental conditions the photocurrent is proportional to the mean value of the intensity (the square of the amplitude) of the light over some time \(T\), determined by the parameters of the electric circuit, with \(T \gg \tau\) (“inertial observation”; \(T\) is the time constant). Let us see what happens if the electrical system whose input is the photoelectric converter is capable of following changes in the photocurrent, even very rapid ones in comparison with \(\tau\). We shall assume that the emission current of the photocathode is proportional to the instantaneous value of the intensity, i.e., to the square of the amplitude of the light oscillation (inertia-free photocurrent). Then

Fig. 2.

the current at the output of the phototransducer will have the same form as curve б in Fig. 2, i.e., it will represent the sum of a constant component, low-frequency noise, and a chaotically modulated oscillation having the carrier frequency \(F\) and correlation time \(\tau\). The spectrum of the photocurrent will have the form shown in Fig. 3. Thus, frequencies \(f_1\) and \(f_2\) will be converted into the difference frequency \(F\). This conversion is a nonlinear process, analogous to that which occurs in heterodyning a signal in a radio receiver. The frequencies \(f_1\) and \(f_2\) are analogous to the signal frequency and the frequency of the local oscillator (heterodyne). The nonlinear element necessary for heterodyning is here the photoelectric transducer: the photocurrent is a nonlinear (quadratic) function of the light amplitude.

If the difference between the frequencies \(f_1\) and \(f_2\), as can easily be achieved in optics, is of the order of \(10^{10}\) Hz, \(F\) falls in the microwave radio range, and modern microwave technology can be used to isolate this frequency. It is possible, by heterodyning light, to convert an optical doublet into radio-frequency oscillations.

Fig. 3.

For simplicity of exposition, we have so far not taken into account the shot effect (or, what here amounts to the same thing, the photon mechanism of electron emission). This is justified only if the number of electrons emitted during a time of order \(1/F\) is very large. As calculations show, such a situation would correspond to a very high—practically unattainable—mean light intensity. In all other cases it is necessary to reformulate our initial reasoning, replacing everywhere the words “emission current,” “photocurrent” by the words “probability of electron emission.” It is precisely this probability, and not the photocurrent itself, that is proportional (on the assumption that the photocurrent is inertia-free) to the instantaneous intensity of the light. With this corrected formulation, curve б in Fig. 2 represents the average statistical course of the change in photocurrent under the action of the specified light oscillation. Superposed on it is the shot noise of the photocathode, which under real experimental conditions greatly exceeds the useful effect.

The difficulty of carrying out heterodyning of light, caused by the shot effect, was already foreseen in Forrester’s note,

Parkinson and Herglotz. It was also indicated there that it can be overcome by means of additional (artificial) periodic modulation of the light, by analogy with the method widely used in radio astronomy.

Let us also note one important circumstance. When the surface of the photocathode is increased, the total current will increase, and at the same time its relative fluctuations caused by the shot effect will decrease. But at the same time the relative magnitude of the useful effect also decreases. Because of the finite size of the light source, as the dimensions of the photocathode are increased the beatings at different points of the photocathode cease to be in phase (the beatings at sufficiently distant points of the photocathode are incoherent). This leads to a smoothing of the pulsations of the total photocurrent: an increase of the intensity at some points of the photocathode compensates its decrease at other points. This effect is the more pronounced, other conditions being equal, the larger the size of the light source. As a simple calculation shows (see \(^{1,2}\)), the pulsations of the photocurrent (more precisely: the pulsations of the probability of electron emission) remain practically in phase over the entire surface of the photocathode if it does not exceed the value \(\lambda^{2}/\Omega\), where \(\lambda\) is the wavelength, and \(\Omega\) is the solid angle under which the source is seen from the photocathode.

Heterodyning of light may also be interpreted as demodulation of a modulated oscillation, whose time dependence is shown in Fig. 2. This is only one of the possible cases of demodulation of light by means of radio-electronic apparatus. Another possible case, already discussed theoretically \(^{2,4}\), is the detection of chaotic modulation of light, the spectrum of which is an unresolved spectral line.

We now turn to the description of the experiment of Forrester, Gudmundsen, and Johnson.

The basic requirements imposed on the apparatus are determined by the following considerations. Light beatings, as was already said, preserve their phase over areas of the order of \(\lambda^{2}/\Omega=A_{0}\). The mean square \(\bar{i}\) of the useful component of the photocurrent at the beat frequency from each such area is, in order of magnitude, equal to the mean square of the dc component of the photocurrent from the same area:

\[ \bar{i}^{\,2}=\bar{i}_{\text{const}}^{\,2} =\left(I_{\text{const}}\frac{A_{0}}{A}\right)^{2} =\left(I_{\text{const}}\frac{\lambda^{2}}{A\Omega}\right)^{2}. \]

Here \(I_{\text{const}}\) is the dc component of the total current, and \(A\) is the area of the photocathode. The beatings on different areas are statistically independent. Therefore the mean square of the useful component of the total photocurrent \(\overline{I}^{\,2}\) will be equal to \(\bar{i}^{\,2}A_{0}/A\), and the spectral density of the useful current is determined as

\[ \overline{I}_{\omega}^{\,2} =\frac{\overline{I}^{\,2}}{\delta} =\frac{I_{\text{const}}^{2}\lambda^{2}}{A\Omega\,\delta}. \]

HETERODYNING OF LIGHT

At the same time, owing to the shot effect in the photocurrent there is noise with a uniform spectral density

\[ \overline{I^2_{\text{shot}}}=2eI_{\text{dc}} \]

(\(e\) is the electron charge). Since the passband of the receiving channel in which the oscillation of frequency \(F\) is amplified can be only much smaller than \(\delta\), the ratio of the useful signal to the noise at the input of the receiving channel will be (in order of magnitude)

\[ \frac{S}{N}= \frac{\overline{i^2_\omega}}{\overline{I^2_{\text{shot}\,\omega}}} = \frac{\lambda^2 I_{\text{dc}}}{2e\delta A\Omega}. \]

Taking into account that \(I_{\text{dc}}=AgB\Omega\), where \(g\) is the sensitivity of the photocell and \(B\) is the brightness of the source, we obtain:

\[ \frac{S}{N}=\frac{\lambda^2 gB}{2e\delta}. \]

It follows from this that increasing the area of the photocathode beyond a value of the order of \(A_0\) gives no gain in \(S/N\). It must be borne in mind, however, that in addition to the shot noise of the photoconverter, the apparatus also contains other noises that do not depend on the magnitude of the photocurrent (for example, the thermal noise of the resonator tuned to the frequency \(F\)). In order to reduce the role of these additional noises, it is necessary to increase \(I_{\text{dc}}\) as much as possible, and consequently the area of the photocathode.

A direct influence on the experimental conditions is exerted by the sensitivity of the photocathode \(g\). Naturally, the choice falls on an antimony–cesium photocathode. Its spectral characteristic is such that the quantity \(\lambda^2 g\) has a maximum near \(\lambda=5300\) Å. The light source must give a line with a wavelength close to this value and with as large as possible a value of the ratio \(B/\delta\)—the spectral density of brightness. These requirements are very well satisfied by the bright green mercury line \(\lambda=5461\) Å. Owing to the large atomic weight of mercury, its Doppler broadening is small. By working with a pure isotope one can get rid of the broadening associated with hyperfine structure. In addition, the green mercury line has the advantage that it is standard for the half-wave plate (see below) and for coated optics. It is easily separated from the other lines by means of a light filter. In a magnetic field the line gives Zeeman splitting, the scheme of which is shown in Fig. 4. The \(\pi\)-components, plotted upward from the horizontal line, are polarized along the field; the \(\sigma\)-components, plotted downward, are polarized transverse to the field.

In the experiments, beats between the \(\sigma\)-components were measured. The light source was an electrodeless tube filled with iso-

topes Hg\(^ {202}\) and excited by a field of frequency 2450 Mc/s. In order to avoid self-absorption of the line, the tube was made flat, 0.1 cm thick; the light passed through its broad sides. At \(\xi=8\cdot10^8\) cgs units, \(A\Omega=0.7\ \text{cm}^2\times\) steradian from the experiment we obtain \(a=3.88\cdot10^{-6}\), which corresponds to \(S/N=10^{-4}\). Consequently, in order to observe the effect of heterodyne detection of light, it is necessary to use a modulating device with a gain of order \(10^4\). The difficulty in creating such a device lies in the fact that it must periodically vary the useful signal, while leaving the shot noise practically constant. The last requirement may be formulated more precisely: the constant component of the photocurrent must vary as a result of modulation by no more than \(10^{-5}\) of its mean value.

Fig. 4.

Labels in Fig. 5: Polaroid; Magnet; Glass plate; Light filter; Phototransducer; Polarization analyzer; Modulator, 120 cycles; Current pulse; Double triode; Heterodyne; Amplifier of the intermediate frequency and detector; Heterodyne control; Horiz.; Vert.; Amplifier at 45 cycles; Phase shifter; Amplifier at 45 cycles and phase rotator; Phase commutator; Light-sensitive surface (\(\sim 5000\) volts); Commutator (phase shifter); Polaroid; Light source; Diaphragm; Plate \(\lambda/2\); Self-writing recorder; After-pulse amplifier; Compensation control; Switch.

Fig. 5.

A modulator satisfying this requirement was implemented in the following way (Fig. 5). The light coming from the source on the left and containing the \(\pi\)- and \(\sigma\)-components passes through a plate-

HETERODYNING OF LIGHT

a \(\lambda/2\) plate rotating at a frequency of \(11\frac{1}{4}\) cps. The light then passes through a polaroid. Linearly polarized light falls on the photocathode, in which the \(\pi\)- and \(\sigma\)-components replace one another at a frequency of \(45\) cps (four times the rotation frequency of the \(\lambda/2\) plate). The useful effect is modulated in the same way as the transmission of the \(\sigma\)-component changes. If the \(\pi\)- and \(\sigma\)-components had strictly identical intensity, the total current would then remain constant. In reality, however, the light emitted by the source proves to be partially polarized, and a component at the modulation frequency appears in the total current. This parasitic modulation is reduced to a harmless magnitude by means of an inclined glass plate and a device consisting of a polaroid placed to the left of the source, a diaphragm, and a spherical mirror reflecting the light through the source to the right.

The parasitic modulation is monitored by an instrument connected at the output of a circuit consisting of a narrow-band amplifier at \(45\) cps, a phase rotator, a phase detector with a large time constant, and a dc amplifier.

The most important part of the apparatus is the photoelectric converter, the creation of which required considerable experimental skill. The converter (Fig. 6) is enclosed in a glass evacuated bulb with a spherical front bottom. A platinum conducting layer is deposited on this bottom; the window in this layer is a semitransparent antimony–cesium cathode. Inside is placed a focusing system made of molybdenum electrodes: spherical and cylindrical. In the cylindrical electrode there is a window for the passage of photoelectrons, covered with a fine tungsten mesh. A third, flat electrode with an aperture in the middle is located near the resonator; its purpose is to prevent extraneous electrons from entering the resonator. The photoelectrons, accelerated in the spherical capacitor formed by the bottom and the first electrode, pass into the aperture of a toroidal resonator having a natural frequency close to the beat frequency and excite it. In order for the excitation conditions to be the best, the resonator must have a large shunt resistance: the gap between the front wall and the central rod must be increased. At the same time, the transit time of the electrons through the gap must remain small compared with the period of oscillation of the resonator. Consequently, the electrons must have high velocities. In the process of depositing the photosensitive layer, cesium settles on the Pyrex glass of the bulb, and the conductivity increases so much that the converter cannot withstand the necessary voltage (5000 V). To avoid this, parts made of lead glass and of platinum, having the same temperature coefficient of expansion as this glass, were installed in the appropriate places.

The resonator had a fixed frequency; tuning was carried out by changing the magnetic field in which the light source is located. This changes the magnitude of the Zeeman splitting, i.e., the beat frequency.

The high-frequency oscillations arising in the resonator are fed, through a coupling loop, a coaxial line, and then a waveguide, to a mixer made in the form of a double tee. After the mixer

Fig. 6.

there follows an intermediate-frequency amplifier at 30 MHz. The amplifier bandwidth is 7 MHz. Since the heterodyne frequency is very high—of the same order as the beat frequency—it may drift by an amount greater than the bandwidth of the resonator. The apparatus has a circuit for monitoring the heterodyne frequency. The intermediate-frequency signal is fed to a second detector, at whose output the useful signal, masked by noise, is obtained. The signal passes through a narrow-band filter at a frequency of 45 Hz and is fed to a phase detector, which is a commutator mounted on the same shaft as a \(\lambda/2\) plate. The rectified voltage, through an \(RC\) filter with a time constant of 250 sec, is fed to a dc amplifier and from there to the instrument. Periodic monitoring of the zero drift of the dc amplifier is provided.

HETERODYNING OF LIGHT

The apparatus described made it possible to obtain at the output of the instrument a signal-to-noise ratio of the order of 2. The results of the measurements are shown in Fig. 7.

Here one can clearly see a maximum corresponding to that field at which the difference frequency \(F\) of the most intense \(\sigma\)-components coincides with the natural frequency of the resonator.

Fig. 7.

The theoretical calculation of the apparatus gave, for the expected effect, the value

\[ \frac{S}{N} = 0.83, \]

which is half the effect obtained experimentally. The authors of the work regard such a degree of agreement as satisfactory, noting at the same time that the cause of the discrepancy may be the nonuniformity of emission from different parts of the photocathode.

In conclusion, the authors of the paper being reviewed consider it necessary to emphasize that the degree of agreement they obtained between calculation and experiment confirms the basic idea of the latter—namely, the idea that the photocurrent is proportional to the square of the instantaneous amplitude of the resultant wave, which means the existence of interference between light waves emitted independently (different \(\sigma\)-components are emitted by different atoms). It is further emphasized that, even if there is a delay between the absorption of a photon and the emission of an electron, it is considerably less than \(10^{-10}\) sec; a relaxation time equal to the beat period would reduce the signal by a factor of 6.4.

References

1. A. T. Forrester, R. A. Gudmundsen and P. O. Johnson, Phys. Rev. 99, 1691 (1955).
2. G. Gorelik, DAN 43, 46 (1947).
3. A. T. Forrester, W. E. Parkins and E. G. Gerjuoy, Phys. Rev. 72, 728 (1947).
4. G. S. Gorelik, UFN 34, 321 (1948).
5. G. S. Gorelik, Oscillations and Waves, Moscow–Leningrad, 1950.