OBSERVATION OF THE SPIN ANGULAR MOMENTUM OF CENTIMETER WAVES

Half a century ago, in 1899, A. A. Sadovsky[^1] showed that a circularly polarized electromagnetic wave must possess an angular momentum with respect to the direction of its propagation, as a result of which a body absorbing such a wave, or changing the character of its polarization, must experience a torque. Ten years later Poynting[^2] refined the theory of this phenomenon.

From the quantum point of view, circular polarization of an electromagnetic wave means that the spins of all the photons constituting this wave are oriented, depending on the direction of polarization, either along the direction of their motion or opposite to it. Since the spin of a photon is equal to 1, the density of the flux of the longitudinal component of the angular momentum of the electromagnetic wave, i.e., the longitudinal component of the angular momentum carried by the electromagnetic wave in unit time through a unit area normal to the direction of its propagation, is equal to

\[ \sigma_z=\pm j\hbar, \tag{1} \]

where \(j\) is the photon-flux density, and the sign \(+\) or \(-\) is chosen depending on the direction of polarization.

Taking into account that the magnitude of the Umov–Poynting vector is

\[ |S|=j\hbar\omega, \tag{2} \]

where \(\omega\) is the angular frequency, we have:

\[ \sigma_z=\pm\frac{|S|}{\omega}, \tag{3} \]

which coincides with the expression obtained within classical electrodynamics (see, for example, ³).

It is not difficult to show that, in the case of an elliptically polarized wave in vacuum,

\[ E_x = E_1 \cos \omega t,\quad E_y = E_2 \sin \omega t \tag{4} \]

the flux density of the longitudinal component of the momentum moment is

\[ \sigma_z=\frac{1}{\omega}\sqrt{\frac{\varepsilon_0}{\mu_0}}\,E_1E_2. \tag{5} \]

When a wave is absorbed, the absorbing body will evidently experience a torque

\[ L=\sigma_z S, \tag{6} \]

where \(S\) is the cross section of absorption of the wave by the body. If, as a result of the passage of an electromagnetic wave through a compensator, the character of its polarization changes—for example, a linearly polarized wave becomes elliptically polarized—then (neglecting reflection and absorption of the wave in the compensator) the compensator must experience a torque

\[ L=(\sigma_{z\text{нач}}-\sigma_{z\text{кон}})S, \tag{7} \]

where \(\sigma_{z\text{нач}}\) and \(\sigma_{z\text{кон}}\) are the longitudinal components of the flux density of the momentum moment of the wave, respectively before and after its passage through the compensator, and \(S\) is the cross section of the compensator.

Experimental observation of this phenomenon is extremely difficult owing to the negligible smallness of \(\sigma_z\). Thus, for light frequencies \((\omega \simeq 4\cdot 10^{15}\ \text{sec}^{-1})\), at an intensity of circularly polarized light equal to the intensity of direct sunlight, the torque experienced by an absorbing screen, referred to unit area of the screen, is about \(\sigma_z=3\cdot 10^{-10}\ \text{dyne}/\text{cm}\). This explains why, until very recently, all attempts to detect the spin moment of an electromagnetic wave in a macroscopic experiment remained unsuccessful, and experimental proofs of the existence of spin in photons were confined to the region of atomic phenomena, where the phenomena of exchange of angular momentum between light and matter are of primary importance (for example, in the acts of emission and absorption of light by an atom). Only in 1935 did Beth⁴ succeed in experimentally measuring the torque experienced by a birefringent quartz plate when light passed through it.

Entirely new possibilities in this direction arose as a result of the development of microwave engineering. Since, according to (3), \(\sigma_z\) is inversely proportional to the frequency, the transition from light waves to the microwave radio-wave range leads to an increase of the effect by many orders of magnitude. Thus, on centimeter waves \((\omega \simeq 10^{10}\ \text{sec}^{-1})\), at a quite attainable value of the Poynting vector of \(1\ \text{W}/\text{cm}^2\), it amounts to a value of the order of \(10^{-3}\ \text{dyne}/\text{cm}\), which is already quite measurable.

The paper under review⁵ is an attempt to measure the spin angular momentum of electromagnetic waves in the centimeter range.

The author used radiation from a magnetron, which gave pulses of duration \(1\ \mu\text{sec}\) in the amount of 1000 pulses per second at

with wavelength \(\lambda = 3.2\ \mathrm{cm}\) (9360 Mc/s). The peak power was \(50\ \mathrm{kW}\); the average power, respectively, was \(50\ \mathrm{W}\). The linearly polarized wave emitted by the magnetron was directed first along a rectangular waveguide, and then entered a vertical circular waveguide of diameter \(20\ \mathrm{mm}\), ending at its upper end in a small horn. Bolometric measurements showed that the energy flux density was constant over the entire cross-section and was approximately equal to \(1\ \mathrm{W/cm^2}\). A part of the circular waveguide, capable of rotating about its axis, was equipped with a paraffin plate (refractive index \(n = 1.47\)) a few millimeters thick. The dimensions of the plate were chosen so that the component of the wave whose electric vector is parallel to the plate lagged by a quarter period.

If the electric vector of the wave traveling in the waveguide was parallel or perpendicular to the plate, then the state of polarization of the wave remained unchanged; if, however, the electric vector made some angle with the plate, then the plate caused the appearance of elliptical polarization (at an angle of \(45^\circ\) the wave was transformed into a circularly polarized one).

A glass tube with a plane-parallel window was placed above the horn. Inside this tube, directly above the mouth of the horn, absorbing polarizing or birefringent screens were suspended on two thin silk threads. The screens were provided with mirrors, toward which a beam of light was directed. The rotation of the screen could be registered by the displacement of the light spot on a scale located at a distance of \(1\ \mathrm{m}\) from the mirror. The glass tube, together with the screens suspended in it, could rotate about a vertical axis, which made it possible to change the orientation of the screen relative to the plane of polarization of the incident wave. The authors carried out three series of experiments with three different types of screens.

In the first case the absorbing screen consisted of two flat disks \(3\ \mathrm{cm}\) in diameter. The lower disk, made of mica, was coated with a layer of resin containing graphite. The upper disk, separated from the lower one by a distance \(\lambda/4\), was made of aluminum.

A wave incident on such a screen is partially reflected from the mica disk and partially penetrates through it. The transmitted wave, in turn, is reflected from the aluminum disk, passes through the mica disk, and interferes with the wave reflected from the mica disk. Since the path difference between the two reflected waves is equal to \(\lambda/2\), the waves mutually extinguish one another, and it appears possible to adjust the system so that the reflected radiation vanishes completely, i.e. the screen completely absorbs all the energy incident upon it. Consequently, according to (6), such a screen must experience a torque \(L = \sigma_z S\). On the contrary, if the screen is an ideal reflector, then \(\sigma_z\) is the same for the incident and reflected waves and \(L = 0\).

Experiment showed that, indeed, a reflecting (for example, aluminum) screen experiences no measurable torque under the action of an electromagnetic wave—the spot on the scale remains at rest. Likewise, an absorbing screen experiences no torque if the wave incident on it is linearly polarized. However, if the wave incident on the absorbing screen was circularly polarized, then the screen turned, the displacement of the spot on the scale being about \(6\ \mathrm{cm}\) (angle of rotation about \(3'\!.5\)).

In another series of experiments the screen consisted of two square mica sheets of size \(3 \times 3\) cm, placed one above the other at a distance of \(\lambda/8\). On each of the sheets a row of parallel metal wires of diameter \(0.2\) mm was fastened, the sheets being oriented so that the directions of the wires on the upper and lower sheets were mutually perpendicular. The wave incident on such a screen was decomposed by it into two linearly polarized components. One of them, with an electric vector parallel to the wires of the lower sheet, was reflected back by this sheet. The other component, polarized in a direction perpendicular to the first, passed through the lower sheet, was reflected from the upper one, and, passing again through the lower sheet, interfered with the component reflected from the lower sheet. Since the path difference between the two reflected components is \(\lambda/4\), a circularly polarized wave incident on such a screen is transformed upon reflection into a linearly polarized one, and conversely, a linearly polarized wave acquires elliptical polarization upon reflection.

When a circularly polarized wave was sent onto such a screen, the screen rotated (to the right or to the left, according to the direction of polarization of the wave), the deflection of the spot on the scale being about \(8\) cm (angle of rotation approximately \(4.5^\circ\)), which, as the author indicates, corresponds to a specific torque of the order of \(10^{-3}\) dyn/cm, expected on the basis of the theoretical estimate given above.

In the case where a linearly polarized wave was incident on the screen, the effect depended substantially on the direction of polarization. When the plane of polarization was parallel to the wires of one of the sheets, the character of the polarization did not change upon reflection and the spot on the scale remained motionless. On the contrary, if the plane of polarization was rotated by \(45^\circ\) relative to the direction of the wires, then the reflected wave was circularly polarized and the screen rotated. Both the direction of rotation and its magnitude were found to be in agreement with the theoretical predictions.

The third series of experiments was carried out with a screen of the same type as the preceding one, but the distance between the sheets was one quarter of the wavelength. The circularly polarized wave was reflected from such a screen likewise as circularly polarized, but with the opposite direction of polarization. The experiment showed that, in accordance with (7), the rotation of the screen was twice as large as in the preceding experiments.

The author emphasizes that the deflections of the spot observed by him were very distinct and constant, so that the presence of a torque of the two-dimensional screen in a circularly polarized wave could be established with complete certainty. At the same time, complete qualitative agreement with the theory was obtained. Quantitatively, however, one can speak only of agreement in order of magnitude, since, although the mechanical measurements could be performed with sufficient accuracy, the value of the Umov–Poynting vector itself was determined only approximately. In addition, since the screens had dimensions close to the wavelength, diffraction, nonuniformity in the distribution of energy density, and a certain spread of phases caused additional and apparently significant errors. There is no doubt that further experiments in this direction will make it possible to carry out a quantitative comparison of theory with experiment.

G. Rosenberg

References

1. A. A. Sadovskii, Acta et comm. Imp. Univ. Jureviensis 7, No. 1–3 (1899); 8, No. 1–2 (1900).
2. J. H. Poynting, Proc. Roy. Soc. 82, 560 (1909).
3. D. Ivanenko and A. Sokolov, Classical Field Theory. Publishing House of Technical-Theoretical Literature, 1949.
4. R. A. Beth, Phys. Rev. 48, 471 (1935), 50, 115 (1936).
5. N. Carrada, Nature 164, No. 4177, 882 (1949).