LIGHT PRESSURE ON A MIRROR IMMERSED IN A REFRACTIVE MEDIUM

The enormous role played in the development of physics by the discovery of the ponderomotive actions of light is well known. The classical works of P. N. Lebedev showed that electromagnetic waves possess momentum, and the investigations of S. A. Sadovskii established the presence in them of their own (spin) angular momentum. These most fundamental discoveries lay, in essence, at the foundation of those ideas with which the emergence of photon concepts of light, and later of the Bohr model of the atom, is associated. However, the experiments of P. N. Lebedev, as well as subsequent direct and indirect measurements of the ponderomotive actions of light, concerned the case of propagation of electromagnetic waves in vacuum. The question of the influence of a medium on light pressure long remained experimentally unexplained. Meanwhile, theory predicted that this pressure should be proportional to the refractive index of the medium. Indeed, the density

) Lannuier, Astronomie, 67*, Oct., 351—371 (1953).

of the impulse of an electromagnetic wave propagating in a medium is determined by the expression¹

\[ g=\frac{1}{4\pi c}\,[DB], \tag{1} \]

and, consequently, the pressure of light on a mirror immersed in the medium must be equal to

\[ p=gv(1+R)\cos^2\varphi, \tag{2} \]

where \(v\) is the velocity of light in the medium, \(R\) is the reflection coefficient of the mirror in the medium, and \(\varphi\) is the angle of incidence of the light on the mirror*).

Taking into account that for an isotropic nonabsorbing medium with refractive index \(n\)

\[ \mathbf{D}=n^2\mathbf{E}\quad \text{and}\quad \mathbf{B}=\mathbf{H}, \]

we have:

\[ p=\frac{n^2v}{c^2}(1+R)S\cos^2\varphi, \tag{3} \]

where

\[ \mathbf{S}=\frac{c}{4\pi}[\mathbf{EH}] \tag{4} \]

is the Umov–Poynting vector in the medium. If the medium possesses dispersion and the irradiating beam is not monochromatic, then

\[ p=\frac{\cos^2\varphi}{c^2}\int_0^\infty n_\lambda^2 v_\lambda S_\lambda(1+R_\lambda)\,d\lambda. \tag{5} \]

As the author asserts, a rigorous calculation shows that by \(v\) one should understand not the group, but the phase velocity of light in the medium. Accepting this assertion and assuming that \(R\) depends only weakly on \(\lambda\), equation (5) may be rewritten in the form

\[ p=\frac{\cos^2\varphi}{c}\,l(1+R)\bar n^{**}), \tag{6} \]

where

\[ l=\int_0^\infty S_\lambda\,d\lambda \quad \text{and}\quad \bar n=\int_0^\infty n_\lambda S_\lambda\,d\lambda. \]

*) Elementary photon representations lead, according to (2), to the conclusion that the impulse of a unit photon in the medium must be equal to

\[ \frac{h}{\lambda} \]

(i.e. \(\frac{h\nu}{v}\) instead of \(\frac{h\nu}{c}\) for vacuum).

**) It is interesting to note that the proportionality of the light pressure to the refractive index of the medium follows directly from Newton’s corpuscular representations of light (the velocity in the medium is greater than in vacuum).

We have already reported \(^{2*}\) on the preliminary results of Jones’s experiments,\(^{3}\) which showed that the pressure of light on a mirror immersed in a medium is proportional to its refractive index. A detailed description has now been published of the more carefully performed subsequent experiments, in which significantly greater accuracy was achieved. In view of the great fundamental importance of the question of the pressure of light in a medium, it seems advisable to return once again to this question and to acquaint the reader with some details of the experiment.

As usual, the method of measurements consisted in irradiating mirror vanes mounted on a bifilar suspension and in determining the torque produced by light pressure. However, to determine the dependence of \(p\) on \(n\), the suspension with the vanes had to be immersed in a liquid, which created a number of substantial difficulties. In contrast to measurements in vacuum, the main source of disturbances is associated with the inevitable occurrence of convective currents, \(^{**}\) which can produce an additional torque of two kinds: a) owing to the asymmetry of the vanes situated in the convective flow, and b) owing to the difference in temperature (and consequently also in convective velocities) on the illuminated and unilluminated sides of a vane. Reduction of these effects was achieved by: a) manufacturing the vanes from a well-reflecting material with high thermal conductivity; b) replacing discontinuous irradiation with continuous irradiation, but with periodic displacement of the illuminating beam over the surface of the vanes; c) careful installation of the vane in the suspension; d) reducing the size of the vessel containing the suspension; and e) reducing the response time of the mechanical system (allowing for the inertia of thermal effects).

The vessel in which the measurements were made was a vertical thick-walled bronze tube, whose internal diameter was \(0.8\ \mathrm{cm}\), fitted with two pairs of glass windows (thickness \(0.1\ \mathrm{cm}\)). The tube was filled with liquid to approximately one half. On a taut (to eliminate the influence of surface forces) bifilar suspension (a strip of gold alloy of cross section \(0.005 \times 0.0005\ \mathrm{cm}\)) there was fastened a thin copper wire carrying the vanes. The vanes were rectangular silver mirrors (\(0.2 \times 0.5 \times 0.01\ \mathrm{cm}\)), coated with rhodium to avoid corrosion and tied to the copper wire with an ordinary sailor’s knot. (Other methods of fastening—gluing, soldering—proved incomparably less convenient.) On the same wire, somewhat higher up (remaining in air when the vanes were immersed in the liquid), a second mirror was placed (\(0.2 \times 0.5 \times 0.02\ \mathrm{cm}\)), serving for reading the angle of rotation of the suspension. Finally, still higher on the wire there was tied a weak magnet \(0.2\ \mathrm{cm}\) long and \(0.03\ \mathrm{cm}\) in diameter, located in the field of an electromagnet placed outside the tube. This magnetic system served simultaneously for calibration and for checking the sensitivity of the suspension, as well as for its additional damping.

The angle of rotation of the suspension was read from the displacement of a light spot reflected from the upper mirror. In view of the smallness of the rotation angles, an optical amplification scheme was used, ensuring sufficient stability of the readings; the zero drift per hour did not exceed \(10^{-8}\) radian (i.e. approximately 1% of the magnitude of the displacements corresponding to the Brownian motion of the suspension), which made it possible confidently to interpret all observed features of the registration trace as motions of the suspension.

Negative feedback between the optical amplification and the control magnetic system damped the system and reduced its response time from \(0.7\ \mathrm{sec}\) to \(0.5\ \mathrm{sec}\). The linearity of the response corresponded to an acc—

*) In \(^{2}\), in formula (3), \(4\pi\) was erroneously placed in the denominator.

**) The radiometric effect in a liquid is practically absent.

precision was ±0.1%; taking into account all possible sources of error, the error in determining the position of the suspension was ±0.3%.

As a light source for irradiating the vanes, a tungsten incandescent lamp of 12 V, 48 W was used, operated in a state of considerable underload: 9.8 V, 30 W. A system of mirrors and lenses produced on the vanes two images of the light source, located one on its front surface, the other on its rear surface at various distances from the axis of rotation (positions \(AA'\)). As a result of the light pressure there arose a couple tending to turn the suspension; moreover, the moment of the couple depended both on the intensity of the light and on the relative position of the images.

By simply rotating the shaped diaphragm disks, both images could be shifted over the surface of the vane so that the moment of the couple, while remaining the same in magnitude, changed sign to the opposite one (positions \(BB'\)). The shaped diaphragms were mounted on an axis rotated by a motor. Thus it was possible periodically to change the sign of the moment of the couple at an arbitrary frequency (transferring the beams from positions \(AA'\) to positions \(BB'\) and back).

Preliminary measurements carried out with a vessel filled with air showed that, when the vanes were illuminated during the first \(\sim 10\) sec, there occurred a process of gradual establishment of the equilibrium position of the suspension, and the magnitude and direction of the displacement depended on the design of the suspension, but practically (with careful adjustment) did not depend on the position of the beams (\(AA'\) or \(BB'\)). Shifting the beams produced an additional displacement that was not accompanied (to an accuracy of 1%) by the appearance of moments of convective origin (as indicated by control experiments in which a couple of forces was produced by means of an electromagnet).

Estimates showed that, when the beams were separated by 0.3 mm, the torque should have been close to \(1.3 \cdot 10^{-6}\) dyne·cm, which corresponded to a rotation of the suspension by \(4 \cdot 10^{-5}\) radian. The observed values lay between \(3 \cdot 10^{-5}\) and \(6 \cdot 10^{-5}\) radian. An estimate of Brownian noise gave the value \(10^{-6}\) radian, i.e. about 2% of the measured quantity. In reality the noise often reached 5%.

When the suspension was immersed in a liquid, the picture changed substantially. After the equilibrium deflection had been established, shifting the beams entailed an initially rapid throw (over a few seconds), followed by a comparatively slow return almost to the previous equilibrium position (5–7 sec), again replaced by a shift in the initial direction and by the gradual establishment of a new equilibrium position; moreover, the equilibrium displacement was approximately twice as large as the initial throw. The authors explained this picture by the influence of convection and, as a result of a detailed analysis, came to the conclusion that the initial throw corresponds to the true value of the light pressure. Therefore they resorted to a rapid periodic shifting of the beams back and forth (50–70 times per minute by rotating the figured diaphragms), assuming that in this way the influence of convective forces would be excluded.

Control experiments with the production of a variable couple of forces by means of an electromagnet confirmed the validity of this procedure; moreover, the influence of convective forces was estimated as less than 0.2% of the measured effect.

The final measurements were carried out as follows. Initially the suspension was in air. A calibrated couple of forces, produced by an electromagnet and approximately equal to that expected as a result of switching the light beams, was applied, and the throw was measured. Then the measured

recoil caused by light pressure. After this the vessel was filled with liquid and the experiment repeated.

In this way, the torque produced by light pressure was compared (both in the liquid and in air) with a constant calibration torque.

As a result, the “experimental” ratio of the pressure of light in the liquid to the pressure of light in air was found. An estimate of the errors showed that the accuracy of determining this ratio in a single experiment was of the order of \(\pm 1.5\%\).

Corrections were then introduced:

1. For the absorption of light in the liquid. In the path of the beam a cuvette with water was placed, serving as a light filter and limiting the wavelength interval to between \(0.4\) and \(1.2\,\mu\), with a maximum near \(0.8\,\mu\). Absorption in the liquids was measured integrally by means of a selenium photoelement, and a correction for attenuation of the light beam was introduced. In addition to attenuation of the illuminating beam, absorption led to the appearance of volume pressure forces in the liquid, which were also taken into account.

2. For the dependence of the reflectivity of the glass window on the refractive index of the liquid. This correction was determined by measuring the transparency of a window immersed in benzene (absorption in the glass), and of a window in air, and then using Fresnel’s formulas to account for the influence of the refractive index of the liquid.

In addition, a correction for multiple reflections was estimated by calculation.

1. For the dependence of the reflection coefficient \(R\) on \(n\)—by direct measurements of this dependence (but without taking into account weak diffuse reflection).

2. For the change in the position of the light beam on the vane and in its angle of incidence—by approximate theoretical estimates.

All these corrections are given in the table in the form of correction factors.

The mean refractive index \(\bar n\) was calculated by measuring \(n_\lambda\) for various \(\lambda\) and graphically evaluating \(\int n_\lambda I_\lambda\,d\lambda\) from the known spectral characteristic of the light source. (It turned out that \(\bar n\) was practically equal to \(n\) at \(\lambda = 0.8\,\mu\).) The results of the measurements are summarized in the table, which includes data obtained over the course of six months for six different liquids. As can be seen from the table, there is excellent agreement between the experimental results and the theoretical formula (6), and the mean error of the measurements does not exceed \(\pm 1.2\%\).

The authors note that in the case of carbon disulfide the group velocity of light at \(\lambda = 0.8\,\mu\) is approximately \(3\%\) less than the phase velocity, and that, despite the smallness of this quantity, the experiment evidently testifies in favor of the assertion made by them that in formula (3) \(v\) denotes not the group velocity but the phase velocity.

G. Rozenberg

References

1. I. E. Tamm, Foundations of the Theory of Electricity, Gostekhizdat, 1949.
2. G. Rozenberg, UFN 44, No. 3, 463 (1951).
3. R. V. Jones, Nature 167, 439 (1951).
4. R. V. Jones and J. C. S. Richards, Proc. Roy. Soc. 221A, No. 1147, 480 (1954).