On the Question of Total Reflection of Waves

(A Historical Essay)

A. A. Korobko-Stefanov

Introduction

Total reflection of waves belongs among the phenomena in which the full diversity of wave optics is manifested, and interest in the study of this phenomenon has not weakened up to the present day. Even Newton¹, to whom the first experimental investigation of this phenomenon belongs, drew attention to the fact that, under total reflection, light penetrates into the optically less dense medium. Subsequently, the study of the phenomenon of total reflection proceeded along the path of experimentally determining the depth of penetration and clarifying its dependence on the angle of incidence, wavelengths, and the position of the plane of polarization of the incident light.

Attempts at a theoretical description of the observed phenomena on the basis of the mechanical theory of light did not yield positive results. A particular difficulty here was posed by the question of the character of the waves propagating in the less dense medium. A correct and exhaustive theoretical description of the process of total reflection of light was given in 1909 by Professor of Moscow University A. A. Eichenwald² on the basis of electromagnetic theory.

In this work A. A. Eichenwald established that, when an unbounded plane wave is reflected from the boundary formed by transparent media, at angles of incidence greater than the limiting angle, “the energy of the incident ray, as it were, dives into the second medium, in order then to appear wholly in the first medium in the reflected ray.”

The attention of subsequent investigators was focused on describing the process of total reflection of waves bounded by a diaphragm. This made it possible, on the one hand, to refine and describe in greater detail the observed displacement of the reflected light beam and, on the other hand, to show the influence of the interaction of the diaphragm with the transmitted wave on the magnitude of the displacement.

NEWTON’S EXPERIMENTS

Newton’s study of the total reflection of light from the transparent boundary of two media was carried out with two rectangular glass prisms.^3 The large face of one of them was a slightly convex surface. By bringing the prisms into contact with their large faces, it was possible to observe transmitted light through the place of contact of the prisms, as through solid glass. In other places, where there is no contact between the surfaces of the prisms, total reflection occurs. If one looks at the place of contact of the prisms from the side of the reflected light, then the place of contact of the prism surfaces will be dark.

Fig. 1. Diagram of the arrangement of prisms in the Newton–Quincke experiments.

The scheme of Newton’s experiment is shown in Fig. 1. This scheme was subsequently used by almost all researchers for observing the penetration of light into a less dense medium under total reflection.

The dimensions of the dark spot, as Newton indicates, vary depending on the angle of incidence in the region of total reflection. This circumstance, as was clarified later, is connected with the dependence of the depth of penetration of light into the less dense medium on the angle of incidence of light on the interface.

Measurement of the geometrical dimensions of the dark spot enabled Newton to make a quantitative judgment about the depth of penetration of light into the less dense medium under total reflection.^4

The depth at which light could be detected in the less dense medium, as determined by Newton, proved to be equal to \(2.5\cdot 10^{-6}\) cm. Wishing to connect the penetration of light into the less dense medium with total reflection, Newton believed that in this case the trajectory of light in the less dense medium is a parabola\(^5\). It is appropriate to note that the experimental proof, given by Newton, of the penetration of light into a less dense medium long remained unknown to adherents of both the corpuscular and wave theories of light, who in one way or another repeated his experiments. But, regardless of views on the nature of light, the results of the experiments led to one conclusion: under total reflection light penetrates into the less dense medium. The quantitative discrepancy in determining the depth of penetration should not be taken into account, since neither side presented serious arguments. Some believed that the depth of penetration is \(1/4\), while others \(1/2\), of the wavelength of the incident light. Thus Newton’s experiments, whose explanation did not fit within the framework of the corpuscular theory of light, laid the foundation for experimental-theoretical investigations of the phenomenon of total reflection. Without dwelling on details, let us point to the works of such researchers as Huygens\(^6\), Young\(^7\), Verdet\(^8\), Biot\(^9\), Fresnel\(^10\), Babinet\(^11\), Stokes\(^12\), and Billet\(^13\). This series of investigations of the problem of total reflection, which extended over almost two centuries, was completed by Quincke’s thorough experimental work\(^14\), which we shall describe in detail.

QUINCKE’S OBSERVATIONS

The scheme of Quincke’s experiment is the same as Newton’s, i.e. two glass prisms placed together with their large faces. For brevity we shall call these faces the diagonal planes. One of the diagonal planes of the prisms had a spherical form with radius of curvature equal to \(1607.7\) mm. When the diagonal planes are brought together until they touch, an air layer is formed between them, the thickness of which increases in all directions as the distance from the point of contact increases. Thus, for light incident on the face of the prism with the unrounded diagonal plane, according to geometrical optics the conditions for total reflection are fulfilled over the entire diagonal plane (see Fig. 1), with the exception of the point of contact.

If one now observes, as Newton did, in the direction of the arrow \(N\), i.e. from the side of the reflected light, then one sees a dark spot on a light background. But if one observes in the direction \(Q\), as Quincke did, then one sees the inverse picture, i.e. a light spot formed as a result of the penetration of light through the point

of contact and near it, in all those places where the distance between the diagonal planes is less than the depth of penetration of light into the second medium.

In order to make sure that light penetrates into the second prism not only at the point of contact, but also near it, it was necessary to create, as Quincke did, a gap between the diagonal planes, i.e. to eliminate the point of contact. The dark spot that Newton observed then disappears, but the bright one, formed by the penetrating light, remains.

Consequently, in all those places where the distance between the diagonal planes is less than the depth of penetration of light into the less dense medium, the light that has entered it under total reflection has the possibility, according to the ordinary laws of refraction, to penetrate into the second prism.

Thanks to the spherical shape of the diagonal plane of the second prism, Quincke achieved a smooth change in the distance between the diagonal planes. From the geometrical dimensions of the light spot, which had the form of an ellipse, one can compute by an elementary method the depth of penetration of light into the less dense medium under total reflection.

Indeed, let us denote by \(2\rho\) the major axis of the ellipse of the light spot. Then the distance \(\varepsilon\) from the periphery of the ellipse to the plane surface of the first prism, or the depth of penetration, is determined from the equation

\[ \varepsilon=\frac{\rho^2}{2R}, \]

where \(R\) is the radius of curvature of the diagonal surface of the second prism.

Determining \(\rho\) at different angles of incidence, Quincke found a dependence of the depth of penetration on the angle of incidence. Quincke considered the cases when the incident light was polarized either in the plane of incidence or perpendicular to it. It turned out that, at one and the same angle of incidence, the penetration of light into the less dense medium depends substantially on the position of the plane of polarization.

The dependence of the depth of penetration on the angle of incidence for the two cases of polarization of light incident under conditions of total reflection on the glass–air interface is presented in Fig. 2. The dashed line indicates the dependence of the depth of penetration into the less dense medium of light polarized in the plane perpendicular to the plane of incidence, and the solid line—parallel to it.

Quincke carried out an investigation of the depth of penetration with many prisms made from various kinds of glass,

filling the interval between them with various transparent liquids. In Fig. 2 is presented the result of an observation when the first prism was made of flint glass, \(n = 1.6160\), and the second of crown glass, \(n = 1.5149\). The interval between them was filled with atmospheric air. The limiting angle \(\vartheta_g\) in this case proved to be equal to \(38^\circ 14'\).

Attention should be drawn to an important circumstance, which was first discovered by Quincke in this very delicate experiment. In Fig. 2 the penetration depth of light polarized in the plane of incidence, at angles of incidence somewhat greater than the limiting angle, is less than for light polarized perpendicular to it.

Fig. 2. Dependence of the penetration depth of polarized light into the second medium on the angle of incidence according to Quincke.

But this difference in penetration depths, as the angle of incidence increases, becomes smaller and smaller, becoming zero at some angle of incidence \(\vartheta_0\). At angles of incidence greater than \(\vartheta_0\), the difference becomes negative, i.e. light polarized in the plane of incidence penetrates into the less dense medium more deeply than light polarized in the plane perpendicular to it. Quincke’s experiments date to 1866.

Much later, in 1902, Quincke’s experiments were repeated in a more perfect manner by Hall\(^ {24}\). Hall’s investigation confirms the dependence, discovered by Quincke, of the penetration depth of light into a less dense medium under total reflection on the angle of incidence, the wavelength, and the position of the plane of polarization of the incident light (Fig. 3).

It should be noted that the course of the curves of the depth of penetration of polarized light into the less dense medium, discovered by Quincke and Hall, is in contradiction with the results of theoretical consideration. A detailed discussion of this interesting fact we shall give below, in connection with the experiments of Goos and Hänchen[^35].

Fig. 3. Dependence of the depth of penetration of polarized light on the angle of incidence into the second, optically less dense medium, according to Hall.

THE INCONSISTENCY OF FORT’S THEORY AND EXPERIMENT

As we have already indicated, the corpuscular theory of light could not explain the penetration of light into a less dense medium under total reflection. As for the wave theory, constructed on a mechanical basis, then, as is clear from Fort’s theoretical investigation[^15], it likewise could not give a correct explanation of the above-described experiments of Quincke. The mechanical theory of light, applied by Fort to the given optical phenomenon, led him to the erroneous assertion of the existence in the second medium of a longitudinal light wave.

From the theory proposed by Fort it followed that in the second, optically less dense, medium there are two waves of unequal amplitudes. One of them propagates along the normal to the interface directed into the second medium, while the other propagates along the interface.

According to Fort’s assertion, the wave that propagates along the normal is transverse, whereas the other, propagating in the second medium along the interface, is longitudinal. And since until then there had not been a single experiment in which the longitudinal character of light waves, possible in the mechanical theory of light, would have been detected, Fort’s assertion became the soil on which

there arose the question of the character of the light waves penetrating into the second medium under total reflection. This longitudinal wave, in Fort’s opinion, could be directly observed under the conditions of an experiment which he proposed^16. Fort’s experimental scheme for observing the longitudinal wave is clear from Fig. 4.

Light falls on the face \(AB\) of a glass prism in such a way that it undergoes total reflection at the faces \(BC\) and \(CD\), which form an obtuse angle with one another.

A longitudinal wave, in Fort’s opinion, propagating along the face \(BC\), cannot follow its sharp bend and at the edge \(C\) becomes separated; as a result of this, if one looks in the direction of the arrow (Fig. 4), one should see the edge \(C\) glowing, which is indeed observed, but for another reason.

Fig. 4. Fort’s experimental scheme.

The unsoundness of the theoretical explanation of Quincke’s experiments which Fort proposed, and of his attempt to observe a longitudinal light wave, are a consequence of his stubborn defense of the mechanical theory of light. It is enough to point out that Fort’s work dates to 1899, i.e. appeared 35 years after Maxwell’s publication of the electromagnetic theory (1864).

Let us note that Fort was not alone in his belated recognition of the electromagnetic theory of light. Maxwell’s theory needed the experiments of P. N. Lebedev^17 in order to gain universal recognition. Thus William Thomson (Lord Kelvin), in a conversation with K. A. Timiryazev^18, said: “You may know that all my life I fought with Maxwell, not recognizing his light pressure, but your Lebedev forced me to surrender before his experiments.”

VARIOUS EXPERIMENTS ESTABLISHING THE PENETRATION OF LIGHT INTO A LESS DENSE MEDIUM

Fort’s experiment was explained by Ketteler^19 as a diffraction phenomenon, as a result of which a glow is observed at the edge \(C\) (see Fig. 4). Such an interpretation of the phenomenon observed by Fort provided the basis for the experiments of Ditscheiner^20, Exner^21, and Edser-Senior^22. The peculiarity of their experiments consisted in the fact that they applied diffraction gratings to the diagonal planes of prisms. Light,

penetrating into an optically less dense medium, fell on a diffraction grating and it was possible to observe the diffracted light. In this way they again succeeded in showing the penetration of light into the less dense medium under total reflection. However, these experiments could not give an answer to the question of whether the wave penetrating into the optically less dense medium is transverse or longitudinal, since the phenomenon of diffraction, generally speaking, is not an experimentum crucis in the question of the transverse character of light waves. The diffraction pattern they observed did not differ in any way from the ordinary one.

Much later Wood \(^{23}\), observing diffraction from the edges of soot particles deposited on the diagonal plane of a prism, confirmed this same result.

Nor did Hall’s experiment \(^{24}\) introduce anything new; instead of a diffraction grating he deposited a light-sensitive layer on the diagonal plane of the prism. The blackening produced by the light penetrating into the emulsion did not differ in any way from the usual one.

All this shows that the very fact of the penetration of light incident on the interface under conditions of total reflection into the second, optically less dense, medium was experimentally proved. But the question of the character of the light waves in the second medium remained open. It remained unclear whether there really is in the second medium a real light wave in which the light oscillations occur in the direction of propagation.

Some authors, such as, for example, Drude \(^{25}\), supposed that if the wave propagating along the interface in the second medium is longitudinal, then this can be permitted, since it is a wave of unequal amplitudes; thus the property of transverse character of light waves applies only to waves of equal amplitudes.

THE THEORY OF A. A. EICHENWALD

The erroneousness of Voigt’s theoretical conclusions, connected with the mechanical theory of light, was revealed by the investigations of A. A. Eichenwald \(^{2}\) on the basis of the electromagnetic theory of light.

Eichenwald’s investigations concern the case when a plane monochromatic wave is incident on a plane boundary separating transparent media under conditions of total reflection. The starting point in his work is Maxwell’s equations, which, under the given boundary conditions, make it possible to determine the wave field on both sides of the interface.

Under these conditions he finds that light also penetrates into the second medium, but, first, the intensity of the light rapidly decreases with depth below the plane of separation and, second, the angle \(\chi\) (the angle of refraction) becomes variable in time.

Regarding the angle \(\chi\), Eichenwald writes: “Thus, although in the case of total reflection one cannot find such a constant direction of the refracted ray that would satisfy the boundary conditions, nevertheless the refracted ray exists, but has a direction changing with time according to the law:

\[ \tan \chi=-\frac{a}{k}\tan \frac{2\pi}{T}(t-ax), \]

where \(a\) is the reciprocal of the phase velocity along the \(x\)-axis, lying in the plane of the interface; \(k\) is a quantity depending on the angle of incidence \(\varphi\), the velocity of propagation of light in the first medium \(v_1\), and the relative refractive index \(n\). The quantity \(k\) is determined by the equation

\[ k=\frac{\sqrt{\sin^{2}\varphi-n^{2}}}{v_1}. \]

The components of the wave field on both sides of the interface, found by Eichenwald for the case of total reflection, make it possible to determine the form of the lines of energy fluxes (rays) entering the second medium. These curves are shown by us in Fig. 5.

If the coordinate plane \(XOZ\) is taken as the plane of incidence, then “the components of the energy fluxes,” writes Eichenwald, “along the axes \(Z\) and \(X\), both for a ray polarized in the plane of incidence and for a ray polarized perpendicular to this plane, will be exactly the same, if the amplitudes of both rays are equal.”

The analytic expression for the components of the energy fluxes in the case of incidence of polarized light in the second medium has the form

\[ f_x=-A^2\frac{\varepsilon}{8\pi k}e^{-\frac{4\pi k}{T}z}\cdot \sin \frac{4\pi}{T}(t-ax), \]

\[ f_z=A^2\frac{\varepsilon a}{8\pi k}e^{-\frac{4\pi k}{T}z}\cdot \left[1-\cos \frac{4\pi}{T}(t-ax)\right]. \]

“The expression \(f_z\),” writes Eichenwald, “shows that the light energy oscillates in the direction \(Z\) with period \(T/2\), i.e., during the time of one full period of the light oscillation \(T\) the energy manages twice to enter from the first medium into the second and twice to go back.”

“However,” Eichenwald points out, “the expression \(f_z\), and in particular our curves (see Fig. 5), show that the entry and exit of energy at different places of the interface plane (for different \(x\)) do not occur at the same time, so that at the time when in some places \(f_z>0\), energy enters from the first medium into the second, in

in other places, at a distance from the first equal to

\[ x = T/4a = \frac{\lambda}{4}. \]

\(f_z < 0\), the energy passes from the second medium into the first.

Fig. 5.

Since the interface considered by Eichenwald is transparent, calculation of the amount of energy flowing from the first medium into the second and flowing from the second medium into the first convinces us that the reflection is indeed total.

“Thus the energy of the incident ray,” Eichenwald concludes, “as it were penetrates into the second medium, only then to appear wholly in the first medium in the reflected ray. But the entry and exit of the energy occur at different places and at different times; and for a given time the entry of the energy into the second medium occurs at points separated from one another by half the distance between identical phases along the \(X\) axis.”

With regard to Focht’s longitudinal wave, Eichenwald remarks that “the expression for \(f_x\) always (for any \(t\)) and everywhere (for any \(x\)) has a positive value, and this circumstance has given rise to the misunderstanding that we are dealing here with an independent ray traveling along the plane of separation. Moreover, this fictitious ray possesses a special property: in it the light oscillations take place in the same direction \(X\), i.e. these oscillations are longitudinal.

Although P. Drude believes that this circumstance does not contradict the transverse character of light oscillations, which supposedly obtains only for constant amplitude, it seems to us that this misunderstanding also disappears if one takes into account that \(f_x\) cannot be regarded as an independent ray of light, but only as the projection of rays onto the \(X\) axis; in reality, the rays of light in the second medium have the curved form shown in the drawing (see Fig. 5) by the thick lines; the oscillations in them are strictly transverse.”

Thus Eichenwald’s theoretical investigation gives an exhaustive description of the experimentally discovered phenomenon of the penetration of light into a less dense transparent medium and explains the misunderstanding concerning Focht’s longitudinal light wave.

THE EXPERIMENT OF SCHAEFER AND GROSS

Experimental confirmation of Eichenwald’s theoretical conclusions is found in the work of Schaefer and Gross \([26]\). Schaefer and Gross experimented with electromagnetic waves. This is understandable, since at that time (1910) it was necessary to confirm the electromagnetic nature of light by as large a number as possible of irrefutable experiments. Following Eichenwald, Schaefer and Gross theoretically determined the magnitude of the effects that could be observed in a concrete experiment, the scheme of which is presented in Fig. 6. \(P_1\) and \(P_2\) represent two prisms not in contact with each other, made of paraffin. On the diagonal plane of prism \(P_1\) there occurs total reflection of the incident radiation, which is an electromagnetic wave of length 15 cm. The wave radiator \(E\) is located at the focus of the concave mirror \(H_1\). The beam reflected by the mirror \(H_1\) passes through the diaphragm \(D\) and falls on the face of prism \(P_1\). On the concave mirror \(H_2\),

in the focus of which a thermoelement is located, there will fall a beam reflected by the diagonal plane \(P_1\). But owing to the fact that under total reflection the electromagnetic wave penetrates into the less dense medium to a certain depth, it has the possibility of entering prism \(P_2\), if the diagonal plane of prism \(P_2\) is separated from the diagonal plane of prism \(P_1\) by a distance \(d\), comparable with the depth of penetration. The radiation penetrating into the prism, upon emerging from it, is focused by a mirror onto the thermoelement.

Fig. 6. Diagram of the experiment of Schaefer and Gross.

The distance between the diagonal planes can be varied by the screw \(G\) and determined with the aid of an indicator on the scale \(S_k\).

As we see, Schaefer and Gross, unlike Quincke, did not round off the diagonal plane of the second prism and did not bring it into contact with the first, i.e., did not create those conditions against which Fort27 quite rightly objected, asserting that total reflection is thereby violated. Instead of visible light, Schaefer and Gross used electromagnetic waves, which Righi28 had used for the same purposes, carrying out the experiment in the form in which it had been arranged by Quincke. We shall not dwell on Righi’s experiments, since he was unable to overcome the technical difficulties of this delicate experiment and his results were erroneous. Schaefer and Gross observed transmitted radiation at distances between the diagonal planes up to \(1.2\lambda\).

The dependence of the magnitude of the transmitted radiation on the distance between the diagonal planes is shown in Fig. 7.

Fig. 8 confirms that in the reflected radiation there was lacking precisely that fraction of the radiation incident on the diagonal plane of prism \(P_1\) which, owing to penetration into the second medium, had the possibility of passing into the second prism, i.e., was detected in the transmitted radiation. In Figs. 7 and 8 the dashed lines indicate the theoretically calculated curves, and the separate points—the results of experimental observation.

Fig. 7.

Fig. 8.

Fig. 9.

which, as can be seen, fit the theoretical curve well.

Thus, the solution of the problem of total reflection obtained by Eichenwald was confirmed by the work of Schäfer and Gross.

Moreover, their theoretical calculations show that Eichenwald’s solution makes it possible to carry out specific computations, taking into account the features of the experimental setup.

Schäfer and Gross found the behavior of the energy fluxes for the case when the second medium borders on a third, and thereby experimentally confirmed the correctness of Eichenwald’s theoretical description of the wave field in the less dense medium.

The behavior of the energy flux in total reflection under the conditions of the experiment of Schäfer and Gross, i.e. under conditions when the thickness of the less dense medium is of the same order as the depth of penetration of the wave into it, is shown in Fig. 9.

THE EXPERIMENT OF L. I. MANDELSTAM

In one of L. I. Mandelstam’s papers \(^{29}\) an experiment is described which very clearly demonstrates the penetration of light into the less dense medium at angles of incidence greater than the limiting angle. Indeed, if a glass prism with its diagonal plane is immersed in a solution of a fluorescing substance—fluorescein—and light is made to fall from the glass onto the boundary with the liquid at an angle exceeding the limiting angle, an intense glow of a small volume of the solution is observed under the action of the light penetrating into it.

Fig. 10. Scheme of Mandelstam’s experiment. Under the action of light penetrating into the fluorescein solution, an intense glow of another spectral composition is observed. Filter \(F_2\) is chosen so that it passes only the fluorescence radiation.

The scheme of Mandelstam’s experiment is presented in Fig. 10. It should be noted that this experiment was arranged not to prove the penetration of light into the less dense medium, but to create a light source situated at a distance from the boundary separating the media comparable with the wavelength of light.

Thus, L. I. Mandelstam laid the foundation for the practical application of optical effects in total ref-

scattering. In this connection we shall point to the work of F. S. Baryshanskaya,^30 who investigated the concentration quenching of fluorescence in a layer whose thickness is comparable with the wavelength of light. The arrangement of Baryshanskaya’s experiment is shown in Fig. 11. It is worth noting that the penetration of light into a less dense medium can be successfully used to study the optical properties of the less dense medium in the case where it possesses absorption. This is especially valuable if the liquid under investigation is only slightly transparent in a thin layer. The production of a thin layer is directly associated with technical difficulties. The method of total reflection removes this difficulty. But the investigation of optical properties by this method has not yet received due development.

fluorescence radiation

Fig. 11. Baryshanskaya’s experiment. The fluorescence radiation excited by the light penetrating into it was projected onto the slit of a spectrophotometer.

development. The reason for this should be sought in the fact that no relationship has been established between the reflection coefficients at angles of incidence greater than the limiting angle and the optical constants.

DIFFRACTION AND THE PHENOMENON OF TOTAL REFLECTION

Let us now dwell on one very important question connected with the experimental observation of the phenomenon of total reflection.

In all practical investigations one always deals with light beams bounded by a diaphragm. From this point of view it is of interest to investigate the incidence of a bounded light beam on the interface under conditions of total reflection. In this case, as we shall see below, it is possible to clarify in somewhat greater detail the process of penetration of the wave into the second medium.

Work of this kind was carried out by Picht.^31 Picht considered the total reflection of a cylindrical wave sent by a long luminous filament through a narrow slit of unlimited

length. The luminous filament was assumed to be located at a finite distance from the plane boundary separating two transparent media. In this case Picht had to consider the phenomenon of diffraction caused by the passage of the light wave through a slit. The necessity of such an investigation was insisted upon by Voigt\({}^{37}\) in his polemic with Eichenwald\({}^{32}\) concerning the latter’s theoretical study and the experiments of Schaefer and Gross, in which the diffraction caused by the diaphragm \(D\) (see Fig. 6) in front of the mirror \(H_1\) was not taken into account.

The study of the diffraction pattern, generally speaking—and all the more so in the case of total reflection—presents considerable difficulties of a mathematical nature. However, the physical aspect of the phenomenon can be clarified, as Neter\({}^{33}\) indicated, by comparatively simple mathematical means.

First of all let us point out that Eichenwald’s description of the total reflection of a plane monochromatic wave from a transparent boundary surface is, in the opinion of Picht\({}^{31}\) and Neter\({}^{33}\), insufficient because it does not make it possible to establish at precisely what point of the boundary surface the wave, observed experimentally, penetrates into the less dense medium. Moreover, according to Eichenwald’s investigations, the time average of the energy flux is zero at every point of the boundary plane, i.e., on the average the energy does not penetrate into the second medium. But this contradicts the experimental data, which reveal energy in the second medium, since every optical measurement is ultimately reduced to the determination of the mean value of the energy flux.

The contradiction is removed, as follows from Picht’s work, if one studies the incidence upon the boundary of a light beam with a finite angle of aperture. The physical essence of this situation can be clarified by considering the incidence of two plane waves with two nearby directions of propagation, as is shown in Fig. 12. It is clear from the figure that the two waves arrive at some point \(O\) of the boundary plane (perpendicular to the plane of the figure) with different phases. As a result, the energy-flux vector, perpendicular to the resulting vectors \(\mathbf{E}\) and \(\mathbf{H}\), will be represented by two terms. The time average of one term in this case proves to be equal to zero, but for the other it is different from zero. The sign indicating the direction of the flux vector along the \(X\)-axis depends essentially on the coordinate \(y\) in the plane of separation of the media, within the region of the geometrical contour of the light beam.

Thus one may imagine that, in a beam bounded by a diaphragm, whose angle of aperture is equal to the difference of the angles of incidence of the two waves indicated, energy enters the second medium on one side of the beam axis and leaves it on the other. The beam itself may be represented as a superposition of plane waves,

directions of propagation lie inside the two indicated directions. Scheffer and Pich1 approach the solution of this problem in a different way. They consider a bounded light beam, in whose cross-section the amplitude does not remain constant. They choose the amplitude distribution in the beam cross-section in the form of an isosceles trapezoid. In doing so they find that the energy flux at the boundary of separation enters the less dense medium from one side of the beam, and leaves it from the other. Nevertheless, even with such a solution, it proved impossible to determine the trajectory of the energy in the second medium, from the point of entry into the less dense medium to the point of exit, in any greater detail than had been done by Eichenwald. If the aperture angle of the light beam is made to approach zero, i.e. if one passes to plane waves,

Labels in the figure:
\(\varepsilon < 1\); \(x\); \(y\); \(O\); \(\alpha_1\); \(\alpha_2\); boundary of separation; 1st phase \(t_1^0/c\); 2nd phase \(t_2^0/c\); \(t_1\); \(t_2\); 1st phase \(0\); 2nd phase \(0\).

Fig. 12.

as was shown by Picht, then we arrive at Eichenwald’s results. Consequently, consideration of total reflection of bounded light beams only refines Eichenwald’s investigation, but introduces nothing essentially new in comparison with it.

THE GOOS AND HÄNCHEN EXPERIMENT

In all experiments connected with proving the penetration of light into a second, optically less dense, medium when it is incident under conditions of total reflection, the process of reflection was in fact disturbed. In these experiments, as we have seen, for a wave penetrating

...entering the second medium, conditions are created under which it has no possibility of completely returning from the less dense medium. This, properly speaking, was the basis for the proof of the penetration of light into the less dense medium.

Indeed, in the experiments of Quincke, Schäfer, and Gross, the wave partly penetrates from the second medium through the second boundary into a third. In the experiments of Hall and Mandelstam, the energy brought by the wave into the second medium is spent either on the decomposition of silver bromide, or on exciting the fluorescence of fluorescein. Consequently, in both kinds of experiments the conditions of total reflection were violated by the fact that part of the energy was withdrawn in order to reveal itself in an optically less dense medium.

Fig. 13. Schematic representation of the process of total reflection.

However, as Goos and Hänchen[^35] established, in proving the penetration of light into a less dense medium, the disturbances inherent in the experiments described above can be eliminated. Their investigation is based on the fact that the entry of the wave into the less dense medium and its exit from it into the denser one under total reflection, as was shown by Eichenwald, occur at points separated from one another by some distance depending on the wavelength of the incident light.

This proposition of Goos and Hänchen is illustrated by the schematic Fig. 13, where the plane perpendicular to the plane of the drawing is the interface between the media. If at the point \(Q\), where the ray \(P\) falls under conditions of total reflection, direct reflection occurred, then the reflected ray would be \(R\); but since the light penetrates into the second, optically less dense, medium, travels there some path indicated by the dotted line, and emerges at the point \(S\), the ray that has actually returned will go in the direction \(T\). Thus, the ray in total reflection is displaced. The magnitude of the displacement \(D\) is the subject of investigation in the work of Goos and Hänchen.

The arrangement by which the scheme shown in Fig. 13 can be realized may be imagined as follows. Let us apply to the diagonal plane of a prism \(P\) (Fig. 14) a silver strip \(efgh\). The diagonal plane of the glass prism will thus border, in the narrow strip, on silver, and at all other points on air. If now a beam of light \(PP'\) falls on the diagonal plane under such conditions that at both boundaries, “glass—air” and “glass—silver,” there must be

place of total reflection, the following phenomenon will be observed.

Above the silver strip and below it (i.e. at the glass–air boundary) the penetration of light into the second medium will take place. The light, after traveling some distance in it, will enter the prism again above the points \(S\) and \(S'\), giving the rays \(T\) (we spoke of the ray \(T\) when discussing Fig. 13).

If it is assumed that at the glass–silver boundary direct reflection takes place, i.e. that it occurs between the points \(Q\) and \(Q'\), giving the ray \(R\), then the incident beam \(PP'\), which initially had a rectangular cross-section, upon leaving the prism after total reflection from its diagonal plane, on which the silver strip is deposited, is split and, falling on the photographic film, produces a darkening that reproduces the shape of its cross-section.

Fig. 14.

The shape of the reflected light beam is shown in Fig. 15, on which the magnitude of the displacement between the rays reflected from the glass–air and glass–silver boundaries is also indicated.

The silver strip deposited on the diagonal plane of the prism plays the role of a device that gives us the undisplaced position of the light ray, the so-called zero ray. The assumption made by the authors, that reflection from silver has a direct character, i.e. that it occurs without penetration of the light into the thickness of the silver, may be justified by the fact that the incidence

ON THE QUESTION OF THE TOTAL REFLECTION OF WAVES

of light occurred at an angle somewhat greater than the limiting angle for the glass—air boundary, but for the glass—silver boundary this angle is greater than the limiting angle by an amount of the order of \(30^\circ\). The depth of penetration of light into silver at such angles of incidence becomes considerably less than the wavelength of light. Nevertheless, penetration into the second medium (silver) takes place and must manifest itself as soon as we increase the number of reflections. Therefore the magnitude of the displacement is, in essence, the difference between the displacements in the total reflection of light from the glass—air and glass—silver boundaries.

The magnitude of the displacement obtained by Goos and Hänchen after a single reflection (see Fig. 13) proved insufficient for it to be measured directly. Resorting to multiple total reflection by means of the device shown in Fig. 16, they obtained a resultant displacement accessible to measurement. In Fig. 16, \(Pl\) is a glass plane-parallel plate to which prisms \(P_1\) and \(P_2\) are cemented. After multiple reflection, the ray emerging from prism \(P_2\) produces a darkening on the photographic film which corresponds to the shape of its cross-section. The silver strips on the plane-parallel plate were applied both on one side and on the other. It should be pointed out that Goos and Hänchen could not increase the number of reflections to such an extent that the displacement of the rays became directly visible, since the intensities of the rays \(R\) and \(T\) after several tens of reflections were so different that the darkening caused by the ray \(R\) became imperceptible.

Fig. 15. Displacement of a light beam after total reflection from the diagonal plane of a prism having a silvered strip.

The difference in the intensities of the rays \(R\) and \(T\) can be explained only by the fact that the glass—air boundary is transparent, whereas the glass—silver boundary is opaque, and light penetrating even to an insignificant depth into the thickness of the silver is noticeably absorbed by it during multiple reflection. This circumstance induced Goos and Hänchen to resort to another method of detecting the displacement of a light ray in total reflection.

As follows from the theoretical investigation of the phenomenon of total reflection, which was carried out by Eichenwald, the displacement (in Eichenwald’s terminology it is called the distance between the entry of the ray from the first medium into the second and its exit from the second medium into the first) is proportional to the wavelength of the light incident on the boundary between the media. Relying on this, Goos and Hänchen, in order to avoid the absorption of light upon reflection from silver, which, as we have seen,

violates total reflection, they measured the displacement between rays of different wavelengths. They called this method differential.

Indeed, if, for example, in the radiation incident on the interface there are two waves whose wavelengths are \(\lambda = 0.587\,\mu\) and \(\lambda = 0.437\,\mu\), then after reflection from the interface under conditions of total reflection there will occur between them a displacement \(\Delta D\), which can be measured if the reflection is made multiple. The intensity of both rays will then be approximately the same, and the number of reflections can thereby be made sufficiently

Fig. 16. Owing to multiple reflection, the displacement of the light beam attains considerable dimensions.

large. Goos and Hänchen succeeded in measuring, after multiple reflection, twice the value of this displacement of the rays in the following way.

Let us imagine that, over the height of the rectangular cross section of the light beam \(PP'\) (see Fig. 14) incident on the interface, the rays are distributed in such a way that one color (for example, yellow—\(\lambda = 0.587\,\mu\)) borders another (blue), i.e. the yellow color is located above and below, and the blue in the middle. After reflection there will occur between them a displacement caused by the difference in wavelengths, and the beam emerging from the prism \(P_2\) will have the form indicated in Fig. 17 on the left, where the dotted line bounds the region cor-

corresponding to blue, and a solid line to yellow. If we now interchange the colored rays so that the blue color borders the yellow, we obtain the image shown in Fig. 17 on the right.

Hence it becomes clear that the difference of the measured distances is a doubled displacement caused by the difference of wavelengths in the incident beam, i.e. \(2\Delta D=b-c\), where \(b\) and \(c\) are the distances indicated in Fig. 17. As we see, the differential method makes it possible to establish the fact that light penetrates into the less dense medium under total reflection, without disturbing the reflection process itself.

The magnitude of the displacement, as the measurement results show, depends on the difference of the angles \(\vartheta-\vartheta_g\), where \(\vartheta\) is the angle of incidence and \(\vartheta_g\) is the limiting angle. Thus, for example, if the difference \(\vartheta-\vartheta_g\) is \(3^\circ23',2\), then the displacement \(D\) is equal to \(0.84\,\mu\), while for an angular difference equal to \(0^\circ13',6\), the displacement is \(3.48\,\mu\).

In processing the experimental results, Goos and Hänchen find that the magnitude of the displacement is related to the function

\[ \sqrt{\sin^2 \vartheta-\left(\frac{n_2}{n_1}\right)^2} \]

in the following way:

\[ D=kn_2\,\frac{\lambda_1}{\sqrt{\sin^3 \vartheta-\left(\frac{n_2}{n_1}\right)^2}}, \]

where \(k\) is a constant factor, \(\lambda_1\) is the wavelength in the denser medium, and \(n_2\) is the refractive index of the less dense medium. Since in the expression given all quantities except \(k\) are accessible to direct measurement, the constant \(k\) can be determined empirically:

\[ k=\frac{D}{\lambda_1}\,\frac{1}{n_2}\sqrt{\sin^3 \vartheta-\left(\frac{n_2}{n_1}\right)^2}. \]

The value of the constant \(k\) obtained from this proves to be \(0.52\pm2\%\). It should be noted here that this result was obtained when measuring the displacements of natural light.

Fig. 17. Displacement of the beam in the differential method.

Measurements of the displacements of light polarized in mutually perpendicular planes give the same value of the constant \(k\). Thus, from the experiment of Goos and Hänchen it follows that \(k_{\parallel}=k_{\perp}\), i.e. the displacement does not depend on the position of the plane of polarization of the incident light. This conclusion, following from the experiment of Goos and Hänchen, contradicts the results of the experiments of Quincke and Hall, since it denies the dependence they found of the depth of penetration on the position of the plane of polarization of the incident light.

light. Indeed, if one turns to the equations determining the components of the wave field in the less dense medium, at angles of incidence greater than the limiting angle, one finds that upon incidence on the boundary of separation of light polarized in mutually perpendicular planes, the attenuation of the amplitude, independently of the position of the plane of polarization of the incident light, is determined by one and the same exponential factor:

\[ \exp \left[-\frac{2\pi}{\lambda_1} z \cdot \sqrt{\sin^2 \vartheta-\left(\frac{n_2}{n_1}\right)^2}\right]. \]

The dependence of the exponent on \(\sqrt{\sin^2 \vartheta-\left(\frac{n_2}{n_1}\right)^2}\) allows Goos and Hänchen to express it through the displacement \(D\) and the constant \(k\) in the following form:

\[ \exp \left[2\pi \cdot \frac{k}{D} \cdot n_2 z\right]. \]

It follows from this that the depth of penetration of the light into the less dense medium and the displacement are proportional to each other. In fact, setting the exponent equal to unity, we find that the penetration depth is expressed through the displacement as follows:

\[ z=\frac{1}{2\pi n_2 k}D. \]

But, according to Goos and Hänchen, the displacement \(D\) does not depend on the position of the plane of polarization of the incident light; consequently, the penetration depth also should not depend on it.

INVESTIGATION OF THE ERROR OF GOOS AND HÄNCHEN

The discrepancy between the theory and the experiment of Goos and Hänchen was pointed out by Artmann\(^{36}\). In his investigation Artmann formulates the diffraction problem on the basis of the particular experiment of Goos and Hänchen.

Fig. 18. Scheme of the Goos and Hänchen experiment analyzed by Artmann.

The scheme of the Goos and Hänchen experiment considered by Artmann is presented in Fig. 18. Considering the incident light beam as

superposition of elementary plane waves whose directions of propagation differ little from one another, Artmann finds that the displacement under total reflection must be equal to

\[ D=-\frac{\lambda_1}{2\pi}\frac{d\varphi}{d\vartheta}, \]

where \(\vartheta\) is the angle of incidence, and \(\varphi=\varphi(\vartheta)\) is the phase shift under total reflection.

If, instead of \(\dfrac{d\varphi}{d\vartheta}\), one substitutes here its expression according to the Fresnel formulas, which determine the phase shift under total reflection as a function of the angle of incidence, then we obtain that the displacement depends on the polarization and on the angle of incidence of the incident light in the following way:

\[ D_{E\perp}=\frac{1}{\pi n_1}\frac{n_2\lambda_1}{\sqrt{\sin^2\vartheta-\sin^2\vartheta_g}}; \]

\[ D_{H\perp}=\left(\frac{n_1}{n_2}\right)^2\frac{1}{\pi n_1}\frac{n_2\lambda_1}{\sqrt{\sin^2\vartheta-\sin^2\vartheta_g}}, \]

where \(D_{E\perp}\) is the displacement of the light beam when the vector \(\mathbf{E}\) is perpendicular to the plane of incidence; \(D_{H\perp}\) is the displacement in the case when the vector \(\mathbf{H}\) is perpendicular to the plane of incidence. Comparing Artmann’s theoretical results with what was obtained empirically by Goos and Hänchen, we see that the constants \(k\) for the two positions of the plane of polarization of the incident light differ from one another:

\[ k_{\parallel}=\frac{1}{\pi n_1}, \]

\[ k_{\perp}=\frac{n_1}{\pi n_2^2}. \]

If one substitutes the numerical values of the refractive indices of the media that were taken by Goos and Hänchen, namely: \(n_1=1\), \(n_2=1.52\), then we obtain:

\[ k_{\perp}=0.48;\qquad k_{\parallel}=0.21, \]

whereas in the experiments with polarized light Goos and Hänchen obtained

\[ k_{\parallel}=k_{\perp}=0.52. \]

It should be pointed out, however, that although Artmann solved the diffraction problem posed within the framework of Kirchhoff’s conceptions, he did not succeed in clarifying the cause of the discrepancy between the theory and the experiment of Goos and Hänchen.

Artmann believes the failure itself to consist in the fact that the description of diffraction according to Kirchhoff does not touch upon the question of the interaction of the edge of the diaphragm and the wave passing by it. Artmann succeeded in considering this interaction in another paper,\(^{37}\) where he specially examines the influence of a diaphragm of finite thickness on the pro-

...wave passing by it. It turns out in this case that the interaction of the edge of the diaphragm depends substantially on the position of the plane of polarization of the passing wave.

If one makes use of the results of Schaefer and Pich, then without particular difficulty, as Frahnstein showed ^38, one can obtain an expression for the displacement of a light beam under total reflection. If one leaves aside the polemic between Artmann ^39, on the one hand, and Schaefer and Frahnstein ^40, on the other, then their theoretical results, obtained by different methods, agree with one another. This is indisputable proof that the experiments of Goos and Hänchen with polarized light are erroneous.

The necessity of repeated experiments became obvious. Repeated experiments were carried out by Goos and Hilda Lindberg-Hänchen ^41. The results of these experiments, performed quite recently, are in full agreement with theory.

ON THE POLARIZATION OF THE REFLECTED RAY

The investigation of the polarization of light reflected from the interface of two transparent media under total reflection presented no difficulties.

Fresnel had already established that reflected light is elliptically polarized. He also proposed a prism in passing through which linearly polarized light becomes circularly polarized.

A special investigation of the polarization of reflected light was carried out by Frölich ^42, whose results are in full agreement with theory.

REFLECTION OF A PULSE

In the experiments and theoretical investigations considered above, the process of reflection of a light beam from an interface formed by transparent media was considered stationarily.

In this case, as Eichenwald showed, the reflection is total. The matter is different if one considers a reflection process that pertains to a finite interval of time. Planck ^43 had already pointed out that the solution of the problem of total reflection would be exhaustive only if processes pertaining to a finite interval of time were considered.

Turning to the study of the reflection process over a finite interval of time, we are compelled to consider the incidence of a wave pulse. Fisher’s work ^44 is devoted to the investigation of the reflection of a pulse from a transparent interface.

Fisher established that the reflection of a wave pulse, incident...

does not become complete when incident on a transparent interface at an angle greater than the limiting angle.

This peculiarity is due to the fact that in this case we have a process limited in time, during which the wave packet has time to spread out. In his work Fischer used the \(\zeta\)-function, by means of which he succeeded in constructing a wave impulse.

BRIEF CONCLUSIONS

In the total reflection of light from an interface formed by transparent media, a light field exists in the optically less dense medium.

The existence of the light field in the less dense medium has been indisputably proved by a whole series of experiments, differing from one another, carried out at different times by various investigators.

The results of experiments which detect light in the less dense medium are in complete agreement with the theory of this phenomenon, which was given by A. A. Eichenwald on the basis of electromagnetic conceptions of the nature of light.

The penetration of light into the second medium under total reflection is not associated with a loss of energy. The energy flux in the reflected wave is equal to the energy flux of the incident wave. And this result, established experimentally, is in complete agreement with theory, since the reflection process in question is stationary. In the case where a wave impulse is reflected, as theory shows, the reflection is not complete. When a bounded light beam is reflected, a displacement is observed. The magnitude of the displacement depends on the angle of incidence, the wavelength, and the polarization of the incident light.

From a theoretical consideration of the reflection of a plane unbounded light wave from a plane boundary formed by transparent media, it follows that in the second medium the field decays along the normal to the interface. In this case the exponent determining the decay is the same for two mutually perpendicular polarizations of the incident wave. Hence it follows that the depth of penetration of light into the less dense medium, defined as the distance over which the field decreases by a factor of \(e\), should not depend on the polarization of the incident light. But the results of measuring the depth of penetration of light into the less dense medium, on the contrary, revealed this dependence.

This discrepancy between experiment and theory is easily explained, for in the experiments bounded light beams were used, whereas the theory refers to a plane unbounded wave.

But even so, a peculiarity in the course of the curves of the penetration depth of polarized light as a function of ...

...on the angle of incidence, to which we pointed when describing the experiments of Quincke and Hall. This question remains open for the time being.

Proceeding from the fact that the depth of penetration of light into the less dense medium is determined by the same factors as the displacement, it should be supposed that the depth of penetration, just like the displacement, depends on the polarization and on the angle of incidence.

The present review is not entirely exhaustive, since otherwise it would inevitably have proved too extensive. But we have tried to illuminate in detail all the major works relating to the question of total reflection. At the same time, we are able properly to assess the contribution made by Russian scientists. The work of Professor A. A. Eichenwald of Moscow University not only gave the correct theoretical solution, but also indicated the paths along which the further study of this phenomenon should proceed. The work of Professor of Moscow University, Academician L. I. Mandelstam, pointed to the possibility of using this subtle physical phenomenon for practical purposes.

In conclusion I express my gratitude to Professor S. N. Rzhevkin for his critical remarks and advice.

REFERENCES CITED

1. Newton, Optics, Book 2, Part I, Obs. 8, p. 160, 1927.
2. Eichenwald, Journal of the Russian Physico-Chemical Society (physical section) 41, 131 (1909).
3. Newton, Optics, Book 2, Part I, Obs. 1, p. 151, 1927.
4. Newton, Optics, Book 3, Part I, Obs. 2, p. 152, Question 29, p. 289, 1927.
5. Newton, Mathematical Principles of Natural Philosophy, Book 1, Pref. 96, p. 253, 1935.
6. Huygens, Traité de la Lumière, p. 38 (1690).
7. Young, Lectures on Nat. Phil. I, 461 (1807); II, 623 (1807); Phil. Trans. 12 (1801).
8. Verdet, Oeuvres, IV, pp. 430–563.
9. Biot, Traité de physique III, pp. 276–290 (1816).
10. Fresnel, Ann. d. Chim. et Phys. 23, 130 (1823), 29, 175 (1825), 46, 263 (1831).
11. Babinet, Comptes Rendus 8, 709 (1839).
12. Stoekes, Phil. Trans. 8, 642 (1849).
13. Billet, Ann. de Chim. 64, 410 (1862).
14. Quincke, Pogg. Ann. 127, 1 (1866).
15. Voigt, Gött. Nachr. 2, 49 (1884).
16. Voigt, Wied. Ann. 67, 185 (1899).
17. Lebedev, Selected Works, pp. 66, 151, 195, 1949.
18. K. A. Timiryazev, Science and Democracy, p. 48, 1920.
19. Ketteler, Wied. Ann. 67, 879 (1899).
20. Ditscheiner, Wien Ber. (2) 60, 584 (1870).
21. Exner, Wien. Ber. IIa, 98, 98 (1889).
22. Fdser and Senior, Phil. Mag. 4, 346 (1902).

1. Wulff, Physical Optics, p. 450, 1936.
2. Hall, Phys. Rev. 15, 73 (1902).
3. Drude, Winkelman’s Handbuch d. Ph. 6, 1275 (1906).
4. Schaefer und Gross, Ann. d. Phys. 32, 648 (1910).
5. Voigt, Ann. d. Phys. 34, 797 (1911).
6. Righi, Optik d. Elektr. Schwing., p. 161, 1898.
7. Mandelstam, Complete Collected Works, vol. 1, p. 261, 1948.
8. Baryshanskaya, Dokl. Akad. Nauk 17, 3 (1937).
9. Picht, Ann. d. Phys. 3, 433 (1929).
10. Eichenwald, Ann. d. Phys. 35, 1035 (1911).
11. Noether, Ann. d. Phys. 11, 141 (1931).
12. Schaefer und Pich, Ann. d. Phys. 30, 245 (1937).
13. Goos und Hänchen, Ann. d. Phys. 1, 333 (1947).
14. Artman, Ann. d. Phys. 2, 87 (1948).
15. Artman, Zeits. f. Phys. 127, 468 (1950).
16. Fragstein, Ann. d. Phys. 4, 271 (1949).
17. Artman, Ann. d. Phys. 7, 209 (1950).
18. Schaefer und Fragstein, Ann. d. Phys. 6, 39 (1949).
19. Goos und Hilda Lindberg-Hänchen, Ann. d. Phys. 5, 251 (1949).
20. Fröhlich, Ann. d. Phys. 63, 900 (1920).
21. Planck, Introduction to Theoretical Physics, vol. IV, p. 30, 1934.
22. Fischer, Ann. d. Phys. 2, 211 (1948).

1. Reference number as printed on the page. ↩