ON MAGNETO-OPTICAL PHENOMENA IN FERROMAGNETS

(Macroscopic Theory)

A. V. Sokolov

The theory of magneto-optical phenomena was developed in its time by Goldhammer, Drude, Focht1, and others.

A shortcoming of these older works is that their authors did not draw a sufficiently sharp distinction between phenomenological theory and model classical theories, which have long since become obsolete. It is therefore quite expedient to purge the phenomenological theory of magneto-optical phenomena of model representations and to present it clearly and accessibly for a broad circle of physicists. In the present work the most general formal macroscopic description of magneto-optical phenomena is carried out without the introduction of any microscopic models.

1. GENERAL INFORMATION ON THE FARADAY AND KERR PHENOMENA IN FERROMAGNETS

Magneto-optical phenomena in ferromagnetic metals, according to the method of observation, may be divided into two main groups: 1) the Faraday effect and 2) the Kerr effect.

By the Faraday effect is meant the rotation of the plane of polarization and the simultaneous appearance of ellipticity when initially linearly polarized light passes through thin magnetized films of a ferromagnetic metal.

By the Kerr effect is meant the influence of the magnetization of a ferromagnetic mirror on the state of polarization of light reflected from its surface. The influence of magnetization on the reflected light depends essentially on the geometrical arrangement of the mirror surface, the plane of incidence, and the direction of the magnetization vector I. The general case reduces to three principal ones, distinguished by the arrangement of the magnetization vector, namely: 1) polar magnetization, in which the magnetization vector is perpendicular to the plane of the mirror but parallel to the plane of incidence; 2) meridional magnetization, in which the vector

A. V. SOKOLOV

magnetization parallel both to the surface of the mirror and to the plane of incidence; 3) equatorial magnetization, in which the magnetization vector is parallel to the surface of the mirror but perpendicular to the plane of incidence (Fig. 1).

Accordingly, one distinguishes the polar, meridional, and equatorial Kerr effects. In order to clarify the essential difference between arrangements I and II, it is useful to consider the case in which, in the incident ray, the oscillations of the electric vector occur parallel or perpendicular to the plane of incidence. Then, according to the laws

Fig. 1.

of ordinary metal optics (without magnetization), this state of oscillation remains unchanged upon reflection.

If the magnetization is directed as in I, then, in addition to the rectilinear oscillation of the electrons in the metal caused by the electric vector of the incident light, the Lorentz force gives rise to another oscillation perpendicular to it.

Accordingly, in the secondary wave there also arises a normal component, called the Kerr component, which, by the laws of superposition of oscillations, causes an elliptical oscillation of the reflected light (which without magnetization should be linear). With meridional magnetization and normal incidence there is no effect, since the magnetic field has rotational symmetry. With oblique incidence of light this effect exists. With equatorial magnetization, likewise on the basis of symmetry, no effect occurs if, in the incident light, the oscillations of the electric vector occur parallel or perpendicular to the plane of incidence. At other angles of the plane of oscillation of the incident light relative to the planes \(P\) and \(S\) (\(P\) and \(S\) are the planes respectively parallel and perpendicular to the plane of incidence), changes occur in the amplitude and phase of the reflected light owing to the magnetization, and a corresponding effect is observed. We shall not dwell on descriptions of the experimental setups used for detecting the effects under consideration, and refer those interested to reviews on this question² and to the original papers.

Experimental study of the phenomena under consideration has shown that the magnitude of the angle of rotation \(\alpha\) and the degree of ellipticity \((B/A)\) in both cases, under given external conditions (temperature, frequency of light, etc.), are proportional to the technical magnetization \(I\) of the ferromagnetic specimen, and not to the magnitude of the external magnetic field. Thus, for example, for the angle of rotation when light is transmitted we have:

\[ \alpha_{\Phi}=K_{\Phi} l I, \tag{1,1} \]

and in the case of the polar effect under normal reflection

\[ \alpha_{\kappa}=K_{\kappa} I, \tag{1,2} \]

where \(l\) is the thickness of the specimen being traversed by the light, \(K_{\Phi}\) is the Schegliaev–Kundt constant and \(K_{\kappa}\) is the Kerr constant. These constants, as experiment shows, depend on the frequency of the light (dispersion) and on the temperature.

The magneto-optical Kerr effect has been found only in ferromagnets, and it has been established that, in addition to special magnetic properties, a necessary condition for its existence is the presence of absorption of light, i.e. a complex refractive index of the substance. As a rule, for a given material the two effects exhibit opposite signs of rotation. Nevertheless, the dispersion curves in both effects, especially in pure metals, reveal a quite analogous course. The curves of rotatory dispersion measured by Du Bois\(^3\) and, in the infrared region of the spectrum, by Ingersoll\({}^{4,5}\), show for iron and cobalt

Fig. 2.

a sufficiently uniform course without sharp maxima or minima. On the contrary, nickel, as well as many ferromagnetic compounds and alloys,\(^6\) show sharply pronounced maxima and minima on the curves of the dependence of rotation on frequency, and even a change in the sign of the rotation. In particular, Ingersoll, and somewhat later Klitting, found that the Kerr effect in nickel in the infrared region gives a positive rotation, whereas in the visible region of the spectrum the rotation is negative (as for iron and cobalt). The position of the zero point is, for the Kerr effect, at \(\lambda_1=1.5\mu\) \((\nu_1=2\cdot10^{-14}\ \mathrm{sec}^{-1})\), and for the Faraday effect at \(\lambda_1=1\mu\) \((\nu_1=3\cdot10^{14}\ \mathrm{sec}^{-1})\) (Fig. 2). Klitting also

It has been established that the position of the zero point, within the accuracy of the measurements, does not depend on temperature and field strength. Du Bois, studying the Kerr effect on a heated nickel mirror, found that the rotation disappears at the Curie point. Measurements of this phenomenon on nickel were also carried out by Hirsch[^7]. M. M. Noskov[^12] conducted experiments on nickel in an atmosphere of hydrogen and showed that the data of the above-mentioned authors are inaccurate, namely: the rotation is preserved even at temperatures several degrees above the Curie point, together with the so-called “true magnetization.” Martin[^6] investigated the temperature dependence of the Kerr rotation in many ferromagnetic compounds and, in every case in which the mirror permitted heating up to the Curie point, established that the rotation decreases with increasing temperature and disappears at the Curie point. An unambiguous connection with magnetization was established experimentally by M. M. Noskov[^12] on alloys of nickel with copper of different concentrations. The proportionality of the Kerr rotation to magnetization is the basis for the application of magneto-optical methods to the study of the magnetic behavior of ferromagnetic substances. These methods also deserve attention because they make it possible to determine the absolute magnetization from optical measurements. A mirror is made from a thin plate of the material under investigation, it is magnetized, and the polar Kerr rotation is measured as a function of the field strength. Between the magnetization \(I\) of the plate, whose demagnetizing factor is \(N = 4\pi\), and the field strength \(H_e\), there exists the relation

\[ I = \frac{\varkappa}{1 + 4\pi \varkappa} H_e . \tag{1,3} \]

In the region of small fields the susceptibility \(\varkappa\) is considerably greater than unity; as a consequence, \(I\), and also the Kerr rotation \(\alpha_k\), are proportional to the external field \(H_e\):

\[ I = \frac{1}{4\pi} H_e \quad \text{and} \quad \alpha_k = K_k I = \frac{1}{4\pi} K_k H_e . \tag{1,4} \]

From the measured values of the rotation \(\alpha_k\) at definite field strengths in this region, the Kerr constant for the material of the mirror is obtained. From the rotation at saturation \(\alpha_\infty\), the saturation magnetization \(I_\infty\) is then obtained by dividing the former quantity by the Kerr constant determined earlier. In this way Du Bois determined the magnetization curve and the saturation magnetization of magnetite. The saturation magnetization according to Du Bois can also be obtained directly from the graphical representation of the Kerr rotation as a function of field strength. The abscissa of the point of intersection of the straight line

\[ \alpha = \frac{K_k}{4\pi} H_e \]

with the asymptote \(\alpha_\infty = K_k I_\infty = \mathrm{const}\) is equal to \(H_B = 4\pi I_\infty\). Fig. 3 illustrates the above in a clear way.

In this way Barker[^8] and Loria[^8] succeeded in showing that the magneto-optical determination of the saturation magnetization agrees very well with the magnetically measured values of \(I_\infty\). These results justify—

find application of the method described to mirrors made of other materials, whose magnetic saturation is unknown. When comparing magneto-optical methods with magnetic ones, it must be remembered that magnetic measurements give the average magnetization of a certain volume of substance, whereas magneto-optical measurements give conclusions about the state of the substance in a thin surface layer from which the reflection of light takes place. In addition to the general experimental difficulties of measuring a very small rotation of the plane of polarization, there are also

Fig. 3.

fundamental difficulties, which have their basis in the imperfection of the surface of the mirror.

Owing to the presence of cracks and pores in the material, the surface of the mirror is divided into small areas, whose demagnetizing factor is less than \(4\pi\). The assumption used in the calculation is then no longer satisfied, and therefore a smaller value is obtained for the saturation magnetization. The above-described feature of the Kerr effect, which makes it possible to obtain information about the state of the surface of a body, suggests that measurements of the Kerr effect may be useful in solving metallographic questions concerning the surface of the machined part.

It is interesting in this connection to note a very recent work \(^{13}\), in which observation of ferromagnetic regions was carried out by means of the Kerr effect on surfaces perpendicular and inclined to the hexagonal axis of cobalt. If the surface intersects the hexagonal axis, then every domain extending to the surface forms a north or south magnetic pole. Linearly polarized light incident perpendicular to the surface \((0001)\) undergoes a rotation of approximately \(1/2^\circ\) upon reflection; the rotation is positive or negative depending on the polarity of the reflecting region.

Using an optical setup consisting of a polarizer, analyzer, and compensator, one can obtain a group of domains of some

polarities, giving darkness, whereas the other group is illuminated; consequently, observation of domains is permissible. It should be noted that, although it is comparatively easy to obtain photographs of patterns of polarized light, visual observation is impossible because of the very low intensity of the light and insufficient contrast.

The condition of the surface of the specimen under investigation has a substantial influence on the magnitude of the Kerr effect.

The rotation of the plane of polarization upon reflection of light from a ferromagnetic mirror magnetized along the normal can change appreciably in magnitude if the metal borders on a medium optically denser than air or vacuum.

M. M. Noskov\(^9\) experimentally showed that a dielectric deposited on the surface of a steel mirror in the form of a film with a thickness of the order of the wavelength of visible light causes the angle of rotation of the plane of polarization to increase severalfold; moreover, \(\alpha/\alpha_0\)—the relative increase of this angle—depends on the thickness of the film and has a first sharp maximum at a thickness of the order of one quarter of the wavelength.

Ya. I. Frenkel\(^ {10}\) proposed a theory of this phenomenon, according to which the change of \(\alpha/\alpha_0\) with the thickness of the film should have a strictly periodic character, without damping, with maxima at thicknesses \(d_m=(2s+1)\lambda/4\). The maximum value of \(\alpha/\alpha_0\) should be equal to the square of the refractive index of the substance of the film.

Further investigations by M. M. Noskov\(^ {11}\) showed that absorbing films (iron oxide, silver iodide, etc.) deviate appreciably from Ya. I. Frenkel’s theory. For absorbing films the values of \(\alpha/\alpha_0\) at the first maxima prove to be considerably larger than follows from Ya. I. Frenkel’s formula \((\alpha_m/\alpha_0=n^2)\). As for the subsequent course of the curves of the dependence of the angle of rotation on the thickness of the film, then, in addition to damping, sharp violations of the law of simple periodicity are observed: a change in the sign of the effect in the case of silver iodide and, in particular, enormous rotations of the opposite sign for specimens with iron-oxide films. A more general theory of the dependence of the magneto-optical Kerr effect on the thickness and optical constants of thin films was constructed by M. M. Noskov and the author\(^ {12}\).

2. RELATION BETWEEN THE AMPLITUDES OF THE INCIDENT AND REFLECTED WAVES IN THE CASE OF A MAGNETIZED FERROMAGNET. MINIMUM AND ZERO ROTATION OF THE ANALYZER AND POLARIZER AND THE LAW OF RECIPROCITY

Between the complex amplitudes of the electric vector \(A_p\) and \(A_s\) of the incident wave and the complex amplitudes \(R_p\) and \(R_s\) of the wave reflected from a magnetized ferromagnet, there exist the relations\(^1\):

\[ \left. \begin{aligned} R_s&=r_{11}A_s+r_{12}A_p,\\ R_p&=r_{21}A_s+r_{22}A_p. \end{aligned} \right\} \tag{2,1} \]

Here \(p\) and \(s\) are two mutually perpendicular directions, namely: parallel and perpendicular to the plane of incidence. The parameters \(r_{ik}\), generally speaking, are complex quantities. Relations (2.1) do not depend on the particular form of the theory and are permissible in all cases when, within each of the media, and also at the boundary between them, linear conditions hold for the components of the oscillations in the light wave. Since the linearity of these equations is based on the experimental fact of the superposition of waves, the above relations may be regarded as requirements following from experiment. Equations (2.1) should be viewed as a generalization of the well-known Fresnel formulas. The usual Fresnel reflection formulas (without allowance for the influence of magnetization) may be written in the following form:

\[ \begin{pmatrix} R_s\\ R_p \end{pmatrix} = \begin{pmatrix} r_{11} & 0\\ 0 & r_{22} \end{pmatrix} \begin{pmatrix} A_s\\ A_p \end{pmatrix}. \tag{2.2} \]

The generalized reflection formulas (with allowance for the influence of magnetization) may, in accordance with (2.1), be written as follows:

\[ \begin{pmatrix} R_s\\ R_p \end{pmatrix} = \begin{pmatrix} r_{11} & r_{12}\\ r_{21} & r_{22} \end{pmatrix} \begin{pmatrix} A_s\\ A_p \end{pmatrix}. \tag{2.3} \]

From a comparison of (2.2) and (2.3) it follows directly that only those components of the reflection matrix which have different subscripts depend on the magnetization*), whereas the components with identical subscripts must be independent of the field. When the magnetization is equal to zero, \(r_{12}\) and \(r_{21}\) vanish, and, consequently, the generalized reflection formulas (2.3) go over into the Fresnel reflection formulas (2.2). Thus, we may say that, together with the appearance of magnetization in the body, Kerr components \(r_{12}A_p\) and \(r_{21}A_s\) arise in the reflected light. The absolute values \(|r_{12}|\) and \(|r_{21}|\) are a measure of the ratio of the amplitudes of the oscillations of either of the two Kerr components to the amplitude of the incident wave that produced it. The quantities \(\left|\dfrac{r_{12}}{r_{11}}\right|\) and \(\left|\dfrac{r_{21}}{r_{22}}\right|\) are, analogously, a measure of their ratio to the corresponding amplitude of the ordinarily reflected component. The tangents of the phase-shift angle of the Kerr component relative to the oscillation component that produced it are determined by the expressions

\[ -\frac{\operatorname{Im}(r_{12})}{\operatorname{Re}(r_{12})} \quad \text{and} \quad -\frac{\operatorname{Im}(r_{21})}{\operatorname{Re}(r_{21})}, \tag{2.4} \]

where \(\operatorname{Im}\) denotes the imaginary part, and \(\operatorname{Re}\) the real part of \(r_{ik}\).

* Strictly speaking, the diagonal elements of the reflection matrix also depend on the magnetization, but this dependence is so small that it may be neglected.

The tangents of the phase-shift angle of the Kerr component relative to the ordinarily reflected component are determined by the expressions:

\[ -\frac{\operatorname{Im}\left(\dfrac{r_{12}}{r_{11}}\right)} {\operatorname{Re}\left(\dfrac{r_{12}}{r_{11}}\right)} \quad \text{and} \quad -\frac{\operatorname{Im}\left(\dfrac{r_{21}}{r_{22}}\right)} {\operatorname{Re}\left(\dfrac{r_{21}}{r_{22}}\right)} . \tag{2,5} \]

a) Minimum—rotation of the analyzer and ellipticity

Among the directly observed quantities in the measurement of the Kerr effect, the minimum-rotation of the analyzer and polarizer play a special role. If the reflected light, as is considered in the general case, is elliptically polarized, then the direction of the oscillations transmitted by the analyzer that gives the least intensity is, obviously, parallel to the minor axis of the oscillation ellipse, while the major axis of the ellipse is perpendicular to the direction of the oscillations transmitted by the analyzer \(A\). If the reflected oscillations are represented in the form

\[ R_p = Fe^{if}, \qquad R_s = Ge^{ig}, \tag{2,6} \]

where \(F\) and \(G\) are real amplitudes, \(f\) and \(g\) are phase constants, then the axis of the oscillation ellipse forms with the \(p\)-direction an angle \(\varphi\), determined by the relation\({}^{14}\)

\[ \tg 2\varphi = \frac{2FG}{F^2-G^2}\cos(f-g), \tag{2,7} \]

where the two values of this angle correspond to the directions of the major and minor axes of the oscillation ellipse. If the oscillations of the incident light occur in the plane of incidence \((A_s=0)\), then in the reflected light the major axis forms a small angle \(\chi_p\) with the \(p\)-direction and, since in this case \(F\) is large in comparison with \(G\), it follows from (2,6) and (2,7) that

\[ \chi_p = \frac{G}{F}\cos(f-g) = \operatorname{Re}\left(\frac{R_s}{R_p}\right) = \operatorname{Re}\left(\frac{r_{12}}{r_{22}}\right), \tag{2,8} \]

The analyzer must be turned by this same angle from the \(s\)-direction in order to transmit only the linear oscillations corresponding to the minor axis, i.e. to establish the minimum intensity of the reflected light. Conversely, if in the incident light the direction of oscillations is parallel to the \(s\)-direction \((A_p=0)\), then \(G\) is large in comparison with \(F\), and the major axis of the reflected oscillations forms with the \(s\)-direction a small angle \(\chi_s\), equal to

\[ \chi_s = -\frac{F}{G}\cos(f-g) = -\operatorname{Re}\left(\frac{R_p}{R_s}\right) = -\operatorname{Re}\left(\frac{r_{21}}{r_{11}}\right). \tag{2,9} \]

To obtain a minimum of intensity, the analyzer must be rotated from the position parallel to the \(p\)-direction through the same angle. Thus, formulas (2.8) and (2.9) determine the minimum rotation of the analyzer, or they determine the angles through which the analyzer must be turned from the position perpendicular to the \(p\)-direction (respectively perpendicular to the \(s\)-direction) in order to obtain the minimum intensity of the reflected light. To determine the ellipticity \(\dfrac{B}{A}=\operatorname{tg}\vartheta\), we shall start from the well-known formula\({}^{14}\)

\[ \sin 2\vartheta=\pm \frac{2FG}{F^2+G^2}\sin(f-g) =\pm \frac{2\dfrac{G}{F}}{1+\dfrac{G^2}{F^2}}\sin(f-g). \tag{2,10} \]

In the first case (\(A_s=0\)) we assume \(G\ll F\) and, consequently, the quantity \(\dfrac{G^2}{F^2}\) may be neglected in comparison with unity; then we shall have:

\[ \frac{B}{A}=\operatorname{tg}\vartheta\simeq \sin\vartheta =\pm \frac{G}{F}\sin(f-g) =\pm \operatorname{Im}\left(\frac{R_s}{R_p}\right) = \]

\[ =\pm \operatorname{Im}\left(\frac{r_{12}}{r_{22}}\right). \tag{2,11} \]

Similarly, for the second case (\(A_p=0\)) we obtain:

\[ \frac{B}{A}=\pm \operatorname{Im}\left(\frac{r_{21}}{r_{11}}\right). \tag{2,12} \]

b) Minimum rotation of the polarizer and the reciprocity law

Let us now consider the case in which linearly polarized light is incident at an angle \(\chi^0\) to the \(p\)-axis, and let us determine \(\chi^0\) so that the absolute values \(R_s\) or \(R_p\) take the smallest value. These two values of \(\chi^0\) represent the minimum rotation of the polarizer. If

\[ A_p=A\cos\chi^0,\qquad A_s=A\sin\chi^0, \tag{2,13} \]

where \(A\) is the real amplitude of the incident oscillation, then the oscillations of the reflected light may be written in the following form:

\[ R=A(r\sin\chi^0+r'\cos\chi^0). \tag{2,14} \]

Putting further \(r=re^{i\varphi}\) and \(r'=r'e^{i\varphi'}\), we obtain:

\[ R=Ae^{i\varphi}\left(r\sin\chi^0+r'e^{i(\varphi'-\varphi)}\cos\chi^0\right). \tag{2,15} \]

The minimum of \(R\) is determined by the minimum of the quantity

\[ \left|r\sin\chi^0+r'e^{i(\varphi'-\varphi)}\cos\chi^0\right|. \]

Since this expression in expanded form has the form:

\[ r^2\sin^2\chi^0 + r'^2\cos^2\chi^0 + 2rr'\cos(\rho' - \rho)\sin\chi^0\cos\chi^0, \]

the quantity \(|R|\) assumes a minimum value for the value of \(\chi^0\) satisfying the condition

\[ \tg 2\chi^0 = \frac{2rr'}{r'^2 - r^2}\cos(\rho' - \rho). \tag{2,16} \]

This formula is similar to formula (2,7) and must be investigated in the same way. In doing so, \(R\) should be considered once as \(R_s\), and a second time as \(R_p\). If we require that \(|R_s|\) have a minimum, then the angle \(\chi^0\) must be small and equal to \(\chi_p^0\). At the same time one should put \(r=r_{11}\), \(r'=r_{12}\), whence it follows that \(r'\) is small compared with \(r\). Formula (2,16) then gives

\[ \chi_p^0 = -\frac{r'}{r}\cos(\rho' - \rho) = -\operatorname{Re}\left(\frac{r_{12}}{r_{11}}\right). \tag{2,17} \]

If we require that \(|R_p|\) have a minimum, then it must be

\[ \chi^0 = \frac{\pi}{2} + \chi_s^0 \]

(where \(\chi_s^0\) is a small angle), and at the same time one should put \(r=r_{21}\), \(r'=r_{22}\), and therefore \(r\) is small compared with \(r'\). In this case formula (2,16) gives

\[ \chi_s^0 = \frac{r}{r'}\cos(\rho-\rho') = \operatorname{Re}\left(\frac{r_{21}}{r_{22}}\right). \tag{2,18} \]

Putting \(r_{12}=r_{21}\) (polar effect) and comparing (2,8) and (2,18) and, respectively, (2,9) and (2,17), we obtain the law of mutual reciprocity for the polar effect, namely:

\[ \chi_s=\chi_p^0 \quad \text{and} \quad \chi_p=\chi_s^0. \tag{2,19} \]

Putting \(r_{12}=-r_{21}\) (meridional effect) and comparing the same formulas, we obtain the law of mutual reciprocity for the meridional effect, namely:

\[ \chi_p^0=-\chi_s \quad \text{and} \quad \chi_s^0=-\chi_p. \tag{2,20} \]

c) Formulas for null-rotation of the analyzer and polarizer

One can always choose such a position of the polarizer near the \(s\)- or \(p\)-direction at which the reflected light, which is, generally speaking, elliptically polarized, will be linearly polarized and, consequently, can be completely extinguished by the analyzer. This is the so-called null-rotation, requiring both rotation

of the analyzer as well as of the polarizer. In experiment, zero settings are possible only for very small angles of rotation; therefore, at the same time the quantities \(\left(\dfrac{R_s}{R_p}\right)\) and \(\left(\dfrac{A_s}{A_p}\right)\), or \(\left(\dfrac{R_p}{R_s}\right)\) and \(\left(\dfrac{A_p}{A_s}\right)\), are small numbers.

In the first case, relations (2.1), neglecting quantities of the second order, may be written as:

\[ R_s=A_s r_{11}+A_p r_{12} \quad \text{and} \quad R_p=r_{22}A_p \]

or

\[ r_{22}\frac{R_s}{R_p}=\left(r_{11}\frac{A_s}{A_p}+r_{12}\right). \tag{2.21} \]

Let the incident light be linearly polarized and make an angle \(\psi_p^0\) with the \(p\)-direction; then \(\dfrac{A_s}{A_p}=\psi_p^0\). Since the reflected light is completely extinguished by the analyzer, it must also be linearly polarized, and, accordingly, we may put \(\dfrac{R_s}{R_p}=\psi_p\), where \(\psi_p\) is the angle determining the position of the plane of oscillations of the reflected light relative to the \(p\)-direction. Taking these considerations into account, formula (2.21) may be written in the following form:

\[ r_{22}\psi_p=r_{11}\psi_p^0+r_{12}; \tag{2.22} \]

because of the complex nature of \(r_{ik}\), (2.22) represents two equations for determining \(\psi_p\) and \(\psi_p^0\). In the second case, relations (2.1) take the form:

\[ R_s=A_s r_{11} \quad \text{and} \quad R_p=r_{21}A_s+r_{22}A_p \]

or

\[ r_{11}\frac{R_p}{R_s}=\left(r_{21}+r_{22}\frac{A_p}{A_s}\right). \tag{2.23} \]

Putting, as before, \(\dfrac{A_p}{A_s}=-\psi_s^0\) and \(\dfrac{R_p}{R_s}=-\psi_s\), we obtain from (2.23):

\[ r_{11}\psi_s=r_{22}\psi_s^0-r_{21}. \tag{2.24} \]

If \(r_{12}=\pm r_{21}\), formulas (2.22) and (2.24) again lead to the law of reciprocity for zero rotations. Taking equation (2.22) and its complex conjugate,

\[ r_{22}^{*}\psi_p=r_{11}^{*}\psi_p^0+r_{12}^{*} \]

and, solving them simultaneously, we obtain:

\[ \psi_p= \frac{r_{11}^{*}r_{12}-r_{11}r_{12}^{*}} {r_{11}^{*}r_{22}-r_{11}r_{22}^{*}} = \frac{\operatorname{Im}\left(\dfrac{r_{12}}{r_{11}}\right)} {\operatorname{Im}\left(\dfrac{r_{22}}{r_{11}}\right)}, \tag{2,25} \]

\[ \psi_p^{0}= \frac{r_{12}^{*}r_{12}-r_{12}^{*}r_{22}} {r_{11}^{*}r_{22}-r_{11}r_{22}^{*}} = -\frac{\operatorname{Im}\left(\dfrac{r_{12}}{r_{22}}\right)} {\operatorname{Im}\left(\dfrac{r_{11}}{r_{22}}\right)}. \tag{2,26} \]

Carrying out analogous calculations with equation (2,24) and its conjugate, we obtain:

\[ \psi_s= -\frac{\operatorname{Im}\left(\dfrac{r_{21}}{r_{22}}\right)} {\operatorname{Im}\left(\dfrac{r_{11}}{r_{22}}\right)}, \tag{2,27} \]

\[ \psi_s^{0}= \frac{\operatorname{Im}\left(\dfrac{r_{21}}{r_{11}}\right)} {\operatorname{Im}\left(\dfrac{r_{22}}{r_{11}}\right)}. \tag{2,28} \]

3. FUNDAMENTALS OF THE MACROSCOPIC DESCRIPTION OF MAGNETO-OPTICAL PHENOMENA IN FERROMAGNETS[^15]

Since Kerr’s discovery of the rotation of the plane of polarization of light upon its reflection from a magnetized ferromagnetic mirror, a number of theories have been proposed to explain this effect1. In all these theories, certain classical microscopic models were explicitly used, on the basis of which the relations for magneto-optical phenomena were derived—in particular, for the description of the Kerr effect. However, precisely in the case of ferromagnets, the classical electron theory is untenable even from an illustrative point of view, since the very phenomenon of ferromagnetism is a purely quantum effect. As was recently shown[^16], only quantum-mechanical theory makes it possible to obtain a correct microscopic description of the magneto-optical Kerr phenomenon. Nevertheless, it can be shown that the most general formal macroscopic description of magneto-optical phenomena does not require the use of any microscopic models. An attempt to avoid invoking the classical electron theory in establishing the basic phenomenological relations of magneto-optics was made by Darwin2. However, Darwin’s work cannot be regarded as a definitive and complete solution of this question, if only because it is restricted to consideration solely of the normal incidence of light under polar magnetization of a ferromagnetic mirror. In this paragraph, a path is indicated for a general phenomenological description of magneto-optical phenomena without invoking micro-

scopic models. For the macroscopic description of magneto-optical effects in ferromagnets one should proceed from the general differential equations of the electromagnetic field

\[ \frac{1}{c}\frac{\partial \mathbf H}{\partial t}=-\operatorname{rot}\mathbf E, \tag{3.1} \]

\[ \frac{1}{c}\frac{\partial \mathbf D}{\partial t}=\operatorname{rot}\mathbf H, \tag{3.2} \]

and the tensor equation

\[ \mathbf D=\varepsilon \mathbf E, \tag{3.3} \]

where \(\varepsilon\) is the dielectric tensor of the magnetized ferromagnet. It should be noted that in macroscopic crystal optics, for the description of light phenomena occurring in crystals, the same system of equations (3.1), (3.2), and (3.3) is used. The dielectric tensor for transparent substances is Hermitian, which can easily be shown by using the law of conservation of energy. The components of the tensor \(\varepsilon\) for transparent crystals are real quantities, whereas for absorbing media they are, generally speaking, complex, and the imaginary part of them must be negative.

The dielectric tensor of a magnetized ferromagnet is constructed on the basis of the theory of forced anisotropy. An isotropic magnetized medium may be regarded, optically, as a birefringent crystal whose optical properties are determined by the dielectric tensor. Without loss of generality it may be assumed that the magnetization vector (the residual magnetization or the magnetization “induced” by an external field) is directed along the \(z\)-axis. All planes passing through this distinguished direction \(z\) are equivalent to one another. Hence it follows that the dielectric tensor of a magnetized ferromagnet must possess cylindrical symmetry, i.e.,

\[ \begin{aligned} \varepsilon_{xx}&=\varepsilon_{yy}=\varepsilon,\\ \varepsilon_{xy}&=-\varepsilon_{yx}=-i\varepsilon Q,\\ \varepsilon_{xz}&=\varepsilon_{yz}=\varepsilon_{zx}=\varepsilon_{zy}=0,\\ \varepsilon_{zz}&=\varepsilon_0, \end{aligned} \tag{3.4} \]

where \(Q\) is the magneto-optical parameter, generally speaking a complex quantity depending on the magnetization of the body. When the magnetization is equal to zero, the off-diagonal elements of the tensor vanish, and the diagonal elements become identical, \(\varepsilon=\varepsilon_0\), i.e. the body becomes optically completely isotropic. Substituting (3.4) into (3.3), we shall have:

\[ \begin{aligned} D_x&=\varepsilon E_x-i\varepsilon Q E_y,\\ D_y&=i\varepsilon Q E_x+\varepsilon E_y,\\ D_z&=\varepsilon_0 E_z. \end{aligned} \tag{3.5} \]

The simultaneous solution of (3.1), (3.2), and (3.5), using the boundary conditions, gives a description of all magneto-optical effects and, in particular, of the Kerr and Faraday effects in ferromagnets. The diagonal components of the dielectric tensor of a ferromagnet, \(\varepsilon_{xx}=\varepsilon_{yy}=\varepsilon\), are functions of the magneto-optical parameter \(Q\). Experiment shows that this parameter is small \((Q \ll 1)\); therefore the function \(\varepsilon(Q)\) can be expanded in a power series in \(Q\) and one may restrict oneself to the first terms of the expansion, i.e.,

\[ \varepsilon(Q)=\varepsilon_0+\left(\frac{\partial \varepsilon}{\partial Q}\right)_{Q=0}Q+ \frac{1}{2}\left(\frac{\partial^2\varepsilon}{\partial Q^2}\right)_{Q=0}Q^2. \tag{3.6} \]

When the direction of magnetization is reversed, expression (3.6) takes the form:

\[ \varepsilon(-Q)=\varepsilon_0-\left(\frac{\partial \varepsilon}{\partial Q}\right)_{Q=0}Q+ \frac{1}{2}\left(\frac{\partial^2\varepsilon}{\partial Q^2}\right)_{Q=0}Q^2. \tag{3.7} \]

Furthermore, we have \(\varepsilon(Q)=\varepsilon(-Q)\), since the physical properties of a body must not change when the direction of magnetization is reversed. Comparing (3.6) and (3.7), we obtain:

\[ \varepsilon(Q)=\varepsilon_0+\frac{1}{2}\left(\frac{\partial^2\varepsilon}{\partial Q^2}\right)_{Q=0}Q^2. \tag{3.8} \]

Let us introduce such a multiplier \(f\) that its product with \(\varepsilon_0\) gives us the value of the coefficient before \(Q^2\) in (3.8), i.e.,

\[ \frac{1}{2}\left(\frac{\partial^2\varepsilon}{\partial Q^2}\right)_{Q=0}=\varepsilon_0 f. \tag{3.9} \]

Then

\[ \varepsilon=\varepsilon_0+\varepsilon_0 fQ^2. \tag{3.10} \]

The description of the polar Kerr effect at normal incidence can be obtained with the aid of equations (3.1), (3.2), and (3.5), whereas for the description of the polar effect at oblique incidence, and also for the meridional and equatorial effects, one must use the additional condition (3.10), obtained here as opposed to Voigt, without any reference to the classical electron theory.

4. PROPAGATION OF A PLANE INHOMOGENEOUS WAVE INSIDE A FERROMAGNETIC METAL

It is known that by a plane homogeneous wave one understands a wave whose amplitude has a constant value in the wave plane and changes in the direction of propagation. Such a wave is written analytically in the form \(Fe^{i\omega\tau}\), where \(\omega\) is the angular frequency,

\[ \tau=t-\frac{\alpha x+\beta y+\gamma z}{v}, \]

and

\[ v=\frac{V}{1-i\chi} \]

is the complex velocity—

...propagation velocity, \(V\) is the material velocity, \(\chi\) is the quantity associated with the absorption coefficient by the relation \(\chi n = k\), and \(\alpha,\beta,\gamma\) are the direction cosines of the wave normal. In what follows we shall have to use an inhomogeneous wave, whose amplitude varies in the wave plane, while the plane of equal phases makes with the plane of equal amplitudes a certain angle, which is the material angle of refraction. The phase of such an inhomogeneous wave may be written in the form

\[ \omega \tau = \omega \left( t - \frac{\alpha_1 x + \beta_1 y + \gamma_1 z - i \chi(\alpha_2 x + \beta_2 y + \gamma_2 z)}{V} \right), \tag{4,1} \]

where \(\alpha_1,\beta_1,\gamma_1\) are the direction cosines of the normal to the plane of constant phases, and \(\alpha_2,\beta_2,\gamma_2\) are the direction cosines of the normal to the plane of constant amplitudes. Thus, the inhomogeneous plane wave will have the form

\[ F e^{i\omega \tau} = F e^{-\frac{\omega \chi}{V}(\alpha_2 x+\beta_2 y+\gamma_2 z)} e^{i\omega\left(t-\frac{\alpha_1 x+\beta_1 y+\gamma_1 z}{V}\right)} . \tag{4,2} \]

This way of writing an inhomogeneous plane wave clearly shows that the plane of equal phases

\[ \alpha_1 x + \beta_1 y + \gamma_1 z = \mathrm{const} \]

and the plane of equal amplitudes

\[ \alpha_2 x + \beta_2 y + \gamma_2 z = \mathrm{const}, \]

generally speaking, do not coincide. If we introduce the abbreviated notation

\[ (\alpha_1-i\chi\alpha_2)^2+(\beta_1-i\chi\beta_2)^2+(\gamma_1-i\chi\gamma_2)^2 = \]

\[ = 1-\chi^2-2i\chi(\alpha_1\alpha_2+\beta_1\beta_2+\gamma_1\gamma_2)=R, \]

\[ \frac{\alpha_1-i\chi\alpha_2}{R}=\alpha^*,\qquad \frac{\beta_1-i\chi\beta_2}{R}=\beta^*,\qquad \frac{\gamma_1-i\chi\gamma_2}{R}=\gamma^* \quad \text{and} \quad \frac{V}{R}=v^*, \]

then \(\alpha^{*2}+\beta^{*2}+\gamma^{*2}=1\), and the phase of a plane inhomogeneous wave can be represented in the form

\[ \omega\tau^*=\omega\left(t-\frac{\alpha^*x+\beta^*y+\gamma^*z}{v^*}\right). \tag{4,3} \]

In the preceding section we established that, in order to describe magneto-optical effects in ferromagnets, one should start from the general differential equations of the electromagnetic field (3,1), (3,2), and (3,5). Taking \(\operatorname{rot}\) of (3,1) and using (3,2), we obtain:

\[ c^2(\Delta E-\operatorname{grad}\operatorname{div} E)=\frac{\partial^2 D}{\partial t^2}. \tag{4,4} \]

Substituting (3.5) into (4.4), we shall have:

\[ \begin{aligned} c^2(\Delta E_x-\operatorname{grad}_x \operatorname{div}\mathbf E)&=\varepsilon \ddot E_x-i\varepsilon Q\ddot E_y,\\ c^2(\Delta E_y-\operatorname{grad}_y \operatorname{div}\mathbf E)&=\varepsilon \ddot E_y+i\varepsilon Q\ddot E_x,\\ c^2(\Delta E_z-\operatorname{grad}_z \operatorname{div}\mathbf E)&=\varepsilon_0 \ddot E_z. \end{aligned} \tag{4.5} \]

Taking the solution in the general form \(E=E_0 e^{i\omega \tau^*}\) and putting \(\dfrac{c}{v^*}=\mathfrak n^*\), where \(\mathfrak n^*\) is the complex refractive index of the magnetized ferromagnetic, we obtain:

\[ \begin{aligned} \varepsilon E_x-i\varepsilon Q E_y &=\mathfrak n^{*2}\left[E_x-\alpha^*(\alpha^*E_x+\beta^*E_y+\gamma^*E_z)\right],\\ i\varepsilon Q E_x+\varepsilon E_y &=\mathfrak n^{*2}\left[E_y-\beta^*(\alpha^*E_x+\beta^*E_y+\gamma^*E_z)\right],\\ \varepsilon_0 E_z &=\mathfrak n^{*2}\left[E_z-\gamma^*(\alpha^*E_x+\beta^*E_y+\gamma^*E_z)\right]. \end{aligned} \tag{4.6} \]

Eliminating \(E_x, E_y\), and \(E_z\), we obtain an equation for \(\mathfrak n^{*2}\):

\[ \left[\varepsilon(\alpha^{*2}+\beta^{*2})+\varepsilon_0\gamma^{*2}\right]\mathfrak n^{*4} -\left[\varepsilon^2(1-Q^2)(\alpha^{*2}+\beta^{*2}) +\varepsilon\varepsilon_0(1+\gamma^{*2})\right]\mathfrak n^{*2} +\varepsilon_0\varepsilon^2(1-Q^2)=0. \tag{4.7} \]

This equation is quadratic with respect to \(\mathfrak n^{*2}\), and therefore in any two mutually opposite directions two waves propagate with pairwise equal velocities. Multiplying equations (4.6), respectively, by \(\alpha^*, \beta^*, \gamma^*\) and adding, we obtain:

\[ (\varepsilon\alpha^*+i\varepsilon Q\beta^*)E_x +(\varepsilon\beta^*-i\varepsilon Q\alpha^*)E_y +\varepsilon_0\gamma^*E_z=0. \tag{4.8} \]

From this equation it is seen that the complex direction of the oscillations of the electric field \(\mathbf E\) and the wave normal \(\mathbf S^*(\alpha^*,\beta^*,\gamma^*)\) are not perpendicular to one another. If the wave is homogeneous, then \(\alpha^*, \beta^*, \gamma^*\) are real and equal to \(\alpha,\beta,\gamma\). The relation \((\mathbf E\cdot \mathbf S)=0\) should also express the transverse character of the oscillations for the complex solution \(E_x,E_y,E_z\). Since it is not satisfied, the oscillations of the electric field even in a homogeneous wave are not transverse. The magnetic field of a light wave behaves differently. From equations (3.2), for a plane inhomogeneous wave there follows the relation \(H_x\alpha^*+H_y\beta^*+H_z\gamma^*=0\), expressing “complex transversality.” Passing to a homogeneous wave, i.e., replacing \(\alpha^*,\beta^*,\gamma^*\) by \(\alpha,\beta,\gamma\), we obtain real transversality.

5. MAGNETIC ROTATION AND ELLIPTICITY IN THE PASSAGE OF LIGHT THROUGH THIN LAYERS OF A FERROMAGNETIC (FARADAY EFFECT)

For longitudinal propagation of light one should put \(\alpha^*=\beta^*=0\) and \(\gamma^*=1\), and then equations (4.6) give:

\[ D^\pm=D_x\pm iD_y=\varepsilon(1\mp Q)(E_x\pm iE_y) =\mathfrak n^{*2}(E_x\pm iE_y), \tag{5.1} \]

\[ D_z=\varepsilon_0E_z=0. \]

Hence we obtain two solutions:

\[ \begin{aligned} & n_+^{*2}=\varepsilon(1-Q) \quad \text{for} \quad E_x=+iE_y,\\ & n_-^{*2}=\varepsilon(1+Q) \quad \text{for} \quad E_x=-iE_y . \end{aligned} \tag{5,2} \]

Parallel to the lines of force there propagate two transverse circularly polarized waves; the refractive index \(n_-^*\) corresponds to the right-polarized wave, and \(n_+^*\) to the left-polarized wave. If one takes into account that the refractive indices \(n_+^*\) and \(n_-^*\) differ from each other, and also from the refractive index \(n\) in the absence of magnetization, only by a small amount, and that the magneto-optical parameter \(Q\) is considerably less than unity, then from (5,2), neglecting small quantities of second order, it follows that

\[ n_-^* - n_+^* = nQ, \tag{5,3} \]

which, together with the relation \(n_+^*+n_-^*=2n\), gives

\[ n_\pm^*=n\left(1\mp\frac{1}{2}Q\right). \tag{5,4} \]

From (5,4) it also follows that:

\[ \left. \begin{aligned} n-n_+^*&=\frac{nQ}{2}=G,\\ n-n_-^*&=-\frac{nQ}{2}=-G . \end{aligned} \right\} \tag{5,5} \]

For transverse propagation of light (for example, parallel to the \(x\)-axis), \(\alpha^*=1\), \(\beta^*=\gamma^*=0\), and then (4,7) gives

\[ (n^{*2}-\varepsilon_0)\,[n^{*2}-\varepsilon(1-Q^2)]=0, \]

or

\[ n_z^{*2}=\varepsilon_0,\qquad n_\perp^{*2}=\varepsilon(1-Q^2). \tag{5,5′} \]

The first relation (5,5′) represents the refractive index of a light wave in which the oscillations of the electric vector occur parallel to the lines of force, whereas the second corresponds to a wave with oscillations perpendicular to them.

Let us now consider a plane-parallel layer of a ferromagnetic metal of thickness \(l\), and suppose that the \(z\)-axis and the direction of magnetization are perpendicular to the plate. If linearly polarized light is incident in the direction of magnetization, while the \(x\)-axis is parallel to the direction of oscillation and lies in the plane of the plate, then the amplitudes of the light oscillations in this plane are equal to:

\[ D_x=1,\quad D_y=0,\ \text{i.e.}\ D^+=1,\ D^-=1. \tag{5,6} \]

Upon emerging from the plate the amplitudes will be equal to:

\[ \left. \begin{aligned} D^{+}&=e^{-ikn_{+}^{*}l}\\ D^{-}&=e^{-ikn_{-}^{*}l}, \end{aligned} \right\} \tag{5,7} \]

where \(k=\dfrac{2\pi}{\lambda}\).

Since

\[ D_x=\frac{D^{+}+D^{-}}{2} \quad\text{and}\quad D_y=\frac{D^{+}-D^{-}}{2i}, \]

then, taking (5,7) into account, we shall have:

\[ \begin{aligned} D_x &=\frac{1}{2}\left(e^{-ikn_{+}^{*}l}+e^{-ikn_{-}^{*}l}\right)=\\ &=\frac{1}{2}e^{-iknl}\left[e^{ik(-n_{+}^{*})l}+e^{+ik(n-n_{-}^{*})l}\right]=\\ &=\frac{1}{2}e^{-iknl}\left(e^{ikGl}+e^{-ikGl}\right)=\\ &=\frac{1}{2}e^{-iknl}\left[e^{ik(G_1-iG_2)l}+e^{-ik(G_1-iG_2)l}\right]=\\ &=\frac{1}{2}e^{-iknl}\left[e^{ikG_1l}e^{kG_2l}+e^{-ikG_1l}e^{-kG_2l}\right]. \end{aligned} \tag{5,8} \]

Introducing the abbreviated notation

\[ \left. \begin{aligned} v&=kG_1l\\ u&=kG_2l, \end{aligned} \right\} \tag{5,9} \]

we can express \(D_x\) and \(D_y\) in the following form:

\[ D_x=e^{-iknl}\operatorname{ch}(u+iv),\qquad D_y=-ie^{-iknl}\operatorname{sh}(u+iv). \tag{5,10} \]

The ratio of the amplitudes

\[ \frac{D_y}{D_x}=-i\,\operatorname{th}(u+iv) \tag{5,11} \]

thus turns out to be a complex quantity, and the linearly polarized light wave that has passed through the layer of ferromagnetic metal becomes elliptically polarized. The fact that the light becomes elliptically polarized can be shown by direct calculations, using the following formulas from the theory of hyperbolic functions:

\[ \left. \begin{aligned} \operatorname{sh}(u+iv)&=a_2e^{i\delta_2},\\ \operatorname{ch}(u+iv)&=a_1e^{i\delta_1},\\ \operatorname{th}(u+iv)&=\frac{a_2}{a_1}e^{i\delta}. \end{aligned} \right\} \tag{5,12} \]

where:

\[ \left. \begin{aligned} a_2&=\sqrt{-\frac{1}{2}(\operatorname{ch}2u-\cos 2v)},& \operatorname{tg}\delta_1&=\frac{\operatorname{tg}v}{\operatorname{th}u},\\ a_1&=\sqrt{\frac{1}{2}(\operatorname{ch}2u+\cos 2v)},& \operatorname{tg}\delta_2&=\operatorname{th}u\,\operatorname{tg}v,\\ \delta&=\delta_2-\delta_1,& \operatorname{tg}\delta&=\frac{\sin 2v}{\operatorname{sh}2u}. \end{aligned} \right\} \tag{5.13} \]

Taking (5.12) into account and introducing the time factor \(e^{i\omega t}\), we obtain:

\[ D_x=a_1 e^{i(\omega t-knz+\delta_1)},\qquad D_y=a_2 e^{i\left(\omega t-knz+\delta_2-\frac{\pi}{2}\right)} . \tag{5.14} \]

In the standard way it can be shown that the components \(D_x\) and \(D_y\) satisfy the equation

\[ \frac{D_x^2}{a_1^2}+\frac{D_y^2}{a_2^2} -2\frac{D_xD_y}{a_1a_2}\sin\hat{\delta} =\cos^2\hat{\delta}, \tag{5.15} \]

where

\[ \hat{\delta}=\delta_2-\delta_1-\frac{\pi}{2}. \]

But this is the equation of an ellipse lying in the plane \(xy\), the center of which coincides with the origin of coordinates, but whose axes do not coincide with the coordinate axes. The angle \(\psi\) through which the major axis of the ellipse is rotated relative to the initial direction of oscillations (the \(x\)-axis) can be found by using the formula

\[ \operatorname{tg}2\psi=\frac{2a_1a_2}{a_1^2-a_2^2}\sin\hat{\delta}. \tag{5.16} \]

Substituting from (5.13) \(a_1\), \(a_2\) and \(\sin\hat{\delta}\) into (5.16) (since \(\operatorname{tg}\delta=\dfrac{\sin 2v}{\operatorname{sh}2u}\), then

\[ \sin\hat{\delta}=\frac{\sin 2v}{\sqrt{\operatorname{sh}^2 2u+\sin^2 2v}} \quad\text{and}\quad \cos\hat{\delta}=\frac{\operatorname{sh}2u}{\sqrt{\operatorname{sh}^2 2u+\sin^2 2v}}), \]

we obtain:

\[ \operatorname{tg}2\psi= \sqrt{\frac{\operatorname{ch}^2 2u-\cos^2 2v} {\operatorname{sh}^2 2u+\sin^2 2v}}\, \operatorname{tg}2v. \]

Since the root is equal to unity, we obtain:

\[ \operatorname{tg}2\psi=\operatorname{tg}2v \]

or

\[ \psi=v=\varkappa G_1 l. \]

Thus, the angle of rotation of the major axis of the ellipse relative to the initial direction of oscillations will be

\[ \alpha_{\Phi}=\psi=v=\operatorname{Re}(kGl)=\operatorname{Re}\frac{\pi}{\lambda}(n_-^*-n_+^*)\,l. \tag{5.17} \]

Let us proceed to the determination of the ellipticity. As is known, the ellipticity is determined by the relation \(\dfrac{B}{A}=\operatorname{tg}\vartheta\).

Starting from the formula \(\sin 2\vartheta=\pm \sin 2\alpha\cdot \cos\delta\) (for \(\delta_2-\delta_1-\dfrac{\pi}{2}=\delta\)), one can show that

\[ \sin 2\vartheta=\pm \operatorname{th}2u . \tag{5,18} \]

Indeed, this is seen from the following transformations:

\[ \sin 2\vartheta=\pm 2\sin\alpha\cos\alpha\, \frac{\operatorname{sh}2u}{\sqrt{\operatorname{sh}^{2}2u+\sin^{2}2v}} = \]

\[ =\pm \frac{2\operatorname{tg}\alpha}{\sqrt{1-\operatorname{tg}^{2}\alpha}}\cdot \frac{1}{\sqrt{1+\operatorname{tg}^{2}\alpha}}\cdot \frac{\operatorname{sh}2u}{\sqrt{\operatorname{sh}^{2}2u+\sin^{2}2v}} = \]

\[ =\pm \frac{2a_{2}}{a_{1}\left(1+\dfrac{a_{2}^{2}}{a_{1}^{2}}\right)} \cdot \frac{\operatorname{sh}2u}{\sqrt{\operatorname{sh}^{2}2u+\sin^{2}2v}} = \]

\[ =\pm \frac{2a_{1}a_{2}}{a_{1}^{2}+a_{2}^{2}}\cdot \frac{\operatorname{sh}2u}{\sqrt{\operatorname{sh}^{2}2u+\sin^{2}2v}} = \]

\[ =\pm \frac{\sqrt{\operatorname{ch}^{2}2u-\cos^{2}2v}}{\operatorname{ch}2u}\cdot \frac{\operatorname{sh}2u}{\sqrt{\operatorname{sh}^{2}2u+\sin^{2}2v}} = \]

\[ =\pm \sqrt{\frac{\operatorname{ch}^{2}2u-\cos^{2}2v}{\operatorname{sh}^{2}2u+\sin^{2}2v}} =\pm \operatorname{th}2u . \]

Relation (5,18) is easily reduced to the form

\[ \frac{2\operatorname{tg}\vartheta}{1+\operatorname{tg}^{2}\vartheta} = \pm \frac{2\operatorname{th}u}{1+\operatorname{th}^{2}u}. \tag{5,19} \]

This relation is a quadratic equation with respect to \(\operatorname{tg}\vartheta\), i.e., with respect to the ellipticity. If we take the sign \(+\), then we obtain two roots

\[ \operatorname{tg}\vartheta_{1}=\frac{1-\operatorname{th}^{4}u}{2\operatorname{th}u} \quad \text{and} \quad \operatorname{tg}\vartheta_{2}=\operatorname{th}u, \]

whereas if we take the sign \(-\), then we obtain:

\[ \operatorname{tg}\vartheta_{3}=-\operatorname{th}u \quad \text{and} \quad \operatorname{tg}\vartheta_{4}=-\operatorname{cth}u. \]

Thus, the ellipticity is equal to:

\[ \left(\frac{B}{A}\right)_{\Phi} = \pm \operatorname{th} u = \pm \operatorname{th}\operatorname{Im}\frac{\pi}{\lambda} \left(n_-^{*}-n_+^{*}\right)l \]

\[ = \pm \operatorname{th}\frac{\pi}{\lambda} \left(n_-\chi_- - n_+\chi_+\right)l . \tag{5,20} \]

The ellipticity in the Faraday effect is thus the result of a change, in a magnetic field, of the transparency of the metal for oppositely circularly polarized components of the light wave (circular magnetic dichroism). Assuming, as usual, \(n=n-ik\) and \(Q=|Q|e^{-iq}\), and using relation (5,3), the rotation and ellipticity in the Faraday effect can be written in the form

\[ \psi=a_{\Phi} = \pm \frac{\pi l}{\lambda}|Q|\,(n\cos q-k\sin q), \tag{5,21} \]

\[ \left(\frac{B}{A}\right)_{\Phi} = \pm \frac{\pi l}{\lambda}Q\,(n\sin q+k\cos q). \tag{5,22} \]

To conclude this paragraph, we shall describe a brief method for determining the rotation and ellipticity in the Faraday effect. Relation (5,11) may be represented in the following form:

\[ \frac{D_y}{D_x} = -i\operatorname{th}(u+iv) = \frac{a_2}{a_1}e^{\,i\left(\delta-\frac{\pi}{2}\right)} = \frac{a_2}{a_1}\sin\delta - i\frac{a_2}{a_1}\cos\delta . \tag{5,23} \]

The real and imaginary parts of the amplitude ratio can be written as:

\[ \operatorname{Re}\left(\frac{D_y}{D_x}\right) = \operatorname{Re}\left(-\operatorname{th}(u+iv)\right) = \frac{a_2}{a_1}\sin\delta = \frac{\sin 2v}{\operatorname{ch}2u+\cos 2v}, \tag{5,24} \]

\[ \operatorname{Im}\left(\frac{D_y}{D_x}\right) = \operatorname{Im}\left(-i\operatorname{th}(u+iv)\right) = -\frac{a_2}{a_1}\cos\delta = -\frac{\operatorname{sh}2u}{\operatorname{ch}2u+\cos 2v}. \tag{5,25} \]

Let us note that the rotation is connected with the variation of the real part of the refractive index, i.e. with the quantity

\[ v=k\,\frac{n_- - n_+}{2}\,l . \]

The ellipticity, on the other hand, is caused by absorption, i.e. by the quantity

\[ u=kG_zl = \frac{\pi l}{\lambda} \left(n_-\chi_- - n_+\chi_+\right). \]

Therefore, putting \(u=0\) in (5,24), we obtain:

\[ \operatorname{Re}\left(\frac{D_y}{D_x}\right)_{u=0} = \frac{\sin 2v}{1+\cos 2v} = \operatorname{tg}v = \operatorname{tg}\operatorname{Re}\frac{\pi l}{\lambda} \left(n_-^{*}-n_+^{*}\right). \]

Hence formula (5,17) follows directly. Putting in (5,25)

\(v=0\), we obtain the ellipticity

\[ \frac{B}{A}=\operatorname{Im}\left(\frac{D_y}{D_x}\right)_{v=0} =-\frac{\operatorname{sh}2u}{\operatorname{ch}2u+1} =-\operatorname{th}u . \]

6. THE POLAR KERR EFFECT AT NORMAL INCIDENCE OF LIGHT

In the important case of perpendicular incidence of light and polar magnetization (in this case the measurements are the most reliable, since the disturbing influence of the surface film is minimal), the formulas for the minimum rotation and ellipticity can be obtained very simply2. In this case the usual formula for normal reflection is applicable to each circular component, so that the reflected light is a superposition of circular waves with amplitudes

\[ \frac{n_- -1}{n_- +1}, \quad \text{and} \quad \frac{n_+ -1}{n_+ +1}. \]

The circular amplitudes of the reflected light can be represented in the form

\[ \begin{aligned} E^- &= E_x+iE_y,\\ E^+ &= E_x-iE_y. \end{aligned} \tag{6,1} \]

From (6,1) it follows directly that

\[ \frac{E_y}{E_x} =\frac{1}{i}\frac{E^- - E^+}{E^- + E^+} =\frac{1}{i}\, \frac{ \dfrac{n_- -1}{n_- +1}-\dfrac{n_+ -1}{n_+ +1} }{ \dfrac{n_- -1}{n_- +1}+\dfrac{n_+ -1}{n_+ +1} }. \tag{6,2} \]

After simple transformations we obtain:

\[ \operatorname{tg}\alpha'_{\mathrm{k}} =\frac{E_y}{E_x} =-i\,\frac{n_- - n_+}{n_+n_- -1}. \tag{6,3} \]

Since the ratio of the amplitudes is a complex quantity, the oscillation after reflection from a magnetized ferromagnet is transformed from linear into elliptical, and the major axis is rotated through a certain angle relative to the original direction of oscillation of the linearly polarized light. Without loss of generality, \(\alpha'_{\mathrm{k}}\) may be regarded as a small quantity, and (6,3) may be written in the form

\[ \alpha'_{\mathrm{k}} =-i\,\frac{n_- - n_+}{n_+n_- -1}. \tag{6,4} \]

The real angle of rotation \(\alpha'_{\mathrm{k}}\) of the major axis of the ellipse of oscillations

It follows from this, as the result of the variation of the imaginary part of the refractive index of a magnetized ferromagnet, which is nonzero only in absorbing bodies,

\[ \alpha_k=\operatorname{Re}(a'_k)=-\operatorname{Im}\frac{n_- - n_+}{n_+ n_- -1}. \tag{6,5} \]

The ellipticity, which is the result of the variation of the real part of the refractive index, will be

\[ \left(\frac{B}{A}\right)_k=\operatorname{Im}(a'_k)=-\operatorname{Re}\frac{n_- - n_+}{n_+ n_- -1}. \tag{6,6} \]

A very interesting conclusion is obtained when these formulas are applied to a transparent medium (dielectric). In this case \(n_-\) and \(n_+\) are real, and from (6,5) and (6,6) it follows that \(\alpha_k=0\), but \(\left(\dfrac{B}{A}\right)_k\ne 0\). Thus the reflected light undergoes no rotation at all of the plane of polarization, but acquires an ellipticity with its major axis along the direction of the plane of polarization of the incident beam. An effect of this kind has not yet been observed, since it is vanishingly small. Indeed, for heavy flint in a field \(H=10^4\) oersted, \(\dfrac{B}{A}\sim 10^{-6}\).

In order to express the rotation and ellipticity in terms of the magneto-optical parameter \(Q\), we could use formula (6,4). However, for the purposes of the subsequent discussion we shall take another route \(^{2,18}\). We shall again start from circular vibrations, which in complex notation are represented in the form of the relation

\[ A_p=\mp iA_s. \tag{6,7} \]

The reflected \(\mp\) circular waves will then have amplitudes

\[ \begin{aligned} R_{s\pm}&=A_s(r_{11}\mp ir_{12}),\\ R_{p\pm}&=A_p(r_{22}\mp ir_{21}), \end{aligned} \left\} \tag{6,8} \right. \]

as is readily verified by substituting (6,7) into (2,1). Further, we have:

\[ \left(\frac{R_s}{A_s}\right)_{\pm}=r_{11}\mp ir_{12}. \tag{6,9} \]

In the absence of magnetization the ordinary law of metallic reflection is valid,

\[ \left(\frac{R_s}{A_s}\right)_0=r_{11}=-r_{22}=\frac{1-n}{1+n}. \tag{6,10} \]

If, however, the ferromagnet is magnetized, then one must put

\[ \left(\frac{R_s}{A_s}\right)_{\pm}=\frac{1-n_{\pm}}{1+n_{\pm}}. \tag{6,11} \]

Substituting (5.4) into (6.11), we shall have:

\[ r_{11}\pm i r_{12}= \frac{1-n(1\mp Q/2)}{1+n(1\mp Q/2)} =\frac{1-n}{1+n}\pm \frac{nQ}{(1+n)^2} \tag{6.12} \]

and, consequently,

\[ r_{12}=r_{21}=-\frac{i n Q}{(1+n)^2}. \tag{6.13} \]

Further, with the aid of (6.13) and (6.10), we find:

\[ \frac{r_{12}}{r_{22}}=-\frac{r_{12}}{r_{11}}= \frac{i n Q}{1-n^2} \tag{6.14} \]

and, neglecting unity in comparison with \(n^2\),

\[ \frac{r_{12}}{r_{22}}=\frac{Q(\varkappa-i)}{n(1+\varkappa^2)}. \tag{6.15} \]

In the same approximation \([n^2(1+\varkappa^2)\gg 1]\), the formulas of ordinary metal optics for the principal azimuth \(\Theta\) and the principal angle of incidence \(\Phi\) are valid:

\[ \varkappa=\operatorname{tg}\Theta \quad \text{and} \quad \sin\Phi\,\operatorname{tg}\Phi=n\sqrt{1+\varkappa^2}. \tag{6.16} \]

From equations (6.16) and \(Q=|Q|e^{-iq}\) it then follows that

\[ \frac{r_{12}}{r_{22}}= \frac{|Q|e^{-i\left(q+\frac{\pi}{2}-2\Theta\right)}}{\sin\Phi\,\operatorname{tg}\Phi} =\zeta e^{-i\delta}. \tag{6.17} \]

The ratio \(\zeta\) of the real amplitude of the Kerr component to the amplitude of the ordinarily reflected light will therefore be

\[ \zeta=\frac{|Q|}{\sin\Phi\,\operatorname{tg}\Phi} \tag{6.18} \]

and the phase lag \(\delta\) of the first relative to the second is

\[ \delta=q+\frac{\pi}{2}-2\Theta. \tag{6.19} \]

Further, the minimum rotation and the ratio of the semiaxes of the vibration ellipse of the reflected light will be

\[ \delta_k=\operatorname{Re}\left(\frac{r_{12}}{r_{22}}\right) =\zeta\cos\delta = \frac{|Q|\cos\left(q+\frac{\pi}{2}-2\Theta\right)} {\sin\Phi\,\operatorname{tg}\Phi}, \tag{6.20} \]

\[ \left(\frac{B}{A}\right)_k =\pm\operatorname{Im}\left(\frac{r_{12}}{r_{22}}\right) =\mp\zeta\sin\delta =\mp \frac{|Q|\sin\left(q+\frac{\pi}{2}-2\Theta\right)} {\sin\Phi\,\operatorname{tg}\Phi}. \tag{6.21} \]

7. THE POLAR KERR EFFECT FOR OBLIQUE INCIDENCE OF LIGHT

In order to consider the general case of the problem of magneto-optical phenomena, we would have to solve equation (4.7). However, the general very complicated result becomes relatively simple if we restrict ourselves to terms of first order with respect to the magneto-optical parameter \(Q\). Substituting (3.10) into (4.7), after simple but rather cumbersome transformations, we obtain:

\[ \mathbf n^{*2}=\varepsilon_0\left[1\mp \gamma^{*}Q+\left(f-\frac{1}{2}(1+f)(1-\gamma^{*2})Q^2\right)\right] \tag{7.1} \]

and also

\[ \mathbf n^{*2}=\varepsilon\left[1\mp \gamma^{*}Q-\frac{1}{2}(1+f)(1-\gamma^{*2})Q^2\right]. \tag{7.2} \]

The double sign refers, as before, to the \(\pm\) circular waves. Neglecting terms of second order with respect to \(Q\), and also taking \(\varepsilon=\varepsilon_0=n^2\)—the square of the refractive index in the absence of magnetization—we obtain

\[ \mathbf n^{*2}\simeq n^2(1\mp \gamma^{*}Q). \tag{7.3} \]

Comparing this formula with (5.2), we see that in the general case (under the restriction introduced) the laws for the complex refractive index are the same as if the wave propagated parallel to the field lines and only the field strength entering \(Q\) were weakened by the value of the complex direction cosine \(\gamma^{*}\) of the wave normal with respect to the field lines. It is easy to show that relation (7.3) leads to a formula analogous to (5.4), namely:

\[ n_{\pm}^{*}=n\left(1\mp \frac{Q'}{2}\right), \tag{7.4} \]

where \(Q'=Q\gamma^{*}\).

For the consideration of the general case of the reflection problem, the results established in § 4 concerning the propagation of an inhomogeneous wave in any direction with respect to the magnetic field lines must be combined with the general boundary conditions of electrodynamics for the passage of electromagnetic waves through the boundary of two media. For simplicity, we shall consider only cases in which the interface between the two media and the plane of incidence of the wave are situated parallel or perpendicular to the field lines. The geometrical laws of reflection and refraction follow, without introducing the exact form of the boundary conditions, already from the fact that these conditions are homogeneous and linear with respect to the components of the electromagnetic-field strength. It follows from this that, if one operates with a solution of the form

\[ F e^{i\omega\left(t-\frac{\alpha^{*}x+\beta^{*}y+\gamma^{*}z}{v^{*}}\right)}, \]

then the quantity

\[ \frac{\alpha^{*}x+\beta^{*}y+\gamma^{*}z}{v^{*}} \]

must take the same value at the boundary for the incident, reflected, and refracted waves. In particular, if the boundary is the \(XY\)-plane, and the plane \(XZ\) is the plane of incidence both for the plane of constant phases and for the plane of constant amplitudes, then \(\frac{\alpha^*}{v^*}\) must have the same value for all three waves, while \(\beta^*\) is equal to zero for them and \(\alpha^{*2}+\gamma^{*2}=1\). We shall consider only the case when the light is incident from an isotropic transparent medium, in particular from vacuum (or air), and when the incident wave is homogeneous. Then, in the incident and reflected waves, \(\alpha^*\) and \(\sigma^*\) are real, and, moreover, \(\sigma^*\) has the same value \(c\) for both waves. By virtue of the constancy of the ratio \(\alpha^*/\sigma^*\) for both waves, the \(\alpha^*\) (i.e., the sines of the actual angles of incidence and reflection) must be equal; the corresponding value we shall denote by \(\alpha\). The angle \(\gamma^*\) must then have opposite signs for the two waves, since otherwise the waves become identical. Thus the complex wave normal \(S^*\) is brought, for the incident and reflected waves, to the form

\[ s_e=\alpha x+\gamma z \quad \text{and} \quad s_r=\alpha x-\gamma z. \]

Under the assumptions made, in the incident and reflected waves the oscillations are transverse. For each of the two waves penetrating into the magnetized medium, one has

\[ r_d=\alpha_d^*x+\gamma_d^*z, \]

where \(\alpha=\alpha_d^* n_d^*\), \(n_d^*=\frac{c}{v_d^*}\), and \(d=1,2\). Analogous relations hold if, instead of the boundary plane \(XY\), one takes the plane \(YZ\) or \(ZX\). To obtain general reflection formulas for polar magnetization, one would have to use the specific boundary conditions requiring the continuity of the corresponding tangential components of the electric and magnetic vector strengths. It is interesting, however, that these formulas can also be obtained by means of simple reasoning. Suppose that on a perpendicularly magnetized ferromagnetic mirror (\(XZ\) is the plane of incidence, \(XY\) the boundary plane), at an angle \(\varphi\), there falls a circularly polarized wave. We decompose the circular oscillation into two, one of which—elliptical—is placed in a plane parallel to the plane of the mirror, and the other—linear—is placed so that the oscillations in it occur along the magnetic lines of force. The oscillation taking place along the magnetic lines of force does not change at all and, consequently, has no effect on the magneto-optical Kerr effect. The elliptical oscillation will be the projection of the circular oscillation onto the plane of the mirror and, approximately, may be represented in the form

\[ A_p=\mp i\,\frac{n\cos\varphi+\cos\psi}{n\cos\psi+\cos\varphi}\,A_s, \tag{7,5} \]

where \(\psi\) is the complex angle of refraction (the coefficient before \(A_s\) is equal to \(\cos(\varphi-\psi)\simeq \cos\varphi\)). At normal incidence \(\varphi=0\), and the elliptic oscillation described by formula (7.5) becomes circular. The reflected waves will then have the amplitudes:

\[ R_{s\pm}=A_s\left(r_{11}\pm i\,\frac{n\cos\varphi+\cos\psi}{n\cos\psi+\cos\varphi}\,r_{12}\right), \]
\[ R_{p\pm}=A_p\left(r_{22}\mp i\,\frac{n\cos\psi+\cos\varphi}{n\cos\varphi+\cos\psi}\,r_{21}\right), \tag{7.6} \]

as is readily verified by substituting (7.5) into (2.1). Further, we have:

\[ \left(\frac{R_s}{A_s}\right)_{\pm} = r_{11}\pm i\,\frac{n\cos\varphi+\cos\psi}{n\cos\psi+\cos\varphi}\,r_{12}. \tag{7.7} \]

In the absence of magnetization, Fresnel’s formulas are valid:

\[ \left(\frac{R_s}{A_s}\right)_{Q=0} = r_{11} = \frac{\cos\varphi-n\cos\psi}{\cos\varphi+n\cos\psi}, \tag{7.8a} \]

\[ \left(\frac{R_p}{A_p}\right)_{Q=0} = r_{22} = \frac{n\cos\varphi-\cos\psi}{n\cos\varphi+\cos\psi}. \tag{7.8b} \]

If, however, the ferromagnet is magnetized, then one must put

\[ \left(\frac{R_s}{A_s}\right)_{\pm} = \frac{\cos\varphi-n_{\pm}\cos\psi}{\cos\varphi+n_{\pm}\cos\psi}, \tag{7.9a} \]

\[ \left(\frac{R_p}{A_p}\right)_{\pm} = \frac{n_{\pm}\cos\varphi-\cos\psi}{n_{\pm}\cos\varphi+\cos\psi}. \tag{7.9b} \]

Substituting (7.4) into (7.9a) and (7.9b), we obtain:

\[ r_{11}\pm i\,\frac{n\cos\varphi+\cos\psi}{n\cos\psi+\cos\varphi}\,r_{12} = \frac{\cos\varphi-n\left(1\mp\frac{Q'}{2}\right)\cos\psi} {\cos\varphi+n\left(1\mp\frac{Q'}{2}\right)\cos\psi} = \]

\[ = \frac{\cos\varphi-n\cos\psi}{\cos\varphi+n\cos\psi} \pm \frac{nQ'\cos\varphi\cos\psi}{(\cos\varphi+n\cos\psi)^2}, \tag{7.10a} \]

\[ r_{22}\mp i\,\frac{n\cos\psi+\cos\varphi}{n\cos\varphi+\cos\psi}\,r_{21} = \frac{n\left(1\mp\frac{Q'}{2}\right)\cos\varphi-\cos\psi} {n\left(1\mp\frac{Q'}{2}\right)\cos\varphi+\cos\psi} = \]

\[ = \frac{n\cos\varphi-\cos\psi}{n\cos\varphi+\cos\psi} \mp \frac{nQ'\cos\varphi\cos\psi}{(n\cos\varphi+\cos\psi)^2}. \tag{7.10b} \]

From these formulas it follows directly that

\[ r_{12}=r_{21}= -i\,\frac{nQ'\cos\varphi\cos\psi} {(\cos\varphi+n\cos\psi)(n\cos\varphi+\cos\psi)}. \tag{7.11} \]

Since for the majority of metals, including ferromagnetic ones,

\[ \mathfrak{n}^{2}>1,\quad \text{then}\quad \gamma^{**}=\cos\psi=\sqrt{1-a^{*2}}=\sqrt{1-\frac{a^{2}}{\mathfrak{n}^{2}}}\simeq 1 . \]

Taking this remark into account, we obtain the formulas for reflection under polar magnetization in the following form:

\[ \left. \begin{aligned} R_s&=r_{11}A_s+r_{12}A_p,\qquad R_p=r_{21}A_s+r_{22}A_p,\\ r_{11}&=\frac{\cos\varphi-\mathfrak{n}}{\cos\varphi+\mathfrak{n}},\qquad r_{22}=\frac{\mathfrak{n}\cos\varphi-1}{\mathfrak{n}\cos\varphi+1},\\ r_{12}&=r_{21}=-\frac{i\mathfrak{n}Q\cos\varphi}{(\cos\varphi+\mathfrak{n})(\mathfrak{n}\cos\varphi+1)} . \end{aligned} \right\} \tag{7,12} \]

To determine the minimum and the null rotation it is first of all necessary to know the ratios \(r_{12}/r_{11}\) and \(r_{12}/r_{22}\). Neglecting unity in comparison with \(\mathfrak{n}^{2}(1+\chi^{2})\), we obtain:

\[ \frac{r_{12}}{r_{11}}= \frac{Q\cos\varphi[-\mathfrak{n}\chi\cos\varphi+i(\mathfrak{n}\cos\varphi+\sin^{2}\varphi)]} {(\sin^{2}\varphi+\mathfrak{n}\cos\varphi)^{2}+\mathfrak{n}^{2}\chi^{2}\cos^{2}\varphi}, \tag{7,13a} \]

\[ \frac{r_{12}}{r_{22}}= \frac{Q\cos\varphi[\mathfrak{n}\chi\cos\varphi-i(\mathfrak{n}\cos\varphi-\sin^{2}\varphi)]} {(\sin^{2}\varphi-\mathfrak{n}\cos\varphi)^{2}+\mathfrak{n}^{2}\chi^{2}\cos^{2}\varphi}. \tag{7,13б} \]

The parameter \(Q\), generally speaking, is a complex quantity; however, from experimental data it is known that for iron (steel) the imaginary part of \(Q\) in the yellow and red regions of the spectrum is very small in comparison with the real part, while for nickel and cobalt it is at least not very significant. Therefore, for a qualitative discussion of the results obtained (in particular, for considering the sign of the minimum and null rotation) the imaginary part of \(Q\) may be neglected. The parameter \(Q\) is considered positive if the lines of force are arranged parallel to the positive direction of the \(z\)-axis. Formulas (7,13a and b) then give, because the denominator is always positive,

\[ \text{for }\operatorname{Re}\left(\frac{r_{12}}{r_{11}}\right)\text{ the sign of the quantity }-\mathfrak{n}\chi\cos\varphi, \]

\[ \text{» }\operatorname{Im}\left(\frac{r_{12}}{r_{11}}\right)\text{ » » }\quad +(\mathfrak{n}\cos\varphi+\sin^{2}\varphi), \]

\[ \text{» }\operatorname{Re}\left(\frac{r_{12}}{r_{22}}\right)\text{ » » }\quad +\mathfrak{n}\chi\cos\varphi, \]

\[ \text{» }\operatorname{Im}\left(\frac{r_{12}}{r_{22}}\right)\text{ » » }\quad -(\mathfrak{n}\cos\varphi-\sin^{2}\varphi). \]

Therefore the sign of the first quantity is always negative, that of the second and third is always positive, whereas the sign of the last quantity is negative at small angles of incidence and positive at large ones. The change in sign of the last quantity takes place at the angle of incidence \(\varphi_1\), determined by the equality

\[ \operatorname{tg}\varphi_1\sin\varphi_1=\mathfrak{n}. \tag{7,14} \]

This special angle of incidence \(\varphi_1\) is completely independent of the magnetization (and consequently also of the magneto-optical parameter), and is determined by the real part of the refractive index. Taking into account the remarks on signs given above, we see that the minimum-rotation of the analyzer and polarizer for \(r_{12}=r_{21}\)

\[ \chi_p=\chi_s^0=\operatorname{Re}\left(\frac{r_{12}}{r_{22}}\right) \]

and

\[ \chi_s=-\chi_p^0=-\operatorname{Re}\left(\frac{r_{12}}{r_{11}}\right), \]

i.e., for positive \(Q\), must be positive. Since the null-rotations of the analyzer and polarizer for \(r_{12}=r_{21}\) are expressed by the formulas (see § 2)

\[ \psi_p=\psi_s^0= \frac{\operatorname{Im}\left(\dfrac{r_{21}}{r_{11}}\right)} {\operatorname{Im}\left(\dfrac{r_{22}}{r_{11}}\right)} \]

and

\[ \psi_s=-\psi_p^0= -\frac{\operatorname{Im}\left(\dfrac{r_{12}}{r_{22}}\right)} {\operatorname{Im}\left(\dfrac{r_{11}}{r_{22}}\right)}, \]

then for positive \(Q\), in view of the fact that for all angles of incidence

\[ \operatorname{Im}\left(\frac{r_{11}}{r_{22}}\right)<0,\quad \text{and}\quad \operatorname{Im}\left(\frac{r_{22}}{r_{11}}\right)>0, \]

we obtain that \(\psi_p=\psi_s^0\) is always positive, while \(\psi_s=-\psi_p^0\) is negative for angles of incidence smaller than the critical angle \(\varphi_1\), and positive for angles of incidence greater than the critical angle.

8. GENERAL FORMULAS FOR REFLECTION UNDER MERIDIONAL MAGNETIZATION

We shall not dwell on the derivation of the reflection formulas under meridional magnetization, but shall give only their final expressions. In the same approximation in which the polar effect was considered, we obtain:

\[ \left. \begin{aligned} R_s&=r_{11}A_s+r_{12}A_p,\qquad R_p=r_{21}A_s+r_{22}A_p,\\ r_{11}&=\frac{\cos\varphi-n}{\cos\varphi+n},\qquad r_{22}=\frac{n\cos\varphi-1}{n\cos\varphi+1},\\ r_{12}&=-r_{21}=-\frac{iQ\sin\varphi\cos\varphi} {(\cos\varphi+n)(n\cos\varphi+1)}. \end{aligned} \right\} \tag{8,1} \]

To determine the minimum- and null-rotations, the expressions for \(\dfrac{r_{21}}{r_{11}}\) and \(\dfrac{r_{21}}{r_{22}}\) are calculated in the former approximation, in which, as unity,

in comparison with \(n^2(1+\chi^2)\) are neglected:

\[ \left. \begin{aligned} \frac{r_{21}}{r_{11}}&= \frac{Q\sin\varphi\cos\varphi\,[\,\chi(2n\cos\varphi+\sin^2\varphi)+i(n(\chi^2-1)\cos\varphi-\sin^2\varphi)\,]} {n(1+\chi^2)\left[(\sin^2\varphi+n\cos\varphi)^2+n^2\chi^2\cos^2\varphi\right]},\\[6pt] \frac{r_{21}}{r_{22}}&= \frac{Q\sin\varphi\cos\varphi\,[\,-\chi(2n\cos\varphi-\sin^2\varphi)-i(n(\chi^2-1)\cos\varphi+\sin^2\varphi)\,]} {n(1+\chi^2)\left[(\sin^2\varphi-n\cos\varphi)^2+n^2\chi^2\cos^2\varphi\right]}. \end{aligned} \right\} \tag{8,2} \]

Formulas (8,2) then give, owing to the fact that the denominator is always positive:

for \(\operatorname{Re}\left(\dfrac{r_{21}}{r_{11}}\right)\), the sign of the quantity \(+(2n\cos\varphi+\sin^2\varphi)\),

for \(\operatorname{Im}\left(\dfrac{r_{21}}{r_{11}}\right)\), the sign of the quantity \(+(n(\chi^2-1)\cos\varphi-\sin^2\varphi)\),

for \(\operatorname{Re}\left(\dfrac{r_{21}}{r_{22}}\right)\), the sign of the quantity \(-(2n\cos\varphi-\sin^2\varphi)\),

for \(\operatorname{Im}\left(\dfrac{r_{21}}{r_{22}}\right)\), the sign of the quantity \(-(n(\chi^2-1)\cos\varphi+\sin^2\varphi)\).

The first expression is always positive, the last always negative, since in it \(\chi>1\). The second expression is positive at small angles of incidence and changes its sign at \(\varphi=\varphi_2\), determined by the equality

\[ \sin\varphi_2\,\operatorname{tg}\varphi_2=n(\chi^2-1). \tag{8,3} \]

The third expression at small angles of incidence is negative and changes its sign at \(\varphi=\varphi_3\), where

\[ \sin\varphi_3\,\operatorname{tg}\varphi_3=2n. \tag{8,4} \]

We relate these results to formulas (2,8), (2,9), (2,17), and (2,18), whence, for \(r_{12}=-r_{21}\), for the minimum rotation of the analyzer \(\chi\) and of the polarizer field \(\chi^0\), we obtain:

\[ \left. \begin{aligned} -\chi_s&=\chi_p^0=\operatorname{Re}\left(\frac{r_{21}}{r_{11}}\right)\\ \text{and}\qquad -\chi_p&=\chi_s^0=\operatorname{Re}\left(\frac{r_{21}}{r_{22}}\right). \end{aligned} \right\} \tag{8,5} \]

Again taking \(Q\) positive, \(+\chi_s\) and \(-\chi_p^0\) will be negative for all angles of incidence, whereas \(+\chi_p\) and \(-\chi_s^0\) for small angles of incidence are positive; for those same angles which exceed the critical angle \(\varphi_3\), they must be negative. For the null rotation of the analyzer and polarizer from (2,25), (2,26), (2,27), and (2,28) (under the premise—

from the condition \(r_{12}=-r_{21}\) there follow the formulas

\[ -\psi_s=\psi_p^0= \frac{\operatorname{Im}\left(\dfrac{r_{21}}{r_{22}}\right)} {\operatorname{Im}\left(\dfrac{r_{11}}{r_{22}}\right)} \]

and

\[ -\psi_p=\psi_s= \frac{\operatorname{Im}\left(\dfrac{r_{21}}{r_{11}}\right)} {\operatorname{Im}\left(\dfrac{r_{22}}{r_{11}}\right)} . \tag{8,6} \]

It follows from this that, for all angles of incidence, \(\psi_s=-\psi_p^0\) are negative, whereas \(\psi_p=-\psi_s^0\) are negative at small angles of incidence and positive for those that exceed the critical angle \(\varphi_2\). For a qualitative discussion of the behavior of the sign of the minimum and zero rotations we neglected the imaginary part of the magneto-optical parameter \(Q\). Dropping this restriction, we obtain relatively simple equations determining the critical angles of incidence \(\varphi_1\), \(\varphi_2\), and \(\varphi_3\). We have seen that in the first two cases a zero-rotation passes through zero, and in the last case a minimum-rotation. Representing the magneto-optical parameter in the form \(Q=|Q|e^{-iq}\), we obtain from (7,13a and 13b) and (8,2) the following expressions for \(\varphi_1\), \(\varphi_2\), and \(\varphi_3\):

\[ \tg \varphi_1 \sin \varphi_1 = n(1+x \tg q) \quad \text{polar effect,} \tag{8,7} \]

\[ \left. \begin{aligned} \tg \varphi_2 \sin \varphi_2 &= \frac{n(x^2-1)-2nx\,\tg q}{1+x\,\tg q},\\[4pt] \tg \varphi_3 \sin \varphi_3 &= \frac{2nx+n(x^2-1)\tg q}{x-\tg q} \end{aligned} \right\} \quad \begin{gathered} \text{meridional}\\ \text{effect.} \end{gathered} \tag{8,8} \]

It has been established empirically that the critical angles of incidence \(\varphi_1\), \(\varphi_2\), and \(\varphi_3\) do not depend on the magnetization. It follows that the quantity \(q\) in the approximation under consideration does not depend on the magnetization, but is determined exclusively by the optical constants. Because of their cumbersomeness, we shall not give the expanded formulas for the minimum and zero rotations, as well as for the ellipticity in the general case of oblique incidence of light with polar and meridional magnetization. Concerning comparison with experiment of the rigorous formulas for minimum and zero rotations, as well as for the phase and amplitude ratio at oblique incidence, one should consult the works of Goldhammer and Seemann \(^{20}\); the agreement is satisfactory.

9. EQUATORIAL KERR EFFECT

Let us now consider the case in which the magnetization vector is directed along the boundary surface, but perpendicular to the plane of incidence \(^{19}\). If the plane \(YZ\) is chosen as the boundary plane and \(YX\) as the plane of incidence, then for all angles of incidence

the direction cosine \(\gamma^*\) is equal to zero:

\[ \alpha^{*2}+\beta^{*2}=1,\qquad \beta_d^*=\beta/n_d^*, \]

where the index \(d\) refers to both refracted waves. In this case from (4,6) we obtain:

\[ \left. \begin{aligned} \varepsilon E_x-i\varepsilon Q E_y&=n^{*2}\beta^*(\beta^*E_x-\alpha^*E_y),\\ i\varepsilon Q E_x+\varepsilon E_y&=n^{*2}\alpha^*(\alpha^*E_y-\beta^*E_x),\\ \varepsilon_0E_z&=n^{*2}E_z . \end{aligned} \right\} \tag{9,1} \]

These equations show that oscillations (inside the ferromagnet) parallel to the lines of force are completely separated from oscillations perpendicular to the lines of force. For the refractive indices the exact formulas (5,5′) are valid, which, when restricted to quantities of first order with respect to \(Q\), take the form:

\[ n_z^{*2}=n_{\perp}^{*2}=\varepsilon=\varepsilon_0=n^2. \tag{9,2} \]

Both refracted waves consequently have identical propagation speeds and directions and coincide. The problem of reflection and refraction now contains the influence of magnetization only through equation (4,8), which for \(\gamma^{**}=0\) will have the form:

\[ (\alpha^{**}+i\beta^{**}Q)E_x+(\beta^{**}-i\alpha^{**}Q)E_y=0. \tag{9,3} \]

Fig. 4.

It follows from this that the components of the oscillations in the plane of incidence are related to one another quite differently than in the absence of magnetization. We now set, in accordance with Fig. 4,

\[ \left. \begin{aligned} E_x^e&=-\beta A_p e^{i\omega\tau_e},\qquad E_y^e=\alpha A_p e^{i\omega\tau_e},\qquad E_z^e=A_s e^{i\omega\tau_e},\\ \tau_e&=t-\frac{\beta y+\alpha x}{V},\\[4pt] E_x^r&=-\beta R_p e^{i\omega\tau_r},\qquad E_y^r=-\alpha R_p e^{i\omega\tau_r},\qquad E_z^r=R_s e^{i\omega\tau_r},\\ \tau_r&=t-\frac{\beta y-\alpha x}{V},\\[4pt] E_x^d&=D^{(1)}e^{i\omega\tau_d},\qquad E_y^d=D^{(2)}e^{i\omega\tau_d},\qquad E_z^d=D^{(3)}e^{i\omega\tau_d},\\ \tau_d&=t-\frac{\beta^* y+\alpha^* x}{v^*}. \end{aligned} \right\} \tag{9,4} \]

In Fig. 4 \(s_e\) and \(P_e\) are the normal and the direction of oscillations in the incident wave, \(s_r\) and \(P_r\)—in the reflected wave. The boundary conditions require continuity of the quantities

\[ E_y,\ E_z,\ \frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}\quad \text{and}\quad \frac{\partial E_z}{\partial x}. \]

Applying the boundary conditions to (9.4), we shall have:

\[ \left. \begin{gathered} \alpha(A_p-R_p)=D^{(2)},\quad A_p+R_p=\mathfrak n^*(\alpha^*D^{(2)}-\beta^*D^{(1)}),\\ A_s+R_s=D^{(3)},\quad \alpha(A_s-R_s)=\mathfrak n^*\alpha^*D^{(3)}. \end{gathered} \right\} \tag{9.5} \]

The components of the oscillations parallel and perpendicular to the plane of incidence therefore remain, just as in the reflection problem, independent of one another. For the refracted components lying in the plane of incidence, (9.3) gives the relation

\[ (\alpha^*+i\beta^*Q)D^{(1)}=-(\beta^*-i\alpha^*Q)D^{(2)} \]

or

\[ \alpha^*D^{(2)}-\beta^*D^{(1)}= \frac{D^{(2)}}{\alpha^*+i\beta^*Q}. \tag{9.6} \]

Substituting (9.6) into (9.5), we obtain:

\[ R_p[(\alpha^*+\mathfrak n\alpha)+i\beta^*Q] =-A_p[(\alpha^*-\mathfrak n\alpha)+i\beta^*Q], \tag{9.7a} \]

\[ R_s(\alpha+\mathfrak n\alpha^*)=-A_s(\alpha-\mathfrak n\alpha^*). \tag{9.7b} \]

From these relations it is seen that the component oscillating perpendicular to the plane of incidence, and consequently polarized in this plane, does not depend on the magnetization. The component with oscillations parallel to the plane of incidence undergoes, owing to the magnetization, a change in amplitude and phase; however, to demonstrate this change directly is very difficult. The object of observation is the fraction \(R_s/R_p\) in the case when linearly polarized light is incident on the boundary, making with the plane of incidence an azimuth within the limits from \(0\) to \(90^\circ\). Since the incident light is linearly polarized, \(A_p\) and \(A_s\) will be real and \(A_s/A_p=\operatorname{tg}\Gamma\), where \(\Gamma\) is the azimuth of the electric vector. From (9.7a) and (9.7b), taking (2.6) into account, we obtain:

\[ \frac{R_s}{R_p}=\frac{G}{F}e^{-i(g-f)}= \]

\[ =-\frac{\cos\varphi-\mathfrak n\alpha^*}{\cos\varphi+\mathfrak n\alpha^*}\, \frac{\alpha^*+\mathfrak n\cos\varphi+i\,\dfrac{Q\sin\varphi}{\mathfrak n}} {\alpha^*-\mathfrak n\cos\varphi+i\,\dfrac{Q\sin\varphi}{\mathfrak n}}\, \frac{A_s}{A_p}, \tag{9.8} \]

where

\[ \mathfrak n=n(1-i\chi)=n\sqrt{1+\chi^2}\,e^{-2i\Theta} =\sin\Phi\,\operatorname{tg}\Phi\,e^{-2i\Theta} \]

and

\[ \alpha^*=\sqrt{1-\frac{\sin^2\varphi}{\mathfrak n^2}}. \]

Restricting ourselves to terms of first order with respect to \(Q\), (9.8) can be represented in the form

\[ \frac{R_s}{R_p} = \frac{G}{F}e^{-i(g-f)} = \]

\[ = -\frac{A_s}{A_p}\, \frac{\cos\varphi-na^{*}}{\cos\varphi+na^{*}}\, \frac{a^{*}+n\cos\varphi}{a^{*}-n\cos\varphi} = -\left(1-\frac{iQ\sin 2\varphi}{a^{*2}-n^{2}\cos^{2}\varphi}\right). \tag{9.9} \]

In experiments on measuring the equatorial effect, the azimuth of the polarizer is set at such a value \(\Gamma\) for which equality of the components \(F\) and \(G\) is achieved (their phases differ only slightly); this is detected bolometrically. Then the magnetic field is switched on and the small rotation \(\partial\Gamma\) of the polarizer is measured, which is necessary in order to restore equality of the reflected components. Using the subscript “0” to denote the ordinary values (without magnetization, i.e. for \(Q=0\)) and putting \(F=G\), in each case we have (if \((g-f)\) is denoted by \(\delta\)):

\[ e^{-i\delta}\operatorname{ctg}\Gamma = -\frac{\cos\varphi-na^{*}}{\cos\varphi+na^{*}}\, \frac{a^{*}+n\cos\varphi}{a^{*}-n\cos\varphi} \left(1-\frac{iQ\sin 2\varphi}{a^{*2}-n^{2}\cos^{2}\varphi}\right), \tag{9.10} \]

\[ e^{-i\delta_{0}}\operatorname{ctg}\Gamma_{0} = -\frac{\cos\varphi-na^{*}}{\cos\varphi+na^{*}}\, \frac{a^{*}+n\cos\varphi}{a^{*}-n\cos\varphi}. \tag{9.11} \]

Dividing the first equation by the second, we obtain:

\[ e^{-i(\delta-\delta_{0})}\frac{\operatorname{tg}\Gamma_{0}}{\operatorname{tg}\Gamma} = 1-\frac{iQ\sin 2\varphi}{a^{*2}-n^{2}\cos^{2}\varphi} = 1-\frac{iQ\sin 2\varphi}{1-\dfrac{\sin^{2}\varphi}{n^{2}}-n^{2}\cos^{2}\varphi}. \tag{9.12} \]

Putting, for brevity, \(n^{2}(1-\chi^{2})=b\) and \(Q=|Q|e^{-iq}\), equation (9.12) can be represented in the form

\[ e^{-i(\delta-\delta_{0})}\frac{\operatorname{tg}\Gamma_{0}}{\operatorname{tg}\Gamma} = 1+ \frac{|Q|\sin 2\varphi\,e^{-i\left(q+\frac{\pi}{2}\right)}} {1-\dfrac{\sin^{2}\varphi\,e^{4i\theta}}{b}-b\cos^{2}\varphi\,e^{-4i\theta}} = \]

\[ = 1+ \frac{|Q|\sin 2\varphi\,e^{-i\left(q+\frac{\pi}{2}+\vartheta\right)}} {\sqrt{\left[1-\dfrac{\cos 4\theta\,(b^{2}\cos^{2}\varphi+1)}{b}\right]^{2} +\sin^{2}4\theta\left[\dfrac{b^{2}\cos^{2}\varphi-1}{b}\right]^{2}}}, \tag{9.13} \]

where

\[ \operatorname{tg}\vartheta = -\operatorname{tg}4\theta\, \frac{b^{2}\cos^{2}\varphi-1}{b^{2}\cos^{2}\varphi+1-b\sec 4\theta}. \]

Instead of \(b^{2}+1\) one may write \(b^{2}\), since in all cases under consideration \(b^{2}>100\). Expression (9.13) can be reduced to the form

\[ \frac{\operatorname{tg}\Gamma_{0}}{\operatorname{tg}\Gamma} = \left[ 1-\frac{|Q|\sin 2\varphi\sin(\vartheta+q)}{y} \right] e^{-i\left[\delta_{0}-\delta+\dfrac{|Q|\sin 2\varphi\cos(\vartheta+q)}{y}\right]}, \tag{9.14} \]

where

\[ y=\sqrt{b^2\cos^4\varphi-2N\cos^2\varphi+1-\frac{2\cos4\Theta}{b}} \]

[here, for convenience, the notation \(N=(b-2\cos4\Theta)\cos4\Theta+1\) has been introduced].

Equating the moduli and arguments in (9.14), we obtain:

\[ \delta-\delta_0=\frac{|Q|\sin2\varphi\cos(\vartheta+q)}{y}, \tag{9.15} \]

\[ \operatorname{tg}\Gamma=\operatorname{tg}\Gamma_0\left[1+\frac{|Q|\sin2\varphi\sin(\vartheta+q)}{y}\right]. \tag{9.16} \]

Replacing \(\Gamma\) in the left-hand side by \(\Gamma_0+\delta\Gamma\), after simple transformations we obtain:

\[ 2\delta\Gamma=\frac{|Q|\sin2\varphi\sin2\Gamma_0\sin(\vartheta+q)}{y}, \tag{9.17} \]

where \(\Gamma_0\) is determined by the formula

\[ \cos2\Gamma_0=\frac{2n\sin\varphi\,\operatorname{tg}\varphi}{\sin^2\varphi\,\operatorname{tg}^2\Phi+b}. \tag{9.18} \]

Snow compared Ingersoll’s measurements, relating to sodium light \((\lambda=5900\,\text{Å})\), with the theory of the equatorial effect set forth here, and found that the theory is in good agreement with the observations for iron (steel), nickel, and cobalt. This comparison is presented in Fig. 5.

Fig. 5.

Theoretical and observed change of azimuth as a function of angle of incidence, for \(\lambda=0.59\mu\). Curves and points are shown for steel, cobalt, and nickel; the data of Ingersoll and Snow are indicated.

The effect and found that the theory is in good agreement with observations for iron (steel), nickel, and cobalt. This comparison is presented in Fig. 5.

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1. UFN, vol. L, issue 2. ↩↩