Magnetic Rotation of the Plane of Polarization in Centimeter Waves

A. L. Mikaelyan

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
I. Magnetic rotation of the plane of polarization (general information) . . . . . . . . . 206
II. Rotation of the plane of polarization in ferrites . . . . . . . . . . . . . . . . . . . . 211
    1. Brief information on ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
    2. Theory of ferromagnetic resonance . . . . . . . . . . . . . . . . . . . . . . . . . 212
    3. Propagation of waves in a ferromagnetic medium . . . . . . . . . . . . . . . . . 214
    4. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
III. Rotation of the plane of polarization in artificial dielectrics . . . . . . . . . . . 231
IV. Rotation of the plane of polarization in an electron plasma . . . . . . . . . . . . . 235
    1. General theory of wave propagation in an ionized medium in the presence of a constant magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
    2. Calculation of some cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
    3. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
V. Application of magnetic rotation of the plane of polarization in centimeter waves . . 248

Introduction

In recent years interest in magneto-optical phenomena at centimeter wavelengths has increased considerably in connection with the possibility of practical use of these phenomena in centimeter-wave radio engineering, and a number of works have appeared (chiefly experimental), the generalization of whose results has become a timely task.

Among the various practical applications, the most important, perhaps, is the possibility of creating an element for centimeter waves that behaves differently for waves propagating in opposite directions. In a particular case this element will behave as a linear “valve,” i.e., it will transmit waves propagating only in one direction. The principle of operation of the “valve” is based on magnetic rotation of the plane of polarization (the Faraday effect) and is considered in detail in Chapter V. The application

such an element would make it possible to solve easily a number of difficult problems in centimeter-wave technology.

Therefore, the study of the Faraday phenomenon at centimeter wavelengths in various materials is of practical interest. Specifically, the problem reduces to the creation of a material that would be “transparent” to centimeter waves and would produce a significant rotation of the plane of polarization.

From optical experiments it is known that appreciable rotation of the plane of polarization is produced only by ferromagnetic materials (thin transparent films). The same may be expected at centimeter wavelengths as well. Therefore, in the present survey the greatest attention is devoted to the investigation of ferrites, i.e., a group of ferromagnetics transparent to centimeter waves. In addition, we shall consider the possibility of using plasma and an artificial dielectric to rotate the plane of polarization.

We shall also give a description of certain technical schemes whose principle of operation is based on magnetic rotation of the plane of polarization.

I. MAGNETIC ROTATION OF THE PLANE OF POLARIZATION

(General information)

It is known that if a polarized wave is passed, in the direction of the field, through a substance placed in a constant magnetic field, then the plane of polarization of the wave will rotate through some angle depending on the dimensions and properties of the substance and on the strength of the magnetic field. In this case the direction of rotation is connected with the direction of the magnetic field and does not depend on the direction of propagation of the wave (in contrast to the natural rotation of the plane of polarization). This means that if a ray emerging from the substance is forced, by reflection, to traverse the same path a second time in the opposite direction, then the total rotation will be twice as great as after a single passage. As we shall see below, the practical use of the phenomenon described is based on this property.

The explanation of this phenomenon, called the Faraday effect, within the framework of classical electrodynamics reduces to the following: in the absence of a magnetic field, the electrons in a substance are in a definite state of motion. According to Larmor’s theorem¹, the imposition on the electrons (which determine the refractive index) of a constant magnetic field is equivalent to a transition to a coordinate system rotating about the direction of the magnetic field with angular velocity

\[ \omega_L=-\frac{e}{2mc}H=2.8\cdot \pi \cdot H \ \text{Mcps}, \tag{1.1} \]

where \(H\) is the intensity of the magnetic field acting on the electron, in oersteds. In the MKS system of units, which we shall use everywhere, formula (1.1) takes the form

\[ \omega_L=-\frac{e}{2m}\mu_0H=\frac{1.78}{2}\cdot 10^{11}\cdot B, \tag{1.1} \]

where \(B\) is expressed in webers per square meter.

Let us mentally decompose a linearly polarized wave of frequency \(\omega\), incident in the direction of the magnetic lines of force, into two circularly polarized waves with opposite directions of rotation. Then the left-polarized wave, i.e., the wave polarized in a circle in the counterclockwise direction, if one looks along the lines of force of the magnetic field, will have, with respect to a coordinate system rotating with angular velocity \(\omega_L\), a frequency smaller by \(\omega_L\), i.e. \(\omega-\omega_L\). A wave with the same frequency \(\omega\), but right-polarized, i.e., polarized in a circle in the opposite direction, has in the rotating coordinate system a larger frequency, namely \(\omega+\omega_L\). Consequently, if \(n(\omega)\) is the refractive index of the substance, then when the magnetic field is switched on the velocity of the right-polarized wave is determined by the value of this function for the frequency \(\omega+\omega_L\), and the velocity of the left-polarized wave by its value for the frequency \(\omega-\omega_L\). In accordance with this, the plane of polarization of the wave, on passing through a distance \(l\) in the substance, rotates through the angle

\[ \psi=(n_+-n_-)\frac{\omega}{2c}\,l=(n_+-n_-)\pi\frac{l}{\lambda_0}, \tag{1.2} \]

where

\[ n_-=n(\omega-\omega_L), \qquad n_+=n(\omega+\omega_L) \tag{1.3} \]

and \(\lambda_0\) is the wavelength in vacuum.

Figure 1 shows the behavior of the refractive index for the two indicated waves (curves 1 and 2). It also follows from the figure that the curve of the change \(n_- - n_+\) is symmetric with respect to the spectral line \((\omega=\omega_0)\), and outside this line the rotation of the plane of polarization retains a positive sign; it coincides with the direction of the current in the coil producing the magnetic field \(H\). The curves in Fig. 1 are valid only for diamagnetics, i.e., for substances whose atoms or molecules, in the absence of an external magnetic field, do not possess a magnetic moment.

If, however, the magnetic moment of the atoms and molecules of the medium in the absence of an external magnetic field is different from zero, i.e., an individual molecule already possesses in advance a preferred direction of rotation of the electrons, as occurs in paramagnetics, then the picture will be different from that described by Fig. 1. Namely, when a constant magnetic field parallel to the direction of

of propagation of the wave, a partial orientation of the molecules will occur, i.e., the number of molecules in which the electrons rotate counterclockwise with respect to the direction of the field will increase, while the number of molecules in which the electrons rotate in the opposite direction will decrease. This will lead to an increase in the refractive index for the left-polarized wave and to a decrease for the right-polarized wave (Fig. 2). In this case the quantity \((n_- - n_+)\), which determines the Faraday effect, becomes asymmetric with respect to the line \(\omega = \omega_0\).

Fig. 1.

Fig. 2.

It should be noted that the dependence of the refractive index on frequency is determined by the motion not only of electrons, but also of molecules. For the Faraday effect, however, only the motion of electrons is essential, since the mass of a molecule is thousands of times greater than the mass of an electron, and, in accordance with formula (I.1), the action of the magnetic field on molecules is correspondingly that many times weaker than on electrons. Consequently, one must take into account only that part of the refrac-

of the refractive index, which is caused by the motion of the electrons. For the case of light waves, when \(\omega_L \ll \omega\), the magnitude of the magnetic rotation of the plane of polarization can be estimated by making use of the fact that

\[ n_{+}-n_{-}=n(\omega+\omega_L)-n(\omega-\omega_L)=\frac{dn}{d\omega}\,2\omega_L . \tag{I.4} \]

Then

\[ \frac{\psi}{l}=\frac{\omega}{2c}\cdot \frac{dn}{d\omega}\cdot 1.4\pi H=\rho\cdot H, \tag{I.5} \]

where \(\rho\) is the Verdet constant, depending on the substance. This is the well-known Becquerel formula\(^1\).

For all nonferromagnetic materials the angle of rotation in fields of the order of thousands of oersteds is measured in minutes. In ferromagnets, where formula (I.5) is invalid, the rotation of the plane of polarization has a magnitude several orders larger. This fact finds its explanation only within the framework of quantum mechanics\(^9\).

At centimeter waves the condition \(\omega_L \ll \omega\) is no longer satisfied, and approximation (I.4) loses its validity. Moreover, in magnetic fields of the order of thousands of oersteds the frequency of the propagating oscillation and the Larmor frequency are very close and may coincide, which leads to a violation of the linear relation between the angle of rotation of the polarization and the magnetic field, i.e. in this case the Verdet constant loses its meaning. Under the condition of equality of the frequencies \(\omega\) and \(\omega_L\), the rotation of the plane of polarization will obviously be equal to:

\[ \frac{\psi}{l}=\frac{\omega}{2c}\,[n(2\omega)-n(0)], \tag{I.6} \]

where \(n(2\omega)\) is the refractive index of the substance at the frequency \(2\omega\), and \(n(0)\) is the refractive index of the substance at zero frequency. Thus, in formula (I.6) the only unknown is the value \(n(2\omega)\). It will be shown below that in the case of equality of the frequencies \(\omega\) and \(\omega_L\) (gyromagnetic orbital resonance) the rotation of the plane of polarization has its greatest magnitude. In general, even at centimeter waves the rotation of the plane of polarization in all materials except ferromagnets is small and is measured in minutes. We shall briefly present the main experimental results confirming this proposition.

Measurements of the angles of rotation of the plane of polarization were carried out at centimeter waves, using a circular waveguide with the \(H_{11}\) wave, a section of which was filled with the substance under investigation.

The magnitude of the magnetic field applied along the axis of the waveguide was of the order of 1350 gauss.

The paramagnetic properties of \( \mathrm{MnCl_2\cdot4H_2O} \) and \( \mathrm{MnSO_4\cdot H_2O} \),\(^{3,4}\) which in the presence of a magnetic field strongly absorb centimeter waves, have been investigated in greatest detail. The results are given in Table I.

Table I

\( \lambda \) (cm)	\( H \) (gauss)	\( l \) (cm)	\( \psi \) (minutes)
\(\mathrm{MnSO_4\cdot H_2O}\)	\(\mathrm{MnSO_4\cdot H_2O}\)	\(\mathrm{MnSO_4\cdot H_2O}\)	\(\mathrm{MnSO_4\cdot H_2O}\)
3.34	1350	9.2	9.5
3.45	1350	9.2	12.5
3.338	1350	9.2	12.5
\(\mathrm{MnCl_2\cdot4H_2O}\)	\(\mathrm{MnCl_2\cdot4H_2O}\)	\(\mathrm{MnCl_2\cdot4H_2O}\)	\(\mathrm{MnCl_2\cdot4H_2O}\)
3.338	1350	9.2	22.7
3.338	1350	9.65	28.7
3.338	1350	23.4	61.5
3.338	1350	30.5	73.3
3.338	920	30.5	55.2
3.338	460	30.5	25.9
3.442	1350	30.5	25.7
3.553	1350	30.5	46.9

Here \(l\) is the path of the wave in the substance, and \(\psi\) is the rotation of the plane of polarization (in minutes).

It follows from the table that the rotation is the greater, the stronger the magnetic field. In addition, a dependence on frequency is noticeable. Measurement of the dependence of the rotation of the plane of polarization on the magnetic-field strength in the region of gyromagnetic resonance was carried out in detail by Riter\(^{6}\) for manganese sulfate \( \mathrm{MnSO_4\cdot4H_2O} \) at a frequency of 9500 MHz and by Gikhard\(^{7}\) at a frequency of 3000 MHz. The form of the measured curve indicates the resonant character of the dependence of the rotation of the plane of polarization on the magnitude of the applied magnetic field. The maximum rotation of the plane of polarization of the wave corresponds to the point of gyromagnetic resonance.

In testing a number of other materials (Table II), no appreciable rotation of the plane of polarization was found. Thus, one may conclude that substances not belonging to ferromagnets possess a very small rotatory power, measured in tens of minutes per centimeter of path.

Table II

Ethyl alcohol (liquid)	Polystyrene (solid)
Nitrobenzene (liquid)	Ammonia (gas)
Methyl alcohol (liquid)	Sodium chloride (cryst.)
Ammonium nitrate (cryst.)	Glycerin (liquid)
Ammonium hydroxide (liquid)	Ferric sulfate (cryst.)
Water (distilled)	Iron chloride (powder)
Chloroform	Carbon tetrachloride (liquid)
Ethylene chloride	Manganese carbonate (solid)
Methylene chloride	Ferric sulfate (solid)
Benzene (liquid)	Nitrous oxide iron (solid)
Cobalt chloride (solid)	Cobalt sulfite (solid)

II. ROTATION OF THE PLANE OF POLARIZATION IN FERRITES

1. Brief information on ferrites

Ferrites constitute a special group of ferromagnetic substances possessing a very high specific resistance (of the order of \(10^2\)—\(10^6\) ohm·cm, as compared with \(10^{-5}\) ohm·cm for ordinary iron), i.e., they are ferromagnetic semiconductors. The possibility of the existence of such materials is connected with the fact that the phenomena of electrical conductivity and ferromagnetism are not determined by the same electrons. The electrons responsible for ferromagnetism make only a small contribution to electrical conductivity\(^{9}\).

Thus, an electromagnetic wave propagating in a ferrite will be weakly absorbed, which makes it possible to use this material also for the purpose of rotating the plane of polarization of a wave.

Without dwelling on the physical theory of the structure of ferromagnetics, we shall merely recall that ferromagnetic bodies have a crystalline structure. Every ferromagnetic material is polycrystalline, i.e., consists of a large number of crystallites, whose orientation in many materials is random. In this case each crystallite behaves like a single crystal, isolated from neighboring crystallites.

The general formula of ferrites may be written as \(\mathrm{Me}^{\mathrm{II}}\mathrm{O}\cdot\mathrm{Fe}_2\mathrm{O}_3\), where \(\mathrm{Me}^{\mathrm{II}}\) is an ion of a divalent metal. Ferrites in which the ions Ni, Co, Fe, Mn, Mg, Ca, Cu\(^{8}\) serve as \(\mathrm{Me}^{\mathrm{II}}\) are well known in engineering. Although ferrites with cubic, hexagonal, and tetragonal lattices are known at the present time, the ferrites that have chiefly been studied are those with cubic structure. Detailed information on the properties of various types of ferrites at centimeter wavelengths can be found in the cited books by S. V. Vonsovskii\(^{9,10}\).

2. Theory of Ferromagnetic Resonance

If a weak high-frequency magnetic field is applied to a ferromagnetic material magnetized to saturation in a constant external magnetic field, in a direction perpendicular to the constant magnetic field, then at a certain frequency \(\omega_0\), determined mainly, as will be shown below, by the frequency of precession of the free electron spin in the external field, a phenomenon is observed which is called ferromagnetic resonance. In particular, a plane wave propagating in a ferromagnet in the direction of the applied constant magnetic field will undergo resonant absorption.

The fundamental possibility of the phenomenon of ferromagnetic resonance was first predicted in the works of V. K. Arkad’ev \(^{11,12}\), and the quantum mechanism of this phenomenon was considered by Ya. G. Dorfman \(^{13}\). A fundamental study in this direction was the work of L. D. Landau and E. M. Lifshitz \(^{14}\), in which a general theory of the behavior of ferromagnetic crystals in alternating magnetic fields was constructed and the resonance effect was investigated quantitatively. The theory of Landau and Lifshitz was developed in detail as applied to new experimental facts in a series of papers by Kittel \(^{15,16,17}\).

When ferromagnets are used for the purpose of rotating the plane of polarization, the phenomenon of ferromagnetic resonance plays an extremely important role, since it determines to a considerable extent the magnitude of the losses. Therefore we shall set forth below the basic propositions of this theory, which is based on the classical model of a ferromagnet first proposed by Landau and Lifshitz in the work cited above.

It is known \(^{18}\) that almost the entire magnetic moment of ferromagnetic materials is associated with the spin of the electrons, and not with their orbital motion. (Measurements of the gyromagnetic ratio, i.e. the ratio of the magnetic moment to the mechanical one, give a value close to \(\dfrac{e}{m}\), which corresponds to spin, whereas for orbital motion it is equal to \(\dfrac{e}{2m}\).) In accordance with this, the magnetization per unit volume of a ferromagnet \(\mathbf{M}\) is related to the resultant internal magnetic field by the equation \(^{2}\)

\[ \frac{d\mathbf{M}}{dt}=\gamma[\mathbf{MH}]\,\mu_0, \tag{II.1} \]

in which \(\gamma=\dfrac{e}{m}=1.78\cdot 10^{11}\ \dfrac{\text{coulomb}}{\text{kg}}\).

If we assume that \(\mathbf{H}\) is equal to the applied constant field \(H_z\), then (for a harmonic time dependence of \(M\)) we obtain, in the first approximation\(^{15}\),

\[ i\omega M_x=\gamma M_y H_z \mu_0, \tag{II.2} \]

\[ i\omega M_y=-\gamma M_x H_z \mu_0, \tag{II.3} \]

whence

\[ \left(-\omega^2+\gamma^2 H_z^2\mu_0^2\right)M_x=0, \]

which leads to the expression for the resonance frequency:

\[ \omega=\gamma H_z\mu_0. \tag{II.4} \]

This means that the absorption of a wave of frequency \(\omega_0\), propagating through a ferromagnet, will pass through a maximum when the field \(H_z\) reaches the value \(\dfrac{\omega_0}{\gamma\mu_0}\). In deriving formula (II.4) we set the internal field equal to the applied one, i.e., we neglected the demagnetizing factors. If these factors are taken into account, then for the case of an ellipsoid with principal axes parallel to the \(x, y, z\) axes, the internal magnetic field \(\mathbf{H}^i\) will be equal to\(^{16}\)

\[ \begin{aligned} H_x^i&=H_x-N_xM_x,\\ H_y^i&=-N_yM_y,\\ H_z^i&=H_z-N_zM_z. \end{aligned} \left. \begin{array}{} \\ \\ \end{array} \right\} \tag{II.5} \]

Here \(H_x\) is the component of the magnetic field of the plane wave (there are no other components), \(H_z\) is the constant magnetic field, with \(H_z\gg H_x\), and \(N_x, N_y, N_z\) are the demagnetizing factors, depending on the shape of the ferromagnet.

Substituting equation (II.5) into (II.3), we obtain, in the same way as above, the expression for the resonance frequency\(^{16}\):

\[ \omega_0=\gamma\{[H_z+(N_y-N_z)M_z][H_z+(N_x-N_z)M_z]\}^{\frac12}\mu_0. \tag{II.6} \]

Let us note several special cases of the shape of the ferromagnet:

1. Sphere \(\left(N_x=N_y=N_z=\dfrac13\right)\)

\[ \omega_0=\gamma H_z\mu_0. \tag{II.6a} \]

1. Plate \(y=0\) \((N_x=N_z=0;\;N_y=1)\)

\[ \omega_0=\sqrt{\frac{B_zH_z}{\mu_0}}\,\mu_0, \tag{II.6б} \]

where

\[ B_z=(H_z+M_z)\mu_0. \]

3. Infinite circular cylinder

\[ \left(N_x=N_y=\frac{1}{2},\; N_z=0\right) \]

\[ \omega_0=\gamma\left(H_z+\frac{1}{2}M_z\right)\mu_0. \tag{II.6b} \]

In deriving equation (II.6) it was assumed that the dimensions of the ferromagnet are much smaller than the penetration depth of the propagating wave.

In addition to their dependence on the shape of the body, the demagnetizing factors also depend on the anisotropy energy, or, as it is sometimes called, the magnetocrystalline energy of the ferromagnetic crystal. The latter is connected with the fact that in the crystal there exist directions (coinciding with definite crystallographic axes) along which the crystal is most easily magnetized. These directions are called directions of easy magnetization. There also exist directions in which it is most difficult to magnetize the crystal. The excess energy required to magnetize the crystal to saturation in a difficult direction, as compared with an easy one, is the anisotropy energy.

Thus, instead of formula (II.6) one must write¹⁶:

\[ \omega_0=\gamma\{[H_z+(N_x+N_x^e-N_z)M_z][H_z+(N_y+N_y^e- \]

\[ -N_z)M_z]\}^{\frac{1}{2}}\mu_0, \tag{II.7} \]

where \(N_x^e,\;N_y^e\) take into account the influence of the energy of magnetic crystallographic anisotropy on the resonance frequency.

The calculation of these coefficients for a number of particular cases, as well as a number of experimental data, is given in Kittel’s work and in works by other authors. From the quantum-mechanical point of view, the derivation of formula (II.7) was given by Van Vleck²².

In the practical use of a ferromagnet it is necessary to create such conditions that the frequency of the propagating wave be far from the resonant one; in this case the absorption of the wave will be the smallest.

We shall now turn to the consideration of the question of wave propagation in a ferromagnetic medium in the presence of a constant magnetic field. The main purpose of this consideration is a quantitative estimate of the angle of rotation of the plane of polarization of a wave propagating in a ferromagnet.

3. Propagation of Waves in a Ferromagnetic Medium

Let us consider the propagation of a plane wave in a ferromagnetic medium (ferrite), magnetized in a definite direction to saturation. If losses (magnetic and dielectric) are neglected, then we must proceed from the equation of motion

of spin, written in the form (II.1)

\[ \frac{d\mathbf{M}}{dt}=\gamma[\mathbf{M}\mathbf{H}]\,\mu_0, \]

with the aid of which the relation is established between the magnetic induction and the magnetic-field strength in a ferromagnetic medium magnetized to saturation. To establish this relation, let us suppose that in an infinite ferromagnetic medium, which is subjected to the action of a constant magnetic field \(H_z\), a wave is propagating, i.e. a variable field \(\mathbf{h}=\mathbf{H}-\mathbf{H}_z\) is acting. If the variable components of the magnetization are denoted by \(\mathbf{m}\), then we may write that \(\mathbf{m}=\mathbf{M}-\mathbf{M}_z\), where \(\mathbf{M}_z\) is the magnetization of the medium in the absence of the variable field. Substituting in (II.1) the values \(\mathbf{M}=\mathbf{m}+\mathbf{M}_z\) and \(\mathbf{H}=\mathbf{h}+\mathbf{H}_z\), we obtain (for a harmonic dependence of \(\mathbf{M}\) on time):

\[ \begin{aligned} i\omega m_x&=\mu_0\gamma(m_yH_z-M_zh_y),\\ i\omega m_y&=\mu_0\gamma(M_zh_x-m_xH_z),\\ i\omega m_z&=0 \end{aligned} \tag{II.8} \]

(we have neglected in this equation the product of small quantities \(\mathbf{h}\) and \(\mathbf{m}\), since \(|h|\ll H_z\)).

Solving this system with respect to \(m_x\) and \(m_y\), we obtain:

\[ \begin{aligned} m_x&=-\frac{\mu_0^2\gamma^2M_zH_z}{\mu_0^2\gamma^2H_z^2-\omega^2}\,h_x -i\,\frac{\mu_0\omega\gamma M_z}{\mu_0^2\gamma^2H_z^2-\omega^2}\,h_y,\\ m_y&=-\frac{\mu_0^2\gamma^2M_zH_z}{\mu_0^2\gamma^2H_z^2-\omega^2}\,h_y +i\,\frac{\mu_0\omega\gamma M_z}{\mu_0^2\gamma^2H_z^2-\omega^2}\,h_x. \end{aligned} \tag{II.9} \]

To find the variable component of the magnetic induction \(\mathbf{b}\), it remains for us to substitute into the equation for \(\mathbf{b}=(\mathbf{h}+\mathbf{m})\mu_0\) the values of the components \(\mathbf{m}\) from (II.9). This gives:

\[ \begin{aligned} b_x&=\mu h_x-ik h_y,\\ b_y&=\mu h_y+ik h_x,\\ b_z&=\mu_0 h_z, \end{aligned} \tag{II.10} \]

where

\[ \frac{\mu}{\mu_0}= \frac{\mu_0\gamma^2B_zH_z-\omega^2}{\mu_0^2\gamma^2H_z^2-\omega^2}, \qquad \frac{k}{\mu_0}= -\frac{\mu_0M_z\gamma\omega}{\mu_0^2\gamma^2H_z^2-\omega^2}, \qquad B_z=(H_z+M_z)\mu_0. \tag{II.11} \]

Thus, equations (II.10) and (II.11) establish the relation between the variable components of the magnetic induction and the magnetic-field strength in a ferromagnetic medium magnetized to

saturation, under the condition that there are no losses in the medium. To take losses into account, one must introduce into the basic equation (II.1) an additional term reducing the energy of the precessional motion of the electron.

If this term is used in the form proposed by Landau and Lifshitz\({}^{14}\), then instead of (II.1) we obtain:

\[ \frac{d\mathbf{M}}{dt} = \gamma[\mathbf{M}\mathbf{H}]\,\mu_0 - \frac{\gamma\delta}{M}[\mathbf{M}[\mathbf{M}\mathbf{H}]]\,\mu_0, \tag{II.1$_1$} \]

where \(\delta\) is a coefficient characterizing the damping and determined experimentally from the width of the resonance absorption curve. For example,\({}^{20}\) for the case of the ferromagnet \(\mathrm{NiOFe_2O_3}\) (nickel ferrite) of spherical form at a frequency of \(24\,000\) MHz, \(\delta = 4.5\cdot 10^{-3}\). If equation (II.1\(_1\)) is solved in the same approximation as equation (II.1), we again obtain the system (II.10), in which only \(\mu\) and \(k\) will be expressed by more complicated formulas.

These formulas have the following form\({}^{23}\):

\[ \mu=\mu'-i\mu'',\quad k=k'-ik'', \tag{II.12} \]

with

\[ \frac{\mu'}{\mu_0}=1+ \frac{ [\mu_0^2\gamma^2 H_z^2(1+\delta^2)-\omega^2][\mu_0^2 M_z\gamma^2 H_z(1+\delta^2)]+2\mu_0^2 M_z\omega^2\gamma\delta^2 H_z }{ [\mu_0^2\gamma^2 H_z^2(1+\delta^2)-\omega^2]^2+4\omega^2\gamma^2\delta^2 H_z^2\mu_0^2 }, \tag{II.13$_1$} \]

\[ \frac{\mu''}{\mu_0}= \frac{ \mu_0 M_z\gamma\omega[\mu_0^2\gamma^2 H_z^2(1+\delta^2)+\omega^2] }{ [\mu_0^2\gamma^2 H_z^2(1+\delta^2)-\omega^2]^2+4\omega^2\gamma^2\delta^2 H_z^2\mu_0^2 }, \tag{II.13$_2$} \]

\[ \frac{k'}{\mu_0}= \frac{ \mu_0 M_z\gamma\omega[\mu_0^2\gamma^2 H_z^2(1+\delta^2)-\omega^2] }{ [\mu_0^2\gamma^2 H_z^2(1+\delta^2)-\omega^2]^2+4\omega^2\gamma^2\delta^2 H_z^2\mu_0^2 }, \tag{II.13$_3$} \]

\[ \frac{k''}{\mu_0}= \frac{ 2\mu_0^2 M_z\omega^2\gamma^2\delta H_z }{ [\mu_0^2\gamma^2 H_z^2(1+\delta^2)-\omega^2]^2+4\omega^2\gamma^2\delta^2 H_z^2\mu_0^2 }. \tag{II.13$_4$} \]

It is easy to see that (II.12) and (II.13) pass into (II.11) if losses are neglected, i.e. if one sets \(\delta=0\).

To solve the problem of the propagation of a plane wave in a ferromagnetic medium, we evidently have to find such a solution of Maxwell’s equations as would not contradict relations (II.10), and in which the field components \(\mathbf{b}\), \(\mathbf{h}\), \(\mathbf{E}\), \(\mathbf{D}\) were proportional to \(e^{i\omega t-\Gamma z}\).

For simplicity of exposition we immediately consider the practically important case when the wave propagates along the \(z\)-axis, i.e. along the constant magnetic field. Eliminating from Maxwell’s equations, written in the form

\[ \operatorname{rot}\mathbf{E}=-\frac{\partial\mathbf{b}}{\partial t},\qquad \operatorname{rot}\mathbf{h}=\frac{\partial\mathbf{D}}{\partial t}, \]

MAGNETIC ROTATION OF THE PLANE OF POLARIZATION

\(\mathbf{E}\) and \(\mathbf{D}\) (we assume that \(\mathbf{D}=\varepsilon \mathbf{E}\)), we obtain:

\[ \nabla^{2}\mathbf{h}=\varepsilon \frac{\partial^{2}\mathbf{b}}{\partial t^{2}} . \tag{II.14} \]

Let us note that \(\varepsilon\) is the complex dielectric permittivity of the medium. Setting \(\mathbf{h}=\mathbf{h}_{0}e^{i\omega t-\Gamma z}\), \(\mathbf{b}=\mathbf{b}_{0}e^{i\omega t-\Gamma z}\), we obtain:

\[ -\Gamma^{2}\mathbf{h}_{0}=\omega^{2}\varepsilon \cdot \mathbf{b}_{0}. \tag{II.15} \]

Substituting now, in place of the quantity \(\mathbf{b}_{0}\), its values from (II.10), we arrive at a system of equations of the following form:

\[ \begin{aligned} (\mu h_{0x}-ikh_{0y})\omega^{2}&=-\Gamma^{2}h_x,\\ (\mu h_{0y}+ikh_{0x})\omega^{2}&=-\Gamma^{2}h_y. \end{aligned} \tag{II.16} \]

The system (II.16) can be satisfied if we set:

\[ h_{0x}=\pm i h_{0y}. \tag{II.17} \]

At the same time, equation (II.17) indicates that the wave has circular polarization, the upper sign referring to a wave polarized in a circle in the clockwise direction, if one looks along the constant magnetic field (a right-polarized wave); the lower sign corresponds to a left-polarized wave. The propagation constant, according to (II.16) and (II.17), will be equal to:

\[ \Gamma_{\pm}=i\omega \sqrt{\varepsilon(\mu \mp k)} . \tag{II.18} \]

Thus, a plane wave propagating in a ferromagnetic medium in the direction of the constant magnetic field is decomposed into two waves of circular polarization, having different propagation constants. It can be shown that if a plane wave propagates at an angle \(\theta\) to the direction of the lines of force of the constant magnetic field \(H_z\), then in this general case it propagates as two elliptically polarized waves, which propagate with different velocities. In the absence of losses, the propagation constants of the two mentioned waves are expressed by the formula\({}^{24}\)

\[ \Gamma_{\pm}=i\omega \sqrt{ \frac{ \varepsilon }{ 2 } \frac{ M\sin^{2}\theta+2\mu \pm \sqrt{M^{2}\sin^{4}\theta+4k^{2}\cos^{2}} }{ \left(\frac{\mu}{\mu_{0}}-1\right)\sin^{2}\theta+1 } }, \tag{II.19} \]

where, for convenience of notation, it has been set

\[ \mu_{0}\left(\frac{\mu^{2}}{\mu_{0}^{2}}-\frac{\mu}{\mu_{0}}-\frac{k^{2}}{\mu_{0}^{2}}\right)=M . \]

In the case \(\theta=\dfrac{\pi}{2}\) we have two linearly polarized waves whose propagation constants are equal to:

\[ \Gamma_{-}=i\omega\sqrt{\mu_0\varepsilon},\qquad \Gamma_{+}=i\omega\sqrt{\frac{\varepsilon}{\mu}\left(\mu^2-k^2\right)}. \tag{II.19a} \]

These formulas indicate the well-known phenomenon of double refraction in a medium subjected to the action of a constant magnetic field (the Cotton–Mouton phenomenon). The minus sign refers to the ordinary wave, the plus sign to the extraordinary one.

In the case \(\theta=0\) or \(\theta=\pi\), we obtain, as was indicated above, two waves of circular polarization with propagation constants

\[ \Gamma_{+}=i\omega\sqrt{\varepsilon(\mu+k)} \quad\text{and}\quad \Gamma_{-}=i\omega\sqrt{\varepsilon(\mu-k)}. \tag{II.19b} \]

In formulas (II.19) losses are not taken into account, and the values of \(\mu\) and \(k\) must be taken from relation (II.11).

Let us return to the investigation of formula (II.18), i.e. of the general case of wave propagation along the direction of the magnetic field in the presence of losses. First of all we note that the effective magnetic permeability for waves of circular polarization is determined by the quantity \(\mu\pm k\), which near resonance varies over wide limits. For what follows it is convenient to separate the real and imaginary parts of equation (II.18). Therefore we shall put

\[ \Gamma_{\pm}=i\omega\sqrt{(\mu\pm k)\varepsilon}=\beta_{\pm}+i\alpha_{\pm}. \tag{II.20} \]

Obviously, \(\beta_{\pm}\) characterizes the attenuation of the wave, while \(\alpha_{\pm}\) is its phase velocity. Solving equation (II.20) with respect to \(\alpha\) and \(\beta\), we obtain:

\[ \alpha_{\pm}=\omega\sqrt{\frac{(\mu'\pm k')\varepsilon'}{2}}\, \sqrt{\sqrt{1+\operatorname{tg}\delta_{\mathrm{M}}\cdot G+\operatorname{tg}^{2}\delta_{\mathrm{d}}}+1+\operatorname{tg}\delta_{\mathrm{M}}\operatorname{tg}\delta_{\mathrm{d}}}, \tag{II.21} \]

\[ \beta_{\pm}=\omega\sqrt{\frac{(\mu'\pm k')\varepsilon'}{2}}\, \sqrt{\sqrt{1+\operatorname{tg}\delta_{\mathrm{M}}\cdot G+\operatorname{tg}^{2}\delta_{\mathrm{d}}}-1-\operatorname{tg}\delta_{\mathrm{M}}\operatorname{tg}\delta_{\mathrm{d}}}, \tag{II.22} \]

where, for convenience of notation, it has been put that

\[ 4\operatorname{tg}\delta_{\mathrm{d}}+\operatorname{tg}^{2}\delta_{\mathrm{M}}\left(1+\operatorname{tg}^{2}\delta_{\mathrm{d}}\right)=G. \]

Here

\[ \operatorname{tg}\delta_{\mathrm{M}}=\frac{\mu''\pm k''}{\mu'\pm k'},\qquad \operatorname{tg}\delta_{\mathrm{d}}=\frac{\varepsilon''}{\varepsilon'},\qquad \varepsilon=\varepsilon'-i\varepsilon''. \tag{II.23} \]

MAGNETIC ROTATION OF THE PLANE OF POLARIZATION

We shall consider two degenerate cases of formulas (II.21) and (II.22), since their investigation in the general form is not possible.

In the first case\(^{23}\) it is assumed that the magnetic losses are negligibly small, i.e. \(\operatorname{tg}\delta_{\mathrm{m}} \simeq 0\). This assumption may be justified when the frequency of ferromagnetic resonance is shifted considerably relative to the frequency of the propagating wave, since the principal component of the magnetic losses is due precisely to the phenomenon of ferromagnetic resonance. Other factors, for example, relaxation of domain boundaries, also increase the magnetic losses; but if the ferromagnet is magnetized to saturation, then the role of these factors is considerably reduced.

Thus, if we put \(\operatorname{tg}\delta_{\mathrm{m}} = 0\), we obtain:

\[ \alpha_{\pm} = \omega \sqrt{\frac{(\mu' \pm k')\varepsilon'}{2}}\, \sqrt{\sqrt{1+\operatorname{tg}^{2}\delta_{\mathrm{d}}}+1}, \tag{II.21a} \]

\[ \beta_{\pm} = \omega \sqrt{\frac{(\mu' \pm k')\varepsilon'}{2}}\, \sqrt{\sqrt{1+\operatorname{tg}^{2}\delta_{\mathrm{d}}}-1}, \tag{II.22a} \]

or

\[ \alpha_{\pm} = \omega \sqrt{\frac{|\varepsilon|+\varepsilon'}{2}}\, \sqrt{\mu' \pm k'}, \tag{II.21б} \]

\[ \beta_{\pm} = \omega \sqrt{\frac{|\varepsilon|-\varepsilon'}{2}}\, \sqrt{\mu' \pm k'}, \tag{II.22б} \]

where

\[ |\varepsilon|=\sqrt{\varepsilon'^2+\varepsilon''^2}. \]

When a wave traverses a path \(l\) in a ferromagnetic medium, the plane of polarization of the wave will rotate through the angle:

\[ \varphi = (n_{+}-n_{-})\frac{\omega}{c}\frac{l}{2} = \]

\[ = \frac{1}{\sqrt{\varepsilon_{0}\mu_{0}}}\, \frac{\omega}{c}\frac{l}{2}\, \sqrt{\frac{|\varepsilon|+\varepsilon'}{2}}\, \left[\sqrt{\mu'+k'}-\sqrt{\mu'-k'}\right]. \tag{I.2a} \]

If, moreover, \(\omega \gg \omega_{\mathrm{res}}\), then, substituting into equation (I.2a) the values from (II.13) or (II.11),

\[ \frac{\mu'}{\mu_0} \simeq 1 \quad\text{and}\quad \frac{k'}{\mu_0} \simeq -\,\frac{\mu_0 M_z\gamma}{\omega}, \]

we obtain:

\[ \frac{\psi}{l} = \frac{\omega}{2c} \sqrt{\frac{|\varepsilon|+\varepsilon'}{2\varepsilon_0}} \left[ \sqrt{1-\frac{\mu_0 M_z\gamma}{\omega}} - \sqrt{1+\frac{\mu_0 M_z\gamma}{\omega}} \right]. \tag{I.26} \]

Taking into account that saturation of ferrites occurs approximately at
\(\mu_0 M_z = 0.2\ \mathrm{wb}/\mathrm{m}^2\) (which corresponds to 2000 gauss), we obtain for centimeter waves\({}^{23}\):

\[ \frac{\mu_0 M_z\gamma}{\omega} \leq \frac{0.2\cdot 1.78\cdot 10^{11}}{2\pi\cdot 10^{10}} = 0.567. \]

In this approximation:

\[ \frac{\psi}{l} = \frac{1}{2c} \sqrt{\frac{|\varepsilon|+\varepsilon'}{2\varepsilon_0}}\, \mu_0 M_z\gamma \tag{I.2v} \]

or

\[ \frac{\psi}{l} = \frac{1.78\cdot 10^{11}}{2\cdot 3\cdot 10^8\cdot \sqrt{2}}\, \sqrt{\frac{|\varepsilon|+\varepsilon'}{\varepsilon_0}}\, \mu_0 M_z\,\frac{180}{\pi} = \]

\[ = 120 \sqrt{\frac{|\varepsilon|+\varepsilon'}{\varepsilon_0}}\, \mu_0 M_z\, \frac{\mathrm{grad}}{\mathrm{cm}}. \tag{I.2g} \]

For example, for \(\varepsilon' = 15\varepsilon_0\), \(\varepsilon'' = 0\), \(\mu_0 M_z = 0.1\ \mathrm{wb}/\mathrm{m}^2\) \((4\pi M_z = 1000\ \text{gauss})\), we obtain \(\psi/l = 65\ \mathrm{grad}/\mathrm{cm}\). It is important to note that, within the limits of the approximations given, the rotation of the plane of polarization does not depend on frequency, as is seen from formula (I.2v).

Let us examine in somewhat more detail the case when magnetic losses are absent, i.e. \(\delta_m = 0\). Then from formula (II.22b) we obtain:

\[ n_{+} = \sqrt{\varepsilon_{\mathrm{eff}}(\mu' + k')}\, \frac{1}{\sqrt{\varepsilon_0\mu_0}}, \tag{II.24} \]

\[ n_{-} = \sqrt{\varepsilon_{\mathrm{eff}}(\mu' - k')}\, \frac{1}{\sqrt{\varepsilon_0\mu_0}}, \tag{II.25} \]

where

\[ \varepsilon_{\mathrm{eff}} = \frac{1}{2}\left(|\varepsilon|+\varepsilon'\right). \]

Since \(\delta_m=0\), we find from (II.11)

\[ \frac{\mu' \pm k'}{\mu_0} = 1+ \frac{\mu_0 M_z\gamma\,[\mu_0\gamma H_z \pm \omega]} {\mu_0^2\gamma^2 H_z^2-\omega^2} \]

and obtain:

\[ \left. \begin{aligned} n_+ &= \sqrt{\varepsilon_{\mathrm{eff}}}\, \sqrt{ 1+ \frac{\mu_0 M_z\gamma}{\mu_0\gamma H_z-\omega}\, \frac{1}{\sqrt{\varepsilon_0\mu_0}} }, \\[6pt] n_- &= \sqrt{\varepsilon_{\mathrm{eff}}}\, \sqrt{ 1+ \frac{\mu_0 M_z\gamma}{\mu_0\gamma H_z+\omega}\, \frac{1}{\sqrt{\varepsilon_0\mu_0}} }. \end{aligned} \right\} \tag{II.26} \]

Graphically, \(n_+\) and \(n_-\) as functions of \(\omega\) are shown in Fig. 3, from which there follows a rather remarkable law of variation of the refractive index \(n_+\) for a right-polarized wave. In a certain region of frequencies \(\omega\) the refractive index \(n_+\) becomes purely imaginary, which means the cessation of the propagation process (analogously to what takes place in a limiting waveguide). In this region the right-polarized wave is strongly absorbed. As for the refractive index of the left-polarized wave, with increasing frequency \(\omega\) it decreases slowly.

Fig. 3.

It is easy to see that the rotation of the plane of polarization, proportional to \((n_- - n_+)\), has a positive sign for \(\omega>\omega_0\), and a negative sign for \(\omega<\omega_0\). The particular case \(\omega \gg \omega_{\mathrm{res}}\), which we considered above (formula (I.2b)), is obtained at once from formulas (II.26). For this case one can also estimate the behavior of the attenuation coefficient \(\beta_{\pm}\). Indeed, comparing formulas (II.22b) and (II.21b), we can come to the conclusion that the graph in Fig. 3 will also characterize the dependence of \(\beta_{\pm}\) on frequency.

Thus the absorption of the right-polarized wave increases with frequency \(\omega\) until \(\omega<\omega_{\mathrm{res}}\); with a further increase of the frequency this wave does not propagate at all. Then, at the frequency \(\omega=\omega_0+\mu_0 M_z\gamma\), the wave again begins to pass, and the losses

slowly increase with frequency. The left-polarized wave does not react at all to the phenomenon of ferromagnetic resonance, and its absorption slowly decreases with increasing frequency.

It should be especially emphasized that Fig. 3 indicates the change in the refractive index due to the change in the effective magnetic permeability. The dependence of the dielectric permittivity on frequency in Fig. 3 is not taken into account, i.e., it is assumed that the dielectric permittivity does not depend on frequency.

In the case where there are only magnetic losses, the formulas for the propagation constant and the attenuation coefficient have the following form:

\[ \alpha_{\pm} = \omega \sqrt{\frac{(\mu' \pm k')\varepsilon'}{2}}\, \sqrt{\sqrt{1+\operatorname{tg}^{2}\delta_{M}}+1}, \tag{II.27} \]

\[ \beta_{\pm} = \omega \sqrt{\frac{(\mu' \pm k')\varepsilon'}{2}}\, \sqrt{\sqrt{1+\operatorname{tg}^{2}\delta_{M}}-1} \tag{II.28} \]

or

\[ \alpha_{\pm} = \omega \sqrt{\varepsilon'\frac{|\mu_{\pm}|+\mu_{1\pm}}{2}}, \tag{II.29} \]

\[ \beta_{\pm} = \omega \sqrt{\varepsilon'\frac{|\mu_{\pm}|-\mu_{1\pm}}{2}}, \tag{II.30} \]

where

\[ \mu_{1\pm}=\mu'\pm k', \qquad |\mu_{\pm}|=\sqrt{(\mu'\pm k')^{2}+(\mu''\pm k'')^{2}}. \]

Then the refractive index will be equal to

\[ n_{\pm} = \sqrt{\varepsilon'\frac{\mu'\pm k'}{2}}\, \sqrt{ 1+ \sqrt{ 1+ \left( \frac{\mu''\pm k''}{\mu'\pm k'} \right)^{2} } }\, \frac{1}{\sqrt{\varepsilon_{0}\mu_{0}}}, \tag{II.31} \]

where \(\mu'\), \(k'\), \(\mu''\), \(k''\) are determined from formulas (II.13), or

\[ n_{\pm} = \sqrt{\frac{1}{2}\varepsilon'\mu_{\mathrm{eff}\,\pm}}\, \sqrt{ 1+ \sqrt{ 1+ \left( \frac{\mu''\pm k''}{\mu_{\mathrm{eff}\,\pm}} \right)^{2} } }\, \frac{1}{\sqrt{\varepsilon_{0}\mu_{0}}}, \tag{II.32} \]

where

\[ \mu_{\mathrm{eff}\,\pm}=\mu'\pm k'. \]

If we substitute the values of \(\mu'\) and \(k'\) from (II.13), we obtain

\[ \frac{\mu_{\mathrm{eff}\,\pm}}{\mu_{0}} = 1+ \frac{\mu_{0}M_{z}\gamma}{\omega}\, \frac{ \left(1+\frac{\omega}{\omega_{0}}\right)^{2} \left(1\pm\frac{\omega}{\omega_{0}}\right) + 2\frac{\omega^{2}}{\omega_{0}^{2}}\delta^{2} }{ \left(1-\frac{\omega^{2}}{\omega_{0}^{2}}\right)^{2} + 4\frac{\omega^{2}}{\omega_{0}^{2}}\delta^{2} }, \tag{II.33} \]

where

\[ \omega_{0}=\gamma H_{z}\mu_{0}. \]

For two values of \(\dfrac{\mu_0 M_z \gamma}{\omega}\) (corresponding to the centimeter-wave region), Figs. 4 and 5 plot the dependences of \(\mu_{\mathrm{eff}\pm}\) on

Fig. 4.

\(\dfrac{\omega}{\omega_0}\). These curves have the same form as in Fig. 3, with the exception of the resonance region, since they take into account damping due to magnetic losses \((\delta \ne 0)\).

In Fig. 5 there is also shown, in reduced form, the difference in the shape of the curves for different values of \(\delta\). Table III characterizes the influence of \(\delta\) on the curves presented in Figs. 4 and 5.

Fig. 5.

As is seen from the table, increasing \(\delta\) one hundredfold has a noticeable effect on the losses only in the region of ferromagnetic resonance.

As follows from formula (II.33), the refractive index is proportional to the square root of \(\mu_{\mathrm{eff}}\). Graphically this dependence is shown in

Table III

$\dfrac{\omega}{\omega_0}$	$\mu_{\varphi\varphi+}$, $\dfrac{\mu_0 M_z\gamma}{\omega_0}=0.62$, $\delta=10^{-4}$	$\mu_{\varphi\varphi+}$, $\dfrac{\mu_0 M_z\gamma}{\omega_0}=0.62$, $\delta=10^{-2}$	$\mu_{\varphi\varphi+}$, $\dfrac{\mu_0 M_z\gamma}{\omega_0}=1.24$, $\delta=10^{-4}$	$\mu_{\varphi\varphi+}$, $\dfrac{\mu_0 M_z\gamma}{\omega_0}=1.24$, $\delta=10^{-2}$	$\mu_{\varphi\varphi-}$, $\dfrac{\mu_0 M_z\gamma}{\omega_0}=0.62$, $\delta=10^{-4}$	$\mu_{\varphi\varphi-}$, $\dfrac{\mu_0 M_z\gamma}{\omega_0}=0.62$, $\delta=10^{-2}$	$\mu_{\varphi\varphi-}$, $\dfrac{\mu_0 M_z\gamma}{\omega_0}=1.24$, $\delta=10^{-4}$	$\mu_{\varphi\varphi-}$, $\dfrac{\mu_0 M_z\gamma}{\omega_0}=1.24$, $\delta=10^{-2}$
0.1	1.69	1.69	2.38	2.38	1.55	1.56	2.13	2.13
0.6	2.55	2.55	4.1	4.1	1.39	1.38	1.775	1.775
0.8	4.1	4.1	7.2	7.2	1.34	1.34	1.69	1.69
0.95	13.4	12.96	25.8	24.9	1.32	1.32	1.64	1.64
0.98	32.0	25.95	63.0	50.9	1.313	1.313	1.63	1.63
1.0	1.31	1.31	1.62	1.62	1.31	1.31	1.62	1.62
1.02	−30.0	−23.8	−61	−48.6	1.307	1.307	1.61	1.61
1.05	−11.4	−10.9	−23.8	−22.8	1.302	1.303	1.605	1.605
1.1	−5.2	−5.13	−11.4	−11.3	1.295	1.295	1.59	1.59
1.5	−0.24	−0.24	−1.48	−1.48	1.248	1.248	1.50	1.50
3.0	0.69	0.38	0.38	0.38	1.155	1.155	1.31	1.31
10	0.931	0.931	0.86	0.86	1.06	1.06	1.11	1.11

Fig. 6, with the ordinate axis representing a quantity proportional to the refractive index. The behavior of the curves has already been explained by us in considering Fig. 3. We shall add only that the rotation of the plane of polarization increases with increasing \(M_z/\omega\) (for the given ratio \(\omega/\omega_0\)). This is easily seen from Fig. 6. The graphs in Figs. 3, 4, 5, 6 make it possible simply to explain the experimental results given in Hogan’s work.\(^{23}\)

Fig. 6.

4. Experimental Results

Magnetic rotation of the plane of polarization at centimeter wavelengths was investigated for the case of wave propagation in a waveguide. Let us note that, for waveguides, the theory presented above will be valid only approximately. However, a study of the magnetic rotation of the plane of polarization for the case of a circular waveguide\(^{25}\) showed that the formula for the rotation of the plane of polarization differs from the same formula in the case of free space only by a constant factor characterizing the type of wave in the waveguide.

The layout of the test chamber is shown in Fig. 7. One of the rectangular waveguides is mounted in such a way that it can be rotated about the longitudinal axis. The ferrite cylinders under investigation are placed in the middle of the circular waveguide.

In addition to measuring the rotation of the plane of polarization, the losses were measured by comparing the transmitted powers in the presence and in the absence of the ferrite cylinder. The ellipticity of the propagating wave was determined by comparing the powers transmitted in the cases when the rectangular waveguide on the detector side was rotated into the positions of maximum and minimum transmission.

MAGNETIC ROTATION OF THE PLANE OF POLARIZATION

Fig. 8 shows the dependence of the rotation of the plane of polarization of the wave on the intensity of the applied magnetic field for manganese-zinc ferrite in the form of a thin disk.

Fig. 7.

The values of the constant magnetic field are such that \(\omega \gg \omega_0\) (see Fig. 6). Consequently, in this case the approximate formula (I.2g) is valid. Indeed, substituting into (I.2g) the measured values \(\varepsilon' = 17\varepsilon_0\), \(\varepsilon'' = 24\varepsilon_0\), \(\psi_0 M_z{}_{\text{sat}} = 0.15 \dfrac{\text{oersted}}{\text{m}^2}\), we find

\[ \frac{\psi}{l} = 121 \ \text{grad}/\text{cm}, \]

which is in complete agreement with the measured values of the angle of rotation (\(123^\circ\)).

Fig. 8.

With a further increase of the magnetic field, the curve of Fig. 8 should go upward, since we shall be approaching the frequency of ferromagnetic resonance; this agrees with the theoretical curves of Fig. 6. (An increase of the magnetic field corresponds to a displacement in Fig. 6 in the region of values \(\dfrac{\omega}{\omega_0} \gg 1\) from right to left.) Indeed, the curves\(^ {26}\) of Fig. 9 confirm this assumption.

Formula (1.2г) also indicates the independence of the rotation from the frequency. The curve in Fig. 10 confirms this only partially, since the frequency difference for the two measurements was taken to be very small (270 MHz).

Fig. 9.

Fig. 10.

The attenuation in ferrite varies within wide limits depending on the material. For the same manganese-zinc ferrite, the principal losses are due to the complex dielectric permittivity. Only in the region close to ferromagnetic resonance do the magnetic losses increase sharply owing to absorption of the right-polarized ...

MAGNETIC ROTATION OF THE PLANE OF POLARIZATION

polarized wave. These magnetic losses can be taken into account if, by measuring the ellipticity of the propagating wave, one calculates the difference between the absorptions of the right- and left-polarized waves. Let us note that the ellipticity of a wave passing through a ferrite is caused mainly by the phenomenon of ferromagnetic resonance, i.e., by the unequal absorption of two waves circularly polarized in opposite directions.

The results obtained in this way for a manganese-zinc ferrite are shown in Fig. 11. This graph confirms

Fig. 11.

the remark we made when considering the curve in Fig. 8 concerning the magnitude of the magnetic field corresponding to the resonance frequency.

With a further increase of the magnetic field, the curve in Fig. 11 will follow the shape of the resonance-absorption curve.

Let us give one more graph (Fig. 12), pertaining to the ferrite “Ferramic G”[^23], which has very small dielectric losses but produces strong absorption at a frequency of 9000 MHz as a result of magnetic losses. What these losses are connected with can be determined experimentally. Indeed, if the losses are connected with the relaxation of domain boundaries, then they affect right- and left-polarized waves equally. If, however, the losses are connected with ferromagnetic resonance, then the right-polarized wave is absorbed.

Fig. 12 illustrates the proposition stated above. With an increase of the magnetic field, the absorption of the left-polarized wave falls rapidly, and at saturation it disappears almost completely. The absorption of the right-polarized wave first falls somewhat, since the losses connected with the relaxation of domain boundaries decrease, and then the absorption rises rapidly; this rise can evidently be explained by the approach to the region of ferromagnetic resonance.

A number of other experimental data[^23], obtained for several ferrites at a frequency of 9000 MHz, are given in Table IV (see p. 230).

Table IV

Material	Dimensions in cm (length × diameter)	Applied field, in oersteds	Rotation per 1 cm of path, in degrees	Losses in dB	Ellipticity in dB	Standing-wave voltage coefficient at input
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	0	0	10,0	50	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	245	15,6	10,3	50	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	490	33,5	10,0	23,2	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	735	58,2	9,2	15,0	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	980	81,6	9,1	12,1	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	1225	107	9,2	10,9	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	1470	120	10	10,4	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	1715	125	11	9,3	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	1960	123	11,2	9,0	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	2206	121	11,3	7,7	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	2450	123	11,4	6,6	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	2695	—	12,4	5,0	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	2940	—	13,0	3,7	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	3185	—	—	3,0	—
Manganese-zinc ferrite $\mathrm{Mn}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$0{,}447 \times 2{,}28$	3675	—	—	1,4	—
$\mathrm{Ni}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$1{,}36 \times 2{,}28$	0	0	0,8	$>40$	—
$\mathrm{Ni}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$1{,}36 \times 2{,}28$	245	25	1,9	$\sim 40$	—
$\mathrm{Ni}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$1{,}36 \times 2{,}28$	490	44	2,7	$\sim 40$	—
$\mathrm{Ni}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$1{,}36 \times 2{,}28$	735	56	2,9	$\sim 40$	—
$\mathrm{Ni}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$1{,}36 \times 2{,}28$	980	61	2,7	40	—
$\mathrm{Ni}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$1{,}36 \times 2{,}28$	1225	68	2,8	—	—
$\mathrm{Ni}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$1{,}36 \times 2{,}28$	1715	82	3,33	—	—
$\mathrm{Ni}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$1{,}36 \times 2{,}28$	1930	85	4,9	—	—
$\mathrm{Ni}_{\delta}\mathrm{Zn}_{1-\delta}\mathrm{Fe}_{2}\mathrm{O}_{4}$	$1{,}36 \times 2{,}28$	2450	118	7,3	0,8	—
Ferramic A	$2{,}54 \times 0{,}635$	0	0	1,1	$>50$	0,7
Ferramic A	$2{,}54 \times 0{,}635$	245	34,9	0,8	$>50$	0,3
Ferramic A	$2{,}54 \times 0{,}635$	490	43,7	0,8	$>50$	0,3
Ferramic A	$2{,}54 \times 0{,}635$	735	48,3	0,8	$>50$	0,3
Ferramic A	$2{,}54 \times 0{,}635$	980	51,1	1,0	$>50$	0,4
Ferramic A	$2{,}54 \times 0{,}635$	1225	54,0	1,1	$>50$	—
Ferramic A	$2{,}54 \times 0{,}635$	1715	57,0	1,1	$>50$	—
Ferramic A	$2{,}54 \times 0{,}635$	1960	60,0	1,9	$>50$	0,4
Ferramic A	$2{,}54 \times 0{,}635$	2450	63,0	3,0	35	—
Ferramic A	$2{,}54 \times 0{,}635$	2695	24,2	3,7	—	—
Ferramic G	$1{,}77 \times 2{,}28$	0	0	23,2	$\gg 30$	—
Ferramic G	$1{,}77 \times 2{,}28$	245	38	21,4	23,0	—
Ferramic G	$1{,}77 \times 2{,}28$	490	77	16,7	7,6	—
Ferramic G	$1{,}77 \times 2{,}28$	735	124	12,4	2,1	—
Ferramic G	$1{,}77 \times 2{,}28$	980	157	9,9	1,4	—
Ferramic G	$1{,}77 \times 2{,}28$	1225	170	7,7	0,7	—
Ferramic G	$1{,}77 \times 2{,}28$	1470	180	6,0	0,7	—
Ferramic G	$1{,}77 \times 2{,}28$	3430	f. r.	7,1	0,0	—

The main conclusions that can be drawn from the data in the table are as follows:

1. Losses in ferrites are, as a rule, considerable. The material “Ferramic A” has minimal losses (0.8 dB).

Fig. 12.

1. In a region not close to ferromagnetic resonance, the wave is approximately linearly polarized (the ellipticity is large).

2. The rotation of the plane of polarization in all ferrites is measured in tens of degrees per centimeter of path length, i.e., it is sufficiently large.

3. In the experiments described, apparently, the influence of the specimen shape on the losses was not investigated. At the same time, one should expect this influence to be significant, since the frequency of ferromagnetic resonance depends on the shape of the body.

II. ROTATION OF THE PLANE OF POLARIZATION IN ARTIFICIAL DIELECTRICS

If metallic particles, insulated from one another, are arranged in a certain order in some volume, then this volume, as is known, will be equivalent to a dielectric (the dimensions of the particles and the distances between them must be much smaller than the wavelength). Just as in an ordinary dielectric molecules are polarized under the action of an external field, i.e., acquire an electric moment, so in the metallic particles of an artificial dielectric

free electrons are displaced, which also leads to the appearance of an electric moment in the metallic particle.

The formula expressing the dielectric permittivity of an artificial dielectric is the same as for an ordinary dielectric:

\[ \varepsilon=\varepsilon_0+aN, \tag{III.1} \]

where \(a\) is the polarizability of the particle, i.e. the ratio of the electric moment \(\mathbf p\) of the particle to the magnitude of the field producing this moment,

\[ a=\frac{\mathbf p}{\mathbf E}, \]

and \(N\) is the number of particles per unit volume. (Formula (III.1) does not take into account the mutual influence of neighboring particles. If this allowance is necessary, one must use the well-known Lorentz–Lorenz formula\(^2\).)

For example, in the case where the particles are metallic disks of radius \(a\ll\lambda\), situated in the plane of the electric vector, the dielectric permittivity is determined by the formula

\[ \varepsilon=\varepsilon_0+\varepsilon_0\frac{16}{3}Na^3, \]

since the polarizability of the disk is equal to\(^ {27}\):

\[ \alpha=\frac{16}{3}\varepsilon_0 a^3. \tag{III.2} \]

In the case of metallic spheres\(^2\), \(\alpha=4\pi a^3\varepsilon_0\). The polarization of the artificial dielectric, i.e. the electric moment of a unit volume, is equal to

\[ \mathbf P=N\mathbf p=N\alpha\mathbf E, \tag{III.3} \]

and the polarization current is

\[ \mathbf J=\frac{\partial \mathbf P}{\partial t} =\alpha N\frac{\partial \mathbf E}{\partial t}. \tag{III.4} \]

Let us now suppose that in the artificial dielectric a wave propagates along the \(z\)-axis, and that a constant magnetic field \(\mathbf B_0\) is applied in the direction of propagation.

In this case, as is known, the Hall effect will be observed in the metallic particles; it consists in the fact that the electrons forming the polarization current, under the action of the magnetic field, are deflected toward one edge of the metallic particle and accumulate there until the electric field they produce balances the deflecting action of the magnetic field. The magnitude and direction of this field are determined by the formula

\[ \mathbf E_H=R[\mathbf B_0\mathbf J], \tag{III.5} \]

where \(R\) is the Hall coefficient, depending on the material of the particle. Со-

accordingly, the electric induction vector will change by the amount

\[ \mathbf{D}_{H}=aN\mathbf{E}_{H}=aNR[\mathbf{B}_{0}\mathbf{J}] =a^{2}N^{2}R\left[\mathbf{B}_{0}\cdot\frac{\partial \mathbf{E}}{\partial t}\right]. \tag{III.6} \]

The resulting vector \(\mathbf{D}\) will be equal to:

\[ \mathbf{D}=(\varepsilon_{0}+aN)\mathbf{E}+\mathbf{D}_{H} =(\varepsilon_{0}+aN)\mathbf{E} +a^{2}N^{2}R\left[\mathbf{B}_{0}\cdot\frac{\partial \mathbf{E}}{\partial t}\right]. \tag{III.7} \]

It now remains to solve Maxwell’s equations under the condition that \(\mathbf{D}\) and \(\mathbf{E}\) are related by formula (III.7).

For simplicity we shall restrict ourselves to the case \(\mu=\mu_{0}\), i.e., the case when the magnetic lines of force of the propagating wave are not distorted in passing through the artificial dielectric under consideration. For this, obviously, it is necessary to make the metallic elements flat and to arrange them perpendicular to the direction of propagation of the wave.

Indeed, in this case the magnetic lines of force of the propagating wave will be tangent to the flat metallic elements and will not be distorted, since the boundary conditions are automatically satisfied.

Taking, therefore, \(\mathbf{B}=\mu_{0}\mathbf{H}\), we obtain:

\[ \nabla^{2}\mathbf{E} =\mu_{0}(\varepsilon_{0}+aN)\frac{\partial^{2}\mathbf{E}}{\partial t^{2}} +\mu_{0}a^{2}N^{2}R \left[\mathbf{B}_{0}\frac{\partial^{3}\mathbf{E}}{\partial t^{3}}\right]. \tag{III.8} \]

For the particular case under consideration of propagation along the \(z\)-axis of a plane wave in an infinite medium, equation (III.8) takes the form:

\[ \frac{\partial^{2}E}{\partial z^{2}} =\frac{1+\rho}{c^{2}}\frac{\partial^{2}E}{\partial t^{2}} +i\frac{\varepsilon_{0}^{2}R B_{0}}{c^{2}} \frac{\partial^{3}E}{\partial t^{3}}, \tag{III.9} \]

where

\[ \rho=\frac{aN}{\varepsilon_{0}};\qquad E=E_{x}+iE_{y}. \tag{III.10} \]

For a harmonic time dependence, i.e. for \(E=Ee^{\pm i\omega t}\), we obtain:

\[ \frac{\partial^{2}E}{\partial z^{2}}+k^{2}_{\pm}E=0, \tag{III.11} \]

where

\[ k^{2}_{\pm} =\frac{\omega^{2}}{c^{2}} \left[1+\rho\mp\omega\varepsilon_{0}\rho^{2}RB_{0}\right]. \tag{III.12} \]

Consequently, the general solution of equation (III.9) will have the form:

\[ E=A\left[ e^{\,i\omega\left(t-\frac{z}{v_{+}}\right)} + e^{-\,i\omega\left(t-\frac{z}{v_{-}}\right)} \right], \tag{III.13} \]

where, according to (III.12),

\[ \frac{1}{v_{\pm}^{2}}=\frac{k_{\pm}^{2}}{\omega^{2}}=\frac{1}{c^{2}}\left[1+\rho \mp \omega \varepsilon_{0}\rho^{2}RB_{0}\right]. \tag{III.14} \]

Equation (III.13) describes two waves, circularly polarized in opposite directions and propagating with different phase velocities in the positive direction of the \(z\)-axis.

This is equivalent to the fact that the plane of polarization of a linearly polarized wave propagating in such a medium will rotate. This becomes immediately clear if (III.13) is rewritten in a somewhat different form, extracting from the exponential factor the part that does not depend on time. Then we obtain:

\[ E=Ae^{i\frac{\omega z}{u}}\cos \omega\left(t-\frac{z}{v}\right), \tag{III.15} \]

where

\[ \frac{1}{v}=\frac{1}{2}\left(\frac{1}{v_{-}}+\frac{1}{v_{+}}\right),\qquad \frac{1}{u}=\frac{1}{2}\left(\frac{1}{v_{-}}-\frac{1}{v_{+}}\right). \tag{III.16} \]

Thus, after traversing a distance \(z\), the plane of polarization of the wave rotates through the angle:

\[ \psi=\frac{\omega z}{u}=\frac{\omega z}{2}\left(\frac{1}{v_{-}}-\frac{1}{v_{+}}\right)=\frac{\omega}{2c}(n_{-}-n_{+})z. \tag{III.17} \]

In the centimeter-wave region, for dielectrics used in practice, the condition is satisfied:

\[ \varepsilon_{0}\rho^{2}RB_{0}\omega \ll 1+\rho. \tag{III.18} \]

Then, expanding (III.14) in a series and retaining the first term of the series, we obtain:

\[ \frac{1}{v_{\pm}}=\frac{1}{c}\sqrt{(1+\rho)\mp\omega\varepsilon_{0}\rho^{2}RB_{0}} \simeq \]

\[ \simeq \frac{1}{c}\sqrt{1+\rho}\left(1\mp\frac{1}{2}\frac{\omega\varepsilon_{0}\rho^{2}RB_{0}}{1+\rho}\right). \tag{III.19} \]

In the same approximation:

\[ \frac{1}{u}=\frac{\rho^{2}}{\sqrt{1+\rho}}\frac{\varepsilon_{0}RB_{0}\omega}{2c} \]

and, consequently,

\[ \psi=\frac{\omega z}{u}=\frac{\rho^{2}}{\sqrt{1+\rho}}\frac{\varepsilon_{0}RB_{0}\omega^{2}}{2c}\,z. \tag{III.20} \]

If the angle of rotation of the plane of polarization per unit path length

if expressed in degrees, then the last formula takes the form

\[ \frac{\psi}{z}=\frac{3RB_0}{\lambda^2}\,\frac{\rho^2}{\sqrt{1+\rho}}\,\frac{\mathrm{grad}}{m}. \tag{III.21} \]

Let us give a number of numerical data.\(^{28}\)

Taking \(B_0=0.1\,\dfrac{vb}{m^2}\) (1000 gauss) and \(\rho=\dfrac{\alpha N}{\varepsilon_0}=8\), we obtain, for the wave \(\lambda=1\ \text{cm}\),

\[ \frac{\psi}{z}=0.61\cdot 10^5 R\,\frac{\mathrm{grad}}{m}. \]

For good conducting metals \(\dfrac{\psi}{z}\sim 0.5\cdot 10^{-5}\,\dfrac{\mathrm{grad}}{m}\), whereas for germanium of type \(N\)

\[ \frac{\psi}{z}=512\,\frac{\mathrm{grad}}{m}. \]

This is explained by the fact that in germanium the resistance is approximately \(10^6\) times greater, as a result of which the Hall constant, which is inversely proportional to the total charge of the conduction electrons in unit volume,\(^{1}\) is very large in germanium.

On the other hand, a large resistance leads to considerable losses in metallic elements. Thus, appreciable rotation of the plane of polarization in all known metals is associated with considerable losses. It should be noted, however, that the resistance and the Hall constant are determined by different factors. Therefore the ratio of the Hall constant to the specific resistance varies within wide limits for different metals. For copper, for example, it is of the order of 0.003, and for bismuth—0.9. This gives some grounds for supposing that a material may be found which would have a large Hall constant and a small resistance.

IV. ROTATION OF THE PLANE OF POLARIZATION IN AN ELECTRON PLASMA

1. General theory of wave propagation in an ionized medium in the presence of a constant magnetic field

We shall use here the same method of investigation as in considering wave propagation in an artificial dielectric. Our problem is therefore reduced to solving Maxwell’s equations

\[ \left. \begin{aligned} \operatorname{rot}\mathbf{E}&=-\frac{\partial \mathbf{B}}{\partial t},\\ \operatorname{rot}\mathbf{H}&=\frac{\partial \mathbf{D}}{\partial t}+\mathbf{J} \end{aligned} \right\} \tag{IV.1} \]

for a definite relation \(\mathbf{D}=\mathbf{D}(\mathbf{E})\) in the medium under consideration, which must first be determined.

Suppose that the wave propagates along the \(z\)-axis, and that the constant magnetic field \(\mathbf{B}_0\) is parallel to the plane \(yOz\) and makes an angle \(\beta\) with the \(z\)-axis (Fig. 13), i.e.,

\[ B_{0z}=B_0\cos\beta,\qquad B_{0y}=B_0\sin\beta . \tag{IV.2} \]

The magnetic permeability of the ionized medium may be taken equal to \(\mu_0\).

Let us first consider the case of a rarefied ionized gas, when the mean free path of the electron is so large that losses may be neglected, i.e., one may put \(\sigma=0\). Under the indicated conditions, equations (IV.1) take the form

\[ \left. \begin{aligned} \operatorname{rot}\mathbf{E}&=-\mu_0\frac{\partial\mathbf{H}}{\partial t},\\ \operatorname{rot}\mathbf{H}&=\frac{\partial\mathbf{D}}{\partial t}. \end{aligned} \right\} \tag{IV.3} \]

Fig. 13.

For a harmonic dependence of \(\mathbf{H}\) and \(\mathbf{D}\) on time we obtain:

\[ \Delta \mathbf{E}+\omega^2\mu_0\mathbf{D}=0 \tag{IV.4} \]

or

\[ \left. \begin{aligned} \frac{\partial^2 E_x}{\partial z^2}+\omega^2\mu_0D_x&=0,\\ \frac{\partial^2 E_y}{\partial z^2}+\omega^2\mu_0D_y&=0. \end{aligned} \right\} \tag{IV.5} \]

To express \(\mathbf{D}\) through \(\mathbf{E}\), we use the equation of motion:

\[ m\mathbf{r}''=-e\mathbf{E}-e[\mathbf{r}'\mathbf{B}_0], \tag{IV.6} \]

where \(\mathbf{r}\) is the displacement of the electrons relative to their initial positions, \(m\) is the mass, and \(e\) is the charge of the electron; primes denote differentiation with respect to time. In doing so we have neglected the magnetic field strength of the propagating wave, since

\[ \mu_0\mathbf{H}\ll \mathbf{B}_0 . \]

Multiplying (6) by \(Ne\), where \(N\) is the number of electrons per unit volume, and taking into account that the electric moment of unit volume is equal to \(\mathbf{P}=-eN\mathbf{r}\), we have:

\[ -\omega^2\mathbf{P}=\frac{Ne^2}{m}\mathbf{E}-i\frac{\omega e}{m}[\mathbf{P}\mathbf{B}_0]. \tag{IV.7} \]

Taking into account the relation \(\mathbf D=\varepsilon_0\mathbf E+\mathbf P\) and formula (IV.7), we obtain\({}^{29}\) the desired relation between \(\mathbf D\) and \(\mathbf E\):

\[ \left. \begin{aligned} D_y&=\varepsilon_0E_y+P_y=-i\varepsilon_0BE_x+\varepsilon_0(1+C)E_y,\\ D_x&=\varepsilon_0E_x+P_x=\varepsilon_0(1+A)E_x+i\varepsilon_0BE_y,\\ D_z&=\varepsilon_0E_z+P_z=0. \end{aligned} \right\} \tag{IV.8} \]

Here

\[ A=p(\omega^2-\omega_0^2);\qquad B=p\,\frac{\omega_z}{\omega}(\omega^2-\omega_0^2); \]

\[ C=p(\omega^2-\omega_0^2-\omega_x^2);\qquad \omega_0^2=\frac{Ne^3}{\varepsilon_0m}=3.22\cdot10^3N; \]

\[ \omega_z=\omega_{\mathrm{res}}\cos\beta=\frac{eB_{0z}}{m};\qquad \omega_y=\omega_{\mathrm{res}}\sin\beta=\frac{eB_{0y}}{m}; \]

\[ p=\frac{\omega_0^2}{(\omega^2-\omega_0^2)(\omega_z^2-\omega^2)+\omega^2\omega_x^2} \quad\text{and}\quad \omega_{\mathrm{res}}=\frac{eB_0}{m} \]

\[ (\omega_{\mathrm{res}}\text{ is the frequency of gyromagnetic resonance}). \]

Substituting the found dependence of \(\mathbf D\) on \(\mathbf E\) from (IV.8) into (IV.5), we obtain:

\[ \left. \begin{aligned} \frac{\partial^2 E_y}{\partial z^2} +\varepsilon_0\mu_0\omega^2[-iBE_x+(1+C)E_y]&=0,\\ \frac{\partial^2 E_x}{\partial z^2} +\varepsilon_0\mu_0\omega^2[(1+A)E_x+iBE_y]&=0. \end{aligned} \right\} \tag{IV.9} \]

We seek the solution of the system in the form:

\[ \left. \begin{aligned} E_x&=E_{mx}e^{-i\frac{\omega}{c}nz},\\ E_y&=E_{my}e^{-i\frac{\omega}{c}nz}, \end{aligned} \right\} \tag{IV.10} \]

where \(n\) is the refractive index of the ionized gas. Substituting (IV.10) into (IV.9), we obtain the system of equations:

\[ \left. \begin{aligned} [1+A-n^2]E_{mx}+iBE_{my}&=0,\\ -iBE_{mx}+[1+C-n^2]E_{my}&=0, \end{aligned} \right\} \tag{IV.11} \]

which will have nontrivial solutions under the condition

\[ \left| \begin{array}{cc} 1+A-n^2 & iB\\ -iB & 1+C-n^2 \end{array} \right|=0 \]

or

\[ n^4-(2+A+C)n^2+(1+A)(1+C)-B^2=0. \tag{IV.12} \]

Solving this equation, we obtain an expression for the refractive index of an ionized gas on which a constant magnetic field is imposed:

\[ n_{\pm}^{2} = 1- \frac{\omega_0^{2}} {\omega^{2} -\dfrac{\omega^{2}\omega_y^{2}}{2(\omega^{2}-\omega_0^{2})} \mp \sqrt{ \left[ \dfrac{\omega^{2}\omega_y^{2}}{2(\omega^{2}-\omega_0^{2})} \right]^{2} +\omega_z^{2}\omega^{2} }} . \tag{IV.13} \]

The polarization of the wave will be characterized by the relation

\[ \left(\frac{E_{mx}}{E_{my}}\right)_{\pm} = i\left[ \frac{\omega_y^{2}\omega}{2(\omega^{2}-\omega_0^{2})\omega_z} \mp \sqrt{ \left[ \frac{\omega_y^{2}\omega}{2\omega_z(\omega^{2}-\omega_0^{2})} \right]^{2} +1 } \right], \tag{IV.14} \]

which is obtained from formulas (IV.11).

Thus, we have obtained a solution indicating that in the medium under consideration two elliptically polarized waves propagate (in opposite directions), and the velocities of propagation of these waves are different.

Let us consider special cases of formulas (IV.13) and (IV.14).

a) The magnetic field is absent \((B_0=0;\ \omega_y=\omega_z=0)\). In this case

\[ n^{2}=1-\frac{Ne^{2}}{\varepsilon_0 m\omega^{2}}, \tag{IV.13a} \]

\[ \begin{aligned} D_x&=\varepsilon_0 n^{2}E_x,\\ D_y&=\varepsilon_0 n^{2}E_y, \end{aligned} \tag{IV.8a} \]

i.e. the medium is isotropic \((\varepsilon_{xx}=\varepsilon_{yy}=\varepsilon_0 n^{2})\).

b) Longitudinal magnetic field \((B_0=B_{0z};\ \omega_y=0;\ \omega_z=-\omega_{\mathrm{res}})\). This is precisely the case when two circularly polarized waves propagate with different velocities. Indeed, from (IV.14) we have:

\[ \left(\frac{E_x}{E_y}\right)_{\pm}=\mp i; \tag{IV.14a} \]

\[ n_{\pm}^{2} = 1-\frac{\omega_0^{2}}{\omega(\omega\mp\omega_z)}, \tag{IV.13b} \]

where the upper sign corresponds to the right-polarized wave, and the lower sign to the left-polarized wave. In Fig. 14 the approximate course of variation of the squares of the refractive indices for right- and left-polarized waves is shown.

Thus, if a plane wave propagates in the medium under consideration, the plane of polarization of the wave will rotate.

Fig. 14.

Fig. 15.

The rotation of the plane of polarization per unit length of the path will be equal to

\[ \frac{\psi}{l}=\frac{\omega}{2c}(n_{-}-n_{+}), \]

i.e.

\[ \frac{\psi}{l} = \frac{\omega}{2c} \left[ \sqrt{1-\frac{Ne^{2}}{\varepsilon_{0}m\omega}\,\frac{1}{\omega+\omega_{\mathrm{res}}}} - \sqrt{1-\frac{Ne^{2}}{\varepsilon_{0}m\omega}\,\frac{1}{\omega-\omega_{\mathrm{res}}}} \right]. \tag{IV.15} \]

This is the final calculation formula. The dependence of the rotation of the plane of polarization of the wave on \(\dfrac{\omega}{\omega_{\mathrm{res}}}\) is shown in Fig. 15.

b) Transverse magnetic field \((B_{0}=B_{0y},\ \omega_{z}=0)\).

This is the case of double refraction

\[ \left. \begin{aligned} \left(\frac{E_{mx}}{E_{my}}\right)_{+}&=\infty,\quad \text{i.e. } E_{1}=E_{mx},\\ \left(\frac{E_{mx}}{E_{my}}\right)_{-}&=0,\quad \text{i.e. } E_{2}=E_{my}. \end{aligned} \right\} \tag{IV.14б} \]

The refractive indices for these two linearly polarized waves are equal to:

\[ \left. \begin{aligned} n_{+}^{2} &= 1- \frac{\omega_{0}^{2}} {\dfrac{\omega^{2}\omega_{y}^{2}}{\omega^{2}-\omega_{0}^{2}}}, \\[6pt] n_{-}^{2} &= 1-\frac{\omega_{0}^{2}}{\omega^{2}} = 1-\frac{Ne^{2}}{\varepsilon_{0}m\omega^{2}}. \end{aligned} \right\} \tag{IV.13в} \]

Thus the wave corresponding to the minus sign does not experience the influence of the magnetic field. This occurs because the indicated wave has a component of the electric field \(E_{y}\) coinciding with the direction of the constant magnetic field.

Under the action of the wave field the electron also moves in the direction of the \(y\)-axis, i.e. in the direction of the constant magnetic field, and consequently does not experience its action. Thus, the phase velocity of the indicated wave does not depend on the constant magnetic field. In optics this wave is called the “ordinary” wave. The other wave—the “extraordinary” wave—propagates with a phase velocity that depends on the external field and has a component along the direction of propagation, i.e. it is not transverse.

Such are the phenomena that occur when waves propagate in an ionized medium in the presence of a constant magnetic field. Let us recall that we have neglected losses caused by collisions of electrons with molecules and ions. If they are taken into account, then in the original

of equations (IV.4), instead of \(\mathbf{D}\) one must write \(\left(\mathbf{D}-i\dfrac{\mathbf{I}}{\omega}\right)\), and in the equation of motion (IV.6) introduce a term proportional to the velocity of the electron \(\mathbf{r}'\).

If we assume that \(\nu\) is the average number of collisions experienced by each electron with neutral molecules in one second, and that at each collision the electron transfers its accumulated momentum to the molecule, then the total change of momentum per second will be determined by the term \(\nu m\mathbf{r}'\).

It is this expression that must be added to the left-hand side of equation (IV.6). The exact value of \(\nu\) can be determined only as a result of a gas-kinetic consideration. It is clear, however, that \(\nu\) is proportional to the number of molecules per unit volume, to the radius of the molecule, and to the mean velocity of the electrons \(\bar{v}\).

We shall give the final formulas only for the case of a longitudinal magnetic field of interest to us \(^{30}\):

\[ \left. \begin{aligned} \varepsilon_{\pm} &= n_{\pm}^{2}-\frac{c^{2}}{\omega^{2}}\beta_{\pm}^{2} = 1-\frac{Ne^{2}}{\varepsilon_{0}m\omega}\, \frac{\omega \mp \omega_{\mathrm{res}}}{(\omega \mp \omega_{\mathrm{res}})^{2}+\nu^{2}}, \\[6pt] \frac{\sigma_{\pm}}{\omega} &= 2n_{\pm}\frac{c}{\omega}\beta_{\pm} = \frac{Ne^{2}}{\varepsilon_{0}m\omega}\, \frac{\nu}{(\omega \mp \omega_{\mathrm{res}})^{2}+\nu^{2}}, \end{aligned} \right\} \tag{IV.16} \]

where \(\beta_{\pm}\) is the attenuation coefficient, since a wave propagating in the direction of the \(z\)-axis contains a factor of the form

\[ e^{\,i[\omega t-(\alpha_{\pm}-i\beta_{\pm})z]} = e^{-\beta_{\pm}z}e^{\,i(\omega t-\alpha_{\pm}z)}. \]

From (IV.16) we find that

\[ \left. \begin{aligned} \alpha_{\pm} &=\frac{\omega}{c}n_{\pm} =\frac{\omega}{c} \sqrt{ \frac{\varepsilon_{\pm}}{2} \left[ \sqrt{1+\left(\frac{\sigma_{\pm}}{\omega\varepsilon_{\pm}}\right)^{2}} +1 \right] }, \\[6pt] \beta_{\pm} &=\frac{\omega}{c} \sqrt{ \frac{\varepsilon_{\pm}}{2} \left[ \sqrt{1+\left(\frac{\sigma_{\pm}}{\omega\varepsilon_{\pm}}\right)^{2}} -1 \right] }, \end{aligned} \right\} \tag{IV.17} \]

where \(\varepsilon_{\pm}\) and \(\sigma_{\pm}\) are given by formulas (IV.16).

2. Calculation of Some Cases

In order to obtain a general idea of what angles of rotation of the plane of polarization and what losses should be expected under practically realizable conditions, we have calculated several variants corresponding to typical regimes in a plasma.

The results obtained are collected in Tables V, VI, and VII below. The calculation was carried out for various \(\omega_{0}\) (the natural frequency of oscillations in the plasma) at two values of \(\nu\).

$\omega_{\mathrm{res}}$	$H$ in oersteds	$a_+$	$a_-$	$a_- - a_+$	$\beta_+$	$\beta_-$
		\multicolumn{5}{c}{$\omega_0 = 2\cdot 10^9,\ \gamma = 4\cdot 10^9$}
0	0	$11237^\circ 15'$	$11237^\circ 15'$	$0^\circ$	0,0077	0,0077
$2\cdot 10^{10}$	1137	$11233^\circ 53'$	$11238^\circ 23'$	$4^\circ 30'$	0,0175	0,0043
$4\cdot 10^{10}$	2270	$11224^\circ 53'$	$11239^\circ 30'$	$14^\circ 37'$	0,0718	0,0027
$5\cdot 10^{10}$	2848	$11208^\circ 2'$	$11240^\circ 38'$	$38^\circ 36'$	0,2824	0,0023
$5{,}5\cdot 10^{10}$	3185	$11195^\circ 46'$	$11240^\circ 38'$	$44^\circ 52'$	0,8637	0,0021
$5{,}8874\cdot 10^{10}$	3350	$11244^\circ 24'$	$11240^\circ 38'$	$-3^\circ 46'$	1,6668	0,00192
$6\cdot 10^{10}$	3415	$11269^\circ 05'$	$11240^\circ 38'$	$-28^\circ 27'$	1,5408	0,00189
$6{,}5\cdot 10^{10}$	3700	$11287^\circ 53'$	$11240^\circ 38'$	$-47^\circ 15'$	0,4962	0,0017
$7\cdot 10^{10}$	3980	$11274^\circ 22'$	$11240^\circ 38'$	$-33^\circ 44'$	0,1903	0,0016
$8\cdot 10^{10}$	4550	$11260^\circ 52'$	$11240^\circ 38'$	$-20^\circ 14'$	0,0576	0,0014
$10\cdot 10^{10}$	5680	$11253^\circ 00'$	$11241^\circ 45'$	$-11^\circ 15'$	0,0156	0,001

$\omega_{\mathrm{res}}$	$H$ in oersteds	$a_+$	$a_-$	$a_- - a_+$	$\beta_+$	$\beta_-$
		\multicolumn{5}{c}{$\omega_0 = 6\cdot 10^{10},\ \gamma = 4\cdot 10^9$}
0	0	$2661^\circ 27'\, i$	$2661^\circ 27'\, i$	$0^\circ$	$29,14\, i$	$29,14\, i$
$2\cdot 10^{10}$	1137	$8470^\circ 6'\, i$	$5370^\circ 8'$	—	$20,86\, i$	8,06
$4\cdot 10^{10}$	2270	$16490^\circ 27'\, i$	$6957^\circ 40'$	—	$43,98\, i$	3,93
$5\cdot 10^{10}$	2848	$25284^\circ 23'\, i$	$7451^\circ 21'$	—	$112,6\, i$	3,05
$5{,}5\cdot 10^{10}$	3185	$32732^\circ 24'\, i$	$7657^\circ 10'$	—	$265,9\, i$	2,67
$5{,}8874\cdot 10^{10}$	3350	$32119^\circ 37'$	$7801^\circ 5'$	$-24318^\circ 32'$	525,11	2,36
$6\cdot 10^{10}$	3415	$35558^\circ 2'$	$7842^\circ 41'$	$-27715^\circ 21'$	439,43	2,37
$6{,}5\cdot 10^{10}$	3700	$32982^\circ 2'$	$8005^\circ 44'$	$-24976^\circ 18'$	152,83	2,21
$7\cdot 10^{10}$	3980	$27528^\circ 41'$	$8155^\circ 16'$	$-19373^\circ 25'$	10,10	2,01
$8\cdot 10^{10}$	4550	$21955^\circ 2'$	$8413^\circ 53'$	$-13541^\circ 9'$	26,61	1,47
$10\cdot 10^{10}$	5680	$17702^\circ 33'$	$8820^\circ 55'$	$-8881^\circ 38'$	14,96	1,09

MAGNETIC ROTATION OF THE PLANE OF POLARIZATION

Table V

\[ \omega_0 = 2\cdot 10^9,\quad \nu = 4\cdot 10^7 \]

\(a_+\)	\(a_-\)	\(a_- - a_+\)	\(\beta_+\)	\(\beta_-\)
\(11237^\circ 15'\)	\(11237^\circ 15'\)	\(0^\circ\)	\(0,00008\)	\(0,77\cdot 10^{-4}\)
\(11233^\circ 53'\)	\(11238^\circ 23'\)	\(4^\circ 30'\)	\(0,00018\)	\(0,43\cdot 10^{-4}\)
\(11223^\circ 46'\)	\(11239^\circ 30'\)	\(15^\circ 44'\)	\(0,00075\)	\(-0,27\cdot 10^{-4}\)
\(11200^\circ 9'\)	\(11239^\circ 30'\)	\(39^\circ 21'\)	\(0,0034\)	\(0,23\cdot 10^{-4}\)
\(11145^\circ 3'\)	\(11239^\circ 30'\)	\(95^\circ 27'\)	\(0,0179\)	\(0,21\cdot 10^{-4}\)
\(15299^\circ 9'\)	\(11239^\circ 30'\)	\(4059^\circ 39'\)	\(166,675\)	\(0,19\cdot 10^{-4}\)
\(11577^\circ 57'\)	\(11239^\circ 30'\)	\(-338^\circ 27'\)	\(0,2040\)	\(0,19\cdot 10^{-4}\)
\(11305^\circ 51'\)	\(11240^\circ 38'\)	\(-65^\circ 13'\)	\(0,0071\)	\(0,17\cdot 10^{-4}\)
\(11277^\circ 44'\)	\(11240^\circ 38'\)	\(-37^\circ 6'\)	\(0,0022\)	\(0,16\cdot 10^{-4}\)
\(11261^\circ 59'\)	\(11240^\circ 38'\)	\(-21^\circ 21'\)	\(0,0006\)	\(0,14\cdot 10^{-4}\)
\(11253^\circ 00'\)	\(11240^\circ 38'\)	\(-12^\circ 22'\)	\(0,0002\)	\(0,1057\cdot 10^{-4}\)

Table VI

\[ \omega_0 = 6\cdot 10^{10},\quad \nu = 4\cdot 10^7 \]

\(a_+\)	\(a_-\)	\(a_- - a_+\)	\(\beta_+\)	\(\beta_-\)
\(2217^\circ 19'\, i\)	\(2217^\circ 24'\, i\)	\(0^\circ\)	\(0,35\, i\)	\(-0,35\, i\)
\(8508^\circ 20'\, i\)	\(5328^\circ 32'\)	—	\(0,21\, i\)	\(0,081\)
\(16826^\circ 39'\, i\)	\(6947^\circ 40'\)	—	\(0,45\, i\)	\(0,040\)
\(27290^\circ 19'\, i\)	\(7444^\circ 39'\)	—	\(1,26\, i\)	\(0,031\)
\(43229^\circ 48'\, i\)	\(7650^\circ 25'\)	—	\(4,16\, i\)	\(0,027\)
\(310962^\circ 56'\)	\(7795^\circ 28'\)	\(303167^\circ 28'\)	\(5423,79\)	\(0,025\)
\(83563^\circ 10'\)	\(7834^\circ 49'\)	\(-75728^\circ 21'\)	\(25,43\)	\(0,024\)
\(37260^\circ 22'\)	\(8001^\circ 14'\)	\(-29259^\circ 8'\)	\(1,93\)	\(0,022\)
\(28657^\circ 35'\)	\(8150^\circ 47'\)	\(-20506^\circ 48'\)	\(0,76\)	\(0,020\)
\(22187^\circ 47'\)	\(8411^\circ 38'\)	\(-13776^\circ 9'\)	\(0,27\)	\(0,017\)
\(17732^\circ 55'\)	\(8818^\circ 40'\)	\(-8914^\circ 15'\)	\(0,09\)	\(0,012\)

Table VII

\(\omega_0 = 3 \cdot 10^{10}, \quad \gamma = 4 \cdot 10^{7}\)

\(\omega_{\mathrm{res}}\)	\(H\), in oersteds	\(\alpha_+\)	\(\alpha_-\)	\(\alpha_- - \alpha_+\)	\(\beta_+\)	\(\beta_-\)
0	0	\(9674^\circ 20'\)	\(9674^\circ 20'\)	\(0^\circ\)	0.020	0.020
\(2 \cdot 10^{10}\)	1137	\(8759^\circ 5'\)	\(10\,095^\circ 59'\)	\(1336^\circ 54'\)	0.051	0.011
\(3 \cdot 10^{10}\)	1710	\(7714^\circ 31'\)	\(10\,170^\circ 12'\)	\(2455^\circ 41'\)	0.105	0.0084
\(4 \cdot 10^{10}\)	2270	\(4002^\circ 26'\)	\(10\,339^\circ 51'\)	\(5437^\circ 25'\)	0.39	0.0066
\(5 \cdot 10^{10}\)	2848	\(9558^\circ 40' i\)	\(10\,424^\circ 19'\)	—	\(-0.90\,i\)	0.0055
\(5.5 \cdot 10^{10}\)	3185	\(19\,297^\circ 25' i\)	\(10\,461^\circ 25'\)	—	\(-2.33\,i\)	0.0050
\(5.8874 \cdot 10^{10}\)	3350	—	\(10\,487^\circ 16'\)	—	—	0.0046
\(6 \cdot 10^{10}\)	3415	\(42\,900^\circ 36'\)	\(10\,496^\circ 16'\)	\(-32\,404^\circ 20'\)	12.39	0.0046
\(6.5 \cdot 10^{10}\)	3700	\(21\,021^\circ 50'\)	\(10\,527^\circ 45'\)	\(-10\,494^\circ 5'\)	0.86	0.0042
\(7 \cdot 10^{10}\)	3980	\(17\,324^\circ 46'\)	\(10\,555^\circ 52'\)	\(-6768^\circ 54'\)	0.31	0.0038
\(8 \cdot 10^{10}\)	4550	\(14\,761^\circ 17'\)	\(10\,606^\circ 28'\)	\(-4154^\circ 39'\)	0.10	0.0033
\(10 \cdot 10^{10}\)	5680	\(13\,168^\circ 58'\)	\(10\,689^\circ 40'\)	\(-2479^\circ 18'\)	0.03	0.0025

The tables confirm the resonant character of the absorption of the right-polarized wave \((\beta_{+})\) and of the rotation of the plane of polarization \((\alpha_{-}-\alpha_{+})\). The maximum values of \(\beta_{+}\) and \((\alpha_{-}-\alpha_{+})\) correspond to the resonant frequency \((\omega=\omega_{\mathrm{res}})\). It also follows from the tables that the rotation of the plane of polarization increases with increasing \(\omega_{0}\). This is understandable, since \(\omega_{0}\) is proportional to the electron concentration. At the same time, however, the losses increase (the magnitude \(\beta_{\pm}\)). At high electron concentrations there exist ranges of values of the external magnetic field in which the refractive index becomes imaginary. This means that the process of wave propagation ceases and the wave penetrates into the plasma only to a small depth.

Of greatest practical interest is the case corresponding to Table V. It follows from this table that, in fields of the order of 1000 oersteds, the angle of rotation of the plane of polarization of the wave is measured in thousands of degrees, with losses of 10–15% in power. Consequently, the required rotations of the plane of polarization of the wave \((45^\circ \div 90^\circ)\) can be achieved with losses not exceeding 1% in power.

Figure 16 shows the dependences of the rotation of the plane of polarization of the wave and of the losses on the external magnetic field in the region of weak fields. The graph makes it possible to trace in greater detail the character of the changes in the quantities \((\alpha_{-}-\alpha_{+})\) and \(\beta_{\pm}\) at various electron concentrations \((\omega_{0})\).

3. Experimental results

We shall briefly reproduce the principal experimental results \(^{31,32}\) concerning the rotation of the plane of polarization of a wave during its propagation in an electron gas to which a constant magnetic field is applied. The experimental arrangement is approximately the same as that described above, in the section on ferrites, with the sole difference that the frequency of the oscillations is now varied within the limits \(4600\text{–}5500\ \mathrm{MHz}\).

The electron gas was obtained by means of a direct-current discharge in an inert gas under a specified pressure. A section of circular waveguide with the wave \(H_{11}\), in which the gas was located, was placed in a solenoid producing a longitudinal magnetic field (Fig. 17). The experimental results reduce to the following:

1. As the gyromagnetic resonance is approached, the angle of rotation of the plane of polarization increases. The polarization of the wave becomes elliptical, and at resonance—circular.

2. On the two sides of the gyromagnetic-resonance frequency, the rotation of the plane of polarization has opposite signs.

On Fig. 18 is shown the dependence of the rotation of the plane of polarization on the magnitude of the magnetic field for the gas Ne + 1% A at a pressure of 1 mm Hg. The oscillation frequency was 5500 MHz. The duration of the discharge pulse was 5 μsec, after which a high-frequency pulse of duration 50 μsec was applied. The amplitude of the discharge pulse was 1050 V at a current of 135 mA.

The explanation of the experimental results is easy to obtain if the wave \(H_{11}\) is decomposed into two waves circularly polarized in opposite directions. We have already done this above, and therefore we shall confine ourselves to a reference to Chapter I. It should be noted that the course of the curves in Fig. 18 completely coincides with the theoretical curve, the approximate course of which we showed in Fig. 15. Measurements using a rectangular waveguide showed that the width of the absorption line increases with pressure and does not depend on the nature of the gas\(^{32}\).

Fig. 17.

As for the dependence of the introduced energy losses on the magnitude of the magnetic field, it, as was already indicated above, has the form of a resonance curve. For the same gas, but at a pressure of 0.1 mm Hg, the curve of the dependence of the introduced energy losses on the applied field is shown in Fig. 19.

Fig. 18.

Fig. 19.

The oscillation frequency was 8200 MHz. The measurements were carried out in a rectangular waveguide, in which only one type of wave \((H_{10})\) could propagate.

V. APPLICATION OF MAGNETIC ROTATION OF THE PLANE OF POLARIZATION IN CENTIMETER WAVES

The interest presented by the phenomenon of magnetic rotation of the plane of polarization for centimeter-wave engineering is due to the fact that devices using this phenomenon violate the principle of reciprocity. We shall describe some of these devices, which are of greatest interest.

An element in which rotation of the plane of polarization occurs, independently of its construction, we shall agree to call a “rotating element,” and we shall depict it as shown in Fig. 20, which should be understood as follows: if the lines of force of the magnetic field go from left to right (Fig. 20,a) (i.e. in the direction of the arrow), then the plane of polarization of the wave is rotated clockwise (relative to the direction of the magnetic field) through an angle \(\varphi^\circ\). In Fig. 20,b the notation is shown for the case when rotation of the plane of polarization occurs in the opposite direction.

Fig. 20.

Fig. 21.

1. Gyrator. A gyrator\(^{33}\) may be defined as a passive four-terminal network in which (Fig. 21):

\[ U_1 = A I_2,\qquad U_2 = - A I_1 . \tag{1} \]

This means that the phase difference between waves passing through the gyrator in opposite directions is \(180^\circ\). How to realize a gyrator practically is shown in Fig. 22. When the wave propagates from right to left the fields are in phase, whereas when it propagates in the opposite direction the phase shift is \(180^\circ\). The notation for the gyrator is shown in Fig. 23.

Fig. 22.

The possibility of creating a gyrator in electroacoustics was considered in the book by V. V. Furduyev\(^{33}\), where it was also shown that

With the aid of a gyrator it is possible to create linear four-terminal networks that do not satisfy the reciprocity principle. On the other hand, it is known that the construction of four-terminal networks is based on the existence of four elements: capacitance, inductance, resistance, and an ideal transformer. Obviously, a fifth element—the gyrator—would make it possible to obtain considerably better solutions to many problems of four-terminal networks.

Fig. 23.

In particular, Tellegen^34 showed that the construction of passive four-terminal networks is considerably simplified when a gyrator is introduced. Namely, the number of elements necessary for constructing a four-terminal network according to specified characteristics is reduced. In addition, the gyrator makes it possible to divide a four-terminal network of order \(n\) (the number of independent elements of such a four-terminal network is \(2n+1\)) into two four-terminal networks of orders \(n-2\) and 2. Questions concerning the application of the gyrator were also studied by Macmillan^35 and Miles^36. The latter showed that with the aid of a gyrator it would be possible to construct a circuit equivalent to that of a class-\(A\) tube amplifier.

Although the power gain in such a circuit does not exceed unity, its use may in some cases be meaningful.

2. Valve—a transmitting system of unilateral action. If rectangular waveguides, situated on both sides of a circular waveguide in which the “rotating element” is located, form an angle of \(45^\circ\), then the wave will be able to propagate in only one direction. This is schematically explained by Fig. 24. Fig. 24, \(a\) shows that the wave from the rectangular waveguide enters the circular one. Before the rotating element, the polarization of the wave does not change (Fig. 24, \(b\)); the rotating element turns the plane of polarization by \(45^\circ\) (Fig. 24, \(v\)), and with this polarization the wave enters the rectangular waveguide, which is turned by \(45^\circ\) relative to the first (Fig. 24, \(g\)).

Fig. 24.

When propagating in the opposite direction, as is seen from Fig. 24, \(d\)—\(z\), the wave arrives with such a polarization that it cannot propagate in the first waveguide. However, if it is not absorbed,

it will evidently be reflected. Having been reflected, the wave will again go from left to right (Fig. 25, \(a\)—\(g\)), will reach the second waveguide, will again be reflected and, arriving at the first waveguide (Fig. 25, \(d\)—\(z\)), will have such a polarization that it can propagate in it. Without dwelling on descriptions of concrete circuits based on the use of a valve, let us point out that the application of this element makes it possible simply to solve a whole series of complicated problems in the technique of centimeter waves.

Fig. 25.

Other possible applications of the rotating element are based on the fact that the magnitude of the rotation of the plane of polarization can be regulated by changing the intensity of the magnetic field. On this principle it is possible to construct waveguide antenna switches, attenuators \(^{38,39}\), and other devices.

Finally, let us also point out the possibility of creating time-adjustable phase shifters for circularly polarized waves. This is based on the dependence, noted above, of the phase velocity of a circularly polarized wave propagating in a rotating element on the magnitude of the longitudinal magnetic field.

In the case of linearly polarized waves, for this purpose one should evidently use the dependence of the phase velocity of the “extraordinary” wave on the magnitude of the transverse magnetic field (see formulas (II.19a) and (IV.13b)).

The author expresses deep gratitude to Corresponding Member of the Academy of Sciences of the USSR A. A. Pistol’kors for guidance of the work.

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