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We analyze resonant magneto-elastic interactions between standing perpendicular spin wave modes (exchange magnons) and longitudinal acoustic phonon modes in free-standing hybrid metal-ferromagnet bilayer and trilayer structures. Whereas the ferromagnetic layer acts as a magnetic cavity, all metal layers control the frequencies and eigenmodes of acoustic vibrations. The here proposed design allows for achieving and tuning the spectral and spatial modes overlap between phonons and magnons that results in their strong resonant interaction. Realistic simulations for gold-nickel multilayers show that sweeping the external magnetic field should allow for observing resonantly enhanced interactions between individual magnon and phonon modes in a broad range of frequencies spanning from tens of GHz up to several hundreds of GHz, which can be finely tuned through the multilayer design. Our results would enable the systematic study and the deep understanding of resonantly enhanced magneto-elastic coupling between individual phonon and magnon modes up to frequencies of great contemporary fundamental and applied interest., 9 pages, 6 figures

Magneto-optical (MO) effects, viz. magnetically induced changes in light intensity or polarization upon reflection from or transmission through a magnetic sample, were discovered over a century and a half ago. Initially they played a crucially relevant role in unveiling the fundamentals of electromagnetism and quantum mechanics. A more broad-based relevance and wide-spread use of MO methods, however, remained quite limited until the 1960s due to a lack of suitable, reliable and easy-to-operate light sources. The advent of Laser technology and the availability of other novel light sources led to an enormous expansion of MO measurement techniques and applications that continues to this day (see section 1). The here-assembled roadmap article is intended to provide a meaningful survey over many of the most relevant recent developments, advances, and emerging research directions in a rather condensed form, so that readers can easily access a significant overview about this very dynamic research field. While light source technology and other experimental developments were crucial in the establishment of today's magneto-optics, progress also relies on an ever-increasing theoretical understanding of MO effects from a quantum mechanical perspective (see section 2), as well as using electromagnetic theory and modelling approaches (see section 3) to enable quantitatively reliable predictions for ever more complex materials, metamaterials, and device geometries. The latest advances in established MO methodologies and especially the utilization of the MO Kerr effect (MOKE) are presented in sections 4 (MOKE spectroscopy), 5 (higher order MOKE effects), 6 (MOKE microscopy), 8 (high sensitivity MOKE), 9 (generalized MO ellipsometry), and 20 (Cotton-Mouton effect in two-dimensional materials). In addition, MO effects are now being investigated and utilized in spectral ranges, to which they originally seemed completely foreign, as those of synchrotron radiation x-rays (see section 14 on three-dimensional magnetic characterization and section 16 on light beams carrying orbital angular momentum) and, very recently, the terahertz (THz) regime (see section 18 on THz MOKE and section 19 on THz ellipsometry for electron paramagnetic resonance detection). Magneto-optics also demonstrates its strength in a unique way when combined with femtosecond laser pulses (see section 10 on ultrafast MOKE and section 15 on magneto-optics using x-ray free electron lasers), facilitating the very active field of time-resolved MO spectroscopy that enables investigations of phenomena like spin relaxation of non-equilibrium photoexcited carriers, transient modifications of ferromagnetic order, and photo-induced dynamic phase transitions, to name a few. Recent progress in nanoscience and nanotechnology, which is intimately linked to the achieved impressive ability to reliably fabricate materials and functional structures at the nanoscale, now enables the exploitation of strongly enhanced MO effects induced by light-matter interaction at the nanoscale (see section 12 on magnetoplasmonics and section 13 on MO metasurfaces). MO effects are also at the very heart of powerful magnetic characterization techniques like Brillouin light scattering and time-resolved pump-probe measurements for the study of spin waves (see section 7), their interactions with acoustic waves (see section 11), and ultra-sensitive magnetic field sensing applications based on nitrogen-vacancy centres in diamond (see section 17)., Despite our best attempt to represent the field of magneto-optics accurately and do justice to all its novel developments and its diversity, the research area is so extensive and active that there remains great latitude in deciding what to include in an article of this sort, which in turn means that some areas might not be adequately represented here. However, we feel that the 20 sections that form this 2022 magneto-optics roadmap article, each written by experts in the field and addressing a specific subject on only two pages, provide an accurate snapshot of where this research field stands today. Correspondingly, it should act as a valuable reference point and guideline for emerging research directions in modern magneto-optics, as well as illustrate the directions this research field might take in the foreseeable future., Journal of Physics D: Applied Physics, 55 (46), ISSN:0022-3727, ISSN:1361-6463

An algorithm for the numerical simulation of the propagation of electromagnetic waves in randomly inhomogeneous magnetic media by the FDTD method has been developed. The formulated algorithm is suitable for analyzing the main timing characteristics, as well as identifying the features of the propagation of various types of signals in both time-independent and time-dependent layered randomly inhomogeneous media. The simulation of the propagation of the sine pulse, sine signal and square wave in time-independent magnetic randomly inhomogeneous media with a various levels of phase contrast of two types - with a “diffuse” distribution of inhomogeneities and their “close packing” is carried out. The influence of the concentration of magnetic granules and the type of their distribution on the characteristics of the transmitted and reflected signals is revealed.

Results are presented from studying the magnetic resonance properties of metal–dielectric (CoFeB + SiO2) and metal–carbon (CoFeB + C) composite films. The effect the concentrations of the magnetic metal, dielectric, and carbon phases of the composite films on the characteristics of ferromagnetic resonance is analyzed.

International audience; We perform a theoretical investigation of the magnetization switching in polycrystalline Ni nanoparticles induced by ultrashort pulses of surface acoustic waves via magnetoelastic interactions. In our numerical simulations, a Ni nanoparticle is modeled as an ellipsoidal disk deposited on a dielectric substrate. The in-plane external magnetic field breaks the symmetry and allows us to adjust the height of the energy barrier between two metastable magnetization states of the free-energy density and dramatically lower the amplitude of elastic strain pulses required for magnetization switching. The switching threshold is shown to depend on the duration of an acoustic pulse, the magnetic shape anisotropy of an elliptical nanoparticle, the amplitude of the external magnetic field, and the magnetostriction coefficient. We introduce the magnetoelastic switching diagram, allowing for the simultaneous visualization of the switching threshold and its characteristic timescale as a function of various physical parameters.

The amplitudes of magnetic and elastic vibrations for Mn0.61Zn0.35Fe2.04O4 spinel crystalline slab are calculated by solving the equations describing the magnetic and elastic dynamics. The anisotropy constants, magnetization, second-order elastic constants and magnetoelastic coupling constants for a studied crystal are expressed as the functions of temperature. The magnetization vector and elastic shear components are found as the functions of the first magnetic anisotropy constant at different values of an external constant magnetic field greater than a saturation field. The procession patterns for normally and tangentially magnetized slabs are displayed for two values of the first anisotropy constant. High absolute values of the first anisotropy constant are shown to refer to reorientation of the magnetization vector.

Features of the nonlinear dynamics of magnetic and elastic subsystems are studied upon the generation of hypersonic oscillations excited by an alternating magnetic field oriented in the film plane.

The micro- and nanostructure of 328–772-nm-thick amorphous granulated (Co45Fe45Zr10) x (ZrO)1 – x (0.27 < x < 0.61) composite films deposited in argon on a lavsan substrate has been examined using atomic force microscopy, magnetic force microscopy, and scanning electron microscopy. It has been shown that the mean sizes of the grain and the metallic phase content govern the conductive and reflective properties of the films in the microwave range.

The nanostructure of amorphous granulated (Co45Fe45Zr10) x (Al2O3)1–x (0.30 < x < 0.75) composite films 2.2-5.8 μm thick deposited onto a pyroceramic substrate is studied by atomic force microscopy, magnetic force microscopy, and scanning electron microscopy. The average grain size and the metallic-phase content are shown to affect the electrical conductivities of these films.

International audience; We investigate generation of exchange magnons by ultrashort, picosecond acoustic pulses propagating through ferromagnetic thin films. Using the Landau-Lifshitz-Gilbert equations we derive the dispersion relation for exchange magnons for an external magnetic field tilted with respect to the film normal. Decomposing the solution in a series of standing spin wave modes, we derive a system of ordinary differential equations and driven harmonic oscillator equations describing the dynamics of individual magnon mode. The external magnetoe-lastic driving force is given by the time-dependent spatial Fourier components of acoustic strain pulses inside the layer. Dependencies of the magnon excita-tion efficiencies on the duration of the acoustic pulses and the external magnetic field highlight the role of acoustic bandwidth and phonon-magnon phase matching. Our simulations for ferromagnetic nickel evidence the possibility of ultrafast magneto-acoustic excitation of exchange magnons within the band-width of acoustic pulses in thin samples under conditions readily obtained in femtosecond pump-probe experiments. $ Fully documented templates are available in the elsarticle package on CTAN.

A comparative study were carried out on composition, nanostructure and conductivity of (Co + Fe + Zr)/(ZrO) amorphous granular composite films deposited on polyethylene terephthalate substrate by ion beam deposition in argon and nitrogen atmosphere. Atomic force microscopy, magnetic force microscopy, energy dispersive analysis and scanning electron microscopy were used. It is shown that the content of the metal phase in the films determines both the average granule sizes and their microwave reflecting and conducting properties. Dependences of granules size, reflection coefficient and conductivity on the metal phase content and film thickness were found. The influence of the metallic phase turns out to be significant in the presence of percolation (flow) processes that arise when the films are deposited.

The work deals with the definition of elastic and magnetic oscillations exited by microwave magnetic field in the three-layer structure. The frequency dependences for the amplitudes of the magnetic and elastic vibrations of the film layers with different values of the material parameters and amplitudes of the external fields are calculated.

By means of consistent solution of the motion equation for magnetization the consideration of energy dissipation for magnetostatic surface waves propagating on in-plane magnetized ferrite plate is regarded. The complex character of the wave number caused by the dissipation is revealed. In this case the dispersion relations for the real and imaginary parts of the wave number are obtained. It is shown that dispersion curves are limited both by the wave number and frequency, and these restrictions tighten with an increase in the damping parameter. Also, it is predicted that there is a new set of branches of the backward dissipative waves. Both for forward and backward waves there is a critical value of the damping parameter and propagation angle, beyond which it does not exist.

The paper considers investigation of hypersound magnetic and elastic oscillations excitation by use of alternating magnetic field in normal magnetized plate. Distinctive features of frequency division and multiplication are studied in the paper. The relation between processes of frequency division and multiplication and structure of the plate as well as parameters of saturation magnetization, anisotropy constant, magnetic and elastic damping has been revealed.

The paper gives a description of hypersound magnetoelastic oscillations excitation along magnetisation reversal of normal magnetized single-layer and multi-layers ferrite plates, also provides definition of distinctive features of magnetic and elastic oscillations development. The relation of oscillations to the parameters of saturation magnetization, magnetic and elastic damping has been revealed.

The maximum amplitudes of magnetic and elastic oscillations are calculated at various material parameters and parameters of external fields. Normally and tangentially magnetized anisotropic magnetic plates were considered. To obtain the maximum amplitudes with additional constraints, the simulated annealing method was used. The behaviour of the magnetic and elastic components of the oscillations was considered. The changes in the position of the magnetization vector and the equilibrium displacements as a function of the first anisotropy constant were revealed.

In magnetostatic approximation of electrodynamics the dispersion relation for magnetostatic back-ward volume waves (MSBVW) in a ferrite film, placed in a constant longitudinal and transverse alternating magnetic field is obtained. The magnetic field, damping and temperature influence on the spectrum and dispersion characteristics of the MSBVW is regarded.

Static conduction and reflection and transmission of electromagnetic waves through thin films of granulated amorphous metal–dielectric zirconium-containing nanocomposites are experimentally studied. It is demonstrated that the dynamic conductivity in the microwave range exceeds by two–three orders of magnitude the static conductivity measured at a dc current. To explain this phenomenon, the mechanism of intraganule currents is suggested. Recommendations for further studies are formulated.

The paper considers the second order precession of the magnetization vector in a perpendicular magnetized anisotropic ferrite plate with magnetoelastic properties. The boundaries of the precession regimes on the frequency and amplitude of the alternating field were defined. The features of the precession of the magnetization vector regimes associated with magnetoelastic properties were revealed.

Monolayer composite, multilayer composite-composite and multilayer composite-semiconductor films were investigated. Dependencies of the internal exchange fields on magnetic metal alloy concentration in nanostructured single and multilayer films at room and liquid nitrogen temperatures were obtained. The calculation is made on the basis of the experimental data and the relationship of the exchange interaction and the broadening of the FMR line.

The present work deals with investigation of features of a magnetization vector of nonlinear precession and elastic displacements close to ferromagnetic resonance in normal magnetized ferrite plate. The system of ordinary differential equations was solved numerically by the Runge-Kutta 7-8 orders method with control of the integration at every step length. The possible excitation of magnetoelastic autooscillations was found out in the paper. Two mechanisms of autooscillations: reorientation and detuning were investigated. The boundaries between the regular and chaotic reorientation autooscillations depending on the magnetic dissipation parameter and magnetoelastic interaction constant were determined.

Excitation of hypersound elastic oscillations in the microwave on the magnetization reversal of the normal magnetized ferrite plate is studied in the paper. Characteristic times of the process of development of magnetic and elastic oscillations were revealed. Their relation to the parameters of the magnetic and elastic damping has been revealed.

Static conduction and reflection of electromagnetic waves from thin films of granulated amorphous metal-insulator nanocomposites are experimentally studied for a wide range of concentrations of the metal phase. It is demonstrated that the dynamic conductivity in the microwave range is several times greater than the static conductivity measured at dc current. Capacitive shunting, effective permittivity, and intracluster currents are used to interpret the results. Recommendations for further experiments are formulated.

The effect of annealing on the structure and microwave magnetic properties of composite single-layer and multilayer films is investigated. It is found that as the threshold annealing temperature is attained, the width of the FMR line increases abruptly, depending on the concentration ratio of metallic alloy and dielectric. It is shown that the FMR line broadening is also associated with the occurrence of and increase in high-frequency modes of absorption. The surfaces of annealed and unannealed films are investigated at various concentrations of metallic alloys and dielectrics using an atomic force microscope. The strong change in the magnetic characteristics after film annealing is associated with a change in surface roughness and inhomogeneity, and in the size distribution of nano- and submicron formations and their spacing.

The problem of hypersound excitation in a structure consisting of three magnetic layers is considered. Motion equations and boundary conditions for the components of magnetization and elastic displacement when there is an arbitrary angle of magnetization vector precession are obtained. The development of oscillations over time in an alternating field is considered.

The second-order precession of the magnetization vector in a normally magnetized magnetic plate with the cubic anisotropy is considered for the [001], [011], and [111] orientations of the axes of a cubic cell along the static field. The precession pattern of the forced oscillation in the form of a large ring and small rings located along the envelope is obtained. A relationship between the observed high- and low-density groups of small rings and the spatial position of the [111] easy magnetization axes is revealed and explained on the basis of the energy model of the potential. It is found that the phenomenon is highly sensitive to the orientation of cubic axes, the anisotropy constant, and the intensities and directions of the static and alternating fields. It is reported that the observed phenomena can be used in practice.

A second-order precession of the magnetization vector in a normally magnetized magnetic plate with a uniaxial anisotropy whose axis is tilted at an angle to the plate normal is considered. Under the conditions for the orientational transition, precession pattern of the forced oscillation in the form of a large ring filled, along the ring generator, with small rings is obtained. A relationship between the observed crowding and sparseness of small rings and the tilt of the anisotropy axis to the normal to the plate plane is established and explained on the basis of the potential-energy model. It is shown that the crowding and sparseness can be compensated by a constant transverse field. It has been found that the phenomenon is critically sensitive to the axis orientation and to the value of the anisotropy constant as well as to the strength of the constant field. It is indicated that the observed phenomena can be put to practical use.

A simple algorithm is proposed for deriving analytic expressions for the amplitudes of reflected, transmitted, and internal waves in individual layers of a multilayer stepwise inhomogeneous structure with an arbitrary number of layers. Functioning of the algorithm is exemplified by the solution of problems of wave propagation in a stepwise inhomogeneous structure with the wave number variation along the structure's length. DOI: 10.1134/S1064226913010026

Transverse-electric (TE) surface plasmons (SPs) are very unusual for plasmonics phenomenon. Graphene proposes a unique possibility to observe these plasmons. Due to transverse motion of carriers, TE SPs speed is usually close to bulk light one. In this work we discuss conditions of TE SPs propagation in cylindrical graphene-based waveguides. We found that the negativity of graphene conductivity’s imaginary part is not a sufficient condition. The structure supports TE SPs when the core radius of waveguide is larger than the critical value Rcr. Critical radius depends on the light frequency and the difference of permittivities inside and outside the waveguide. Minimum value of Rcr is comparable with the wavelength of volume wave and corresponds to interband carriers transition in graphene. We predict that use of multilayer graphene will lead to decrease of critical radius. TE SPs speed may differ more significantly from bulk light one in case of epsilon-near-zero core and shell of the waveguide. Results may open the door for practical applications of TE SPs in optics, including telecommunications.

The forced nonlinear precession of magnetization vector in the normally magnetized magnetic plate is examined. Investigations are performed under orientational transition conditions at the asymmetric orientations of transverse and longitudinal ac fields. It has been revealed that the precession of equilibrium position exhibits pulse-like behavior under the action of two types of field. This feature manifests itself as the nonuniform bunchings of phase trajectories in the precession pattern. It is demonstrated that the nonuniformity of precession can be compensated by applying the asymmetric dc field. It should be emphasized that the compensation is highly sensitive to the dc field amplitude and reaches several hundredths of a percent of its total value. Recommendations are given to observe the discussed phenomena in the experiment. The possibility of their practical application is discussed.

A simple algorithm is proposed for derivimg analytical expressions for the amplitudes of reflected and transmitted waves in the case when two 1D counterpropagating waves are incident on a multilayer stepwise inhomogeneous structure. Wave amplitudes are analytically expressed for structures consisting of three, four, and five layers. The proposed algorithm is applied to solve numerically the problem of wave propagation in an inhomogeneous structure where the wave number linearly changes along the structure’s length. The algorithm involves approximation of a smooth inhomogeneity by a step one. A criterion and conditions are obtained for the applicability of the step approximation for the investigation of wave propagation through inhomogeneous media with smoothly varying parameters. It is shown that the algorithm can be applied to problems on propagation of electromagnetic waves in the cases of the normal and oblique incidence.

The forced nonlinear precession of the magnetization vector in the normally magnetized magnetic plate is analyzed under the conditions for the orientational transition in the presence of the asymmetric static field. The excitation conditions for the precession of the equilibrium position of the magnetization vector are determined. It is demonstrated that the precession pattern of oscillations represents a large circular ring filled with small rings with the nonuniform densening whose position depends on the field asymmetry. The precession of the equilibrium position substantially depends on the degree of asymmetry. When the field exceeds a certain threshold level, the precession of equilibrium is changed by the conventional circular precession with a relatively large amplitude. The domains of existence with respect to dc and ac fields are analyzed for the precession of the equilibrium position. The dependence of the precession period on the degree of asymmetry of the dc field is studied. The main features of the above effects are qualitatively interpreted using complementary (energy and vector) models.

The electric conductivity of amorphous nanogranular composite films has been experimentally studied on direct current and under microwave reflection conditions. It is established that the dynamic conductivity exceeds the static value by two to four orders of magnitude. Suggestions concerning the nature of this difference are formulated.

Fe and (Fe)X(BaF2)Y films 10- to 160-nm-thick are grown on a polymer substrate by vacuum evaporation. The surface relief and nanostructure of the films are examined. The dependence of the conductivity on the film thickness is obtained, and a correlation between the conductivity and surface relief is found. The influence of the BaF2 dielectric phase on the conductivity and reflectivity of the Fe films is studied. It is shown that the BaF2 layer to some extent slows down oxidation and maintains the reflectivity of the Fe films compared with pure iron films exposed to air.

The direct and averaging methods are applied to analyze the propagation of 1D and electromagnetic waves through three different media separated by plane-parallel interfaces. It is shown that application of the direct method for calculation of the amplitudes of propagating waves is ideologically simpler but the averaging method yields simpler final expressions. Universal boundary conditions are obtained for the averaging method. These conditions are suitable for the solution of problems on the normal and oblique incidence of vertically and horizontally polarized electromagnetic waves. It is demonstrated that vector electrodynamic problems and the corresponding scalar 1D problems have the same mathematical contents and that a solution to an electrodynamic problem can be obtained from a solution to a 1D problem with the help of certain changes of coefficients and variables.

The nonlinear dynamics of an ensemble of single-domain interacting particles upon excitation by radio-frequency pulses of a magnetic field is investigated. Reorientation of magnetic moments of ensemble particles under the effect of the pulses of the radio-frequency field is observed. The effect of varying the magnetic structure on the frequency dependences of susceptibility is studied. Temporal characteristics of precession in separate particles under the conditions of an orientation transition are considered.

With the use of the 1D wave equation as an example, the wave incidence on one interface and on two interfaces of media with arbitrary complex parameters is analyzed. It is noted that, when the energy reflection, transmission, and absorption coefficients are calculated by means of traditional techniques, their sum exceeds unity if the wave number in the first medium is complex. The complex character of the energy balance on the media interfaces is revealed. Definitions are introduced and the properties of complex reflection, transmission, localization, reactivity, and absorption are considered.

The nonlinear problem of excitation of hypersound by an alternating magnetic field in a normally magnetized ferrite plate is considered. The plate is characterized by uniaxial and cubic anisotropies and exhibits magnetostriction. The equations of motion and boundary conditions for the magnetization components and for the elastic displacement are obtained for the case of the arbitrary angle of the magnetization-vector precession. The obtained system is solved by means of the Runge-Kutta method. It is shown that, in the nonlinear regime, the amplitude of the excited hypersound may exceed the amplitude observed in the linear regime by three or four orders of magnitude and that, simultaneously, the range of effectively excited frequencies extends by a factor of one and a half or two.

Electromagnetic wave propagation through a layer with finite conductivity is analyzed. The phase-plane method is demonstrated to be applicable to the analysis of propagation. The high efficiency of the method, which makes it possible to visually reveal the features of transition from a dielectric layer to a metallic one, is demonstrated in the case of a layer whose conductivity depends on its thickness.

Penetration of electromagnetic waves into a medium, as well as reflection, transmission, and absorption of waves by a plate with an arbitrary conductivity, is analyzed under the condition of transition from dielectric propagation to metallic propagation. Two types of transition criteria are proposed: a criterion based on the penetration depth and a criterion based on the behavior of the reflection coefficient as a function of the plate depth. It is found that the critical values of conductivity corresponding to different types of criteria differ by a factor of 2.7 ± 0.3. High reflectivity of metal films with a thickness substantially less than the penetration depth is qualitatively explained. It is shown that the frequency of transition from the metallic type of wave propagation to the dielectric type increases as the prescribed value of the reflection coefficient decreases and the conductivity increases.

The work is devoted to the research of the magnetic resonance properties of composite (metal-dielectric) films and revealing their relationship with the nanostructure characteristics and degree of crystallinity. The FMR line width and resonance position for two series of composite films have been investigated. First films series has a small metal crystal phase content. The second one contains metal oxide and amorphous. The results of work shows that oxidation of a metal phase displaces percolation bound to area of the big metal concentration. Strong changes of FMR properties of the films are observed below the percolation bound. The significant changes of FМR properties are connected to nanostructural transformations in the ferromagnetic granules topology and presence of metal phase oxide.

A set of equations in the Gilbert form that describes the motion of the magnetization vector in an ensemble of interacting magnetic nanoparticles is numerically solved for the case of high-amplitude rf pulses. Based on the numerical solution, the magnetic structure of spherical particle ensembles showing cubic anisotropy at certain parameters of the variable field is studied. This phenomenon is shown to have a threshold. The dependence of the field threshold amplitude on the acting pulse repetition rate and amount of magnetic interaction is determined. It is demonstrated that a change in the magnetic structure of the interacting particle ensembles causes a change in the spectrum of the response. This fact can be used for pulsed rf writing and readout of information based on ferromagnetic resonance.

The averaging and direct methods are applied to calculate the characteristics of propagation of an electromagnetic wave through a parallel-sided plate with an arbitrary conductance in the case when the wave is obliquely incident on the plate’s plane. On the basis of averaged electrodynamic boundary conditions, simplified expressions for the reflection, transmission, and absorption coefficients are derived for the arbitrary conductance of the plate. It is shown that the averaging method facilitates calculations and yields simplified expressions owing to reduction of the order of the system of equations to be solved and to elimination of exponents and trigonometric functions. The error of the averaging method relative to the error of the direct method is several percent for the incidence angle ranging from 0° to 90° and for a plate as thick as 0.35 of the wavelength in the medium.

The averaging method is used for calculation of the reflection and transmission coefficients of a 1D longitudinal electromagnetic wave and a 1D transverse electromagnetic wave that are incident along the normal onto a plane-parallel plate. The averaged electrodynamic boundary conditions are derived for an arbitrary conductivity of the plate. It is demonstrated that the averaging method enables one to simplify the calculations and resulting expressions owing to a decrease in the order of the system of equations and elimination of trigonometric functions. The error of this method is several percent for planes with thicknesses up to 0.35 of the wavelength in the medium.