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In recent years, the microwave processes in artificial multiferroic media such as electromagnonic crystals have attracted increased research interest due to their potential applications for voltage-controlled spintronic devices. In contrast to the conventional magnonic crystals, the artificially created periodic structures are characterised by electrically and magnetically tunable band gaps in the wave spectrum where the propagation of the electromagnons is forbidden. In this study, an experimental realisation of an electromagnonic crystal based on a ferrite–ferroelectric–ferrite multilayer has been proposed. The authors have demonstrated for the first time a band-gap splitting, which arises from an interaction of three fundamental modes of two ferrite films separated by a ferroelectric layer. This splitting manifested itself as an additional stop-band appearance in the frequency response of the electromagnonic crystal. The obtained band structures are confirmed by numerical modelling using the coupled-mode approach and the transfer-matrix method. The authors expect that their results allow exploiting the electromagnonic crystal for enhanced logic control as well as for tunable microwave devices.

Introduction. The distinctive feature of a coplanar transmission line with thin ferrite and ferroelectric films is the absence of undesirable irregularities in dispersion for relatively low frequencies when the wavelength approaches the thickness of ferroelectric layer, in contrast to the open ferrite-ferroelectric wave-guiding structure without metallization. Aim. The purpose of this paper is twofold: (i) to develop a theory of the wave spectra in the multiferroic structures based on the coplanar lines; (ii) using this theory to find ways to enhance the electric tuning range. Materials and methods . The dispersion relation for spin-electromagnetic waves was derived through analytical solution of the full set of the Maxwell's equations utilizing a method of approximate boundary conditions. Results. A theory of spin-electromagnetic wave spectrum has been developed for the thin-film ferrite-ferroelectric structure based on a coplanar transmission line. According to this theory, dispersion characteristics of the spin-electromagnetic waves were described and analyzed for different parameters of the structure. The obtained results show that the investigated structure demonstrates a dual electric and magnetic field tunability of wave spectra. Its efficiency increases with an increase in the thicknesses of the ferrite and ferroelectric films and with a decrease in the width of the central metal strip. Conclusion. The distinctive features of the proposed coplanar waveguides are the thin-film planar topology and dual tunability of the wave spectra. All these advantages make the proposed structures perspective for development of new microwave devices.

A nonlinear spin-wave (magnonic) logic gate based on the ferrite-film Mach-Zehnder interferometer was realized. The gate has two inputs and one output. The principle of operation of the device is based on the power-dependent phase shift of spin waves propagating in an yttrium iron garnet film. A XOR logic functionality for two digital data streams supplied to the gate inputs is shown. No control signal is needed for the gate operation. The performance characteristics of the device are discussed. Reliable functionality in the pulsed regime at 50 and 250 kb/s is demonstrated.

A theory of spin-electromagnetic wave spectrum has been developed for the thin-film ferrite-ferroelectric structure based on a coplanar transmission line (TL). The dispersion relation was derived through analytical solution of the full set of Maxwell’s equations utilizing a method of approximate boundary conditions. The theory was used for numerical simulation of the transmission characteristics of the periodic multiferroic heterostructures containing the coplanar TL with a periodic variation of its width. We named these heterostructures “electromagnonic crystals.” The transmission characteristics of the electromagnonic crystals were calculated by the transfer-matrix method. The obtained results show that a high microwave signal rejection of more than 30 dB appears in the electromagnonic crystals due to the Bragg reflections. The distinctive features of the proposed electromagnonic crystals are the thin-film planar topology and dual tunability of the electromagnonic band-gap positions.

The present work focuses on the effect of vanadium dioxide (VO2) films exhibiting a metal–insulator transition (MIT) on the performance characteristics of the magnetic multilayers. It has been shown that the MIT provides a novel mechanism for controlling the microwave spin-wave dynamics in the yttrium iron garnet (YIG) films. In particular, the low and high levels of microwave attenuation of spin waves transmitted through the YIG–VO2 bilayer has been observed due to a variation of the VO2 conductivity within a narrow temperature range. This effect has been utilized to realize fully reconfigurable magnonic crystals composed of the thickness-modulated YIG and regular VO2 films. A promising functionality of the proposed waveguiding structures arises from a controllability of wave intensity, which provides an altering of the frequency response from an original band structure to a full rejection of spin waves. Numerical simulations taking into account both the YIG film saturation magnetization and the VO2 film conductivity have confirmed the experimentally observed spin-wave dynamics. An interest in ferrite–VO2 bilayers arises not only from possible practical applications but also from a variety of fundamental scientific problems devoted to the physics of wave phenomena in planar thin-film magnetic multilayers. Post-print / Final draft

The interest in artificial magnetic media such as magnonic crystals increased substantially in recent years due to their potential applications in information processing at microwave frequencies. The main features of these crystals are the presence of band gaps in the spin-wave spectra, usually formed due to Bragg reflections of spin-waves on the artificially created periodic structures. Here, we study spin-wave propagation in longitudinally magnetized width- and thickness-modulated yttrium iron garnet waveguides by means of Brillouin light scattering and microwave spectroscopy techniques. It is found that the width modulated crystal does not manifest noticeable Bragg reflections, but still demonstrates a pronounced band gap in its transmission characteristic. The phenomenon can be explained by the destructive interference between different frequency-degenerated spin-wave modes excited by the crystal. Such a reflection-less crystal is promising for future design of multi-element magnonic devices. By modulating the dimensions of magnonic crystals, their transmission properties in the microwave regime may be tuned and so applications including information processing are anticipated. The authors here show reflection-less spin-wave propagation may be realised in a width-modulated waveguide.

Transmission characteristics of the periodic multiferroic heterostructure consisting of two ferrite films separated by a thin ferroelectric layer have been studied for the first time. This structure is characterized by a high microwave signal rejection of 25 dB due to the Bragg reflections. An influence of the various parameters on the behavior of bandgaps has been analyzed. A distinctive features of the proposed structure are the thin-film planar topology and dual tunability of the band-gap positions by both magnetic and electric fields. By analogy with known magnonic crystal, periodic multiferroic structures were named as electromagnonic crystals.

The paper reports experimental results on development of a broadband microwave dynamical chaos in a microwave photonic generator. An influence of a bias point of Mach-Zehnder modulator on scenario of dynamical chaos development is investigated in details. It is shown that increase of a ring gain coefficient leads to changing of generation regimes. A rich set of the generation regimes is observed for the first time. It includes the continuous wave, rectangular periodic pulses, period doubled pulses, period doubled chaotic pulses, chaotic pulses, “weak” chaos, chaos, broadband chaos, and noise. The threshold values of a ring gain coefficient corresponding to changing of the generation regimes are determined. A shifting of the bias point to the maximum of transmission characteristics leads to a consistent disappearance of the regimes.

One of the main challenges for the modern magnonics, which, as opposed to the conventional electronics, operates with quanta of spin waves in magnetically ordered materials—magnons—is energy efficient control of magnon transport on small time and space scales. The magnon propagation in a time-dependent periodic spatial potentials—dynamic magnonic crystals—paves a way to this aim. To date, dynamic manipulation of the magnonic crystals has been realized with electric current and optic control influence. However, both approaches show limited potential for reduction in energy consumption and miniaturization of magnonic circuits. Voltage (or electric field) control of magnon currents promises to be fast and low energy consuming. It can be achieved in ferrite-ferroelectric (multiferroic) heterostructures, where strong coupling of magnons and microwave photons constitutes new quasiparticles called electromagnons. Here, we present an experimental realization of a voltage-controlled dynamic electromagnonic crystal operating with electromagnons at microwave frequencies. Fast and energy-efficient control of magnon transport in magnonic crystals is one of the main challenges of modern magnonics. High performances for voltage control are expected but have so far only been predicted theoretically and investigated numerically. In this paper, the authors report of the first experimental realization of voltage-controlled magnon currents in a dynamic electromagnonic crystal.

A method for analysis of dispersion characteristics of guided optical modes propagating in the optical waveguides with small cross-sections is proposed. The method is based on introduction of a correction factor for a longitudinal wavenumber of propagating modes. The correction factor arises when a cross-section of the basic rectangular waveguide is subjected to perturbation. The electromagnetic field distributions along with the mode longitudinal wavenumber are found by means of variable separation method. The longitudinal wavenumber correction factor is analytically calculated in terms of coupled mode theory. The combined use of the complete set of equations of electrodynamics together with the concept of effective sources gives rise to the correction factor in the form of an intermodal coupling coefficient. It is pointed out that the coupling coefficient consists of two components, namely bulk and surface, owing to accurate account of the electrodynamics boundary conditions. Using the method proposed, the dispersion characteristics of the fundamental modes propagating in the practically employed optical waveguides having a trapezoidal cross-section are calculated. An impact of the waveguide cross-section shape to cladding dielectric constant ratio on the mode dispersion characteristics is analyzed. The necessity to take into consideration an imperfection of the waveguide cross-section in a wide range of operating wavelengths is demonstrated.

A general theory describing the dispersion of surface spin waves has been developed for ferrite-dielectric-metal multilayered structures that takes into account the finite values of the layer permittivity and conductivity. The influence of different structure parameters on the dispersion characteristics was analyzed. A change of the conductivity of the conducting film layer manifests itself in a strong wavenumber variation. A novel miniature spin-wave phase shifter based on ferrite-dielectric-vanadium dioxide multilayered structures is proposed. A structure 1 mm in length gives a phase shift of 5.27 rad and an insertion loss of 2.1 dB at a frequency of 8.99 GHz. The introduction of a dielectric layer between the ferrite and vanadium dioxide films preserves the wavenumber variation and minimizes the thermal effect induced by heating of the vanadium dioxide film.

The microwave bistability in active ring resonators (RRs) with dual spin-wave and optical nonlinearities has been investigated. The theory for the nonlinear transfer function of the active RRs consisting of the spin-wave delay line and highly nonlinear optical fiber was developed taking into account both the dispersion and nonlinear properties of the ferrite film as well as the nonlinear properties of the optical fiber. The bistable behavior of the transmission characteristics of the active RR was analyzed using the developed theory for two types of spin-wave delay lines based on forward volume spin waves (FVSWs) and surface spin waves (SSWs). It was shown that an introduction of highly nonlinear optical fiber in the active RR based on the FVSW delay line provides compensation of the spin-wave nonlinearity. However, exactly the same nonlinear optical fiber in the active RR based on the SSW delay line enhanced the bistability. The transmission characteristics of the particular case of the active RRs consisted of the nonlinear FVSW or SSW delay lines, and linear optical single-mode fiber was experimentally measured. The obtained theoretical results are in good agreement with the experiment.

We demonstrate theoretically, and confirm experimentally, that nonlinear spin waves excited in thin yttrium iron garnet films are good candidates for squeezing vacuum quantum noise. The experimental demonstration is in the form of a measurement of spin-wave induced modulation instability (IMI) conducted in the classical regime. The experiment evidences strong phase locking of an idler wave parametrically generated in the film with a deterministic small-signal wave launched into the film from an external source. The theory predicts that the same behaviour will be observed for vacuum quantum noise, resulting in squeezing of the noise.

Novel Mach-Zender-type interferometer based on thin-film multiferroic structure has been proposed and described. The performance of the main element of such device, a tunable phase shifter, has been analysed. The distinctive features of the proposed interferometer are high efficiency and small device area in comparison with other spin-wave logic gates. We expect that our results allow exploiting spin-electromagnetic wave phenomena for enhanced logic functionality and efficient data transfer.

A theory of the transmission characteristic formation for the generalized configuration of the multi-ring resonator is developed. The configuration contains an arbitrary number of the delay lines connected in parallel. Furthermore, each delay line is composed of an arbitrary number of the dispersive waveguides and coupled ring resonators. The numerical simulations of the transmission characteristics for the common cases of the multi-ring configurations are carried out using the developed theory. The obtained theoretical results are proved by the experimental investigation of the tunable spin-wave optoelectronic multi-loop resonator having parallel configuration.

The results of a study of the nonlinear frequency response of a silicon micro-ring resonator (MRR) manufactured using silicon-on-insulator technology are presented. Appearance of optical bistability caused by the frequency shift of the MRR resonant spectrum under continuous wave pumping is experimentally demonstrated. The experimental results are explained in the framework of an original theory, which accounts for the linear loss, Kerr nonlinearity, two-photon absorption, free-carrier-induced absorption and dispersion, and the thermo-optic effect. It is shown that the bistability of the frequency response originates from competition between frequency shifts of opposite signs that appear due to the presence of the thermo-optic and free-carrier effects, with the latter one being dominant.

In this paper we investigate in details the scenario of a transition to chaos in the spin-wave optoelectronic oscillator. The feature of the investigated circuit is the presence of two nonlinearities with different nature – modulation instability of spin waves and nonlinearity of the transmission characteristic of the electro-optical modulator. The influence of a position of the modulator bias point on the scenario of chaos development is experimentally investigated.

A microwave electrically tunable photonic crystal based on a periodically width-modulated slot transmission line with a ferroelectric film is investigated for the first time. The dispersion and transmission characteristics of electromagnetic waves propagating in such a structure are calculated. The electric control over photonic crystal characteristics is simulated.

A theory is developed to describe the wave processes that occur in waveguide media having several types of nonlinearity, specifically, multinonlinear media. It is shown that the nonlinear Schrodinger equation can be used to describe the general wave process that occurs in such media. The competition between the electric wave nonlinearity and the magnetic wave nonlinearity in a layered multinonlinear ferrite–ferroelectric structure is found to change a total repulsive nonlinearity into a total attractive nonlinearity.

The transition to chaos in a microwave photonic generator has been experimentally studied for the first time, and the generated broadband chaotic microwave signal has been analyzed. The generator represented a ring circuit with the microwave tract containing a low-pass filter and a microwave amplifier. The optical tract comprised a fiber delay line. The possibility of generating chaotic oscillations with uniform spectral power density in a 3–8 GHz range is demonstrated.

Influence of an optical fiber length on nonlinear dynamics of a spin-wave optoelectronic oscillator is investigated. An increase of the signal power level leads to transition from a stable monochromatic oscillations to a chaotic signal generation through a number of bifurcations. Generation of the rectangular pulses, periodic nonlinear pulses, chaotic pulses, and dynamic chaos are observed for the different fiber lengths. Numerical characterization demonstrates increase of the fractal dimension with increasing of the optical fiber length.

A theory of spin-electromagnetic wave (SEW) spectrum has been developed for a thin-film ferrite-ferroelectric structure based on a coplanar transmission line. The dispersion relation was derived through analytical solution of the full set of Maxwell`s equations utilizing a method of approximate boundary conditions. The distinctive features of the investigated structures are the thin-film planar topology and dual tunability of the SEW spectrum.

In this paper we investigate in details the influence of the noise intensity on the result of the fractal dimension calculation. The fractal dimension of multidimensional torus is considered as an analysis object. Sum of a number of harmonic signals with irrational ratio of their frequencies and noise signal allows to construct partially destructed torus. It is shown that the presence of the noise leads to increase of the fractal dimension. Value of the increase depends on the relation between noise intensity and harmonics intensity.

Formation of losses in the dielectric optical micro-waveguides of rectangular cross section is analytically studied. It is shown that the two processes are involved in the formation of the frequency characteristic of the attenuation constant: the redistribution of energy in the waveguide-environment system and the group velocity dispersion. The group velocity dispersion determined by the geometry of the waveguiding structure leads to the formation of a local maximum of the attenuation constant in the frequency response.

The nontrivial feature and penetration depth of the topological surface states (TSS) in ${\mathrm{SmB}}_{6}$ were studied via spin pumping. The experiments used ${\mathrm{SmB}}_{6}$ thin films grown on the bulk magnetic insulator ${\mathrm{Y}}_{3}{\mathrm{Fe}}_{5}{\mathrm{O}}_{12}$ (YIG). Upon the excitation of magnetization precession in the YIG, a spin current is generated in the ${\mathrm{SmB}}_{6}$ that produces, via spin-orbit coupling, a lateral electrical voltage in the film. This spin-pumping voltage signal becomes considerably stronger as the temperature decreases from 150 to 10 K, and such an enhancement most likely originates from the spin-momentum locking of the TSS and may thereby serve as evidence for the nontrivial nature of the TSS. The voltage data also show a unique film thickness dependence that suggests a TSS depth of $\ensuremath{\sim}32\text{ }\text{ }\mathrm{nm}$. The spin-pumping results are supported by transport measurements and analyses using a tight binding model.

Active ring resonators (ARRs) in the form of a feedback loop consisting of a spin-wave delay line and microwave amplifier are widely used for investigation of nonlinear dynamics in driven damped systems (see, e.g. [1–5]). Recently, generation of spectrally pure and low noise microwave (MW) signal was observed using optoelectronic feedback loops with optical delay lines [6–8]. An advantage of such schemes is due to attain an ultrahigh Q-factor, which is defined by the length of optical fiber [9]. It is physically clear that a connection in the closed-loop of a ferrite waveguide and a high-Q optical storage element opens new opportunities to investigate nonlinear effects. The aim of this work is to study the foldover effect that appears due to the spin-wave nonlinearity enhanced by the high-Q optical element in a combined spin-wave/optical ARR. The considered ARR scheme consists of a spin-wave delay line, a microwave attenuator, a microwave amplifier, an electro-optic modulator, an optical fiber, and a microwave photodetector, as it is shown in Fig. 1. In this scheme, the light from the external laser is introduced into the modulator, closing the circuit to the ring. Note that this ring, like an optoelectronic oscillator [9], may start up into MW self-generation in case a microwave amplification compensates for losses. In the regime preceding the self-generation, the amplitude of the signal circulating in the high-Q ring increases significantly at the resonant frequencies, and exceeds the nonlinear phenomena threshold, which is determined by the nonlinear properties of the spin waves. It manifests itself as a resonant frequency shift and foldover effect. The important feature of the investigated ARR in comparison with a typical spin-wave ARR is much higher Q-factor. It means that the optical delay line affects the nonlinear wave properties of the spin-wave ARR. Hence, the “hybrid nature” of the proposed ARR proves itself. To investigate the foldover effect in the spin-wave ARR with high-Q optical element the experiments were carried out. The spin-wave delay line consisted of an yttrium-iron garnet 5-μm thick film and two microstrip transducers separated by 4 mm. The YIG film was magnetized perpendicularly to its surface by a magnetic field of 2630 Oe. The optical part of the ARR consisted of the commercially available elements: a laser giving optical radiation of 1.55 μm, a 10-GHz electro-optical Mach–Zehnder modulator, a single mode fiber with length of 100 m, and a 10-GHz photodetector. It was found out that an increase in the amplifier gain resulted in a nonlinear uppershift by 110 kHz of all resonant frequencies. Note that for a bigger nonlinear frequency shift the foldover instability occurred. In order to illustrate a development of foldover effect the resonant peak shapes measured for −9 dB and −1 dB below the self-oscillation threshold are shown in Fig. 2 by solid blue line and solid red line, respectively. To better understand the physics underlying the foldover effect in the ARR, a theoretical model was developed as follows. First, a nonlinear dispersion relation for the carrier spin waves was derived from the linear dispersion relation taking into account the dependence of the static magnetization on the amplitude of circulating signal. This amplitude was determined as the product of the amplitude of the input signal and the transmission coefficient of the investigated ARR. Then, the wave-number as a function of frequency was numerically calculated from the nonlinear dispersion relation. Its substitution into the linear transmission coefficient resulted in the nonlinear transmission characteristic (NTC) of the ARR. The obtained NTC was used for a comparison with the experimental results (see Fig. 2). One can see that the theoretical and experimental results have a good agreement. In conclusion, the theoretical model of the resonance spectra for the spin-wave optoelectronic ARR was developed taking into account both the dispersion and nonlinear properties of the ferrite film. Results of the numerical simulation have a good agreement with the transmission characteristic measured in the experiment. The developed theory gives a possibility to investigate the foldover effect evolving due to the nonlinear spin-wave properties of the ferromagnetic film included in the ARR containing the high-Q optical elements. The work was supported by the Ministry of Education and Science of Russian Federation (agreement 14.575.21.0157, unique identifier RFMEFI57517X0157).

Artificial magnetic materials with a spatial periodic modulation of their physical properties or geometry, known as magnonic crystals, are promising for new microwave devices such as phase shifters [1, 2], various spin-wave logic elements [3, 4], high-sensitivity magnetic sensors [5, 6], and others [7, 8]. Recent advances related to the thin-film technology have resulted in fabrication of the multilayered multiferroic structures that combine advantages of the ferrites and ferroelectrics. Owning to the dual tunability of wave spectra by both electric and magnetic fields, such structures were widely used in microwave devices [9]. Different kinds of ferrite-ferroelectric all-thin-film structures were suggested based on coplanar or slot transmission lines [10–12]. In these structures, spin-electromagnetic waves (SEW) are originated from the electrodynamic coupling of the electromagnetic wave propagating in the transmission line with the spin waves (SW) propagating in the ferrite film. However, up to now the periodic multiferroic structures based on coplanar waveguides (CPW) were investigated only experimentally [10]. The purpose of this work is to develop a theory of hybridized spin-electromagnetic waves in the thin-film regular and periodic multiferroic waveguiding structures based on CPW. In order to distinguish the periodical multiferroic waveguides from known ferrite magnonic crystals, as well as from photonic crystals, below we name them as electromagnetic crystals (EMC). A studied EMC structure is shown in Fig. 1. It is composed of several layers enumerated with index j, namely, a sapphire substrate (j = 1), a barium strontium titanate (BST) ferroelectric film (j = 2), an epitaxial yttrium iron garnet (YIG) film (j = 3), and a gadolinium gallium garnet substrate (j = 4). The periodic segments of the CPW form the thin-film EMC. Here the central metal strip of width h and two conducting ground planes are positioned in the plane z between ferroelectric and ferrite layers. The segments of narrow $w_{1}$ and wide $w_{2}$ slots has the period λ. Application of control voltage U to the periodic CPW electrodes provides a reduction of the ferroelectric film permittivity $\varepsilon _{2}$ and so maintains an electric tunability. The hybridized waves are considered to propagate along x-axis, i.e. along CPW, which is magnetized to saturation by a uniform magnetic field H along z-axis. Due to the symmetry of the fundamental CPW mode its dispersion relation can be found through analytical solution of the full set of Maxwell’s equations utilizing the method of approximate boundary conditions described in details in Ref. [12]. The obtained dispersion relation for a regular CPW was used for a numerical calculation of the transmission characteristics of the EMC using the transfer-matrix method [13]. Note that this method takes into account the insertion losses and is suitable for the finite-length periodic structures. Following the outlined theory, Fig. 2(a) illustrates the effect of hybridization of two principal electromagnetic modes for the regular CPW and regular slot-line structure. One can clearly see that the area of the maximum hybridization of SEWs in the CPW shifts to the higher wavenumbers in comparison to the slot-line structure. This behavior is determined by a sufficient reduction of the phase velocity of the microwave electromagnetic waves due to additional “magnetic wall” boundary condition applied at the central metal strip. Note that the calculations were carried out for the typical parameters of YIG and BST films commonly used in microwave devices (see, e.g., Ref. [9]). A reduction of the CPW slot width w shifts the SEW dispersion characteristic to the higher wave numbers. Consequently, the SEW formed in the EMC accumulate the different phase shifts in different segments of the periodic structure at a fixed frequency (see Fig. 1). The band-gaps appear at the frequencies where this phase shift is a multiple of π. As a result, a major part of the spin-electromagnetic wave power will be reflected from the electrodes of EMC at the frequencies that satisfy to the Bragg condition. This effect is visible on the transmission characteristic shown in Fig. 2(b). The characteristic was calculated for the EMC with number of periods $N \quad =10$ and $\Lambda \quad =1$ mm with the use of the transfer matrix method. In this case, the width of the first band gap (denoted by I in Fig. 2(b)) is 24.6 MHz at a level of 3 dB from the maximum loss at 33 dB. In addition, this figure illustrates electric tuning of the EMC band gap positions for a reduction of the ferroelectric film permittivity $\varepsilon _{3}$ from 1500 to 750. In summary, a novel EMC based on CPW were studied. In particular, it was found that a high microwave signal rejection of more than 30 dB appears for the periodic structures. Furthermore, the electric tuning for the first band-gap reaches values of 10.45 MHz by the dielectric permittivity reducing of the ferroelectric film by half. All these advantages make this kind of EMC perspective for development of new microwave devices. The work in SPbETU was supported in part by the Russian Science Foundation, Grant 14-12-01296P. The work in LUT was supported by the Academy of Finland.

Exact fractals of nonlinear waves that rely on strong dispersion and nonlinearity and arise spontaneously out of magnetic media were observed for the first time. The experiments make use of a microwave to excite a spin wave in a quasi-one-dimensional magnonic crystal. When the power of the input microwave (${P}_{\text{in}}$) is low, the output signal has a power-frequency spectrum that consists of a single peak. When ${P}_{\text{in}}$ is increased to a certain level, new side modes are generated through modulational instability, resulting in a comblike frequency spectrum. With a further increase in ${P}_{\text{in}}$, each peak in the frequency comb can evolve into its own finer comb through the modulational instability. As ${P}_{\text{in}}$ is increased further, one can observe yet another set of finer frequency combs. Such a frequency-domain fractal manifests itself as multiple layers of amplitude modulation in the time-domain signal.

A theoretical model of an electronically tunable multi-loop optoelectronic oscillator (OEO) based on photonic and microwave spin-wave delay lines has been developed. The theory takes into account a dispersion of electromagnetic waves in optical fibers and a dispersion of spin waves in ferrite films. An expression for transfer function of the OEO circuitry was derived. The theory enables one to calculate transmission characteristics and suppression of spurious harmonics of the multi-loop OEOs.

Envelope solitons of hybrid spin electromagnetic waves have been experimentally observed. These solitons are formed at the self-generation of spin electromagnetic waves in an active ring cavity, where a ferrite-ferroelectric layered multiferroic structure serves as a nonlinear waveguide dispersive medium. The effect of the permittivity of the structure on the nonlinear properties of spin electromagnetic waves has been demonstrated. The hybrid wave nature of solitons has been experimentally and theoretically justified.

Electric and magnetic tuning of phase, frequency, and delay time of electromagnetic-spin waves in thin-film layered multiferroic structure (metal-dielectric gap-ferroelectric material-hexaferrite-dielectric substrate) is numerically simulated. The results are used to demonstrate prospects for application of layered multiferroic hexaferrite-ferroelectric structures in the construction of basically new devices for generation and processing of microwave signals under dual (electric and magnetic) tuning of the working characteristics in the subterahertz frequency range.

A theoretical model describing the resonant frequencies of a tunable optoelectronic oscillator on the optical and spin-wave delay lines is presented. In the theoretical model, the optical delay line has two configurations. The first configuration was formed by parallel connection of optical waveguides having various lengths. The second one is the serial connection of optical waveguides closed in rings. The transmission characteristics for the generators of both configurations are obtained. The paper presents the results of numerical simulations of transmission characteristics for circuits containing one, two and three optical delay lines. The results are compared for cases of parallel and serial connection.

A theoretical model of the tunable multi-loop optoelectronic oscillators based on photonic and microwave spin-wave delay lines was developed. The theory takes into account a dispersion of electromagnetic waves in optical fibers and a dispersion of spin waves in ferrite films. An expression for transfer function of the multi-loop circuitry was derived. Numerical simulations of the resonant frequency spectra were carried out. Theoretical results are in good agreement with experiment.

A logic gate based on a Mach-Zehnder microwave interferometer is developed. The interferometer has a bridge circuitry with two arms. Each arm comprises a multiferroic phase shifter providing the control of the phase of carrier spin-electromagnetic waves. A voltage applied to the multiferroic structure serves as a logical input. Performance characteristics of the device is presented.

An increased demand to frequency-agile materials for microwave devices has led to appearance of artificial multiferroics [1].

At present an increased research interest to study microwave stationary and chaotic signal generation in magnetic film active ring oscillators is evident [1,2].

The foldover effect of nonlinear eigenmodes in feedback ring systems was observed. Experiments made use of a system that consisted of a magnetic thin film strip, which supported the propagation of spin waves, and a microwave amplifier, which amplified the signal from the output of the magnetic strip and then fed it back to the input of the magnetic strip. The signal amplitude vs frequency response in this ring system showed resonant peaks which resulted from ring eigenmodes. With an increase in the resonance amplitude, those resonant peaks evolved from symmetric peaks to asymmetric ones and then folded over to higher frequencies. The experimental observations were reproduced by theoretical calculations that took into account the nonlinearity-produced frequency shift of the traveling spin waves.

A theory has been constructed that strictly describes the spectrum of dipole-exchange spin waves in an arbitrarily magnetized anisotropic ferrite waveguide with a rectangular cross section. The theory takes into account the spatial inhomogeneity of the internal magnetic field in the waveguide cross section. The influence of parameters of the ferrite waveguide on the distribution of the internal magnetic field in the waveguide cross section is analyzed. The dispersion characteristics of two waveguide types most widely used in practice are investigated. The dipole-exchange spin wave spectra calculated for a transversely magnetized waveguide are presented and the distributions of the dynamic magnetization in the waveguide cross section for several types of volume and localized spin-wave modes are constructed.

Microwave properties of ferrite-ferroelectric magnonic crystals are investigated for the first time. The electric and magnetic rearrangement of the band gaps in the spectrum of proper electromagnetic spin waves in such structures is demonstrated.

Amplitude, frequency, and time domain characteristics have been mapped for short spin wave pulses inside a magnonic crystal. A space- and time-resolved magnetoinductive probing system has been used to detail the spin wave spectral, propagation, and evolution characteristics in a geometrically structured yttrium iron garnet film. Experiments have been performed using magnetostatic surface spin waves excited in a chemically-etched magnonic crystal, ultrafast pulsed excitation of the spin waves, and direct spin wave detection using a scannable magnetoinductive probe connected to a synchronized fast oscilloscope. The results show how the frequency discriminating effect of a magnonic bandgap decreases as the excitation pulse width decreases. They also show how the use of rectangular pulses compromise the magnonic crystal performance because of the high frequency components of such pulses. Space and time maps show how these components are transmitted without additional damping.

The nonlinear wave properties of an artificial multiferroic medium consisting of a ferroelectric layer and a ferromagnetic layer have been studied. The simultaneous effect of wave nonlinearities of both layers on the formation and propagation of envelope solitons has been analyzed. It has been shown that bright envelope solitons of electromagnetic spin waves can appear in such a medium owing to the wave nonlinearity of ferroelectric layer. In order to confirm the theoretical results, the coefficients of the nonlinear Schrodinger equation have been calculated and this equation has been solved numerically.

The complex transmission coefficient of active ring resonators based on spin-wave delay lines was investigated both theoretically and experimentally. Influence of the parameters of the delay line on the transmission coefficients was studied and analyzed. It was shown that the resonant frequencies that define the bandpasses of the ring resonator depend on the ferrite film thickness and the distance between spin-wave antennae. These dependences give possibility to reconfigure the transmission coefficient that in combination with magnetic tuning provide flexibility for microwave applications.

A theoretical model of a microwave active ring resonator based on a ferrite-ferroelectric layered structure serving as a waveguide for hybrid electromagnetic-spin waves is developed. The resonator is made in the form of a closed ring consisting of a ferrite-ferroelectric delay line, a microwave amplifier, and a variable attenuator. The resonance properties of this system are studied theoretically. The feasibility is demonstrated of effectively controlling the resonance frequencies of the ring by varying the permittivity of the ferroelectric layer.

The emergence of the self-modulation instability of monochromatic microwave spin waves excited in a periodic magnetic film structure, a magnonic crystal, has been observed. The magnonic crystal was fabricated by chemically etching the surface a single-crystal yttrium-iron garnet film. The self-modulation instability has been observed at the frequencies corresponding to the band gaps of the spin-wave spectrum of the magnonic crystal caused by Bragg resonances. Above a certain threshold value of the input microwave signal, the time profile of the output pulses corresponds to envelope solitons.

Properties of the high-frequency spin-electromagnetic wave chaos developed in active ring oscillators have been investigated. A multiferroic structure composed of yttrium iron garnet film and barium strontium titanate (BST) slab served as a nonlinear dispersive medium of the oscillator. Dual control of the fractal dimension of the chaotic signal attractor was realized by variation of the ring gain and dielectric permittivity of the BST slab.