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A nonlinear phase shift of low-power spin waves (SWs) induced by a high-power pumping SW excited both inside and outside the magnonic band-gaps of a magnonic crystal has been studied. The magnonic crystal with spatially periodic thickness is fabricated from an yttrium iron garnet film by chemical etching. The results show that the phase shift of the low-power SWs can be effectively controlled by variation of power level of the pumping SW. This induced nonlinear phase shift is weakened if the pump frequency lies in the magnonic bandgap. The data obtained are well explained by contradirectional coupling of the high-power forward and reflected spin waves. A theoretical model for this effect is presented. Our findings are important for the further progress in SW computing.

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.

We report on the use of passive micro-ring resonator fabricated with silicon-on-insulator technology for memory applications. We show that the minimal pulse length ensuring the memory operation is limited by 260 ns.

Self-generation of microwave nonlinear waveforms in the magnonic-optoelectronic oscillator (MOEO) was investigated. Nonlinear dynamics of the MOEO was due to both optical and magnonic paths of the oscillator circuit. Four-magnon parametric interactions in the magnonic path and cosine transfer function of the electro-optical modulator caused double nonlinearity of the MOEO. Gain coefficient was used as a control parameter. We found that during a route from regular to chaotic dynamics, the oscillator generates two unusual waveforms: symmetry-breaking soliton-like modes of Möbius type and periodic pulses with chaotic amplitude modulation. Nonlinear waveforms were characterized using a time series analysis. Peculiarities of the signals and their spectra in regular and chaotic regimes of self-generation are discussed. We expect that the multiple nonlinearity of the MOEO may be useful for investigation of various fundamental effects in complex time-delayed systems and for development of novel circuits for neuromorphic computing.

The dispersion bistability in the silicon micro-ring resonators (MRRs) manifests itself due to the competition between the thermo-optic effect and the free-carrier dispersion (FCD) induced by two-photon absorption [1] . The experimental observation of optical bistability induced by the FCD was reported in work [2] for short pulse regime. In this work we demonstrate, for the first time to our knowledge, the carrier-induced bistability in the silicon MRRs under the continuous wave (CW) pumping.

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.

The present work focuses on experimental investigations of a bistabile silicon-on-insulator (SOI) micro-ring resonator (MRR). The resonator exploits a continuous-wave operation of the carrier-induced bistability demonstrating a stable hysteresis response at the through and drop ports when the input power exceeds the threshold value. Flipping the optical input power provides a convenient mechanism for a switching of the MRR output characteristics between two steady states having a long holding time. The transition of the resonator output between these states is experimentally investigated. It is shown that the switching speed is limited by a low-to-high transition of 188 ns. Obtained results shows an application of the passive SOI MRR as an all-optical memory cell with two complementary outputs.

The self-generation of a chaotic microwave signal in a spin-wave optoelectronic generator has been experimentally investigated for the first time. The generator was a ring circuit, in the microwave path of which there was a spin-wave delay line on the film of an yttrium iron garnet (YIG) and a microwave amplifier. The optical path contained a fiber optic delay line. The nonlinear element of the generator was a film of YIG, which worked under the conditions of four-wave parametric interaction of spin waves. It has been shown that it is possible to generate monochromatic, periodic, quasi-periodic, and chaotic microwave signals by controlling the gain coefficient of the ring. The parameters characterizing the chaotic signal that are generated by the ring have been determined.

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 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.

An induced nonlinear phase shift of spin waves excited in an yttrium iron garnet film waveguide has been studied. Experiments were performed with two microwave magnonic signals, being operating and pumping waves, co-propagated at different frequencies. The results show that a nonlinear phase shift up to 180° for the low-power operating wave is induced when increasing the pump-wave power. A theory describing the induced nonlinear phase shift is developed taking into account four-wave parametric coupling and nonlinear damping of two waves. The observed phenomenon paves the way to elaborate different types of nonlinear magnonic logic circuits.

Single crystals of aluminum-substituted barium lead hexaferrite were grown from PbO/B 2 O 3 flux. The stable chemical flux composition presents a lower degree of flux component evaporation than pure lead oxide and provides more stable conditions for crystallization. This leads to good crystal quality coupled with increased substitution levels. At 1533 K substituted barium hexaferrite crystals with compositions Ba 0.84 Pb 0.16 Fe 12– x Al x O 19 and Ba 0.15 Pb 0.85 Fe 12– x Al x O 19 with x ~ 2 were obtained as hexagonal plates with approximately 5 mm in diameter. The unit cell parameters drop substantially upon substitution of Fe 3+ and Ba 2+ with smaller Al 3+ and Pb 2+ , respectively. Millimeter-wave characterization in the frequency range of 75–100 GHz of Ba 0.84 Pb 0.16 Fe 10.1 Al 1.9 O 19 and Ba 0.15 Pb 0.85 Fe 9.9 Al 2.1 O 19 were carried out with a WR-10 waveguide, showing zero field resonance frequencies of about 79 GHz and 90 GHz, respectively, which are larger than for pure barium hexaferrite due to larger anisotropy of the aluminum substituted samples. Magnetic parameters of the materials are determined and discussed.

Theoretical investigation of dual (electric and magnetic) tuning of dispersion characteristics has been studied for the doubly hybridized spin-electromagnetic waves (SEWs) propagating perpendicularly to the direction of static magnetization in tangentially magnetized all-thin-film multiferroic multilayers. The multilayers consisted of two thin ferrite films separated by a thin ferroelectric film. The SEW spectrum was formed as the result of the double hybridization among one electromagnetic mode and two spin-wave modes. Such hybridization took place if the initial dispersion characteristics of these modes were degenerated. The electric tuning was realized owing to the changing of the magneto-dipole interaction between the magnetic films caused by a variation of dielectric constant of the ferroelectric film. It was shown that a decrease in permittivity of the ferroelectric film of micrometer thickness by a factor of two induces change of the SEW wavenumber up to tens of radian per centimeter.

Dynamic modes of microwave signal autogeneration in a radio photonic generator have been investigated. The generator is a ring circuit with a low-pass filter and microwave amplifier in its microwave path. The optical path contains an optical fiber delay line. The generator demonstrates the periodical, chaotic, and noise dynamics. It has been shown that the correlation dimensionality of the random signal attractor in the chaotic generation mode saturates with increasing phase space dimensionality. Saturation is not observed in the noise-generation mode.

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.

A spin-wave delay-line active-ring oscillator has recently been proposed as a suitable substrate to implement the physical reservoir computing model. The concept displays the required properties of fading memory and nonlinearity characteristic to the model. In this paper, we improve the concept by increasing the signal delay time in the yttrium-iron garnet film by more than four times, and we examine further the improved system by evaluating experimentally the performance on two benchmark classification tasks. The short-term memory (STM) task evaluates the linear memory characteristics of the RC, while the parity-check (PC) task evaluates the nonlinear computing capability. Adequate performance on both is achieved, and the linear memory is shown to be strongly dependent on the synchronization between the reservoir computer (RC) inputs and the active-ring circulation time. The extended delay time and other major improvements result in STM and PC capacities reaching maximum values of 4.68 and 1.74, respectively.

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 interferometer using spin-electromagnetic waves was developed for the first time. Theoretical and experimental investigation of its characteristics was performed. The interferometer is a bridge circuit with a phase shifter in one of its arms, which is based on a planar multiferroic structure consisting of ferrite and ferroelectric layers. The device can be regarded as a prototype of spin-electromagnetic logic.

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.

Artificial extrinsic and intrinsic multiferroic structures were developed in the form of yttrium-iron garnet/barium-strontium titanate bi-layers and manganese-doped barium-strontium titanate films with a relatively high doping rate in the range of 10–20 mol%. For the experimental investigations, the planar capacitors were formed on the surfaces of the structures and the capacitance–voltage characteristics were measured. Structural characterization of the samples under study was based on X-ray diffraction analysis and film–substrate interfaces were investigated by X-ray photoelectron spectroscopy. The relative dielectric permittivity, dielectric loss tangent, and electric field tunability of the samples were measured in a temperature range from 77 to 300 К. The additional 6 and 3 % tuning of the dielectric permittivity by external magnetic field of 2000 Oe was demonstrated for the extrinsic and intrinsic structures, respectively.

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.

A dynamic magnonic crystal based on a planar multiferroid ferrite–ferroelectric structure is suggested and studied. A space-periodic electric field applied to the ferroelectric layer converts a regular multi-ferroid waveguide to a periodic structure. An electrodynamic model of a dynamic magnonic crystal is constructed, and its dispersion and transfer characteristics are studied.

Single crystals of barium hexaferrites with compositions BaFe 12 O 19 and BaFe 11.5 Al 0.5 O 19 were grown from carbonate flux. The Al substitution results in a reduction of the unit cell volume from 697.54(7) A 3 for unsubstituted hexaferrite to 695.93(3) A 3 for the Al-substituted phase and the Curie temperature from 450 to 420 °C. The magnetization of BaFe 11.5 Al 0.5 O 19 samples was measured between 3 and 300 K in external magnetic fields of 0.001 and 0.05 T. The temperature dependence of the magnetization at a magnetic field of 0.001 T demonstrates irreversibility, which disappeared at 0.05 T field. Millimeter-wave characterization of BaFe 11.5 Al 0.5 O 19 slabs loaded in a WR-10 waveguide was carried out in the frequency range of 75 to 100 GHz. In this range electromagnetic resonances were observed at the frequencies of 78.6 GHz and 96.3 GHz. The resonant microwave absorption level was controlled by value of bias magnetic field.

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.

A novel method of digital data transmission is developed and experimentally investigated. The main idea of the method is to convert digital data into a chaotic spectrum, mix it with a background signal and transmit to receiver using TCP/IP protocol. The hyper-chaotic Lorenz system of differential equations is used as a source for carriers of the chaotic spectrum. Chaotic synchronization is not used in the method, that allows to improve stability of data transmission. Human voice is used as the background signal. Therefore, the method allows one to use single communication channel for transmission of different signals without losses. Numerical investigations approved that the chaotic signal transformation does not change chaotic nature of the Lorenz system. Thus, developed method demonstrates high privacy level.

The creation of devices based on the electrocaloric effect requires the methods for assessing pyroelectric currents at the different external parameters. Two indirect methods for calculating pyroelectric currents in ferroelectrics are considered. The first one based on the temperature-field dependences of the dielectric hysteresis loops. The second one based on the temperature-field dependences of the dielectric constant. The original method for directly estimating the magnitude of the pyroelectric currents by the charge change with thermal flux of laser radiation pulses is proposed.

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.

Propagation of spin waves in a one-dimensional magnonic crystal to a finite distance have been studied theoretically and experimentally. It was shown that effective spectrum of spin waves at frequencies close to the Bragg resonances strongly depends on the magnonic crystal length and magnetic dissipation. The emergence of the four serial dispersion regions with positive and negative dispersion close to magnonic crystal stop bands was experimentally confirmed. A comparison of the experimental and theoretical results demonstrated their good qualitative and quantitative coincidence.

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.

The characteristics of a spin-wave optoelectronic microwave oscillator are experimentally studied for the first time. The oscillator represents a ring structure the microwave circuit of which contains a spinwave delay line based on the yttrium–iron-garnet film. The optical part of the oscillator contains a fiberoptic delay line with a length of 100 m. It is demonstrated that the spin-wave delay line makes it possible to tune the output frequency in wide ranges at a relatively low level of the phase noise. In the gigahertz frequency range, the level of the phase noise is no greater than–110 dBc/Hz at a detuning of 10 kHz from the carrier frequency.

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.