Foldover Effect in Spin-Wave Optoelectronic Active Ring Resona-tors

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).