Investigation of Caustic Spin Wave Beams in Soft Thin Films

Abstract

In thin-films-based experiments, spin waves (SWs) are often investigated in confined geometries or as plane waves in full films. There are exceptions, such as SW channeling along domain walls [1], which are appealing in terms of SW steering. However, this approach relies on the preparation of a specific magnetic texture and transferring the SW out of it, back into the domains, seems challenging. An alternative route towards the steering of strong, channeled SWs exists, namely caustic SW beams (CSWBs).Caustic phenomena in anisotropic wave propagation correspond to sharply-defined lines/surfaces with large intensities [2]. They have been investigated in a variety of fields [2], and can be expected in micromagnetism. In thin films, caustics can arise due to the marked anisotropy of the SW dispersion relation (DR), illustrated in Fig.1.From such a plot, the direction θv of the group velocity vg=▽kω can be determined from the local normal to the curve. It can be seen in Fig.1 that a portion of the red curve (at 8.5 GHz) is almost straight, leading to a stationary θv around a value θvc for this range of wavefront angles φ. Thus, an excitation of SWs with a broad wave vector spectrum would lead to strong emission in the direction θvc. The corresponding anisotropic enhancement defines a CSWB. Such beams have so far been studied notably upon scattering of a confined SW into a wide film [4,5,6].In the present work, we tackle theoretically the emission of CSWBs in full, soft thin films; the latter could be stimulated by the tip of a microstrip antenna. We restrict ourselves to applied fields below ferromagnetic resonance (FMR). After dimensionless rewriting of the DR, only three free parameters are left: the ratio of dipolar-exchange length to film thickness η=lex/d, the normalized SW frequency ν=ω/(γ*Js) (Js: saturation induction), and the reduced applied field h=Ba/Js. We then investigate the properties of caustic points numerically and analytically, using suitable approximations.An experimental situation corresponds to fixed film thickness; we therefore record the caustic wavenumbers (WNs) kc*d, beam directions θvc and wavefront directions φc as a function of ν and h, at fixed η. As the example in Fig.2 shows, caustic points can exhibit a rich behaviour. Starting with the caustic WNs, one can see a low-frequency, low-field pocket in the dipolar-dominated regime. Above its sharp boundary, a relatively wide band appears, corresponding to kc*d varying slowly in the dipolar-exchange regime. A final transition brings kc*d into the exchange-dominated regime; the latter has an upper boundary above which no caustic point exists (blue region in Fig.2).The map of φc(h,ν) displays a similar aspect. In the bottom pocket, the caustic wavefront angle is almost constant, with a numerically found limit of about 54.7°. Our analytical approximation yields a value of arccos(1/√3)≈54.7°, in excellent agreement. The corresponding limit for θvc is numerically 109° and 103° from analytics. In the dipolar-exchange band, by contrast with kc*d, φc varies significantly and saturates at 90° in the exchange-dominated region. This saturation is in stark correlation to that of the caustic beam direction θvc, which also goes to 90°. All the above properties can be understood from the geometrical evolution of the DR curve when ν increases, as illustrated in Fig.1. The markers on the curves correspond to their caustic point (2.5 GHz omitted for clarity).At the bottom pocket's boundary, the tangent to the DR at kc*d passes close to the origin: hence, ∂kc/∂φc is large. In the exchange-dominated regime, the curve flattens near φ=π/2. This accelerates the increase of kc*d (cf. curves at 16 and 18.5 GHz) while bringing θvc and φc to π/2 at a certain threshold frequency. Beyond the latter, the curve rounds more and more: θv can no longer be stationary, thus, no caustic point exist.From this investigation, we can thus predict two very different types of CSWBs. At the low-kc pocket's upper boundary, due to the very straight and oblique DR curve, the CSWB should feature a broad wavelength spectrum. By contrast, at the upper boundary of the caustic band, the CSWB's wavelength spectrum should be much narrower. Indeed, the DR at kc*d is tangent to a circle k*d=cst. In addition, since this CSWB consists of two beams with overlapping directions, its amplitude should be correspondingly larger. This phenomenon has been studied theoretically by Kim et al. [7].The first type of CSWB should be invisible in magneto-optical microscopy, but Brillouin light scattering should reveal the broadband SW beam. The second type of CSWB would require time-resolved Scanning Transmission X-ray Microscopy due to the higher kc. **