Spin-wave Talbot effect in a thin ferromagnetic film

The Talbot effect has been known in linear optics since the 19th century and has found various technological applications. In this paper, with the help of micromagnetic simulations, we demonstrate the self-imaging phenomenon for spin waves in a thin, out-of-plane and in-plane magnetized ferromagnetic film whose propagation is described by the Landau-Lifshitz nonlinear equation. We show that the main features of the obtained Talbot carpets for spin waves can be described, to a large extent, by the approximate analytical formulas yielded by the general analysis of the wave phenomena. Our results indicate a route to a feasible experimental realization of the Talbot effect at low and high frequencies and offer interesting effects and possible applications in magnonics.

Figure 1

Diagram of self-images arising behind the diffraction grating with indicated Talbot length $z_T$, diffraction grating constant $d$, aperture width $s$, wave vector $k_R$ as a function of incident angle $\theta$, and wave vector $k_z$ propagating along the $z$ axis.

Figure 2

Talbot carpets obtained from Eqs. (12) and (13) for 31 sum terms and (a) the aperture width $s=\lambda =1\left[\text{arb.}\text{units}\right]$ and diffraction grating constant $d=3\lambda$, and (b) $s=\lambda =0.25\left[\text{arb.}\text{units}\right], d=12\lambda$. The grating is located at the left edge of each plot.

Figure 3

Spin-wave intensity distribution at frequency of 40 GHz after passing through a one-dimensional diffraction grating with period of (a) 80, (b) 200, and (c) 280 nm. The grating is located at the left edge of each plot.

Figure 4

Spin-wave intensity distribution field at frequency of 3 GHz after passing through a one-dimensional diffraction grating with period of (a) 400, (b) 1000, and (c) 1400 nm. The grating is located at the left edge of each plot.

Figure 5

Functions representing the Talbot length depending on the diffraction grating period from Eq. (1) together with the data taken from micromagnetic simulations for (a) 40 and (b) 3 GHz.

Figure 6

SW spectra with marked first peaks responsible for the propagation direction wavelength. On the left panel there are spectra from simulations performed for 40 GHz for period $d$, which is equal to (a) 80, (b) 200, and (c) 280 nm. On the right panel, we have analogous results for 3 GHz for period of (d) 400, (e) 1000, and (f) 1400 nm.

Figure 7

Reciprocal-space maps of SW amplitude distribution at 40 GHz for the grating period equal to (a) 80, (b) 200, and (c) 280 nm, and at 3 GHz for the period equal to (d) 400, (e) 1000, and (f) 1400 nm. Isofrequency contours are presented as red circles. The quantized wave-vector lines are shown in green.

Figure 8

Spin-wave Talbot effect in thin Py film with damping constant $\alpha =0.005$ for 3 GHz and diffraction grating period $d=1000$ nm as (a) a 2D intensity plot with mapped two lines (red and green) along which (b) 1D intensity plots were created. Diffraction grating is located at the left edge, and the line colors are analogous on both graphs.

Figure 9

Spin-wave intensity distribution fields after passing through a diffraction grating $d=500$ nm with corresponding representations in reciprocal spaces at frequency of 15 GHz for (a) BV and (b) DE geometry. Analytical isofrequency contours are presented as red ellipses, and quantized wave-vector lines separated by $\delta k_x=2\pi /d$ are shown in green.

Figure 10

Diagram showing the situation described in this section together with relevant markings–a diffraction grating consisting of $N$ sources distant from the analyzed point $P$ by $r_n$, where $n$ is a number indicating the given aperture. The center of the coordinate system is in the aperture $n=0$.

Figure 11

Propagation scheme of two diffraction orders of the plane wave fronts behind a diffraction grating, in accordance with the rule described by Eq. (B1). Gray lines are the wave fronts with equal phases, while the black arrows indicate the wave-vector directions. Wavelength representations on the axes are marked below.

Figure 12

Graph of the Talbot length as a function of the diffraction grating constant. The general form and the form derived in this paper are shown by blue and green lines. The discrepancy function is shown by the red line.

Figure 13

The dispersion relation for 5-nm-thick Py film in the presence of a 1.1-T out-of-plane magnetic field. The two marked values of the wave vector correspond to 3- and 40-GHz frequencies.