1. Introduction

Phenomenon of the lateral shift at the reflection of an electromagnetic wave from the interface of two media in contrast to the laws of the geometric optics was predicted and observed for the first time by F. Goos and H. Hänchen at the glass-air interface in the forties of the 20th century [1] and nowadays is known as the Goos-Hänchen (GH) effect. Since that, various applications of this effect have been proposed, including chemical and temperature sensors [2–4], optical switches [5] and de/multiplexers [6]. The GH effect is well studied in linear optics [7,8], including magneto-optics where this phenomenon has been investigated theoretically and numerically [9–17] and observed in experiments [18]. The lateral beam shift was also studied for acoustic waves (Schoch effect) [19] and spin waves [20,21] (or their quasi-particle dual equivalents, phonons and magnons, respectively). The GH effect is common for the wave processes not only in optics, acoustics and spin wave dynamics, but also for reflection of electrons [22] and neutrons [23] because these particles can be presented as electronic and neutron waves.

In the case of three-wave interaction, GH effect has been studied for optical second harmonic generation [24,25] and for inelastic, or Brillouin light scattering (BLS) by acoustic phonons [26]. Nowadays BLS is used as very powerful tool for the study of different kind of elementary excitations in condensed matter physics including magnetic excitations, or spin waves [27–33]. Interaction between the electric field of an electromagnetic wave and the magnetization vector M of the medium in the phenomenological approach can be described via magnetization-dependent part of the dielectric permittivity tensor [34]. Contribution of the static part of M into permittivity leads to such magneto-optic (MO) phenomena as Faraday and Kerr effects (rotation of the polarization plane of the transmitted or reflected electromagnetic wave, respectively), as well to the magnetic birefringence [34], whereas the dynamic (time-dependent) part of the magnetization Δm is responsible for the BLS (or inelastic light scattering) by spin waves.

It should be expected that the GH effect will take place in the case of BLS by spin waves. The description of this effect can be done similarly to approach used in our paper [26] where we described the GH shift at BLS by sound (acoustic phonons). However, in contrast to acousto-optics, for MO interaction we can consider contributions of magnetic subsystem to the polarization which are linear and quadratic on magnetization [34].

In this paper, we investigate the GH effect which occurs at BLS by spin waves at the interface between a magnetic film of yttrium-iron garnet (YIG) and a dielectric substrate of gadolinium-gallium garnet (GGG). We study the case of transversal MO configuration when the static magnetization is perpendicular to the incidence plane [34] and a spin wave propagates in YIG close to the YIG/GGG interface, which corresponds to the Damon-Eshbach configuration. In this configuration, the spin-wave propagation direction is perpendicular to the static magnetization orientation. This type of spin waves has surface character and its amplitude localization, at the top or bottom surface of the magnetic film, changes with the change of the propagation direction [35–37].

2. General equations

We assume that an electromagnetic wave of fundamental frequency ω and fundamental wavelength λ₀ (in vacuum) is incident from YIG on GGG under the incidence angle ω. The interface between the media is parallel to the x-axis, and (xz) is the incidence plane. A spin wave of the angular frequency Ω propagates along the x-axis, as shown in Fig. 1. The incident light wave interacts with the spin wave, and the scattered electromagnetic wave of the frequency ω ± Ω undergoes a lateral GH shift Δx.

 

Fig. 1. Schematics of the BLS by a spin wave. Here s (S) and p (P) denote the incident (scattered) s- and p-polarized light, ω and Ω are the fundamental light angular frequency and the angular frequency of the spin wave, respectively, and Δx is the GH shift of the scattered electromagnetic wave. The violet and yellow arrows show the static m0 and the dynamic Δm parts of the magnetization vector M (black dashed arrow) which precesses around the y-axis.

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Propagation of an electromagnetic wave with the angular frequency ω in YIG is described by the solution of the wave equation [38]

Taking into account both linear and quadratic MO interactions, electric polarization vector has the following form [34]:

It should be noted, that to simplify the analysis, we neglect the dependence of the magnetic permeability of YIG on the magnetization and assume µ₁ = µ₀. Indeed, the value of linear MO tensor element (gyromagnetic coefficient) of YIG is one order of magnitude less than the gyroelectric coefficient f [39].

The reduced magnetization vector m = M/|M| in YIG is decomposed into the static part m₀ and the dynamic part Δm(r, t) as:

We will consider transverse MO configuration, when (xz) is the incidence plane, and the static magnetization vector m₀ = {0, m_0y, 0} is along the positive direction of the y-axis, i.e. m_0y > 0. Thus, according to Eq. (4), the dynamic magnetization vector can be written as Δm(r, t) {Δm_x, 0, Δm_z}. In what follows, we assume {Δm_x, Δm_z} << m_0y.

In this case, the polarization of the incident and reflected light, which depends on the static magnetization, writes:

The amplitude of the electromagnetic wave modulated by the spin wave due the MO interaction can be found as a solution of wave equation with the polarization at the combined frequencies:

The polarization modulated by the magnetization oscillations has the following components (neglecting the terms containing (Δm_x)² and (Δm_z)² since they are much less than ${({m_{0y}})^2}$):

Taking the Fourier transform of Eq. (7) with respect to time and using the slowly varying envelope approximation [38], where ${{\mathbf K}_{\omega \pm \Omega }} \cdot \nabla {\mathbf E}({{\mathbf r},\omega \pm \Omega } )> > \Delta {\mathbf E}({{\mathbf r},\omega \pm \Omega } )$, we obtain the following reduced form of the wave equation for the BLS process:

The electric field components $E_i^{\textrm{sc}}({\omega \pm \Omega } )$ (i = x,y,z) of the scattered electromagnetic wave can be obtained from the corresponding polarization terms through the following equation:

Using Eq. (1) and boundary conditions, the amplitudes of the reflected s- and p-polarized fundamental electromagnetic wave E^{ref (s,p)} can be related to those of the incident wave E^{inc (s,p)} using the corresponding reflection coefficients:

Here ${k_x}$ is the component of the wave vector along the x-axis, $k_{1z,\omega }^{(s,p)}$ and k_2z are the z-components of wave vectors in YIG and GGG, respectively:

Using Eqs. (9)– (11), one can relate the amplitudes of the scattered electromagnetic wave to those of the incident.

The components of P-polarized scattered wave write:

In what follows, we assume the amplitudes of the spin wave to be equal: Δm_x = Δm_z = Δm. In this case, the amplitude of the scattered P-polarized electromagnetic wave $E_{}^{\textrm{sc}(P)} = \sqrt {{{({E_x^{(\textrm{sc})}} )}^2} + {{({E_z^{(\textrm{sc})}} )}^2}}$ is related to the amplitude of the s-polarized incident electromagnetic wave $E_y^{\textrm{inc}} = E_{}^{\textrm{inc}(s)}$:

The component of the S-polarized scattered electromagnetic wave $E_y^{\textrm{sc}} = E_{}^{\textrm{sc}(s)}$ has the following form:

The components of the reflected electromagnetic wave $E_x^{\textrm{ref}}$ and $E_z^{\textrm{ref}}$ can be related to the amplitude of the incident electromagnetic wave using Maxwell’s equations in the MO medium Eq. (1). Thus, the amplitude of S-polarized scattered electromagnetic wave is related to the amplitude of p-polarized incident fundamental wave as:

Using Eqs. (14) and (16), one can write the scattered functions ${\tilde{R}^{(S,P)}}({k_x})$ which relates the amplitudes of the scattered S and P-polarized electromagnetic wave to those of the p- and s-polarized incident electromagnetic wave as:

Assuming an incident Gaussian beam of waist w₀, the spatial profiles of the reflected ${E^{\textrm{ref}}}(x)$ and scattered ${E^{\textrm{sc}}}(x)$ beams can be calculated as:

3. Results of numerical calculations

For the numerical calculations, we take the following values of the parameters of YIG at the fundamental wavelength in vacuum λ₀ = 1.15 µm: ε₁ = 4.58, f = −2.47·10⁻⁴, g₁₁ = 2.89·10⁻⁴, g₁₂ = 10⁻⁴, 2g₄₄ = 2.31·10⁻⁴ [39]. The relative dielectric permittivity of GGG is ε₂ = 3.76 [39]. Thus, the refractive indices of YIG along the s and p directions $n_1^{(s)} = |k_\omega ^{(s)}|/{k_0}$ and $n_1^{(p)} = |k_\omega ^{(p)}|/{k_0}$ are larger than the refractive index of GGG ${n_2} = \sqrt {{\varepsilon _2}} ,$ and thus the total internal reflection of electromagnetic waves takes place at YIG/GGG interface at the incidence (critical) angles $\theta _{cr}^{(s)} = \arcsin ({n_2}/n_1^{(s)}) = 65.0225^\circ$ and $\theta _{cr}^{(p)} = \arcsin ({n_2}/n_1^{(p)}) = 65.0243^\circ$ for s- and p-polarization states, respectively. We limit our consideration to the total internal reflection geometry only, since it has been shown that at the light reflection at a single interface the GH shift is maximal at the incidence angles around the critical angle [1].

The static component of the magnetization vector in YIG, according to the approximation used here, is m_0y ≈ 1, which is provided by the external magnetic field applied along the y-axis with the amplitude equal to the saturation magnetization of YIG: H₀ = M₀ = 0.194·10⁶ A/m [39]. It should be noted that to saturate a thin film in-plane (if there is no anisotropy, as in the case considered here), even smaller external field is sufficient, so that in practice H₀ can be less than M₀. The dynamic magnetization components in YIG are Δm_z = Δm_x = 0.1. The angular frequency of the spin wave is then Ω = 2π·20 rad·GHz.

The waist of the incident electromagnetic beam is taken w₀ = 100λ₀. This choice is related to one of the characteristics of a Gaussian beam – the Rayleigh length ${z_R} = {k_0}w_0^2/2$. At the distances below z_R, the divergence of the beam can be neglected. With the chosen values of w₀ and λ₀, z_R is about 3.6 cm, which is a realistic dimension for experimental setups.

First, we analyze the beam envelopes of the reflected and scattered light as functions of the incidence angle θ, as shown in Fig. 2. The shape and the amplitude of the reflected beam envelope is different for s- and p-polarized light when the incidence angle is less than the critical angles (see Figs. 2(a) and 2(b)). The GH shift, defined as the position of the maximum of that envelope relatively to the position of the maximum of the incident beam (x = 0), reaches the maximum around the critical angle whatever the polarization state of light is. The amplitudes of the scattered light are four orders of magnitude less than those of the reflected one. The amplitude of the P-polarized scattered light is about five times larger than that of the S-polarized scattered light. It should be noted that the shapes of the beam envelopes of s-polarized (p-polarized) reflected and P-polarized (S-polarized) scattered beams are similar due to the MO properties of YIG (see Eqs. (13)– (15) and discussion between). Thus, the GH shift takes place for both reflected and scattered electromagnetic waves.

 

Fig. 2. The absolute value of the electric field (in V/m), or beam envelope, of the reflected light (top panel) and scattered light (bottom panel) of s-polarization (panels (a) and (c)) and p-polarization (panels (b) and (d)) as a function of the incidence angle θ and the x-coordinate. The white lines show the position of the maximum of the field relatively to the center of the incident beam (at x = 0).

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Figure 3 compares the GH shifts of the reflected and scattered light. The lateral shift of s-polarized reflected light Δx_s^(ref) ≈ 6 µm (or 5.2 wavelength λ₀ in vacuum) is smaller than the GH shift of the S-polarized scattered light Δx_S^(sc) ≈ 7.5 µm (6.5λ₀), as shown in Fig. 3(a). On the contrary, the GH shift of the P-polarized scattered electromagnetic wave Δx_P^(sc) ≈ 6 µm (5.2λ₀) is smaller than that of the p-polarized reflected beam Δx_p^(ref) ≈ 7.3 µm (6.35λ₀), see Fig. 3(b).

 

Fig. 3. The GH shift of s-polarized (a) and p-polarized (b) reflected light (blue lines) and scattered light (orange lines).

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In magnetic systems, the optical properties depend on the direction of the external magnetic field H₀. The reversal of H₀ (and thus the reversal of the static magnetization m₀, so that it is directed antiparallel to the y-axis in the transverse MO configuration, i.e. m_0y = −1) does not affect the reflection coefficient R^(s) of the s-polarized light because it depends on the square of the magnetization component, ${({m_{0y}})^2}$ (see Eqs. (11a), (11c), and (12a)). The scattered P-polarized light field demonstrated a similar dependence on m_0y. On the contrary, the static magnetization component is included in the reflected p-polarized and scattered S-polarized field in the form of the first power, i.e. m_0y (Eqs. (11b), (11d), and (16)). The change of the intensity of the reflected p-polarized light with the magnetization reversal manifests itself in the so-called transverse MO Kerr effect [34]. It has been shown that the lateral shift of p-polarized light reflected from a YIG/GGG bilayer changes by less than 1% with the magnetization switching in the transverse MO configuration at the incidence angles around the reflectivity minimum [12]. At the reflection from the YIG/GGG interface around $\theta _{cr}^{(p)}$, the decrease of Δx_p^(ref) is of the same order of magnitude. Similarly, upon m₀ reversal, the scattered S-polarized field changes, and so does the GH shift. Indeed, upon the switch from m₀ parallel to antiparallel to the y-axis, the maximum of Δx_S^(sc) decreases by 2%.

It should be noted that in the materials characterized by high values of gyromagnetic coefficients (such as, for instance, Bi-dopped YIG), the dependence of the tensor components of magnetic permeability on the magnetization cannot be neglected, which will lead to change of the s-polarized reflected light and P-polarized scattered light and the corresponding GH shifts with the magnetization reversal.

In addition to the magnetic field, the parameters of the spin wave also can be a controlling tool of the optical response of a magnetic system and thus can change the lateral shift of the scattered electromagnetic wave. The scattered field is proportional to the dynamical magnetization Δm (see Eqs. (14) and (16)), thus, the variation of the spin wave amplitude will only affect the amplitude of the scattered light and will not have an impact on the GH shift. On the contrary, the change of the frequency Ω of the spin wave will change both the amplitude and the shape of the scattered beam. Our calculations show that, for instance, at BLS by a spin wave of frequency Ω = 2π·40 rad·GHz the GH shifts Δx_S^(sc) and Δx_P^(sc) increase by 1.4% and 0.8%, respectively, in comparison to the scattering by a spin wave of Ω = 2π·20 rad·GHz discussed above. Such a different polarization-dependent tendency for the GH shift of the scattered light is related to the interplay between the linear and quadratic (by magnetization) elements of the MO tensors which affect the behavior of s- and p-polarized electromagnetic waves (see, for instance, Eq. (12)).

We can expect, that in multilayered systems, for instance, in photonic-magnonic crystals [15], the GH shift of the electromagnetic wave inelastically scattered by a spin wave can be increased up to several tens of the wavelength and can be efficiently controlled by the frequency of the spin wave.

4. Conclusions

In conclusion, we have shown theoretically and numerically that at the Brillouin light scattering by a spin wave it is possible to observe the Goos-Hänchen effect — a spatial lateral shift of the reflected electromagnetic beam at the frequencies shifted by the frequency of the spin wave. We have shown that this effect is sensitive to the polarization state and the incidence angle of the light beam, namely, the Goos-Hänchen shift of the scattered light is larger for the S-polarization state than for the P-polarization state, and, similarly to the reflected light at the fundamental frequency, both these shifts reach maxima (about ten wavelengths) around the critical angles of incidence. Moreover, the Goos-Hänchen shift of the S-polarized scattered electromagnetic wave can be varied by the reversal of the static magnetization in the magnetic medium. Additionally, the change of the spin wave frequency leads to a small enhancement of the lateral shift, which can be increased in multilayered photonic systems. We hope that investigated phenomenon of lateral shift at the inelastic light scattering by magnons will be useful for the study of dynamical properties of magnetic materials and complex magnetic superstructures.