1. Introduction

Magneto-medium (MO) effect mainly describes the phenomena and theories of the interaction between light and magnetic substances. Magnetic substance has an intrinsic magnetic moment, so under the action of a bias magnetic field, the magnetization intensity will be formed, leading to various MO effects that will modify the transport behavior of light wave. When a beam of light is incident on the interface of the substance, the propagation characteristics of light, including polarization, phase and scattering characteristics, will change to a certain extent due to the MO effect. And the MO effect includes Faraday spin effect [1], Kerr effect [2], Ketton-Morton effect, Zeeman effect [3], and magnetically excited light scattering. After the discovery of the magnetic domain structure in yttrium iron garnet (YIG) [4,5] single-crystal materials by Dillon at Bell Labs in 1956, the applications of MO effects began to be everywhere, such as MO deflectors, MO switches and modulators [6].

Ferrimagnetic material is a typical MO medium. There are many fascinating phenomena in + MO medium, such as negative refraction [7,8], nonreciprocal Goos-Hänchen (GH) shift [9], unidirectional electromagnetic windmill scattering [10] and so on. Recently, it has become an excellent platform to investigate the fundamental physics underlying topological photonic state (TPS) [11–23]. Many authors focus on changing lattice structure in order to observe various topological protect states, thus the general solving method for transmission of electromagnetic wave and the basic physical image in MO medium have been less discussed. For the transmission of electromagnetic wave in MO medium, many authors use the basic solution method of electromagnetic field. As written in textbook [6,24], start from the Maxwell's equations, the propagation formula of electromagnetic field is obtained by solving the continuous boundary conditions and material equations. Other works also make contributes to the solving of electromagnetic field in MO medium, Zak gives the calculation method based on general electromagnetic field theory, but introduces many physical parameters which make the expression of the formula too messy and not clear enough [25,26]. Abdulhalim presents the propagation matrix method for MO media with multilayer structure, the evolution of electromagnetic field in MO medium is well exhibited [27]. But they didn’t do a further discussion for the physical image and dig the interesting physical phenomena may existent.

In this paper, we consider a TE (transverse electric) plane wave incident on MO medium and the schematic diagram is shown in Fig. 1. We utilize a simple and rigorous electromagnetic solution approach [28] to obtain explicit formulations for all relevant physical quantities, such as the electromagnetic field distribution, energy flux, zero-reflection angle, reflection/transmission coefficients of intensity, and GH shift, by using this approach. We use YIG crystal for numeric calculation, which is magnetized by the applied bias field parallel with YIG crystal's surface. Analyzing all these equations and performing numerical calculations with an incident plane wave frequency of 14 GHz, we find that the electromagnetic field within the MO crystal evolves helically. The phase change for incident wave reflection/transmission is also discussed, which is different from isotropic medium. Besides, it is found that the MO effect can lead to the disappearance of Brewster angle for TE wave incident from air to MO medium. Finally, we discovered that the direction and amplitude of the GH shift may be adjusted by varying the direction and value of the external magnetic field.

 

Fig. 1. Schematic of electromagnetic field reflection/transmission excited by TE polarized plane wave at air-YIG interface. The direction of bias magnetic field is paralleled to the surface of YIG medium and perpendicular to the incident plane wave. The incident and reflection angles are both denoted as $\alpha $, and the refraction angle of wave wavefront and energy flux are denoted as $\varphi $ and $\beta $ respectively.

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2. Solution for electromagnetic wave through magneto-optical medium

We consider a plane electromagnetic wave incident from a homogeneous, isotropic medium (medium 1, e.g. air, with dielectric constant ${\varepsilon _1}$ and refractive index ${n_1}$) into MO medium (medium 2) with an incident angle $\alpha $. Reflection and refraction occur at the surface of the boundary. We present the rigorous analytical solution in MO medium [yttrium-ion-garnet (YIG)]. Notice that when a stable direct current magnetic field is placed in the direction parallel to YIG crystal's surface (along the z-axis direction), the permeability tensor of magnetized MO medium is of the following antisymmetric form as

Then we start from the following Maxwell's equations to solve the relationship of electric field:

We consider the incident electromagnetic wave being in either the TE or TM plane wave. For the case of TE wave incident on medium 2, the magnetic field is affected by permeability in x and y direction, while the electromagnetic field evolution of TM state is similar to the one in homogeneous medium. Thus, we focus on the TE state that exhibits much more complicated transmission behavior. The TE state has electric field along the z-axis direction, so that $({{E_z},{H_x},{H_y}} )\ne 0,$ while $({{H_z},{E_x},{E_y}} )= 0.$ The electric field for the incident, reflection, and transmission electromagnetic waves are expressed as

Here ω is the angle frequency of wave, ${\boldsymbol{k}_i} = (k_{ix}^{}, - k_{iy}^{})$, ${\boldsymbol{k}_r} = (k_{rx}^{},k_{ry}^{})$, and ${\boldsymbol{k}_t} = (k_{tx}^{}, - k_{ty}^{})$ are the wave vectors for the incident, reflection, and transmission wave. The electric field of the incident wave is ${E_i} = (c{\mu _0}/\sqrt {{\varepsilon _1}{\mu _1}} ){H_i}$, with c being the light speed in vacuum. Then Eq. (2) is expanded as

2.1 Wave vector relationship for transmission wave

According to the stringent electromagnetic field boundary condition, the wave vector has a continuum relation that ${k_{ix}} = {k_{rx}} = {k_{tx}} = {n_1}{k_0}\sin \alpha $, where ${k_0} = \omega /c$ is the wave number of microwave in vacuum. In air, we have $k_{ix}^2 + k_{iy}^2 = k_{rx}^2 + k_{ry}^2 = n_1^2k_0^2$. Now we apply the transmission wave [i.e. Equation (5)] into the Maxwell’s equations [Eq. (6)], and we can get the similar formula MO medium:

We define ${\mu _{_{eff}}} = \frac{{\mu _r^2 - \mu _k^2}}{{{\mu _r}}}$ for simplification, and thus

The form of Eq. (8b) is the same as the dispersion equation of microwave in isotropic media, where ${\mu _{eff}}$ is the effective permeability. So far, we have obtained the dispersion relation of wave vector for the microwave propagation in the MO medium, which describes the propagation characteristics of electromagnetic wave.

2.2 Reflection and transmission coefficient of field

The critical step is utilizing boundary conditions matching where the horizontal components of the electric and magnetic field are continuous at the interface, i.e. ${E_{iz}}{|_{y = 0}} + {E_{rz}}{|_{y = 0}} = {E_{tz}}{|_{y = 0}}$ and ${H_{ix}}{|_{y = 0}} + {H_{rx}}{|_{y = 0}} = {H_{tx}}{|_{y = 0}}$. Taking Eqs. (3)–(5) into Eq. (6), we can get the reflection and transmission coefficient of amplitude (See Supplement 1 for detailed process), as follow,

From Eq. (8b) we can further get ${k_{ty}} = \sqrt {k_0^2{\varepsilon _2}{\mu _{eff}} - k_0^2{\varepsilon _1}{\mu _1}{{\sin }^2}\alpha } $, besides, where ${n_1} = \sqrt {{\varepsilon _1}{\mu _1}}$. Thus, we can further simplify the reflection and transmission coefficient formula about the incident angle $\alpha $. The results are

We can see from Eq. (10) that there exists imaginary part in both the reflection and transmission coefficients, thus they are sensitively dependent on the permeability tensor of the MO medium. This feature is very different from the isotropic dielectric media, and also quite different from in other anisotropic media like uniaxial crystal [28]. The imaginary part in Eq. (10) might lead to the change of phase when the reflection and transmission of microwave occur on the surface of MO medium.

2.3 Energy flux and reflection/transmission coefficient of intensity

The next important step is to determine the energy flux direction, so that we can find out the physical details and construct a clear physical picture of wave transport and field evolution in the MO medium. In order to achieve this purpose, we calculate the time-averaged Poynting vector as $\boldsymbol{S} = \frac{1}{2}\textrm{Re} [{\boldsymbol{E}^\ast } \times \boldsymbol{H}]$ for evaluating the energy flux:

We go further to derive the incident-angle-dependent reflection and transmission coefficient of light intensity. They can be evaluated by the energy flux ratio $R = {{|{S_{ry}}|} / {|{S_{iy}}|}}$ and $T = {{|{S_{ty}}|} / {|{S_{iy}}|}}$. Finally, we obtain the reflection and transmission (See Supplement 1 for detailed process) as follows,

2.4 Refraction angle for wavefront and energy flux

The angle of refraction for the transmission wave is the central issue of the optics of birefringent crystals. Our previous work has shown that in birefringence crystal, the refraction angle of wavefront is different from the refraction angle of energy flux [28]. So, it is interesting to see when the medium is turned to MO, what happens to the relationship of refraction angle of wavefront and energy flux and whether it is the same as for an isotropic medium or for a birefringent crystal. For this purpose, we characterize the electromagnetic wave transport with two quantities, one is the wave vector k, denoting the wavefront motion and phase evolution, and the other is the energy flux S, denoting the electromagnetic energy flux motion. We will analyze in great details these two quantities in current problem. Obviously both the incident and reflection wave have identical wavefront angle and energy flux angle as $\alpha $, because they are transporting and evolving in a simple isotropic dielectric medium (such as air).

The wavefront angle $\varphi $ for the refraction wave is determined by the wave vector ${\boldsymbol{k}_t}$ and given explicitly as

The refraction angle of energy flux $\beta $ determines the true transport direction of transmission wave with respect to the normal of surface (See Fig. 1). This angle is related with the energy flux vector ${\boldsymbol{S}_t}$ and given by

From Eq. (13b) and Eq. (14b), it is clearly that the refraction angles of wave vector and energy flux are equal, that is to say now the MO medium behaves like an isotropic medium rather than a more complicated birefringent crystal. On the other hand, if we change the bias direction of external magnetic field to x- or y-axis, then the result will be $\sin \varphi \ne \sin \beta $ which means there is a phenomenon of birefringence for this new configuration of magnetization.

2.5 Reduction to isotropic medium

In the case of that medium 2 is a homogeneous isotropic medium with refractive index ${n_2}$, the solution can be simplified greatly. Then we find $\tan \varphi = \tan \beta = {n_1}\sin \alpha /{[{n_2^2 - n_1^2{{\sin }^2}\alpha } ]^{1/2}}$, or equivalently ${n_2}\sin \varphi = {n_2}\sin \beta = {n_1}\sin \alpha $, which is just the well-known Snell’s law. The reflection/transmission coefficient for field is reduced from Eq. (10) and yields

The reflection/transmission coefficient for intensity is reduced from Eq. (12) and yields

Equation (16) and Eq. (17) are the well-known Fresnel formula for field and intensity, which describes the reflection and transmission of light at the interface between two homogeneous and isotropic media. This confirms the self-consistency of the developed approach.

2.6 Vanishment of zero-reflection Brewster angle

As we all know, zero-reflection Brewster angle is a typical phenomenon for light in optics. Meanwhile, there should be the same phenomena for electromagnetic wave. First discovered about 200 years ago, now this phenomenon has found broad applications such as laser science, optical detection and other optical engineering [29,30]. For TE wave, the Brewster angle only exists in the material with magnetic response, i.e. $\mu \ne 1$[30,31]. Moreover, there are still certain restrictions on the value of the permeability. For an isotropic medium, its reflection coefficient can be obtained by simplifying Eq. (12) with ${\mu _k} = 0$ and ${\mu _{eff}} = {\mu _2}$. Additionally, when $R = 0$, Eq. (12) is simplified as ${\mu _2}\cos\alpha - {\mu _1}\sqrt {{\varepsilon _2}{\mu _2}/n_1^2 - {{\sin }^2}\alpha } = 0$. Then the zero-reflection Brewster angle is finally expressed as

Or equivalently

We analyze formula (17a) and (17b) and obtain the following conditions for the existence of Brewster angle:

Thus, we can obtain ${\mu _2} \ge {\mu _1}{\varepsilon _2}/{\varepsilon _1}$ or $0 \le {\mu _2} \le {\mu _1}{\varepsilon _1}/{\varepsilon _2}$ (See Supplement 1 for detailed process). When the above conditions are met, Brewster angle exists.

Interestingly, MO medium manifests magnetic response when an out-of-plane magnetic field is applied, however, it does not support the existence of Brewster angle at air-YIG interface. As shown in Eq. (12), the reflection coefficient is the sum of two square terms. The additional term $\mu _1^2\mu _k^2{\sin ^2}\alpha /\mu _r^2$ is generated by MO effect and proportional to the square of gyromagnetic ratio ${({\mu _k}/{\mu _r})^2}$. Consequently, zero reflection only occurs for a normal incident with rigorous impedance matching $\sqrt {{\mu _{eff}}/{\varepsilon _2}} = \sqrt {{\mu _1}/{\varepsilon _1}} $. Now, we proceed to reveal how MO effect hampers the existence of Brewster angle. Generally, as an incident electromagnetic wave is linear polarization, its magnetic field H_x has zero phase difference against its electric field E_z. On the contrary, for a magnetized YIG, the imaginary part of the ratio of H_x to E_z is related to ${\mu _k}$, as shown in Eq. (7). This means that there exists an additional phase delaying between H_x and E_z, owing to the existence of MO effect. Hence, the transmission wave in YIG is not a linear polarization (We will show this specific polarization behaviors in the following section 3.1), and the mismatching of polarization between the incident and transmission wave leads to the inevitable reflection on the interface. More specifically, the MO effect produces the phase delay between electric field and magnetic field at the air-interface, which causes the unavoidable reflection, i.e. the disappearance of Brewster angle.

3. Numerical calculation of YIG

In the above section, we have obtained the explicit formulations for a series of relevant physical quantities. The reflection and transmission waves of TE state within the MO medium are also well described and represented by these formulations. Indeed, we have found several interesting phenomena such as spiraling magnetic field and zero-reflection Brewster angle. In this section, we explore the physical quantities when a plane wave of $\omega = 14 \times 10^{9}$ rad/m is incident on a prominent MO medium as YIG, and YIG is under a certain bias magnetic field. Notice that the bias magnetic field discussed in this paper always refers to the static magnetic field inside the material.

3.1 Spiraling magnetic field

Firstly, we notice that Eq. (7) shows the magnetic field involving imaginary part, that is to say, ${H_x}$ and ${H_y}$ should illustrate a phase difference. According to the expressions of microwave magnetic field, we perform numerical calculation and visualize the evolution of microwave magnetic field. The calculation results indicate that the synthetic field of ${H_x}$ and ${H_y}$ presents an interesting and prominent spiral moving track as shown in Fig. 2. In the XOY plane, we can see the spiral evolution of the magnetic field in YIG, which is very different from the typical sine form in the air. Such spiral magnetic fields can carry the transversal spin angular momentum, which can widely appear in various physical scenes, such as tightly focused optical beams, surface plasmon polaritons or the other evanescent waves [32–34]. While YIG, as a natural material, intrinsically supports bulk mode with the transversal spin angular momentum without complex artificial structure.

 

Fig. 2. (a) Magnetic field of the electromagnetic wave transmitted from air to the YIG medium; (b) electric and magnetic field in the YIG medium.

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3.2 Reflection/refraction coefficients of intensity and vanishment of Brewster angle

To demonstrate the influence of MO effect on reflection and refraction, we compare the reflection and transmission coefficients of YIG with an isotropic medium of identical permittivity and effective permeability, so that the effective permeability of YIG can be calculated in advance. Due to the gyromagnetic feature of YIG, there are different permeability constants under different bias magnetic fields. Taking ${\mu _r} = 1 + {\omega _0}{\omega _m}/(\omega _0^2 - {\omega ^2})$ and ${\mu _k} = \omega {\omega _m}(\omega _0^2 - {\omega ^2})$ into the effective permeability, we have:

The variation of ${\mu _{eff}}$ with the bias magnetic field is calculated and shown in Fig. 3.

 

Fig. 3. Variation curve of equivalent permeability with bias magnetic field H₀.

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As revealed in Eq. (19), there is a magnetic field intensity (H₀= 304 Gauss) which satisfies ${\omega _0} = (\sqrt {4{\omega ^2} + {\omega _m}^2} - {\omega _m})/2$, resulting in that the denominator in Eq. (19) is equal to zero which leads to an infinity effective permeability. In Fig. 3, when H₀ varies from 100 Gauss to 304 Gauss, the effective permeability changes from –7 ∼ –∞. After the bias magnetic field exceeding the threshold value of 304 Gauss, the effective permeability changes from + ∞ ∼ 1. Therefore, YIG has remarkable ability in the modulation of permeability. Now, we apply ${\mu _2} = {\mu _{eff}}$ for the compared isotropic medium. Note that the permittivities of YIG and isotropic medium are 15.26. The Brewster angle will exist in the condition of isotropic medium with ${\mu _2} > 15.26$. Nevertheless, the Brewster angle will vanish in the presence of MO effect.

Reflection/transmission coefficient of intensity are important physical quantities, as they describe the response of a material to the incident electromagnetic wave. They can help us understand the properties of this material more deeply. We can calculate the reflection/transmission coefficient of intensity when the incident angle varies at the range of 0° ∼ 90° through Eq. (12). In Fig. 4, we present reflection/transmission coefficient of intensity at air-YIG interface under H₀ = 350 Gauss (${\mu _{eff}} = {\mu _2} = 34.8 > 15.26$) and H₀ = 500 Gauss (${\mu _{eff}} = {\mu _2} = 9.0 > 15.26$).

 

Fig. 4. Reflection and transmission coefficient of intensity at air-YIG interface for (a) bias H₀ = 350 Gauss and (b) bias H₀ = 500 Gauss. The results of air-isotropic medium with identical permittivity and effective permeability are also illustrated as dash lines.

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In Fig. 4(a), there is a zero-reflection Brewster angle at 48.6° for the isotropic medium (black dash line). However, for the YIG with identical permeability, the Brewster angle disappears. This phenomenon corresponds to the result of Eq. (12). As we discussed in section 2.6, the MO effect causes the spiral polarization in YIG, which further leads to the inevitable mismatching between the incident and transmission wave. This gives rise to the non-zero reflection and results in the vanish of Brewster angle. Additionally, although there is an interesting spiral propagation for the evolution of microwave magnetic field, the changes of reflection and transmission still follow the law of conservation of energy, namely, $R + T = 1$. Under different bias, the reflection becomes 1 at $\alpha = {90^ \circ }$ simultaneously. However, the reflections of H₀ = 350 Gauss (0.09) and H₀ = 500 Gauss (0.21) are quite different near $\alpha = {60^ \circ }$, which manifests MO effect has potential on modulating electromagnetic wave.

3.3 Phase change for reflection and transmission

From Eq. (10) we know that there is an imaginary part in the reflection/transmission coefficients, that is to say when microwave is incident on the surface of YIG medium, there will be a phase change for both the reflection and transmission electromagnetic field. For more details, we calculate the phase change of reflection/refraction microwave according to Eq. (10) (See Supplement 1 for detailed derivation):

We display the phase change of reflection and refraction in Figs. 5(a) and 5(b), respectively. Meanwhile, we also give the phase change of different polarizations when a plane wave incident from air to glass.

 

Fig. 5. Phase change of (a) reflection and (b) refraction at H₀ = 350 Gauss (black solid line) and H₀ = 500 Gauss (red solid line) with incident angle increasing from 0° to 90°. The phase changes of TE (blue dash line) and TM (magenta dash line) electromagnetic wave at air-glass interface are also demonstrated.

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In Fig. 5(a), the reflection phase of TE wave maintains at π when it transports from air to glass at an arbitrary angle. While the reflection phase of TM wave transits from 0 to π when its incident angles are larger than Brewster angle (56.3°). Thus, for isotropic medium, the reflection phase has only two values, 0 or π. However, the reflection phase at the interface between air and magnetized YIG can be continuously varied from 0 to π. For example, for the YIG applied with an external magnetic field of H₀= 350 Gauss, when the incident angle increases from 0 to 90°, the reflection phase moves from 0 to π smoothly. The physical mechanism can be revealed by the analytic formula of reflection phase in Eq. (20), i.e. the MO effect allows the reflection phase to change gradually with the increase of incident angle, while the reflection coefficient at air-glass interface must be a positive or negative real number, which corresponds to the reflection phase of 0 or π. More remarkably, for H₀= 500 Gauss, with the increase of incident angle, the reflection phase first drop from π, and then continue to increase to π, which is completely difference from the case of H₀= 350 Gauss (evolved from 0 to π). This is because that the effective permeability between magneto-optical media applied with different magnetic fields, resulting in that the sign of their reflection coefficient is reversed. Moreover, Fig. 5(b) shows that the transmission phase of air-YIG interface grows gradually with the increase of incident angle, while the transmission phase of air-glass keeps zero. Additionally, it also can be seen that the transmission phase under H₀= 350 Gauss is larger than that under H₀= 500 Gauss. This is because that the transmission phase is positively correlated to the gyromagnetic ratio ${\mu _k}/{\mu _r}$ [See Eq. (21)], i.e. the gyromagnetic ratio under H₀= 350 Gauss is larger than that under H₀= 500 Gauss.

3.4 Goos-Hänchen shift

When a light beam is totally reflected at the interface, within an ideal model, the incident point upon the surface is at the same position with the reflection point. But in practice, the incident light will have a small transverse shift at the surface of medium, which is a phenomenon called GH shift and discovered at the experiment in 1947 [35]. Since it was discovered, researchers have given reasonable theoretical explanations and performed numerical calculation of GH shift. There are two main theoretical methods: stationary-phase method [36] and energy-flux method [37]. In this work, we use stationary-phase method to calculate this interesting physical quantity. The formula is given as follows:

 

Fig. 6. (a) GH shift under H₀= 320 Gauss, H₀= 330 Gauss and H₀= 340 Gauss when the bias magnetic field is in the positive direction of z-axis; (b) GH shift under H₀= 320 Gauss, H₀= 330 Gauss and H₀= 340 Gauss when the bias magnetic field is in the negative direction of z-axis.

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In general, the GH shift is positive. Interestingly, in MO system, we can decide the negative or positive GH shift by converting the direction of the applied bias magnetic field. One can see that the GH shifts are opposite when the YIG medium are applied with opposite external magnetic fields, as shown in Figs. 6(a) and 6(b). This is because that the flip of external magnetic field causes the sign change of ${\mu _k}$, as indicated in Eq. (20) and Eq. (22). Therefore, the direction of GH shifts can be arbitrarily tuned by controlling the external magnetic field direction. In addition, the magnitude of the maximum displacement is different under different bias magnetic fields. Additionally, the magnitude GH shift is nonzero at normal incident, which is also observed in the previous study [38,39]. The results in Fig. 6 simply shows that the maximum value of GH shift slightly decreases in the process of increasing the bias magnetic field from 320 Gauss to 340 Gauss. Therefore, both the value and direction of GH shift can be modulated by the external magnetic bias. According to the calculation, the maximum GH shift under this condition is about 8 wavelengths.

4. Conclusion

We have developed a simple but rigorous electromagnetic solution approach for application of electromagnetic wave transport in MO medium system, which allows us to easily derive and obtain the explicit formulae of all relevant physical quantities, such as electromagnetic field distribution, energy flow, reflection and transmission coefficient, reflection and transmission phase. We have used microwave transport in a prominent MO material YIG as an actual calculation example, which can be magnetized in multiple directions by external magnetic field.

We have first discovered an interesting spiral evolution of electromagnetic field when microwave propagates within the YIG material, and this unique transport behavior is perfectly represented by the explicit formulation of electromagnetic field distribution. Then, we found the transmission wave in YIG manifest spiral polarization, which is generated by the MO effect. Moreover, the mismatching between spiral polarized transmission wave and linear polarized incident wave leads to the unavoidable reflection and further causes the disappearance of Brewster angle. We have disclosed the existence of phase change for both the reflection and transmission wave occurring right at the surface of YIG medium, and found that such a phase change behavior is very different from that in an ordinary isotropic medium. Finally, we have investigated the GH shift for a beam impinging upon the surface of YIG medium under different external magnetic fields. We have found that the direction and magnitude of GH shift can be adjusted by changing the direction and value of applied bias magnetic field.

All these results have clearly shown that there still exist many interesting and novel physical and optical phenomena that can occur when electromagnetic wave transports in MO medium and pass across the interface between ordinary dielectric medium and MO medium. The developed simple electromagnetic theory can offer a very powerful tool to understand, predict, and explain all these novel phenomena based on a series of explicit analytical formulations. Surely this theory can help to deepen and broaden our physical understanding of basic electromagnetics, optics, and electrodynamics in application to gyromagnetic and MO homogeneous medium and microstructures, and might help to disclose and develop new ways and routes to high technologies in optics and microwave.