Optical Propagation in Magneto-Optical Materials

Abstract

Magneto-optical materials present anisotropy in the electrical permittivity controlled by a magnetic field, which affects the propagation characteristics of light and stands out in the design of nonreciprocal devices, such as optical isolators and circulators. Based on Maxwell’s equations, this chapter focuses on the wave propagation in magneto-optical media. The following cases are covered: The propagation of a plane wave in an unbounded magneto-optical medium, where the phenomenon of Faraday rotation is discussed, and the guided propagation in planar magneto-optical waveguides with three and five layers, highlighting the phenomenon of nonreciprocal phase shift and its potential use on the design of nonreciprocal optical devices.

1. Introduction

A material is classified as magneto-optical (MO) if it affects the propagation characteristics of light when an external magnetic field is applied on it. For ferromagnetic materials, which are composed by magnetically ordered domains, MO phenomena may also occur in the absence of an external magnetic field. A great number of magneto-optical phenomena are the direct or indirect outcome of the splitting of energy levels in an external or spontaneous magnetic field [1].

The MO effect depends on the polarization of the magnetic field. It also depends on the polarization of the light and on its propagation direction, so it is an anisotropic phenomenon, which has attracted great attention from researchers in optical devices. The MO materials can have their anisotropy controlled by a magnetostatic field (H_DC), and this behavior can be exploited on the design of nonreciprocal devices. By nonreciprocal devices or structures, it means that waves or guided modes supported by them have their propagation characteristics altered when the wave propagation sense is reversed. Optical isolators and circulators can be highlighted as examples of such devices. Isolators are designed to protect optical sources from reflected light and are present in optical amplification systems. The circulators are employed as signal routers and act in devices that extract wavelengths in WDM systems.

The design of optical devices with MO materials is addressed in several works such as [2, 3, 4, 5]. The challenges for the design of such devices are the development of MO materials with high-induced anisotropy and high transparency at the optical spectrum. Therefore, research activities on the improvement of MO materials and structures have also great relevance and are covered in works such as [6, 7, 8, 9, 10]. Integration of MO materials and structures with other optical system components, with reduction of insertion losses, is also a target for researches in optical devices. Research of MO effects in optical structures such as photonic crystals has also been addressed [11, 12, 13].

This chapter presents analytical formalisms derived from Maxwell’s and wave equations to analyze the propagation characteristics of transverse electromagnetic (TEM) waves in unbounded magneto-optical material. The guided propagation characteristics of transverse magnetic (TM) modes in three- and five-layered planar magneto-optical waveguides are also formalized and discussed. The analytical formalism is versatile so that each layer can be set as magneto-optical or isotropic in the mathematical model.

2. Wave propagation characteristics

This section focuses on the optical propagation analysis in magneto-optical media using Maxwell’s equations as starting point. In a magnetized MO media, cyclotron resonances occur at optical frequencies, if the wave is properly polarized. This physical phenomenon induces a coupling between orthogonal electric field components in the plane perpendicular to the applied magnetostatic field H_DC, which affects the wave polarization. Depending on the orientation of the magnetostatic field, the configuration of the electric permittivity tensor changes. If H_DC is oriented along one of the Cartesian axes, the relative electric permittivity assumes the form

$${\overline{\overline{\epsilon }}}_r=\left[\begin{array}{lll}{n}^2& 0& 0\\ 0& n^2& \mathit{j\delta }\\ 0& -\mathit{j\delta }& n^2\end{array}\right],\text{for}{H}_{\mathit{DC}}\Vert x-\text{axis};$$ E1

$${\overline{\overline{\epsilon }}}_r=\left[\begin{array}{lll}{n}^2& 0& \mathit{j\delta }\\ 0& n^2& 0\\ -\mathit{j\delta }& 0& n^2\end{array}\right],\text{for}{H}_{\mathit{DC}}\Vert y-\text{axis};$$ E2

$${\overline{\overline{\epsilon }}}_r=\left[\begin{array}{lll}{n}^2& \mathit{j\delta }& 0\\ -\mathit{j\delta }& n^2& 0\\ 0& 0& n^2\end{array}\right],\text{for}{H}_{\mathit{DC}}\Vert z-\text{axis}.$$ E3

where n is the refractive index of the material and δ is the magneto-optical constant. The MO constant is proportional to H_DC. If the sense of H_DC is reversed, δ(-H_DC) = −δ(H_DC), and for H_DC = 0, the off-diagonal components of the electric permittivity tensor are zero [14, 15].

2.1 TEM wave in an unbounded magneto-optical medium

Let us consider a TM wave propagating in an unbounded MO medium, as shown in Figure 1.

Figure 1.

TEM wave in an unbounded magneto-optical medium.

From Maxwell’s equations, the vectorial Helmholtz equation for anisotropic media and for the electric field $\overline{E}\left(x,y,z\right)$ can be written as

$${\omega }^2{\mu }_0{\epsilon }_0{\overline{\overline{\epsilon }}}_r\overline{E}+{\nabla }^2\overline{E}-\nabla \left(\nabla \cdot \overline{E}\right)=\overline{0},$$ E4

where ω is the angular frequency in rad/s, μ₀ is the magnetic permeability of the vacuum in H/m, and ε₀ is the electric permittivity of the vacuum in F/m.

To develop a plane wave solution for MO media, it is assumed that H_DC is parallel to the y-axis and ${\overline{\overline{\epsilon }}}_r$ is given by Eq. (2) from now on. This assumption does not imply on lack of generality because it is assumed that the wave propagates at an arbitrary direction, with the electric field vector given by

$$\overline{E}={\overline{E}}_0\text{exp}\left(\mathit{j\omega t}\right)\text{exp}\left[-j\left({\gamma }_{x}x+{\gamma }_{y}y+{\gamma }_{z}z\right)\right].$$ E5

where $\overline{\gamma }={\gamma }_x\overrightarrow{\mathbf{i}}+{\gamma }_y\overrightarrow{\mathbf{j}}+{\gamma }_z\overrightarrow{\mathbf{k}}$ is the propagation constant vector.

From Gauss’ law for a medium with equilibrium of charges, $\nabla .\left({\epsilon }_0{\overline{\overline{\epsilon }}}_r\overline{E}\right)=0$ , we obtain:

$$\nabla \cdot \overline{E}=j\frac{\delta }{n^2}\left(\frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x}\right).$$ E6

Substituting Eq. (6) into Eq. (4) leads to

$${\omega }^2{\mu }_0{\epsilon }_0{\overline{\overline{\epsilon }}}_r\overline{E}+{\nabla }^2\overline{E}-j\frac{\delta }{n^2}\nabla \left(\frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x}\right)=\overline{0}.$$ E7

Expanding Eq. (7) in the Cartesian coordinates results in

$${\omega }^2{\mu }_0{\epsilon }_0\left(n^2{E}_x+\mathit{j\delta }{E}_z\right)+\frac{{\partial }^2{E}_x}{\partial x^2}+\frac{{\partial }^2{E}_x}{\partial y^2}+\frac{{\partial }^2{E}_x}{\partial z^2}-j\frac{\delta }{n^2}\left(\frac{{\partial }^2{E}_x}{\partial x\partial z}-\frac{{\partial }^2{E}_z}{\partial x^2}\right)=0,$$ E8

$${\omega }^2{\mu }_0{\epsilon }_0{n}^2{E}_y+\frac{{\partial }^2{E}_y}{\partial x^2}+\frac{{\partial }^2{E}_y}{\partial y^2}+\frac{{\partial }^2{E}_y}{\partial z^2}-j\frac{\delta }{n^2}\left(\frac{{\partial }^2{E}_x}{\partial y\partial z}-\frac{{\partial }^2{E}_z}{\partial x\partial y}\right)=0,$$ E9

$${\omega }^2{\mu }_0{\epsilon }_0\left(-\mathit{j\delta }{E}_x+n^2{E}_z\right)+\frac{{\partial }^2{E}_z}{\partial x^2}+\frac{{\partial }^2{E}_z}{\partial y^2}+\frac{{\partial }^2{E}_z}{\partial z^2}-j\frac{\delta }{n^2}\left(\frac{{\partial }^2{E}_x}{\partial z^2}-\frac{{\partial }^2{E}_z}{\partial x\partial z}\right)=0.$$ E10

The spatial derivatives in Eqs. (8)–(10) are now calculated by considering Eq. (5):

$$\left({\omega }^2{\mu }_0{\epsilon }_0{n}^2-{\left|\gamma \right|}^2-j\frac{\delta }{n^2}{\gamma }_x{\gamma }_z\right)E_x+\mathit{j\delta }\left({\omega }^2{\mu }_0{\epsilon }_0-\frac{1}{n^2}{\gamma }_x^2\right)E_z=0$$ E11

$$\left({\omega }^2{\mu }_0{\epsilon }_0{n}^2-{\left|\gamma \right|}^2\right)E_y+j\frac{\delta }{n^2}\left({\gamma }_y{\gamma }_z{E}_x-{\gamma }_x{\gamma }_y{E}_z\right)=0,$$ E12

$$-\mathit{j\delta }\left({\omega }^2{\mu }_0{\epsilon }_0-\frac{1}{n^2}{\gamma }_z^2\right)E_x+\left({\omega }^2{\mu }_0{\epsilon }_0{n}^2-{\left|\gamma \right|}^2-j\frac{\delta }{n^2}{\gamma }_x{\gamma }_z\right)E_z=0,$$ E13

where $\left|\gamma \right|=\sqrt{{\gamma }_x^2+{\gamma }_y^2+{\gamma }_z^2}$ .

2.1.1 TEM wave with electric field vector parallel to H_DC

By observing Eqs. (11–13), we note that when the electric field of the electromagnetic wave is polarized along the y-axis and is parallel to H_DC, so that E_x = E_z = 0, the magneto-optical constant δ related to H_DC will have no effect on the propagation characteristics of the wave. In this case, from Eq. (12), the propagation constant modulus would be

$$\left|\gamma \right|=\mathit{n\omega }\sqrt{{\mu }_0{\epsilon }_0},$$ E14

which is the same expression for a traveling wave in an isotropic material. Note that when the electric field is polarized along the y-axis, the wave is traveling in the plane xz, so that γ_y = 0.

2.1.2 The general expression for the propagation constant

In a general case, by solving the system formed by Eqs. (11) and (13), we obtain the following equation

$${\left({\omega }^2{\mu }_0{\epsilon }_0{n}^2-{\left|\gamma \right|}^2-j\frac{\delta }{n^2}{\gamma }_x{\gamma }_z\right)}^2-{\delta }^2\left({\omega }^2{\mu }_0{\epsilon }_0-\frac{1}{n^2}{\gamma }_x^2\right)\left({\omega }^2{\mu }_0{\epsilon }_0-\frac{1}{n^2}{\gamma }_z^2\right)=0.$$ E15

Solving Eq. (15) for |γ|, we obtain:

$$\left|\gamma \right|=\sqrt{{\omega }^2{\mu }_0{\epsilon }_0{n}^2-j\frac{\delta }{n^2}{\gamma }_x{\gamma }_z\pm \delta \sqrt{\left({\omega }^2{\mu }_0{\epsilon }_0-\frac{1}{n^2}{\gamma }_x^2\right)\left({\omega }^2{\mu }_0{\epsilon }_0-\frac{1}{n^2}{\gamma }_z^2\right)}}.$$ E16

Note that when the MO constant δ = 0, Eq. (16) reduces to Eq. (14).

The parameters γ_x and γ_z are projections of the propagation constant vector along the x and the y-axis, respectively.

2.1.3 TEM wave propagating parallel to H_DC

If the TEM wave is propagating along the H_DC direction (y-axis), so that γ_x = γ_z = 0, Eq. (16) assumes the simpler form:

$$\left|\gamma \right|=\omega \sqrt{{\mu }_0{\epsilon }_0\left(n^2\pm \delta \right)},$$ E17

and from Eq. (5), the electric field vector becomes

$$\overline{E}=\left(E_{0x}\overrightarrow{\mathbf{i}}+E_{0z}\overrightarrow{\mathbf{k}}\right)\text{exp}\left[j\left(\mathit{\omega t}-\mathit{y\omega }\sqrt{{\mu }_0{\epsilon }_0\left(n^2\pm \delta \right)}\right)\right].$$ E18

From Eq. (11), we see that the electric field components are connected by

$$E_{0x}=-\mathit{j\delta }\frac{{\omega }^2{\mu }_0{\epsilon }_0}{\left({\omega }^2{\mu }_0{\epsilon }_0{n}^2-{\left|\gamma \right|}^2\right)}{E}_{0z}.$$ E19

Substituting Eq. (17) in Eq. (19), we obtain:

$$E_{0x}=\pm {\mathit{jE}}_{0z}.$$ E20

Therefore, substituting Eq. (20) in Eq. (18), and given that ±j = exp(±jπ/2), the electric field components can be written as

$$E_x=E_{0z}\text{exp}\left[j\left(\mathit{\omega t}-\mathit{y\omega }\sqrt{{\mu }_0{\epsilon }_0\left(n^2\pm \delta \right)}\pm \pi /2\right)\right],$$ E21

$$E_z=E_{0z}\text{exp}\left[j\left(\mathit{\omega t}-\mathit{y\omega }\sqrt{{\mu }_0{\epsilon }_0\left(n^2\pm \delta \right)}\right)\right].$$ E22

Eqs. (21) and (22) represent a circular polarized wave, which can be dismembered into two circular polarized eigenmodes propagating along the y-axis with different propagation constants. If the plus sign (in “±”) is adopted for Eqs. (21) and (22), we obtain a counterclockwise (CCW) circular polarized eigenmode. Otherwise, if the minus sign is adopted, we obtain a clockwise (CW) circular polarized eigenmode, as shown in Figure 2. From Eq. (17), it is possible to associate an equivalent refractive index to each eigenmode:

Figure 2.

Decomposition of a linear polarized TEM wave into two circular polarized components. The circular polarized components travel with distinct propagation constants in a MO medium.

1. $n^{+}=\sqrt{n^2+\delta },$ for the CCW circular polarized eigenmode;

2. $n^{-}=\sqrt{n^2-\delta },$ for the CW circular polarized eigenmode.

A linear polarized wave propagating along the y-axis may be decomposed into two opposite circular polarized waves in the xz plane, as shown in Figure 2. Since these eigenmodes propagate with distinct propagation constants, the linear polarization will rotate in the xz plane as the wave propagates along the y-axis, in a phenomenon known as Faraday rotation, which is depicted in Figure 3.

Figure 3.

Faraday rotation of a linear polarized TEM wave in a MO medium. The propagation direction is parallel to the magnetostatic field H_DC.

When the sense of the magnetostatic field H_DC is reversed, the magneto-optical constant δ changes its signal, and the values of n⁺ and n⁻ are interchanged, and the sense of rotation of a linear polarized wave in the MO media will change.

The Faraday rotation angle (ϕ_F) may be calculated (in radians) as a function of the propagation distance y by

$${\varphi }_F=\frac{1}{2}\left({\varphi }^{+}-{\varphi }^{-}\right)=\frac{1}{2}\left(n^{+}\frac{2\pi }{{\lambda }_0}y-n^{-}\frac{2\pi }{{\lambda }_0}y\right)=\frac{\pi }{{\lambda }_0}\left(\sqrt{n^2+\delta }-\sqrt{n^2-\delta }\right)y,$$ E23

where λ₀ is the optical wavelength in vacuum. The Faraday rotation effect is responsible for a periodic power transfer between the transverse components, in this case, E_x and E_z. This phenomenon in MO materials may be exploited for the design of optical isolators based on Faraday rotation.

When a MO waveguide, with H_DC applied along its longitudinal direction, supports degenerate orthogonal quasi TEM modes, the power transfer between these modes will be maximized. Figure 4 shows a MO rib waveguide [16], where layers 1 and 2 are composed of bismuth yttrium iron garnet (Bi-YIG) grown on top of a gadolinium gallium garnet (GGG) substrate with n_SR = 1.94. For the Bi-YIG layers, the relative permittivity tensor has the form of Eq. (2), with δ = 2.4 × 10⁻⁴, n₁ = 2.19, and n₂ = 2.18. The waveguide dimensions are w = 8 μm, h = 0.5 μm, t₁ = 3.1 μm, and t₂ = 3.4 μm. The optical wavelength is λ₀ = 1.485 μm.

Figure 4.

Magneto-optical rib waveguide.

Figure 5 shows numerical results for the power transfer between the transverse components along the propagation direction. These results were obtained using a finite difference vectorial beam propagation method (FD-VBPM) [17]. We observe that the length for maximum energy transfer is around 6800 μm. In practice, as observed in [16], the device length must be set at half that length (∼3400 μm) so that a 45° rotation is achieved at the output port. Therefore, if a reflection occurs at this point, the reflected field will complete a 90° rotation at the input port, which can then be blocked with a polarizer without affecting the input field, so that an optical isolator is obtained.

Figure 5.

Normalized intensity evolution of the transverse field components along the propagation direction (y-axis) of the MO waveguide.

In Eq. (23), by adopting δ = 2.4 × 10⁻⁴, n = n₁ = 2.19, λ₀ = 1.485 μm, and ϕ_F = π/4 (45°), we obtain y = 3388 μm, which is a propagation length that converges with the FD-VBPM result.

2.1.4 TEM wave propagating along the diagonal of an imaginary cube

Before finishing this section, let us consider another particular case of propagation direction—suppose, in Figure 1, that γ_x = γ_y = γ_z = γ_u, with γ_u ≠ 0. This case corresponds to a TEM wave propagating along the diagonal of an imaginary cube, adjacent to the Cartesian axes. From Eq. (16), we obtain:

$$\left|\gamma \right|=\sqrt{{\omega }^2{\mu }_0{\epsilon }_0{n}^2-j\frac{\delta }{n^2}{\gamma }_u^2\pm \delta \left({\omega }^2{\mu }_0{\epsilon }_0-\frac{1}{n^2}{\gamma }_u^2\right)}.$$ E24

From the relation $\left|\gamma \right|=\sqrt{{\gamma }_x^2+{\gamma }_y^2+{\gamma }_z^2}$ we can also obtain:

$$\left|\gamma \right|={\gamma }_u\sqrt{3}.$$ E25

Equaling Eqs. (24)–(25) and solving for γ_u result in

$${\gamma }_u=\omega \sqrt{\frac{{\mu }_0{\epsilon }_0\left(n^2\pm \delta \right)}{3\pm \frac{\delta }{n^2}+j\frac{\delta }{n^2}}}.$$ E26

Substituting Eq. (26) in Eq. (25), we obtain the propagation constant:

$$\left|\gamma \right|=\omega \sqrt{\frac{{\mu }_0{\epsilon }_0\left(n^2\pm \delta \right)}{1\pm \frac{\delta }{3n^2}+j\frac{\delta }{3n^2}}}.$$ E27

The corresponding electric field vector can be retrieved by substituting the results of Eqs. (26)–(27) in Eq. (11) to obtain

$$E_x=\pm {\mathit{jE}}_z.$$ E28

However, for the considered propagation direction, the E_y component is not zero. From Eq. (12) we obtain:

$$E_y=-\frac{n^2\pm \delta +j3\left(\delta \pm n^2\right)}{5n^2}{E}_z.$$ E29

By using the results of Eqs. (26)–(29) in Eq. (5), we can express the electric field vector for this particular case by

$$\overline{E}=\left(\pm j\overrightarrow{\mathbf{i}}-\frac{n^2\pm \delta +j3\left(\delta \pm n^2\right)}{5n^2}\overrightarrow{\mathbf{j}}+\overrightarrow{\mathbf{k}}\right)E_{0z}\text{exp}\left[j\left(\mathit{\omega t}-\omega \sqrt{\frac{{\mu }_0{\epsilon }_0\left(n^2\pm \delta \right)}{3\pm \frac{\delta }{n^2}+j\frac{\delta }{n^2}}}\left(x+y+z\right)\right)\right],$$ E30

where i, j, and k are the unit vectors along the x-, y-, and z-axis, respectively.

As in the previous case of propagation, Eq. (30) provides two eigenmodes for TEM propagation. From Eq. (28) we can observe that, when projected in the xz plane, the electric field vector of each eigenmode is circular polarized. The combination of these eigenmodes will result in a wave with linear polarization progressively rotated as it propagates. The E_y component has the role of projecting the Faraday rotation to the plane perpendicular to the propagation direction (the diagonal of the cube), since the wave is TEM regarding this propagation direction. Figure 6 shows a simulation of the TEM wave eigenmodes along the diagonal of an imaginary cube.

Figure 6.

TEM eigenmodes for diagonal propagation where γ_x = γ_y = γ_z. The trajectory of the electric field vector is represented by red lines.

The simulations presented in Figure 6 were performed for f = 193.4145 THz, n = 2, and δ = 0.2. Note that both eigenmodes present losses as they propagate. This is due the complex characteristic of the propagation constant expressed by Eq. (27), where the imaginary part depends on the magneto-optical constant δ. It was observed that increasing δ enhances the Faraday rotation but also increases the losses for diagonal propagation.

Equivalent refractive indexes for the circular polarized eigenmodes can be obtained from Eq. (27), which leads to the following equation to compute the Faraday rotation for diagonal propagation:

$${\varphi }_F=\frac{\pi }{{\lambda }_0}\text{Re}\left(\sqrt{\frac{n^2+\delta }{1+\frac{\delta }{3n^2}+j\frac{\delta }{3n^2}}}-\sqrt{\frac{n^2-\delta }{1-\frac{\delta }{3n^2}+j\frac{\delta }{3n^2}}}\right)d,$$ E31

where d is the propagation distance along the diagonal.

For n = 2, δ = 0.2, and λ₀ = 1.55 μm, we obtain ϕ_F/d = 0.27046 rads/μm. Comparing with the case for propagation along the y-axis (parallel to H_DC), by using Eq. (23), we obtain ϕ_F/y = 0.40549 rads/μm. These results show that we can obtain a better Faraday rotation when the propagation direction is aligned with the magnetostatic field, when considering TEM waves.

2.2 TM mode in a planar magneto-optical waveguide

Figure 7 presents a planar MO waveguide, which is composed by three MO layers. The magnetostatic field H_DC is applied along the y-axis. The propagation direction is now the z-axis. The planar waveguide supports transversal electric, TE, modes (H_x, E_y, H_z components) and transversal magnetic, TM, modes (E_x, H_y, E_z components). As discussed in Section 2.1.1, if H_DC is parallel to the electric field vector of the wave, then MO constant δ does not affect the propagation characteristics of the mode. Therefore, for the TE modes, no MO effect will be observed. For TM modes, however, the electric field components are perpendicular to H_DC, and nonreciprocal propagation characteristics will take place. In this section, mathematical expressions to calculate the propagation constants for TM modes in a MO planar waveguide will be derived. For the occurrence of guided modes in the structure shown in Figure 7, n₁ > n₂ and n₁ > n₃.

Figure 7.

Longitudinal section of a planar MO waveguide.

Defining $\overline{\overline{\xi }}$ as the inverse of the electric permittivity tensor of Eq. (2), we have:

$$\overline{\overline{\xi }}={\overline{\overline{\epsilon }}}_r^{-1}=\left[\begin{array}{lll}\frac{n^2}{n^4-{\delta }^2}& 0& -\frac{\mathit{j\delta }}{n^4-{\delta }^2}\\ 0& \frac{n^2}{n^4-{\delta }^2}& 0\\ \frac{\mathit{j\delta }}{n^4-{\delta }^2}& 0& \frac{n^2}{n^4-{\delta }^2}\end{array}\right]=\left[\begin{array}{lll}{\xi }_{\mathit{xx}}& 0& -{\xi }_{\mathit{zx}}\\ 0& {\xi }_{\mathit{yy}}& 0\\ {\xi }_{\mathit{zx}}& 0& {\xi }_{\mathit{zz}}\end{array}\right].$$ E32

From Maxwell’s equations at the frequency domain, considering TM modes (E_x, H_y, E_z components) and no field spatial variations along the y-axis, we obtain:

$$\mathit{j\omega \mu }{H}_y=\mathit{j\beta }{E}_x+\frac{\partial E_z}{\partial x},$$ E33

$$E_x=\frac{1}{{\epsilon }_0}\left({\xi }_{\mathit{xx}}\frac{\beta }{\omega }{H}_y+j\frac{{\xi }_{\mathit{zx}}}{\omega }\frac{\partial H_y}{\partial x}\right),$$ E34

$$E_z=\frac{1}{{\epsilon }_0}\left({\xi }_{\mathit{zx}}\frac{\beta }{\omega }{H}_y-j\frac{{\xi }_{\mathit{zz}}}{\omega }\frac{\partial H_y}{\partial x}\right),$$ E35

where β is the propagation constant of the guided TM mode in radians per meter.

Substituting Eqs. (34)–(35) in Eq. (33), we obtain the following wave equation for nonreciprocal media in terms of the H_y component:

$$\frac{{\partial }^2{H}_y}{\partial x^2}+\left(\frac{k_0^2-{\xi }_{\mathit{xx}}{\beta }^2}{{\xi }_{\mathit{zz}}}\right)H_y=0,$$ E36

where $k_0=\omega \sqrt{{\mathit{\mu \epsilon }}_0}$ .

The solution for H_y is expressed for each waveguide layer as.

$$H_y=C\text{exp}\left(-\mathit{\zeta x}\right),\text{for}x\ge 0.$$ E37

$$H_y=C\text{cos}\left(\mathit{\kappa x}\right)+D\mathrm{sen}\left(\mathit{\kappa x}\right),\text{for}–\mathrm{d}\le \mathit{x}\le 0.$$ E38

$$H_y=\left[C\text{cos}\left(\mathit{\kappa d}\right)-D\mathrm{sen}\left(\mathit{\kappa d}\right)\right]\text{exp}\left[\gamma \left(x+d\right)\right],\text{for}x\le -\mathrm{d}.$$ E39

The solution for the component E_z at each layer is obtained by substituting the corresponding solution for H_y in Eq. (35), resulting in.

$$E_z=\frac{C}{{\mathit{\omega \epsilon }}_0}\left({\xi }_{\mathit{zx}}^{\left(3\right)}\beta +j{\mathit{\zeta \xi }}_{\mathit{zz}}^{\left(3\right)}\right)\text{exp}\left(-\mathit{\zeta x}\right),\text{for}x\ge 0.$$ E40

$$E_z=\frac{1}{{\mathit{\omega \epsilon }}_0}\left\{C\left[{\xi }_{\mathit{zx}}^{\left(1\right)}\beta \text{cos}\left(\mathit{\kappa x}\right)+j{\mathit{\kappa \xi }}_{\mathit{zz}}^{\left(1\right)}\mathrm{sen}\left(\mathit{\kappa x}\right)\right]+D\left[{\xi }_{\mathit{zx}}^{\left(1\right)}\beta \mathrm{sen}\left(\mathit{\kappa x}\right)-j{\mathit{\kappa \xi }}_{\mathit{zz}}^{\left(1\right)}\text{cos}\left(\mathit{\kappa x}\right)\right]\right\},\text{for}–\mathrm{d}\le \mathit{x}\le 0.$$ E41

$$E_z=\frac{C\text{cos}\left(\mathit{\kappa d}\right)-D\mathrm{sen}\left(\mathit{\kappa d}\right)}{{\mathit{\omega \epsilon }}_0}\left({\xi }_{\mathit{zx}}^{\left(2\right)}\beta -j{\mathit{\gamma \xi }}_{\mathit{zz}}^{\left(2\right)}\right)\text{exp}\left[\gamma \left(x+d\right)\right],\text{for}x\le -\mathrm{d}.$$ E42

The superscripts between parentheses on the inverse permittivity tensor elements identify the corresponding waveguide layer, as specified in Figure 7. The continuity of E_z at x = 0 and at x = −d leads to the following system:

$$C\left[\left({\xi }_{\mathit{zx}}^{\left(3\right)}-{\xi }_{\mathit{zx}}^{\left(1\right)}\right)\beta +j{\mathit{\zeta \xi }}_{\mathit{zz}}^{\left(3\right)}\right]+D\left(j{\mathit{\kappa \xi }}_{\mathit{zz}}^{\left(1\right)}\right)=0,$$ E43

$$C\left\{\left[\left({\xi }_{\mathit{zx}}^{\left(1\right)}-{\xi }_{\mathit{zx}}^{\left(2\right)}\right)\beta +j{\mathit{\gamma \xi }}_{\mathit{zz}}^{\left(2\right)}\right]\text{cos}\left(\mathit{\kappa d}\right)-j{\mathit{\kappa \xi }}_{\mathit{zz}}^{\left(1\right)}\mathit{sen}\left(\mathit{\kappa d}\right)\right\}+D\left\{\left[\left({\xi }_{\mathit{zx}}^{\left(2\right)}-{\xi }_{\mathit{zx}}^{\left(1\right)}\right)\beta -j{\mathit{\gamma \xi }}_{\mathit{zz}}^{\left(2\right)}\right]\mathrm{sen}\left(\mathit{\kappa d}\right)-j{\mathit{\kappa \xi }}_{\mathit{zz}}^{\left(1\right)}\text{cos}\left(\mathit{\kappa d}\right)\right\}=0.$$ E44

After solving this system formed by Eqs. (43)–(44), we obtain:

$$\text{tan}\left(\mathit{\kappa d}\right)=\frac{{\mathit{\kappa \xi }}_{\mathit{zz}}^{\left(1\right)}\left[{\mathit{\zeta \xi }}_{\mathit{zz}}^{\left(3\right)}+{\mathit{\gamma \xi }}_{\mathit{zz}}^{\left(2\right)}-j\left({\xi }_{\mathit{zx}}^{\left(3\right)}-{\xi }_{\mathit{zx}}^{\left(2\right)}\right)\beta \right]}{{\left({\mathit{\kappa \xi }}_{\mathit{zz}}^{\left(1\right)}\right)}^2-\left[\left({\xi }_{\mathit{zx}}^{\left(3\right)}-{\xi }_{\mathit{zx}}^{\left(1\right)}\right)\beta +j{\mathit{\zeta \xi }}_{\mathit{zz}}^{\left(3\right)}\right]\left[\left({\xi }_{\mathit{zx}}^{\left(2\right)}-{\xi }_{\mathit{zx}}^{\left(1\right)}\right)\beta -j{\mathit{\gamma \xi }}_{\mathit{zz}}^{\left(2\right)}\right]}.$$ E45

The constants ζ, κ, and γ can be determined by substituting Eq. (37), Eq. (38), or Eq. (39), respectively, in Eq. (36), resulting in

$$\zeta =\sqrt{\frac{{\xi }_{\mathit{xx}}^{\left(3\right)}{\beta }^2-k_0^2}{{\xi }_{\mathit{zz}}^{\left(3\right)}}},$$ E46

$$\kappa =\sqrt{\frac{k_0^2-{\xi }_{\mathit{xx}}^{\left(1\right)}{\beta }^2}{{\xi }_{\mathit{zz}}^{\left(1\right)}}},$$ E47

$$\gamma =\sqrt{\frac{{\xi }_{\mathit{xx}}^{\left(2\right)}{\beta }^2-k_0^2}{{\xi }_{\mathit{zz}}^{\left(2\right)}}},$$ E48

where k₀ = 2π/λ₀, and λ₀ is the optical wavelength.

From the roots of Eq. (45) for β, the dispersion curve for TM modes in MO waveguides can be retrieved. Assuming that n₁ = 2.26, n₂ = 2.0, n₃ = 2.23, d = 1 μm, and only the layer 3 is magneto-optical with δ = 0.019, the dispersion curve for the fundamental and a superior TM mode is shown in Figure 8. We observe that the effective index profile changes when the propagation direction is reversed, which opens the possibility to the design of nonreciprocal devices. This phenomenon is known as nonreciprocal phase shift. If the magnetostatic field is not applied (δ = 0), the effective index profile becomes reciprocal and converges to the dashed line shown in Figure 8. The TM modes reach cutoff for optical wavelengths at which the effective index reaches the minimum value of 2.23. For greater optical wavelengths, the mode becomes irradiated and escapes through layer 3.

Figure 8.

Dispersion curves of the fundamental TM₀ mode and the superior TM₁ mode.

Figure 9 shows the transversal distributions of the H_y component at two distinct optical wavelengths. For this waveguide design, λ₀ = 1.55 μm is near cutoff, and the mode is highly distributed in the MO layer, which increases the nonreciprocal phase shift. Note from Figure 8 that the difference between the effective indexes of the counter propagating TM modes are greater for optical wavelengths near cutoff, but as the wavelengths decreases, the mode becomes more confined at the waveguide core, and its interaction with the MO layer decreases, resulting in a decrease of the nonreciprocal phase shift effect, considering this waveguide configuration.

Figure 9.

Transversal distribution of the H_y component of the fundamental TM₀ mode at λ₀ = 1.31 μm and at λ₀ = 1.55 μm.

2.3 TM mode in a planar magneto-optical directional coupler

Now let us consider a five-layered MO planar structure as shown in Figure 10.

Figure 10.

Longitudinal section of the five-layered MO planar structure.

The solutions for Eq. (36) in each layer, making use of the proper radiation conditions, are [18]

$$\begin{array}{l}{H}_y=A_1\text{exp}\left[-{\gamma }_1\left(x-S_3-d_2\right)\right],\text{for}x\ge S_3+d_2,\\ H_y=A_2\text{cos}\left[{\kappa }_2\left(x-S_3-d_2/2\right)\right]+A_3\text{sin}\left[{\kappa }_2\left(x-S_3-d_2/2\right)\right]\text{for}{\mathit{\text{S}}}_3\le x\le S_3+d_2,\\ H_y=A_4\text{exp}\left(-{\gamma }_{3}x\right)+A_5\text{exp}\left({\gamma }_{3}x\right),\text{for}–S_3\le x\le S_3,\\ H_y=A_6\text{cos}\left[{\kappa }_4\left(x+S_3+d_4/2\right)\right]+A_7\text{sin}\left[{\kappa }_4\left(x+S_3+d_4/2\right)\right],\text{for}–S_3–d_4\le x\le –S_3,\\ H_y=A_8\text{exp}\left[{\gamma }_5\left(x+S_3+d_4\right)\right],\text{for}\mathit{\text{x}}\le –S_3–d_{4,}\end{array}$$

where A₁ through A₈ are constants to be determined, κ_i and γ_j are given by.

$${\kappa }_i=\sqrt{\frac{k_0^2-{\xi }_{\mathit{xx}}^{\left(i\right)}{\beta }^2}{{\xi }_{\mathit{zz}}^{\left(i\right)}}},i=2,4,$$ E49

$${\gamma }_j=\sqrt{\frac{{\xi }_{\mathit{xx}}^{\left(,j\right)}{\beta }^2-k_0^2}{{\xi }_{\mathit{zz}}^{\left(,j\right)}}},j=1,3,5,$$ E50

where k₀ = 2π/λ₀, and λ₀ is the optical wavelength.

The electric field components E_x and E_z can be directly obtained with Eq. (34) and Eq. (35), respectively. Applying the boundary conditions for the tangential components H_y and E_z, one obtains a system of eight equations and eight unknowns, which can be conveniently written in matrix form as follows:

$$\left[M\left(\beta \right)\right]\mathbf{A}=\mathbf{0}.$$ E51

Here, [M(β)] is an 8×8 matrix that depends on the unknown longitudinal propagation constant β and A = [A₁ A₂ … A₈]^T. The propagation constant can be easily found by solving the equation Det([M(β)]) = 0. The nonzero elements of the matrix [M(β)] are listed below:

$$\begin{array}{l}{M}_{11}=1;M_{12}=-\text{cos}\left({\kappa }_2{d}_2/2\right);M_{13}=-\text{sin}\left({\kappa }_2{d}_2/2\right);\\ M_{21}=-j{\xi }_{\mathit{zx}}^{\left(1\right)}\beta +{\gamma }_1{\xi }_{\mathit{zz}}^{\left(1\right)};M_{22}=j{\xi }_{\mathit{zx}}^{\left(2\right)}\beta \text{cos}\left({\kappa }_2{d}_2/2\right)-{\kappa }_2{\xi }_{\mathit{zz}}^{\left(2\right)}\text{sin}\left({\kappa }_2{d}_2/2\right);\\ M_{23}=-j{\xi }_{\mathit{zx}}^{\left(2\right)}\beta \text{sin}\left(k_2{d}_2/2\right)+k_2{\xi }_{\mathit{zz}}^{\left(2\right)}\text{cos}\left(k_2{d}_2/2\right);\\ M_{32}=\text{cos}\left({\kappa }_2{d}_2/2\right);M_{33}=-\text{sin}\left({\kappa }_2{d}_2/2\right);M_{34}=-\text{exp}\left(-{\gamma }_3{S}_3\right);M_{35}=-\text{exp}\left({\gamma }_3{S}_3\right);\\ M_{42}=-j{\xi }_{\mathit{zx}}^{\left(2\right)}\beta \text{cos}\left({\kappa }_2{d}_2/2\right)-{\kappa }_2{\xi }_{\mathit{zz}}^{\left(2\right)}\text{sin}\left({\kappa }_2{d}_2/2\right);\\ M_{43}=j{\xi }_{\mathit{zx}}^{\left(2\right)}\beta \text{sin}\left({\kappa }_2{d}_2/2\right)-{\kappa }_2{\xi }_{\mathit{zz}}^{\left(2\right)}\text{cos}\left({\kappa }_2{d}_2/2\right);\\ M_{44}=\left(j{\xi }_{\mathit{zx}}^{\left(3\right)}\beta -{\xi }_{\mathit{zz}}^{\left(3\right)}{\gamma }_3\right)\text{exp}\left(-{\gamma }_3{S}_3\right);M_{45}=\left(j{\xi }_{\mathit{zx}}^{\left(3\right)}\beta +{\xi }_{\mathit{zz}}^{\left(3\right)}{\gamma }_3\right)\text{exp}\left({\gamma }_3{S}_3\right);\\ M_{54}=\text{exp}\left[{\gamma }_3{S}_3\right];M_{55}=\text{exp}\left[-{\gamma }_3{S}_3\right];M_{56}=-\text{cos}\left({\kappa }_4{d}_4/2\right);M_{57}=-\text{sin}\left({\kappa }_4{d}_4/2\right);\\ M_{64}=-\left(j{\xi }_{\mathit{zx}}^{\left(3\right)}\beta +{\xi }_{\mathit{zz}}^{\left(3\right)}{\gamma }_3\right)\text{exp}\left[{\gamma }_3{S}_3\right];M_{65}=-\left(j{\xi }_{\mathit{zx}}^{\left(3\right)}\beta +{\xi }_{\mathit{zz}}^{\left(3\right)}{\gamma }_3\right)\text{exp}\left[-{\gamma }_3{S}_3\right];\\ M_{66}=j{\xi }_{\mathit{zx}}^{\left(4\right)}\beta \text{cos}\left({\kappa }_4{d}_4/2\right)-{\xi }_{\mathit{zz}}^{\left(4\right)}{\kappa }_4\text{sin}\left({\kappa }_4{d}_4/2\right);\\ M_{67}=j{\xi }_{\mathit{zx}}^{\left(4\right)}\beta \text{sin}\left({\kappa }_4{d}_4/2\right)+{\xi }_{\mathit{zz}}^{\left(4\right)}{\kappa }_4\text{cos}\left({\kappa }_4{d}_4/2\right);\\ M_{76}=\text{cos}\left({\kappa }_4{d}_4/2\right);M_{77}=-\text{sin}\left({\kappa }_4{d}_4/2\right);M_{78}=-1;\\ M_{86}=-j{\xi }_{\mathit{zx}}^{\left(4\right)}\beta \text{cos}\left({\kappa }_4{d}_4/2\right)-{\xi }_{\mathit{zz}}^{\left(4\right)}{\kappa }_4\text{sin}\left({\kappa }_4{d}_4/2\right);\\ M_{87}=j{\xi }_{\mathit{zx}}^{\left(4\right)}\beta \text{sin}\left({\kappa }_4{d}_4/2\right)-{\xi }_{\mathit{zz}}^{\left(4\right)}{\kappa }_4\text{cos}\left({\kappa }_4{d}_4/2\right);\\ M_{88}=j{\xi }_{\mathit{zx}}^{\left(5\right)}\beta +{\xi }_{\mathit{zz}}^{\left(5\right)}{\gamma }_5;\end{array}$$

As an example, Table 1 shows the material parameters and layer thicknesses for each layer. Layers 1 and 5 are unbounded, and their thicknesses are theoretically infinite for the analytical model. The optical wavelength is λ₀ = 1.55 μm.

Layer	Parameters
	n	δ	Thickness (μm)
1	2.23	−0.019	∞
2	2.26	0	1.20
3	2.00	0	0.75
4	2.26	0	1.23
5	2.23	−0.019	∞

Table 1.

Material and geometric parameters of the MO directional coupler.

Figure 11 shows a plot of guided supermodes that occurs in the planar structure for forward propagation (along +z). The guided propagation along the five-layered structure, as well the periodical energy exchange of light between the two waveguides, can be expressed as a linear combination of these supermodes. The coupling length for the structure is given by L_π = π/|β₁ – β₂|, where β₁ and β₂ are the propagation constants of the supermodes obtained from the roots of Det([M(β)]) = 0. The computed coupling length, which refers to the propagation along the +z axis, is $L_{\pi }^{+}=1389.84\mathrm{\mu m}$ .

Figure 11.

Transversal distribution of the supermodes (H_y component) for forward propagation (+z).

Figure 12 shows the plot of the supermodes, now considering backward propagation of the TM mode (along -z). The computed coupling length, which refers to the backward propagation along the z-axis, is $L_{\pi }^{-}=689\mathrm{\mu m}$ .

Figure 12.

Transversal distribution of the supermodes (H_y component) for backward propagation (−z).

Considering both propagation senses, when the condition $L=L_{\pi }^{+}=2L_{\pi }^{-}$ for the length of the directional coupler is achieved, we obtain an optical isolator calibrated for the given optical wavelength. The operation of the optical isolator is depicted in Figure 13. If an optical source is placed at the port 1 of the waveguide A, all optical power will be coupled into port 3 of the waveguide B, if the length of the directional coupler is $L=L_{\pi }^{+}$ . If some light is reflected at port 3, since $L=2L_{\pi }^{-}$ , all optical power is directed to the port 4. Therefore, the optical source at port 1 becomes isolated from the reflected light. Figures 14, 15 show simulations of the forward and backward optical propagation in the MO directional coupler via a propagation projection of a linear combination of the corresponding supermodes.

Figure 13.

Operation of an optical isolator based on nonreciprocal phase shift.

Figure 14.

Forward propagation simulation of the TM mode component H_y excited at port 1 (P₁) of the five-layered structure. The light exits through port 3 (P₃). The starting transversal H_y field was supermode 1 plus supermode 2 of Figure 11.

Figure 15.

Backward propagation simulation of the TM mode component H_y excited at port 3 (P₃) of the five-layered structure. The light exits through port 4 (P₄). The starting transversal H_y field was supermode 1 minus supermode 2 of Figure 12.

The MO directional coupler of Figure 10 also acts as an optical circulator, considering the following sequence of input and output ports: 1 to 3; 3 to 4; 4 to 2; and 2 to 1.

3. Conclusions

The propagation characteristics of optical waves in magneto-optical media and in planar waveguides with three and five MO layers were exposed. The effects of Faraday rotation and nonreciprocal phase shift were discussed with mathematical background to support the analyses. The propagation of TEM waves in unbounded MO media was discussed, where it was shown that the Faraday rotation is maximized when the propagation occurs in the same direction of the applied magnetostatic field. It was also mathematically shown that if there is no such alignment, losses may be added to the wave propagation. A planar MO waveguide and a directional coupler were also analyzed in the context of their nonreciprocity. For these structures, nonreciprocity is observed for TM-guided modes. The theoretical analyses confirm that magneto-optical materials have great potential to be employed on the design of nonreciprocal optical devices, such as isolators and circulators.