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 (HDC), 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 [2–5]. 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 [6–10]. 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 [11–13].

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 HDC, which affects the wave polarization. Depending on the orientation of the magnetostatic field, the configuration of the electric permittivity tensor changes. If HDC is oriented along one of the Cartesian axes, the relative electric permittivity assumes the form

$$
\overline{\overline{\varepsilon}}\_r = \begin{bmatrix} n^2 & 0 & 0 \\ 0 & n^2 & j\delta \\ 0 & -j\delta & n^2 \end{bmatrix}, \text{for } H\_{DC} \parallel \infty - \text{axis}; \tag{1}
$$

where ω is the angular frequency in rad/s, μ<sup>0</sup> is the magnetic permeability of the

To develop a plane wave solution for MO media, it is assumed that HDC is parallel to the y-axis and ε<sup>r</sup> 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

E ¼ E<sup>0</sup> exp ð Þ jωt exp �j γxx þ γyy þ γzz

δ n2

E � j δ <sup>n</sup><sup>2</sup> <sup>∇</sup>

∂2 Ex ∂y<sup>2</sup> þ

∂2 Ey ∂y<sup>2</sup> þ

From Gauss' law for a medium with equilibrium of charges, <sup>∇</sup>: <sup>ε</sup>0εrE � � <sup>¼</sup> 0, we

∂Ex <sup>∂</sup><sup>z</sup> � <sup>∂</sup>Ez ∂x

∂2 Ex <sup>∂</sup>z<sup>2</sup> � <sup>j</sup>

∂2 Ey <sup>∂</sup>z<sup>2</sup> � <sup>j</sup>

is the propagation constant vector.

∂Ex <sup>∂</sup><sup>z</sup> � <sup>∂</sup>Ez ∂x � �

> δ n2

> > ∂2 Ex <sup>∂</sup>y∂<sup>z</sup> � <sup>∂</sup><sup>2</sup>

δ n2 ∂2 Ex ∂x∂z

h i � � : (5)

� �: (6)

� ∂2 Ez ∂x<sup>2</sup>

Ez ∂x∂y

� �

� �

¼ 0: (7)

¼ 0, (8)

¼ 0, (9)

vacuum in H/m, and ε<sup>0</sup> is the electric permittivity of the vacuum in F/m.

arbitrary direction, with the electric field vector given by

∇ � E ¼ j

where γ ¼ γ<sup>x</sup> i

obtain:

Figure 1.

ω2

233

μ0ε<sup>0</sup> n<sup>2</sup>

ω2 μ0ε0n<sup>2</sup>

! þγ<sup>y</sup> j ! þγzk !

TEM wave in an unbounded magneto-optical medium.

Optical Propagation in Magneto-Optical Materials DOI: http://dx.doi.org/10.5772/intechopen.81963

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

<sup>μ</sup>0ε0εrE <sup>þ</sup> <sup>∇</sup><sup>2</sup>

∂2 Ex ∂x<sup>2</sup> þ

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

ω2

Ex þ jδEz � � <sup>þ</sup>

> Ey þ ∂2 Ey ∂x<sup>2</sup> þ

$$
\overline{\overline{\varepsilon}}\_r = \begin{bmatrix} n^2 & 0 & j\delta \\ 0 & n^2 & 0 \\ -j\delta & 0 & n^2 \end{bmatrix}, \text{for } H\_{DC} \parallel y-\text{axis};\tag{2}
$$

$$
\overline{\overline{\varepsilon}}\_r = \begin{bmatrix} n^2 & j\delta & 0 \\ -j\delta & n^2 & 0 \\ 0 & 0 & n^2 \end{bmatrix}, \text{for } H\_{DC} \parallel z-\text{axis.} \tag{3}
$$

where n is the refractive index of the material and δ is the magneto-optical constant. The MO constant is proportional to HDC. If the sense of HDC is reversed, δ(-HDC) = �δ(HDC), and for HDC = 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.

From Maxwell's equations, the vectorial Helmholtz equation for anisotropic media and for the electric field E xð Þ ; y; z can be written as

$$
\alpha^2 \mu\_0 \varepsilon\_0 \overline{\overline{\varepsilon}} \overline{E} + \nabla^2 \overline{E} - \nabla \left( \nabla \cdot \overline{E} \right) = \overline{\mathbf{0}},\tag{4}
$$

Optical Propagation in Magneto-Optical Materials DOI: http://dx.doi.org/10.5772/intechopen.81963

spectrum. Therefore, research activities on the improvement of MO materials and structures have also great relevance and are covered in works such as [6–10]. 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

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

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 HDC, which affects the wave polarization. Depending on the orientation of the magnetostatic field, the configuration of the electric permittivity tensor changes. If HDC is oriented along one of the Cartesian axes, the relative electric permittivity assumes

> 3 7

3 7

> 3 7

where n is the refractive index of the material and δ is the magneto-optical constant. The MO constant is proportional to HDC. If the sense of HDC is reversed, δ(-HDC) = �δ(HDC), and for HDC = 0, the off-diagonal components of the electric

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

From Maxwell's equations, the vectorial Helmholtz equation for anisotropic

<sup>5</sup>,for HDC <sup>k</sup> <sup>x</sup> � axis; (1)

<sup>5</sup>,for HDC <sup>k</sup> <sup>y</sup> � axis; (2)

<sup>5</sup>,for HDC <sup>k</sup> <sup>z</sup> � axis: (3)

<sup>E</sup> � ∇ ∇ � <sup>E</sup> � � <sup>¼</sup> <sup>0</sup>, (4)

addressed [11–13].

Electromagnetic Materials and Devices

the mathematical model.

the form

in Figure 1.

232

2. Wave propagation characteristics

ε<sup>r</sup> ¼

ε<sup>r</sup> ¼

ε<sup>r</sup> ¼

permittivity tensor are zero [14, 15].

2 6 4

2 6 4

> 2 6 4

2.1 TEM wave in an unbounded magneto-optical medium

media and for the electric field E xð Þ ; y; z can be written as

<sup>μ</sup>0ε0εrE <sup>þ</sup> <sup>∇</sup><sup>2</sup>

ω2

n<sup>2</sup> 0 0 0 n<sup>2</sup> jδ <sup>0</sup> �j<sup>δ</sup> <sup>n</sup><sup>2</sup>

n<sup>2</sup> 0 jδ 0 n<sup>2</sup> 0 �j<sup>δ</sup> <sup>0</sup> <sup>n</sup><sup>2</sup>

n<sup>2</sup> jδ 0 �j<sup>δ</sup> <sup>n</sup><sup>2</sup> <sup>0</sup> 0 0 n<sup>2</sup>

Figure 1. TEM wave in an unbounded magneto-optical medium.

where ω is the angular frequency in rad/s, μ<sup>0</sup> is the magnetic permeability of the vacuum in H/m, and ε<sup>0</sup> is the electric permittivity of the vacuum in F/m.

To develop a plane wave solution for MO media, it is assumed that HDC is parallel to the y-axis and ε<sup>r</sup> 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

$$\overline{E} = \overline{E}\_0 \exp\left(jat\right) \exp\left[-j\left(\chi\_x x + \chi\_y y + \chi\_z x\right)\right].\tag{5}$$

where γ ¼ γ<sup>x</sup> i ! þγ<sup>y</sup> j ! þγzk ! is the propagation constant vector.

From Gauss' law for a medium with equilibrium of charges, <sup>∇</sup>: <sup>ε</sup>0εrE � � <sup>¼</sup> 0, we obtain:

$$\nabla \cdot \overline{E} = j \frac{\delta}{n^2} \left( \frac{\partial E\_x}{\partial \mathbf{z}} - \frac{\partial E\_x}{\partial \mathbf{x}} \right). \tag{6}$$

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

$$
\rho \alpha^2 \mu\_0 \varepsilon\_0 \overline{\overline{e}} \overline{E} + \nabla^2 \overline{E} - j \frac{\delta}{n^2} \nabla \left( \frac{\partial E\_\mathbf{x}}{\partial \mathbf{z}} - \frac{\partial E\_\mathbf{z}}{\partial \mathbf{x}} \right) = \overline{\mathbf{0}}.\tag{7}$$

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

$$
\rho \alpha^2 \mu\_0 \varepsilon\_0 \left( n^2 E\_\mathbf{x} + j \delta E\_\mathbf{z} \right) + \frac{\partial^2 E\_\mathbf{x}}{\partial \mathbf{x}^2} + \frac{\partial^2 E\_\mathbf{x}}{\partial \mathbf{y}^2} + \frac{\partial^2 E\_\mathbf{x}}{\partial \mathbf{z}^2} - j \frac{\delta}{n^2} \left( \frac{\partial^2 E\_\mathbf{x}}{\partial \mathbf{x} \partial \mathbf{z}} - \frac{\partial^2 E\_\mathbf{z}}{\partial \mathbf{x}^2} \right) = \mathbf{0},\tag{8}
$$

$$
\alpha^2 \mu\_0 \varepsilon\_0 n^2 E\_y + \frac{\partial^2 E\_y}{\partial x^2} + \frac{\partial^2 E\_y}{\partial y^2} + \frac{\partial^2 E\_y}{\partial z^2} - j \frac{\delta}{n^2} \left( \frac{\partial^2 E\_x}{\partial y \partial z} - \frac{\partial^2 E\_x}{\partial x \partial y} \right) = 0,\tag{9}
$$

$$\rho \alpha^2 \mu\_0 \varepsilon\_0 \left( -j \delta E\_x + n^2 E\_x \right) + \frac{\partial^2 E\_x}{\partial x^2} + \frac{\partial^2 E\_x}{\partial y^2} + \frac{\partial^2 E\_x}{\partial z^2} - j \frac{\delta}{n^2} \left( \frac{\partial^2 E\_x}{\partial z^2} - \frac{\partial^2 E\_x}{\partial x \partial z} \right) = 0. \tag{10}$$

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

$$\left(\left\|\boldsymbol{\alpha}^{2}\mu\_{0}\boldsymbol{\varepsilon}\_{0}\boldsymbol{\eta}^{2} - \left|\boldsymbol{\eta}\right|^{2} - j\frac{\delta}{n^{2}}\boldsymbol{\chi}\_{\mathrm{x}}\boldsymbol{\chi}\_{\mathrm{z}}\right)\boldsymbol{E}\_{\mathrm{x}} + j\delta\left(\boldsymbol{\alpha}^{2}\mu\_{0}\boldsymbol{\varepsilon}\_{0} - \frac{1}{n^{2}}\boldsymbol{\chi}\_{\mathrm{x}}^{2}\right)\boldsymbol{E}\_{\mathrm{z}} = \mathbf{0} \tag{11}$$

$$\left(\alpha^2 \mu\_0 \varepsilon\_0 n^2 - |\boldsymbol{\gamma}|^2\right) E\_\boldsymbol{\gamma} + j \frac{\delta}{n^2} \left(\boldsymbol{\gamma}\_\circ \boldsymbol{\gamma}\_x E\_\mathbf{x} - \boldsymbol{\gamma}\_x \boldsymbol{\gamma}\_\circ E\_\mathbf{z}\right) = \mathbf{0},\tag{12}$$

$$-j\delta \left( \alpha^2 \mu\_0 \varepsilon\_0 - \frac{1}{n^2} \chi\_x^2 \right) \mathbf{E}\_x + \left( \alpha^2 \mu\_0 \varepsilon\_0 n^2 - |\boldsymbol{\gamma}|^2 - j\frac{\delta}{n^2} \chi\_x \chi\_x \right) \mathbf{E}\_x = \mathbf{0},\tag{13}$$

where j j γ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ2 <sup>x</sup> þ γ<sup>2</sup> <sup>y</sup> þ γ<sup>2</sup> z q .
