2.1.1 TEM wave with electric field vector parallel to HDC

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 HDC, so that Ex = Ez = 0, the magneto-optical constant δ related to HDC will have no effect on the propagation characteristics of the wave. In this case, from Eq. (12), the propagation constant modulus would be

$$|\chi| = n\alpha \sqrt{\mu\_0 \varepsilon\_0},\tag{14}$$

j j <sup>γ</sup> <sup>¼</sup> <sup>ω</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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

ω2μ0ε0n<sup>2</sup> � j j γ

Therefore, substituting Eq. (20) in Eq. (18), and given that �j = exp(�jπ/2), the

<sup>E</sup>0<sup>x</sup> ¼ �j<sup>δ</sup> <sup>ω</sup>2μ0ε<sup>0</sup>

Ex <sup>¼</sup> <sup>E</sup>0<sup>z</sup> exp <sup>j</sup> <sup>ω</sup><sup>t</sup> � <sup>y</sup><sup>ω</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>n</sup><sup>2</sup> <sup>þ</sup> <sup>δ</sup> <sup>p</sup> , for the CCW circular polarized eigenmode;

<sup>n</sup><sup>2</sup> � <sup>δ</sup> <sup>p</sup> , for the CW circular polarized eigenmode.

Ez <sup>¼</sup> <sup>E</sup>0<sup>z</sup> exp <sup>j</sup> <sup>ω</sup><sup>t</sup> � <sup>y</sup><sup>ω</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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

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

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.

exp <sup>j</sup> <sup>ω</sup><sup>t</sup> � <sup>y</sup><sup>ω</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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

! þE0<sup>z</sup>k � �!

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

electric field components can be written as

equivalent refractive index to each eigenmode:

<sup>n</sup><sup>þ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>n</sup>� <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

Figure 2.

235

E ¼ E0<sup>x</sup> i

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

<sup>μ</sup>0ε<sup>0</sup> <sup>n</sup>ð Þ <sup>2</sup> � <sup>δ</sup> <sup>p</sup> , (17)

μ0ε<sup>0</sup> nð Þ <sup>2</sup> � δ h i � � <sup>p</sup> : (18)

<sup>2</sup> � � <sup>E</sup>0z: (19)

E0<sup>x</sup> ¼ �jE0z: (20)

<sup>μ</sup>0ε<sup>0</sup> <sup>n</sup>ð Þ <sup>2</sup> � <sup>δ</sup> <sup>p</sup> � <sup>π</sup>=<sup>2</sup> h i � � , (21)

μ0ε<sup>0</sup> nð Þ <sup>2</sup> � δ h i � � <sup>p</sup> : (22)

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<sup>y</sup> = 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

$$\left(\left\|\boldsymbol{\alpha}^{2}\boldsymbol{\mu}\_{0}\boldsymbol{\varepsilon}\_{0}\boldsymbol{n}^{2}-\left|\boldsymbol{\gamma}\right|^{2}-j\frac{\delta}{n^{2}}\boldsymbol{\gamma}\_{x}\boldsymbol{\gamma}\_{x}\right\}^{2}-\delta^{2}\left(\boldsymbol{\alpha}^{2}\boldsymbol{\mu}\_{0}\boldsymbol{\varepsilon}\_{0}-\frac{1}{n^{2}}\boldsymbol{\gamma}\_{x}^{2}\right)\left(\boldsymbol{\alpha}^{2}\boldsymbol{\mu}\_{0}\boldsymbol{\varepsilon}\_{0}-\frac{1}{n^{2}}\boldsymbol{\gamma}\_{x}^{2}\right)=\mathbf{0}.\tag{15}$$

Solving Eq. (15) for |γ|, we obtain:

$$|\gamma| = \sqrt{\alpha^2 \mu\_0 \varepsilon\_0 n^2 - j\frac{\delta}{n^2} \chi\_x \chi\_x \pm \delta \sqrt{\left(\alpha^2 \mu\_0 \varepsilon\_0 - \frac{1}{n^2} \chi\_x^2\right) \left(\alpha^2 \mu\_0 \varepsilon\_0 - \frac{1}{n^2} \chi\_x^2\right)}}.\tag{16}$$

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

The parameters γ<sup>x</sup> and γ<sup>z</sup> are projections of the propagation constant vector along the x and the y-axis, respectively.

#### 2.1.3 TEM wave propagating parallel to HDC

If the TEM wave is propagating along the HDC direction (y-axis), so that γ<sup>x</sup> = γ<sup>z</sup> = 0, Eq. (16) assumes the simpler form:

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

ω2

Eq. (5):

<sup>μ</sup>0ε<sup>0</sup> �jδEx <sup>þ</sup> <sup>n</sup><sup>2</sup>

Electromagnetic Materials and Devices

ω2

�jδ ω<sup>2</sup>

where j j γ ¼

<sup>μ</sup>0ε0n<sup>2</sup> � j j <sup>γ</sup>

ω2

<sup>μ</sup>0ε<sup>0</sup> � <sup>1</sup>

γ2 <sup>x</sup> þ γ<sup>2</sup>

q

constant modulus would be

plane xz, so that γ<sup>y</sup> = 0.

the following equation

<sup>μ</sup>0ε0n<sup>2</sup> � j j <sup>γ</sup>

<sup>2</sup> � <sup>j</sup> δ <sup>n</sup><sup>2</sup> <sup>γ</sup>xγ<sup>z</sup>

Solving Eq. (15) for |γ|, we obtain:

δ

<sup>n</sup><sup>2</sup> <sup>γ</sup>xγ<sup>z</sup> � <sup>δ</sup>

ω<sup>2</sup>μ0ε0n<sup>2</sup> � j

along the x and the y-axis, respectively.

2.1.3 TEM wave propagating parallel to HDC

γ<sup>x</sup> = γ<sup>z</sup> = 0, Eq. (16) assumes the simpler form:

� �<sup>2</sup>

ω2

j j γ ¼

234

� �

Ez � � <sup>þ</sup>

� �

∂2 Ez ∂x<sup>2</sup> þ

<sup>2</sup> � <sup>j</sup> δ <sup>n</sup><sup>2</sup> <sup>γ</sup>xγ<sup>z</sup>

<sup>μ</sup>0ε0n<sup>2</sup> � j j <sup>γ</sup> <sup>2</sup> � �

> <sup>y</sup> þ γ<sup>2</sup> z

2.1.1 TEM wave with electric field vector parallel to HDC

2.1.2 The general expression for the propagation constant

.

<sup>n</sup><sup>2</sup> <sup>γ</sup><sup>2</sup> z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

∂2 Ez ∂y<sup>2</sup> þ ∂2 Ez <sup>∂</sup>z<sup>2</sup> � <sup>j</sup>

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

Ey þ j

Ex <sup>þ</sup> <sup>ω</sup><sup>2</sup>

Ex <sup>þ</sup> <sup>j</sup>δ ω<sup>2</sup>

δ

<sup>μ</sup>0ε0n<sup>2</sup> � j j <sup>γ</sup>

By observing Eqs. (11–13), we note that when the electric field of the electro-

magnetic wave is polarized along the y-axis and is parallel to HDC, so that Ex = Ez = 0, the magneto-optical constant δ related to HDC will have no effect on the propagation characteristics of the wave. In this case, from Eq. (12), the propagation

j j¼ γ nω ffiffiffiffiffiffiffiffiffi

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

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

<sup>μ</sup>0ε<sup>0</sup> � <sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

If the TEM wave is propagating along the HDC direction (y-axis), so that

<sup>ω</sup><sup>2</sup>μ0ε<sup>0</sup> � <sup>1</sup>

� �

vuut : (16)

� �

<sup>n</sup><sup>2</sup> <sup>γ</sup><sup>2</sup> x

> <sup>n</sup><sup>2</sup> <sup>γ</sup><sup>2</sup> x

ω2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s � �

<sup>μ</sup>0ε<sup>0</sup> � <sup>1</sup>

<sup>ω</sup><sup>2</sup>μ0ε<sup>0</sup> � <sup>1</sup>

� �

<sup>n</sup><sup>2</sup> <sup>γ</sup><sup>2</sup> z

> <sup>n</sup><sup>2</sup> <sup>γ</sup><sup>2</sup> z

¼ 0: (15)

� <sup>δ</sup><sup>2</sup> <sup>ω</sup><sup>2</sup>

Note that when the MO constant δ = 0, Eq. (16) reduces to Eq. (14). The parameters γ<sup>x</sup> and γ<sup>z</sup> are projections of the propagation constant vector

μ0ε<sup>0</sup>

δ n2

<sup>μ</sup>0ε<sup>0</sup> � <sup>1</sup>

<sup>n</sup><sup>2</sup> <sup>γ</sup>yγzEx � <sup>γ</sup>xγyEz � �

� �

<sup>2</sup> � <sup>j</sup> δ <sup>n</sup><sup>2</sup> <sup>γ</sup>xγ<sup>z</sup>

� �

<sup>n</sup><sup>2</sup> <sup>γ</sup><sup>2</sup> x

p , (14)

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

Ez ∂x∂z

¼ 0: (10)

Ez ¼ 0 (11)

¼ 0, (12)

Ez ¼ 0, (13)

� �

$$|\chi| = a\sqrt{\mu\_0 \varepsilon\_0 (n^2 \pm \delta)},\tag{17}$$

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

$$\overrightarrow{E} = \left(E\_{0x}\overrightarrow{\mathbf{i}} + E\_{0x}\overrightarrow{\mathbf{k}}\right)\exp\left[j\left(\alpha t - y\omega\sqrt{\mu\_0\varepsilon\_0(n^2 \pm \delta)}\right)\right].\tag{18}$$

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

$$E\_{0x} = -j\delta \frac{\alpha^2 \mu\_0 \varepsilon\_0}{\left(\alpha^2 \mu\_0 \varepsilon\_0 n^2 - |\gamma|^2\right)} E\_{0x}. \tag{19}$$

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

$$E\_{0x} = \pm jE\_{0x}.\tag{20}$$

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

$$E\_{\mathbf{x}} = E\_{0\mathbf{x}} \exp\left[j\left(a\mathbf{t} - \mathbf{y}a\sqrt{\mu\_0 \varepsilon\_0(n^2 \pm \delta)} \pm \pi/2\right)\right],\tag{21}$$

$$E\_x = E\_{0x} \exp\left[j\left(at - \gamma a\nu\sqrt{\mu\_0 \varepsilon\_0 (n^2 \pm \delta)}\right)\right].\tag{22}$$

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:

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

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

#### 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.

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.

n<sup>1</sup> = 2.19, and n<sup>2</sup> = 2.18. The waveguide dimensions are w = 8 μm, h = 0.5 μm,

ϕ<sup>F</sup> = π/4 (45°), we obtain y = 3388 μm, which is a propagation length that

Before finishing this section, let us consider another particular case of propagation direction—suppose, in Figure 1, that γ<sup>x</sup> = γ<sup>y</sup> = γ<sup>z</sup> = γu, with γ<sup>u</sup> 6¼ 0. This case corresponds to a TEM wave propagating along the diagonal of an imaginary

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffi 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ0ε<sup>0</sup> nð Þ <sup>2</sup> � δ <sup>3</sup> � <sup>δ</sup> <sup>n</sup><sup>2</sup> <sup>þ</sup> <sup>j</sup> <sup>δ</sup> n2

<sup>u</sup> � δ ω<sup>2</sup>μ0ε<sup>0</sup> � <sup>1</sup>

we can also obtain:

<sup>s</sup> � �: (24)

δ <sup>n</sup><sup>2</sup> <sup>γ</sup><sup>2</sup>

2.1.4 TEM wave propagating along the diagonal of an imaginary cube

cube, adjacent to the Cartesian axes. From Eq. (16), we obtain:

ω<sup>2</sup>μ0ε0n<sup>2</sup> � j

γ2 <sup>x</sup> þ γ<sup>2</sup>

q

Equaling Eqs. (24)–(25) and solving for γ<sup>u</sup> result in

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

γ<sup>u</sup> ¼ ω

<sup>y</sup> þ γ<sup>2</sup> z

j j γ ¼ γ<sup>u</sup>

s

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

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

, n = n<sup>1</sup> = 2.19, λ<sup>0</sup> = 1.485 μm, and

<sup>n</sup><sup>2</sup> <sup>γ</sup><sup>2</sup> u

<sup>p</sup> : (25)

: (26)

t<sup>1</sup> = 3.1 μm, and t<sup>2</sup> = 3.4 μm. The optical wavelength is λ<sup>0</sup> = 1.485 μm.

optical isolator is obtained.

In Eq. (23), by adopting <sup>δ</sup> = 2.4 � <sup>10</sup>�<sup>4</sup>

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

converges with the FD-VBPM result.

j j γ ¼

From the relation j j γ ¼

Figure 5.

237

the MO waveguide.

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

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

$$\phi\_F = \frac{1}{2}(\phi^+ - \phi^-) = \frac{1}{2}\left(n^+ \frac{2\pi}{\lambda\_0}y - n^- \frac{2\pi}{\lambda\_0}y\right) = \frac{\pi}{\lambda\_0}\left(\sqrt{n^2 + \delta} - \sqrt{n^2 - \delta}\right)y,\tag{23}$$

where λ<sup>0</sup> is the optical wavelength in vacuum. The Faraday rotation effect is responsible for a periodic power transfer between the transverse components, in this case, Ex and Ez. This phenomenon in MO materials may be exploited for the design of optical isolators based on Faraday rotation.

When a MO waveguide, with HDC 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 nSR = 1.94. For the Bi-YIG layers, the relative permittivity tensor has the form of Eq. (2), with <sup>δ</sup> = 2.4 � <sup>10</sup>�<sup>4</sup> ,

Figure 3.

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

Figure 4. Magneto-optical rib waveguide.

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

polarization will rotate in the xz plane as the wave propagates along the y-axis, in a

When the sense of the magnetostatic field HDC is reversed, the magneto-optical constant δ changes its signal, and the values of n<sup>+</sup> and n� are interchanged, and the

The Faraday rotation angle (ϕF) may be calculated (in radians) as a function of

where λ<sup>0</sup> is the optical wavelength in vacuum. The Faraday rotation effect is responsible for a periodic power transfer between the transverse components, in this case, Ex and Ez. This phenomenon in MO materials may be exploited for the

When a MO waveguide, with HDC 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 nSR = 1.94. For the Bi-YIG layers, the relative permittivity tensor has the form of Eq. (2), with <sup>δ</sup> = 2.4 � <sup>10</sup>�<sup>4</sup>

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

¼ π λ0

ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>n</sup><sup>2</sup> <sup>þ</sup> <sup>δ</sup> <sup>p</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffi

n<sup>2</sup> � δ � � <sup>p</sup> y, (23)

,

phenomenon known as Faraday rotation, which is depicted in Figure 3.

sense of rotation of a linear polarized wave in the MO media will change.

<sup>y</sup> � <sup>n</sup>� <sup>2</sup><sup>π</sup> λ0 y

� �

the propagation distance y by

<sup>2</sup> <sup>ϕ</sup><sup>þ</sup> � <sup>ϕ</sup>� ð Þ¼ <sup>1</sup>

Electromagnetic Materials and Devices

<sup>2</sup> <sup>n</sup><sup>þ</sup> <sup>2</sup><sup>π</sup> λ0

design of optical isolators based on Faraday rotation.

<sup>ϕ</sup><sup>F</sup> <sup>¼</sup> <sup>1</sup>

Figure 3.

Figure 4.

236

Magneto-optical rib waveguide.

the magnetostatic field HDC.

n<sup>1</sup> = 2.19, and n<sup>2</sup> = 2.18. The waveguide dimensions are w = 8 μm, h = 0.5 μm, t<sup>1</sup> = 3.1 μm, and t<sup>2</sup> = 3.4 μm. The optical wavelength is λ<sup>0</sup> = 1.485 μm.

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.

In Eq. (23), by adopting <sup>δ</sup> = 2.4 � <sup>10</sup>�<sup>4</sup> , n = n<sup>1</sup> = 2.19, λ<sup>0</sup> = 1.485 μm, and ϕ<sup>F</sup> = 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<sup>x</sup> = γ<sup>y</sup> = γ<sup>z</sup> = γu, with γ<sup>u</sup> 6¼ 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:

$$|\chi| = \sqrt{\alpha^2 \mu\_0 \varepsilon\_0 n^2 - j\frac{\delta}{n^2} \chi\_u^2 \pm \delta \left(\alpha^2 \mu\_0 \varepsilon\_0 - \frac{1}{n^2} \chi\_u^2\right)}.\tag{24}$$

From the relation j j γ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ2 <sup>x</sup> þ γ<sup>2</sup> <sup>y</sup> þ γ<sup>2</sup> z q we can also obtain:

$$|\chi| = \chi\_u \sqrt{3}.\tag{25}$$

Equaling Eqs. (24)–(25) and solving for γ<sup>u</sup> result in

$$\gamma\_u = \alpha \sqrt{\frac{\mu\_0 \varepsilon\_0 (n^2 \pm \delta)}{3 \pm \frac{\delta}{n^2} + j\frac{\delta}{n^2}}}.\tag{26}$$

#### Figure 5.

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

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

$$|\chi| = o\sqrt{\frac{\mu\_0 \varepsilon\_0 (n^2 \pm \delta)}{1 \pm \frac{\delta}{3n^2} + j\frac{\delta}{3n^2}}}.\tag{27}$$

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

$$E\_x = \pm jE\_x.\tag{28}$$

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

$$E\_{\gamma} = -\frac{n^2 \pm \delta + j\Im(\delta \pm n^2)}{\Im n^2} E\_x. \tag{29}$$

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

$$\overrightarrow{E} = \left( \pm j \stackrel{\rightarrow}{\mathbf{i}} - \frac{n^2 \pm \delta + j3(\delta \pm n^2)}{5n^2} \stackrel{\rightarrow}{\mathbf{j}} + \stackrel{\rightarrow}{\mathbf{k}} \right) E\_{0\mathbf{r}} \exp\left[ j \left( \alpha \mathrm{t} - \alpha \sqrt{\frac{\mu\_0 \varepsilon\_0 (n^2 \pm \delta)}{3 \pm \frac{\delta}{n^2} + j \frac{\delta}{n^2}}} (\mathbf{x} + \mathbf{y} + \mathbf{z})} \right) \right],\tag{30}$$

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 Ey 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.

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:

$$\phi\_F = \frac{\pi}{\lambda\_0} \text{Re}\left(\sqrt{\frac{n^2 + \delta}{1 + \frac{\delta}{3n^2} + j\frac{\delta}{3n^2}}} - \sqrt{\frac{n^2 - \delta}{1 - \frac{\delta}{3n^2} + j\frac{\delta}{3n^2}}}\right) d,\tag{31}$$

2.2 TM mode in a planar magneto-optical waveguide

Longitudinal section of a planar MO waveguide.

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

Figure 6.

Figure 7.

239

represented by red lines.

Figure 7 presents a planar MO waveguide, which is composed by three MO layers. The magnetostatic field HDC is applied along the y-axis. The propagation direction is now the z-axis. The planar waveguide supports transversal electric, TE, modes (Hx, Ey, Hz components) and transversal magnetic, TM, modes (Ex, Hy, Ez components). As discussed in Section 2.1.1, if HDC 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 HDC, and

TEM eigenmodes for diagonal propagation where γ<sup>x</sup> = γ<sup>y</sup> = γz. The trajectory of the electric field vector is

where d is the propagation distance along the diagonal.

For n = 2, δ = 0.2, and λ<sup>0</sup> = 1.55 μm, we obtain ϕF/d = 0.27046 rads/μm. Comparing with the case for propagation along the y-axis (parallel to HDC), by using Eq. (23), we obtain ϕF/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.

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

#### Figure 6.

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

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ0ε<sup>0</sup> nð Þ <sup>2</sup> � δ

<sup>3</sup>n<sup>2</sup> <sup>þ</sup> <sup>j</sup> <sup>δ</sup> 3n<sup>2</sup>

: (27)

Ex ¼ �jEz: (28)

<sup>5</sup>n<sup>2</sup> Ez: (29)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ0ε<sup>0</sup> nð Þ <sup>2</sup> � δ <sup>3</sup> � <sup>δ</sup> <sup>n</sup><sup>2</sup> <sup>þ</sup> <sup>j</sup> <sup>δ</sup> n2

" # !

ð Þ x þ y þ z

(30)

,

s

<sup>1</sup> � <sup>δ</sup>

The corresponding electric field vector can be retrieved by substituting the

However, for the considered propagation direction, the Ey component is not

Ey ¼ � <sup>n</sup><sup>2</sup> � <sup>δ</sup> <sup>þ</sup> <sup>j</sup><sup>3</sup> <sup>δ</sup> � <sup>n</sup><sup>2</sup> ð Þ

By using the results of Eqs. (26)–(29) in Eq. (5), we can express the electric field

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 Ey 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

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

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

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n<sup>2</sup> þ δ

<sup>3</sup>n<sup>2</sup> <sup>þ</sup> <sup>j</sup> <sup>δ</sup> 3n<sup>2</sup>

For n = 2, δ = 0.2, and λ<sup>0</sup> = 1.55 μm, we obtain ϕF/d = 0.27046 rads/μm. Comparing with the case for propagation along the y-axis (parallel to HDC), by using Eq. (23), we obtain ϕF/y = 0.40549 rads/μm. These results show that we can obtain a better Faraday rotation when the propagation direction is aligned with the

�

! s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n<sup>2</sup> � δ

<sup>3</sup>n<sup>2</sup> <sup>þ</sup> <sup>j</sup> <sup>δ</sup> 3n<sup>2</sup> d, (31)

<sup>1</sup> � <sup>δ</sup>

<sup>1</sup> <sup>þ</sup> <sup>δ</sup>

s

where d is the propagation distance along the diagonal.

magnetostatic field, when considering TEM waves.

E0<sup>z</sup> exp j ωt � ω

j j γ ¼ ω

results of Eqs. (26)–(27) in Eq. (11) to obtain

zero. From Eq. (12) we obtain:

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vector for this particular case by

� <sup>n</sup><sup>2</sup> � <sup>δ</sup> <sup>þ</sup> <sup>j</sup><sup>3</sup> <sup>δ</sup> � <sup>n</sup><sup>2</sup> ð Þ

! � �

<sup>5</sup>n<sup>2</sup> <sup>j</sup>

! þ k

E ¼ �j i

!

imaginary cube.

238

losses for diagonal propagation.

Faraday rotation for diagonal propagation:

<sup>ϕ</sup><sup>F</sup> <sup>¼</sup> <sup>π</sup> λ0 Re

TEM eigenmodes for diagonal propagation where γ<sup>x</sup> = γ<sup>y</sup> = γz. The trajectory of the electric field vector is represented by red lines.

Figure 7. Longitudinal section of a planar MO waveguide.

#### 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 HDC is applied along the y-axis. The propagation direction is now the z-axis. The planar waveguide supports transversal electric, TE, modes (Hx, Ey, Hz components) and transversal magnetic, TM, modes (Ex, Hy, Ez components). As discussed in Section 2.1.1, if HDC 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 HDC, 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, n<sup>1</sup> > n<sup>2</sup> and n<sup>1</sup> > n3.

Defining ξ as the inverse of the electric permittivity tensor of Eq. (2), we have:

$$
\overline{\overline{\xi}} = \overline{\overline{e}}\_r^{-1} = \begin{bmatrix} n^2 & 0 & -\frac{j\delta}{n^4 - \delta^2} \\ \overline{n^4 - \delta^2} & -\frac{n^2}{n^4 - \delta^2} \\ 0 & \frac{n^2}{n^4 - \delta^2} & 0 \\ \frac{j\delta}{n^4 - \delta^2} & 0 & \frac{n^2}{n^4 - \delta^2} \end{bmatrix} = \begin{bmatrix} \xi\_{xx} & 0 & -\xi\_{xx} \\ 0 & \xi\_{yy} & 0 \\ \xi\_{xx} & 0 & \xi\_{zz} \end{bmatrix}.\tag{32}
$$

From Maxwell's equations at the frequency domain, considering TM modes (Ex, Hy, Ez components) and no field spatial variations along the y-axis, we obtain:

$$
\dot{j}\alpha\mu H\_{\text{y}} = \dot{j}\beta E\_{\text{x}} + \frac{\partial E\_{\text{x}}}{\partial \mathbf{x}},\tag{33}
$$

C ξð Þ<sup>3</sup>

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

> zx � <sup>ξ</sup>ð Þ<sup>1</sup> zx � �<sup>β</sup> � <sup>j</sup>γξð Þ<sup>2</sup>

zx � <sup>ξ</sup>ð Þ<sup>2</sup> zx � �<sup>β</sup> <sup>þ</sup> <sup>j</sup>γξð Þ<sup>2</sup>

tan ð Þ¼ <sup>κ</sup><sup>d</sup> κξð Þ<sup>1</sup>

κξð Þ<sup>1</sup> zz � �<sup>2</sup>

Eq. (39), respectively, in Eq. (36), resulting in

C ξð Þ<sup>1</sup>

Figure 8.

241

<sup>þ</sup> <sup>D</sup> <sup>ξ</sup>ð Þ<sup>2</sup>

zx � <sup>ξ</sup>ð Þ<sup>1</sup> zx � �<sup>β</sup> <sup>þ</sup> <sup>j</sup>ζξð Þ<sup>3</sup>

zz h i cosð Þ� <sup>κ</sup><sup>d</sup> <sup>j</sup>κξð Þ<sup>1</sup>

n o

zz h isenð Þ� <sup>κ</sup><sup>d</sup> <sup>j</sup>κξð Þ<sup>1</sup>

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

zz ζξð Þ<sup>3</sup>

zx � <sup>ξ</sup>ð Þ<sup>1</sup> zx � �<sup>β</sup> <sup>þ</sup> <sup>j</sup>ζξð Þ<sup>3</sup> zz � � <sup>ξ</sup>ð Þ<sup>2</sup>

ζ ¼

κ ¼

γ ¼

where k<sup>0</sup> = 2π/λ0, and λ<sup>0</sup> is the optical wavelength.

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

� <sup>ξ</sup>ð Þ<sup>3</sup>

zz h i <sup>þ</sup> D jκξð Þ<sup>1</sup>

n o <sup>¼</sup> <sup>0</sup>:

zz <sup>þ</sup> γξð Þ<sup>2</sup>

zz � <sup>j</sup> <sup>ξ</sup>ð Þ<sup>3</sup>

zx � �<sup>β</sup> � �

The constants ζ, κ, and γ can be determined by substituting Eq. (37), Eq. (38), or

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ξð Þ<sup>3</sup> xx <sup>β</sup><sup>2</sup> � <sup>k</sup><sup>2</sup> 0

ξð Þ<sup>3</sup> zz <sup>s</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 <sup>0</sup> � <sup>ξ</sup>ð Þ<sup>1</sup> xx β<sup>2</sup> ξð Þ<sup>1</sup> zz <sup>s</sup>

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ξð Þ<sup>2</sup> xx <sup>β</sup><sup>2</sup> � <sup>k</sup><sup>2</sup> 0

ξð Þ<sup>2</sup> zz <sup>s</sup>

From the roots of Eq. (45) for β, the dispersion curve for TM modes in MO waveguides can be retrieved. Assuming that n<sup>1</sup> = 2.26, n<sup>2</sup> = 2.0, n<sup>3</sup> = 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

zz � � <sup>¼</sup> <sup>0</sup>, (43)

zz � � : (45)

, (46)

, (47)

, (48)

(44)

zz senð Þ κd

zx � <sup>ξ</sup>ð Þ<sup>2</sup>

zz cosð Þ κd

zx � <sup>ξ</sup>ð Þ<sup>1</sup> zx � �<sup>β</sup> � <sup>j</sup>γξð Þ<sup>2</sup>

$$E\_{\rm x} = \frac{1}{\varepsilon\_0} \left( \xi\_{\rm xx} \frac{\partial}{\partial \nu} H\_{\rm y} + j \frac{\xi\_{\rm xx}}{\alpha \nu} \frac{\partial H\_{\rm y}}{\partial \mathbf{x}} \right), \tag{34}$$

$$E\_x = \frac{1}{\varepsilon\_0} \left( \xi\_{xx} \frac{\beta}{\alpha} H\_\text{y} - j \frac{\xi\_{xx}}{\alpha} \frac{\partial H\_\text{y}}{\partial \alpha} \right), \tag{35}$$

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 Hy component:

$$\frac{\partial^2 H\_{\mathcal{Y}}}{\partial \mathbf{x}^2} + \left(\frac{k\_0^2 - \xi\_{\text{xx}} \beta^2}{\xi\_{\text{xx}}}\right) H\_{\mathcal{Y}} = \mathbf{0},\tag{36}$$

where k<sup>0</sup> ¼ ω ffiffiffiffiffiffiffi με<sup>0</sup> p .

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

$$H\_{\mathcal{Y}} = \mathbb{C} \exp \left( -\zeta \mathfrak{x} \right), \text{ for } \mathfrak{x} \ge \mathbf{0}. \tag{37}$$

$$H\_{\mathcal{Y}} = \mathcal{C}\cos\left(\kappa\mathbf{x}\right) + D\sin(\kappa\mathbf{x}), \text{ for-d} \le \mathbf{x} \le \mathbf{0}.\tag{38}$$

$$H\_{\mathcal{Y}} = [\mathcal{C}\cos(\kappa d) - D\,\text{sen}(\kappa d)]\,\exp\left[\boldsymbol{\gamma}(\kappa + d)\right], \text{ for } \kappa \le -d.\tag{39}$$

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

$$E\_x = \frac{C}{a\varkappa\_0} \left( \xi\_{\text{xx}}^{(3)} \beta + j\zeta \xi\_{\text{xx}}^{(3)} \right) \exp\left( -\zeta \varkappa \right), \text{ for } \varkappa \ge 0. \tag{40}$$

$$E\_x = \frac{1}{\alpha \varepsilon\_0} \left\{ C \left[ \xi\_{\text{xx}}^{(1)} \beta \cos(\kappa \mathbf{x}) + j \kappa \xi\_{\text{xx}}^{(1)} \text{sen}(\kappa \mathbf{x}) \right] + D \left[ \xi\_{\text{xx}}^{(1)} \beta \text{sen}(\kappa \mathbf{x}) - j \kappa \xi\_{\text{xx}}^{(1)} \cos(\kappa \mathbf{x}) \right] \right\}, \text{ for } -\mathbf{d} \le x \le 0. \tag{41}$$

$$E\_x = \frac{C\cos\left(\kappa d\right) - D\text{sen}(\kappa d)}{\alpha \varkappa\_0} \left(\xi\_{xx}^{(2)}\beta - j\gamma \xi\_{xx}^{(2)}\right) \exp\left[\wp(\varkappa + d)\right], \text{for } x \le -\text{d.}\tag{42}$$

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

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

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

Defining ξ as the inverse of the electric permittivity tensor of Eq. (2), we have:

<sup>n</sup><sup>4</sup> � <sup>δ</sup><sup>2</sup>

¼

∂Ez

ξzx ω

ξzz ω

� �

� �

∂Hy ∂x

∂Hy ∂x

Hy ¼ C exp ð Þ �ζx , for x≥ 0: (37)

Hy ¼ Ccosð Þþ κx D senð Þ κx , for–d≤x≤0: (38)

zx <sup>β</sup>senð Þ� <sup>κ</sup><sup>x</sup> <sup>j</sup>κξð Þ<sup>1</sup>

Hy ¼ ½ � Ccosð Þ� κd D senð Þ κd exp ½ � γð Þ x þ d , for x≤ � d: (39)

The solution for the component Ez at each layer is obtained by substituting the

<sup>þ</sup> <sup>D</sup> <sup>ξ</sup>ð Þ<sup>1</sup>

n o h i

zx <sup>β</sup> � <sup>j</sup>γξð Þ<sup>2</sup> zz

The superscripts between parentheses on the inverse permittivity tensor elements identify the corresponding waveguide layer, as specified in Figure 7. The

� �

2 6 4

ξxx 0 �ξzx 0 ξyy 0 ξzx 0 ξzz

<sup>∂</sup><sup>x</sup> , (33)

Hy ¼ 0, (36)

exp ð Þ �ζx ,for x≥ 0: (40)

zz cosð Þ κx

exp ½ � γð Þ x þ d ,for x≤ � d: (42)

,for–d ≤x≤0:

(41)

, (34)

, (35)

3 7

<sup>5</sup>: (32)

<sup>n</sup><sup>4</sup> � <sup>δ</sup><sup>2</sup>

Hy, Ez components) and no field spatial variations along the y-axis, we obtain:

jωμHy ¼ jβEx þ

ξxx β <sup>ω</sup> Hy <sup>þ</sup> <sup>j</sup>

ξzx β <sup>ω</sup> Hy � <sup>j</sup>

k2 <sup>0</sup> � <sup>ξ</sup>xxβ<sup>2</sup> ξzz !

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

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

Substituting Eqs. (34)–(35) in Eq. (33), we obtain the following wave equation

From Maxwell's equations at the frequency domain, considering TM modes (Ex,

structure shown in Figure 7, n<sup>1</sup> > n<sup>2</sup> and n<sup>1</sup> > n3.

<sup>n</sup><sup>4</sup> � <sup>δ</sup><sup>2</sup> <sup>0</sup> � <sup>j</sup><sup>δ</sup>

<sup>n</sup><sup>4</sup> � <sup>δ</sup><sup>2</sup> <sup>0</sup> <sup>n</sup><sup>2</sup>

Ex <sup>¼</sup> <sup>1</sup> ε0

Ez <sup>¼</sup> <sup>1</sup> ε0

for nonreciprocal media in terms of the Hy component:

∂2 Hy ∂x<sup>2</sup> þ

corresponding solution for Hy in Eq. (35), resulting in.

ξð Þ<sup>3</sup>

zz senð Þ κx

ξð Þ<sup>2</sup>

continuity of Ez at x = 0 and at x = �d leads to the following system:

zx <sup>β</sup> <sup>þ</sup> <sup>j</sup>ζξð Þ<sup>3</sup> zz

� �

Ez <sup>¼</sup> <sup>C</sup> ωε<sup>0</sup>

h i

zx <sup>β</sup> cosð Þþ <sup>κ</sup><sup>x</sup> <sup>j</sup>κξð Þ<sup>1</sup>

Ez <sup>¼</sup> <sup>C</sup>cosð Þ� <sup>κ</sup><sup>d</sup> <sup>D</sup>senð Þ <sup>κ</sup><sup>d</sup> ωε<sup>0</sup>

<sup>n</sup><sup>4</sup> � <sup>δ</sup><sup>2</sup> <sup>0</sup>

<sup>0</sup> <sup>n</sup><sup>2</sup>

n2

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jδ

ξ ¼ ε �1 r ¼

meter.

Ez <sup>¼</sup> <sup>1</sup> ωε<sup>0</sup>

240

where k<sup>0</sup> ¼ ω ffiffiffiffiffiffiffi

C ξð Þ<sup>1</sup>

με<sup>0</sup> p .

$$\mathbf{C}\left[\left(\xi\_{\mathbf{x}\mathbf{x}}^{(3)} - \xi\_{\mathbf{x}\mathbf{x}}^{(1)}\right)\boldsymbol{\upbeta} + \mathbf{j}\zeta\xi\_{\mathbf{x}\mathbf{x}}^{(3)}\right] + D\left(\mathbf{j}\kappa\xi\_{\mathbf{x}\mathbf{x}}^{(1)}\right) = \mathbf{0},\tag{43}$$

$$\begin{split} &C\left\{ \left[ \left( \xi\_{\rm{xx}}^{(1)} - \xi\_{\rm{xx}}^{(2)} \right) \beta + j\chi \xi\_{\rm{xx}}^{(2)} \right] \cos\left(\kappa d\right) - j\kappa \xi\_{\rm{xx}}^{(1)} \sin(\kappa d) \right\} \\ &+ D\left\{ \left[ \left( \xi\_{\rm{xx}}^{(2)} - \xi\_{\rm{xx}}^{(1)} \right) \beta - j\chi \xi\_{\rm{xx}}^{(2)} \right] \sin(\kappa d) - j\kappa \xi\_{\rm{xx}}^{(1)} \cos(\kappa d) \right\} = 0. \end{split} \tag{44}$$

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

$$\tan\left(\varkappa d\right) = \frac{\kappa \xi\_{\text{xx}}^{(1)} \left[\zeta \xi\_{\text{xx}}^{(3)} + \chi \xi\_{\text{xx}}^{(2)} - j \left(\xi\_{\text{xx}}^{(3)} - \xi\_{\text{xx}}^{(2)}\right) \beta\right]}{\left(\kappa \xi\_{\text{xx}}^{(1)}\right)^2 - \left[\left(\xi\_{\text{xx}}^{(3)} - \xi\_{\text{xx}}^{(1)}\right) \beta + j \zeta \xi\_{\text{xx}}^{(3)}\right] \left[\left(\xi\_{\text{xx}}^{(2)} - \xi\_{\text{xx}}^{(1)}\right) \beta - j \chi \xi\_{\text{xx}}^{(2)}\right]}.\tag{45}$$

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

$$
\zeta = \sqrt{\frac{\xi\_{\text{xx}}^{(3)} \rho^2 - k\_0^2}{\xi\_{\text{xx}}^{(3)}}},\tag{46}
$$

$$\kappa = \sqrt{\frac{k\_0^2 - \xi\_{\text{xx}}^{(1)} \rho^2}{\xi\_{\text{xx}}^{(1)}}},\tag{47}$$

$$\gamma = \sqrt{\frac{\xi\_{\text{xx}}^{(2)} \beta^2 - k\_0^2}{\xi\_{xx}^{(2)}}} \tag{48}$$

where k<sup>0</sup> = 2π/λ0, and λ<sup>0</sup> 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 n<sup>1</sup> = 2.26, n<sup>2</sup> = 2.0, n<sup>3</sup> = 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

Figure 8. Dispersion curves of the fundamental TM0 mode and the superior TM1 mode.

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.

The solutions for Eq. (36) in each layer, making use of the proper radiation

Hy ¼ A<sup>2</sup> cos½κ2ð Þ x � S<sup>3</sup> � d2=2 � þ A<sup>3</sup> sin ½ � κ2ð Þ x � S<sup>3</sup> � d2=2 for S<sup>3</sup> ≤x≤S<sup>3</sup> þ d2,

Hy ¼ A<sup>6</sup> cos½κ4ð Þ x þ S<sup>3</sup> þ d4=2 � þ A<sup>7</sup> sin ½ � κ4ð Þ x þ S<sup>3</sup> þ d4=2 , for –S3–d<sup>4</sup> ≤x≤–S3,

where A1 through A8 are constants to be determined, κ<sup>i</sup> and γ<sup>j</sup> are given by.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 <sup>0</sup> � <sup>ξ</sup>ð Þ<sup>i</sup> xxβ<sup>2</sup> ξð Þ<sup>i</sup> zz <sup>s</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ξð Þ<sup>j</sup> xx <sup>β</sup><sup>2</sup> � <sup>k</sup><sup>2</sup> 0

ξð Þ<sup>j</sup> zz <sup>s</sup>

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

Here, [M(β)] is an 8�8 matrix that depends on the unknown longitudinal

easily found by solving the equation Det([M(β)]) = 0. The nonzero elements of the

zx <sup>β</sup> cosð Þ� <sup>κ</sup>2d2=<sup>2</sup> <sup>κ</sup>2ξð Þ<sup>2</sup>

zx <sup>β</sup> <sup>þ</sup> <sup>ξ</sup>ð Þ<sup>3</sup> zz γ<sup>3</sup> � � exp <sup>γ</sup>ð Þ <sup>3</sup>S<sup>3</sup> ;

zx <sup>β</sup> <sup>þ</sup> <sup>ξ</sup>ð Þ<sup>3</sup> zz γ<sup>3</sup> � � exp �<sup>γ</sup> ½ � <sup>3</sup>S<sup>3</sup> ;

zz cosð Þ k2d2=2 ; M<sup>32</sup> ¼ cosð Þ κ2d2=2 ; M<sup>33</sup> ¼ � sin ð Þ κ2d2=2 ; M<sup>34</sup> ¼ � exp �γ ð Þ <sup>3</sup>S<sup>3</sup> ; M<sup>35</sup> ¼ � exp γð Þ <sup>3</sup>S<sup>3</sup> ;

zz sin ð Þ κ2d2=2 ;

M<sup>54</sup> ¼ exp γ½ � <sup>3</sup>S<sup>3</sup> ; M<sup>55</sup> ¼ exp �γ ½ � <sup>3</sup>S<sup>3</sup> ; M<sup>56</sup> ¼ � cosð Þ κ4d4=2 ; M<sup>57</sup> ¼ � sin ð Þ κ4d4=2 ;

zz cosð Þ κ2d2=2 ;

zz κ<sup>4</sup> sin ð Þ κ4d4=2 ;

zz κ<sup>4</sup> cosð Þ κ4d4=2 ;

zz κ<sup>4</sup> cosð Þ κ4d4=2 ;

zz κ<sup>4</sup> sin ð Þ κ4d4=2 ;

T

, i ¼ 2, 4, (49)

, j ¼ 1, 3, 5, (50)

. The propagation constant can be

zz sin ð Þ κ2d2=2 ;

½ � Mð Þ β A ¼ 0: (51)

conditions, are [18]

Hy ¼ A<sup>1</sup> exp �γ ½ � <sup>1</sup>ð Þ x � S<sup>3</sup> � d<sup>2</sup> , for x≥S<sup>3</sup> þ d2,

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

Hy ¼ A<sup>8</sup> exp γ½ � <sup>5</sup>ð Þ x þ S<sup>3</sup> þ d<sup>4</sup> , for x≤–S3–d4,

Hy ¼ A<sup>4</sup> exp �γ ð Þþ <sup>3</sup>x A<sup>5</sup> exp γð Þ <sup>3</sup>x , for –S<sup>3</sup> ≤x≤S3,

κ<sup>i</sup> ¼

γ<sup>j</sup> ¼

where k<sup>0</sup> = 2π/λ0, and λ<sup>0</sup> is the optical wavelength.

propagation constant β and A = [A<sup>1</sup> A<sup>2</sup> … A8]

zx <sup>β</sup> sin ð Þþ <sup>k</sup>2d2=<sup>2</sup> <sup>k</sup>2ξð Þ<sup>2</sup>

zx <sup>β</sup> cosð Þ� <sup>κ</sup>2d2=<sup>2</sup> <sup>κ</sup>2ξð Þ<sup>2</sup>

zx <sup>β</sup> sin ð Þ� <sup>κ</sup>2d2=<sup>2</sup> <sup>κ</sup>2ξð Þ<sup>2</sup>

zx <sup>β</sup> cosð Þ� <sup>κ</sup>4d4=<sup>2</sup> <sup>ξ</sup>ð Þ <sup>4</sup>

zx <sup>β</sup> sin ð Þþ <sup>κ</sup>4d4=<sup>2</sup> <sup>ξ</sup>ð Þ <sup>4</sup>

zx <sup>β</sup> sin ð Þ� <sup>κ</sup>4d4=<sup>2</sup> <sup>ξ</sup>ð Þ <sup>4</sup>

zz γ5;

zx <sup>β</sup> <sup>þ</sup> <sup>ξ</sup>ð Þ<sup>5</sup>

zx <sup>β</sup> cosð Þ� <sup>κ</sup>4d4=<sup>2</sup> <sup>ξ</sup>ð Þ <sup>4</sup>

M<sup>11</sup> ¼ 1; M<sup>12</sup> ¼ � cosð Þ κ2d2=2 ; M<sup>13</sup> ¼ � sin ð Þ κ2d2=2 ;

zz ; M<sup>22</sup> <sup>¼</sup> <sup>j</sup>ξð Þ<sup>2</sup>

� � exp �<sup>γ</sup> ð Þ <sup>3</sup>S<sup>3</sup> ; M<sup>45</sup> <sup>¼</sup> <sup>j</sup>ξð Þ<sup>3</sup>

� � exp <sup>γ</sup>½ � <sup>3</sup>S<sup>3</sup> ; M<sup>65</sup> ¼ � <sup>j</sup>ξð Þ<sup>3</sup>

M<sup>76</sup> ¼ cosð Þ κ4d4=2 ; M<sup>77</sup> ¼ � sin ð Þ κ4d4=2 ; M<sup>78</sup> ¼ �1;

matrix [M(β)] are listed below:

zx <sup>β</sup> <sup>þ</sup> <sup>γ</sup>1ξð Þ<sup>1</sup>

zx <sup>β</sup> � <sup>ξ</sup>ð Þ<sup>3</sup> zz γ<sup>3</sup>

> zx <sup>β</sup> <sup>þ</sup> <sup>ξ</sup>ð Þ<sup>3</sup> zz γ<sup>3</sup>

<sup>M</sup><sup>21</sup> ¼ �jξð Þ<sup>1</sup>

<sup>M</sup><sup>23</sup> ¼ �jξð Þ<sup>2</sup>

<sup>M</sup><sup>42</sup> ¼ �jξð Þ<sup>2</sup>

<sup>M</sup><sup>43</sup> <sup>¼</sup> <sup>j</sup>ξð Þ<sup>2</sup>

<sup>M</sup><sup>44</sup> <sup>¼</sup> <sup>j</sup>ξð Þ<sup>3</sup>

<sup>M</sup><sup>64</sup> ¼ � <sup>j</sup>ξð Þ<sup>3</sup>

<sup>M</sup><sup>66</sup> <sup>¼</sup> <sup>j</sup>ξð Þ <sup>4</sup>

<sup>M</sup><sup>67</sup> <sup>¼</sup> <sup>j</sup>ξð Þ <sup>4</sup>

<sup>M</sup><sup>86</sup> ¼ �jξð Þ <sup>4</sup>

<sup>M</sup><sup>87</sup> <sup>¼</sup> <sup>j</sup>ξð Þ <sup>4</sup>

<sup>M</sup><sup>88</sup> <sup>¼</sup> <sup>j</sup>ξð Þ<sup>5</sup>

243

Figure 9 shows the transversal distributions of the Hy component at two distinct optical wavelengths. For this waveguide design, λ<sup>0</sup> = 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.

#### 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 9.

Transversal distribution of the Hy component of the fundamental TM0 mode at λ<sup>0</sup> = 1.31 μm and at λ<sup>0</sup> = 1.55 μm.

Figure 10. Longitudinal section of the five-layered MO planar structure.

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,

Figure 9 shows the transversal distributions of the Hy component at two distinct optical wavelengths. For this waveguide design, λ<sup>0</sup> = 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

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

nonreciprocal phase shift effect, considering this waveguide configuration.

Transversal distribution of the Hy component of the fundamental TM0 mode at λ<sup>0</sup> = 1.31 μm and at

2.3 TM mode in a planar magneto-optical directional coupler

Figure 9.

λ<sup>0</sup> = 1.55 μm.

Figure 10.

242

Longitudinal section of the five-layered MO planar structure.

the mode becomes irradiated and escapes through layer 3.

Electromagnetic Materials and Devices

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

$$\begin{aligned} H\_{\mathcal{Y}} &= A\_1 \exp\left[-\gamma\_1 (\mathbf{x} - \mathbf{S}\_3 - d\_2)\right], \text{ for } \mathbf{x} \ge \mathbf{S}\_3 + d\_2, \\ H\_{\mathcal{Y}} &= A\_2 \cos\left[\kappa\_2 (\mathbf{x} - \mathbf{S}\_3 - d\_2/2)\right] + A\_3 \sin\left[\kappa\_2 (\mathbf{x} - \mathbf{S}\_3 - d\_2/2)\right] \text{ for } \mathbf{S}\_3 \le \mathbf{x} \le \mathbf{S}\_3 + d\_2, \\ H\_{\mathcal{Y}} &= A\_4 \exp\left(-\gamma\_3 \mathbf{x}\right) + A\_5 \exp\left(\gamma\_3 \mathbf{x}\right), \text{ for } -\mathbf{S}\_3 \le \mathbf{x} \le \mathbf{S}\_3, \\ H\_{\mathcal{Y}} &= A\_6 \cos\left[\kappa\_4 (\mathbf{x} + \mathbf{S}\_3 + d\_4/2)\right] + A\_7 \sin\left[\kappa\_4 (\mathbf{x} + \mathbf{S}\_3 + d\_4/2)\right], \text{ for } -\mathbf{S}\_3 - d\_4 \le \mathbf{x} \le -\mathbf{S}\_3, \\ H\_{\mathcal{Y}} &= A\_8 \exp\left[\gamma\_5 (\mathbf{x} + \mathbf{S}\_3 + d\_4)\right], \text{ for } \mathbf{x} \le -\mathbf{S}\_3 - d\_4. \end{aligned}$$

where A1 through A8 are constants to be determined, κ<sup>i</sup> and γ<sup>j</sup> are given by.

$$\kappa\_i = \sqrt{\frac{k\_0^2 - \xi\_{xx}^{(i)} \beta^2}{\xi\_{xx}^{(i)}}}, i = 2, 4,\tag{49}$$

$$\gamma\_j = \sqrt{\frac{\xi\_{\text{xx}}^{(j)} \beta^2 - k\_0^2}{\xi\_{\text{xx}}^{(j)}}}, j = 1, 3, 5,\tag{50}$$

where k<sup>0</sup> = 2π/λ0, and λ<sup>0</sup> is the optical wavelength.

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

$$[\mathbf{M}(\boldsymbol{\beta})] \mathbf{A} = \mathbf{0}.\tag{51}$$

Here, [M(β)] is an 8�8 matrix that depends on the unknown longitudinal propagation constant β and A = [A<sup>1</sup> A<sup>2</sup> … 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:

M<sup>11</sup> ¼ 1; M<sup>12</sup> ¼ � cosð Þ κ2d2=2 ; M<sup>13</sup> ¼ � sin ð Þ κ2d2=2 ; <sup>M</sup><sup>21</sup> ¼ �jξð Þ<sup>1</sup> zx <sup>β</sup> <sup>þ</sup> <sup>γ</sup>1ξð Þ<sup>1</sup> zz ; M<sup>22</sup> <sup>¼</sup> <sup>j</sup>ξð Þ<sup>2</sup> zx <sup>β</sup> cosð Þ� <sup>κ</sup>2d2=<sup>2</sup> <sup>κ</sup>2ξð Þ<sup>2</sup> zz sin ð Þ κ2d2=2 ; <sup>M</sup><sup>23</sup> ¼ �jξð Þ<sup>2</sup> zx <sup>β</sup> sin ð Þþ <sup>k</sup>2d2=<sup>2</sup> <sup>k</sup>2ξð Þ<sup>2</sup> zz cosð Þ k2d2=2 ; M<sup>32</sup> ¼ cosð Þ κ2d2=2 ; M<sup>33</sup> ¼ � sin ð Þ κ2d2=2 ; M<sup>34</sup> ¼ � exp �γ ð Þ <sup>3</sup>S<sup>3</sup> ; M<sup>35</sup> ¼ � exp γð Þ <sup>3</sup>S<sup>3</sup> ; <sup>M</sup><sup>42</sup> ¼ �jξð Þ<sup>2</sup> zx <sup>β</sup> cosð Þ� <sup>κ</sup>2d2=<sup>2</sup> <sup>κ</sup>2ξð Þ<sup>2</sup> zz sin ð Þ κ2d2=2 ; <sup>M</sup><sup>43</sup> <sup>¼</sup> <sup>j</sup>ξð Þ<sup>2</sup> zx <sup>β</sup> sin ð Þ� <sup>κ</sup>2d2=<sup>2</sup> <sup>κ</sup>2ξð Þ<sup>2</sup> zz cosð Þ κ2d2=2 ; <sup>M</sup><sup>44</sup> <sup>¼</sup> <sup>j</sup>ξð Þ<sup>3</sup> zx <sup>β</sup> � <sup>ξ</sup>ð Þ<sup>3</sup> zz γ<sup>3</sup> � � exp �<sup>γ</sup> ð Þ <sup>3</sup>S<sup>3</sup> ; M<sup>45</sup> <sup>¼</sup> <sup>j</sup>ξð Þ<sup>3</sup> zx <sup>β</sup> <sup>þ</sup> <sup>ξ</sup>ð Þ<sup>3</sup> zz γ<sup>3</sup> � � exp <sup>γ</sup>ð Þ <sup>3</sup>S<sup>3</sup> ; M<sup>54</sup> ¼ exp γ½ � <sup>3</sup>S<sup>3</sup> ; M<sup>55</sup> ¼ exp �γ ½ � <sup>3</sup>S<sup>3</sup> ; M<sup>56</sup> ¼ � cosð Þ κ4d4=2 ; M<sup>57</sup> ¼ � sin ð Þ κ4d4=2 ; <sup>M</sup><sup>64</sup> ¼ � <sup>j</sup>ξð Þ<sup>3</sup> zx <sup>β</sup> <sup>þ</sup> <sup>ξ</sup>ð Þ<sup>3</sup> zz γ<sup>3</sup> � � exp <sup>γ</sup>½ � <sup>3</sup>S<sup>3</sup> ; M<sup>65</sup> ¼ � <sup>j</sup>ξð Þ<sup>3</sup> zx <sup>β</sup> <sup>þ</sup> <sup>ξ</sup>ð Þ<sup>3</sup> zz γ<sup>3</sup> � � exp �<sup>γ</sup> ½ � <sup>3</sup>S<sup>3</sup> ; <sup>M</sup><sup>66</sup> <sup>¼</sup> <sup>j</sup>ξð Þ <sup>4</sup> zx <sup>β</sup> cosð Þ� <sup>κ</sup>4d4=<sup>2</sup> <sup>ξ</sup>ð Þ <sup>4</sup> zz κ<sup>4</sup> sin ð Þ κ4d4=2 ; <sup>M</sup><sup>67</sup> <sup>¼</sup> <sup>j</sup>ξð Þ <sup>4</sup> zx <sup>β</sup> sin ð Þþ <sup>κ</sup>4d4=<sup>2</sup> <sup>ξ</sup>ð Þ <sup>4</sup> zz κ<sup>4</sup> cosð Þ κ4d4=2 ; M<sup>76</sup> ¼ cosð Þ κ4d4=2 ; M<sup>77</sup> ¼ � sin ð Þ κ4d4=2 ; M<sup>78</sup> ¼ �1; <sup>M</sup><sup>86</sup> ¼ �jξð Þ <sup>4</sup> zx <sup>β</sup> cosð Þ� <sup>κ</sup>4d4=<sup>2</sup> <sup>ξ</sup>ð Þ <sup>4</sup> zz κ<sup>4</sup> sin ð Þ κ4d4=2 ; <sup>M</sup><sup>87</sup> <sup>¼</sup> <sup>j</sup>ξð Þ <sup>4</sup> zx <sup>β</sup> sin ð Þ� <sup>κ</sup>4d4=<sup>2</sup> <sup>ξ</sup>ð Þ <sup>4</sup> zz κ<sup>4</sup> cosð Þ κ4d4=2 ; <sup>M</sup><sup>88</sup> <sup>¼</sup> <sup>j</sup>ξð Þ<sup>5</sup> zx <sup>β</sup> <sup>þ</sup> <sup>ξ</sup>ð Þ<sup>5</sup> zz γ5;

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 λ<sup>0</sup> = 1.55 μm.

Considering both propagation senses, when the condition L ¼ L<sup>þ</sup>

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

projection of a linear combination of the corresponding supermodes.

Transversal distribution of the supermodes (Hy component) for backward propagation (�z).

Operation of an optical isolator based on nonreciprocal phase shift.

coupler is L ¼ L<sup>þ</sup>

and 2 to 1.

Figure 12.

Figure 13.

245

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

<sup>π</sup> . If some light is reflected at port 3, since L ¼ 2L�

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

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;

<sup>π</sup> ¼ 2L�

<sup>π</sup> , all optical

<sup>π</sup> for the

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<sup>π</sup> = π/|β<sup>1</sup> – β2|, where β<sup>1</sup> and β<sup>2</sup> 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 Lþ <sup>π</sup> ¼ 1389:84 μm.

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� <sup>π</sup> ¼ 689μm.


Table 1. Material and geometric parameters of the MO directional coupler.

Figure 11. Transversal distribution of the supermodes (Hy component) for forward propagation (+z).

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

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

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<sup>π</sup> = π/|β<sup>1</sup> – β2|, where β<sup>1</sup> and β<sup>2</sup> 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

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

 2.23 �0.019 ∞ 2.26 0 1.20 2.00 0 0.75 2.26 0 1.23 2.23 �0.019 ∞

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

<sup>π</sup> ¼ 689μm.

n δ Thickness (μm)

infinite for the analytical model. The optical wavelength is λ<sup>0</sup> = 1.55 μm.

Lþ

Table 1.

Figure 11.

244

<sup>π</sup> ¼ 1389:84 μm.

Electromagnetic Materials and Devices

the backward propagation along the z-axis, is L�

Material and geometric parameters of the MO directional coupler.

Layer Parameters

Considering both propagation senses, when the condition L ¼ L<sup>þ</sup> <sup>π</sup> ¼ 2L� <sup>π</sup> 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 ¼ L<sup>þ</sup> <sup>π</sup> . If some light is reflected at port 3, since L ¼ 2L� <sup>π</sup> , 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.

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.

Figure 12. Transversal distribution of the supermodes (Hy component) for backward propagation (�z).

Figure 13. Operation of an optical isolator based on nonreciprocal phase shift.

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

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

Cyberspatial Institute, Federal Rural University of Amazon, Belém, Brazil

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: licinius@ufra.edu.br

provided the original work is properly cited.

circulators.

Author details

247

Licinius Dimitri Sá de Alcantara

#### Figure 14.

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

Figure 15.

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

#### 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

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

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.
