Open access peer-reviewed chapter - ONLINE FIRST

Optical Propagation in Magneto-Optical Materials

By Licinius Dimitri Sá de Alcantara

Submitted: July 15th 2018Reviewed: October 11th 2018Published: November 9th 2018

DOI: 10.5772/intechopen.81963

Downloaded: 170

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.

Keywords

  • magneto-optical media
  • light propagation
  • Faraday rotation
  • nonreciprocal phase shift
  • 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 (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, 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 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

ε¯¯r=n2000n20n2,forHDCxaxis;E1
ε¯¯r=n200n200n2,forHDCyaxis;E2
ε¯¯r=n20n2000n2,forHDCzaxis.E3

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.

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 E¯xyzcan be written as

ω2μ0ε0ε¯¯rE¯+2E¯E¯=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 HDC is parallel to the y-axis and ε¯¯ris 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

E¯=E¯0expjωtexpjγxx+γyy+γzz.E5

where γ¯=γxi+γyj+γzkis the propagation constant vector.

From Gauss’ law for a medium with equilibrium of charges, .ε0ε¯¯rE¯=0, we obtain:

E¯=jδn2ExzEzx.E6

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

ω2μ0ε0ε¯¯rE¯+2E¯jδn2ExzEzx=0¯.E7

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

ω2μ0ε0n2Ex+Ez+2Exx2+2Exy2+2Exz2jδn22Exxz2Ezx2=0,E8
ω2μ0ε0n2Ey+2Eyx2+2Eyy2+2Eyz2jδn22Exyz2Ezxy=0,E9
ω2μ0ε0Ex+n2Ez+2Ezx2+2Ezy2+2Ezz2jδn22Exz22Ezxz=0.E10

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

ω2μ0ε0n2γ2jδn2γxγzEx+ω2μ0ε01n2γx2Ez=0E11
ω2μ0ε0n2γ2Ey+jδn2γyγzExγxγyEz=0,E12
ω2μ0ε01n2γz2Ex+ω2μ0ε0n2γ2jδn2γxγzEz=0,E13

where γ=γx2+γy2+γz2.

2.1.1. TEM wave with electric field vector parallel to HDC

By observing Eqs. (1113), 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

γ=μ0ε0,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 γy = 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

ω2μ0ε0n2γ2jδn2γxγz2δ2ω2μ0ε01n2γx2ω2μ0ε01n2γz2=0.E15

Solving Eq. (15) for |γ|, we obtain:

γ=ω2μ0ε0n2jδn2γxγz±δω2μ0ε01n2γx2ω2μ0ε01n2γz2.E16

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

The parameters γx and γz 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 γx = γz = 0, Eq. (16) assumes the simpler form:

γ=ωμ0ε0n2±δ,E17

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

E¯=E0xi+E0zkexpjωtμ0ε0n2±δ.E18

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

E0x=ω2μ0ε0ω2μ0ε0n2γ2E0z.E19

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

E0x=±jE0z.E20

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

Ex=E0zexpjωtμ0ε0n2±δ±π/2,E21
Ez=E0zexpjωtμ0ε0n2±δ.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+=n2+δ,for the CCW circular polarized eigenmode;

  2. n=n2δ,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 HDC.

When the sense of the magnetostatic field HDC 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 (ϕF) may be calculated (in radians) as a function of the propagation distance y by

ϕF=12ϕ+ϕ=12n+2πλ0yn2πλ0y=πλ0n2+δn2δy,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, 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 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 ϕF = 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 γx = γy = γz = γu, with γu ≠ 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:

γ=ω2μ0ε0n2jδn2γu2±δω2μ0ε01n2γu2.E24

From the relation γ=γx2+γy2+γz2we can also obtain:

γ=γu3.E25

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

γu=ωμ0ε0n2±δ3±δn2+jδn2.E26

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

γ=ωμ0ε0n2±δ1±δ3n2+jδ3n2.E27

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

Ex=±jEz.E28

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

Ey=n2±δ+j3δ±n25n2Ez.E29

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

E¯=±jin2±δ+j3δ±n25n2j+kE0zexpjωtωμ0ε0n2±δ3±δn2+jδn2x+y+z,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 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.

Figure 6.

TEM eigenmodes for diagonal propagation where γx = γy = γz. 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:

ϕF=πλ0Ren2+δ1+δ3n2+jδ3n2n2δ1δ3n2+jδ3n2d,E31

where d is the propagation distance along the diagonal.

For n = 2, δ = 0.2, and λ0 = 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.

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, n1 > n2 and n1 > n3.

Figure 7.

Longitudinal section of a planar MO waveguide.

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

ξ¯¯=ε¯¯r1=n2n4δ20n4δ20n2n4δ20n4δ20n2n4δ2=ξxx0ξzx0ξyy0ξzx0ξzz.E32

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:

jωμHy=Ex+Ezx,E33
Ex=1ε0ξxxβωHy+jξzxωHyx,E34
Ez=1ε0ξzxβωHyjξzzωHyx,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 Hy component:

2Hyx2+k02ξxxβ2ξzzHy=0,E36

where k0=ωμε0.

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

Hy=Cexpζx,forx0.E37
Hy=Ccosκx+Dsenκx,fordx0.E38
Hy=CcosκdDsenκdexpγx+d,forxd.E39

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

Ez=Cωε0ξzx3β+jζξzz3expζx,forx0.E40
Ez=1ωε0Cξzx1βcosκx+jκξzz1senκx+Dξzx1βsenκxjκξzz1cosκx,fordx0.E41
Ez=CcosκdDsenκdωε0ξzx2βjγξzz2expγx+d,forxd.E42

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:

Cξzx3ξzx1β+jζξzz3+Djκξzz1=0,E43
Cξzx1ξzx2β+jγξzz2cosκdjκξzz1senκd+Dξzx2ξzx1βjγξzz2senκdjκξzz1cosκd=0.E44

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

tanκd=κξzz1ζξzz3+γξzz2jξzx3ξzx2βκξzz12ξzx3ξzx1β+jζξzz3ξzx2ξzx1βjγξzz2.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

ζ=ξxx3β2k02ξzz3,E46
κ=k02ξxx1β2ξzz1,E47
γ=ξxx2β2k02ξzz2,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 Hy 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 Hy 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]

Hy=A1expγ1xS3d2,forxS3+d2,Hy=A2cosκ2xS3d2/2+A3sinκ2xS3d2/2forS3xS3+d2,Hy=A4expγ3x+A5expγ3x,forS3xS3,Hy=A6cosκ4x+S3+d4/2+A7sinκ4x+S3+d4/2,forS3d4xS3,Hy=A8expγ5x+S3+d4,forxS3d4,

where A1 through A8 are constants to be determined, κi and γj are given by.

κi=k02ξxxiβ2ξzzi,i=2,4,E49
γj=ξxxjβ2k02ξzzj,j=1,3,5,E50

where k0 = 2π/λ0, and λ0 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:

MβA=0.E51

Here, [M(β)] is an 8×8 matrix that depends on the unknown longitudinal propagation constant β and A = [A1 A2A8]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:

M11=1;M12=cosκ2d2/2;M13=sinκ2d2/2;M21=jξzx1β+γ1ξzz1;M22=jξzx2βcosκ2d2/2κ2ξzz2sinκ2d2/2;M23=jξzx2βsink2d2/2+k2ξzz2cosk2d2/2;M32=cosκ2d2/2;M33=sinκ2d2/2;M34=expγ3S3;M35=expγ3S3;M42=jξzx2βcosκ2d2/2κ2ξzz2sinκ2d2/2;M43=jξzx2βsinκ2d2/2κ2ξzz2cosκ2d2/2;M44=jξzx3βξzz3γ3expγ3S3;M45=jξzx3β+ξzz3γ3expγ3S3;M54=expγ3S3;M55=expγ3S3;M56=cosκ4d4/2;M57=sinκ4d4/2;M64=jξzx3β+ξzz3γ3expγ3S3;M65=jξzx3β+ξzz3γ3expγ3S3;M66=jξzx4βcosκ4d4/2ξzz4κ4sinκ4d4/2;M67=jξzx4βsinκ4d4/2+ξzz4κ4cosκ4d4/2;M76=cosκ4d4/2;M77=sinκ4d4/2;M78=1;M86=jξzx4βcosκ4d4/2ξzz4κ4sinκ4d4/2;M87=jξzx4βsinκ4d4/2ξzz4κ4cosκ4d4/2;M88=jξzx5β+ξzz5γ5;

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.

LayerParameters
nδThickness (μm)
12.23−0.019
22.2601.20
32.0000.75
42.2601.23
52.23−0.019

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 Lπ = π/|β1β2|, 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 Lπ+=1389.84μm.

Figure 11.

Transversal distribution of the supermodes (Hy 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 Lπ=689μm.

Figure 12.

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

Considering both propagation senses, when the condition L=Lπ+=2Lπfor 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=Lπ+. If some light is reflected at port 3, since L=2Lπ, 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 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.

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.

Download

chapter PDF

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite and reference

Link to this chapter Copy to clipboard

Cite this chapter Copy to clipboard

Licinius Dimitri Sá de Alcantara (November 9th 2018). Optical Propagation in Magneto-Optical Materials [Online First], IntechOpen, DOI: 10.5772/intechopen.81963. Available from:

chapter statistics

170total chapter downloads

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Access personal reporting

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

More About Us