Material and geometric parameters of the MO directional coupler.
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.
- magneto-optical media
- light propagation
- Faraday rotation
- nonreciprocal phase shift
- optical devices
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 .
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 (
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
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 can be written as
To develop a plane wave solution for MO media, it is assumed that
where is the propagation constant vector.
From Gauss’ law for a medium with equilibrium of charges, , we obtain:
Expanding Eq. (7) in the Cartesian coordinates results in
2.1.1 TEM wave with electric field vector parallel to HDC
By observing Eqs. (11–13), we note that
which is the same expression for a traveling wave in an isotropic material. Note that when the electric field is polarized along the
2.1.2 The general expression for the propagation constant
Solving Eq. (15) for |
2.1.3 TEM wave propagating parallel to HDC
and from Eq. (5), the electric field vector becomes
From Eq. (11), we see that the electric field components are connected by
Eqs. (21) and (22) represent a circular polarized wave, which can be dismembered into two circular polarized eigenmodes propagating along the
for the CCW circular polarized eigenmode;
for the CW circular polarized eigenmode.
A linear polarized wave propagating along the
When the sense of the magnetostatic field HDC is reversed, the magneto-optical constant δ changes its signal, and the values of
The Faraday rotation angle (
When a MO waveguide, with
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) . We observe that the length for maximum energy transfer is around 6800 μm. In practice, as observed in , 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
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
From the relation we can also obtain:
However, for the considered propagation direction, the
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
The simulations presented in Figure 6 were performed for
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:
For n = 2,
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
Defining as the inverse of the electric permittivity tensor of Eq. (2), we have:
From Maxwell’s equations at the frequency domain, considering TM modes (
The solution for
The solution for the component
The superscripts between parentheses on the inverse permittivity tensor elements identify the corresponding waveguide layer, as specified in Figure 7. The continuity of
From the roots of Eq. (45) for
Figure 9 shows the transversal distributions of the
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.
where A1 through A8 are constants to be determined,
The electric field components
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
Figure 11 shows a plot of guided supermodes that occurs in the planar structure for forward propagation (along +
Figure 12 shows the plot of the supermodes, now considering backward propagation of the TM mode (along -
Considering both propagation senses, when the condition 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 . If some light is reflected at port 3, since , 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.
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.
Antonov V, Harmon B, Yaresko A. Electronic Structure and Magneto-Optical Properties of Solids. Dordrecht: Springer; 2004. 528 p. DOI: 10.1007/1-4020-1906-8
Levy M. The on-chip integration of magnetooptic waveguide isolators. IEEE Journal of Selected Topics in Quantum Electronics. 2002; 8(6):1300-1306. DOI: 10.1109/JSTQE.2002.806691
Ando K. Waveguide optical isolator: A new design. Applied Optics. 1991; 30(9):1080-1095. DOI: 10.1364/AO.30.001080
Bahlmann N, Chandrasekhara V, Erdmann A, Gerhardt R, Hertel P, Lehmann R, et al. Improved design of magnetooptic rib waveguides for optical isolators. Journal of Lightwave Technology. 1998; 16(5):818-823. DOI: 10.1109/50.669010
Li TF, Guo TJ, Cui HX, Yang M, Kang M, Guo QH, et al. Guided modes in magneto-optical waveguides and the role in resonant transmission. Optics Express. 2013; 21(8):9563-9572. DOI: 10.1364/OE.21.009563
Bolduc M, Taussig AR, Rajamani A, Dionne GF, Ross CA. Magnetism and magnetooptical effects in Ce-Fe oxides. IEEE Transactions on Magnetics. 2006; 42(10):3093-3095. DOI: 10.1109/TMAG.2006.880514
Fratello VJ, Licht SJ, Brandle CD. Innovative improvements in bismuth doped rare-earth iron garnet Faraday rotators. IEEE Transactions on Magnetics. 1996; 32(5):4102-4107. DOI: 10.1109/20.539312
Pedroso CB, Munin E, Villaverde AG, Medeiros Neto JA, Aranha N, Barbosa LC. High Verdet constant Ga:S:La:O chalcogenide glasses for magneto-optical devices. Optical Engineering. 1999; 38(2). DOI: 10.1117/1.602080
Kalandadze L. Influence of implantation on the magneto-optical properties of garnet surface. IEEE Transactions on Magnetics. 2008; 44(11):3293-3295. DOI: 10.1109/TMAG.2008.2001624
Nomura T, Kishida M, Hayashi N, Ishibashi T. Evaluation of garnet film as magneto-optic transfer readout film. IEEE Transactions on Magnetics. 2011; 47(8):2081-2086. DOI: 10.1109/TMAG.2011.2123103
Inoue M, Arai K, Fujii T, Abe M. One-dimensional magnetophotonic crystals. Journal of Applied Physics. 1999; 85(8):5768-5770. DOI: 10.1063/1.370120
Koerdt C, Rikken GL, Petrov EP. Faraday effect of photonic crystals. Applied Physics Letters. 2003; 82(10):1538-1540. DOI: 10.1063/1.1558954
Zvezdin AK, Belotelov VI. Magnetooptical properties of two-dimensional photonic crystals. The European Physical Journal B–Condensed Matter and Complex Systems. 2004; 37:479-487. DOI: 10.1140/epjb/e2004-00084-2
Haider T. A review of magneto-optic effects and its application. International Journal of Electromagnetics and Applications. 2017; 7(1):17-24. DOI: 10.5923/j.ijea.20170701.03
Zak J, Moog ER, Liu C, Bader SD. Magneto-optics of multilayers with arbitrary magnetization directions. Physical Review B. 1991; 43:6423-6429. DOI: 10.1103/PhysRevB.43.6423
Wolfe R, Lieberman R, Fratello V, Scotti R, Kopylov N. Etch-tuned ridged waveguide magneto-optic isolator. Applied Physics Letter. 1990; 56:426-428. DOI: 10.1063/1.102778
Alcantara LDS, Teixeira FL, Cesar AC, Borges BH. A new full-vectorial FD-BPM scheme: Application to the analysis of magnetooptic and nonlinear saturable media. Journal of Lightwave Technology. 2005; 23(8):2579-2585. DOI: 10.1109/JLT.2005.850811
Alcantara LDS, De Francisco CA, Borges BH. Analytical model for magnetooptic five-layered planar waveguides. In: SBMO/IEEE MTT-S International Microwave and Optoelectronics Conference (IMOC’17); 27–30 August 2017; Águas de Lindóia, Brazil. pp. 1-5. DOI: 10.1109/IMOC.2017.8121106