This chapter discusses the propagation of TM and TE waves in the one-dimensional gyrotropic magnetophotonic crystals with ferrite and plasma-like layers. Elements of the transfer matrix are calculated in closed analytical form on the base of electrodynamic problem rigorous solution for arbitrary location of the gyrotropic elements on the structure period. Dispersion equation of the layered periodic structure with gyrotropic elements is obtained. Dispersion properties of the structure for TE and TM modes are analyzed for different configurations of magnetophotonic crystals (ferrite and plasma-like layers). Existence areas of transmission bands for surface and bulk waves are obtained. The effect of problem parameters on the dispersion properties of magnetophotonic crystals for TM and TE modes is investigated. Regimes of complete transmission of wave through limited magnetophotonic crystal are analyzed for bulk and surface waves.
- magnetophotonic crystal
- gyrotropic media
- dispersion diagrams
- TE and TM modes
- bulk and surface waves
Photonic crystals (PCs) are artificial periodic structures with spatially modulated refractive index in one or more coordinates [1, 2]. Their outstanding optical properties are due to the existence of frequency band gaps where the propagation of electromagnetic waves is impossible. Application of these structures became very attractive for modern optoelectronics which uses the various waveguides, resonators, sensors, and other devices on the basis of PC [3, 4]. Moreover, the control of the PC structure characteristics is the important problem that is usually solved using external electric or magnetic fields. These methods of providing controllability are based on the variation of refractive index of special materials such as liquid crystals and magnetic materials [5, 6]. Since these sensitive materials are anisotropic, then theoretical analysis of their properties is more complicated.
When at least one of the PCs’ unit cell components is a magnetically sensitive (gyrotropic) material, they exhibit unique magneto-optical properties and identified as magnetophotonic crystals (MPCs). Investigations of the MPCs are begun for simplest one-dimensional structures [7, 8]. However, one-dimensional MPCs are the basis elements for various active field-controlling applications so far [9, 10]. Changing of the permeability by external magnetic field is one of the main phenomena that allow developing electronically tuned devices in different frequency bands: filters, circulators and so on [11, 12, 13].
Along with the properties inherent in conventional PCs, these structures have additional optical and magneto-optical properties which considerably expand their functionality. Kerr effect, Faraday rotation and optical nonlinearity can be enhanced in MPC due to light localization within magnetic multilayer. Magneto-optical system with large Faraday or Kerr rotation can be used for effective optical isolators [14, 15], spatial light-phase modulators  and magnetic field and current sensor  development. Furthermore, one can obtain stronger enhancement of the magneto-optical phenomena due to resonant effects in the MPCs , which characterized by specific polarization properties. Using PCs with magneto-optical layers provides possibility of control of optical bistability threshold in structure based on graphene layer . It should be noted that not only magnetic materials are suitable for MPC. Namely, one-dimensional PC with plasma layers can be tunable by external magnetic field .
A number of applications of the MPCs are inspired by their nonreciprocal properties. For example, special spatial structure of the MPC layers provides the asymmetry of dispersion characteristics and, as a result, the effect of unidirectional wave propagation . This phenomenon allows enhancing field amplitude in the MPC without any periodicity defects. In this case, the so-called frozen mode regime occurs instead the defect mode one.
One of the unique properties of gyrotropic materials is the possibility of negative values of material parameters under the certain conditions. Usually, these are so-called single-negative media that are divided into epsilon-negative media (plasma) and mu-negative ones (gyrotropic magnetic materials). The term “double-negative media” or “left-handed materials” is used for media with negative values of both permittivity and permeability and often replaced by term “metamaterials.” Application of metamaterials in one-dimensional PC systems results in unusual regularities of bulk and surface wave propagation and is the subject of experimental and theoretical research [22, 23].
Theoretical description of the various types of one-dimensional PCs is usually based on the transfer-matrix method of Abelès  that was applied by Yeh et al. to periodic layered media . This method cannot be applied in general case for anisotropic multilayer structures because of mode coupling. However, this is possible in special cases, namely, in two-dimensional model of wave propagation in periodic layered media . This case is considered in this chapter. Such an approach makes it possible to simplify significantly the analysis of physical phenomena in complex layered media with various combinations of gyrotropic and isotropic elements. Moreover using well-known permutation duality principle of Maxwell’s equations results in a reduction of unique combinations number. In turn, this allows better understanding of regularities of bulk and surface wave propagation in one-dimensional MPC and finding new modes for applications in modern microwave, terahertz and optical devices.
2. Formulation and solution of the problem for modes of gyrotropic periodic structures
2.1. Basic relationships
We study electromagnetic wave propagation in periodic structure in general case with bigyrotropic layers (one-dimensional MPC) (Figure 1). Each of two layers on the structure period
For plasma media, the value of the permittivity is tensor, whereas the value of the permeability is scalar. Such media are called electrically gyrotropic. It is opposite for the ferrite media; the permeability is tensor, whereas permittivity is scalar. Such media are usually called magnetically gyrotropic. If the permittivity and permeability are simultaneously described by the tensors (Eq. (1)), such media are called gyrotropic or bigyrotropic. The material parameters included in tensors and are defined by the value of the external bias magnetic field , which is directed along
The study of the general case of gyrotropic media with material parameters of form (Eq. (1)) is reasonable, primarily because it allows using the permutation duality principle when obtaining main equations for fields and characteristic Eqs. . According to this principle generalized for gyrotropic media, namely, when simultaneously the substitution of fields and material parameters is done, the receiving of general equations, from which the equations for magneto-gyrotropic media (ferrite), electro-gyrotropic media (plasma), and gyrotropic media (bianisotropic media) can be obtained easily, turns to be more simple than in each of the mentioned particular cases separately.
Indeed, it follows from Maxwell equations:
where is inductance vector of electric field and is inductance vector of magnetic field. Components of these vectors can be written as:
In general case, one can obtain two connected differential equations for longitudinal components of electromagnetic fields
Using the relation between field components
Eq. (3) describes TM waves (
This analysis shows the important conclusion that in the general case of a gyrotropic medium, the fields of TM and TE waves with respect to the direction of the bias magnetic field in the two-dimensional case are divided into two independent solutions of the Maxwell equations. Moreover, Eq. (3) and the expressions for the tangential components of fields
To determine the eigenvalues and the corresponding eigenfunctions of the two-layer gyromagnetic MPC, we consider Helmholtz equation for the
The solution of the Helmholtz equation for TM waves for both tangential components of the fields
Let us note one feature in the expressions for the electromagnetic fields (Eqs. (4) and (5)). The presence of gyrotropy in the layers (
To find the dispersion equation that relates the longitudinal wave number
Eliminating the coefficients
An important property of the
The resulting matrix equation (Eq. (11)), which determines the relationship of the unknown coefficients in two identical layers of different periods of the periodic structure and the Floquet-Bloch theorem, allows us to find the characteristic (dispersion) equation for determining the previously introduced unknown longitudinal wave number
The phase factor
The unknown real values of the roots of the characteristic equation have the form:
It is easy to show that this expression is transformed to the well-known solution of the dispersion equation for the case of two magnetodielectric layers (
Note that the longitudinal wave number enters into equation as
Using the permutation duality principle, we found the solutions of the electrodynamic problem for TE wave propagation in the gyrotropic MPC. For this case, we change the material parameters according to the rule in the transfer-matrix elements (Eqs. (12)–(15)), in the dispersion relation, and in the solution (Eq. (18)). Then, we obtain
Here and .
It is apparent that in this case, one can write
In the absence of gyrotropy (
The given elements of the transfer-matrix
2.2. Eigen regimes of MPCs
We perform the analysis of the features for the propagation of the electromagnetic waves in different MPCs for various eigenmodes. We identified 10 variants of such regimes.
Taking into account the permutation duality principle, we obtain equations analogous to variant 2.
Taking into account the permutation duality principle, we obtain equations analogous to variant 6.
Taking into account the permutation duality principle, we obtain equations analogous to variant 8.
3. Analysis of the propagation of TE and TM waves in MPCs
3.1. General aspects of wave propagation in MPCs
Let us do a physical analysis of the obtained results and determine general rules of electromagnetic wave propagation in MPC. If the wave number
3.2. Analysis of dispersion characteristics
Let us consider first the case of a MPC in the absence of gyrotropy (
The solutions of the dispersion equations for the magnetodielectric periodic structures written out in this section include all possible combinations of the signs and magnitudes of the material parameters. As an example, let us consider dispersion characteristics of one-dimensional magnetodielectric PCs for several combinations of the parameters.
Figure 2 shows dispersion curves for the value of the longitudinal wave number
If we consider a MPC consisting of two different magnetodielectrics, then there are no fundamental differences.
Figure 3 shows the dispersion diagrams for PC with two dielectric layers (
The existence of surface waves at the boundaries of PC layers is illustrated in the dispersion diagram (Figure 3) for TE waves (
With the advent of new artificial media (metamaterials) for which the permittivity and permeability are simultaneously negative, optoelectronics devices with new functionalities are developed. In this connection, it is expedient to consider a MPC, one of whose layers on the structure period is a metamaterial (e.g.,
In Figure 4, dispersion diagrams are calculated for the one-dimensional PC with alternative layers of magnetodielectric and metamaterial (
It is clear that there are transmission bands only for surface slow waves (modified Zenneck-Sommerfeld waves) that are located below the light line . For bulk waves (
Dispersion curves for real
We now turn to an analysis of the propagation of waves in MPC consisting of a magnetodielectric and a semiconductor plasma layer . It is advisable to consider two cases in the presence of gyrotropy of the medium (
If the above conditions are satisfied, then the existence of real values of the Floquet-Bloch wave number other than zero is possible when the condition |cos
The results of calculations of the dispersion diagram are shown in Figure 6. The calculation was carried out with the following parameters:
In the case
Figure 7 shows the dispersion diagrams for both polarizations in the case
The case of MPC with a magnetodielectric and ferrite layer is analogous to that considered earlier by virtue of the permutation duality principle. Let us now consider the following case, when both layers on the structure period are ferrite with different material parameters. In this case, modes of bulk wave’s existence in the transmission bands can be observed when two conditions and are fulfilled. Regime of surface waves is realized when opposite conditions (, ) and also the additional condition are fulfilled:
Note that this condition is equally suitable for both positive and negative values of the effective magnetic permeability of ferrite
In Figure 8, we represent the dispersion diagrams of TM waves for the considered above MPC with two ferrite layers at the period of the structure. In the calculation, the following task parameters were chosen:
The solid line in the figure, below which the solutions of the dispersion equation are in the regime of surface waves, is determined by the equation
Figure 9 illustrates the case of the existence of a modified Zenneck-Sommerfeld wave for the case when the effective magnetic permeability of one of the layers is negative. Here, the dispersion diagrams are calculated with the following parameters of the problem:
Figure 10 illustrates the evolution of dispersion diagrams at change of ferrite effective magnetic permeability
Figure 11 shows dispersion diagrams for the modified Zenneck-Sommerfeld surface wave of MPC with two ferrite layers on the structure period at change of width of the layer
Increase of the second-layer width
Let us move on the dispersion diagrams for two bigyrotropic layers on the structure period. Taking into account the complete symmetry of the dispersion for TE and TM waves, the case of polarization indifference can be realized in this case. We will show this by example.
Figure 12 shows the dispersion diagrams for TE and TM waves for MPC consisting of two bigyrotropic layers on the period of the structure. Figure 12a corresponds to the following values of the parameters:
The solid lines in the figures distinguish the area of fast (upper part of figures) and slow wave (the lower part of figures). The complete identity of the dispersion diagrams for the transmission bands of both surface and bulk waves follows from the figures and formulas (18) and (23). By changing the bias magnetic field, it is possible to control the width and location of transmission bands for both polarizations.
Dispersion diagrams in Figure 13 correspond to the case when only the surface wave transmission bands for both polarizations are realized. The parameters of the problem were chosen as follows:
Therefore, we have considered the main features of the propagation of TE and TM waves in various magnetophotonic gyrotropic crystals. Important application of this one-dimensional PC theory is the problem of the electromagnetic waves scattering by a structures with limited number of periods.
4. Scattering of a plane wave by a MPC
In this section, the scattering of the plane wave on gyromagnetic MPC with
There are three transmission bands in the frequency range under consideration. Each of these bands contains resonances which observed with respect to the frequency of the complete transmission of the wave (Figure 14b and c). Frequency resonances correspond to different modes of the periodic structure. The number of modes is determined by the number of periods of the structure (
Figure 15 depicts the frequency dependences of the transmission coefficient modulus in the regime of the surface waves. In this case the incident angle of wave is greater than the angle of total internal reflection. We can see one transmission band in this case. Inset in Figure 15 shows enlarged frequency dependence within this band. Complete propagation is observed for each resonant frequency of modes of the limited periodic structure.
The electrodynamic problem is solved for the proper TE and TM waves of a MPC with two gyrotropic layers. The elements of the transmission matrix, the dispersion equation, and its solution are obtained analytically. An analysis of the dispersion properties of TE and TM waves for MPC is carried out, and features of the existence of fast and slow waves are revealed. Different regimes of gyrotropic surface waves are found. The conditions for the existence of surface waves are established for positive and negative values of the permittivity and permeability. Analytic expressions for the reflection and transmission coefficients for a limited MPC are obtained, and their analysis is performed for the regime of bulk and surface waves. Complete transmission of the wave through this structure is realized at resonant frequencies that correspond to different spatial distributions of the mode field in limited MPC.