## 1. Introduction

Increasing attention has been paid to magnetic photonic crystals (MPCs) because the properties of the MPCs can be modulated not only with the change of their structure (including components, layer thickness or thickness ratio) but also with the external magnetic field. MPCs are capable of acting as tunable filters [1] at different frequencies, and that controllable gigantic Faraday rotation angles [2-6] are simultaneously obtained. The nonmagnetic media in MPCs generally are ordinary dielectrics, so the electromagnetic wave modes are just magnetic polaritons. The effect of magnetic permeability and dielectric permittivity of two component materials in MPCs on the photonic band groups were discussed, where the permeability and permittivity were considered as scalar quantities [7].

Recently, our group investigated the optical properties of antiferromagnetic/ ion-crystal (AF/IC) PCs [8-11]. It is well known that the two resonant frequencies of AFs, such as, FeF2 and MnF2, fall into the millimeter or far infrared frequencies regions and some ionic semiconductors possess a very low phonon-resonant frequency range like the AFs. Especially, these frequency regions also are situated the working frequency range of THz technology, so the AF/IC PCs may be available to make the new elements in the field of THz technology. Note that in ICs, including ionic semiconductors, when the frequencies of the phonon and the transverse optical (TO) phonon modes of ICs are close, the dispersion curves of phonon and TO phonon modes will be changed and a kind of coupled mode called phonon polariton will be formed. Therefore, in the AF/IC PCs, the TO phonon modes of ICs can directly couple with the electric field in an electromagnetic wave and this coupling generates the phonon polaritons, however, the magnetization’s motion in magnets can directly couple with the magnetic field, which is the origin of magnetic polaritons. Thus in such an AF/IC PCs, we refer to collective polaritons as the magneto-phonon polaritons (MPPs). In the presence of external magnetic field and damping, MPPs spectra display two petty bulk mode bands with negative group velocity. It is worthy of mentioning that many surface modes emerge in the vicinity of two petty bulk mode bands, and that some surface modes bear nonreciprocality [11]. The optical properties of the AF/IC PCs can be modulated by an external magnetic field.

In addition, we have concluded that there is a material match of an AF and an IC, for which a common frequency range is found, in which the AF has a negative magnetic permeability and the IC has negative dielectric permittivity [10]. Consequently, the AF/IC structures are thought to be of the left-handed materials (LHMs) which have attracted much attention from the research community in recent years because of their completely different properties from right-handed materials (RHMs). In a LHM, the electric field, magnetic field and wave vector of a plane electromagnetic wave form a left-handed triplet, the energy flow of the plane wave is opposite in direction to that of the wave vector [12-17]. LHMs have to be constructed artificially since there is no natural LHM. Several variations of the design have been studied through experiments [18-20]. Up to now, scientists have found some LHMs available in infrared and visible ranges [21-25], but each design has a rather complicated structure. We noticed a work that discussed the left-handed properties of a superlattice composed of alternately semiconductor and antiferromagnetic (AF) layers, where the interaction between AF polaritons and semiconductor plasmons lead to the left-handedness of the superlattice [26]. However the plasmon resonant frequency sensitively depends on the free charge carrier’s density, or impurity concentration in semiconductor layers, so if one wants to see a plasmon resonant frequency near to AF resonant frequencies, the density must be very low since AF resonant frequencies are distributed in the millimeter to far infrared range. In the case of such a low density, the effect of the charge carriers on the electromagnetic properties may be very weak [27] so that there is not the left-handedness of the superlattice. According the discussion above, we propose a simple structure of multilayer which consists of AF and IC layers. An analytical condition under which both left-handeness and negative refraction phenomenon appear in the film is established by calculating the angle between the energy flow and wave vector of a plane electromagnetic wave in AF/IC PCs and its refraction angle.

## 2. Magneto-phonon polaritons (MPPs) in AF/IC PCs

Polaritons in solids are a kind of electromagnetic modes determining optical or electromagnetic properties of the solids. Natures of various polaritons, including the surface and bulk polaritons, were very clearly discussed in Ref. [28]. Recent years, based on magnetic multilayers or superlattices, where nonmagnetic layers are of ordinary dielectric and their dielectric function is a constant, the polaritons in these structures called the MPCs were discussed [29-34]. On the other hand, ones were interested in the phonon polaritons [35-36], where the surface polariton modes could be focused by a simple way and probably possess new applications. In this part, the collective polaritons, MPPs in a superlattice structure comprised of alternating AF and IC layers, will be discussed. In the past, for simplicity, the damping was generally ignored in the discussion of dispersion properties regarding the polaritons[30,32-34]. Actually, most materials are dispersive and absorbing. Therefore, it is also necessary to consider the effect of damping.

### 2.1. MPPs in one-dimension AF/IC PCs

An interesting configuration in experiment is the Voigt geometry as illustrated in Fig.1, where the polariton wave propagates in the *x-y* plane and the magnetic field of an electromagnetic wave is parallel to this plane, but the wave electric field aligns the *z* direction. We concentrate our attention on the case where the external magnetic field and AF anisotropy axis both are along the *z* axis and parallel to layers. The *y* axis is perpendicular to layers in the structure. The semi-space (*μm*. Thus, as long as the thicknesses of AF and IC layers are less than 10*μm*, the wavelength

#### 2.1.1. EMM for one-dimensional AF/IC PCs

We first present the permeability of the AF film. In the external magnetic field

Where

We assume that there are an effective relation

And for those components discontinuous at the interface, one assumes

where the AF volume fraction

with the elements

On the similar principle, we can find that the effective dielectric permittivity tensor is diagonal and its elements are

On the base of these effective permeability and permittivity, one can consider the AF/IC PCs as homogeneous and anisotropical AF films or bulk media. The similar discussions can be found in the Chapter 3 of the book “Propagation of Electromagnetic Waves in Complex Matter” edited by Ahmed Kishk [39].

#### 2.1.2. Dispersion relations of surface and bulk MPP with transfer matrix method (TMM)

The wave electric fields in an AF layer and IC layer are written as

respectively. *y* wave-vector components. *j=A, B, C or D*) denotes the amplitudes of the electric fields. Additionally, the corresponding magneticfields can be found with the relation

We see from the wave equation that

with *c* is the light velocity in vacuum, and

Employing the well-known TMM, together withthe boundary conditions of *n*th and *n+*1th bi-layers

where *T* is the transfer matrix expected and its components are

with

Based on matrix relations (14) and (17), we obtain the polariton dispersion equation

For a semi-infinite structure, it is interesting in physics that *x* axis. We need the electromagnetic boundary conditions at the surface of this structure to find another necessary equation for the surface polariton modes. This equation is just

where

Numerical simulations based on FeF_{2}/TlBr will be performed with TMM. The reason is that their resonant frequencies lie in the far infrared range and are close to each other. The physical parameters here applied are

The MPP spectra are displayed in Fig.2, 3, and 5. In these spectra for dimensionless reduced f

The MPP spectra are displayed in Fig.2, 3, and 5. In these spectra for dimensionless reduced frequency*k*, the shaded regions stand for the bulk bands whose boundaries are determined by Eq. (18) or (19) with _{1} and L_{2}.Fig.2 shows the bulk bands and surface modes for the ratio

Fig.4 displays the bulk bands and surface modes for ratio

#### 2.1.3. Limiting case of small period (EMM)

To examine the limiting case of small period or long wavelength is meaningful in physics. We let

for the bulk modes with

for the surface polaritons. If the external magnetic field implicitly included in Eqs.(20) and (21) is equal to zero, the dispersion relations can be reduced to those in our earlier paper [10].Hence equations (20) and (21) also can be considered as the results achieved by the EMM.

Fig.5 shows the bulk bands and surface modes for the ratio

### 2.2. MPPs in two-dimension AF/IC PCs

In this part, we consider such an AF/IC PCs constructed by periodically embedding cylinders of ionic crystal into an AF, as shown in Fig.6. We focus our attention on the situation where the external magnetic field and the AF anisotropy axis both are along the cylinder axis, or the *z*-axis. The surface of the MPC is parallel to the *x-z* plane. L and R indicate the lattice constant and cylindrical radius, respectively. We introduce the AF filling ratio,

#### 2.2.1. EMM for the two-dimensional AF/IC PCs

When the AF/IC PCs cell size is much shorter than the wavelength of electromagnetic wave, an EMM can be established for one to obtain the effective permeability and permittivity of the AF/IC PCs. The principle of this method is in a cell, an electromagnetic-field component continuous at the interface is assumed to be equal in the two media and equal to the corresponding effective-field component in the MPC, but one component discontinuous at the interface is averaged in the two media into another corresponding effective-field component [30,33,40-41]. Because the interface between the two media is of cylinder-style, before establishing an EMM, a TMM should be introduced. This matrix is

Thus, we find the expression of the permeability in the cylinder coordinate system

with

where the field components on the left side of Eqs. (24)-(27) are defined as the effective components in the AF/IC PCs and those on the right side are the field components in the AF and IC media within the cell. In the AF, the relation between * b* and

*is determined by (23) in the*

**h**After defining the relation between the effective fields in the AF/IC PCs,

with

Formula (29) is the expression of the effective magnetic permeability in the

(33) |

If one applies directly this form into the Maxwell equations, the resulting wave equation will be very difficult to solve. Thus, a further approximation is necessary. We think that if the wavelength of an electromagnetic wave is much longer than the cell size, then the wave will feel very slightly the structure information of the AF/IC PCs. Here, the averages of some physics quantities are important. Hence,

This means*xx* and *yy* elements of the final permeability should be equal and its xy element equal to -*yx* element.

By a similarprocedure, the effective dielectric permittivity can be easily found. According to the principle of EMM, we present the equations for the electric-field and electric-displacement components as follows,

with

With

Transforming (37) into the form for the *xyz* system, we see

(39) |

Then, its average value with respect to angle

We see_{2} with its resonant frequency about*μm*. When the cell size is taken as

#### 2.2.2. Dispersion equations of surface and bulk MPP

The effective permittivity (40) and permeability (34) are applied to determine the dispersion equations of surface and bulk MPP in the AF/IC PCs. In the geometry of Fig.6, if the magnetic field of a plane electromagnetic wave is along the *z*-axis, the sublattice magnetizations in the AF do not couple with it, so the AF plays a role of an ordinary dielectric. Thus, we propose the electric fields of polariton waves in the AF/IC PCs are along the *z*-axis and the magnetic field is in the *x-y* plane. For a surface polariton, its electric field decaying with distance from the surface can be written as

in the AF/IC PCs, where *k* is the wave-vector component along the *x*-axis, but

which lead to two relations

withwhere *x-y* plane is

The bulk polariton bands are just such regions determined by (46). One can calculate directly the dispersion curves of the surface polariton from (45).

FeF2 and TlBr are utilized as constituent materials in the AF/IC PCs, which the parameters have been introduced in the last section. We place the AF/IC PCs into an external field of

For comparison, we first present the polariton dispersion figures in the AF FeF2 and IC TlBr, as indicated in Fig. 7, respectively. For the AF, there exist three bulk bands and two surface modes. The surface modes appear in a nonreciprocal way and have a positive group velocity (

For the AF/IC PCs with

Figure 9 shows the bulk bands and surface modes for

The two mini bulk bands possess a special interest, corresponding to the negative effective magnetic permeability and negative effective dielectric permittivity of the AF/IC PCs. We present Fig. 10 for *x-y* plane, the refraction index is negative and the left-handedness can exist in the two mini bulk bands. When electromagnetic waves propagate along the *z*-axis, there is no coupling between AF magnetizations and electromagnetic fields, so the electromagnetic waves cannot enter this range where

## 3. Presence of left-handedness and negative refraction of AF/IC PCs

In the previous section, we have discussed MPPs in AF/IC PCs with the TMM and EMM for one- and two-dimension. Based on FeF_{2}/TIBr, there are a number of surface and bulk polaritons in which the negative refraction and left-handedness can appear. In order to investigate the formation mechanism of LHM in AF/IC PCs, the external magnetic field and magnetic damping is set to be zero. In this case, according Eqs.(7) and (8), the effective permeability

where

Let us consider an incident plane electromagnetic wave propagating in the x-y plane as shown in Fig.11. Such a wave can be divided into two polarizations, a TE mode with its electric field parallel to axis z and a TM mode with its magnetic field parallel to axis z. According to Maxwell’s equations, these wave vectors and frequencies of the two modes inside the film satisfy the following expressions

Since

where

and the corresponding magnetic field can be given by

The radiation in the film consists of two parts, one is the forward light (refraction light) related to amplitude

The inner product between a wave vector (

where *j*=*A* or *B*. It is obvious that

It can be seen from the expression (57) of *x* component is negative and *y* component is positive when both

FeF_{2} and TlBr are used as constituent materials where the AF resonant frequency

As shown in Fig.13(a) for

It can be seen from Fig.13(b) in comparing with Fig.13(a) that the frequency region of negative refraction is obviously narrower and the negative refraction angle becomes smaller. Numerical simulations also show both positive and negative refraction angles are in the spectral range of approximate left-handed feature shown in Fig.12.

## 4. Transmission, refraction and absorption properties of AF/IC PCs

In this section, we shall examine transmission, refraction and absorption of AF/IC PCs, where the condition of the period much smaller than the wavelength is not necessary. The transmission spectra based on FeF2/TIBr PCs reveal that there exist two intriguing guided modes in a wide stop band [11]. Additionally, FeF2/TIBr PCs possess either the negative refraction or the quasi left-handedness, or even simultaneously hold them at certain frequencies of two guided modes, which require both negative magnetic permeability of AF layers and negative permittivity of IC layers. The handedness and refraction properties of the system can be manipulated by modifying the external magnetic field which will determine the frequency regimes of the guided modes.

The geometry is shown in Fig. 1. We assume the electric field solutions in AF and IC layers as

where

with

*N*. Then transmission and reflection coefficients of the AISL can be written as

(68) |

Note that incident wave amplitude is taken as 1. Therefore, the transmission and reflection coefficients can be determined with Eq. (68), and then the transmission ratio is

As described in Ref. [8], magnetic superlattices possess two mini-bands with negative group velocity. When the incident wave is located in the frequency regions corresponding to the two mini-bands, what are the optical properties of the AF/IC PCs? In the preceding section, the expressions of transmission and absorption to be used have been derived. To grasp handedness and refraction properties of the AF/IC PCs, the refraction angle and propagation direction need to be determined. Therefore, subsequently we give the expression of the refraction angle. However, this structure possibly possesses a negative refraction, and generally the directions of the energy flow of electromagnetic wave and the wave vector misalign. We start with the definition of energy flow (

The magnetic field components of the forward-going wave in the adjacent IC layers are

The amplitudes of two neighboring layers satisfy

According to boundary conditions, the electric and magnetic fields of every layer are acquired when the incident wave is known. Then the expressions of refraction energy flow in all layers are written as

What needs to be emphasized is that we here concentrate only on the refraction, so only the forward-going wave corresponding to the first term in Eq.(61) is considered and the backward-going wave is ignored. Owing to refraction angles being different in various layers, the refraction angle of the AF/IC PCs should be effective one. The angle between the energy flow and wave vector, and the refraction angle of the AF/IC PCs are defined as

with

Numerical calculations based on FeF_{2}/TlBr PCs. We take the AF layer thickness

corresponding to the band gap of magneto-phonon polariton in Ref. [8]. Here the most interesting may be that guided modes arise in the forbidden band. The two guided modes lie in the proximity of

As already noted, the damping is included and then the absorption appears. We are more interest in the two guided modes, so only the absorption corresponding to two guided modes will be considered in Fig. 15 (a) and (c) display the absorption spectra in the case of right incidence, but (b) and (d) illustrate the absorption spectra for incident angle

To capture the handedness and refraction behaviors of the AF/IC PCs, the angle of refraction and the angle between the energy flow and wave vector are illustrated. Fig. 16 shows the angle

Figure 17 shows the refraction angle _{2}/TlBrsuperlattices have the natures of either negative refraction or quasi left-handedness, or even simultaneously bear them at the certain frequencies of two guided modes.

To have a deeper understanding of the negative refraction and quasi- left-handedness of the AF/IC PCs, subsequently the expressions of the dielectric function

the dielectric function

## 5. Summary

This chapter aims to discover optical properties of AF/IC PCs in the presence of external static magnetic field. First, within the effective-medium theory, we investigated dispersion properties of MPPs in one- and two-dimension AF/IC PCs. The ATR (attenuated total reflection) technique should be powerful in probing these MPPs. Second, there is a frequency region where the negative refraction and the quasi left-handedness appear when the AF/IC PCs period is much shorter than the incident wavelength. Finally, an external magnetic field can be used to modulate the optical properties of the AF/IC PCs.

### Acknowledgement

This work was financially supported by the National Natural Science Foundation of China with Grant no.11084061, 11104050, and the Natural Science Foundation of Heilongjiang Province, with no. ZD200913.