Optical Properties of Antiferromagnetic/Ion-Crystalic Photonic Crystals

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].


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][3][4][5][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][9][10][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][13][14][15][16][17]. LHMs have to be constructed artificially since there is no natural LHM. Several variations of the design have been studied through experiments [18][19][20]. Up to now, scientists have found some LHMs available in infrared and visible ranges [21][22][23][24][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 lefthandedness 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.

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][30][31][32][33][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][33][34]. Actually, most materials are dispersive and absorbing. Therefore, it is also necessary to consider the effect of damping.

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 ( 0 y  ) is of vacuum, where a d and i d are thicknesses of AF and IC layers, respectively. For the far infrared wave, the order of the wavelength is about 100μm. Thus, as long as the thicknesses of AF and IC layers are less than 10μm, the wavelength  will much longer than the period of AF/IC PCs. With this condition, the AF/IC PCs will become a uniform film by means of an effective-medium method (EMM).

EMM for one-dimensional AF/IC PCs
We first present the permeability of the AF film. In the external magnetic field 0 H  , the magnetic permeability is well-known, with its nonzero elements [33,37]  for IC layers. These fields are local fields in the layers. For the components of magnetic induction and field continuous at the interface, one assumes And for those components discontinuous at the interface, one assumes , . x where the AF volume fraction / where T is the transfer matrix expected and its components are 2 11 12 (1 ) ( The Bloch's theorem implies another relation Based on matrix relations (14) and (17), we obtain the polariton dispersion equation (18) Q is the Bloch wave number for an infinite structure and is a real number, and then equation (18) describes the bulk polariton modes.
For a semi-infinite structure, it is interesting in physics that Q i  is an imaginary number.
Thus equation (18) can be used to determine the surface modes traveling along the 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  [10,33,34].
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 r   versus k, the shaded regions stand for the bulk bands whose boundaries are determined by Eq. (18) or (19) with 0 Q  and 0 k  , respectively. Meanwhile, the surface modes are obtained by Eqs. (18), (19) and (21). The thick curves with the serially numbered

Limiting case of small period (EMM)
To examine the limiting case of small period or long wavelength is meaningful in physics. We let 0 a d  and 0 i d  in Eqs. (18) , (19) and then find 2 e e 2 e 2 xx yy λ 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. For the bulk bands, we see that the results obtained within the two methods almost are equal. However, the surface modes obtained by the EMM start from the photonic lines and are continuous, but those achieved by the TMM do not. It is because the interfacial effects are efficiently considered within the TMM, but the EMM neglects these. For the surface polaritons, their many main features attained by the two methods are still analogous. 12 surface mode branches are seen from Fig.2 within the TMM, but 11 surface modes from Fig.5 within the EMM. 10 surface mode branches arise in the common vicinities of two AF resonant frequencies and TO phonon frequency.Except branches 1 and 2, all surface modes are nonreciprocal and their non-reciprocity results from the magnetic contribution in AF layers. Surface polaritons 1 and 2 should be called the quasi-phonon polaritons since the contribution of the magnetic response to the polaritons is very weak in their frequency range. Another interesting feature is that many surface modes possess negative group velocities ( 0 d dk   ), which is due to the combined contributions of magnetic damping and phonon damping.

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 / a f R L    , and the IC filling ratio, Our aim is to determine general characteristics of the surface and bulk polaritons with an effective-medium method under the condition of L   (λ is the polariton wavelength).

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 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 h is determined by (23) in the r z  system, but in the IC, the relation is After defining the relation between the effective fields in the AF/IC PCs, , the effective permeabilityresulting from (24)-(27) Formula (29) is the expression of the effective magnetic permeability in the r z  system.
When we discuss the surface polaritons, the AF/IC PCs will be considered as a semi-infinite system with a single plane surface, so the corresponding permeability in the rectangular coordinate system (orthe xyz system) will be used. Applying the transformation matrix (22) 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, e    is averaged with respect to angle  and one determines the averaged effective magnetic permeability, This means e e e xx yy      . In physics, this AF/IC PCs should be gyromagnetic and be of C4symmetry, as shown by Fig.1, which leads to the 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 electricdisplacement components as follows, Transforming (37) Then, its average value with respect to angle  is   (34) and (40) are reasonable.

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 where  and 0  are determined by (45)  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 ( 0 d dk   ). For the IC, its surface modes and bulk bands are depicted in Fig. 7(b).
The surface modes are of reciprocity. Comparing Fig. 7(b) with the previous results without phonon damping [42], it is different that the two surface modes bear bended-back and end on the vacuum light line, due to the phonon damping.  For the AF/IC PCs with a 0.9 f  , Fig. 8 illustrates the dispersion features of magneto-phonon polaritons. Four bulk bands and 13 surface mode branches are found, where the surface modes are nonreciprocal (meaning the surface modes are changed when reversing their propagation directions). Two mini bulk bands and 11 surface mode branches exist in the vicinities of two AF resonant frequencies, where they are neither similar to those of the AF nor to those of the IC. Due to the combined contributions of the magnetic damping and phononic damping, the surface-mode group velocities become negative in some frequency ranges. For frequencies near the higher AF resonant frequency, the top bulk band bears a resemblance in nature to the top one of the AF, but the bottom band is analogous to the bottom one of the IC for frequencies close to the IC resonant frequency. Figure 9 shows the bulk bands and surface modes for a 0.8 f  . The bulk bands in this figure are characteristically similar to those in Fig. 8, but the two mini bands are narrowed and the top one risesstrikingly. For the surface modes, mode 6 in Fig. 8 splits into two surface modes in Fig. 9. Modes 5 and 11 in Fig. 8 disappear from the field of view. The surface modes are still nonreciprocal.
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 a 0.8 f  to display the relevant effective permeability and permittivity.
One can see that dielectric permittivity, e zz  , is negative in a large frequency range. The two AF resonant frequencies lie in this range and the magnetic permeability, e  , is negative in the vicinities of the AF resonant frequencies. Thus, for electromagnetic waves traveling in the 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 0 e   (see Eq. (1-30b)). The negative refraction and left-handedness were predicted in a one-dimension structure composed of identical materials [10].

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 FeF2/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) 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   (56) Figure 11. Draft for incidence, reflection, refraction and transmission rays. After Wang & Song, 2009. The radiation in the film consists of two parts, one is the forward light (refraction light) related to amplitude 0 A and the other is the backward light (reflection light) related to amplitude 0 B . Here, y k is defined as a negative number, otherwise the refraction wave corresponds to amplitude 0 B . These two situations are equivalent in essence. According to the definition of energy flow density of electromagnetic wave * Re( ) / 2   S E H , the flow densities of the two lights can be given by The inner product between a wave vector ( [ , ,0] FeF2 and TlBr are used as constituent materials where the AF resonant frequency r  is closer to the phonon resonant frequency T  and located in the far infrared regime. Fig.12 shows this angle as a function of frequency for 0.8 a f  and 0.6. It can be seen from Fig.12 that for most of the frequency range occupied by the curves, angle  is at least bigger than 160 o for various incident angles. So the wave vector, electric and magnetic fields form an approximate left-handed triplet. The operation frequency range becomes narrow as a f decreases, as shown in Fig.12b. As shown in Fig.13(a)

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 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 2 t and reflection ratio is 2 r . Additionally, absorption ratio is represented with Other quantities in Eq. (68) are 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 The magnetic field components of the forward-going wave in the adjacent IC layers are The amplitudes of two neighboring layers satisfy 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 0.943 r    and 1.064 r    , corresponding to the mini-bands with negative group velocity in Ref. [8]. At the same time, the magnetic permeability of AF layers and the dielectric function for IC layers are both negative. To distinctly observe two guided modes, the partially enlarged Fig. 14(b) corresponding to Fig. 14(a) is exhibited. Seen from Fig. 14(b), the maximum transmission of the guided mode with lower-frequency is 40% and that of higher mode is 28.4%. As is well known, the optical thicknesses of films are determined by the frequency-dependent magnetic permeability and the dielectric function. Then the optical thicknesses of thin films are varied with the frequency of incident wave. The optical path of wave in media can also be altered by changing the incident angle. 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) Fig. 17. Namely, the frequency regime of negative refraction near to 0.943 r    is strikingly greater than that occupied by the quasi-lefthandedness in Fig.17. However, the result is opposite in the vicinity of 1.064 r    .
Therefore, we here reckon the left-handedness is not always accompanied by negative refraction. FeF2/TlBrsuperlattices have the natures of either negative refraction or quasi lefthandedness, 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   . This completely covers the frequency range of AF resonance, so the dielectric function must be negative in the region of negative magnetic permeability v  . It is found from Fig.18 that the magnetic permeability v  is negative in certain regions, where the dielectric function i  is also negative. In our previous work [8], utilizing the effective medium theory, we verified the effective dielectric function and magnetic permeability are both negative in the long wavelength limit when v  and i  are negative. In other words, the AF/IC PCs is of negative refraction in the limit of long wavelength. Regarding the results arising from the two methods mentioned above, we conclude that the necessary condition of negative refraction or left-handedness is that v  and i  are both negative in this PCs.

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.