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  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 .
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 . 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 . 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 . 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  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. . 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
2.1.1. EMM for one-dimensional AF/IC PCs
Where and are the high- and low-frequency dielectric constants, but is the TO resonant frequency of and is the phonon damping coefficient. The IC is nonmagnetic, so its magnetic permeability is taken as.
We assume that there are an effective relation between effective magnetic induction and magnetic field, and an effective relation between effective electric field and displacement, where these fields are considered as the wave fields in the structures. Butand in any layer, where is given in Eqs.(1) for AF layer and 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
where the AF volume fraction and the IC volume fraction with the period.Thus the effective magnetic permeability is achieved from Eqs. (4),(5) and it is definite by,
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 .
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. is the wave-vector component along x axis. and are the decay coefficients when they are real, otherwise they correspond to the
We see from the wave equation that
Employing the well-known TMM, together withthe boundary conditions of and continuous at the interfaces, we can find a matrix relation between wave amplitudes in any two adjacent bi-layers, or the relation between amplitudes in the
with and. The Bloch’s theorem implies another relation
Based on matrix relations (14) and (17), we obtain the polariton dispersion equation
For a semi-infinite structure, it is interesting in physics that is an imaginary number. Thus equation (18) can be used to determine the surface modes traveling along the
where with.is the decay coefficient in vacuum and must be positive. It needs to be emphasized that the existence of surface modes requires Re() >0.Eqs. (18) and (19) will be applied to determine the bulk polariton bands and surface polaritons.
Numerical simulations based on FeF2/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 frequencyversus
Fig.4 displays the bulk bands and surface modes for ratio. The top bulk band is distinctly ascended to the high frequency direction, together with surface mode 1and 2. The bottom bulk band is widened conspicuously. Contrary to the previous situation, the two mini bands get significantly narrower. Compared with Fig.2, the slopes of surface modes 7 and 12 diminish appreciably, meaning their group velocity dwindles as the AF volume fraction () decreases.
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 and in Eqs.(18),(19) and then find
for the bulk modes with, and
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 .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 (), which are obtained 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 (), which is due to the combined contributions of magnetic damping and phonon damping.
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
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 and. The theoretical processes of obtaining effective magnetic permeability, , and electric permittivity, , are presented as follows. According to the principle,we can introduce the following equations:
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
After defining the relation between the effective fields in the AF/IC PCs, , the effective permeabilityresulting from (24)-(27) is
Formula (29) is the expression of the effective magnetic permeability in the 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) again, we find
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, is averaged with respect to angle and one determines the averaged effective magnetic permeability,
This means. In physics, this AF/IC PCs should be gyromagnetic and be of C4-symmetry, as shown by Fig.1, which leads to the
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 in the AF or IC. After using the definition, , the effective dielectric permittivity of the AF/IC PCs in the system is determined as
Transforming (37) into the form for the
Then, its average value with respect to angle is
We see. Now, this AF/IC PCs can be considered as an effective medium with effective electric permittivityand magnetic permeability. If the AF medium is taken as FeF2 with its resonant frequency about, proper wavelengths should be between 170 and 190
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
in the AF/IC PCs, where
which lead to two relations
where and are determined by (45) with the conditions, and. Of course, it is very easy to find the dispersion relation of bulk polaritons. For the infinite AF/IC PCs, we find from the wave equation (44) that the dispersion relation in the
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 along the z-axis and employ a dimensionless reduced frequencyin figures. Surface mode curves are plotted against wave vector k along the x-axis. Bulk modes form some continuous regions shown with shaded areas.
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 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 , 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, 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. 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 to display the relevant effective permeability and permittivity. One can see that dielectric permittivity,, is negative in a large frequency range. The two AF resonant frequencies lie in this range and the magnetic permeability, , is negative in the vicinities of the AF resonant frequencies. Thus, for electromagnetic waves traveling in the
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 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), the effective permeability and dielectric permittivity will be described as
where is the AF filling ratio, and is IC filling ratio with as a bi-layer thickness.
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 is a positive real number, relation (51) corresponds to the case of an ordinary optical (when) or an opaque (when, contributing to an imaginary) film, and so, we no longer consider the TM case, but deal with only the case of TE incident mode, and found the left-handed feature and negative refraction behavior. For the TE mode, the presence of left-handed feature (or negative refraction) needs the satisfaction of the prerequisite condition that and andcan not be simultaneously positive [i.e., at least one of andis negative, see (50)]. According to expressions (48) and (49),
where and. Frequency region (52) is very large and covers regions (53) and (54) for the selected physical parameters. It is noted that both and can occur simultaneously only when AF layers are thicker than IC layers, which corresponds to spectral domain.The wave electric field in the film can be written at
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 and the other is the backward light (reflection light) related to amplitude. Here, is defined as a negative number, otherwise the refraction wave corresponds to amplitude. These two situations are equivalent in essence. According to the definition of energy flow density of electromagnetic wave, the flow densities of the two lights can be given by
The inner product between a wave vector (or) and its corresponding energy flow is given by expression, or. Thus the angle between energy flow and wave vector can be expressed as
It can be seen from the expression (57) of that its
FeF2 and TlBr are used as constituent materials where the AF resonant frequency is closer to the phonon resonant frequency and located in the far infrared regime. Fig.12 shows this angle as a function of frequency for 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 thanfor various incident angles. So the wave vector, electric and magnetic fields form an approximate left-handed triplet. The operation frequency range becomes narrow as decreases, as shown in Fig.12b.
As shown in Fig.13(a) for, the refraction angle is positive on the left side and negative on the right side of the intersection point of the curves. This corresponds to the following critical frequency obtained under the condition of and:
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 . 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 signify AF or IC layers, respectively. The corresponding magnetic field solutions are also achieved via. According to the boundary conditions of and continuous at interfaces, the relation between wave amplitudes in the two same layers of the two adjacent periods can be shown as a transfer matrix . The matrix elements are expressed by the following equations:
with and. In AF layers, there are the relation with, andthe dielectric constant. The magnetic permeability tensor components of AF layers are represented by
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 and reflection ratio is. Additionally, absorption ratio is represented with. Other quantities in Eq. (68) are, , ,.
As described in Ref. , 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 () with a view to achieving refraction properties. Based on wave solutions of the electric field and Maxwell equations, the magnetic field components of the forward-going wave in AF layers are shown as
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 and, where volume fractions are and, respectively.
Numerical calculations based on FeF2/TlBr PCs. We take the AF layer thickness and the thickness of IC layers. The stacking number is. Figure 14 shows the transmission spectra with specific angles of the incidence in an external magnetic field. As illustrated in Fig. 14(a), the forbidden band ranges from to
corresponding to the band gap of magneto-phonon polariton in Ref. . Here the most interesting may be that guided modes arise in the forbidden band. The two guided modes lie in the proximity of and, corresponding to the mini-bands with negative group velocity in Ref. . 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. Fig. 14(c) shows the transmission spectrum with incident angle and other parameters are the same as those in 14(a). The partially enlarged Fig. 14(d) corresponding to Fig. 14(c) is given. Compared with the normal incidence case, for the forbidden band becomes wide and their maximum transmissions are reduced to 27.1% and 21.9%, but two guided modes keep their frequency positions unaltered.
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. We see that the absorption has a great influence on the transmission spectra. In the absorbing band at, relative tiny absorption corresponds exactly to the maximum transmission. In the absorbing band at, the absorption is obviously strengthened with enhancing the 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 between the energy flow and wave vector of forward-going wave varies with frequency for and. As illustrated in Fig. 16, the angles in the vicinities of and are greater than, but less than. It indicates that the AF/IC PCs possesses a quasi left-handedness in these frequency regions.
Figure 17 shows the refraction angle versus frequency under the same condition as Fig. 16. We find the refraction angles are negative in the neighborhood of and. The two frequency ranges for negative angle do not completely coincide with those of the quasi-left-handedness in Fig. 17. Namely, the frequency regime of negative refraction near tois strikingly greater than that occupied by the quasi-left-handedness in Fig.17. However, the result is opposite in the vicinity of. 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 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 functionof IC layers and the magnetic permeability of AF layers are analyzed. We find that when frequency lies in
the dielectric function is negative, namely the range. This completely covers the frequency range of AF resonance, so the dielectric function must be negative in the region of negative magnetic permeability. It is found from Fig.18 that the magnetic permeability is negative in certain regions, where the dielectric function is also negative. In our previous work , utilizing the effective medium theory, we verified the effective dielectric function and magnetic permeability are both negative in the long wavelength limit when and 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 and are both negative in this PCs.
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.
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.
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