1. Introduction

Recently, there have been many studies of a novel class of artificial materials which exhibit negative refraction of electromagnetic waves (EMW) in certain frequency ranges (see, for example, Refs. [1–4] and citations therein). These materials, termed as “left-handed”, are characterized by a negative permittivity and simultaneously a negative permeability, and lead to a variety of unusual optical phenomena [5]. In addition, in such materials the subwavelength focusing is possible which can be used to fabricate a perfect lens [6]. More recently, the predictions of Veselago and Pendry stimulated strong theoretical and experimental efforts to study negative refraction in various systems such as two-dimensional photonic-gap crystals [7], multilayer structures involving anisotropic materials [8], semiconductor superlattices in the presence of Bloch oscillations [9]. Experimental success has been demonstrated in ferromagnet-superconductor superlattices for millimeter waves [10].

In this paper, the problem of anomalous refraction is studied in a periodic ferromagnetsemiconductor multilayer composite, and the differences between the phase and group refracted indices are discussed. We show that in certain frequency range, left-handed behavior in such a structure is possible. Analytical expressions for both phase and group refractive indices are obtained. It is shown that the group index can be both positive and negative depending on the wave frequency and material parameters while the phase index is always positive.In section 2, we describe the physical picture of the problem and find the effective permittivity and permeability tensors in the presence of an external magnetic field. In section 3, the regions of propagation and dispersion curves of TE-wave are studied and the conditions of existence for the backward mode are obtained. In section 4, the pecularities of the phase and group refractive indices are examined and the problem of reflection at negative refraction is considered.Section 5 contains a summary of the principal results obtained in this paper.

2. Effective permittivity and permeability tensors

Consider a superlaffice (SL) or a periodic multilayer composite which consists of alternating layers of a ferromagnetic insulator (for example, ZnMnO) and nonmagnetic semiconductor (ZnO), in the presence of an external magnetic field B ₀ parallel to the plane of the layers. Let’s choose a Cartesian coordinate system with z-axis along B ₀ and y-axis along the optical axis of SL, so as the layers are parallel to the xz-plane; the space region y<0 is taken to be a vacuum (see Fig. 1).

 

Fig. 1. Geometry of the problem. A semiinfinite SL-medium is in the space region y>0; the space y<0 is taken to be a vacuum.

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An individual semiconductor layer of thickness l ₁ is described by dielectric permittivity tensor [11]

where

ω_c=eB ₀/m is the cyclotron frequency, ω_p=(Ne ²/mε ₀ ε _∞)^1/2 is the plasma frequency, e,m and N are the charge, effective mass and density of the free charge carriers, ε _∞ is the high-frequency dielectric constant and ε ₀ is the permittivity of the free space. We assume that semiconductor layers contain only one type of free charge carriers, and that the angular frequency ω of EMW is much larger compared to the frequency of the carrier collisions and is far from the excitonic frequencies so as the damping of the waves can be ignored.

In the same coordinate system, the magnetic permeability tensor of an individual ferromagnetic layer of thickness l ₂ is given by [12]

where

ω_s=γM ₀, ω ₀=γH ₀ is the ferromagnetic resonance frequency, γ is the magnetomechanical ratio and M ₀ is the saturation magnetization. For simplicity, we assume that dielectric constant of the ferromagnetic layers is equal to ε _∞.

It is well known that if the period of SL d=l ₁+l ₂ is much smaller than the wavelength of EMW, the method of effective anisotropic homogeneous medium can be used to obtain the permittivity and permeability of a periodic layered composite [13–16]. Using this method, after simple calculations we obtain the constitutive relations for SL, in the form

where the relative effective permittivity tensor is given by

and the relative permeability tensor has a form

where

In the following, we shall consider the case of the Voigt configuration, when EMW propagates in the xy-plane perpendicular to the external magnetic field. Then, from the Maxwell equations

for the plane waves E,H~exp[i(k_xx+k_yy-ωt)] we obtain two solutions corresponding to the transverse electric (TE) and transverse magnetic (TM) waves. Dispersion relation for TMwave is given by

and practically depends only on the parameters of the effective permittivity tensor. In this wave, the magnetic field H||B ₀ and does not produce any interesting effects. That is why, in the following we shall restrict ourselves to the consideration of TE-wave for which the electric field is parallel to B ₀ : E={0,0,E}, and the magnetic field is parallel to the xy-plane:

where

is the Voigt permeability,

corresponds to zero of µ_v(ω).

3. Regions of propagation and dispersion curves of TE-wave. Conditions of existence for the backward mode

The dispersion relation of TE-wave is given by

αnd describes the coupled magnon-plasmon polaritons with two different frequency regions of propagation:

if the effective plasma frequency ω_{p 1}<ω_{s 3}, and

if ω_{p 1}>ω_{s 3} Here

is the resonance frequency at which the wave number k →∞,β is the angle between the wave vector k and the+y-axis; ω=ω_{p 1} and ω=ω_{s 3} are the cutoff frequencies at which k=0.

The corresponding dispersion curves are shown schematically in Fig. 2, for three different values of the effective plasma frequency: ω_{p 1}<ω_r [Fig. 2(a)], ω_r<ω_{p 1}<ω_{s 3} [Fig. (2b)] and ω_{p 1}>ω_{s 3} [Fig. (2c)].

 

Fig. 2. Dispersion curves of TE-wave for the cases: a. ω_{p 1}<ω_r, b. ω_r<ω_{p 1}<ω_{s 3}, c. <ω_{p 1}>ω_{s 3}.

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One can see that the dispersion of the high-frequency branch is always normal: dω/dk >0 and the angle φ between the phase and group velocities is acute while the behavior of the low-frequency branch depends on the ratio δ=ω_{p 1}/ω_r. For δ<1, the dispersion of the branch is normal while for δ>1, it is anomal: dω/dk<0 and φ>π/2. To be convinced of such a behavior of the wave, let us examine the sign of the scalar product kS, where S is the time averaged Poynting vector. For TE-wave under consideration, vector S has the following form:

Using Eqs. (10) and (13), we obtain

according to which the wave is only forward (kS>0, φ<π/2) for ε ₃>0, that is, in the frequency region ω>ω_{p 1}. Thus, in the high-frequency regions of the wave propagation in Figs. 2(a,b,c), the SL behaves as a regular right-handed medium.

On the other side, the power flow in the wave is in the backward direction with respect to the phase velocity, that is, kS<0, φ>π/2, only if the following conditions are fulfilled:

After a simple analysis we conclude that conditions (15) can only be fulfilled simultaneously in the frequency range

that is, for the low-frequency branchs in Figs. 2(b,c).

Note that conditions (15) are quite different from those in an isotropic case when scalar ε and µ should be negative simultaneously [5]. Indeed, in (15) only one of the components of effective permittivity tensor is to be negative, namely, the diagonal component ε ₃ along the direction of the external magnetic field. Also, instead of the scalar µ, the Voigt permeability µ_v must be negative. Moreover, an additional condition restricting the direction of the backward mode propagation appears. For instance, if µ ₁<0 and µ ₂>0, the third condition (15) gives tan² β<µ ₂/|µ ₁|.

4. Negative refraction and problem of reflection. Phase and group refractive indices

Consider now a TE-type polarized EMW which is incident from the vacuum on the plane surface of the medium, at the angle of incidence α. In this case, the refracted wave is TE polarized too and is described by Eq.(10). According to the causality principle, the energy flux of the wave is always directed away from the interface: S_y>0. Then, using Eq.(13) we conclude that in the region (16), the normal component of the wave vector is negative, that is, the wave vector of the refracted wave is directed to the interface. In other words, with respect to the low-frequency branch, the medium behaves as a left-handed (LH) one.

Using Eq. (14), for the angle φ between the vectors k and S we obtain

Where

and

are the phase and group refractive indices, β is the angle of refraction and γ is the angle between the group velocity of the refracted wave V _g=∂ω/∂ k (or the Poynting vector S) and the+y-axis. Both n_p and n_g depend on the angle of incidence and are given by

Note that in Eq.(20a) n_p is positive while after Veselago[5] it is accepted to use negative sign for it. However, a careful examination of the wave refraction problem shows (for the details see Ref.[16]) that the phase refractive index determined as the ratio of the light speed in vacuum to the modulus of the phase velocity (ck/ω) is always positive while the group refractive index can be both positive and negative. The confusion in the sign of n_p appears because of in an isotropic LHM, one has n_p=|n_g|.

For the backward wave under consideration, n_g changes its sign at ω=ω_σ, where ω_σ is given in Eq.(5c) and corresponds to zero of µ ₁(ω). Thus, the frequency region of existence for the backward mode (16) must be divided in two parts:

and

where the symbols ω_σ, ω_r, ω_{p 1} and ω_{s 3} are key model parameters the meaning of which is explained above [see also Eqs. (4c), (9a) and (12)].

In part I, µ ₁<0, S_x>0 and consequently, n_g is positive; in other words, SL behaves as a positive index LH-medium [see Fig. 3(a)]. Note that in this case, the refraction occurs at an arbitrary value of the angle of incidence.

 

Fig. 3. Refraction of TE-wave in the case of positive index (a) and negative index (b) LHM. k ₀, k _R and k are wave vectors for incident, reflected and refractive waves, respectively.

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(a)µ ₁<0, S_x>0, n_g>0. (b) µ ₁>0, S_x<0, n_g<0. In both cases ε ₃,µ_v<0.

In part ??, we have µ ₁>0, S_x<0 and n_g<0, that is, SL behaves as negative index LH (or absolute left-handed) medium. The corresponding picture of refraction is shown in Fig. 3(b). In this case, the refraction is only possible for α<α_c, where the critical angle of incidence is given by

We have to note again that for both cases shown in Figs. 3(a) and 3(b), the medium is left-handed: the refracted wave vector is directed to the interface, and the angle between the vectors k and S is obtuse: π/2<φ<π. These vectors can only be exactly opposite in part II, and only for α=α_c. However, in this case k_y=0, the wave is propagated along the interface in the direction parallel to the+x-axis. The main difference between Figs. 3(a) and 3(b) is the different orientation of the Poynting vector with respect to the normal to the interface: for the case n_g<0, it is in the same side of the normal as in the incident wave while for the case n_g>0 it is in the opposite side, as in a regular medium.

It is important to note the possibility of tuneability of both regions of anomalous refraction (21a) and (21b) by means of external magnetic field variation. Using Eqs. (5c), (9a) and (12) for the characteristic frequencies ω_σ,ω_{s 3} and ω_r, it is easy to see that with increasing magnetic field the low frequency region (21a) shifts toward the high frequency side without any change of its width. The tuneability of the negative index region (21b) is more interesting and promising from experimental point of view: for a given value e of ω_{p 1}>ω_{s 3}, the lower limit of the region increases monotonically with increasing magnetic field while the interval of the region increases initially, reaches a maximum value at ω_{s 3}=ω_{p 1}, then decreases rapidly and vanishes at ω_σ=ω_{p 1}.

Finally, let us consider briefly the problem of reflection from the surface of LHM. Using the standard continuity conditions for the tangential field components at the interface, we obtain the reflection coefficient for our TE-polarization case in the form

where n_p is given by Eq.(18a). It is easily to see that the function R(α) has a minimum value R_m at α=α_m, where

For a given value of ω in the range (16), all the parameters A, B,C are positive. The minimum exists only if the condition

is fulfilled. Normal component of the refracted wave vector at α=α_m is given by

Note that in different frequency ranges (21a,b), angle dependence of the function R(α) is different [see Figs. 4(a,b)].

 

Fig. 4. Angle dependence of the reflection coefficient from the surface of LHM with positive (a) and negative (b) group refractive index.

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In these figures, the coefficient of reflection at the normal incidence is given by

and the angle α_c at which the total reflection occurs, is given by Eq.(22).

5. Conclusions

The obtained results may be summarized as follows. The dispersion, refraction and reflection of electromagnetic waves are studied analytically in ferromagnet-semiconductor multilayer in the presence of an external magnetic field. It is shown in terms of effective medium approach and Voigt configuration that such a multilayer or a superlattice can behave as a left-handed medium for a TE-polarized wave.

General necessary conditions are found for anomalous refraction [see Eq.(15)] which are quite different from those in an isotropic case considered in [5]. It is shown that a backward mode exists only if the effective plasma frequency, at which the wave has a cutoff, exeeds the resonance frequency given in Eq.(12). Unlike the isotropic case, only one of the diagonal components of the effective permittivity tensor, namely, the component along the external magnetic field direction should be negative. Also, instead of the scalar µ(ω) in an isotropic case, the Voigt permeability µ_v(ω) given in Eq.(9) must be negative. Finally, an additional condition restricting the direction of the refracted wave propagation must be fulfilled.

A frequency range is found in which all the necessary conditions are fulfilled simultaneously. It is shown that the range can be devided in two parts, in one of which (low frequency part) the group refractive index is positive while in the second one it is negative. These parts are separated by the characteristic frequency ω_σ given in Eq.(5c). The main difference between these parts is the different orientation of the Poynting vector of the refracted wave with respect to the normal to the interface [see Figs. 3(a) and 3(b)]. In both these regions, the medium is left-handed: the refracted wave vector is directed to the interface and makes an obtuse angle φ>π/2 with the Poynting vector.

It is shown that with increasing magnetic field, the positive index region shifts toward the high-frequency side without any change of its width while the negative index region becomes wider initially, reaches a maximum value and then decreases monotonically. Application of this result is promising from experimental point of view, taking into account the possibility of tuneability of the regions by means of external magnetic field variation.