The THz and infrared photonic dispersion relations of superlattices composed of polaritonic material and bismuth (Bi) layers are theoretically investigated. The semimetal Bi presents a far-infrared response described by a Drude-like formula with a large high-frequency background dielectric constant. We have considered three different polaritonic materials: NaBr, LiCl and LiH, such that the Bi plasma frequency lies above, in-between and below the characteristic frequencies of the polaritonic materials, respectively. Investigating the photonic band structure of each superlattice, we have found that when the Bi plasma frequency lies below the polaritonic gap, appears a photonic pass band of negative dispersion for TM modes just above the longitudinal-optical phonon frequency, attributed to the frequency dependence of the permittivity for the polaritonic material and the large high-frequency background dielectric constant for Bi, that can be relevant to the metamaterials community.
© 2015 Optical Society of America
The study of the interaction of electromagnetic waves with periodic multilayer systems goes back at least a couple of centuries ago, with the pioneering results of Lord Rayleigh in 1888, showing the existence of a photonic band-gap in one dimensional multilayer dielectric stacks. In recent years, the interest in the field has grown with the seminal works of Yablonovitch  and John  both published in 1987, extending the study not only to one dimensional systems. Yablonovitch showed that it is possible to inhibit totally the spontaneous emission of atoms by modifying its environment, opening the possibility of wide technological applications that we appreciate nowadays. On the other hand, John showed that for specific disordered dielectric superlattices it is possible to strongly localize light, which is known as Anderson localization . The applications of such periodic systems, named presently as photonic crystals, are focused mainly on one dimensional periodic systems in the thin-films field, used to eliminate undesired reflections in mirrors and lenses, as well as coatings in paint and ink industries. The fabrication of one dimensional systems can be as easy as simply stacking layers onto a substrate. Drilling holes in a suitable substrate can be a way of fabricating two dimensional photonic crystals, and in fact holey fibers are commercially available at present. However, the situation is remarkable different for three dimensional structures, being still far from commercial applications. Among the variety of fabrication techniques highlights photolithography , interfering two pulsed lasers, and self-assembly colloidal crystals, such as inverse opals [5, 6].
In the last fifteen years, the interest in the field of photonic crystals has been renewed with the appearance of the photonic metamaterials or simply called metamaterials, obtaining the first experimental realization in 2000 by Smith et. al. , showing the existance of a band where both the dielectric permittivity and the magnetic permeability become negative at GHz frequencies. The main properties of such a system were first studied by the pioneering work of Veselago in 1968 , and since then, the topic has attracted the attention of a large number of researchers. The potential applications of such systems, not only in electromagnetism but also in acoustic  and seismic  fields, and recently in medicine , exploit their properties such as perfect lensing  and invisibility [13–16].
As it is well known, metal is the main functional component of most of today’s metamaterials, basing their fascinating properties and possible applications mostly on the negative permittivity response of the metal, resulting from the resonant free electron currents [7,17]. In order to reduce losses, it has been proposed the use of high-index dielectrics instead of metals achieving both negative permeability (like artificial magnetism) and negative permittivity [18–20]. A suitable alternative can be the use of polaritonic materials, which combine the advantages of both metallic and high-index dielectric materials. Polaritonic materials are polar crystals where an incident electromagnetic wave can excite optical phonons (lattice vibrations) within the crystal, occurring this coupling typically in the THz regime . The electromagnetic response of this type of materials can be described by a Lorentzian resonant electric permittivity response function, characterized by both strong positive and negative permittivity regimes, making polaritonic materials a perfect replacement of metals in metamaterials working at THz frequencies. By tailoring properly such polaritonic inclusions, it is possible to achieve metamaterial properties like negative effective permeability  or negative refractive index [19, 22].
With the recent developing technology of THz sources, such as Quantum Cascade Lasers (QCL) , the interest in exploring and exploiting the THz regime becomes more appealing, due to the huge technological applications, ranging from telecommunications , security and sensing , tissue imaging , and even in astronomy . Since the conventional optical devices are not suitable in general to manipulate THz beams, there is the need to develop new kind of materials to be used as lenses, beam splitters, polarizers, collimators, etc. A good candidate to fulfill the requirements are the so-called THz metamaterials, that is, metamaterials with polaritonic inclusions showing resonances at THz regime, a topic that has attracted the attention of a growing scientific community (see, for example, [28–33] and references therein). Indeed, as shown in previous works [29,30], two-dimensional photonic crystals composed of dielectric-polaritonic or polaritonic-polaritonic material constituents behave as THz uniaxial metamaterials with hyperbolic dispersion relation. Moreover, the THz photonic band structure for superlattices of metal and polar material is characterized by flat Fabry-Perot resonance bands outside the polaritonic gap . Such Fabry-Perot resonance bands turn out to be noticeably altered by the metal spatial dispersion, which is well manifested in the THz range [34, 35]. Another interesting conducting material to be used in THz metamaterials is the semimetal bismuth (Bi), since its frequency-dependent permittivity is well described by the local Drude model with an infrared plasma frequency and a large high-frequency background dielectric constant [36, 37]. The use of Bi as one of the material constituents in polaritonic superlattices might lead to a strong plasma-phonon polariton coupling, which could manifest itself in the photonic dispersion, as well as optical spectra.
In this paper, we present a study of a one-dimensional periodic stack-slab system, using a polaritonic slab material inserted in between bismuth slabs. We explore the consequences in the THz spectrum of the whole system by changing the position of the polaritonic gap of the polaritonic material with respect to the Bi plasma frequency, showing the appearance of frequency bands with negative dispersion. In Sec. 2, we present the general analytical formulas for the photonic dispersion relation for transverse-electric (TE) and transverse-magnetic (TM) modes in a superlattice composed of alternating layers of polaritonic material and Bi. In Sec. 3, explicit formulas for the calculation of the principal values of the macroscopic permittivity tensor are applied in order to analyze the photonic dispersion relation at long wavelengths for polaritonic-material/Bi superlattices, as well as their optical characterization, named reflection and transmission spectra. Finally, the size quantization of polaritons in superlattices, having relatively thick layers of both polaritonic material and Bi (i.e. when the long wavelength approach does not hold in the THz range), are studied in Sec. 4. Our findings are summarized in Sec. 5.
2. Photonic dispersion
We consider a binary superlattice composed of metallic and resonant-dielectric layers, which are alternated along the z-axis (Fig. 1), both materials being non-magnetic materials. The infrared response of the polaritonic material (a-layer) is described with a resonant permittivity of the form
Using the permittivities (1) and (2), Maxwell equations, the boundary conditions for the electric and magnetic fields (i.e. the continuity of their tangential components at the interfaces of the superlattice layers), and the Bloch theorem, one can derive the dispersion equation for the plasma-phonon polaritonic modes as [38–41]
Here, kz is the Bloch wave vector, , , and kx is given by the x-component of the wave vector of incident light on the superlattice (in practice, a multilayer periodic stack of finite size L ≫ Λ), and c is the velocity of light in vacuum.
3. Effective medium approach
In the case of wavelengths much larger than the superlattice period Λ, |kz|Λ ≪ 1, the optical response of a binary superlattice can be described by using an effective permittivity tensor εij = εiδij (i, j = x,y,z) with diagonal components given by 
Bi is characterized by a dielectric tensor corresponding to a uniaxial crystal, which has two basic dielectric tensor components corresponding to the ordinary and extraordinary axes. Here εb corresponds to the ordinary axis since the z-axis is oriented parallel to the growth direction of the superlattice. Thus, in the case of s-polarization the Bi optical axis should be oriented along the x-axis, whereas for p-polarization should be oriented along y-axis.
The use of Rytov formulas (5) for superlattices of metal components can be questioned [43–46], because they were derived by assuming smooth variations of the electromagnetic field inside each layer,46]:
Using the diagonal effective permittivity tensor, εiδij (i = x,y,z), the dispersion relation for TE and TM electromagnetic modes with zero y-component of the wave vector (ky = 0) can be written as
We equate kx with the x-component of the wave vector of the incident light on the superlattice: kx = (ω/c)sin θ, where θ is the incidence angle. Then, the dispersion relations (9) can be rewritten as
In the next subsections, the dispersion curves for TE and TM modes in NaBr/Bi, LiCl/Bi, and LiH/Bi superlattices, for an incidence angle θ = 45°, will be calculated by using the exact expressions (3) and (4). Such superlattices are interesting because the plasma frequency for Bi is in the THz range (ωP = 60.77 × 1012rad/s (0.04 eV), taken from ) and close to the frequencies of the longitudinal and transverse optical phonon modes (at Γ point) of the polaritonic materials (see Table 1 where the high-frequency permittivity ε∞ values are also indicated). Other Bi parameters used here are [36, 37]: its relatively-large background permittivity εb,∞ = 102 and a damping parameter ħνb = 9 × 10−3eV. The chosen damping constant for a polaritonic material is νa = 10−2ωTO. Besides, the thicknesses of the superlattice layers are: da = 0.3Λ and db = 0.7Λ (Λ = da + db = 0.15µm). Consequently, at THz frequencies, the photonic band structures for such superlattices can also be obtained by applying the effective medium approaches discussed above. According to the chosen parameters, the conditions for applying Rytov formulas (5), i.e. the smooth variation of the “microscopic” field inside the layers (6), are well satisfied at frequencies ω of the order of the Bi plasma frequency ωP except, perhaps, near the resonance frequency ωTO for the polaritonic material. In addition, in using the other formula for the effective permittivity components εx = εy [i.e. Eq. (7)], the condition of smallness of the parameter |, Eq. (8), is fulfilled within a wide frequency interval (see Fig. 2). Notice that the quantity is independent of the resonant THz response of the a-layers.
Below, we will also show s- and p-polarization reflectivity and transmissivity spectra for NaBr/Bi, LiCl/Bi, and LiH/Bi multilayer stacks, having 50 bilayers and being embedded in air. The optical spectra are calculated by applying the transfer matrix method  and compared with the photonic dispersion curves for polaritonic-material/Bi superlattices.
3.1. NaBr/Bi superlattice (ωLO < ωP)
In the case of NaBr/Bi superlattice, the plasma frequency ωP of Bi is above the reststrahlen band for NaBr. In panels (a) and (c) of Fig. 3, we present the frequency dependences of the quantities η1 and η2, whose square root determines the wave vector kz for TE and TM modes according to Eqs. (10) and (11). The curves η1(ω) and η2(ω) were calculated with applying both Eq. (5) and Eq. (7). However, the difference between their predictions is not discernible in the panels. Below ωP, both η1 and η2 have a negative real part, as well as a large imaginary one. For this reason, the values of kz for TE and TM modes are complex numbers with relatively-large imaginary parts at ω < ωP, whereas they become positive real numbers at ω > ωP. So, there is a low-frequency stop band, whose width Δ is determined by an effective plasma frequency ωP,eff (Δ ≡ ωP,eff), which can be estimated by using Rytov formulas. Indeed, employing Eqs. (2) and (5), we can write the effective permittivity components εx and εy as
In deriving Eq. (12), we have used εa = εa,∞ which is valid at frequencies ω ≳ ωP > ωLO. From Eqs. (10)–(14), it follows that ℜη1 [ℜη2] vanishes at ω ≈ ωP,eff ≈ ωP in good agreement with the results shown in panel (a) [(c)] of Fig. 3.
Subfigures 3(b) and 3(d) respectively show the s– and p–polarization reflectivity and transmissivity spectra for a stack of 50 NaBr/Bi bilayers in air. The angle of the incident light is θ = 45°. These results confirm the existence of a stop band below the effective plasma frequency ωP,eff (13). Indeed, at ω < ωP,eff ≈ ωP, the transmissivities Ts and Tp are extremely small. The fact that the reflectivity spectra (Rs and Rp) are smaller than the unity, implies that the absorption (A = 1−R−T) inside the multilayer system is considerable. Unlike the Rs spectrum, having only one resonance at ω = ωTO < ωP, the reflectivity Rp exhibits an additional resonance at ω = ωLO < ωP. Above the effective plasma frequency ωP,eff, there is a pass band where both transmissivity and reflectivity spectra have oscillations associated with Fabry-Perot resonances occurring within the whole multilayer stack.
3.2. LiCl/Bi superlattice (ωTO < ωP < ωLO)
In Fig. 4 (panels (a) and (c)), the frequency dependence of η1 (10) [η2 (11)] for TE [TM] modes in a LiCl/Bi superlattice at θ = 45° is presented. The curves η1(ω) and η2(ω) were calculated by using both effective-medium approaches, namely Eqs. (5) and (7). However, their difference is almost undistinguishable. In the case of the LiCl/Bi superlattice, the Bi plasma frequency ωP is between the frequencies of the transverse (ωTO) and longitudinal (ωLO) optical phonons for LiCl. Since the wave vector kz is given by the square root of η1 and η2 for TE and TM photonic modes, respectively, a low-frequency stop band should appear below the Bi plasma frequency ωP where both η1 and η2 have negative real parts together with large imaginary parts. Our numerical results for the s- and p-polarizaion reflectivity and transmissivity spectra (see panels (b) and (d) of Fig. 4) confirm the presence of the low-frequency gap at ω < ωP. So, the bottom of the first pass band for both TE and TM modes is below ωLO, i.e. inside the LiCl polaritonic gap. As in NaBr/Bi superlattice system, we can introduce an effective plasma frequency ωP,eff, which limits the low-frequency stop band. Using Rytov formulas (2) and considering the relatively-large Bi background permittivity (εb,∞ ≫ |εa(ω)| at ω ≳ ωP > ωTO), the components εx and εy of the effective permittivity tensor can be written asFig. 4). The photonic dispersion for TM modes also exhibits a narrow stop band near ωLO (compare subfigures 4(c) and 4(d)) and at the bottom of their second pass band (at ω = ωU), the real part of the factor in η2 (11) vanishes.
3.3. LiH/Bi superlattice (ωP < ωTO)
For the LiH/Bi superlattice, the LiH reststrahlen band is above the Bi plasma frequency ωP and, consequently, the dispersion curves for both TE and TM modes have a low-frequency band gap at ω < ωP where η1 (10) [η2 (11)] have negative real parts and large imaginary parts (see subfigures 5(a) and 5(c)). It should be commented that the frequency dependences η1(ω) and η2(ω) were calculated by using formulas (5) and (7) for the principal values of the effective permittivity tensor. As above (panels (a) and (c) in Figs. 3 and 4), there is no visible difference between the predictions of Rytov formula (5) and those of (7) since the superlattice layers are rather thin. The low-frequency band gap is well seen in both reflectivity an transmissivity spectra shown in panels (b) and (d) of Fig. 5. The first two pass bands for TE modes are separated by a narrow gap near the frequency of transverse optical phonons (ωTO). On the other hand, the band structure for the TM photonic modes has an additional narrow gap around ωLO. These results are interesting because the propagation of photonic modes in the superlattice can occur within the LiH polaritonic gap despite the fact that the photon modes in the polaritonic material a-layers are evanescent waves.
The effective plasma frequency, determining the width of the low-frequency photonic gap, can be analytically calculated with the Rytov formula (5). The frequency dependence of the components εx and εy at low frequencies (ω < ωTO) is given by
Using Eqs. (10), (11), (5) (this last for εz) and the approximate formula (16), it follows that the real parts of the quantities η1 and η2 vanish at frequencies very close to ωP,eff (17), in good agreement with the curves shown in the panels (a) and (c) of Fig. 5.
To determine the zeros ω1 and ω2 of η1(ω) and η2(ω), respectively, within the polaritonic gap (ωP < ωTO ≲ ω < ωLO), it is convenient to write the effective components εx and εy [Eq. (5)] as
Since the Bi background dielectric permittivity is large in comparison with that of LiH (εb,∞ ≫ εa,∞), the zeros of the real parts of η1(ω) (i.e εy − sin2 θ) and η2(ω) (i.e. εx) in the polaritonic gap are close to ωTO:
Substituting the parameters for LiH and Bi into Eqs. (20) and (21), we get ω1 ≈ ω2 = 1.02ωTO. The quantity ℜ[η2] for TM modes has another zero at ω = ωU (the bottom of the third TM pass band, see subfigures 5(c)) because of the vanishing of the factor in (11) at that frequency.
4. Size-quantized polaritons
Let us calculate the photonic band structure of a LiH/Bi superlattice (ωP < ωTO) with layers thicker than those considered in Subsec. 3.3, such that the conditions (6) and (8) for the applicability of the effective medium approaches are not satisfied in the THz and far-infrared ranges. In Fig. 6, we present the photonic dispersion for both TE and TM modes calculated by using the general expressions (3) and (4). The thicknesses of LiH and Bi layers are, respectively, da = 0.9Λ and db = 0.1Λ, where Λ = 5.914µm (the reason for choosing such a Λ value is explained below) and the incidence angle is set to θ = 45°. Fig. 6 exhibits dispersion curves, kz(ω), corresponding to the solutions of Eqs. (3) and (4), whose imaginary part ℑ[kz] is positive. As it is seen in Fig. 6, the band structures for both TE and TM modes below the polaritonic gap (ω < ωTO) are characterized by narrow pass bands, associated with Fabry-Perot resonances arising in the polaritonic layer because of the high contrast between LiH and Bi impedances (|α| ≫ 1). In such a case, the dispersion equation (3) for the electromagnetic modes has solutions for the Bloch wave number kz with |ℜ[kz]| < π/Λ and 0 < ℑ[kz] < |ℜ[kz]| in slightly-dispersive bands, including the frequencies ωj (j = 1,2,3,…) where the conditionFig. 7, the blue circles indicate the frequencies where the value of is precisely an integer. In fact, Eq. (22) was used to choose the thickness da of LiH layer such that the first Fabry-Perot resonance band appear at ħωj=1 = 0.03eV which is smaller than ωP (ω1 < ωP). The thickness db of the Bi layer was assumed to be equal to da/9 and, consequently, Λ = da + db = 5.914µm. At the frequency ω1 < ωP, Bi has a negative permittivity with large absolute value . On the other hand, at frequencies ωj > ωP (j > 1), the Bi layers possess a large positive permittivity (εb ≈ εb,∞). According to Figs. 6 and 7, the Fabry-Perot resonance bands are highly concentrated just below the polaritonic gap.
Because of the frequency dependence of the LiH permittivity, εa(ω) (1), the Fabry-Perot resonance condition (22) is also satisfied at frequencies ωj′ (j′ = 1,2,3,…), being above the polaritonic gap (ωj′ > ωLO) (see Figs. 6 and 7). It should be noted that at such frequencies the contrast between the impedances of the a- and b-layers is rather large and, therefore, the Fabry-Perot resonance bands are very narrow. The photonic band structures for both TE and TM modes also exhibit an almost-flat pass band inside the polaritonic gap (Fig. 6). The band corresponds to the lowest Fabry-Perot resonance in the Bi layer, at the frequency ωm=1, where the conditionFig. 7).
In the case of TM modes, there is another pass band at ω = ω* > ωLO (see panel (c) of Fig. 6). This peculiar band does not correspond to any size-quantized polariton (photon) mode in the LiH (Bi) layers. Its origin is related with the frequency dependence of the parameter α(ω) in Eqs. (3) and (4). In general, the absolute value of α is large because of the high contrast between the impedances of the LiH and Bi layers. However, the function α(ω) has a zero at ω = ω*, where the solution kz for the dispersion equation (3) may be a real number. Within the leading approximation in the parameter εa,∞/εb,∞ ≪1, the frequency ω* can be calculated by using the formula
Substituting the LiH parameters into Eq. (24), we obtain ω* = 1.057ωLO.
The photonic band structure shown in Fig. 6 corresponds to modes whose wave vector has a positive imaginary part (ℑ[kz] > 0), i.e. modes decaying along the positive direction of the z-axis. The pass bands of negative dispersion are those where ℜ[kz] [|ℜ[kz]| ≫ℑ[kz]] and the derivative dω/dkz have opposite sign. Interestingly, the Fabry-Perot resonance band (m = 1) inside the polaritonic gap and the photonic band just above ωLO (at ω = ω*) possess negative dispersion.
Panels (a) and (b) of Fig. 8 show the s- and p-polarization reflectivity and transmissivity spectra for a multilayer stack of 50 LiH/Bi bilayers. The thicknesses of LiH and Bi layers are equal to those of the superlattice analyzed in previous paragraphs. The calculated optical spectra agree with the predicted photonic band structure for the LiH/Bi superlattice (Fig. 6). Indeed, both s- and p-polarization reflectivities exhibit resonances at frequencies corresponding to the pass bands associated with size-quantized polariton and photon modes in LiH and Bi layers, respectively (compare Figs. 6 and 8). As can be seen, the transmissivity is rather small implying considerable absorption in the superlattice.
We have studied the effect of the plasma-phonon polariton coupling upon the photonic band structure of superlattices composed of Bi and polaritonic material layers. We have analyzed the photonic dispersions for TE and TM plasma-phonon polaritons in three cases: i) the Bi plasma frequency is above the LO phonon frequency (ωLO < ωP) as for NaBr/Bi superlattices; ii) the Bi plasma frequency is inside the polaritonic gap (ωTO < ωP < ωLO) as for LiCl/Bi heterostructures; and iii) the Bi plasma frequency is below the TO phonon frequency (ωP < ωTO) as for LiH/Bi superlattices. Due to the plasma-like behavior of the semimetal Bi in the infrared, in the three cases investigated there is a low-frequency band gap similar to that observed in metal-dielectric superlattices [35, 44, 46]. We have generated reflectivity and transmissivity spectra for each of the three cases investigated to help experimentalists to compare results with our theoretical predictions.
In the long wavelength limit, we have calculated the effective permittivity tensor for homogenized polaritonic-material/Bi superlattices, which behave as uniaxial crystals. The frequency dependence for the transverse components of the effective permittivity tensor is described by a Drude-like formula, but with an effective plasma frequency, ωP,eff. Thanks to the high-frequency background dielectric constant of Bi, ωP,eff is very close to the Bi plasma frequency ωP in all the cases. We have shown that the effective medium theory developed by Rytov  and a more accurate theory  coincides for the three cases investigated, thus enabling the use of the simpler theory to model these kind of superlattice systems.
For superlattices with relatively thick layers, their infrared photonic band structures exhibit narrow pass bands associated to Fabry-Perot resonances either in the polaritonic layers or in the Bi ones. These resonance bands are owing to the high-dielectric contrast between the polar material and the semimetal Bi. In the photonic dispersion for both TE and TM modes propagating in a LiH/Bi superlattice, we have observed a Fabry-Perot resonance band inside the polaritonic gap of the polaritonic material. Interestingly, for TM modes there is, additionally, a pass band just above the LO phonon frequency that presents a negative dispersion, attributed to the frequency dependence of the permittivity for the polaritonic material and the large high-frequency background dielectric constant for Bi, that can be relevant to the metamaterials community.
This work was partially supported by SEP-CONACyT (grant CB-2011-01-166382), VIEP-BUAP (grants 71-EXC13-I and ZEMB-ING13-I), PIFI 2012, and PIFCA 2013. A.R.-C. thanks the financial support from PROMEP (grant 103.5/12/4367) and PAPIIT-UNAM (grant IA105015).
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