Effective Parameters for 1D Photonic Crystals with Isotropic and Anisotropic Magnetic Inclusions: Coherent Wave Homogenization Theory

A homogenization theory that can go beyond the regime of long wavelengths is proposed, namely, a theory that is still valid for vectors of waves near the edge of the first zone of Brillouin. In this paper, we consider that the displacement vector and the magnetic induction fields have averages in the volume of the cell associated with the values of the electric and magnetic fields in the edges of the cell, so they satisfy Maxwell’s equations. Applying Fourier formalism, explicit expressions were obtained for the case of a photonic crystal with arbitrary periodicity. In the case of one-dimensional (1D) photonic crystals, the expressions for the tensor of the effective bianisotropic response (effective permittivity, permeability and crossed magneto-electric tensors) are remarkably simplified. Specifically, the effective permittivity and permeability tensors are calculated for the case of 1D photonic crystals with isotropic and anisotropic magnetic inclusions. Through a numerical calculation, the dependence of these effective tensors upon the filling fraction of the magnetic inclusion is shown and analyzed. Our results show good correspondence with the approach solution of Rytov’s effective medium. The derived formulas can be very useful for the design of anisotropic systems with specific optical properties that exhibit metamaterial behavior.


Introduction
The concept of metamaterial was initially introduced to explain the striking physical properties of photonic crystals, composed of resonant elements or having a very large dielectric contrast. For example, recently in [1], the design of a bilayer device (1D photonic crystal) made of dielectric materials on a substrate was reported. This membrane-type device enhances the Faraday rotation in comparison with plasmonic or metal-dielectric periodic structures, as well as showing good efficiency as a waveguide for transverse magnetic and transverse electric polarizations. Staude and

Finite Fourier Transform
In this section, we shall derive an expression for the effective tensors based on the principle of wavelengths comparable with the lattice period for characterizing the bulk optical response of a homogenized magneto-dielectric photonic crystal. The most general form for such a response is the so-called bianisotropic response, generally written in the EH-(or Tellegen) representation, using the permittivity ↔ ε and permeability ↔ µ tensors and two crossed magneto-electric dyadics, ↔ ζ and ↔ ξ , which depend on the position ( → r ) in the photonic crystal. Such tensorial representation relates the electric E and magnetic H fields with the displacement vector D and the magnetic induction B: The behavior of these fields is governed by Maxwell equations: ·B( → r ) = 0.
Now, with Equations (1) and (2), we rewrite the laws of Ampere-Maxwell (in Equations (3) and (4)) and Faraday in (Equations (5) and (6)) in matrix form: In this latter equation, the dyadic unit and zero are introduced ( ↔ I and ↔ 0 , respectively), as well as the vectors EH and DB (v( → r ) and w( → r ), respectively), given by: Due to the periodicity of the bianisotropic response tensors, we write the matrix  Moreover, we can expand the vectors in Bloch waves due to the periodicity along the plane parallel to one of the surfaces of the photonic crystal:

v(
where → G || is a vector of the reciprocal lattice of the plane perpendicular to the z-direction. Substituting Equation (11) into Equation (7), we get: Now, we will apply the finite Fourier transform in an interval along the z-axis of length equal to the lattice constant a: where G z = 2πn z a , n z =−∞, . . . , ∞. The inverse transform is given by the series: From Equations (12) and (13), we obtain: Rewriting this equation and considering that: here, = A is the matrix of the bianisotropic response. Therefore, Equation (15) can be rewritten as: where: The amplitudes v( Materials 2020, 13, 1475 Using Equations (17)-(19), a homogeneous system is obtained for the amplitudes v( and v( → G || , z = a). Therefore, the amplitudes for → G || 0 can be expressed in terms of one of them, for example, v( → G || = 0, z = 0). Now, let us determine the effective response. From Equations (17) and (19), we have: This expression can be written as: Evaluating this equation in → G || = 0, we obtain an equation for the coherent component: In the bulk of the photonic crystal, the Bloch theorem must be satisfied, v( → G || = 0, z = a) = e iq z a v( → G || = 0, z = 0), so the term on the left side in Equation (23) can be written as: Substituting this into Equation (23), we arrive at the equations for the coherent amplitude: which correspond to the Maxwell equations for the amplitude of the field E-H, propagating in the z-direction, in a homogeneous medium: where  (21)). Therefore, we have explicit expressions to calculate the effective tensors of the bianisotropic response of a homogenized photonic crystal without any restrictions on the wave vector q z .

1D Photonic Crystals
Let us calculate the effective parameters for a homogenized binary (bilayer) 1D photonic crystal. (16) are given by Equation (28); the crossed magnetoelectric tensors are assumed to be equal to 3 × 3 zero matrices:

The Fourier coefficients
effecting the integral on the unit cell, the following expression is obtained: The matrix = D (Equation (18)), acquires the simplest form: and it satisfies On the other hand, the effective matrix takes the form: so, the effective matrix (Equation (33)) acquires the simplest form: Note that this expression depends on q z , which in general is non-local, i.e., the effective parameters depend on both frequency ω and the wave vector q z .

Implementation and Results
In this section, we apply the Coherent Wave Homogenization theory developed in the previous section for calculating the effective permeability and permittivity tensors versus the filling-fraction of 1D magneto-dielectric photonic crystals. Here, it is convenient to indicate that in the case of alternating layers being sufficiently thin in comparison with the characteristic wavelength, we can treat the whole system as an anisotropic medium with effective response tensors (we consider that the Bloch wavelength is much greater than the lattice parameter of the photonic crystal, q z a << 1). As follows from Figure 1 and according to the previous conditions for wavelength frequencies in THz, a lattice parameter a = 0.15 µm is proposed, d is the thickness of the inclusion layer and for the case of a superlattice with two alternating layers, the filling fraction f = d/a, and is valid for values of d in the range 0 ≤ d ≤ a. In this work, the interest in the THz regime becomes more attractive due to the huge technological applications, e.g., telecommunications, sensing and astronomy. In general, the periodicity of the photonic structure will be proportional to the wavelength of the electromagnetic radiation to be controlled. Therefore, for homogenized photonic crystals of arbitrary thickness, the effective electromagnetic magnitudes of magnetic permeability and dielectric permittivity that characterize the propagation of electromagnetic waves in the medium can always be determined. Here, we will study photonic crystals whose bilayer unitary cell is composed of a layer of dielectric material with inclusions of magnetic-isotropic and magnetic-anisotropic media. The constituents of the unit cell will be considered in a usual manner-that is, medium "A" will correspond to the inclusion and medium "B" to the background or host (see Figure 1).

1D Photonic Crystals with Isotropic Inclusion
Now, using Equation (35), we have numerically calculated the 6 × 6 matrix ̿ of the effective tensor of the bianisotropic response, in the local limit ( →0) with a total number of nz = 101 waves. As an initial application, we consider photonic crystals composed of a homogeneous host and isotropic inclusions. Specifically, in Figure 2, we present the results for a superlattice whose unit cell is composed of a magnetic material, in this case isotropic ferrite, considered as medium "A", and a dielectric material homogeneous to silicon, as medium "B"; the parameters considered for this calculation were: εA = 13ε0, μA = 8μ0 and εB = 12.25ε0, μB = μ0, respectively (ε0 and μ0 are the permittivity and permeability of the vacuum, respectively). In this figure we show the permittivity and the effective permeability as a function of the filling fraction f, note that εx = εy and μx = μy, since there is isotropy in the x − y plane. Figure 3 corresponds to a unit cell formed by ferrite in air; the parameters considered were: εA = 13ε0, μA = 8μ0 and εB = ε0, μB = μ0.

1D Photonic Crystals with Isotropic Inclusion
Now, using Equation (35), we have numerically calculated the 6 × 6 matrix = A e f f of the effective tensor of the bianisotropic response, in the local limit (q z →0) with a total number of n z = 101 waves. As an initial application, we consider photonic crystals composed of a homogeneous host and isotropic inclusions. Specifically, in Figure 2, we present the results for a superlattice whose unit cell is composed of a magnetic material, in this case isotropic ferrite, considered as medium "A", and a dielectric material homogeneous to silicon, as medium "B"; the parameters considered for this calculation were: ε A = 13ε 0 , µ A = 8µ 0 and ε B = 12.25ε 0 , µ B = µ 0 , respectively (ε 0 and µ 0 are the permittivity and permeability of the vacuum, respectively). In this figure we show the permittivity and the effective permeability as a function of the filling fraction f, note that ε x = ε y and µ x = µ y , since there is isotropy in the x − y plane. Figure 3 corresponds to a unit cell formed by ferrite in air; the parameters considered were: ε A = 13ε 0 , µ A = 8µ 0 and ε B = ε 0 , µ B = µ 0 .
Next, we considered photonic crystals with high dielectric and magnetic contrast in their unit cell. Figure 4 corresponds to a system of common glass in a homogeneous dielectric as the host, the parameters used were: ε A = 100ε 0 , µ A = 200µ 0 and ε B = 2.25ε 0 , µ B = µ 0 . Finally, in Figure 5, a medium of common glass in air is considered, whose parameters are: ε A = 100ε 0 , µ A = 200µ 0 and ε B = ε 0 , µ B = µ 0 .   Next, we considered photonic crystals with high dielectric and magnetic contrast in their unit cell. Figure 4 corresponds to a system of common glass in a homogeneous dielectric as the host, the parameters used were: εA = 100ε0, μA = 200μ0 and εB = 2.25ε0, μB = μ0. Finally, in Figure 5, a medium of common glass in air is considered, whose parameters are: εA = 100ε0, μA = 200μ0 and εB = ε0, μB = μ0.  Next, we considered photonic crystals with high dielectric and magnetic contrast in their unit cell. Figure 4 corresponds to a system of common glass in a homogeneous dielectric as the host, the parameters used were: εA = 100ε0, μA = 200μ0 and εB = 2.25ε0, μB = μ0. Finally, in Figure 5, a medium of common glass in air is considered, whose parameters are: εA = 100ε0, μA = 200μ0 and εB = ε0, μB = μ0.  Figures 2-5 indicate the averages of the permittivity and the permeability in the unit cell and it is observed that the increase in the values of εx (=εy) and μx (=μy), as well as in εz and μz in the function of the filling fraction present a linear and non-linear behavior, respectively. Therefore, our results describe the effective medium approach proposed by Rytov [15]-if the materials of the superlattice layers are local and isotropic, the permittivity tensor is diagonal with principal values [15]: Here, the metal and dielectric layers are characterized by their permittivity, m and d; f is the metal filling fraction and z denotes the direction perpendicular to the planes of the layers in the superlattice. Equations similar to the previous ones are fulfilled for the magnetic media:  2-5 indicate the averages of the permittivity and the permeability in the unit cell and it is observed that the increase in the values of ε x (=ε y ) and µ x (=µ y ), as well as in ε z and µ z in the function of the filling fraction present a linear and non-linear behavior, respectively. Therefore, our results describe the effective medium approach proposed by Rytov [15]-if the materials of the superlattice layers are local and isotropic, the permittivity tensor is diagonal with principal values [15]: Here, the metal and dielectric layers are characterized by their permittivity, εm and εd; f is the metal filling fraction and z denotes the direction perpendicular to the planes of the layers in the superlattice. Equations similar to the previous ones are fulfilled for the magnetic media: Evidently, when f →0, the medium "B" predominates and as the filling fraction increases, namely f →1, the material will correspond to the inclusion. On the other hand, in Figures 4 and 5 it is observed how for the effective electric permittivity and the effective magnetic permeabilities in the z-component for filling fractions near 0.8, their values remain very close to the host, later ascending rapidly before reaching the value of inclusion. This is due to the high contrast dielectric and magnetic qualities of the materials in the unit cell, the contrast being defined as the ratio of medium "A" and medium "B"-that is, εA/εB and µA/µB, respectively. Note that this behavior is not presented in Figures 2 and 3 due to the low dielectric and magnetic contrast between the host and the inclusion.

1D Photonic Crystals with Anisotropic Inclusion
In the applications that follow, we calculate the effective parameters of a photonic crystal whose unit cell is formed by an anisotropic inclusion and isotropic background in the dielectric constant and magnetic permeability. Figure 6a shows the results for a superlattice where the inclusion presents anisotropy in the dielectric constant; as can be observed, as the amount of the anisotropic medium increases (medium "A"), the effective response becomes anisotropic, creating three well-defined values-that is, when the filling fraction tends towards zero (f →0), the system is an isotropic homogenous medium with permittivity ε B . In the opposite case, when f →1 the system is an anisotropic homogenous medium with permittivity ε A . Note that ε x and ε y maintain a linear behavior determined by Equation (36), while the component ε z is represented by a curve defined by the result of the inverse relationship of Equation (37). In the case of effective permeability (see Figure 6b), a linear behavior is observed, as determined by Equation (38), which indicates an isotropic behavior for any filling fraction of the inclusion. Moreover, from the inspection of Figure 6b, it is evident that µ x = µ y = µ z , considering that both the inclusion and the host have a relative permeability approximately equal to one (vacuum permeability).
On the other hand, the effective parameters of a superlattice whose inclusion presents anisotropy in the magnetic permeability are determined. Therefore, the effective permittivity turns out to be isotropic throughout the filling fraction of the inclusion (see Figure 7a) and shows a linear behavior (ε x = ε y = ε z ), as determined in the first approximation by Equation (36). However, in Figure 7b, the situation is somewhat different because as the amount of the anisotropic medium increases, the effective response becomes anisotropic-that is, when f →0 the homogeneous medium is isotropic and when f →1 the homogeneous medium is clearly anisotropic. However, it should be noted that µ x and µ y maintain a linear behavior determined by Equation (38), while the µ z component is represented by a curve which is defined by Equation (39).
Finally, Figure 8 shows the results of the effective response for a photonic crystal whose inclusion in its unit cell presents anisotropy in the dielectric permittivity and magnetic permeability. As can be seen in both graphs, the physical situation is similar to that already discussed in Figures 6a and 7b for the effective permittivity and permeability, respectively, where it is basically noted that when increasing the filling fraction of the anisotropic medium, the effective response is anisotropic.
Materials 2020, 13, x FOR PEER REVIEW 10 of 13 In the applications that follow, we calculate the effective parameters of a photonic crystal whose unit cell is formed by an anisotropic inclusion and isotropic background in the dielectric constant and magnetic permeability. The parameters used were: εAx = 2.5ε0, εAy = 5ε0, εAz = 7.5ε0, μA = μ0 (anisotropic media) and B = 12.250, μB = μ0 (isotropic media). Figure 6a shows the results for a superlattice where the inclusion presents anisotropy in the dielectric constant; as can be observed, as the amount of the anisotropic medium increases (medium "A"), the effective response becomes anisotropic, creating three well-defined values-that is, when the filling fraction tends towards zero (f →0), the system is an isotropic homogenous medium with permittivity B. In the opposite case, when f →1 the system is an anisotropic homogenous medium with permittivity A. Note that εx and εy maintain a linear behavior determined by Equation (36), while the component εz is represented by a curve defined by the result of the inverse relationship of Equation (37). In the case of effective permeability (see Figure 6b), a linear behavior is observed, as determined by Equation (38), which indicates an isotropic behavior for any filling fraction of the inclusion. Moreover, from the inspection of Figure 6b, it is evident that μx = μy = μz, considering that both the inclusion and the host have a relative permeability approximately equal to one (vacuum permeability). The parameters used were: ε Ax = 2.5ε 0 , ε Ay = 5ε 0 , ε Az = 7.5ε 0 , µ A = µ 0 (anisotropic media) and ε B = 12.25ε 0 , µ B = µ 0 (isotropic media).  On the other hand, the effective parameters of a superlattice whose inclusion presents anisotropy in the magnetic permeability are determined. Therefore, the effective permittivity turns out to be isotropic throughout the filling fraction of the inclusion (see Figure 7a) and shows a linear behavior (εx = εy = εz), as determined in the first approximation by Equation (36). However, in Figure 7b, the situation is somewhat different because as the amount of the anisotropic medium increases, the effective response becomes anisotropic-that is, when f →0 the homogeneous medium is isotropic and when f →1 the homogeneous medium is clearly anisotropic. However, it should be noted that μx and μy maintain a linear behavior determined by Equation (38)  The parameters used were: ε A = 13ε 0 , µ Ax = 4µ 0 , µ Ay = 8µ 0 , µ Az = 12µ 0 and ε B = 12.25ε 0 , µ B = µ 0 .
isotropic throughout the filling fraction of the inclusion (see Figure 7a) and shows a linear behavior (εx = εy = εz), as determined in the first approximation by Equation (36). However, in Figure 7b, the situation is somewhat different because as the amount of the anisotropic medium increases, the effective response becomes anisotropic-that is, when f →0 the homogeneous medium is isotropic and when f →1 the homogeneous medium is clearly anisotropic. However, it should be noted that μx and μy maintain a linear behavior determined by Equation (38), while the μz component is represented by a curve which is defined by Equation (39). The parameters used were: ε Ax = 2.5ε 0 , ε Ay = 5ε 0 , ε Az = 7.5ε 0 , µ Ax = 4µ 0 , µ Ay = 8µ 0 , µ Az = 12µ 0 and ε B = 12.25ε 0 , µ B = µ 0 .

Conclusions
In this work, a homogenization theory for 1D magneto-dielectric photonic crystals was presented. The theory developed reports of analytical expressions that allow us to calculate the effective components of the permittivity and permeability tensors. In particular, we have analyzed the case of photonic crystals whose bilayer unitary cell is composed of a layer of dielectric material with inclusions of magnetic-isotropic and magnetic-anisotropic media, in the local limit, when the wavelength of the work frequency of the incident radiation on the medium is large compared to the period of the unit cell. The numerical implementation of the formulas provides results for new types of homogenized photonic metamaterials; in particular, we have studied theoretically the anisotropy of effective magnetic permeability and effective dielectric permittivity for homogenized magneto-dielectric binary superlattices versus the filling fraction. We demonstrated that the principal values for the components of the permeability and permittivity effective tensors describe a regime where Rytov's formulas are valid. Our results may be useful for the better comprehension and design of metamaterials, due to the anisotropy that they present in electro-magnetic modes, which can be manufactured even with structures as simple superlattices.