1. Introduction
In many photonic systems, we commonly focus on the interaction between electromagnetic waves and materials concerning the electric field component. This perspective is biased due to the relatively easy liberation of free electric charges, with magnetic interactions occurring only at a second-order level. Nevertheless, in some cases, because of the high mobility of the free charge carriers in conducting media, their electrodynamic parameters might strongly depend on external magnetic fields. Owing to this, various characteristics of optical devices containing conducting constituents can be altered with the use of an external magnetic field [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11]. Among various optical systems, optical superlattices, regular structures of alternating layers with different optical parameters, attract great attention. Due to this, it is of interest to analyze how the external magnetic field can alter the photonic characteristics of a superlattice consisting of alternating dielectric and conducting layers. Thus, the study of using magnetic fields in superlattices can be seen as a possible way to design and generate new optical devices for future all-optical technologies. Recently, even the influence of randomness has been studied in similar optical devices [
12], and nonlinear modes such as ferroelectric solitons in epitaxial bismuth ferrite superlattices have been successfully observed in experimental setups [
13]. In a standard approach, it is assumed that such an external magnetic field points in one constant direction, and then one studies how the strength of this field affects the way that light behaves in the superlattice system. However, in this study, we look at how changing the direction of the external magnetic field can alter the way that the superlattice interacts with light. One notable thing about this work is the consideration of anisotropy. We study the case when the conducting layers of the superlattice become electrodynamically anisotropic in the presence of the external magnetic field. Therefore, the behavior of the conducting layers in the superlattice may change, depending on which way the direction of the external magnetic field is applied. It is well-known how to obtain the photonic spectrum of the isotropic superlattices, which are described in terms of
unit-cell transfer matrix; see, for example, [
14]. As for the anisotropic scenario, the formalism of the
transfer matrix becomes inapplicable, and it has to be considerably modified. Because of the anisotropy, the dimensions of the unit-cell transfer matrix become
, which considerably complicates the theoretical analysis of the problem [
15]. For instance, to describe the eigenstates of the electromagnetic field in the superlattice, one must determine the eigenvalues of the unit-cell transfer matrix. For this, one has to find all roots of an algebraic equation of the fourth degree. In the present study, we show that finding the four roots problem can be reduced to finding the solution of a quadratic equation, whose roots are described by a very well-known formula.
Another critical question is how many new branches in the photonic spectrum of the superlattice can be generated by applying an external magnetic field. On the one hand, as is also well-known, in the absence of the external magnetic field, there is one dispersion relation; for example, ref. [
14] shows a degenerated Bloch electromagnetic eigenmode corresponding to propagation of the electromagnetic wave along the axis of the superlattice. On the other hand, the number of eigenstates of the electromagnetic field can be associated with the number of the eigenvalues of the unit-cell transfer matrix, whose value for the
unit-cell transfer matrix is equal to four. Although the unit-cell transfer matrix generally has four different eigenvalues, it turns out that the photonic spectrum of the superlattice in the presence of the external magnetic field is described by two dispersion equations for two different Bloch phases.
It is worth mentioning that there are, in fact, numerical methods that can be used to simulate the photonic properties of optical systems; see, for instance, [
16,
17]. It is essential to underline that such numerical algorithms can be applied quite effectively to isotropic optical structures. However, their effectiveness and practicality become significantly limited when it comes to anisotropic systems. Anisotropic materials possess unique properties and behaviors that set them apart from their isotropic counterparts, presenting a distinct challenge for conventional approaches. They have to be modified in a general case where tensors with all non-zero components describe the anisotropy, as takes place in the problem under consideration. The analytical results that we obtain in the present study significantly simplify the problem of describing the photonic properties of the magneto-optical superlattices because we can readily find frequency bands where the superlattice operates as a conductor of electromagnetic waves and where it does as an insulator.
It is worth mentioning that, in paper [
18], we studied the impact of the external magnetic field on the propagation of electromagnetic waves in general cases, where both the direction of the wave propagation and the orientation of the magnetic field are arbitrary. To do this, we numerically analyzed the eigenstates of the electromagnetic field within the framework of the transfer matrix method. However, we adopt a pure analytical methodology to unveil the intricate dispersion relations in this current study. These relations are essential in understanding how electromagnetic waves travel through materials with layered structures and distinctive magnetic properties. Our paper is organized as follows: in the next section, we describe the tensorial model used in our anisotropic superlattice system based on classical electromagnetic theory; next, in
Section 3, we obtain the corresponding analytical dispersion relations that characterize the photonic spectrum; then, in
Section 4, we apply the analytical results for describing a superlattice consisting of dielectric and semiconductor cells, showing the possibility to switch between pure photo-isolating and pure photo-conducting state; finally, in
Section 5, we offer the conclusions and point out future work. It is important to remark that, in this work, we are really focusing on just the theoretical aspects. Still, we hope our results can motivate future experimental works to design low-cost anisotropic superlattice photonic systems to control light for future all-optical technologies. In principle, the results obtained here can be used to help build and create an optical diode-like device due to the switching between conducting and non-conducting light in the system.
2. Materials and Methods: Anisotropic Superlattices
We start by establishing basic relations describing electromagnetic waves propagating along the axis of a superlattice, which is an array composed of identical unit-cells, in the presence of static and homogenous magnetic field
of arbitrary direction, as shown in
Figure 1. Without loss of generality, we consider that the direction of the magnetic field
lies in the
x–
y plane. The unit-cell contains two layers: the dielectric
a-slab and the magneto-optical plasma-containing
b-slab, with constant thicknesses
and
, respectively. Below, we assume that the magneto-optical
b-slab is a semiconductor and the superlattice is the sequence of the alternating layers of the isolator and semiconductor. The width of each unit
cell
is the spatial period of the superlattice. The dielectric
a-slabs are specified by permittivity
, refractive index
, wave number
, and phase shift
. Quantity
is
, where
is the frequency of the electromagnetic wave.
In the presence of the external magnetic field
, the magneto-optical
b-slabs are specified by the conductivity tensor
. When the direction of the external magnetic field
lies in the
x–
y plane, the conductivity tensor
of the magneto-optical
b-layers has the following elements [
18]:
where
is the plasma frequency,
n is the concentration of the charge carriers in the plasma,
m is the effective mass of the charge carrier with charge
,
is the cyclotron frequency of the charge carrier in the external magnetic field
, and quantities
and
are
and
. Expressions in Equations (
1)–(
6) are obtained within the Drude model. Note that angle
, which specifies the direction of the external magnetic field, enters the conductivity tensor through the quantities
and
:
The electromagnetic field in the superlattice is a superposition of the static external magnetic field
, the alternating electric
field, and the alternating magnetic
field. Directing the
x-axis normally to the layers of the superlattice, the alternating electromagnetic field reads as
As follows from the Maxwell equations,
inside the dielectric
a-layer, while, inside the magneto-optical
b-slab,
According to the conductivity tensor
, we have introduced the permittivity tensor
of the magneto-optical
b-layer in the presence of the external magnetic field:
Here, quantity
is the permittivity of the lattice of the magneto-optical layer.
3. Results: Anisotropic Dispersion Relations
From the anisotropic electromagnetic relations obtained in the past section, we proceed to look for their corresponding analytical dispersion relations. Note that, contrary to the magneto-optical slab, the dielectric layer is isotropic, and the form of the electromagnetic field inside it can be easily established. Thus, inside the dielectric
a-layer of the
nth unit-cell, the
x-dependent part of the electromagnetic field is
where
, and
and
are the coordinates of the left-hand side and right-hand side of the layer, respectively; quantities
,
,
, and
are the four complex amplitudes of the electromagnetic field inside the dielectric layer of the
nth unit-cell.
The
x-dependent part of the electromagnetic field inside the magneto-optical
b-slab is a superposition of four modes with four different wave vectors. Thus, inside the magneto-optical
b-slab of the
nth unit-cell, the components of the electric and magnetic fields are
where
and index
can be
x,
y, or
z;
and
are, respectively, the positions of the left-hand side and right-hand side of the magneto-optical
b-layer within the
nth unit-cell. The quantities
,
,
, and
in Equations (
18) and (
19) are the complex amplitudes of the electromagnetic field inside the magneto-optical
b-layer of the
nth unit-cell; the quantities
,
,
, and
are the
x-components of the wave vectors of the four electromagnetic modes inside the magneto-optical
b-layer.
Let us consider the equations for the electric and magnetic fields within the magneto-optical
b-slab. From Equation (
10), we obtain that
The first equation in Equation (
20) gives three scalar equations:
Here, the prime means the derivative with respect to
x. After substitution of Equation (
21) into Equations (
22) and (
23), we eliminate quantity
and obtain a system of two equations for
and
. Such a system can be written as one matrix equation for two-component quantity
:
where elements of 2 × 2 matrix
are related to the components of the permittivity tensor of the magneto-optical
b-layer, which is given by Equation (
11):
It can be checked that
.
The matrix equation in Equation (
24) has solutions that depend on
x as
, where
k is the
x-component of the wave vector within the magneto-optical
b-layer. One can see that a general solution of Equation (
24) can be formulated in terms of matrix
:
where
is the identity matrix. Thus, all possible values of
k—
,
,
, and
—can be determined from the condition
, which leads to a biquadratic equation:
From Equation (
27), we obtain the four possible components of the wave vector:
where the two refractive indexes
and
of the magneto-optical
b-layer are
The quantities
and
, where
, in the electric field components
and
are the nontrivial solutions of the homogeneous system of equations
The values are easily found. Using them with Equations (
21) and (
18), we also find the coefficients
, so that, for quantities
,
, and
, where
, we have
Using the second equation in Equations (
20) and Equation (
18), one finds that coefficient
of the magnetic field is zero, whereas the remaining ones,
and
, are expressed through quantities
and
, respectively:
We have obtained the general expressions that completely define the electromagnetic field distribution within the dielectric a- and magneto-optical b-layers. Inside the dielectric layer, the expressions for the components of the electromagnetic field contain four complex amplitudes: , , , and . Within the magneto-optical layer, the expressions for the components of the electromagnetic field contain four complex amplitudes as well: , , , and . With the use of the expressions describing the electromagnetic field inside the dielectric a- and magneto-optical b-layers, we can obtain the unit-cell transfer matrix, which describes the wave transmission through the unit-cell of the superlattice. Since, inside the layers of the superlattice, the electromagnetic field is defined by the four amplitudes, the unit-cell transfer matrix has dimensions .
In the superlattice composed of the
a- and
b-layers, the complex amplitudes corresponding to the
a-layer and those corresponding to the
b-layer are not independent; they are related to each other by the continuous boundary conditions for the tangential
y- and
z-components of the electromagnetic field. Using the boundary conditions at the left (
) and right (
) boundaries of the magneto-optical
b-layer within the
nth unit-cell, we obtain two matrix relations for quantities
,
, and
C:
Quantities
and
define the state of the electromagnetic field within the dielectric layers of the
nth and
th unit-cells, respectively; quantity
C defines the state of the electromagnetic field within the magneto-optical
b-layer of the
nth unit-cell.
One can obtain that the conditions expressing the continuity of the tangential components of the electromagnetic field at the left (
) and right (
) boundaries of the magneto-optical
b-layer within the
nth unit-cell are
Matrices
,
,
, and
are given by
Here, quantities
and
are
From Equation (
37), we find the following recurrent relation for the complex amplitudes of the electromagnetic field inside the dielectric
a-layers of the
th and
nth unit-cells:
Matrix
has the meaning of the unit-cell transfer matrix since it describes the wave transmission through the entire
unit-cell.
To obtain the dispersion relations describing the eigenstates of the electromagnetic field, we have to find the eigenvalues of the unit-cell transfer matrix
. They can be determined from algebraic equation
for unknown quantity
, where
is the identical matrix. Since the dimensions of the unit-cell transfer matrix
are
, it has the four eigenvalues
,
,
, and
, satisfying a polynomial equation of the fourth degree
Coefficients
u and
r are expressed through the parameters of the problem as follows:
Here, quantities
are given by
As a main result of our derivation, it is important to note that, due to the symmetry of the coefficients in the quartic equation in Equation (
43), the problem can be reduced to a pure quadratic equation
by means of the following change in variables:
. As a result, the roots
,
,
, and
of Equation (
43) are
From Equations (
47) and (
48), one can see that only two out of four eigenvalues are independent, since
Therefore, if, for the independent eigenvalues, we choose
and
, then the remaining two,
and
, are expressed through
and
, respectively:
Owing to Equation (
51), we can express the four eigenvalues
,
,
, and
of the unit-cell transfer matrix through two independent quantities
and
:
Here,
and
are the Bloch phases of the electromagnetic eigenmodes of the superlattice in the presence of the external magnetic field. From these equations, one can see that the two Bloch phases
and
satisfy the relations
or, equivalently,
After substitution of Equations (
44), (
45), and (
49) into Equation (
55), one finds the equations for the Bloch phases
. Such a form of the equations is not final, and they can be simplified: after some algebra, one can obtain that quantity
can be represented as a perfect square:
This allows us to remove the irrationality
in Equation (
55); one can use the expression in the curly brackets in Equation (
56) as
. Then, substituting Equation (
44) into Equation (
55), we obtain the final form of the desired dispersion equations:
Remarkably, we have obtained the following essential result: we find that, in the presence of the external magnetic field, the photonic spectrum of the superlattice is, therefore, characterized by two dispersion relations for two Bloch modes: and . This result, considering anisotropy, is contrary to the case of the isotropic superlattice, where only one Bloch mode can exist.
It is not difficult to check that, in the limit of the vanishing external magnetic field
, the Bloch phases
and
go to the Bloch phase
of the isotropic superlattice
Indeed, in the absence of the external magnetic field, the refractive indexes
and
of the magneto-optical
b-layer become equal to each other:
where
is the Drude permittivity of the magneto-optical
b-layer. Due to this,
Consequently, as follows from Equations (
57) and (
58), when the value of the external magnetic field decreases, the photonic spectrums
and
merge into one
, which satisfies the dispersion equation
where
and
. This is a well-known dispersion equation [
14] that describes propagation of the electromagnetic waves in the superlattice composed of the isotropic
a- and
b-slabs with refractive indexes
and
, respectively. Therefore, the equations of the photonic spectrums obtained in this work, considering anisotropy, can be considered as a generalization of the pure isotropic scenario.
4. Discussion: The Anisotropic Superlattice
Next, we apply the analytical results obtained in the preceding sections to a superlattice consisting of the dielectric and semiconductor slabs. We consider the following physical and representative parameters: the dielectric
a-slab is made of quartz with a refractive index
(see paper [
19]), while the semiconductor
b-slab is made of indium antimonide, with concentration of the charge carriers
cm
and permittivity of the lattice
. Additionally, for simplicity, we assume that there is no absorption in the layers of the superlattice, so that the relaxation frequency
in the plasma of
b-layers can be set to zero:
The thicknesses of the dielectric
a- and semiconductor
b-layers are
and
, where
and
. The numerical values of the quantities listed above are
THz,
mm,
mm, and
mm.
Figure 2 displays the photonic spectrum of the superlattice in the absence of the external magnetic field,
, within frequency range
. The passbands are highlighted in blue, while the spectral gaps are displayed in white.
Below we consider a frequency region located inside the first lower photonic stop band, beneath frequency
, where, in the absence of the external magnetic field,
, the superlattice cannot conduct light since there are no propagating states of the electromagnetic field within region
(see
Figure 2). Note that frequency
corresponds to wavelength of 0.26
in vacuum. Next, we show how the photonic band structure of the superlattice changes in the presence of the external magnetic field of magnitudes
T and
T, within the frequency range
and for angles of the external magnetic field within the region
. Angle
defines the direction of the external magnetic field
relatively to the
x-axis; the direction of the external magnetic field
lies in the
x–
y plane.
In the absence of absorption, the two refractive indexes
of the
b-layer are either pure imaginary or pure real. Therefore, the right-hand sides of the dispersion equations in Equations (
57) and (
58) are real. Therefore, within some frequency bands where
there are pure real solutions for the Bloch phase
or for the Bloch phase
. It is apparent that such solutions with real
or
define the photonic pass bands where the propagating states of the electromagnetic field exist.
Figure 3a shows the photonic band structure when the strength of the external magnetic field is 0.1 T. The white area corresponds to the points
where there are no propagating states of the electromagnetic field. At such values of angle
and frequency
, the superlattice cannot conduct light. The blue area displays points
where the propagating states of the electromagnetic field exist, i.e., where the superlattice can conduct light. It is pretty remarkable that the external magnetic field results in the existence of the propagating states of the electromagnetic field within the photonic stop band of the superlattice in the absence of the external magnetic field. Thus, this result reported here could be helpful in designing and generating optical diode-like devices. A distinctive feature of the band structure shown in
Figure 3a is that the size of the photonic pass band strongly depends on the direction
of the external magnetic field. Indeed, as one can see, the size of the lower photonic pass band varies from approximately
at
to zero at
. The size of the next photonic pass band strongly depends on the angle
too.
Figure 3b displays the photonic band structure when the strength of the external magnetic field is 0.4 T. Unlike the previous figure, one can see the region from the frequency which approximately equals
(the black horizontal dashed line) to the frequency
, which is completely filled with the propagating states of the electromagnetic field, regardless of the direction
of the external magnetic field. Therefore, and remarkably, we found that at any orientation
of the external magnetic field, the superlattice can conduct light, providing that the magnetic field is apparently strong enough.