3.1. Electrodynamic Statement of the Problem
A monochromatic TE-polarized electromagnetic wave
where
is a circular frequency,
are complex amplitudes, and
is (unknown) real spectral parameter, propagates in a plane dielectric layer
sandwiched between two half-spaces
and
. The boundary
is open, and at the boundary
, there is a graphene layer, which causes a surface current at this boundary.
Waveguide
is filled with a homogeneous isotropic medium characterized by a constant permittivity
; half-spaces
and
are filled with homogeneous isotropic mediums characterized by constant permittivities
and
, respectively, such that
. Everywhere, permeability
, where
is the magnetic constant. The geometry of the problem is presented in
Figure 1.
Field (
1) satisfies Maxwell’s equations
where
is the dielectric constant, and
Since the boundaries of the waveguide are open, and the electromagnetic field penetrates into half-spaces
and
, it is essential to impose on (
1) the condition of decaying at
. The tangential component
of the electric field is continuous at both boundaries, due to the absence of a surface charge there. The tangential component of the magnetic field is also continuous at the boundary
, due to the absence of a surface current there; however, it undergoes a jump at the boundary
, due to the surface current of charge carried in graphene (the current is induced by the electromagnetic wave), and the jump is equal to surface current density
. Thus, at
, the discussed component satisfies the following condition
where
is a unit vector of the normal directed along the
x axis,
and
are the values of magnetic field above and below the surface
, respectively,
is the vector product, and
is the surface conductivity of graphene.
As was previously noted, the electric conductivity
of graphene depends on the electric field coupling to the charge carriers in graphene; due to the central symmetric structure of graphene, this dependence has the form
where
and
are generally some complex numbers depending on frequency
[
7,
8,
9,
19,
20].
The linear part
of graphene’s electric conductivity is determined by formulas [
21,
22]. We assume that
and
, i.e.,
Such a restriction on
is fair in THz range, where graphene has a strong plasmonic response and much less loss [
23,
24]. For determining
in (
5), there exist several formulas [
7,
19,
20]; for example, one can apply the formula in [
7]; in accordance with this formula, quantity
is purely imaginary and
, i.e.,
The main problem — we call it problem
— is to find such values of parameter
, where there exists field (
1) satisfying Maxwell’s equations (
3) and all the above conditions and exponentially decaying at
.
Substituting (
2) into Maxwell’s equations (
3), one obtains
The third equation in the obtained system provides the relationship between the tangential components of the electric and magnetic fields expressed by the formula
Expressing
and
from the second and the third equations of system (
8), respectively, and substituting them into the first equation, one obtains
Passing to dimensionless variables in the obtained equation by virtue of the formulas
where
, and, omitting the tilde, we obtain the equation
Solving (
10) in half-spaces
,
and using the condition at infinity, one finds
where
and
are constants. Note that
must be real-valued; from here, taking into account inequality
, it follows that
must satisfy
From the above conditions imposed on field (
1) and Formula (
9), expressing the relationship between the tangential components of the magnetic and electric fields, it follows that
must satisfy the conditions
and
where
which results from (
5), taking into account the form of field (
2) and the realness of parameter
.
Then, one can reformulate the problem
in the following way. Problem
is to find such values
satisfying inequality (
12), where there exists the solution
to Equation (
10), satisfying the boundary conditions (
13) and (
14).
3.2. An Eigenvalue Problem
Introducing the notation
, Equation (
10) in the layer can be written in the form
Using conditions (
13) and (
14), the solutions in half-spaces (
11), and Formula (
15), one can write the boundary conditions for function
as
Besides this, we need one more condition in order to obtain a discrete set of solutions to the problem. It is in accordance with the physical process of the electromagnetic wave propagation in waveguiding structures. We use the following form of the additional condition
where
is a constant.
It is easy to determine that when a couple
satisfies Equation (
16),
satisfies it as well. For this reason, we consider positive
.
So, problem
is equivalent to the boundary value problem; we call it problem
, which is to find the positive
satisfying inequality (
12), such that there exists a twice continuously differentiable function
that is a solution to Equation (
16) satisfying conditions (
17)–(
19). A number
is called an
eigenvalue of problem
, and function
is called an
eigenfunction of problem
.
Note that condition (18) is nonlinear with respect to the unknown function ; so, in fact problem is nonlinear. If one sets in (18), then problem degenerates into a linear problem, which we call problem .
Although the statement of problem
is given above, and problem
is its special case, we present the statement of problem
here as well. So, problem
is to find the positive
satisfying inequality (
12), such that there exists a function
, which is a solution to the equation
satisfying the following boundary conditions
the number
is called an eigenvalue of problem
, and function
is called an eigenfunction of problem
.
We stress that problem
does not require (
19).
Problem and problem serve as mathematical models of the monochromatic TE-wave propagation in a plane waveguide having a graphene layer on one of its boundaries, but in the case of problem graphene is characterized by , i.e., the nonlinear response of graphene is neglected. Note that such a model allows one to obtain results close to the experimental data only at intensities of electromagnetic radiation.
3.3. Dispersion Equation of Problem
The solution to Equation (
16) has the form
where
and
are constants. Using condition (
17), one can express
by
and then, using condition (
19), determine
; doing this, one obtains
Using condition (18), one obtains the following equation
Let us introduce notation
,
,
,
,
; taking into account Formulas (
6) and (
7), it is clear that
and
are real,
, and
. Using this notation, Equation (
20) can be written in the form
Expression (
21) can be considered as the characteristic equation of problem
. This means that any solution
to Equation (
21) corresponds to an eigenvalue
of problem
, and any eigenvalue
of problem
corresponds to the solution
of Equation (
21).
From a physical point of view, relation (
21) is the so-called dispersion equation, as it provides the relationship between the thickness of the waveguide and its propagation constants.
Setting
in (
21), one obtains the dispersion equation for problem
in the form
Further, setting
in the previous formula, one obtains the classical dispersion equation of the form
for the problem of the electromagnetic TE-wave propagation in a plane dielectric layer sandwiched between two half-spaces; the above equation is given in [
25] and in [
26] (only for the case
).
3.4. Solvability of Problem
Problem
might have eigenvalues in the interval
, as well as in the unbounded domain
(unless the contrary is proved). The former kind of eigenvalues corresponds to solutions of Equation (
21) belonging to the interval
, where
; the latter one corresponds to solutions of Equation (
21) having the form
with
.
Statement 1. Equation (21) does not have solutions of the form , if , where Proof. So, let
, where
. Substituting
into (
21), one obtains the equation, with respect to
, of the form
It is clear that the right-hand side of (
25) is negative for all
, whereas the left-hand side is positive starting with some
. This means that Equation (
25) does not have solutions in the domain
.
Let us obtain the estimate (
24) for
. It is clear that the left-hand side is positive as soon as both terms in brackets are positive. Taking into account the inequality
and the inequality
, which holds for all
, where
one finds that the expression in the first bracket is positive for all
. Further, the inequality
together with
, which holds for all
, where
, imply that the expression in the second bracket is positive for all
. Combining the obtained results, one obtains Formula (
24). □
Now, let us pass to the case
. For the further analysis, it is convenient to rewrite Equation (
21) in the following way
where
Since functions
and
in the right-hand side of (
26) do not have accumulation points of zeros, then (
26) can have only a finite number of solutions
.
The following result provides a sufficient condition for the existence of at least one solution to Equation (
26).
Statement 2. If inequalitiesandare fulfilled, then Equation (26) has at least one solution , where Proof. Let us consider the right-hand side of (
26). It is clear that function
is continuous for all
. Function
is continuous for all
, but generally, it can change the sign. Nevertheless, one can check that under condition (
27), function
preserves the sign and, to be more precise, is positive for
, where
is defined in (
29). Indeed, taking into account the chain of simple inequalities
and condition (
27), one can see that
Combining this result with the inequality
taking place for
, one comes to conclusion that
for such
. Thus, the right-hand side in (
26) is continuous for
.
Condition (
28) implies that, firstly,
takes all values from zero to
, and secondly, inequality
is valid. Equation (
26) is defined for
, and the right-hand side of (
26) is continuous for such
. This means that a graph of the function in the right-hand side of (
26) has at least one intersection with
in the left-hand side of (
26), for
. □
The sufficient condition for the existence of one solution to Equation (
26) given in Statement 2 can easily be generalized to the case of the existence of
solutions. Indeed, the following result takes place.
Statement 3. If inequalities (27) andare fulfilled, then Equation (26) has at least n solutions , , where is defined in (29).
Proof. The proof of this statement repeats the proof of Statement 2. We have
, where
is defined in (
29). Further,
takes all values from zero to
. Thus, Equation (
26) is defined for
, and the right-hand side of (
26) is continuous in this interval. This means that a graph of the function in the right-hand side of (
26) has at least
n intersections with
in the left-hand side of (
26) in interval
. □
Taking into account Statements 1–3 and the equivalence between Equation (
21) and problem
, one obtains the following result.
Statement 4. If conditions (27) and (30) are fulfilled, then problem has at least n eigenvalues , , wherehere, is defined in (24).
In fact, Statement 3 gives a sufficient condition for the existence of eigenmodes supported by the waveguide under consideration.
3.5. Numerical Results
In the calculations below, we used the following parameters: , , , . The value of amplitude A of the electric field is given in figures’ captions.
It is worth giving a comment about the chosen values of the parameters. The linear part of the electric conductivity of graphene was calculated by virtue of the formula given in [
22]. In this formula, there are three optional parameters: the wave frequency
, the chemical potential
, and the absolute temperature
T; for calculating
, we used
,
, and
; we stress that we neglected the real part of
. Quantity
was calculated by virtue of the formula presented in [
7] using the same parameters as for calculating
.
In
Figure 2, the dispersion curves of problems
(blue curves) and
(red curves) are presented. The dispersion curves were plotted as the dependence of a wave number (a propagation constant) on either the wave frequency
or thickness
h of the waveguide. Since the statement of the problem does not involve
explicitly due to the normalization by
, we plotted the dispersion curves as
vs.
h.
The vertical line
in
Figure 2 corresponds to the waveguide of thickness
. The intersection points of the dispersion curves with this line denoted by diamonds are the eigenvalues of the corresponding problems, and these eigenvalues, in turn, correspond to the propagation constants
of the waveguide in the problems of electromagnetic wave propagation.
In
Figure 3, we plotted the eigenfunction
of problem
and the eigenfunction
of problem
. The eigenvalue
of problem
is a perturbation of eigenvalue
of problem
, and it can be shown that
. Due to the closeness of the eigenvalues
and
, it is natural to expect the closeness of the corresponding eigenfunctions
and
;
Figure 3 demonstrates this clearly; in addition, it can be shown that
. In
Figure 3, one can see that in the nonlinear case, the absolute value of the eigenfunction (tangential component of the electric field) at the boundary
was significantly smaller than in the linear case. This means that the nonlinearity arising in graphene led to a greater localization of the electromagnetic field inside the waveguide. At the same time, the maximum and minimum values of the eigenfunction (tangential component of the electric field) in the nonlinear case were smaller in absolute value than in the linear case, and the extremum points shifted to the left relative to their positions in the linear case.
In
Figure 4, we plotted the eigenfunction
of problem
and the eigenfunction
of problem
. The eigenvalue
of problem
is a perturbation of eigenvalue
of problem
, and it can be shown that
.
Figure 4 shows that the eigenmode corresponding to the nonlinear case was more localized than its linear counterpart. In addition, the maximum and minimum values of the nonlinear eigenmode were smaller in absolute value than the maximum and minimum values of the linear eigenmode, and the extremum points shifted to the left relative to their positions in the linear case.
It also seems interesting to learn at which conditions the discussed nonlinear effect in graphene became significant. In
Figure 5, as well as in
Figure 2, we plotted the dispersion curves of problems
(blue curves) and
(red curves); however, in the calculations, we used a smaller value for the amplitude of the electric field, namely
. In this case, the dispersion curves of the (nonlinear) problem
were no longer strongly different from the dispersion curves of (linear) problem
. In
Figure 6 and
Figure 7, the eigenfunctions of the problems
and
, corresponding to the eigenvalues denoted in
Figure 5, are presented. It can be seen that the eigenmode corresponding to the nonlinear case was more localized within the waveguide than its linear counterpart; however, this effect was much weaker than the one demonstrated in
Figure 3 and
Figure 4, and the reason is that the amplitude of the electric field was smaller. It is worth noting that the strength of the nonlinearity in graphene depends on the value of the nonlinearity coefficient
in the formula as well as on the amplitude of the incident wave; in accordance with the formula given in [
7], the coefficient
was proportional to
. Thus, in order to make the considered nonlinear effect more significant, one can either increase the amplitude of the incident wave or decrease its frequency.