Modification of transfer-matrix method for electromagnetic waves in layered superconductor in presence of dc magnetic field методу трансфер-матриць для електромагнітних хвиль надпровіднику наявності постійного магнітного поля

In the present paper, we modify the transfer-matrix method to study the dissipation-free transition of electromagnetic waves of terahertz range through a plate of layered superconductor embedded in the dielectric environment in the presence of external direct current (dc) magnetic field. In this work, we сonsider TM-polarized electromagnetic waves. The setup is arranged in such a way that the dielectric and superconducting layers in the plate are perpendicular to its interface, and the external magnetic field is directed along the plate and parallel to the layers. We consider the case of a weak external dc field at which magnetic vortices do not penetrate the plate. Due to the nonlinearity of the Josephson plasma formed in the layered superconductor, the dc magnetic field penetrates non-uniformly into the plate and affects the electromagnetic wave. Hence, the magnitude of the external dc magnetic field can be used as a variable parameter to tune various phenomena associated with the propagation of an electromagnetic waves in layered superconductors. In the presence of the external homogeneous dc magnetic field, linear electromagnetic waves in the layered superconductor turn out to be non-exponential. Therefore we cannot directly apply the transfer matrix method, in which the amplitudes of the corresponding exponents are compared. However, in the present paper, it is shown that for a sufficiently thick plate, the matrices describing the wave transfer through the plate can be introduced. The analytical expressions for these matrices are derived explicitly in terms of special Legendre functions. The obtained transfer-matrices can be used for the further study of the wave transfer through the layered superconductor in the presence of an external dc magnetic field.

In the present paper, we modify the transfer-matrix method to study the dissipation-free transition of electromagnetic waves of terahertz range through a plate of layered superconductor embedded in the dielectric environment in the presence of external direct current (dc) magnetic field.
In this work, we сonsider TM-polarized electromagnetic waves. The setup is arranged in such a way that the dielectric and superconducting layers in the plate are perpendicular to its interface, and the external magnetic field is directed along the plate and parallel to the layers. We consider the case of a weak external dc field at which magnetic vortices do not penetrate the plate.
Due to the nonlinearity of the Josephson plasma formed in the layered superconductor, the dc magnetic field penetrates nonuniformly into the plate and affects the electromagnetic wave. Hence, the magnitude of the external dc magnetic field can be used as a variable parameter to tune various phenomena associated with the propagation of an electromagnetic waves in layered superconductors.
In the presence of the external homogeneous dc magnetic field, linear electromagnetic waves in the layered superconductor turn out to be non-exponential. Therefore we cannot directly apply the transfer matrix method, in which the amplitudes of the corresponding exponents are compared. However, in the present paper, it is shown that for a sufficiently thick plate, the matrices describing the wave transfer through the plate can be introduced. The analytical expressions for these matrices are derived explicitly in terms of special Legendre functions. The obtained transfer-matrices can be used for the further study of the wave transfer through the layered superconductor in the presence of an external dc magnetic field.

Introduction
Layered superconductors are periodic structures that consist of thin alternating superconducting and insulating layers. Natural crystals Therefore, the various non-trivial electromagnetic phenomena are predicted for layered superconductors [3,4,5]. Also, these materials are of particular interest due to possibility of flexible tuning their electromagnetic properties by an external direct current (dc) magnetic field [6,7]. The additional interest is related to the operating frequencies of the Josephson plasma waves that are of terahertz (THz) range. By present, there is still a gap in controllable and high-power THz-devices, which are, meanwhile, considered to be promising for many areas starting from basic science to medicine or homeland security [8,9].
To study the transfer of electromagnetic waves it is convenient to use the transfer-matrix method (see, e.g., book [10]). In the absence of dc magnetic field, the electromagnetic properties of the layered superconductor can be described by the effective permittivity tensor [11], and, therefore, the transfer-matrix method can be directly applied (see, e.g., the recent paper [4]). However, in the presence of an external dc magnetic field, the problem becomes more complicated because the electromagnetic field inside the plate is described not by harmonic (exponential) functions, but by special Legendre functions [6].
In this paper, we modify the transfer-matrix method for the electromagnetic wave propagation through a plate of layered superconductor in the presence of an external dc magnetic field and calculate the corresponding transfer-matrices.

Problem Formulation
We study a dissipation-free propagation of an electromagnetic wave through the system that consists of a layered superconductor plate of thickness s placed in the dielectric environment as shown in Fig.1. Dielectric and superconducting layers are considered perpendicular to the interface. The coordinate system is chosen in such a way that the x -axis is directed perpendicular to the plate, the z -axis is orthogonal to the superconducting layers.
The external dc magnetic field 0 H is supposed to be directed parallel to the plate and to the layers, i.e. along the y -axis. We consider TM-polarized wave. In the chosen coordinate system, its electric ( , , , ) E x y z t and magnetic ( , , , ) H x y z t components can be written as follows: where  is the wave frequency, z k is z -projection of the wave vector.
It is worth noting that the plate is supposed to be sufficiently thick: where c  is the London penetration depth along the layers of superconductor. On this assumption, the dc magnetic field deeply inside the layered superconductor is absent. Also, we assume that the external magnetic field magnitude is less than the critical value

Main Equations for the Electromagnetic Field
The expressions for the electromagnetic field components in a dielectric medium can be obtained from the system of Maxwell's equations. At the left and right interfaces, respectively, for magnetic components we have: where d k is the x -projection of the wave vector of the incident wave: For the corresponding electric field components, we have: The field inside the plate which obeys the coupled sine-Gordon equations [3]. For the wavelengths that are greater than the thickness d (i.e. in the continuous limit), it can be represented as follows:   First, we construct the solution for the right and left part of the plate independently. The interaction between magnetic vortices from the opposite sides can be neglected due to the assumption (2), then we neglect the first or second component with cosh in the expression (12). Then the solution of Eq. (11) can be found in terms of associated Legendre functions [13]. We present the solution in the form of superposition for the right half of the plate: and for the left one: (1 ) The specific form of () f   allows us interpret these functions as non-exponential running waves inside layered superconductor. Indeed, for 11 z  there is an asymptotic expression [13] for Legendre functions: The approximation (15) can be applied for  ).
If the external dc field tends to zero, the expressions (17) turn out to be harmonic, and the wave transfer through the investigated system could be described by the matrices of passing through the boundaries and free propagation in the medium [4] in a similar way to the dielectric case. Otherwise, the magnetic field () y Hx can be considered as a plane wave superposition only in the center of the plate.

Transfer matrices
The transfer-matrix T that corresponds to the wave transfer through the plate connects the amplitudes of outgoing and incoming waves for the magnetic field () y Hx . According to (4): = .

RL T RL
Since the field in the layered superconductor can be described by exponential functions only in the center, we can present the matrix T as The boundary conditions are the matching of the tangential components of the electromagnetic field: In accordance with the expressions for the electromagnetic field (4), (5), and (17), these conditions can be rewritten as: therefore, the symmetry of the problem is not broken.

Conclusions
In this theoretical work, we have modified the transfer-matrix method for TM-polarized electromagnetic waves propagating through a plate of layered superconductor taking into account the interaction of Josephson plasma with an external dc magnetic field. It was shown that although the electromagnetic field inside the plate cannot be described by harmonic (exponential) functions, far from the boundaries, it can be considered as a superposition of a running and reflected waves. Then, for sufficiently thick plate, the transfer-matrices can be obtained analytically in terms of special Legendre functions. The received matrices can be used in the further studies related to the transfer of electromagnetic waves through layered superconductors.