A Rigorous Solution to Electromagnetic Wave Diffraction by Junctions of Impedance and Semitransparent Surfaces

Recent studies of surface-wave-based waveguides for terahertz (THz) imaging and spectroscopy have displayed some interest in investigating surface-wave diffraction by junctions of surfaces with different boundary conditions. Generally, many practically applicable surfaces for THz surface waveguides can be described using nontransparent or semitransparent impedance boundary conditions. In this communication, we find rigorous solutions to electromagnetic wave diffraction by the junction of a nontransparent impedance wedge and a semitransparent half-plane, as well as the junction of two semitransparent half-planes. Generalized semitransparent boundary conditions characterize these semitransparent surfaces.

Nontransparent impedance boundary conditions can be used to simulate such nontransparent surfaces as lossy metal (with thickness much larger than skin-layer depth), metal covered by a dielectric layer, or various metasurfaces-based THz plasmonic waveguides [7]. An impedance wedge is a classic structure supporting surface-wave propagation (see Fig. 1). The wedge has an opening angle equal to 2. The wedges' surfaces at ϕ = ± are characterized by the impedance boundary conditions with the impedances Z ± . The diffraction of surface waves by the edge of the wedge has been considered in [6], [11], [12], [13], [14], [15], and [16], including the particular cases of a half-plane in which = π [13], [17], a plane with a discontinuity in impedance in which = π/2 [13], [18], [19], and the case in which there is a step between two half-planes [20], [21].
Generalized semitransparent boundary conditions [22] can simulate other surfaces supporting surface-wave propagation. Manuscript  A semitransparent surface is characterized by discontinuities in the tangential magnetic and electric fields equivalent to electric and magnetic currents. Well-known simplified semitransparent boundary conditions [23] only consider electric current. Introducing electric and magnetic currents extends the class of analyzed structures. For example, metal layers with a thickness from much smaller to many times greater than the skin-layer depth [24] can be analyzed using generalized semitransparent boundary conditions. This technique corresponds to the transition from a highly transparent resistive surface to an almost opaque surface with finite conductivity. Using the generalized semitransparent boundary conditions substantially reduces the restriction on the thickness of an analyzed dielectric layer [25]. The restriction is typical for the simple semitransparent boundary conditions approximation. The accuracy of modeling various gratings formed by conductors with a finite thickness (e.g., circular-or rectangular-shaped) can be increased via an approximation using the generalized semitransparent boundary conditions. The diffraction of surface waves propagating along various semitransparent planes, half-planes, or junctions of surfaces with different boundary conditions has been considered in [26], [27], [28], [29], and [30]. These surfaces are dielectric slabs [26], [27], [28], resistive sheets [27], a particular case of a semitransparent surface with magnetic current only [29], and a surface with generalized semitransparent boundary conditions [30].
This communication considers two structures that appear at surface-wave-based THz waveguides. The first structure is the junction of a wedge with the impedance boundary conditions and a halfplane with the generalized semitransparent boundary conditions. The wedge has an opening angle equal to 2 and the same impedance on the top and bottom surfaces [see Fig. 2(a)]. Such structures simulate, for example, a junction of lossy metal or dielectric surfaces and can represent a surface wave's power splitter/divider.
The second structure is the junction of two semitransparent halfplanes with the generalized semitransparent boundary conditions [see transparent, we investigated the diffraction by the edge of a semitransparent half-plane using this structure. Using the Sommerfeld-Maliuzhinets' integral and symmetry methods, we found rigorous solutions to electromagnetic wave diffraction by these structures. These solutions include the case of surface-wave diffraction by the junctions.

II. BOUNDARY CONDITIONS
Let us consider a case of separate incidence of waves in the xy plane (∂/∂z = 0) with electric (E) or magnetic (H) polarization onto the structures in Fig. 2(a) and (b). The waves propagate in the upper half-space when y > 0. The case of E-or H-polarized waves corresponds to H z = 0 or E z = 0, respectively, in which E z (H z ) is the z-component of the electric (magnetic) field intensity vector in the Cartesian coordinates (x, y, z).

A. Impedance Boundary Conditions
Let us consider the impedance wedge in Fig. 2(a). The impedance boundary conditions on the surface of the wedge are as follows: where E and H are, respectively, electric and magnetic field intensity vectors on the wedge's surface; n is the unit vector in the direction of the normal to the illuminated surface. Impedance Z is a complex number with an argument to be within the range of [−π/2; π/2], which characterizes the properties of the wedge's surface. Let us write the boundary conditions (1) using scalar functions u ± E,H , which represent E z or H z on the wedge's surfaces at ϕ = ± for electric (E) or magnetic (H) polarization, respectively. In the polar coordinates (r , ϕ), the conditions (1) are as follows: where k = 2π/λ, λ is the radiation wavelength in a vacuum, and i is the imaginary unit. The parameters ϑ E,H are the Brewster angles. They characterize the properties of the surface. The reflection coefficient of a wave incident onto an infinite impedance plane at the Brewster angle vanishes. The Brewster angles are associated with the impedance Z as sin ϑ E,H = (Z/Z 0 ) −ξ E,H , in which Z 0 is the wave impedance of the incident wave and ξ E,H is equal to 1 or −1 for the E or H polarization, respectively.

B. Generalized Semitransparent Boundary Conditions
Let us consider an infinitely thin semitransparent half-plane at ϕ = 0 [see Fig. 2(a)]. The generalized boundary conditions on the surface of the semitransparent half-plane are as follows: where E ± and H ± are the electric and magnetic field intensity vectors on the surface of the half-plane when ϕ = ±0, respectively. An electric current j e on the half-plane's surface equals a discontinuity of the tangential component of the magnetic field intensity vector. A magnetic current j m on the half-plane's surface equals a discontinuity of the tangential component of the electric field intensity vector. The impedance (admittance) tensor Z (Y) associates the tangential component of the electric (magnetic) field intensity vector with the electric (magnetic) current j e (j m ).
For the E-or H-polarized waves, we can write the boundary conditions (3) using scalar functions u ± E,H , which represent E z or H z on the surfaces at ϕ = ±0 [30] ⎧ where Z x x (Y x x ) and Z zz (Y zz ) are the components of the impedance (admittance) tensor. They characterize the properties of the surface. These tensor components are complex numbers. Their arguments are within the range of [−π/2; π/2]. The conditions in (4) become the conditions for the perfectly electric conducting half-plane or a perfectly magnetic conducting half-plane when Z x x,zz = 1/Y x x,zz = 0 or 1/Z x x,zz = Y x x,zz = 0, respectively. The conditions in (4) can also be written in the following form: in which impedances Z 1,2 are connected with the tensors' components Z x x , Y x x , Z zz , and Y zz as follows: The conditions in (4) correspond to the general case of an anisotropic surface when impedances for E-and H-polarized waves are different. The conditions in (5) conform to the isotropic surface when there are equal impedances for different polarizations.
The difference between (4) and (5) is negligible for the 2-D structures under investigation because we independently solved diffraction using different polarizations. Therefore, we can set the parameters Z 1,2 independently for different polarizations. However, the form (5) of the semitransparent boundary conditions is shorter and more convenient for finding the solution of the diffraction problem than form (4). This communication uses the generalized semitransparent boundary conditions in the form (5). Note that when Z 1 = Z 2 , the generalized semitransparent boundary conditions (5) describe a case in which the magnetic current on the surface is absent.
Analogous to the structure in Fig. 2(a), the boundary conditions on the semitransparent surface at y = 0, x ≥ 0 for the structure in Fig. 2(b) are the same as (5). On the semitransparent surface at y = 0, x < 0 in Fig. 2(b), the boundary conditions are similar to (5)

III. RIGOROUS SOLUTIONS FOR THE STRUCTURES UNDER STUDY
This section considers the case of a scalar plane-wave diffraction by the structures in Fig. 2. We assume that the time dependence is e −iωt , in which ω is the angular frequency and t is time. An incident plane wave has the form u inc E,H (r, ϕ) = e −ikr cos(ϕ−ϕ 0 ) , in which ϕ 0 is the angle of incidence. The incident field representation enables the consideration of the excitation of the structure by a surface wave propagating along one of its surfaces. For an incident surface wave, the angle of incidence should be a complex number (see Appendix). We assume that the solution to the diffraction problem satisfies the following.
1) The Helmholtz equation u E,H + k 2 u E,H = 0, in which scalar functions u E and u H represent E z and H z , respectively, in free space.
2) The corresponding boundary conditions on the surfaces of the considered structures.
3) The radiation conditions at infinity, including the one preventing any incoming wave except the incident one. 4) The condition of finite power concentrated when r →0, which is equivalent to the Meixner condition at an edge. Radiation conditions for a structure containing semi-infinite guiding surfaces determine the field at infinity (r → ∞). In addition to a standard volumetric cylindrical wave propagating from the point r = 0, this field contains surface waves propagating along guiding surfaces from the point r = 0 to infinity. The field of surface waves is proportional to exp(iγ 0 r ), in which γ 0 is a surface-wave propagation constant.

A. Junction of the Impedance Wedge and the Semitransparent Half-Plane
Conditions (2) and (5) reveal that the analyzed structure is symmetric relative to the xz plane. This symmetry allows us to represent the incident field as a sum of even u inc_e The symmetry also allows us to determine the total field as the sum of the solutions for the even and odd excitations [30]. Let us now derive boundary conditions, which are satisfied by the even u e± E,H and odd u o± E,H total fields at ϕ = ± and ±0. For the deriving, we use the following relations: Substituting (8) into (2) reveals that the even and odd total fields satisfy the impedance boundary conditions (2) at ϕ = ±. Substituting (8) into (5) helps to obtain one impedance boundary condition at ϕ = 0 for the even and odd fields Thus, the solution of the diffraction problem is reduced to the sum of two solutions of the diffraction of plane waves using impedance wedges with an opening angle equal to 2π − [see Fig. 3(a)], which are in the upper and bottom halfspaces at 0 ≤ ϕ ≤ and − ≤ ϕ ≤ 0. Applying the symmetry method allows us to consider only one wedge [in Fig. 3(b)] excited by the even and odd incident fields and to extend the solution for the wedge to the initial structure in Fig. 2(a). The rigorous solution for the wedge is the well-known Maliuzinets' solution (see Appendix): u imp E,H (r, ϕ, ϕ 0 , , ϑ + E,H , ϑ − E,H ). Therefore, the rigorous solution of the diffraction problem using the junction of the impedance wedge and the semitransparent half-plane is expressed using a sum of the rigorous solutions for the wedge as follows: Maliuzinets' solution satisfies conditions 1)-4) stated above. Therefore, the total solution (10) also satisfies these conditions.

B. Junction of Two Semitransparent Half-Planes
To find the rigorous solution for the diffraction problem by the two semitransparent half-planes in Fig. 2(b), let us again use the symmetry method and represent the incident field as the sum of the even and odd components (7). The semitransparent boundary conditions (5) are transformed into (9) at ϕ = 0, and the semitransparent boundary conditions (6) are transformed into the following impedance boundary condition at ϕ = π for the even and odd fields: By analogy with the analysis provided in Section III-A, we can demonstrate that the rigorous solution of the diffraction problem using the considered structure is expressed through the sum of the rigorous solutions for a wedge with an opening angle equal to π (see Fig. 4) when excited by the even and odd incident fields. The solution is as follows: Each term in solution (12) satisfies conditions 1)-4) at the beginning of Section III. When parameters Z 1,2 tend to infinity, the right semitransparent half-plane in Fig. 2 becomes fully transparent. Then, the diffraction by the structures under investigation is reduced to the diffraction by the edge of the impedance wedge and the edge of the semitransparent half-plane characterized by parameters Z 3,4 . In this case, solution (10) for the impedance wedge coincides with the solution from [11] and [14]. Solution (12) for the semitransparent half-plane coincides with the solution from [30]. Note that the diffraction of an H-polarized surface wave by a junction of two semitransparent half-planes is considered in [28]. In that paper, a particular case of the semitransparent boundary conditions (5) is used when Z 1 = Z 2 = Z x x,zz /2 and 1/Y x x,zz = 0. Solution (12) and the solution from [28] for the case of H-polarized surface-wave diffraction by a halfplane coincide for the indicated particular case of the semitransparent boundary conditions.

IV. CONCLUSION
In this communication, we obtained the rigorous solution to the problem of electromagnetic wave diffraction by the junctions of the impedance wedge and the semitransparent half-plane, as well as two semitransparent half-planes with different impedances. This solution is based on the Sommerfeld-Maliuzhinets' integral method.

APPENDIX
As we know, the solution of the Helmholtz equation u E,H + k 2 u E,H = 0 in the form of was used in [11] to solve the plane-wave diffraction problem on an impedance wedge (see Fig. 1) with the impedance boundary conditions (2). In (A.1), γ is the Sommerfeld integration path in the complex plane α; the integrand S E,H corresponding to the impedance boundary conditions is in which is expressed using Maliuzhinets' function (α) In this case, sin ϑ ± E,H = (Z ± /Z 0 ) −ξ E,H . The general form of Maliuzhinets' function (α), which enables the calculation of this function for any argument α, is as follows: Maliuzhinets' function can also be presented using an infinite product of the gamma functions as follows: in which gd(x) = arcsin(tanhx) is the Gudermannian function.