Metamaterials based on wedge-shaped electrodynamic structures

The paper studies a possibility of simulation of artificial composite media with negative values of the real part of the equivalent dielectric (magnetic) permittivity, by the use of segments of hollow composite waveguides with cylindrical guided waves in evanescent mode. Reactive evanescent fields of wedge-shaped waveguide eigenmodes are formed in the evanescent region before the critical section of the waveguide which separates the quasistatic field region from the distributing field of the evanescent waveguide mode. The possibility of simulation is determined by the equivalence of dispersion equation of the eigenmode propagation constant and the dispersion equation for the electric (magnetic) permittivity of plasma-like medium if cut-off frequency and electric (magnetic) plasma frequency of the medium are equal.


Introduction
In recent years, developers of microwave devices and antennas have been attracted by metamaterials and their use in microwave engineering [1][2][3][4][5][6][7][8][9].Different combinations of infinitesimal electric and magnetic dipoles (wire structures, posts, bars, frames of various configurations, rings, etc.) are used for implementation of metamaterials.Often, waveguide sections operating in cutoff mode (evanescent mode) are used in combination with the above mentioned dipoles.Artificial composite medium proposed in [11] was made by placing a regular array of metallic posts between two parallel metallic plates.Further simplification of such model was achieved in [12] by using only one waveguide due to its dispersion properties.

Metamaterials based on wedge-shaped electrodynamic structures
In hollow metallic waveguides, mn E and mn H eigenmodes satisfy dispersion relation [10]: where Г is eigenmode propagation constant; -phase constant of empty space with εo and μo parameters -dielectric permittivity and magnetic permeability of vacuum; ω -the angular frequency; λ -the wavelength in vacuum; μ -the magnetic permeability (for hollow waveguide μ= 1); εeff -the effective dielectric constant given by where f=ω/2π is the operating frequency; fcr -the cutoff frequency of the considered type of eigenmode, e.g., Нmn.
Considering that the dispersion relation ( 1) with account of ( 2) is identical of that of a lossless plasma medium model with plasma frequency fpe = fcr [10], a hollow metallic waveguide can be regarded as onedimensional plasma with respect to EM wave propagation along the axial direction of the waveguide.Moreover, the equivalence between waveguide field spread below cutoff frequency of the eigenmode and EM wave propagation in artificial plasma in the form of evanescent field is theoretically established in [13] and experimentally confirmed in [14].The equivalence exists between waveguide and electric plasma for evanescent Нmn modes of rectangular and circular waveguides.The discussed equivalence shows that medium with negative dielectric permittivity can be simulated by means of Нmn modes at frequencies below cutoff, while medium with negative magnetic permeability can be simulated by means of Еmn modes at frequencies below the corresponding cutoff.
With such approach double-negative medium is implemented by placing a periodic array of split ring resonators inside a rectangular or circular waveguides operating at frequencies below corresponding cutoff frequencies of H10 and H11 modes [3,4,6].The periodic array of resonators is in fact artificial anisotropic magnetic plasma with permeability tensor [13]: where fpm is magnetic plasma frequency (when Lorenz model of the considered medium is used); fo is resonant frequency; γm -magnetic plasma losses.
In such double-negative medium, the EM wave propagates with propagation constant In this structure, Нmn modes can propagate at frequencies at which both brackets in the radical expression have the same sign.Backward wave is formed at frequencies lower than both fcr and fpm.Forward waves propagate at all frequencies that are higher than both fcr and fpm.Between the latter, there is a stopband.When fcr = fpm.all types of waves become propagating.
For Еmn modes of the rectangular waveguide, dispersion relation can be expressed as eff where μeff is effective relative magnetic permeability defined by where fcr is the cutoff frequency of the considered Еmn mode; for hollow waveguide, ε=1.Considering that the dispersion relation (3) with account of ( 4) is identical of that of a lossless plasma medium model with plasma frequency fpm = fcr [14], a hollow metallic waveguide can be regarded as one-dimensional plasma with respect to EM wave propagation along the axial direction of the waveguide.In [14], the authors propose to simulate artificial medium with negative magnetic permeability by using a hollow square waveguide operating at frequencies below cutoff frequency of the lowest E11 mode of the Еmn eigenmode family.Double-negative metamaterial medium is implemented by placing a periodic array of wires inside the square waveguide operating at frequencies below the cutoff frequency of the E11 mode [13,14].The array structure is in fact artificial anisotropic magnetic plasma with dielectric permittivity tensor where fpe is electric plasma frequency (when Lorenz model of the considered medium is used [6]); fo is resonant frequency; e γ -electric plasma losses.
In such double-negative medium, the EM wave propagates with propagation constant The Еmn modes propagate at frequencies at which round parenthesis in the radical expression have the same sign.Backward wave is formed at frequencies lower than both fcr and fpe.Forward wave propagates at frequencies that are higher than both of the above.Regions with negative dielectric permittivity and magnetic permeability in nonhomogenious waveguides with spherical and cylindrical guided waves are implemented by means of evanescent reactive field of the corresponding eigenmodes before critical section of the waveguide separating the quasistatic field region from the distributing field of the evanescent waveguide mode [1,10,19].
In wedge-shaped waveguides with cylindrical guided waves (in ρ, φ, y coordinates) (Figure 1), Maxwell equations can be solved in the form of superposition of "electric' and "magnetic" waves, with their components determined via U and V potential functions [10].Terms "electric" («Emo») and "magnetic" («Hmo») are quoted.It means that such splitting does not comply with the accepted waveguide classification, i.e. y coordinate does not correspond to propagation direction.The waves are cylindrical and propagate in the direction of the ρ 0 radial coordinate.The U and V functions satisfy Helmholtz equations, with particular solutions given by where A and B are constant coefficients determined by excitation conditions; Zm(kρ) is the radial dependence of the field satisfying boundedness condition at kρ → 0, radiation condition at kρ → ∞, and being continuous at the critical section [15]: where pm = mπ/(2γ) are eigenvalues determined from boundary conditions; m -whole numbers; 2γ -wedge angle.
Considering the above conditions, radial dependence should be written as: where Jpm(kρ) are cylindrical Bessel functions of the first kind and pm order, Hpm (2) are cylindrical Hankel functions of the second kind (exp(iωt) time dependence of the field).
Transverse (relative to ρ 0 direction) ⊥ E , ⊥ H field components constitute superposition of transverse EM field components for «Emo» and «Hmo » waves: where Here, φ 0 , y 0 are unit vectors along the φ and y coordinates.The Г propagation "constant" of the eigenmodes is determined by means of logarithmic derivative of radial dependence [16]: where prime mark means full derivative of the argument.Since Г depends on ρ, The word "constant" is quoted.Using asymptotic representation of Hankel function at kρ → ∞, it is possible to show that at narrow wedge angles where coinciding with ( 1) -( 7) for regular waveguides.It allows to use the above methods for regular waveguides in this case too.
In the first example, let us consider a «E10» mode, for which EM field components are given by: ( ) sin 2 Impedance for this type of wave is Z0 is free space characteristic impedance with μa and εa parameters.
Comparing ( 20) and ( 1) at μ=1, we get eff ( )/ ( ) From which If kρ ≤ (kρ)cr, If kρ ≥ (kρ)cr, expressions ( 7) and ( 1), given μ=1, with account of relation ( 13) for propagation of the considered wave type after critical section, give where Npm(kρ) are Neumann functions.The relation gives for kρ ≥ (kρ)cr Note that (kρ) variable in expression ( 28) and (30) can be expressed as: { } The dependence is shown in the diagram in Figure 2. As we see, evanescent region of the wedge-shaped waveguide has negative dielectric permittivity.Doublenegative medium can be implemented by placing an array of split-ring resonators inside the wedge-shaped region according to the structure of the «E10» (22) EM field, i.e. resonator lattice plane should be perpendicular to the magnetic field vector lines.
If kρ ≥ (kρ)cr, then, by comparing expressions ( 6) and ( 19), given ε=1 with account of relation (13) for propagation of the considered wave type after critical section, we get expression (30) for μeff if εeff is substituted by μeff.
The effective dielectric permittivity for «H10» mode of the wedge-shaped waveguide with 2γ=π/2 wedge angle proves to be equal to (33) if εeff is substituted by μeff.This dependence is also shown in the diagram in Fig. 2. As we see, evanescent region of the wedge-shaped waveguide has negative magnetic permeability.Double-negative medium can be implemented by placing a periodic array of thin wires inside the wedge-shaped region according to the structure of the «H10» (34) EM field, i.e. wires direction should coincide with the direction of the electric field.

Conclusion
The research that has been carried out allows to make conclusions as follows: -a region with negative dielectric permittivity can be implemented based on hollow wedge-shaped waveguide at frequencies below the cutoff frequency of the «E10» mode.
-a region with negative magnetic permeability can be implemented based on hollow wedge-shaped waveguide at frequencies below the cutoff frequency of the «H10» mode.
The research work was supported by Ministry of Education and Science of the Russian Federation under state task No 8.4868.2017/8.9.