Introduction To Modern Planar Transmission Lines. Anand K. Verma

(5.4.14)

      The reflection and transmission coefficients of both the TE and TM‐polarized incident waves are identical in form, except that parameters P and Q are different.

      5.4.2 Normal Incidence

For the case of the normally incident TE‐polarized wave, the angles of incidence and refraction are θ₁ = θ₂ = 0^∘. Furthermore, ϕ = k₂d, (k_2x = k₂), and P = η₂/η₁. Under, this condition, equation (5.4.13a,b) for the reflection and transmission coefficients are reduced to the following expressions:

(5.4.15)

      The above expressions are valid for the normally incident TM‐polarized waves also.

      Figure (5.6b) shows the equivalent transmission line model of the three‐layered medium. The reflection coefficients at both interfaces can be obtained from the expression, Γ = (Z_L − Z₀)/(Z_L + Z₀). At the first interface, corresponding medium impedances are Z_L → η₂ and Z₀ → η₁, while at the second interface, these are Z_L → η₁ and Z₀ → η₂. The reflection coefficients Γ₁₂ and Γ₂₃ are defined at the first and second interfaces. The above equations are rewritten as follows:

(5.4.16)

The approximate expression (5.4.16c) is used for the small reflection at the interfaces, i.e. for |Γ₁₂Γ₂₃| < < 1.

The above expressions for total reflection and transmission of a slab can also be obtained from the theory of multiple reflections [B.5, J.2]. Using the relation (5.4.3), equation (5.4.16) can be recast in the following format also:

      (5.4.17)

In the case of the obliquely incident TE waves, in the above equations, we replace the wavevectors k₁ and k₂ as k₁ → k_1x → k₁ cos θ₁ and k₂ → k_2x → k₂ cos θ₁ respectively; where the waves propagate in the x‐direction. The waves are confined in the y‐direction. As a special limiting case, if the electrical parameters of the slab of thickness d and the host medium are identical, the k₂ = k₁, μ₂ = μ₁. It leads to Γ^Nor = 0, i.e. no reflection at the interface, and , i.e. the slab provides a complete transmission with a lagging phase k₂d. This is the case of the matched layer seen in equation (5.4.15a) for η₁ = η₂. The matching applies even to the interface of electrically dissimilar media given by the condition in equation (5.1.4b). The expressions obtained in the present subsection are useful to characterize the metamaterials slab. It is discussed in subsection (21.4.1) of chapter 21.

5.5 EM‐Waves in Metamaterials Medium

      The electromagnetic properties of metamaterials, propagation of EM – waves in the metamaterials media, and circuit models of metamaterials are presented in this section. Some applications of the metamaterials are also discussed. The realization of engineered metamaterials and their further modeling are discussed in chapter 21.

      5.5.1 General Introduction of Metamaterials and Their Classifications

The materials existing in nature, ignoring losses, are characterized electromagnetically by the relative permittivity and relative permeability. Normally, these are positive quantities. Such a medium is called the double‐positive (DPS) medium. It has a positive real refractive index n. It is placed in the first quadrant of the (μ_r, ε_r)‐plane as shown in Fig (5.7). The conductivity or resistivity is added to a medium to account for the losses. Some noble metals, such as silver and gold, have negative permittivity in the visible frequency bands. Drude model, based on the plasma model of metal, is used to model such behavior. It is discussed in chapter 6, section (6.5.2). An identical phenomenon exists in the ionospheres. Below the so‐called plasma frequency mentioned briefly in section (3.3.1) of chapter 3, the permittivity of a plasma medium becomes negative. In this case, the medium does not support wave propagation. The waves become oscillatory decaying evanescent mode. The permeability of the plasma medium remains a positive quantity. Under the evanescent mode condition, the plasma medium below the cut‐off frequency is called the epsilon‐negative (ENG) medium. It is shown in the second quadrant in Fig (5.7). Thus, below the plasma frequency, the medium reflects the wave incident on it from the DPS medium; while inside the ENG medium only decaying evanescent mode exists. However, above the plasma frequency, the ENG plasma medium becomes a normal DPS medium. Likewise, a gyrotropic medium, such as a ferrite medium, has negative permeability over a lower microwave frequency range. However, its permittivity is a positive quantity. Such a medium is called the mu‐negative (MNG) medium. It is shown in the fourth quadrant of Fig (5.7). Both the ENG and MNG media have imaginary refractive index n and support only decaying evanescent mode, as shown in Fig (5.7). As a single negative medium, both the ENG and MNG belong to the group of the single negative, i.e. (SNG) medium. Both the ENG and MNG are described by the following forms of the Drude model:

      (5.5.1)

      where ω_ep and ω_mp are the electric plasma and the magnetic plasma frequency respectively. The magnetic plasma is conceptual as no magnetic charge is available in nature.

      The natural dielectric and magneto‐dielectric media support the slow‐wave propagation as for these media ε_r > 1, μ_r > 1. However, it is possible to engineer a DPS metamaterial for 0 < ε_r < 1, μ_r ≥ 1 and 0 < μ_r < 1, ε_r ≥ 1. The first medium is called epsilon near zero (ENZ) medium, while the second one is called mu near‐zero (MNZ) medium. They belong to the group called near‐zero (NZ) medium. These media support the fast‐wave. The DNG medium also supports the ENZ and MNZ cases. The NZ medium is shown