Introduction To Modern Planar Transmission Lines. Anand K. Verma

in DPS‐DNG Composite Medium

Figure (5.9a) shows the DPS‐DNG composite medium. The incident ray is in the third quadrant of the DPS medium #1. For the DPS medium #2, the refracted ray comes out in the first quadrant; as the angle of refraction (θ₂) is positive. The refraction in the DPS‐DPS composite medium follows Snell's law given by equation (5.2.7c). However, if the medium #2 is DNG‐type then the angle of refraction (−θ₂) is negative due to the negative refractive index. It follows from Snell's law:

      (5.5.10)

      Figure (5.9a) shows that due to the negative angle of refraction (−θ₂), the refracted ray in the DNG medium #2 emerges from the fourth quadrant. It shows the reversal of Snell's law in the DPS‐DNG composite medium, as compared to Snell's law in the DPS‐DPS composite medium. Figure (5.9b) shows that the wavevector in medium #2 must be in the reverse direction to meet the phase‐matching condition, , at the interface of the composite medium.

In the case of the TE‐polarized incident waves, the wavevectors of the incident, reflected, and refracted rays are given by equations (5.2.1a–d), whereas their Poynting vectors are given in equations (5.2.5a–c). For the DPS‐DPS composite medium, both vectors are in the same direction after refraction, showing the presence of the forward‐wave in medium #2. However, for the DPS‐DNG medium, these vectors of the transmitted wave in the DNG medium #2 are given as

Figure 5.9 Refraction of the obliquely incident EM‐wave at the interface of the DPS‐DPS and DPS‐DNG composite medium.

      (5.5.11)

      The wavevector , satisfying the phase‐matching at the interface, of the above expressions, is obtained from Fig (5.9b). The Poynting vector is written from the wavevector diagram shown in Fig (5.9a). It is also obtained by using equation (5.2.4) to compute the Poynting vector in the DNG medium for the angle of refraction (−θ₂). The above expressions further show that the DNG medium #2 supports the backward‐wave propagation because the vectors and are anti‐parallel.

      5.5.3 Basic Transmission Line Model of the DNG Medium

      The unbounded DPS medium and a transmission line both support the forward wave propagation in the TEM mode, so Fig (3.28a) of chapter 3 models a DPS medium by the LC transmission line. The equivalence between the material parameters ε, μ and the circuit parameters C, L is discussed in subsection (3.4.2) of chapter 3. The characteristics impedance and propagation constant of the DPS medium and equivalent transmission lines are summarized below:

      (5.5.12)

      The equivalent LC transmission line is an analog of the DPS medium. Therefore, the medium parameters μ and ε could be treated through the analogous series inductance L, and shunt capacitance C, i.e. L → μ, C → ε. The expressions L = Z(ω)/jω and C = Y(ω)/jω for series inductance and shunt capacitance, respectively, model the DPS medium parameters as follows:

(5.5.13)

In equation (5.5.13), Z(ω) and Y(ω) are the series impedance and shunt admittance of the equivalent line.

      Following the above discussion, a DNG medium, supporting the backward‐wave propagation, could be modeled through the CL‐line, shown in Fig (3.28b) of chapter 3. Using the above expressions, the material parameters of a DNG medium could be modeled as follows:

      (5.5.14)

      The above expressions show that the CL‐line model provides negative values for the permeability and permittivity, as needed for a DNG medium. The characteristic impedance and the propagation constant of the EM‐wave in a DNG medium follows from the above equations:

(5.5.15)

Expression (5.5.15b) shows that even a lossless DNG medium is highly dispersive due to the presence of a term ω² in the denominator. It is unlike a DPS medium which is nondispersive for the lossless case. However, a dispersive medium is always associated with loss to meet the Kramer–Kronig condition of causality. It is discussed in subsection (6.5.4) of chapter 6. The causality fails for a fictitious nondispersive DNG medium. The discussion shows that the DNG medium parameters and wavenumber, i.e. ε_r, μ_r, n, k, are complex quantities [J.8].

Figure 5.10 Circuit models of four kinds of the medium on the (μ, ε)‐plane.

      The DPS transmission line LC‐model can also be extended to the ENG and MNG media. Both these media do not support any EM‐wave propagation. Only the decaying evanescent mode is supported by them, as these media are reactive. The ENG medium is obtained for −ε_r and +μ_r that correspond to negative shunt capacitance and positive series inductance. Likewise, the MNG medium is obtained for −μ_r and +ε_r, i.e. for negative series inductance and positive shunt capacitance. However, the negative capacitance and