Introduction To Modern Planar Transmission Lines. Anand K. Verma

alt="images"/>. It is shown in the first quadrant of Fig (5.7). The phase of the propagating EM‐wave in the DPS medium lags while traveling in the ‐direction.

      In the case of a DNG medium, both permittivity and permeability are negative. Using Maxwell's equations (5.5.2c,d), and replacing ∇ → − jk the wavevector triplet‐ relations for the DNG medium are written as follows:

      (5.5.3)

      However, in the above expression reversal of the direction of the magnetic field involves the reversal of the direction of the power flow toward the source. Physically, it is not possible, so the above equations are rearranged as follows by associating the negative sign with the wavevector :

      (5.5.4)

      The above wavevector triplet‐ is shown in Fig (5.8b), i.e. in the left‐hand (LH) coordinate system, so the DNG medium is also called the left‐handed, LH‐medium. Both the power‐vector and wavevector triplets and their combination are further shown in Fig (5.8d). The field vectors are rotated to maintain the power flow in a positive direction. Figure (5.8d) shows that the phase and group velocities are opposite to each other () as for a DNG medium the vectors and are antiparallel, i.e. . The DNG occupies the third quadrant in the (μ_r, ε_r)‐plane as shown in Fig (5.7).

      In conclusion, a DNG medium supports the backward wave propagation, whereas a forward wave is supported by the DPS medium. As the phase velocity travels toward the source, while energy is traveling from the source to a load, a propagating EM‐wave in a DNG medium, in the direction of the vector , has a leading phase. This is a unique property of the DNG medium. It significantly influences the EM‐wave characteristics of the DNG medium [J.8, B.6, B.10].

Figure 5.8 RH and LH‐coordinate systems for the DPS and DNG media.

      Refractive Index of DNG Medium

      The above discussion shows that Maxwell's equations in the DNG medium are written in the LH‐coordinate system. However, the wave equation (4.5.32) of chapter 4 for the DPS medium remains valid for a lossless (σ = 0) DNG medium. It provides the following expressions for the propagation constant β = k^DPS and refraction index of a DPS medium:

      (5.5.5)

      The evaluation of the square root of negative permeability and negative permittivity is a critical issue in the DNG medium. The negative number (−1) is exp(±jπ). However, to meet the physical condition, discussed in subsection (5.5.3), we take {−1 = exp(−jπ)} [J.8, J.9]. Therefore, the square roots of negative permeability and negative permittivity are obtained as follows:

Using the above relations, the refractive index of a DNG medium, and also the propagation constant, are obtained as follows:

(5.5.6)

      It is interesting to note that the refractive index for a DPS medium n^DPS is a positive quantity, whereas for a DNG medium n^DNG is a negative quantity. So the metamaterials are also known as the negative refractive index materials, i.e. the NIM. Snell's law of refraction for a DNG medium is also modified accordingly. The negative refractive index also shows the reversal of the direction of the phase velocity of the EM‐wave. However, first let us discuss the intrinsic impedance, i.e. the wave impedance for the DNG and SNG media.

      Wave Impedance of DNG and SNG Media

      Following equation (4.5.26b) of chapter 4, the wave impedance η^DNG in a DNG medium is written below:

      (5.5.7)

      Like the wave impedance in a DPS medium η^DPS, the wave impedance in the DNG medium η^DNG is a positive quantity; showing the outward power flow from the source into a DNG medium. However, the wave impedances of the ENG and MNG media are reactive due to nonpropagating evanescent mode:

      (5.5.8)

      The inductive/capacitive reactive wave impedances of the ENG and MNG media create the reflecting surfaces. The circuit model of the metamaterials, discussed in section (5.5.3), elaborates on the nature of the RIS. Further details of the artificial RIS surface is discussed in section (20.2) of chapter 20. It is noted that the ENG/MNG medium is realized through the nonpropagating evanescent wave. Such an environment is provided by a rectangular waveguide below the cut‐off region. It is commented in subsection (7.4.1) of chapter 7.

      The propagation constants of EM‐waves in the ENG and MNG media are obtained below:

(5.5.9)

Expressions (5.5.9 a,b) show that the ENG and MNG media do not support wave propagation. Figure (5.7) shows that these media are placed in the second and fourth quadrants in the (μ_r, ε_r)‐plane. They only support the decaying evanescent mode.