Introduction To Modern Planar Transmission Lines. Anand K. Verma

It is shown below from their reactance:

      (5.5.16)

      Figure (5.10) shows the equivalent T‐network unit cell for all four kinds of media in the (ε_r, μ_r)‐plane, i.e. in the corresponding (C, L)‐plane. The ε_r(C)‐axis and μ_r(L)‐axis show the capacitive shunt element and the inductive series element of the equivalent T‐network of material media. The DPS medium in the first quadrant is changed to the ENG medium by taking a negative value for shunt capacitance that corresponds to a shunt inductor. The L‐L network in the second quadrant corresponds to the ENG medium. The MNG medium is obtained by taking a negative value for the series inductance of the DPS medium. It results in the C‐C network in the fourth quadrant. The characteristics impedance of the ENG and MNG equivalent lines are inductive and capacitive respectively. Of course, by taking negative values for both the series inductance and shunt capacitance of the DPS medium, the DNG medium is created. It is shown as a CL unit cell in the third quadrant of Fig (5.10). Individually, the ENG and MNG media do not support EM‐wave propagation. However, jointly they form a transparent medium, and EM‐wave propagates through the joint medium. It is known as the tunneling phenomenon [J.10, J.11].

      The above description of circuit modeling is also applicable to a waveguide below the cut‐off frequency. The TE‐mode waveguide below cut‐off frequency provides the inductive load. So it could be viewed as an ENG medium, while the TM‐mode waveguide below the cut‐off frequency provides the capacitive load and it could be viewed as the MNG medium [B.6, B.8]. The behavior of the modal wave impedance of a rectangular waveguide below the cut‐off frequency is discussed in subsection (7.4.1) of chapter 7.

      5.5.4 Lossy DPS and DNG Media

      A lossy DPS medium is characterized by the complex permittivity and complex permeability and , also and . The complex wavenumber for the EM‐wave in a lossy DPS medium is obtained as follows:

      (5.5.17)

      (5.5.18)

      On separating the real and imaginary parts of a complex wavenumber in the DPS medium, k^*DPS = k′ − jk^″, the following expressions are obtained:

(5.5.19)

      The DPS medium has . So using the above equations, it is obvious that k^' > 0, k^″ > 0.

The above expressions are also valid for the DNG medium with some modifications. In the case of a DNG medium, the medium parameters are . However, even in the DNG medium, the electric and magnetic losses are positive quantities, i.e. . Using equation (5.5.19), the complex wavenumber in the DNG is given as follows:

      (5.5.20)

      The complex DPS medium is also described by the complex refractive index:

      (5.5.21)

      However, in the case of a complex DNG medium, the complex refractive index is n^*DNG = − (n′ + jn″); as Re(n^*DNG) is a negative quantity and Im(n^*DNG) is still a positive quantity. The above discussion is for the propagating waves in the DPS and DNG media. However, in case the waves are nonpropagating (evanescent) in both media, the real part of the wavenumber is an imaginary quantity i.e. k′ = − jα′. The evanescent wave behaves differently in the DPS and DNG media. It is examined below.

      The electric fields of the x‐directed propagating and nonpropagating EM‐waves in the unbounded lossy DPS and DNG media could be expressed as follows:

      (5.5.22)

      The propagating EM‐wave is attenuated while traveling in both the DPS and DNG lossy media k^″ ≠ 0. However, the DPS medium offers a lagging phase, whereas the DNG medium offers a leading phase to the wave traveling in the positive x‐direction. Poynting vector decides the direction of the EM‐wave propagation. The λ_g/2‐line resonator could be designed in the DPS‐DNG composite, with a length λ_g/4 in each medium, without any phase‐shift at the output. The classical λ_g/2‐line resonator, in a DPS medium, has 180° phase at the output. It is further seen from the above equation that the evanescent wave is decaying with distance x while traveling in the DPS medium. The enhancement of amplitude by the DNG medium could be viewed as the step‐up transformer action, whereas it is increasing in amplitude while traveling in the DNG medium. This property is more clearly seen in a lossless medium with k^″ = 0.

      5.5.5 Wave Propagation in DNG Slab

The EM‐wave propagating through a normal DPS slab provides the lagging phase of the propagating part of the field at the output end of the slab. It is noted that the nonpropagating decaying evanescent wave also exists, along with the propagating wavefield components, inside the slab. However, for a propagating wave, the DNG slab provides the leading phase at its output. The increasing evanescent wave exists inside a DNG slab, along with the propagating wavefield components. Thus, the DNG slab can act both as (i) a phase‐compensator and (ii) as an amplitude‐compensator, i.e. as a field amplifier, without any power amplification. The field amplifier is like a voltage step‐up transformer without any power amplification. The first property, in the form of the DGS‐DNG composite slab, is helpful in the design of compact sub‐wavelength resonator [J.12, B.6]. A normal DPS resonator is half‐wavelength long. The second property offers an opportunity to design subwavelength resolution superlens [J.2, J.13–J.21]. This subsection explains both properties of a DNG slab. The superlens is discussed briefly in the next subsection.

Figure 5.11 EM‐wave propagation through the DNG and composite DPS‐DNG slabs.

      Phase – Compensation in the DPS‐DNG Slab

Figure (5.11a) shows a DNG slab, μ₂ = − |μ₂|, ε₂ = − |ε₂|,