Introduction To Modern Planar Transmission Lines. Anand K. Verma

in the DPS host medium, μ₁ = + |μ₁|, ε₁ = + |ε₁|, η₁. The TE‐polarized wave is normally incident at the first interface. The DNG slab supports the backward wave with the wavevector k₂ in the opposite direction, as power flows from the left to right. This has important consequences.

      Let us assume that in Fig (5.11a), the slab is a DPS type, and it is impedance matched with the host DPS medium, i.e. η₁ = η₂. The reflection and transmission coefficients are obtained from equation (5.4.15):

      (5.5.23)

However, if the DPS slab is replaced by a matched DNG slab, then the following expressions are obtained, on using from equation (5.5.6b):

      (5.5.24)

      A complete transmission of waves occurs through the DPS/DNG slab. However, the DPS slab provides the lagging phase (ϕ = − |k₂|d) at the output of the slab, whereas the DNG slab provides the leading phase (ϕ = + |k₂|d). This property of the DNG slab is useful in compensating the lagging phase of a DPS slab in a DPS‐DNG composite slab.

      Figure (5.11b) of the composite DPS‐DNG slab illustrates the application of a DNG slab as a phase compensator. Again, the impedance matching of both slabs with the host medium is assumed, i.e. η₁ = η₂ = η₀, such that the total reflection coefficient is zero, . The total transmission coefficient at the out of the DNG slab is given as follows:

      (5.5.25)

      To get the compensated phase, i.e. the zero phase, at the output of the DPS‐DNG slab, the following condition, obtained from the above equation, must be met:

      (5.5.26)

      where and are refractive indices of the DPS and DNG slabs. The thicknesses of the slabs need not be equal. The dimension of thicknesses could be even in the sub‐wavelength. It is useful in designing of very compact sub‐wavelength cavity resonators and parallel plate waveguides [J.12]. Such resonance can be developed even in the compact composite ENG‐MNG slab [B.6]. The present concept also finds application in designing properly matched electrically small dipole antenna [B.6].

      Amplitude‐Compensation in the DNG Slab

The resolving power of an optical lens is restricted by the wavelength of a source. This is known as the diffraction limit of the lens. Fourier optics, i.e. the wave optics adequately explain it in terms of the propagating waves and decaying evanescent waves generated by the object source shown in Fig (5.12a). The object source located at x = 0 generates an arbitrary wave field that can be decomposed in terms of the plane wave spectrum. Their superposition reconstructs the wave field. Thus, the wavefield in the real DPS space, propagating in the x‐direction, is composed of 2D Fourier plane wave components in the Fourier domain, i.e. in the k‐space (propagation vector space). The wave field is expressed through the following 2D Fourier integral:

Figure 5.12 Creation of image using wave optics.

(5.5.27)

The wave propagates in the x‐direction and the spectral field components are in the (k_y − k_z)‐plane. The spectral field amplitudes with phase distribution within the bracket [ ] are Fourier components of the electric field . The propagation constant of the plane wave, given by equation (5.5.27b), is obtained from the dispersion relation (4.5.29d) of chapter 4. The propagation constant of propagating plane waves must be a real quantity, whereas imaginary values of provide decaying evanescent waves in the x‐direction. Thus, Fourier components of the wave originated by an object could be divided into two groups: propagating waves and nonpropagating evanescent waves. The grouping of waves is shown in Fig (5.12b). In a DPS medium, the propagation constants for both groups of waves, from equation (5.5.27b), could be written as follows:

(5.5.28)

The electric fields of both the propagating and evanescent waves in the DPS medium could be written from the above relations as follows:

(5.5.29)

      Figure (5.12a) shows both the propagating and exponentially decaying evanescent waves in the DPS medium, generated by an object. The evanescent waves decay fast within a distance under λ. The higher value of transverse components of the wavevector (k_y, k_z) corresponds to the finer spatial details of the object. So at the image-plane, finer spatial details of an image are lost. The DPS based lens cannot recover the lost spatial details in the evanescent waves. The maximum value of the transverse wavevector at the cut‐off wavenumber k_x = 0 determines the limit of the finer spatial details, i.e. the diffraction limit:

      (5.5.30)

      The maximum resolution Δ is the minimum adjacent distance of the finer spatial details of the object. It called the diffraction limit. The relation Δ × k_t,max = 2π is obtained in subsection (19.1.1) of