limit. It is noted that the evanescent waves still propagate at the interface along the y‐axis in the (y − z)‐plane as the surface waves. The surface wave is bounded in the x‐direction. The amplification of the evanescent wave is also treated through the concept of coupled quasi‐particles called plasmon‐polariton, especially in the optical frequency range [J.13, B.11].
HyperLens
The anisotropic DNG medium‐based hyperlens converts the evanescent waves to the propagating waves by reducing the values of the transverse wavenumber below the wavenumber in the medium. It is realized by the hyperbolic dispersion relation of the uniaxial anisotropic medium discussed in section (4.7.5) of chapter 4. Thus, a hyperlens projects the high‐resolution image in the far‐field region. It is shown in Fig (5.12d). The hyperlens has been realized in the cylindrical geometry using very thin alternate curved layers of conducting and dielectric films. The object is placed at the center and the propagating waves are available at the flat end of the cylindrical medium in which the hyperlens is embedded. The normal optical lens, attached with the hyperlens, carries out the optical processing to create the high‐resolution image in the far‐field region [J.17, J.18].
5.5.7 Doppler and Cerenkov Radiation in DNG Medium
The DNG medium acts inversely on Doppler and Cerenkov radiations. It is examined below.
Figure 5.14 Doppler effect in the DPS and DNG media. The source receding the receiver.
Doppler Effect
Doppler effect is related to the change in frequency of a source due to the relative motion of the source and receiver. If a source is moving away, i.e. receding, from the receiver in a DPS medium, the received frequency is less than the stationary frequency of the source. However, if the source is moving toward the receiver, the received frequency is increased.
Let us examine the receding case of a moving source. Figure (5.14a) shows that a source #S is moving away from receiver #R with velocity Vs along the positive x‐axis. The source occupies positions P1 (x = d1) and P2(x = d2) at time t1 and t2 (t2 > t1), respectively. The source S radiates frequency fs and the radiated spherical wave propagates with the phase velocity Vp in the DPS medium. Figure (5.14a) shows the spherical wavefronts at time t1,…,t4. The phase of the received wave at time t1 and t2 could be written as
The received frequency fR of the wave at the stationary receiver is the rate of change in phase, i.e. ωR = 2πfR = LtΔt → 0(Δϕ/Δt). The received frequency, from the above expression (5.5.38c) is obtained as
Inverse Doppler Effect
The DNG medium, shown in Fig (5.14b), supports the backward wave propagation with opposite phase velocity. In this case, the spherical wavefronts move in the backward direction. Equation (5.5.39) is used to get the frequency of the received waves from a source moving away from the receiver in the DNG medium:
(5.5.40)
The received frequency of the radiated waves from a source, moving towards the receiver, is reduced at the stationary receiver. In the case, the source is moving away from the receiver, the received frequency increases as
Due to the reversal of the change in the received frequency, the DNG medium supports the inverse Doppler Effect [J.6]. It has been experimentally confirmed both in the microwave and optical frequency ranges. The inverse Doppler Effect could be used to design tunable and multifrequency radiation sources [J.23–J.25].
Cerenkov Radiation
The Cerenkov radiation, also called the Cherenkov (or Cherenkov) radiation, in the DPS medium is generated from the charged particles traveling with a velocity faster than the velocity of the EM‐wave in that medium [B.12]. The radiation from a leaky‐wave antenna closely follows the radiation mechanism of Cerenkov radiation. Likewise, the planar transmission lines radiate the Cerenkov type radiation within a substrate resulting in high substrate loss [B.13]. It is discussed in subsection (9.7.3) of chapter 9. In a DPS medium the directions of the radiated power, i.e. the direction of the Poynting vector, and the wavevector are in the same direction. However, in a DNG medium, these directions are opposite to each other, giving inverse Cerenkov radiation [J.3]. Both Cerenkov radiation and inverse Cerenkov radiation are similar to the shock waves of supersonics generating the radiation cone.
Figure (5.15a) shows the generation of the radiation cone of the spherical wavefronts by a moving charge, with velocity Vc, in a DPS medium. The charged particle velocity Vc is greater than the phase velocity Vp of the EM‐wave in a DPS medium, i.e. Vc > Vp( = c/n). The positive refractive index is n, and c is the velocity of EM‐wave in free space. The locations P0–P4, on the positive x‐axis, are successive positions of the moving charge at time t0–t4. During the time interval T = (t4 − t0), the spherical wavefront acquires radius P0N = VpT = (c/n)T, while the charged particle travels a distance P0P4 = VcT. Figure (5.15a) also shows other spherical wavefronts for the time sequence for t < T. Following the Huygens principle, the inclined tangential wavefront of the secondary radiation is NP4. The wavevector (k), i.e. P0N, is normal to the wavefront NP4. It forms a Cerenkov angle ϕc at P0 or angle θc at P4. It is obtained from the triangle P0N P4:
(5.5.41)
