Introduction To Modern Planar Transmission Lines. Anand K. Verma

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 V_s 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 f_s and the radiated spherical wave propagates with the phase velocity V_p in the DPS medium. Figure (5.14a) shows the spherical wavefronts at time t_1,…,t₄. The phase of the received wave at time t₁ and t₂ could be written as

(5.5.38)

The received frequency f_R of the wave at the stationary receiver is the rate of change in phase, i.e. ω_R = 2πf_R = Lt_{Δt → 0}(Δϕ/Δt). The received frequency, from the above expression (5.5.38c) is obtained as

(5.5.39)

      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 V_c, in a DPS medium. The charged particle velocity V_c is greater than the phase velocity V_p of the EM‐wave in a DPS medium, i.e. V_c > V_p( = 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 = V_pT = (c/n)T, while the charged particle travels a distance P₀P₄ = V_cT. 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 ϕ_c at P₀ or angle θ_c at P₄. It is obtained from the triangle P₀N P₄:

      (5.5.41)

      Figure (5.15b) shows the directions of the wavevectors and Poynting vectors of the upper and lower radiation from the charged particle moving in the positive x‐direction. Cerenkov radiation is a linearly polarized EM‐wave with an electric field component parallel to the path of the charged particle, i.e. the x‐axis. Cerenkov radiation is directive in the forward direction within the cone angle 2ϕ_c. Similarly, a leaky‐wave antenna is highly directive. The beam becomes narrower for the higher value of the refractive index n.

      Inverse Cerenkov Radiation

      Figure (5.15c) shows that the charged particle moving with a velocity V_c (V_c > V_p) in a DNG medium with a negative refractive index −|n|. It creates backward radiation with the negative Cerenkov angle,

      (5.5.42)