Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 234

Figure (5.9a) shows the DPS‐DNG composite medium. The incident ray is in the third quadrant of the DPS medium #1. For the DPS medium #2, the refracted ray comes out in the first quadrant; as the angle of refraction (θ₂) is positive. The refraction in the DPS‐DPS composite medium follows Snell's law given by equation (5.2.7c). However, if the medium #2 is DNG‐type then the angle of refraction (−θ₂) is negative due to the negative refractive index. It follows from Snell's law:

(5.5.10)

Figure (5.9a) shows that due to the negative angle of refraction (−θ₂), the refracted ray in the DNG medium #2 emerges from the fourth quadrant. It shows the reversal of Snell's law in the DPS‐DNG composite medium, as compared to Snell's law in the DPS‐DPS composite medium. Figure (5.9b) shows that the wavevector in medium #2 must be in the reverse direction to meet the phase‐matching condition, , at the interface of the composite medium.

In the case of the TE‐polarized incident waves, the wavevectors of the incident, reflected, and refracted rays are given by equations (5.2.1a–d), whereas their Poynting vectors are given in equations (5.2.5a–c). For the DPS‐DPS composite medium, both vectors are in the same direction after refraction, showing the presence of the forward‐wave in medium #2. However, for the DPS‐DNG medium, these vectors of the transmitted wave in the DNG medium #2 are given as

Figure 5.9 Refraction of the obliquely incident EM‐wave at the interface of the DPS‐DPS and DPS‐DNG composite medium.

(5.5.11)

The wavevector , satisfying the phase‐matching at the interface, of the above expressions, is obtained from Fig (5.9b). The Poynting vector is written from the wavevector diagram shown in Fig (5.9a). It is also obtained by using equation (5.2.4) to compute the Poynting vector in the DNG medium for the angle of refraction (−θ₂). The above expressions further show that the DNG medium #2 supports the backward‐wave propagation because the vectors and are anti‐parallel.