Anand K. Verma - Introduction To Modern Planar Transmission Lines

At first, the characteristic impedance of equivalent transmission lines, corresponding to wave impedance of both media, is obtained for both the TE and TM‐polarized waves. Next, the relations between reflection/transmission coefficient at the interface of the physical media and reflection/transmission coefficient at the junction of equivalent lines are obtained.

The wave impedance of the incident, and transmitted TE waves in the medium #1 and medium #2, as shown in Fig (5.2a), with respect to the direction of propagations k 1and k 2are given below:

(5.2.20)

However, the interface in the (y − z)‐plane views both the above‐given wave impedances differently due to the oblique incidence. The left (x = 0 −) and right (x = 0 +) side faces view the following x‐directed wave impedances, and , due to the (E z, H y) field components:

(5.2.21)

Figure (5.2b)shows the equivalent transmission line model of the obliquely incident TE‐polarized wave. The wave impedances , and seen by the interface are taken as the characteristic impedance of the equivalent lines#1 and #2, respectively.

Similarly, the wave impedances viewed by the interface with the obliquely incident TM‐polarized wave are obtained with reference to Fig (5.3a):

(5.2.22)

(5.2.23)

The superscripts k 1and k 2,used in equations (5.2.22)and (5.2.23)are dropped in further discussion.

The reflection and transmission coefficients of both the TE and TM‐polarized obliquely incident waves at the interface (x = 0) of physical media as taken as follow:

(5.2.24)

The reflection and transmission coefficients of TE‐polarized at the junction of two equivalent lines, shown in Fig (5.2b), are obtained from E z‐field components of the incident, reflected, and transmitted waves using equations (5.2.2)– (5.2.4):

(5.2.25)

Thus, for the TE‐polarized obliquely incident EM‐wave, the line reflection, and transmission coefficients correspond to the reflection and transmission coefficients at the interface of two physical media. However, for the TM‐polarized case, shown in Fig (5.3b), change occurs for the transmission coefficient. The E y‐field components of the incident, reflected, and transmitted waves using equations (5.2.12)– (5.2.14)are considered to define line junction reflection and transmission coefficients of the TM‐polarized case:

(5.2.26)

At this stage, the reflection and transmission coefficients of the TE/TM‐polarized waves at the interface of two physical media could be computed. Using the above discussion, the reflection coefficient and the transmission coefficient of an obliquely incident TE (perpendicular) – polarized wave are computed from the transmission line model as follows:

(5.2.27)

Equation (5.2.27a,b)are identical to equation (5.2.8c,d), The reflection coefficient and the transmission coefficient of an obliquely incident TM‐polarized wave are further computed using the transmission line model, shown in Fig (5.3b), as follows:

(5.2.28)

Equations (5.2.23)and (5.2.26)are used to get the above expressions. Equation (5.2.28a, b)are identical to equation ( 5.2.16c,d). The transmission line models are also used to obtain the reflection and transmission coefficients of both normal and oblique incident plane waves on a multilayered slab medium [B.1, B.3, B.4]

There are two special cases of the angle of incidence: one for the complete transmission of waves at the interface, and another for the total reflection of the wave at the interface of two media. These are known as Brewster angle and critical angle. Brewster angle is the angle of incidence at which the reflection coefficient is zero and the incident wave is fully transmitted from one medium to another with refraction . So the Brewster angle corresponds to the impedance matching condition under which reflection at the interface, in the medium #1, is zero, and the incident power is completely transmitted to the medium #2. At the critical angle of incidence or the incidence angle greater than the critical angle, the incident wave is completely reflected by the interface of two dielectric media . This kind of total reflection is superior to that of the total reflection from any metallic surface that always introduces loss. We consider these two special cases of the angle of incidence for both the TE and TM polarizations.