Page 448 - Engineering Electromagnetics, 8th Edition

430 ENGINEERING ELECTROMAGNETICS

3
polarization, or is said to be s-polarized. E is also parallel to the interface, and so
the case is also called transverse electric, or TE polarization. We will find that the
reflection and transmission coefficients will differ for the two polarization types, but
that reflection and transmission angles will not depend on polarization. We only need
to consider s- and p-polarizations because any other field direction can be constructed
as some combination of s and p waves.
Our desired knowledge of reflection and transmission coefficients, as well as how
the angles relate, can be found through the field boundary conditions at the interface.
Specifically, we require that the transverse components of E and H be continuous
across the interface. These were the conditions we used to find and τ for normal
incidence (θ 1 = 0), which is in fact a special case of our current problem. We will
consider the case of p-polarization (Figure 12.7a) first. To begin, we write down the
incident, reflected, and transmitted fields in phasor form, using the notation developed
in Section 12.4:
+ − jk · r
E = E e + 1 (51)
+
s1
10
− − jk · r
E = E e − 1 (52)
−
s1
10
E s2 = E 20 e − jk 2 · r (53)
where
k = k 1 (cos θ 1 a x + sin θ 1 a z ) (54)
+
1
k = k 1 (− cos θ a x + sin θ a z ) (55)
−

1
1
1
k 2 = k 2 (cos θ 2 a x + sin θ 2 a z ) (56)
and where
(57)
r = x a x + z a z
√ √
The wavevector magnitudes are k 1 = ω r1 /c = n 1 ω/c and k 2 = ω r2 /c =
n 2 ω/c.
Now, to evaluate the boundary condition that requires continuous tangential elec-
tric field, we need to find the components of the electric fields (z components) that
are parallel to the interface. Projecting all E fields in the z direction, and using (51)
through (57), we find
+
E zs1 = E + e − jk · r = E cos θ 1 e − jk 1 (x cos θ 1 +z sin θ 1 ) (58)
+
+
1
z10
10
jk 1 (x cos θ −z sin θ )
−
E − = E − e − jk · r = E cos θ e 1 1 (59)
−
1
1
zs1
10
z10
E zs2 = E z20 e − jk 2 · r = E 20 cos θ 2 e − jk 2 (x cos θ 2 +z sin θ 2 ) (60)
3 The s designation is an abbreviation for the German senkrecht, meaning perpendicular. The p in
p-polarized is an abbreviation for the German word for parallel, which is parallel.