Page 307 - Electromagnetics

˜
This process is repeated until reaching R 1 , whereupon all of the global reflection coeffi-
cients are known. We then find the amplitudes beginning with a 1 , which is the known
˜
incident field amplitude. From (4.315) we find b 1 = a 1 R 1 , and from (4.313) we find
˜ ˜
(1 + 1 )P 1
a 2 = a 1 .
˜ ˜ ˜ 2
1 + 1 R 2 P
1
This process is repeated until all field amplitudes are known.
We note that the process outlined above holds equallywell for parallel polarization as
long as we use the parallel wave impedances
k z,n η n
Z n =
k n
when computing the interfacial reflection coefficients. See Problem ??.
As a simple example, consider a slab of material of thickness sandwiched between
two lossless dielectrics. A time-harmonic uniform plane wave of frequency ω = ˇω is
normallyincident onto interface 1, and we wish to compute the amplitude of the wave
reflected byinterface 1 and determine the conditions under which the reflected wave
vanishes. In this case we have N = 2, with two interfaces and three regions. By(4.316)
˜
˜
we have R 2 = 2 , where R 2 = R 2 ( ˇω), 2 = 2 ( ˇω), etc. Then by(4.315) we find
1 + R 2 P 1 2 1 + 2 P 1 2
R 1 = 2 = 2 .
1 + 1 R 2 P 1 + 1 2 P
1 1
Hence the reflected wave vanishes when
2
1 + 2 P = 0.
1
Since the field in region 0 is normallyincident we have
√
k z,n = k n = β n = ˇω µ n n .
2
If we choose P =−1, then 1 = 2 results in no reflected wave. This requires
1
Z 1 − Z 0 Z 2 − Z 1
= .
Z 1 + Z 0 Z 2 + Z 1
2
Clearing the denominator we find that 2Z = 2Z 0 Z 2 or
1

Z 1 = Z 0 Z 2 .
2
This condition makes the reflected field vanish if we can ensure that P =−1.To do
1
this we need
e − jβ 1 2 =−1.

The minimum thickness that satisfies this condition is β 1 2 = π. Since β = 2π/λ, this
is equivalent to

= λ/4.

A layer of this type is called a quarter-wave transformer. Since no wave is reflected from
the initial interface, and since all the regions are assumed lossless, all of the power carried
bythe incident wave in the first region is transferred into the third region. Thus, two
regions of differing materials maybe “matched” byinserting an appropriate slab between

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