Page 298 - Electromagnetics

So the reflection coefficients are purelyreal, with signs dependent on the constitutive
parameters of the media. We can write

˜
˜
⊥ = ρ ⊥ e jφ ⊥ , = ρ e jφ ,
where ρ and φ are real, and where φ = 0 or π.
Under certain conditions the reflection coefficients vanish. For a given set of constitu-
˜
tive parameters we mayachieve = 0 at an incidence angle θ B , known as the Brewster
or polarizing angle. A wave with an arbitrarycombination of perpendicular and paral-
lel polarized components incident at this angle produces a reflected field with a single
component. A wave incident with onlythe appropriate single component produces no
reflected field, regardless of its amplitude.
˜
For perpendicular polarization we set ⊥ = 0, requiring
η 2 cos θ i − η 1 cos θ t = 0
or equivalently

µ 2 2 µ 1 2
(1 − sin θ i ) = (1 − sin θ t ).
2 1
By(4.297) we mayput

2 µ 1 1 2
sin θ t = sin θ i ,
µ 2 2
resulting in

2 µ 2 2 µ 1 − 1 µ 2
sin θ i = 2 2 .
µ − µ
1
1 2
The value of θ i that satisfies this equation must be the Brewster angle, and thus

θ B⊥ = sin −1 µ 2 2 µ 1 − 1 µ 2 .
2
2
1 µ − µ
1 2
When µ 1 = µ 2 there is no solution to this equation, hence the reflection coefficient cannot
vanish. When 1 = 2 we have

θ B⊥ = sin −1 µ 2 = tan −1 µ 2 .
µ 1 + µ 2 µ 1
˜
For parallel polarization we set = 0 and have
η 2 cos θ t = η 1 cos θ i .

Proceeding as above we find that

θ B = sin −1 2 1 µ 2 − 2 µ 1 .
2
2
−
µ 1
1 2
This expression has no solution when 1 = 2 , and thus the reflection coefficient cannot
vanish under this condition. When µ 1 = µ 2 we have

θ B = sin −1 2 = tan −1 2 .
1 + 2 1

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