Page 148 - Principles and Applications of NanoMEMS Physics

136 Chapter 3

(2) The fraction that is reflected from ε /ε , is phase-shifted while
2 1
traversing (right-to-left) ε of length d, and then is reflected again
2
at ε /ε , phase-shifted left-to-right the region ε of length d, and
1
2
2
so on. This is the amplitude for transmission after two reflections,
and so on.
The frequency selectivity originates as follows [58]. At frequencies where
k2d is an even multiple of π 2 , we have,

§ π · 2 4
t Total ¨ k 2 d = even number ⋅ ¸ = (1 ' tt + ' r + ' r + ... ), (182)
© 2 ¹

that is, every term inside the parenthesis is exactly in phase and there is
constructive interference; this results in maximum transmission.
On the other hand, if k2d is an odd multiple of π 2 , we have,

§ π ·
2
4
t k ¨ d = odd number⋅ ¸ = (1 ' tt − ' r + ' r + ... ), (183)
Total 2
© 2 ¹
that is, every term inside the parenthesis alternates in sign and there is
destructive interference, which results in minimum transmission. With
µ
Z = i
i ε i representing the characteristic impedance of region i , we obtain
ε
the complex reflection and transmission coefficients as follows,

Z − Z ε − ε
' r = 2 1 = 1 2
Z + Z ε + ε , (184)
2 1 1 2

t '= 2 Z 2 = 2 ε 1
Z + Z ε + ε
2 1 1 2 . (185)

The real reflection and transmission coefficients are given by,

=
Rr' 2
, (186)

and