Page 147 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 135

d d d d d d
1 1 1 1 1 1
d
d d
d
2 2 2 d d d 2 2 d d 1 d 2 2 2 d d d 1 1 1 d d 2 2 2 2 2 2
2
1 1
ε ε
ε
ε ε 1 1 1 ε ε ε 2 2 2 ε 1 1 1 ε ε ε 2 2 2 ε 1 1 1 ε ε ε 2 2 2 ε ε 1 1 1 . . .
ε
ε ε
. . .
Incoming Wave Exiting Wave
Incoming Wave
Exiting Wave
Figure 3-24 Sketch of one-dimensional PBC.
Focusing on a unit cell, see Fig. 3-25, we notice that if a wave impinges
from the left on this unit cell, it will in general, undergo multiple reflections
and trasmissions at two places, namely, t, r at the first ε /ε , discontinuity,
1 2
and r’, t’ at the ε /ε discontinuity.
1
2
d d d 1 1 1 d d d 1 1 1
d d d 2 2
2 2 2 2 2 2 2
ik ik ik z z z t t t e e ik 1 z z
ik 1
e e e 1 1 1 ε 1 ε 1 ε 1 ε 2 ε 2 ε 2 ε 1 ε 1 ε 1 To
Total ltal
Tota
ik z z
r r Totatal e e − − ik 1 1 Un it C e
r Total lTo
Un it C e ll ll
(a)
ε ε ε 1 1 1 ε ε ε 2 2 2 ε ε ε 1 1 1
z z
0 0 d d
tt ik ik 2 2 d d
tt e ' e '
+ + 2 2 2
ik 2 d
ik 2 d
tt e e' ' tt ik 2 d r r r ' e' e' e ik 2 d d
ik 2
+ + 4 4 4
ik 2 d
ik 2 d
tt
tt e e' ' ik 2 d r r r ' e' e' e ik 2 d d
ik 2
. .
+ . . .
. . + . . .
. . = t
= t
To t a l l
To t a
(b)
Figure 3-25 (a) PBC unit cell. (b) Transmission/reflection analysis. k = ω µε is the
i i
wave vector in region i.
Then, the amplitude of the transmitted wave will be given by the sum of the
following terms [58]:
(1) The fraction that is transmitted through the ε /ε interface, is
1 2
phase-shifted while traversing (left-to-right) the region ε of length
2
d, and then is transmitted through the ε /ε discontinuity, namely,
2 1
'
te ik 2 d t . This is the amplitude for direct transmission through two
discontinuities.