Post on 20-Jan-2020
Advanced Topics in Photonic Devices (14/1) W.-Y. Choi
Lect. 4: Total Internal Reflection
i r
t
1 1.0n
2 1.44n
inE rE
tE
12 2 2
12 2 2
21
2 2 2 1
cos sin
cos sin
2cos ( )cos sin
i ir
ii i
t i
ii i
nErE n
E nt nE nn
12 2 22
|| 12 2 22
|| 12 2 22
sin cos
sin cos
2 cos
sin cos
i i
i i
i
i i
n nr
n n
ntn n
Brewster angletan( ) p n
?21 nn
?21 nn
,0 For i
(n>1)
Advanced Topics in Photonic Devices (14/1) W.-Y. Choi
Lect. 4: Total Internal Reflection
i r
t
1 1.44n
2 1.0n
inE rE
tE
What happens when i > c?
12 2 2
12 2 2
cos sin
cos sin
i ir
ii i
nErE n
12 2 22
|| 12 2 22
sin cos
sin cos
i i
i i
n nr
n n
r becomes complex when sin ni (only when n < 1)
(n < 1)
jerr
Advanced Topics in Photonic Devices (14/1) W.-Y. Choi
Lect. 4: Total Internal Reflection
2n
1ni
t
x
z
Consider k-vector diagram:
- Direction: Direction of wave propagation
- Magnitude: - As i changes, ki, kr, kt traces on a circle
1 2sin sini tn n
, ,z r z t zk k k From Lect. 4,
1 0 2 0sin sini tn k n k
1 2If , t in n
With increasing, reaches 90 firstbeyond which no tranmission is possible.
i t
1 12
1
(critical angle) sin sincn nn
Advanced Topics in Photonic Devices (14/1) W.-Y. Choi
Lect. 4: Total Internal Reflection
, 1 0( sin )t z z ik k n k
2n
1ni
t
x
z
2 2 2, ,t t z t xk k k
0
tx tzjk x jk zt iE tE e e
,t xk j
2 2 2, ,t x t t zk k k
What happens if ?i c
2 22 0 1 0( ) ( sin )in k n k
,( )( ) t zjk zxitE e e
12 2 2
0 1 2[( sin ) ]ik n n
Attenuation in x-direction!
Total internal reflection
, From2
,
2
,
2
xtztt kkk 2201
202 sin -α)θk(n)k(n i
202
201
2 sin )k-(n)θk(nα i
?
Advanced Topics in Photonic Devices (14/1) W.-Y. Choi
Lect. 4: Total Internal Reflection
1 2 1 22 , and : from 0 to 90Incident and Transmitted Waves for perpendicular polarization
i
Advanced Topics in Photonic Devices (14/1) W.-Y. Choi
Lect. 4: Total Internal Reflection
12 2 2
12 2 2
cos sinFrom
cos sin
i i
i i
nr
n
For , use or 11
i r tt E E E r tt r
What is , ?r t
sin ( sin sin )i i cn n
1 r
1 12 2 2 22 2 Let sin sini in j n
12 2 2
12 2 2
cos sin
cos sin
i i
i i
j nr
j n
jr e
1 12 2 2 22 2-1 -1(sin ) (sin ) =tan ( ) ( tan ( ))
cos cosi i
i i
n n
12 2 2-1 (sin )2tan ( )
cosi
i
n
for total internal reflection
Advanced Topics in Photonic Devices (14/1) W.-Y. Choi
Lect. 4: Total Internal Reflection
12 2 22
|| 12 2 22
sin cosFrom
sin cos
i i
i i
n nr
n n
|| ||What is , ?r t
From 1 tri r t
i i
HHH H HH H
1|| ||
2
Since and (Lect. 4)tr
i i
HHr tH H
|| ||1 r nt
1 12 2 2 22 2 sin ( sin sin ) Let sin sini i c i in n n j n
||
||
12 2 22
|| 12 2 22
sin cos
sin cos
ji i
i i
j n nr r e
j n n
1 1 12 2 2 2 2 22 2 2-1 -1 -1
|| || 2 2 2
(sin ) (sin ) (sin )1 and = - + tan ( ) ( tan ( )) - + 2tan ( )cos cos cosi i i
i i i
n n nrn n n
2|| || ||
1
1 r t nt
for total internal reflection
Advanced Topics in Photonic Devices (14/1) W.-Y. Choi
Lect. 4: Total Internal Reflection
Frustrated TotalInternal Reflection
Goos-Hanchen phase shift
jr e
Advanced Topics in Photonic Devices (14/1) W.-Y. Choi
Lect. 4: Total Internal Reflection
Beam Splitter
Advanced Topics in Photonic Devices (14/1) W.-Y. Choi
Lect. 4: Total Internal Reflection
Exercise Problem