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