Optoelectronics (16/2) W.-Y. Choi Lect. 4: Dielectric Interface i E i i H t θ t E t H r θ r E r H 1 1 , μ ε 2 2 , μ ε 20 and , 4 2 1 2 1 i ( Cheng 8-10)
Optoelectronics (16/2) W.-Y. Choi
Lect. 4: Dielectric Interface
iE
i
iH
tθ
tE
tHrθ
rE
rH11,με 22 ,με
20 and ,4 2121 i
( Cheng 8-10)
Optoelectronics (16/2) W.-Y. Choi
Lect. 4: Dielectric Interface
i r
t
1n2n
x
y zinE rE
tE
(Assume Ein has only y-component: Perpendicular Polarization)
x zjk x jk zin iE yE e e
xk
trtr EE ,,,
tx tzjk x jk zt tE yE e e
rx rzjk x jk zr rE yE e e
rrzrrx knkknk sin,cos 0101
ttzttx knkknk sin,cos 0202
,cos01 ikn ikn sin01zk Unknowns?
How to solve for Unknows?
Optoelectronics (16/2) W.-Y. Choi
Lect. 4: Dielectric Interface
Boundary Conditions: Constraints on E,H fields at a boundary. Each Maxwell’s Eq. provides one constraint on E or H.
, D
,0 B
11, 22,
na
,dtBdE
snn EE ,11,22snn DD ,1,2
0,1,2 tt EE
,dtDdJH
0 ,1,2 nn BB 0,11,22 nn HH
sn JHHa )( 12 stt JHH ,1,2
Optoelectronics (16/2) W.-Y. Choi
Lect. 4: Dielectric Interface
i r
t
1n2n
x
y zinE rE
tE
Applying BC on at 0,E x 1 0
1 0
2 0
sinsinsin
z i
rz r
tz t
k n kk n kk n k
zjkxjktt
zjkxjkrr
zjkxjkiin
tztx
rzrx
zx
eeEyE
eeEyE
eeEyE
tzz rz jk zjk z jk zi r tE e E e E e
i r
z rz tzk k k
1 1 2sin sin sini r tn n n
(Snell's Law)1 2siand n n sii tn n
and i r tE E E Continuity of ktan
Optoelectronics (16/2) W.-Y. Choi
Lect. 4: Dielectric Interface
i r
t
1n2n
x
y z
inE rE
tE
x=0
inHrH
tHzjkxjk
tt
zjkxjkrr
zjkxjkiin
tztx
rzrx
zx
eeEyE
eeEyE
eeEyE
inH
Applying BC on H at x=0
1 1 2
cos cos cosi tri i tE EE
1
sin cos rx rzjk x jk zri iEx z e e
1
sin cos x zjk x jk zii iEx z e e
2
sin cos tx tzjk x jk ztt t tEH x z e e
rH
Optoelectronics (16/2) W.-Y. Choi
Lect. 4: Dielectric Interface
i r
t
1n2n
x
y z
inE rE
tE
x=0
inHrH
tH
Two simultaneous equations:
r
i
EE
1 1 2
cos cos cos
i r t
i tri i t
E E EE EE
12 2 2
12 2 2
cos sin
cos sin
i i
i i
n
n
r
t 212 2 2 1
2cos ( )cos sin
i
i i
nnnn
t
i
EE
solved! all are ,,, trtr EE
Optoelectronics (16/2) W.-Y. Choi
Lect. 4: Dielectric Interface
i r
t
1n2n
x
y z
inErE
tE
x=0
inHrH
tH
Parallel Polarization
iE , , , ???r t r tH H
x zjk x jk zi iH yH e e
rx rzjk x jk zr rH yH e e
tx tzjk x jk zt tH yH e e
rE 1sin cos x zjk x jk zi i ix z H e e
2sin cos tx tzjk x jk zt t t tE x z H e e 1sin cos rx rzjk x jk zr r rx z H e e
1 0
1 0
2 0
sinsinsin
z i
rz r
tz t
k n kk n kk n k
Optoelectronics (16/2) W.-Y. Choi
Lect. 4: Dielectric Interface
i r
t
1n2n
x
y z
inErE
tE
x=0
inHrH
tH
Parallel Polarization x z
rx rz
tx tz
jk x jk zi i
jk x jk zr r
jk x jk zt t
H yH e e
H yH e e
H yH e e
Applying BC on at 0,H x
1 0
1 0
2 0
sinsinsin
z i
rz r
tz t
k n kk n kk n k
and z rz tz i r tk k k H H H
1 2si and (Snell's n sin Law)i r i tn n
1 1 2sin sin sini r tn n n
Optoelectronics (16/2) W.-Y. Choi
Lect. 4: Dielectric Interface
i r
t
1n2n
x
y z
inErE
tE
x=0
inHrH
tH
Parallel Polarization
Applying BC on at 0,E x
1
1
2
sin cos
sin cos
sin cos
x z
rx rz
tx tz
jk x jk zin i i i
jk x jk zr i i r
jk x jk zt t t t
E x z H e e
E x z H e e
E x z H e e
1 1 2cos cos cosi i i r t tH H H
Optoelectronics (16/2) W.-Y. Choi
Lect. 4: Dielectric Interface
x=0
Two simultaneous equations:
12 2 22
|| 12 2 22
2 2|| 1
2 2 221 1
sin cos
sin cos
2 cos ( )sin cos
i ir
ii i
t i
ii i
n nHrH n n
H n nt nH nn n
i r
t
1n2n
x
y z
inErE
tE
x=0
inHrH
tH 1 1 2
cos cos cosi r t
i i i r t t
H H HH H H
Optoelectronics (16/2) W.-Y. Choi
Lect. 4: Dielectric Interface
i r
t
1n2n
inE rE
tE
inHrH
tH
12 2 2
12 2 2
cos sin
cos sin
i i
i i
n
n
r
t 12 2 2
2cos
cos sini
i in
i r
t
1n2n
inErE
tE
inHrH
tH
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
Perpendicular Parallel
||
||
For 0,11
21
i
nr rn
t tn
Optoelectronics (16/2) W.-Y. Choi
Lect. 4: Dielectric Interface
Homework