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