22-theory of multilayer filmsoptics.hanyang.ac.kr/~shsong/22-theory of multilayer... ·...

22. Theory of Multilayer Films22. Theory of Multilayer Films

Transfer Matrix

Reflectance at Normal Incidence

Anti-reflecting Films

High-Reflectance Films

r

t

11 12

21 22

...a N N N

a N N N

E E E Em mB B B Bm m

1 2 3 N TM M M M M

Transfer Matrix for a Single Film on a Substrate: TE Mode


1 2

1 01 1 1

2 02 2 2

:

, cos

, cos

Consider two waves E and E that have the same frequency

E r t E k r t

E r t E k r t

n0 n2

0E

1rE

n1

2tE

1tE2iE

1iE2rE

0 ˆ (TE-mode)zE E z

0 y (TE-mode)x yB B x B

t is order

0 ˆ (TE-mode)zE E z

0 y (TE-mode)x yB B x B

n0

1 10

mr r

mE E

0E

11r

E

21r

E

mr

E1

11t

E

21t

E

12iE

11iE

22iE

12rE

21iE

22rE

miE 1

mrE 2

mt

E1

miE 2

12tE

22tE

mtE 2

0B

n1n2=ns

1

22m

mtt EE

1

11m

mii EE

1

11m

mtt EE

1

22m

mii EE

1

22m

mrr EE

0E

1

22m

mtt EE

1 10

mr r

m

E E

1

0r

E



Tangential components of E and B-fields are continuous across the interface.

On (a) : 0 1 1 1

0 0 1 0 1 1 1 1

2 2 2

2 1 2 1 2 2

cos cos cos cos

cos cos cos

a r t i

a r t t i t

b i r t

b i t r t t t

E E E E EB B B B B

E E E EB B B B

On (b) :

(a) (b)

0E

2tE

1rE

0 0

0 0 0 0 0

1 1 0 0 1

2 0 0 2

Note :

:

cos

cos

cost

s t

E nB E n Ev c

Define

n

n

n

0 0 1 1 1 1

1 2 2 2

a r t i

b i r s t

B E E E E

B E E E



0 0 1 0 1 1 1 1

2 1 2 1 2 2

( ) cos cos cos cos( ) cos cos cos

a r t t i t

b i t r t t t

a B B B B Bb B B B B

0 1 1 10

0 1

2 2 1

1

2

1

2 1 1

1

exp exp

2 cos

c

exp

os s co

b i r t

i t i

t t

r

t

From the geometry of the problem we can see that :

E E i and E E i

where

Using these ex

E E E E i

press

k n t

k

ions we find :

n t k t

1 2

1 2 2 1 1 1 2

exp

exp exp

i t

b i r t i s t

E i E

B E E E i E i E



t

(7-35)

11

1

11

1

1 1

exp2

exp2

:

exp exp2

b bt

b bi

a t i

t1

b

i1 b bWe can solve the last two equations for :

E BE i

E BE i

we then obtain

i i

At the first bou

E and E in terms of E and

E E E

ndary

B

E

1

1

1 1 1 1 11

1

exp exp2

sincos

exp exp exp exp2 2

sin cos

b

b b

a t i b b

b b

i iB

iE B

i i i iB E E E B

E i B



1

1

11 12

21 22

sincos

sin cos

2 2 :

:

...

a b

a b

a

a

This can be written in matrix form :

iE EB B

i

The x transfer matrix is given by

m mm m

Generalizing to a system with N layers

EB

1 2 3

M

M M M M 11 12

21 22

N N N

N N N

E E Em mB B Bm m

N NM



0 0 0 0 0

1 1 0 0 1

2 0 0 2

cos

cos

cost

s t

n

n

n

for a multilayer

11 121

21 221

sincos

sin cos

im mm m

i

1

1

sincos

sin cos

a b b

a b b

iE E B

B E i B

r

t

,a aE B

,N NE B

1 1,E B

2 2,E B

M1

M2

MN

0 1 2

0 0 1 2

0 1 2 211 121

0 0 1 2 221 221

:

sincos

sin cos

a r b t

a r b s t

r t t

r s t s t

0 r1 t2 :

E E E E EB E E B E

In matrix form we obtain

iE E E Em m

E E E Em mi

The am

In terms of E , E , and E

li

p t

11 12

0 1

21

0

2

0

2 2

:

1

:

1s

s

tr

reflection and transmission coefficients

E

ude are given by

In terms of r and t we obtai

r m t m t

Er tE E

r m t t

n

m

0

0 11 0 12 21 22

0 11 0 12 21 22

0 11 0 12 21 22

2

.

:

s s

s s

s s

tm m m m

m m m mrm

Solving for r and t we have

These equations allow us to evaluate the properties of multilayer

m m

ms

m

fil

Reflection coefficient (r) and transmission coefficient (t)Reflection coefficient (r) and transmission coefficient (t)

11 12

21 22

...a N N N

a N N N

E E E Em mB B B Bm m

1 2 3 N TM M M M M

21 0 0 1

21 0 0 1

11 121 0 0

21 1 0 0 22

cos si

:sincos

sin

n

cos

cos sins s

s s

At normal incidence the matrix elements becomeim m

n

m i n m

At normal incidence the re

n n n i n n n

flectance

rn n

coefficient becomes :

The powe

n i n n n

222 2 2 21 0 0 1

222 2 2 21 0 0 1

cos sin

cos sins s*

s s

r reflectance coefficient is given by :

n n n n n nR = r r

n n n n n n

11 121

21 221

sincos

sin cos

im mm m

i

0 0 0 0 0

1 1 0 0 1

2 0 0 2

cos

cos

cost

s t

n

n

n

19-2. Reflectance at Normal Incidence: Single-Layer Film

19-2. Reflectance at Normal Incidence: Single-Layer Film

1 1 10 0

2 2cos tn t n t

(Reflectance)

0 1 0 1k n t k

1 1n t : optical path difference

1 0/

0

1

2

1 (air)

1.52 (glass)s

nn nn n

1.52sn n 4.3%

(Reflectance at normal incidence)

0

14t

n

0

12t

n

Reflectance at Normal Incidence: Single-Layer Quarter-Wave Film

Reflectance at Normal Incidence: Single-Layer Quarter-Wave Film

010 1

0 1

11 121 0 0 1 0 0 1

121 1 0 0 1 0 0

0

22

1

:

2 cos =0, sin =14 2

sincos 0 0

0sin co

4 4

s 0

For the quarter wave thickness

nk n tn

i i im mn n

im i n i n m

At normal incidence the f

tn

re

2 20 1 0 1

2 20 1 0 1

220 1

220 1

s

s

*

s

s

s

s

lectance coefficient becomes :

The power reflectance coefficient

n n nR = r

i n n n n n nr

i n n n

is give

n n

rn n n

n

n

by :

0RA single /4 film can produce perfect antireflection!

1

0

44 nt

1 0 sIf n n n

Reflectance at Normal Incidence: Two-Layer Quarter-Wave Films


1

1

2

11 2

11 2

2

0:

0:

00 0

00 0

two quarter wa

iThe transfer matrix for a quarter wave thickness film is given by

iFor

i i

i i

The reflectance coefficient

ve fi

be

lms

1

1 2

M

M = M M

2 10 2 2 2 2

1 20 11 0 12 21 22 2 0 1 2 0 12 2 2 2

0 11 0 12 21 22 2 0 1 2 0 12 10

1 2

ss s s s

s s s ss

*

comes :

m m m m n n n nrm m m m n n n n

The power reflectance coefficient is given by :

R = r r

22 22 0 12 22 0 1

2 22 0 1

2

1 0s

s

s

s

n n n nn n n n

Zero reflectance occurs when nnn n

n n n n



01

2

nn

nn s

Although the reflectance is greater, it remains less values over the broad rangeby introducing a /2 layer!

Reflectance at Normal Incidence: Two-Layer Films

Reflectance at Normal Incidence: Two-Layer Films

19-4. Reflectance at Normal Incidence: Three-Layer Antireflecting Films

19-4. Reflectance at Normal Incidence: Three-Layer Antireflecting Films

Reflectance at Normal Incidence: Three-Layer Antireflecting Films

Reflectance at Normal Incidence: Three-Layer Antireflecting Films

1 30

2s

For the quarter - wave film,zero reflectance occurs when :n n n nn

CVI Antireflecting FilmsCVI Antireflecting Films

CVI Broadband Antireflecting FilmsCVI Broadband Antireflecting Films

19-5. High-Reflectance Layers19-5. High-Reflectance Layersby reversing the order of

deposition of the low - and high refractivA high - reflectance film rather than an anti - reflecting film is made

The transfer matrix for a quarter wave t

e

hi

index la

ckness fi

yers

lm

.

is gi0

:0

, :

00 0

00 0

L

HH LHL H L

HH L

L

iven by

iFor two quarter wave films with the high refractive index layer on top

i i

i i

The power reflectance coefficient fothe two l ye

ra

M

M = M M

22 20

2 20

* L s H

L s H

r film at normal incidence

n n n nR =

is giv

r rn n n

en by :

n

High-Reflectance stack of N Double-LayersHigh-Reflectance stack of N Double-Layers

1 1 2 2 ...

00

0 0

N NH L H L HN LN H L HL

NNLL

HHN

HH

LL

For a hig N pairs of high - reflectance film with h - and low re :

The amplitude and p

fractive index layers

o

M M M M M M M M M M

M

2

00

2

00

0

1

1

NN N

LL Hs

s HH LN N N

L H Ls

H L s

L

H

s H*

wer reflectance coefficients for the multi layer film at normal incidenceare given by :

n nn nn nn nn n

rnn n nn n

n n n n

n n n nR = r r

22

20

11

N

Ns L Hn n n n

High-Reflectance LayersHigh-Reflectance Layers

N=2

N=2 +1(high)

N=6

N-pairs of H/L

Reflectance of high-low /4 LayersReflectance of high-low /4 Layers

220

20

11

Ns L H*

Ns L H

n n n nR = r r

n n n n

CVI High-Reflectance CoatingsCVI High-Reflectance Coatings

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