22-theory of multilayer filmsoptics.hanyang.ac.kr/~shsong/22-theory of multilayer... ·...
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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
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
Transfer Matrix for a Single Film on a Substrate: TE Mode
Transfer Matrix for a Single Film on a Substrate: TE Mode
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
Transfer Matrix for a Single Film on a Substrate: TE Mode
Transfer Matrix for a Single Film on a Substrate: TE Mode
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
Transfer Matrix for a Single Film on a Substrate: TE Mode
Transfer Matrix for a Single Film on a Substrate: TE Mode
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
Transfer Matrix for a Single Film on a Substrate: TE Mode
Transfer Matrix for a Single Film on a Substrate: TE Mode
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
Transfer Matrix for a Single Film on a Substrate: TE Mode
Transfer Matrix for a Single Film on a Substrate: TE Mode
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
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
Reflectance at Normal Incidence: Two-Layer Quarter-Wave Films
Reflectance at Normal Incidence: Two-Layer Quarter-Wave Films
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
CVI High-Reflectance CoatingsCVI High-Reflectance Coatings