22.
22. Theory
Theory of
of Multilayer
Multilayer Films
Films
r
Transfer Matrix
Reflectance at Normal Incidence
Anti-reflecting Films
High-Reflectance Films
 Ea 
 B   M1 M 2 M 3 ...MN
 a
 EN 
 EN 
M

T 
B 

B
 N
 N
 m11
m
 21
m12 
m22 
 EN 
B 
 N
t
Transfer
Transfer Matrix
Matrix for
for aa Single
Single Film
Film on
on aa Substrate:
Substrate:
TE
TE Mode
Mode


t is  order
Consider two waves E1 and E2 that have the same frequency  :
 
 

E1  r , t   E01 cos k1  r   t  1
n0
n1
n2
 
 

E2  r , t   E02 cos k2  r   t   2




Er1
Et 2
Ei1

E 0  Ez zˆ (TE-mode)

B 0  Bx 
x  By 
y (TE-mode)
E0
Er 2
Ei 2
Et1
E rm1
n0
n2=ns
n1
Etm2
2

Er1   Erm1
E r1
Eim1
m0
m
1
E r1
Et1
E
Ei21
E
Et21
0
Er 1
Erm2
Ei11
m
i2
2
r2
2
t2
E

E0

Et 2   Etm2
m1

Et1   Etm1
m 1
1
t2
E
2
i2
E

Ei 2   E
1
r2
E
m 1
m
i2
1
i2
E
1
Et1
E0

Er 2   Erm2
m 1
B0

E 0  Ez zˆ (TE-mode)

B 0  Bx 
x  By 
y (TE-mode)

Er1   Erm1
m0

Ei1   Eim1
m 1

Et 2   Etm2
m 1
Transfer
Transfer Matrix
Matrix for
for aa Single
Single Film
Film on
on aa Substrate:
Substrate:
TE
TE Mode
Mode
Tangential components of E and B-fields are continuous across the interface.
On (a) : Ea  E0  Er1  Et1  Ei1
Ba  B0 cos  0  Br1 cos  0  Bt1 cos t1  Bi1 cos t1
On (b) : Eb  Ei 2  Er 2  Et 2
Bb  Bi 2 cos t1  Br 2 cos t1  Bt 2 cos t 2
Er1
Et 2
E0
(a)
(b)
Transfer
Transfer Matrix
Matrix for
for aa Single
Single Film
Film on
on aa Substrate:
Substrate:
TE
TE Mode
Mode
Note : B 
E n
   E  n  0 0 E
v c
Define :
 0  n0  0 0 cos  0
 1  n1  0 0 cos t1
 s  n2  0 0 cos t 2
(a ) Ba  B0 cos  0  Br1 cos  0  Bt1 cos t1  Bi1 cos t1
(b) Bb  Bi 2 cos t1  Br 2 cos t1  Bt 2 cos t 2
Ba   0  E0  Er1    1  Et1  Ei1 
Bb   1  Ei 2  Er 2    s Et 2
Transfer
Transfer Matrix
Matrix for
for aa Single
Single Film
Film on
on aa Substrate:
Substrate:
TE
TE Mode
Mode
From the geometry of the problem we can see that :
Ei 2  Et1 exp  i 
and Ei1  Er 2 exp  i 
 2 
where   k0 1  
 n1 t cos t1

 0 
 k0 n1 t cos t1  k1t cos t1
(7-35)
Using these expressions we find :
Eb  Ei 2  Er 2  Et1 exp  i   Ei1 exp  i   Et 2
Bb   1  Ei 2  Er 2    1  Et1 exp  i   Ei1 exp  i     s Et 2
t
Transfer
Transfer Matrix
Matrix for
for aa Single
Single Film
Film on
on aa Substrate:
Substrate:
TE
TE Mode
Mode
We can solve the last two equations for Et1 and Ei1 in terms of Eb and Bb :
  E  Bb 
Et1   1 b
 exp  i 
 21 
  E  Bb 
Ei1   1 b
 exp  i 
 21 
At the first boundary we then obtain :
 exp  i   exp  i  
 exp  i   exp  i  
Ea  Et1  Ei1  Eb 

  Bb 
2
21




 i sin  
 Eb cos   Bb 

 1 
 exp  i   exp  i  
 exp  i   exp  i  


Ba   1  Et1  Ei1    1 Eb 
B


1 b 
2
21




 Eb   1 i sin    Bb cos 
Transfer
Transfer Matrix
Matrix for
for aa Single
Single Film
Film on
on aa Substrate:
Substrate:
TE
TE Mode
Mode
 i sin  
Ea   Eb cos   Bb 

 1 
Ba  Eb   1 i sin    Bb cos 
This can be written in matrix form :

 Ea   cos 
B   
 a  i sin 
 1
 0  n0  0 0 cos  0
i sin  
E 
1   b 
 B
cos    b 
 1  n1  0 0 cos t1
 s  n2  0 0 cos t 2
m12 
m22 
 m11

 m21

m12   cos 

m22  
 1 i sin 
i sin  
1 

cos  
M1
 E1 , B1 
 EN 
 EN 

M
N
B 
B  
 N
 N
 E2 , B2 
MN
 EN , BN 
Generalizing to a system with N layers :
 Ea 
 B   M1 M 2 M 3 ...M N
 a
 Ea , Ba 
M2
The 2 x 2 transfer matrix is given by :
m
M   11
 m21
r
 m11
m
 21
m12 
m22 
 EN 
B 
 N
t
for a multilayer
In terms of E0 , Er1 , and Et2 :
Ea  E0  Er1
Ba   0  E0  Er1 
Eb  Et 2
Bb   s Et 2
In matrix form we obtain :

 E0  Er1   cos 
 E  E   
r1  
 0 0
 1 i sin 
i sin  
 E  m
 1   t 2    11
  E
m
cos    s t 2   21
m12   Et 2 
m22   s Et 2 
The amplitude reflection and transmission coefficients are given by :
r
Er 1
E0
t
Et 2
E0
In terms of r and t we obtain :
1  r  m11 t  m12  s t
 0 1  r   m12 t  m22  s t
Reflection
Reflectioncoefficient
coefficient(r)
(r)and
andtransmission
transmissioncoefficient
coefficient(t)
(t)
Solving for r and t we have :
t
2 0
 0 m11   0  s m12  m21   s m22
r
 0 m11   0  s m12  m21   s m22
 0 m11   0  s m12  m21   s m22
These equations allow us to evaluate the properties of multilayer films.
 Ea 
 B   M1 M 2 M 3 ...MN
 a
 EN 
 EN 
 B   MT  B  
 N
 N
 m11
m
 21
m12 
m22 
 EN 
B 
 N
19-2.
19-2. Reflectance
Reflectance at
at Normal
Normal Incidence:
Incidence:
Single-Layer
Single-Layer Film
Film
At normal incidence the matrix elements become :
m11  cos 
m12 
m21  i n1  0 0 sin 
 m11
m
 21
i sin 
n1  0 0
i sin  
1 

cos  
 0  n0  0 0 cos  0
m22  cos 
 1  n1  0 0 cos t1
At normal incidence the reflectance coefficient becomes :
r

m12   cos 

m22  
 1 i sin 
n1  n0  ns  cos   i  n0 ns  n12  sin 
 s  n2  0 0 cos t 2
 2 
 2

n
t
cos

 1

t1
 0 
 0

n1  n0  ns  cos   i  n0 ns  n12  sin 

 n1 t

  k0 n1 t  k0 1
The power reflectance coefficient is given by :
R = r r* 
n  n0  ns  cos    n0 ns  n
2
1
2
2
n12  n0  ns  cos 2
2
1  n1 t : optical path difference
 sin 
   n n  n  sin 
0 s
2 2
1
2
2 2
1
2
(Reflectance)
(Reflectance at normal incidence)
t
0
n0  1 (air)
4n1
t
n1  n
0
2n1
n2  ns  1.52 (glass)
n  ns  1.52
4.3%
 1 / 0 
Reflectance
Reflectance at
at Normal
Normal Incidence:
Incidence:
Single-Layer
Single-Layer Quarter-Wave
Quarter-Wave Film
Film
For the quarter  wave thickness t 
  k0 n1 t 
2  n1  0  


0  4 n1  2

4

0
4 n1
:
 cos =0, sin =1
m11  cos   0
m12 
m21  i n1  0 0 sin   i n1  0 0
i sin 
n1  0 0

i
n1  0 0
m22  cos   0

0
 
 i
 1
i 
1 

0 
At normal incidence the reflectance coefficient becomes :
r
i  n0 ns  n12 
i  n0 ns  n12 
n n  n 
n n  n 

0 s
2
1
0 s
2
1
The power reflectance coefficient is given by :
R = r r* 
 n0 ns  n12 
 n0 ns  n

2
2 2
1
t

4

0
4n1
If n1  n0 ns
R0
A single /4 film can produce perfect antireflection!
Reflectance
Reflectance at
at Normal
Normal Incidence:
Incidence:
Two-Layer
Two-Layer Quarter-Wave
Quarter-Wave Films
Films

0
The transfer matrix for a quarter  wave thickness film is given by : M1  

i  1
i 
1 

0 
For two quarter  wave films :

0
M = M1 M2  

i  1
i 
0
1  

0  i  2
 
i   2

2    1
 
0   0


0 

1 
 
2 
The reflectance coefficient becomes :
r
 0 m11   0  s m12  m21   s m22
 0 m11   0  s m12  m21   s m22
 2 
 1 
  s  
2
2
2
2

  2    2  0   s  1  n2 n0  ns n1
  1
n22 n0  ns n12
 22  0   s  12
  
  
0  2   s  1 
 1 
 2 
0 
The power reflectance coefficient is given by :
 n22 n0  ns n12 
R = rr   2
2 
 n2 n0  ns n1 
2
*
Zero reflectance occurs when n22 n0  ns n12 
n2

n1
ns
n0
Reflectance
Reflectance at
at Normal
Normal Incidence:
Incidence:
Two-Layer
Two-Layer Quarter-Wave
Quarter-Wave Films
Films
ns
n2

n1
n0
Although the reflectance is greater, it remains less values over the broad range
by introducing a /2 layer!
Reflectance
Reflectance at
at Normal
Normal Incidence:
Incidence:
Two-Layer
Two-Layer Films
Films
19-4.
19-4. Reflectance
Reflectance at
at Normal
Normal Incidence:
Incidence:
Three-Layer
Three-Layer Antireflecting
Antireflecting Films
Films
Reflectance
Reflectance at
at Normal
Normal Incidence:
Incidence:
Three-Layer
Three-Layer Antireflecting
Antireflecting Films
Films
For the quarter - wave film,
zero reflectance occurs when :
n1 n3
 n0 ns
n2
CVI
CVI Antireflecting
Antireflecting Films
Films
CVI
CVI Broadband
Broadband Antireflecting
Antireflecting Films
Films
19-5.
19-5. High-Reflectance
High-Reflectance Layers
Layers
A high - reflectance film rather than an anti - reflecting film is made by reversing the order of
deposition of the low - and high refractive index layers.

0
The transfer matrix for a quarter  wave thickness film is given by : M  

i 
For two quarter  wave films, with the high  refractive index layer on top :
MHL

0
= MH ML  

i  H
i 
0
H 

0  i  L
 
i   L

L    H
 
0   0

The power reflectance coefficient for
the two  layer film at normal incidence
is given by :
 nL2 n0  ns nH2 
*
R = rr   2
2 
 nL n0  ns nH 
2

0 

 
 H
L 
i


0 
High-Reflectance
High-Reflectance stack
stack of
of N
N Double-Layers
Double-Layers
For a high - reflectance film with N pairs of high - and low refractive index layers :
N
M   MH 1 ML1  MH 2 ML 2  ...  MHN MLN    MH ML   MHL
N
 L
 
M H

 0

N
   N

  L 
0 

   H 
 

 H
0
L 




0

N
 H  

 
 L  
The amplitude and power reflectance coefficients for the multi  layer film at normal incidence
are given by :
N
2N
N
 n0   nL 
 n 
 n 
n0   L   ns   H 
  
  1
n
n
n
n
 L    s  H 
r   H N
N
2N
 n0   nL 
 nL 
 nH 
n0  
  
  ns  

  1
n
n
n
n
 H
 L
 s  H 
  n0 ns  nL nH 2 N  1 
*
R = rr  

2N
  n0 ns  nL nH   1 
2
High-Reflectance
High-Reflectance Layers
Layers
N=6
N=2 +1(high)
N=2
N-pairs of H/L
Reflectance
Reflectance of
of high-low
high-low /4
/4 Layers
Layers
2N

 1
n
n
n
n

0
s  L
H 
*
R = rr  

2N
  n0 ns  nL nH   1 
2
CVI
CVI High-Reflectance
High-Reflectance Coatings
Coatings
CVI
CVI High-Reflectance
High-Reflectance Coatings
Coatings