EE 430.423.001 2016. 2nd Semester
2016. 10. 13.
Changhee Lee
School of Electrical and Computer Engineering Seoul National Univ.
[email protected]
Chapter 4. Multiple-beam interference
Part 2
EE 430.423.001 2016. 2nd Semester
Optical surfaces having virtually any desired reflectance and transmittance characteristics may be produced by means of thin film coatings.
Examples: camera lens, high-reflecting mirrors, high-transmitting mirrors, one-way mirrors, optical filters, etc.
4.4 Theory of multilayer films
EE 430.423.001 2016. 2nd Semester
Boundary conditions
0 1 1
0 1
1 0
0 0
0
0 1
0 0 0 0
0 1
0 0
2 2
layer, dielectric
the of reflection of
index
1 cos
sin
sin ' cos
1 1
) cos sin
' (
) sin ' (cos
1
n
k n
E E kl n
kl in
n kl i n kl
E E n n
E kl E n
kl E in
n E n
E kl E n
i n E kl
E
T T
T
T T
T T
4.4 Theory of multilayer films
EE 430.423.001 2016. 2nd Semester
matrix er
transf cos
sin
sin cos
t coefficien ission
transm
t coefficien reflection
'
1 1
1
1
1 0
0 0
0 0
kl kl
in
n kl i n M kl
E t E
E r E
n t M n r
n
T T
T
4.4 Theory of multilayer films
matrix transfer
overall
film multilayer
for the 1
1 1
1
3 2 1
3 2 1 0
0
D C
B M A
M M
M M
n t M n t
M M
M M n r
n
N
T T
N
EE 430.423.001 2016. 2nd Semester
4.4 Theory of multilayer films
nce transmitta
e reflectanc
t coefficien ission
transm 2
t coefficien reflection
2 2 0
0
0 0 0
0 0
t T
r R Dn
C n
Bn An
t n
Dn C
n Bn An
Dn C
n Bn r An
T T
T T
T T
Antireflecting films
Suppose a single film of index n1 and thickness l is placed on a glass substrate of index nT.
T T
T
T T
T T
n n
R
kl n l
n
n r n
R
kl n
n i kl n
n
kl n
n i kl n
r n
1
2 2 1
2 2 2 1
2 1 1
2 1 1
if 0
2 4 , if
) (
) (
e reflectanc
sin ) (
cos ) 1
(
sin ) (
cos ) 1
t ( coefficien
reflection
EE 430.423.001 2016. 2nd Semester
Antireflecting films
MgF2, index nMgF2 =1.35, glass index nT~1.5
R can be reduced to zero by using multilayer films of low/high/…low/high index with the thickness /4.
EE 430.423.001 2016. 2nd Semester
Antireflecting with L/H-index films
0 2
0 2
2
2 0
2
2 0
2 2
2 0
2
2 0
2
0 0
0
if 0
) (
) (
) (
) (
n n n
n n n n
n R
n n n
n
n n n
r n R
n n n
n
n n n
n n n
n n n
n
n n n n
n n r
T L
H T
L H
T L H
T L H
T L H
T L H
T H
L L
H
T H
L L
H
H L L
H
H
H L
L
n n n
n
in
n i in
n i M
0
0 0
0 0
0
e reflectanc
t coefficien reflection
2 0
0
0 0
r R Dn C
n Bn An
Dn C
n Bn r An
T T
T T
EE 430.423.001 2016. 2nd Semester
Antireflecting with L/H/L-index films
T H
L L T
H
T H
T H
T H
T H
H T
H
H T
H
n n n
n n n
n n
n n R
n n n
n n
n n n
n r n
R
n n n
n n
n n n
n n n
n n in
n n n
in
n n n in
n n n
in r
0 2
2 1 2 2 1 0
2
2
2 2 2 1 0
2
2 2 2 1 0
2 2
2 2 2 1 0
2
2 2 2 1 0
2
2 1 0
2 1
2 1 0
2 1
0
if 0
) (
) (
) (
) (
0 0
0
0 0
0 0
0 0
0 0
0
2 1
2 1
2 2
1
1 2
2 1
1
H
H
H H
H
H
n n in
n n
in
n n n
n
in
n i in
n i in
n i in
n i M
EE 430.423.001 2016. 2nd Semester
High-reflectance films
A high-reflectance film is made by reversing the order of deposition of the low - and high refractive index layers compared to an anti-reflecting film .
R can be high by using multilayer films of high/low/…/high/low index.
N H
L N
L H N
H L L
H
H L L
H
H
H L
L
n n n
n
n n n
n M
n n n
n
in
n i in
n i
) (
0
0 )
( 0
0
0
0 0
0 0
0
EE 430.423.001 2016. 2nd Semester
High-reflectance films
2
2 2 2
2
1 )
(
1 )
( )
( )
(
) (
) (
N L H
N L H
N H N L
L H
N H N L
L H
n n
n n
n n n
n
n n n
n r
R R 1 for large N
The reflectance approaches unity for large N.
Example:
• An 8-layer stack (N=4) of ZnS (nH=2.3) and MgF2 (nL=1.35)
gives a reflectance of ~0.97, which is higher than the reflectance of pure Ag in the visible region of the spectrum.
• A 30-layer stack results in a reflectance of better than 0.999.
EE 430.423.001 2016. 2nd Semester
Fabry-Perot Interference Filter
A Fabry-Perot type of filter (Fabry-Perot etalon with a very spacing) consists of a layer of dielectric having a thickness of /2 (for some wavelength 0) and bounded on both sides by partially reflecting surfaces.
http://www.dfisica.ubi.pt/~hgil/fotometria/HandBook/ch03.html
EE 430.423.001 2016. 2nd Semester
Fabry-Perot Interference Filter
A Fabry-Perot type of filter (Fabry-Perot etalon with a very spacing) consists of a layer of dielectric having a thickness of /2 (for some wavelength 0) and bounded on both sides by partially reflecting surfaces.
EE 430.423.001 2016. 2nd Semester
