lecture about tmm
TRANSCRIPT
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Layered structures:transfer matrix formalism
Petr Kuel Interfaces between LHI media
Transfer matrix formalism Practically only one formula is to be known in order to calculate any
structure
Applications:
Antireflective coatings
Dielectric mirrors, Chirped mirrors
Laser output couplers
Beam-splitters Beam-splitting mirrors
Interference filters
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Transfer matrix formalism
( )
( )=+=
=+=
11120
1112
11010
1101
ri
ri
st
st
EEH
EEE
EEH
EEE
=
=
irs
iti
eEE
eEE
11
11
==
=
11111
11
cos2
cos
dnkd
n
z
n0
n1
k
0 0 0
1
E
E
E
H
H
H
Ei0 Er0
Et1 Es1
Ei1 Er1
n2
x
y
z
E01
E12
k+0
k1 k
+1
(tangential)
(tangential)
TE polarization
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Introduction of transfer matrix
=
12012
010
01
cossin
sincos
H
E
i
i
H
E
=
=
jjj
j
j
j
i
i
mm
mmM
cossin
sincos
2221
1211
Transfer matrix connects tangential fields on both ends of a layer
Forj-th layer:
For the whole structure:
=
=
+
+
+
+
1,0
1,
1,0
1,
21010
01
NN
NN
totNN
NN
N
H
EM
H
EMMM
H
E!
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Reflection and transmission
coefficients
=
=
+=
+
+1,0
1,
000
00
010
01
)( NN
NNtot
tt
ttot
ri
ri
HEM
EEM
EEEE
HE
2221120110
0
2221120110
2221120110
2
mmmm
t
mmmm
mmmmr
tt
tt
tt
+++
=
++++
=
polarisation TE: jjj n = cos cdn jjjj /cos=
polarisation TM: jjj n = cos cdn jjjj /cos=
normal incidence: jj n= cdn jjj /=
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GeneralizationThe formalism is also valid for
absorbing layers;j-th layer absorbs:Nj=njij
j
j
jN
nN 02202 sincos =
layers where total reflection occurs; total reflection onj-th layer:
j
j
jn
nni
20
20 sin
cos
=
jand
jbecome imaginary; one introduces:
j= i
jand
j= i
j,
where jand jare real. The transfer matrix becomes:
=jjj
j
j
j
ji
iM
chsh
shch
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Antireflective single layerLets try /4-layer (then waves with /2 phase delay will interfere)
MgF2
BK7 glass
/4
ns=1.51
n1=1.38n0=1
=0
0
1
1
iiM
21
2
1210
2
10
nnnnr
s
s
s
s
+=+ =
snn =1
400 500 600 700 800
Wavelength (nm)
6
4
2
0
R(%) Antireflective coating
(for 550 nm)
/4
ns= 1.51
glass (not treated)
Broadband
Less efficient (no degreeof freedom for n1)
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Antireflective bilayer quarter-wave(/4-/4) bilayer
=
21
12
00M
21
22
21
22
210
22
210
22
nnn
nnn
rs
s
s
s
+
=+
=
snn
n =1
2
400 500 600 700 800
Wavelength (nm)
6
4
2
0
R(%)
Antireflective coating
(for 550 nm)
/4/4-/4
ns= 1.51
glass (not treated)
n1= 1.65
ZrO2
BK7 glass
CeF3/4
/4
ns=1.51
n2= 2.1
V-like shape (narrow
frequency range)
More efficient (one degree
of freedom for n1, n2)
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Broadband AR coating trilayer structure(/4-/4-/4)
=0
0
2
31
31
2
i
iM
snn
nn
=2
31
23
21
22
23
21
22
23
210
22
23
210
22
nnnn
nnnnr
s
s
s
s
+
=+
=
n3=1.8
BK7 glass
n2=2.0
/4
/4
ns=1.51
n1=1.38
/4
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Broadband AR coating trilayer structure(/4-/2-/4): similar to a quarter-wave bilayer at
the resonant wavelength
half-wave layer helps to extend the antireflective range
snn
n=
1
3
= 3113
0
0M
2
1
2
3
23
21
2
10
2
3
210
23
nnn
nnnr
s
s
s
s
+
=
+
+=
n3=1.7
BK7 glass
n2=2.2
/4
/4
ns=1.51
n1=1.38
/2
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Antireflective coating: summary
400 500 600 700 800Wavelength (nm)
10
8
6
4
2
0
R(%)
Antireflective coating
for 550 nm
/4/4-/4/4-/2/4-/4-/4
/4-/2-/4
ns= 1.51
glass (not treated)
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Dielectric mirrors
nH
nL
/4
ns
nH
/4
nL
nH
nL
1 bilayer (nL
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Dielectric mirrors: example
400 600 800 1000 1200 1400 1600
Wavelength (nm)
1
0.8
0.6
0.4
0.2
0
R
0.9104 1.1104 1.3104 1.5104
Wavenumber (cm-1)
1
0.6
0.2
-0.2
-0.6
-1
Phasechang
e
Dielectric mirrors for 800 nm
Number of bilayers (ZnS/MgF2):
20
8
3
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Chirped dielectric mirrorsThe resonant wavelength is linearly tuned along the stack of bilayers
Different wavelengths are reflected at different depths differentoptical paths
Adding or compensating of a chirp of the pulses
1.1104 1.2104 1.3104 1.4104
Wavenumber (cm-1)
0.3
0.2
0.1
0
-0.1
-0.2D
eparturefrom
the
linearphaseshift Mirrors for 800 nm (12500 cm
-1)
(20 bilayers)
Standard dielectric mirror
Chirped mirror (10 nm step)
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Laser output coupler
R01 Rx< 1
=514.5 nm
Rx
AR coating
Reflective (~ 90%) coating
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Beamsplitter
80%
20%
45 ns
nR, dR
n1, d1
n2
, d2
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Interference band-pass filtersContain:
Stacks of high-reflecting bilayers
Antireflective coatings
Fabry-Prot cavities
Detuning of the resonant wavelength is also often used for
smoothing of the interferences