phys 408 applied optics (lecture 10) jan-april 2016 edition jeff young ampel rm 113
DESCRIPTION
Bottom line Consider M net as M, then S net (A,B,C,D) yields t 02,t 20,r 02,r 20 Or don’t be lazy, and just solve for and fromTRANSCRIPT
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PHYS 408Applied Optics (Lecture 10)JAN-APRIL 2016 EDITIONJEFF YOUNGAMPEL RM 113
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Quick review of key points from last lectureS and M matricies are associated with the transfer of fields across each interface, and their propagation through uniform films.
The matrix elements of S for going across interfaces are obtained from the Fresnel reflection and transmission coefficients.
The matrix elements of S for propagating through a uniform film include diagonal phase accumulation terms only.
The S matricies are straight forward to figure out, and the associated M matricies come from transforming the S matrices using linear algebra.
The net r and t for a stack of thin films is obtained by multiplying all M matricies sequentially to obtain Mnet for the entire structure, and then using linear algebra to either solve for r and t, or using the transformation properties from Mnet to Snet.
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Bottom line
Consider Mnet as M, then Snet(A,B,C,D) yields t02,t20,r02,r20
Or don’t be lazy, and just solve for and from
0
2
UU
0
0
UU
02
0
0 UUU
M net
![Page 4: PHYS 408 Applied Optics (Lecture 10) JAN-APRIL 2016 EDITION JEFF YOUNG AMPEL RM 113](https://reader036.vdocuments.mx/reader036/viewer/2022062600/5a4d1b5a7f8b9ab0599aac51/html5/thumbnails/4.jpg)
Can anyone think of another way to circumvent transforming from Mnet to Snet?
02
0
0 UUU
M net
What prevents you from, once you find Mnet, putting in values forand just multiplying it by Mnet to get the transmission?
0
0
UU
Would this help?
021
0
0 UM
UU
net
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Going “inside” the structureThere is a very significant advantage to this approach.
02
0
0 UUU
M net
02
0
0011112
UUU
MMMor for instance, in our anti-refection example:
021
0
00111 12
UM
UU
MM
What does the right hand side of the following equation give you?
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Internal Field Distributions (d2 infinite)
)()(
0)(
)0()0(
11
11121
0
00111 12 dzU
dzUdzUM
zUzU
MM
n1
d1
n2
d2
)(0U
)(0U
)(1U
)(1U )(
2U
z0
?)(?)(
0)(
)0()0(
?
?121111
0
001 12 zU
zUdzUMM
zUzU
M
?)(?)(
0?)(
?
?21111
101 12 zU
zUzUMMM
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And what can you do with all the intermediate values?
0)(
)()( 121
11
1112
dzUM
dzUdzU
)(11
)(111#
1111 exp)(exp)(|)( dzikdziklayerinside dzUdzUzE
You should verify this agrees with the previous result:
)()(
0)(
)0()0(
11
11111
121111
0
001 12 dzU
dzUM
dzUMM
zUzU
M
11
11
~
~
11 00din
din
ee
M
Recall
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Generalize
n1
d1
n1
d1
n2
d2
n1
d1
n2
d2
n3
d3
n1
d1
n2
d2
… …
nlayers-1 nlayers-1
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Uniform periodic multilayer stack
n1
d1
n2
d2
n1
d1
n2
d2
n1
d1
n2
d2
n1
d1
n2
d2
……
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Bragg reflection n1=1.3; n2=1.4; n3=n2
d1=400 nm; d2=200 nm; d2=d3
21 periods
0 2000 4000 6000 8000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8X: 6284Y: 0.7877
Wavenumber 1/
Ref
lect
ivity
0 200 400 600 800 1000 12000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
X: 297.8Y: 1.147e-006
Wavenumber 1/
Ref
lect
ivity
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Bragg reflection n1=1.3; n2=1.4; n3=n2
d1=580 nm; d2=20 nm; d2=d3
21 periods
0 2000 4000 6000 8000 100000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
X: 6522Y: 0.1349
Wavenumber 1/
Ref
lect
ivity
0 200 400 600 800 10000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
X: 304.7Y: 7.109e-007
Wavenumber 1/R
efle
ctiv
ity
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Bragg reflection n1=1.3; n2=1.4; n3=n2
d1=580 nm; d2=20 nm; d2=d3
200 periods
0 2000 4000 6000 8000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavenumber 1/
Ref
lect
ivity
6250 6300 6350 6400 6450 6500 65500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavenumber 1/
Ref
lect
ivity
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Bragg reflection n1=2; n2=sqrt(12); n3=n2
d1=300 nm; d2=173 nm; d2=d3
10 periods
0 2000 4000 6000 8000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavenumber 1/
Ref
lect
ivity
0 2000 4000 6000 8000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavenumber 1/
Tran
smis
sion
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Add a “defect”
n1
d1
n1
d1
n2
d2
n1
d1
n2
d2
n3
d3
n1
d1
n2
d2
……
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What is this? n1=2; n2=sqrt(12); n3=4d1=300 nm; d2=173 nm; d3=0.15*d2
10 periods
0 2000 4000 6000 8000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavenumber 1/
Tran
smis
sion
-5 -4 -3 -2 -1 0 1
x 10-4
0
10
20
30
40
50
60
70
Z (cm)
|E|2
-5 -4 -3 -2 -1 0 1
x 10-4
0
0.5
1
1.5
2
2.5
3
3.5
4
X: -0.0004499Y: 1
Z (cm)
|E|2
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And this? n1=2; n2=sqrt(12); n3=4d1=300 nm; d2=173 nm; d3=2*d2
10 periods
0 2000 4000 6000 8000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavenumber 1/
Tran
smis
sion
-5 -4 -3 -2 -1 0 1
x 10-4
0
10
20
30
40
50
60
70
Z (cm)
|E|2
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Cavity Modes! n1=2; n2=sqrt(12); n3=4d1=300 nm; d2=173 nm; d3=10*d2
10 periods
0 2000 4000 6000 8000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavenumber 1/
Ref
lect
ivity
-7 -6 -5 -4 -3 -2 -1 0 1
x 10-4
0
5
10
15
20
25
30
35
40
45
50
Z (cm)
|E|2
-7 -6 -5 -4 -3 -2 -1 0 1
x 10-4
0
5
10
15
20
25
30
Z (cm)
|E|2
-7 -6 -5 -4 -3 -2 -1 0 1
x 10-4
0
10
20
30
40
50
60
70
Z (cm)
|E|2
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Cavity Modes! n1=2; n2=sqrt(12); n3=4d1=300 nm; d2=173 nm; d3=10*d2
10 periods SYMMETERIZED
3500 4000 4500 5000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavenumber 1/
Tran
smis
sion
-7 -6 -5 -4 -3 -2 -1 0 1
x 10-4
0
10
20
30
40
50
60
70
80
90
100
Z (cm)
|E|2
-7 -6 -5 -4 -3 -2 -1 0 1
x 10-4
0
20
40
60
80
100
120
140
160
180
200
Z (cm)
|E|2
-7 -6 -5 -4 -3 -2 -1 0 1
x 10-4
0
5
10
15
20
25
30
35
Z (cm)
|E|2