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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

http://lecture5.pdf/http://lecture5.pdf/

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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