Description of Transfer Matrix Method - Journal of the Royal Society ...

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Nov 7, 2013 (3 years and 10 months ago)

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Transfer Matrix Method

The transfer matrix method is a widely used method for calcu
lations
light
-
matter interaction in a
stack of
parallel
dielectric interf
aces. The method treats the Fresnel reflection and transmission at the interface of
two media as a matrix, and the propagation of the light in a particular medium as another matrix. T
his
allows for the multiplication of a stack of matrices to obtain the transmission and reflection by a stack of
dielectric layers.

Since the mathematical operation is a multiplication of matrices, the method lends itself
easily to numerical analysis using

programming languages such as Matlab
,

Python, or any of a large
number of
other programming languages
.

Below, we describe the principles of the transfer matrix method
as implemented for the measurements described in the manuscript.

Figure
1

shows a schematic of a multi
-
layer dielectric structure with alternating high and low index
layers.
m
A

and
'
m
A

are the amplitudes of the forward and backward propagating wave in medium
m
.
m

=

0 is
the medium where the light is incident, and the subscript
s

represents the substrate.


Figure
1
. A schematic of the forward and backward propagating fields in a multi
-
layer dielectric structure.

For the above setup, the fields are

related in the following manner
:


0
''
0
s
s
A A
M
A A
   

   
   


(
1
)

where

the matrix M describes the behavior of all the intervening layers and

is:




1 1
0
1
Q
m m m s
m
M D D P D D
 

 

 
 



(
2
)

where
Q

is the number of dielectric layers. For the specific implementation,
Q

=

2
N
, where
N

is the
numb
er of pairs of layers of alternating high and low refractive index
media
.

And
,


0
, where
0
m
m
i
m m zm m
i
e
P k d
e




 
 
 
 


(
3
)


cos for TE
1 1
, where
cos
for TM
m m
m m
m
m m
m
n
D
n



 

 

 

 

 




(
4
)

where
zm
k

is the normal projection of the wave
-
vector in medium
m
,
m
d
is the thickness of lay
er
m
,
and
m

is the angle if light travelling in medium
m
.

Note that all the information necessary to
compute matrix M is given by the material parameters of the layers, assuming that the incident
angle
m

is k
nown.

A
ssuming:

'
0
s
A


since there is no backward propagating wave in the medium into which the light exits, and
setting the amplitude of the incident wave
0
1
A


gives the final version of equation
(
1
)

as:


'
0
1
0
s
A
M
A
 
 

 
 
 
 


(
5
)

allowing one to solv
e for
s
A
and
'
0
A

as a simple linear equation once the matrix M is solved for
.

For the purpose of the computations described in the manuscript,

the incident angle is assumed to be
normal, i.e.

0
0



(and consequently all
0
m


)
; thus both the transverse electric (TE) and transverse
magnetic (TM) polarizations are degenerate and give the same result.


Matlab Implementation

The above was implemented in Matlab

as a function.

The assumptions were:

For all layer where m = 2, 4, 6, . . . Q
, t
he refractive index was set to
n
L

=

1.34, and the thickness
was set to
d
L
.

For all layers where m = 1, 3, 5, . . . Q
-
1 the refractive index was set to
n
H

(a relatively high
refractive index
), and the thickness was set to
d
H
.

The residual of the difference between the spectrum resulting from initial values of
N
,
n
H
,
d
L
, and
d
H

and the measured spectrum was minimized
by varying
n
H
,
d
L
, and
d
H

using Matlab’s Least Squares
curve fitting routine
lsqcurvefit

for a range of values of
N

from 2 to 11. Subsequently, the spectrum
corresponding to the
N

with the smallest residual was considered to be the final fit, and the
corresponding fitting parameters recorded. This was done because
N

can have only i
nteger values,
and the
lsqcurvefit

routine cannot handle variables with only integral values.