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.
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