Department of Physics and Astronomy
The University of Sheffield
1
4
x
4
Transfer Matrix and Reflectivity Calculations
Study the effect of using a thick substrate
(incoherent back reflections)
2
The aims of this work are
To derive expression of
4
×
4
Transfer matrix at a normal
incidence of light for a model of circularly birefringent
materials.
To calculate the reflectivity spectra in the case of
circularly polarised light for these structures.
To calculate the reflectance magneto

circular dichroism
(RMCD) , the Kerr and Faraday rotations.
To study the effect of using a thick substrate
(incoherent back reflections)
.
3
In recognizing real experimental magneto

optical data.
Magneto photonic structures play a key role in controlling
the optical properties and in enhancing the magneto optical
effect
(
Lourtioz
et al.
,
2008
).
4
Magneto optical studies have importance in understanding
the electronic structure of magnetic media
(
Reim
and
Schoenes
,
1990
).
In forming novel structures that utilise the optical property
sensitivity of photonic crystal to small variations in the
refractive index of the material from which it is fabricated.
5
Electromagnetic wave propagation inside multilayer structures
obeys Maxwell's equations.
in source free
J
=
0
and =
0
It is composed of periodic layers which have varied refractive
index or dielectric constant in one

dimension (
1
D).
The layer thickness is a quarter

wavelength
(
Joannopoulos
et al.
,
2008
)
6
and Kerr rotation as
Sato (
1981
) defined
the reflectance magneto

circular dichroism (RMCD) as
http://www.enzim.hu/~szia
/cddemo/edemo
16
.htm
7
The T

matrix matrix links E and B fields in different layers of
the structure
(Whittaker and
Culshaw
,
1999
), (Hecht,
2002
)
For a number of layers (multilayer film), the T

matrix is computed
as the product of the matrix for every layer, which means,
(
Whittaker and Culshaw,
1999
)
Hecht (
2002
)
8
The constitutive relation at a normal incidence for lossless
media that display a circular birefringence in an applied
magnetic field is given in matrix form by
(
Orfanidis
,
2008
).
9
The superscripts indicate to two values of
q
.
The eigenvector components are
circularly polarised state:
Starting from Maxwell's equations, the magnitude of wave vectors
are calculated
at normal incidence
In addition,
the expression of
4
x
4
transfer matrix is derived
for
these media
M
where
M
is a
4
x
4
transfer matrix of a single layer, and includes
2
x
2
block
.
matrices , are given by
(
1
)
10
11
For multilayer structures such as quarter wave stack and by applying
the boundary conditions at an interface between couple of layers,
equation
(
1
)
can be written as
M
here the superscripts
1
and
N
refer to the initial and final layers, respectively. The
resultant matrix
M
is
4
×
4
matrix.
This matrix is used to calculate the reflectivity spectra for both
right and left circularly polarised lights using computational
codes, which are written by FORTRAN
program
.
12
The reflectivity spectra for both left, and right circularly polarised light at
normal incidence
13
was taken from
(Dong et. al.,
2010
)
The reflectivity spectrum,
14
was taken from
(Dong et. al.,
2010
)
The RMCD against the wavelength
15
The Kerr and Faraday Rotations against the wavelength
16
17
the structure was taken
from (Dong et. al.,
2010
)
Reflectivity Spectrum for cavity structure
18
19
The RMCD against the wavelength
20
The Kerr and Faraday Rotations against the wavelength
At
629
nm
, the maximum is
4.73
compared with
0.0192
for film,
in Kerr rotation
Simulated Spectra for
Simulated Spectra
Dong et al. (
2010
)
Simulated Spectra
(this work)
21
, here we set n
s
=
1.0
Question has been raised
about the effect of use a
thick substrate
22
Those studies considered the coherent and incoherent
multiple reflections and transmissions for
isotropic
structures
to deal with this situations
23
As Previous studies pointed out that the spectra with a fine
Fabry

Perot fringes result, when one layer has a thicker
thickness than others. The resulted spectra are not realistic .
e.g.
(Harbecke,
1986
) ;(
Whittaker and
Gehring
2010
)
24
The total R for fully polarisation are given by
Whittaker and
Gehring
(
2010
)
front
back
(
Whittaker and
Gehring
,
2010
)
25
26
The reflectivity spectra for left circularly polarised light at normal incidence
27
The RMCD against the wavelength
28
3
.
multiple
incoherent back
reflections
2
.
Single
incoherent back
reflections
1
.
without incoherent
back reflections
a thick substrate
The equations of total and are calculated individually as
for x

polarised state
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In a similar way, for y

polarised state
where
30
and are the matrices of linear x and y polarisations, respectively
(
Pedrotti
and
Pedrott
,
1993
)
The Kerr rotation is found as following
31
32
The Kerr Rotation against the wavelength
At
629
nm
, the maximum is
4.73
without
incoherent back reflections compared with
1.368
with incoherent back reflections
33
The Faraday Rotation against the wavelength
A multilayer structure of photonic crystal was modelled for anisotropic
materials that display a circular birefringence
Maxwell's equations were used to derive expression of
4
x
4
T

matrix
for these media
In circularly birefringent media, the reflectivity spectra and magneto

optical effect (RMCD, Kerr and Faraday rotations) were calculated.
There was a significant contribution of incoherent back reflections
….
from substrate . A thick substrate should be studied in real system.
34
35
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