1. No category

Department of Physics and Astronomy
The University of Sheffield
1
4x4Transfer 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
Magneto optical studies have importance in understanding
the electronic structure of magnetic media (Reim and Schoenes, 1990).
Magneto photonic structures play a key role in controlling
the optical properties and in enhancing the magneto optical
effect (Lourtioz et al., 2008).
In recognizing real experimental magneto-optical data.
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.
4
Electromagnetic wave propagation inside multilayer structures
obeys Maxwell's equations.
in source free J=0 and =0
5
It is composed of periodic layers which have varied refractive
index or dielectric constant in one-dimension (1D).
The layer thickness is a quarter-wavelength
(Joannopoulos et al., 2008)
6
http://www.enzim.hu/~szia
/cddemo/edemo16.htm
Sato (1981) defined
the reflectance magneto-circular dichroism (RMCD) as
and Kerr rotation as
7
(Whittaker and Culshaw, 1999)
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,
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
Starting from Maxwell's equations, the magnitude of wave vectors
are calculated at normal incidence
The superscripts indicate to two values of q. The eigenvector components are
circularly polarised state:
In addition, the expression of 4x4 transfer matrix is derived for
these media
(1)
M
where M is a 4x4 transfer matrix of a single layer, and includes 2x2 block
.
matrices , are given by
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
was taken from
(Dong et. al.,2010)
13
The reflectivity spectra for both left, and right circularly polarised light at
normal incidence
was taken from
(Dong et. al.,2010)
14
The reflectivity spectrum,
15
The RMCD against the wavelength
16
The Kerr and Faraday Rotations against the wavelength
the structure was taken
from (Dong et. al.,2010)
17
18
Reflectivity Spectrum for cavity structure
19
The RMCD against the wavelength
At 629 nm, the maximum is 4.73
compared with 0.0192 for film,
in Kerr rotation
20
The Kerr and Faraday Rotations against the wavelength
Simulated Spectra
(this work)
21
Simulated Spectra for
Simulated Spectra
Dong et al. (2010)
, here we set ns=1.0
Question has been raised
about the effect of use a
thick substrate
22
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)
Those studies considered the coherent and incoherent
multiple reflections and transmissions for isotropic structures
to deal with this situations
23
24
front
back
(Whittaker and Gehring, 2010)
The total R for fully polarisation are given by Whittaker and Gehring (2010)
25
26
The reflectivity spectra for left circularly polarised light at normal incidence
27
The RMCD against the wavelength
1. without incoherent
back reflections
2.Single
incoherent back
reflections
3.multiple
incoherent back
reflections
a thick substrate
28
The equations of total
and
for x-polarised state
In a similar way, for y-polarised state
29
are calculated individually as
where
and
are the matrices of linear x and y polarisations, respectively
(Pedrotti and Pedrott, 1993)
30
The Kerr rotation is found as following
31
At 629 nm, the maximum is 4.73 without
incoherent back reflections compared with
1.368 with incoherent back reflections
32
The Kerr Rotation against the wavelength
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 4x4 T-matrix
for these media
In circularly birefringent media, the reflectivity spectra and magnetooptical 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