Department of Physics and Astronomy
The University of Sheffield
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4x4Transfer Matrix and Reflectivity Calculations
Study the effect of using a thick substrate
(incoherent back reflections)
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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).
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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.
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Electromagnetic wave propagation inside multilayer structures
obeys Maxwell's equations.
in source free J=0 and =0
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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)
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http://www.enzim.hu/~szia
/cddemo/edemo16.htm
Sato (1981) defined
the reflectance magneto-circular dichroism (RMCD) as
and Kerr rotation as
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(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)
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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).
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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
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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.
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was taken from
(Dong et. al.,2010)
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The reflectivity spectra for both left, and right circularly polarised light at
normal incidence
was taken from
(Dong et. al.,2010)
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The reflectivity spectrum,
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The RMCD against the wavelength
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The Kerr and Faraday Rotations against the wavelength
the structure was taken
from (Dong et. al.,2010)
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Reflectivity Spectrum for cavity structure
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The RMCD against the wavelength
At 629 nm, the maximum is 4.73
compared with 0.0192 for film,
in Kerr rotation
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The Kerr and Faraday Rotations against the wavelength
Simulated Spectra
(this work)
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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
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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
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front
back
(Whittaker and Gehring, 2010)
The total R for fully polarisation are given by Whittaker and Gehring (2010)
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The reflectivity spectra for left circularly polarised light at normal incidence
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The RMCD against the wavelength
1. without incoherent
back reflections
2.Single
incoherent back
reflections
3.multiple
incoherent back
reflections
a thick substrate
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The equations of total
and
for x-polarised state
In a similar way, for y-polarised state
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are calculated individually as
where
and
are the matrices of linear x and y polarisations, respectively
(Pedrotti and Pedrott, 1993)
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The Kerr rotation is found as following
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At 629 nm, the maximum is 4.73 without
incoherent back reflections compared with
1.368 with incoherent back reflections
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The Kerr Rotation against the wavelength
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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.
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