Modelling Multilayer Structures with Circularly Birefringent Materials
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Modelling Multilayer Structures with Circularly Birefringent
Materials
Entesar Ganash , David Whittaker and Gillian GehringDepartment of Physics and Astronomy
The University of Sheffield
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Outline4x4Transfer 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).
Aims of Work
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In recognizing real experimental magneto-optical data.
Aims of WorkMotivation
Magneto photonic structures play a key role in controlling the optical properties and in enhancing the magneto optical effect (Lourtioz et al., 2008).
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.
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Electromagnetic wave propagation inside multilayer structures obeys Maxwell's equations.
Maxwell’s Equations
in source free J=0 and =0
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Quarter-wave stack
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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Magneto-Optical properties
and Kerr rotation as
Sato (1981) defined the reflectance magneto-circular dichroism (RMCD) as
http://www.enzim.hu/~szia/cddemo/edemo16.htm
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General Idea of Transfer Matrix
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)
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Circularly Birefringent Materials
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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Circularly Birefringent Materials
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 4x4 transfer matrix is derived for these media
Mwhere M is a 4x4 transfer matrix of a single layer, and includes 2x2 block . matrices , are given by
(1)
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Circularly Birefringent Materials
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Circularly Birefringent MaterialsFor 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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The reflectivity spectra for circularly polarised light
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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14 The reflectivity spectrum ,
The reflectivity spectrum for linearly polarised light
was taken from (Dong et. al.,2010)
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RMCD
The RMCD against the wavelength
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16 The Kerr and Faraday Rotations against the wavelength
Kerr and Faraday Rotations
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Cavity Structure
the structure was taken from (Dong et. al.,2010)
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Reflectivity spectrum
Reflectivity Spectrum for cavity structure
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RMCD
The RMCD against the wavelength
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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
Kerr and Faraday Rotations
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Comparison
Simulated Spectra for
Simulated Spectra Dong et al. (2010)
Simulated Spectra (this work)
, here we set ns=1.0
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Circularly birefringent materials on a thick substrateQuestion has been raised about the effect of use a
thick substrate
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Circularly birefringent materials on a thick Substrate
Those studies considered the coherent and incoherent multiple reflections and transmissions for isotropic structures to deal with this situations
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)
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The reflectivity for multilayer structure
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The total R for fully polarisation are given by Whittaker and Gehring (2010)
front back
Circularly Birefringent Materials on a thick substrate
(Whittaker and Gehring, 2010)
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The reflectivity spectra
The reflectivity spectra for left circularly polarised light at normal incidence
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27The RMCD against the wavelength
RMCD
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Circularly birefringent materials on a thick substrate
3.multiple incoherent backreflections
2.Single incoherent back reflections
1. without incoherent back reflections
a thick substrate
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The equations of total and are calculated individually as
for x-polarised state
In a similar way, for y-polarised state
Circularly Birefringent Materials on a thick substrate
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where
Circularly Birefringent Materials on a thick substrate
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
Circularly Birefringent Materials on a thick substrate
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Kerr Rotation
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
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Faraday Rotation
The Faraday Rotation against the wavelength
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A multilayer structure of photonic crystal was modelled for anisotropic materials that display a circular birefringence
Conclusions
Maxwell's equations were used to derive expression of 4x4 T-matrixfor 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.
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AcknowledgmentThank you