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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)

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

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

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

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