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Department of Physics and Astronomy The University of Sheffield 1

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4x4Transfer Matrix and Reflectivity Calculations Study the effect of using a thick substrate (incoherent back reflections) 2

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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). 3

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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.

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5 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) 6

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and Kerr rotation as Sato (1981) defined the reflectance magneto-circular dichroism (RMCD) as http://www.enzim.hu/~szia /cddemo/edemo16.htm 7

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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

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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). 9

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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 M where M is a 4x4 transfer matrix of a single layer, and includes 2x2 block. matrices, are given by (1) 10

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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(1) 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

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The reflectivity spectra for both left, and right circularly polarised light at normal incidence 13 was taken from (Dong et. al.,2010)

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The reflectivity spectrum, 14 was taken from (Dong et. al.,2010)

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The RMCD against the wavelength 15

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The Kerr and Faraday Rotations against the wavelength 16

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17 the structure was taken from (Dong et. al.,2010)

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Reflectivity Spectrum for cavity structure 18

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19 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

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Simulated Spectra for Simulated Spectra Dong et al. (2010) Simulated Spectra (this work) 21, here we set n s =1.0

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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)

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The total R for fully polarisation are given by Whittaker and Gehring (2010) front back ( Whittaker and Gehring, 2010) 25

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26 The reflectivity spectra for left circularly polarised light at normal incidence

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27 The RMCD against the wavelength

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28 3.multiple incoherent back reflections 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 29 In a similar way, for y-polarised state

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where 30 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 31

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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

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33 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 Maxwell's equations were used to derive expression of 4x4 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

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