#### Transcript Modelling multilayer structures with circularly birefringent materials

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