Modelling Multilayer Structures with Circularly Birefringent Materials

1. Modelling Multilayer Structures with Circularly Birefringent Materials EntesarGanash , David Whittaker and Gillian Gehring Department of Physics and Astronomy The University of Sheffield

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

3. Aimsof Work 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).

4. Motivation Aims of Work 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 (Lourtiozet 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.

5. Maxwell’s Equations Electromagnetic wave propagation inside multilayer structures obeys Maxwell's equations. in source free J=0 and =0

6. 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 Quarter-wave stack (Joannopoulos et al., 2008)

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

8. General Idea of Transfer Matrix (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)

9. 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).

10. Circularly Birefringent Materials 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

11. Circularly Birefringent Materials

12. Circularly Birefringent Materials 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 FORTRANprogram.

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

14. The reflectivity spectrum for linearly polarised light was taken from (Dong et. al.,2010) The reflectivity spectrum,

15. RMCD The RMCD against the wavelength

16. Kerr and Faraday Rotations The Kerr and Faraday Rotations against the wavelength

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

18. Reflectivity spectrum Reflectivity Spectrum for cavity structure

19. RMCD The RMCD against the wavelength

20. Kerr and Faraday Rotations At 629 nm, the maximum is 4.73 compared with 0.0192 for film, in Kerr rotation The Kerr and Faraday Rotations against the wavelength

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

22. Circularly birefringent materials on a thick substrate Question has been raised about the effect of use a thick substrate

23. Circularly birefringent materials on a thick Substrate 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 Gehring2010) Those studies considered the coherent and incoherent multiple reflections and transmissions for isotropic structures to deal with this situations

24. The reflectivity for multilayer structure

25. Circularly Birefringent Materials on a thick substrate front back (Whittaker and Gehring, 2010) The total R for fully polarisation are given by Whittaker and Gehring (2010)

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

27. RMCD The RMCD against the wavelength

28. Circularly birefringent materials on a thick substrate 1. without incoherent back reflections 2.Single incoherent back reflections 3.multiple incoherent back reflections a thick substrate

29. Circularly Birefringent Materials on a thick substrate The equations of total and are calculated individually as for x-polarised state In a similar way, for y-polarised state

30. Circularly Birefringent Materials on a thick substrate where and are the matrices of linear x and y polarisations, respectively (Pedrotti and Pedrott, 1993)

31. Circularly Birefringent Materials on a thick substrate The Kerr rotation is found as following

32. Kerr Rotation At 629 nm, the maximum is 4.73 without incoherent back reflections compared with 1.368 with incoherent back reflections The Kerr Rotation against the wavelength

33. Faraday Rotation The Faraday Rotation against the wavelength

34. Conclusions 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.

35. Acknowledgment Thank you