# Optical Effects Induced by Bloch Surface Waves in One-Dimensional Photonic Crystals
## 5 The BSW-Induced Magneto-Optics
The polarization rotation angle of the reflected wave appears from the phase difference between \(s\)- and \(p\)-polarized reflected light, which explains the change in the sign of the Faraday rotation near the resonance of the BSW. Theoretically, the Faraday rotation of light reflected from the MPC at the wavelength of BSW resonance can reach the value of \(\pi/2\)[56]. Besides the Faraday rotation, Ref. [53] also considers other transformations of light polarization, such as transition of the linear polarization to the elliptical one. To describe these polarization changes, the spectral dependencies of Stokes parameters were numerically studied. Obtained Stokes parameter spectra are shown in Figure 8. Three typical cases were considered. In the first case, the position of the BSW resonance is blue-shifted for 30 nm relative to the resonance of the WGM (Figure 8a). In the second case, the position of the BSW resonance is shifted 30 nm to the higher wavelengths (Figure 8c). The third case corresponds to the situation when the BSW and WGM have the same spectral position (Figure 8b). The parameter \(S_{0}\) is the reflectance of light from the sample. The angle of Faraday rotation \(\varphi\) can be achieved from the first and second Stokes parameters as \(\varphi=(1/2)\arctan(S_{2}/S_{1})\). The Stokes parameter \(S_{3}\) corresponds to the ellipticity of reflected light. The shapes of spectra \(S_{1}/S_{0}\) and \(S_{2}/S_{0}\) show that Faraday rotation of reflected light becomes higher while the BSW coupling is more effective. The reflected light ellipticity increases simultaneously (Figure 8, \(S_{3}/S_{0}\) spectrum). When the WGM and BSW have the same wavelength of excitation, the ellipticity equals 1, so the reflected light is circularly polarized. While the BSW coupling becomes less effective, the values of Stokes parameters significantly decrease (Figure 8a,c). The influence of the WGM and its spectral position relative to the BSW manifest themselves in the spectral behavior of the parameter \(S_{2}/S_{0}\). If the BSW excites at shorter wavelength than the WGM, the \(S_{2}/S_{0}\) spectrum has the shape of an asymmetric Fano resonance (Figure 8a). For the BSW excited at a longer wavelength than the WGM, the Fano resonance  
Figure 8: Numerical spectra of the Stokes parameter \(S_{0}\) and normalized Stokes parameters \(S_{1}/S_{0}\), \(S_{2}/S_{0}\) and \(S_{3}/S_{0}\) of light reflected by the magnetophotonic crystals in the presence of a magnetic field. The incident light is an \(s\)-polarized plane wave. Data are presented for the cases when the BSW resonance is (**a**) blue-shifted by 30 nm, (**b**) red-shifted by 30 nm relative to the waveguide mode (WGM) resonance, and (**c**) has the same spectral position as the WGM resonance. Reprinted with permission from [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], Phys. Rev. B 96, 081401(R) (2017). Copyright (2017) by the American Physical Society [53]. in the \(S_{2}/S_{0}\) spectrum takes an inverted asymmetric shape (Figure 8c). Therefore, the Faraday rotation angle changes sign when the dispersion curves of the BSW and WGM resonances intersect each other. In fact, the polarization of light can be controlled by a magnetic field applied to the MPC structures supporting the excitation of the BSW. Due to the excitation of the BSW, studied MPC structures have a great potential for applications as a magnetically-controlled wave-plate of variable thickness.