## 2 Phase Modulation Principle Based on Spectral Evolution of State of Polarization (SOP)
At Set II\({}_{\text{n}}\), \(O_{\text{II}}(\lambda_{0})\), i.e., the \(\text{SOP}_{\text{out}}\) of PMF 2 at \(\lambda_{0}\), becomes \(C^{\prime\prime}\) distant from \(O_{\text{I}}(\lambda_{0})\) by an angular displacement that corresponds to \(\Delta\lambda/8\) and also goes round the entire \(SE_{out}\) trace starting from \(C^{\prime\prime}\) about the \(S_{2}\) axis in the CW direction, as the wavelength increases from \(\lambda_{0}\) to \(\lambda_{0}+\Delta\lambda\). Here, \(O_{\text{I}}(\lambda_{0})\) or \(O_{\text{II}}(\lambda_{0})\) is the initial point of the spectral evolution of \(O(\lambda)\). The position difference between \(O_{\text{I}}(\lambda_{0})\) and \(O_{\text{II}}(\lambda_{0})\) originates from the fact that, while the WOA set goes from Set I\({}_{\text{n}}\) to Set II\({}_{\text{n}}\), the \(SE_{in}\) trace at Set I\({}_{\text{n}}\) is rotated CCW by \(45^{\circ}\) not only about \(\mathbf{p_{1}}\) on the \(\text{SOP}_{\text{in}}\) plane but also about \(\mathbf{p_{2}}\) on the \(\text{SOP}_{\text{out}}\) plane. This means that simultaneously rotating the \(SE_{in}\) trace CCW about both \(\mathbf{p_{1}}\) and \(\mathbf{p_{2}}\) can change the position of \(O(\lambda_{0})\) along the \(SE_{out}\) trace. On the \(SE_{out}\) trace at Set II\({}_{\text{n}}\), therefore, \(O_{\text{II}}(\lambda_{0}+\Delta\lambda/8)\) is located at \(C^{\prime}\), and \(\lambda_{0}+\Delta\lambda/8\) becomes \(\lambda_{peak}\) because \(C^{\prime}\) is nearest to LHP. This implies that the narrowband transmission spectrum moves towards a longer wavelength region by \(\Delta\lambda/8\), that is, \(\phi\) is modulated by \(45^{\circ}\) making \(t_{n}\) be redshifted by \((45^{\circ}/360^{\circ})\Delta\lambda=\Delta\lambda/8\). Similarly, for remaining Sets III\({}_{\text{n}}\)-VIII\({}_{\text{n}}\), \(\phi\) is modulated by \(90^{\circ}\) to \(315^{\circ}\) with an increment of \(45^{\circ}\) making \(t_{n}\) be redshifted by \(\Delta\lambda/4\) to \(7\Delta\lambda/8\) with a step of \(\Delta\lambda/8\), and \(\lambda_{peak}\) changes from \(\lambda_{0}+\Delta\lambda/4\) to \(\lambda_{0}+7\Delta\lambda/8\) with an increment of \(\Delta\lambda/8\). Figure 2: (a) Poincare sphere representation of the spectral evolution of \(\text{SOP}_{\text{in}}\) of PMF 2 at Set I\({}_{\text{n}}\) (\(\phi=0^{\circ}\)), which is denoted by \(SE_{in}\) (indicated by pink circles) and (b) Poincare sphere representation of the spectral evolution of \(\text{SOP}_{\text{out}}\) of PMF 2, which is denoted by \(SE_{out}\) (indicated by pink circles), and two \(O(\lambda_{0})\) positions at two chosen WOA sets (Sets I\({}_{\text{n}}\) and II\({}_{\text{n}}\)). All the SOP traces are obtained in the CW path of Figure 1b over a wavelength range from \(\lambda_{0}\) to \(\lambda_{0}+\Delta\lambda\). In the case of a GTF, or an arbitrary transmittance function, obtained in the proposed filter structure, the \(\text{SOP}_{\text{out}}\) trace of PMF 2, designated as \(SE_{out,GTF}\) trace, is presumed to be a completely different trajectory in comparison with the \(SE_{out}\) trace shown in Figure 2b. However, it can be deduced from the aforementioned conclusions of the previous works that a location change of \(O(\lambda_{0})\) along the \(SE_{out,GTF}\) trace, accompanied by a shift in \(\lambda_{peak}\), is responsible for a phase modulation in the GTF, resulting in a wavelength shift in the transmission spectrum. For a given \(SE_{out,GTF}\) trace, the corresponding \(\text{SOP}_{\text{in}}\) trace of PMF 2, designated as \(SE_{in,GTF}\) trace, should be a circle on the Poincare sphere as well, since a wavelength-dependent BE through which light has passed is just one, PMF 1. However, in terms of the radius and the normal vector (\(\mathbf{p_{1}}\)) of the circular trace, this \(SE_{in,GTF}\) trace will be quite different from the \(SE_{in}\) trace shown in Figure 2a.