## 2 Phase Modulation Principle Based on Spectral Evolution of State of Polarization (SOP)
The vector \(\mathbf{AB}\) connecting \(A\) (\(2\varepsilon=0^{\circ}\), \(2\psi=-135^{\circ}\)) and \(B\) (\(2\varepsilon=0^{\circ}\), \(2\psi=45^{\circ}\)), which lie on the Poincare sphere, points the direction of the slow axis of PMF 2 (\(\theta_{P2}=22.5^{\circ}\)) and is denoted by the vector \(\mathbf{p_{2}}\). Here, \(2\varepsilon\) (\(-90^{\circ}\leq 2\varepsilon\leq 90^{\circ}\)) and \(2\psi\) (\(-180^{\circ}\leq 2\psi\leq 180^{\circ}\)) are the latitude and longitude of the Poincare sphere, respectively. For convenience, the plane whose normal vector is \(\mathbf{p_{1}}\), that is, the plane of the \(SE_{in}\) trace, is designated as the SOP\({}_{\text{in}}\) plane, and another plane with its normal vector of \(\mathbf{p_{2}}\) is designated as the SOP\({}_{\text{out}}\) plane. Actually, \(\mathbf{p_{2}}\) is an axis of revolution, around which the \(SE_{in}\) trace moves as \(\phi\) varies. Hence, \(\mathbf{p_{1}}\) and the SOP\({}_{\text{out}}\) plane are parallel, and \(\mathbf{p_{2}}\) and the SOP\({}_{\text{in}}\) plane are parallel, ending up with \(\mathbf{p_{1}}\) and \(\mathbf{p_{2}}\) being perpendicular. At Set II\({}_{\text{n}}\) (\(\phi=45^{\circ}\)), the \(SE_{in}\) trace shown in Figure 2a rotates CCW by \(45^{\circ}\) around both \(\mathbf{p_{1}}\) and \(\mathbf{p_{2}}\), and thus \(\mathbf{p_{1}}\) shown in Figure 2a rotates \(45^{\circ}\) CCW about \(\mathbf{p_{2}}\) as well. Analogously, for the remaining six sets (III\({}_{\text{n}}\)-VIII\({}_{\text{n}}\)), the \(SE_{in}\) trace at Set I\({}_{\text{n}}\) rotates CCW by \(90^{\circ}\)-\(315^{\circ}\) (with an increment of \(45^{\circ}\)) around both \(\mathbf{p_{1}}\) and \(\mathbf{p_{2}}\). Likewise, the \(\text{SOP}_{\text{out}}\)'s of PMF 2 at Sets I\({}_{\text{n}}\) (\(\phi=0^{\circ}\)) and II\({}_{\text{n}}\) (\(\phi=45^{\circ}\)) spectrally evolve on the Poincare sphere over the same wavelength range of \(\lambda_{0}\) to \(\lambda_{0}+\Delta\lambda\), as shown in Figure 2b. This spectral evolution also indicated by pink circles is designated as \(SE_{out}\). This \(SE_{out}\) trace is embodied by making a CW rotation of the wavelength-dependent \(\text{SOP}_{\text{in}}\) of PMF 2 about \(\mathbf{p_{2}}\) by an angle \(\varphi(\lambda)\) given by \(360^{\circ}(\lambda-\lambda_{0})/\Delta\lambda\). When PMF 2 is passed through by light at \(\lambda_{0}\), the \(\text{SOP}_{\text{in}}\) of PMF 2 at \(\lambda_{0}\), \(C\) (\(2\varepsilon=0^{\circ}\), \(2\psi=0^{\circ}\)) shown in Figure 2a, rotates CW around \(\mathbf{p_{2}}\) by \(\varphi(\lambda_{0})=0^{\circ}\), and the \(\text{SOP}_{\text{out}}\) of PMF 2 at \(\lambda_{0}\) becomes \(C^{\prime}\) (\(2\varepsilon=0^{\circ}\), \(2\psi=0^{\circ}\)) shown in Figure 2b without a change in the position on the Poincare sphere. Similarly, the \(\text{SOP}_{\text{in}}\) of PMF 2 at \(\lambda_{0}+\Delta\lambda/4\) rotates CW around \(\mathbf{p_{2}}\) by \(\varphi(\lambda_{0}+\Delta\lambda/4)=90^{\circ}\) during its passage through PMF 2. If we put the \(\text{SOP}_{\text{out}}\) of PMF 2 at a certain wavelength \(\lambda\) as \(O(\lambda)\), \(O(\lambda_{0})\) at Set I\({}_{\text{n}}\), marked by \(O_{\text{I}}(\lambda_{0})\) in the figure, becomes \(C^{\prime}\). Owing to the wavelength-dependent transformation of the \(\text{SOP}_{\text{in}}\) of PMF 2, \(O(\lambda)\) forms a non-circular trajectory \(SE_{out}\) shaped like a droplet as \(\lambda\) increases from \(\lambda_{0}\) to \(\lambda_{0}+\Delta\lambda\). During the wavelength increase of \(\Delta\lambda\) starting from \(\lambda_{0}\), \(O(\lambda)\) goes round this entire \(SE_{out}\) trace starting from \(C^{\prime}\) in the CW direction around the \(S_{2}\) axis. If we assume the wavelength at which \(t_{n}\) is maximized as \(\lambda_{peak}\), \(\lambda_{peak}\) can be regarded as \(\lambda_{0}\) at Set I\({}_{\text{n}}\) because \(t_{n}\) reaches the maximum when \(O(\lambda)\) is nearest to LHP (\(2\varepsilon=0^{\circ}\), \(2\psi=0^{\circ}\)) on the Poincare sphere.