## 2 Phase Modulation Principle Based on Spectral Evolution of State of Polarization (SOP)
After both \(\theta_{H2}\) and \(\theta_{Q2}\) are elucidated, the SOP\({}_{\text{out}}\) of HWP 1, which is displayed as an SOP location on the Poincare sphere, can simply be determined using the SOP\({}_{\text{out}}\) trace of PMF 1 and \(T_{PMF1}\)\({}^{-1}\), as shown in Figure 3f. In particular, the validity of \(\theta_{H2}\) and \(\theta_{Q2}\) determined above can be verified by checking whether this SOP\({}_{\text{in}}\) of PMF 1 is a single SOP point or not. That is to say, if \(\theta_{H2}\) and \(\theta_{Q2}\) predetermined abiding by the above constraint for \(\mathbf{p_{1}}\) are suitable for the WOA set at Set II\({}_{\text{GTF1}}\), the SOP\({}_{\text{in}}\) of PMF 1 should be unchanged even with variations in the wavelength from \(\lambda_{0}\) to \(\lambda_{0}+\Delta\lambda\). As \(\theta_{H2}\) and \(\theta_{Q2}\) are found from the \(SE_{in,GTF1}\) trace at Set II\({}_{\text{GTF1}}\), this SOP\({}_{\text{in}}\) of PMF 1 (i.e., the SOP\({}_{\text{out}}\) of HWP 1) is the starting point to unveil \(\theta_{H1}\) and \(\theta_{Q1}\). By using the SOP\({}_{\text{out}}\) of HWP 1 (in Figure 3f) and \(T_{HWP1}\)\({}^{-1}\), the SOP\({}_{\text{out}}\) of QWP 1 can be obtained, if we set \(\theta_{H1}\) as a fixed value (unknown yet). Moreover, for a fixed value (unknown either) of \(\theta_{Q1}\), a corresponding SOP\({}_{\text{in}}\) of QWP 1 can be drawn out from the above SOP\({}_{\text{out}}\) of QWP 1 and \(T_{QWP1}\)\({}^{-1}\). Then, to find \(\theta_{H1}\) and\(\theta_{Q1}\), we can harness the fundamental SOP constraint that the SOP\({}_{\text{in}}\) of QWP 1 should be LHP because light emerging from port 2 of the PBS is linear horizontally polarized. Through this reverse tracing algorithm described above, (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), \(\theta_{H2}\)) at Set II\({}_{\text{GTF1}}\) could be found to be (108.49\({}^{\circ}\), 132.67\({}^{\circ}\), 201.92\({}^{\circ}\), 134.71\({}^{\circ}\)), and it was also confirmed from additional calculations, although not suggested here, that several degenerate WOA sets could also exist for \(t_{GTF1}\) at Set II\({}_{\text{GTF1}}\). It is figured out from the SOP\({}_{\text{out}}\) trace of PMF 2 at Set II\({}_{\text{GTF1}}\), the \(SE_{out,GTF1}\) trace in Figure 3g, that \(O(\lambda_{0})\) at Set II\({}_{\text{GTF1}}\) differs from \(O(\lambda_{0})\) at Set I\({}_{\text{GTF1}}\), although the shape of the \(SE_{out,GTF1}\) trace at Set II\({}_{\text{GTF1}}\) remains unchanged in comparison with the shape of the \(SE_{out,GTF1}\) trace at Set I\({}_{\text{GTF1}}\). From \(O(\lambda_{0})\) at Set I\({}_{\text{GTF1}}\), \(O(\lambda_{0})\) at Set II\({}_{\text{GTF1}}\) is distant by an angular displacement that corresponds to \(\Delta\lambda/8\) along this \(SE_{out,GTF1}\) trace towards decreasing wavelength. This implies that \(O(\lambda_{0}\) + \(\Delta\lambda/8)\) at Set II\({}_{\text{GTF1}}\) is the same as \(O(\lambda_{0})\) at Set I\({}_{\text{GTF1}}\) and also as \(O(\lambda_{peak})\) at Set II\({}_{\text{GTF1}}\), and \(\lambda_{peak}\) becomes \(\lambda_{0}+\Delta\lambda/8\) at Set II\({}_{\text{GTF1}}\), resulting in a wavelength shift of \(\Delta\lambda/8\) in \(t_{GTF1}\) (i.e., a phase modulation of 45\({}^{\circ}\)). Figure 3h shows two transmission spectra of \(t_{GTF1}\), which are calculated at Sets I\({}_{\text{GTF1}}\) and II\({}_{\text{GTF1}}\) and displayed by blue and red solid lines, respectively. It is apparent that the transmission spectrum at Set II\({}_{\text{GTF1}}\) is redshifted by \(\Delta\lambda/8\) (= 0.1 nm) compared with that at Set I\({}_{\text{GTF1}}\). In short, for a phase modulation of \(\eta\), or a wavelength tuning of (\(\eta/360^{\circ}\))\(\Delta\lambda\), in \(t_{GTF1}\), one should find a new \(SE_{in,GTF1}\) trace first, obtained by revolving the \(SE_{in,GTF1}\) trace at Set I\({}_{\text{GTF1}}\) CCW by \(\eta\) in angle about \(\mathbf{p_{1}}\) and \(\mathbf{p_{2}}\).