## 2 Phase Modulation Principle Based on Spectral Evolution of State of Polarization (SOP)
Based on the foregoing description on the phase modulation of the GTF using the \(SE_{in,GTF}\) and \(SE_{out,GTF}\) traces, another \(SE_{in,GTF1}\) trace shown in Figure 3c can be obtained by rotating the \(SE_{in,GTF1}\) trace at Set I\({}_{\text{GTF1}}\) CCW by 45\({}^{\circ}\) around both \(\mathbf{p_{1}}\) and \(\mathbf{p_{2}}\), for the phase modulation of 45\({}^{\circ}\) (i.e., the wavelength tuning of \(\Delta\lambda\)/8 = 0.1 nm) in \(t_{GTF1}\). It is easily expected that this new \(SE_{in,GTF1}\) trace will be obtained at another WOA set (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), \(\theta_{H2}\)) designated later as Set II\({}_{\text{GTF1}}\). Figure 3: Wavelength-dependent variation of SOP’s in \(t_{\text{{GTF1}}}\) obtained at Set I\({}_{\text{{GTF1}}}\): (**a**) SOP\({}_{\text{out}}\) and (**b**) SOP\({}_{\text{in}}\) traces of PMF 2 at Set I\({}_{\text{{GTF1}}}\). Spectral evolution of SOP’s in the phase-modulated version of \(t_{\text{{GTF1}}}\), obtained at Set II\({}_{\text{{GTF1}}}\): (**c**) SOP\({}_{\text{in}}\) trace of PMF 2, (**d**) SOP\({}_{\text{out}}\) trace of QWP 2, (**e**) SOP\({}_{\text{out}}\) trace of PMF 1, (**f**) SOP\({}_{\text{out}}\) of HWP 1, and (**g**) SOP\({}_{\text{out}}\) trace of PMF 2. (**h**) Two calculated transmission spectra obtained at Sets I\({}_{\text{{GTF1}}}\) and II\({}_{\text{{GTF1}}}\), indicated as blue and red solid lines, respectively. What we should focus on is a way to obtain Set II\({}_{\text{GTF1}}\), another WOA set described above. Basically, the SOP\({}_{\text{out}}\) of one component of the filter is equivalent to the SOP\({}_{\text{in}}\) of the following component in any propagation path shown in Figure 1b. As our discussion is confined to the CW path, as an example, the SOP\({}_{\text{out}}\) of PMF 1 is equivalent to the SOP\({}_{\text{in}}\) of QWP 2. The same thing can be said for SOP traces. For example, the SOP\({}_{\text{in}}\) trace of HWP 1 is the same as the SOP\({}_{\text{out}}\) trace of QWP 1. A QWP and an HWP rotate input polarization CCW by 90\({}^{\circ}\) and 180\({}^{\circ}\) about their slow axes on the Poincare sphere, respectively. If an SOP trace, composed of multiple different SOP's distributed on the Poincare sphere, enters a waveplate, therefore, not its shape but its position is altered by the waveplate. For a given \(SE_{in,GTF1}\) trace at Set II\({}_{\text{GTF1}}\) (in Figure 3c), a certain SOP\({}_{\text{out}}\) trace of QWP 2 can be obtained by using the \(SE_{in,GTF1}\) trace (i.e., the SOP\({}_{\text{out}}\) trace of HWP 2) and the inverse matrix of \(T_{HWP2}\), or \(T_{HWP2}\)\({}^{-1}\), if \(\theta_{H2}\) is fixed as one value (unknown yet). With this SOP\({}_{\text{out}}\) trace of QWP 2, shown in Figure 3d, a corresponding SOP\({}_{\text{out}}\) trace of PMF 1, which is transformed into the SOP\({}_{\text{out}}\) trace of QWP 2 after passing through QWP 2, can also be derived similarly by using the predetermined SOP\({}_{\text{out}}\) trace of QWP 2 and \(T_{QWP2}\)\({}^{-1}\), if \(\theta_{Q2}\) is fixed as another value (unknown either), as shown in Figure 3e. Because the SOP rotation due to a QWP or an HWP applies equally to all points comprising an SOP trace, the shape of the SOP trace created by PMF 1 remains the same after passage through HWP 2 or QWP 2, and thus the trajectory shape of the SOP\({}_{\text{out}}\) trace of QWP 2 or PMF 1 is equal to that of the \(SE_{in,GTF1}\) trace. In terms of the SOP\({}_{\text{out}}\) trace of PMF 1, however, \(\mathbf{p_{1}}\), the normal vector of the trace, coincides with the \(S_{1}\) axis on the Poincare sphere as the orientation angle \(\theta_{P1}\) of PMF 1 is 0\({}^{\circ}\), as shown in Figure 3e. Then, \(\theta_{H2}\) and \(\theta_{Q2}\) can be determined by finding the SOP\({}_{\text{out}}\) trace of PMF 1 to satisfy this constraint for \(\mathbf{p_{1}}\).