## 2 Phase Modulation Principle Based on Spectral Evolution of State of Polarization (SOP)
Because it is assumed that all waveplates used here are independent of wavelength, they do not distort the shape of the \(\text{SOP}_{\text{in}}\) or the \(\text{SOP}_{\text{out}}\) trace. Thus, the \(\text{SOP}_{\text{out}}\) trace of PMF 1 also has a circular shape identical to that of the \(SE_{in,GTF}\) trace. Like the relationship between the \(SE_{in}\) and \(SE_{out}\) traces, the increase in \(\lambda_{peak}\) results from the CCW revolution of the \(SE_{in,GTF}\) trace about both \(\mathbf{p_{1}}\) on the \(\text{SOP}_{\text{in}}\) plane and \(\mathbf{p_{2}}\)on the \(\text{SOP}_{\text{out}}\) plane. If the \(SE_{in,GTF}\) trace is revolved CCW about both \(\mathbf{p_{1}}\) and \(\mathbf{p_{2}}\) by an amount of \(\xi\), the transmission spectrum makes a redshift of (\(\xi\)/360\({}^{\circ}\))\(\Delta\lambda\), resulting in an increase of (\(\xi\)/360\({}^{\circ}\))\(\Delta\lambda\) in \(\lambda_{peak}\). That is to say, \(\xi\) can be regarded as \(\phi\), or the additional phase delay difference. Thus, as \(\xi\) varies from 0\({}^{\circ}\) to 360\({}^{\circ}\), the phase of the GTF also changes from 0\({}^{\circ}\) to 360\({}^{\circ}\), which means that the GTF is able to be phase-modulated continuously by varying \(\xi\). Briefly, one simultaneous CCW rotation of the \(SE_{in,GTF}\) trace about both \(\mathbf{p_{1}}\) and \(\mathbf{p_{2}}\), implemented by controlling four waveplates contained within the proposed filter, enables its transmission spectrum to be redshifted by an FSR (\(\Delta\lambda\)), in other words, its phase to be modulated by 360\({}^{\circ}\). Finding Algorithm of Waveplate Orientation Angles (WOA's) for Phase Modulation of General Transmittance Function (GTF)  
Considering the aforementioned polarization conditions required for arbitrarily modulating the phase of a GTF or continuously tuning its wavelength, the WOA's (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), \(\theta_{H2}\)) for this arbitrary phase modulation (i.e., continuous wavelength tuning) are investigated with our unique WOA-finding algorithm. This algorithm is suggested as it is quite sophisticated to draw out an exact mathematical expression of a GTF having an application-specific unique spectrum. Moreover, although the mathematical expression \(t_{GTF}\) of a GTF is given, it is cumbersome to solve (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), \(\theta_{H2}\)) from trigonometric simultaneous equations obtained from direct comparison of \(t_{GTF}\) with \(t_{filter}\) in (2). To explain our angle-finding approach, let us choose a specific GTF \(t_{GTF1}\), which is procured at a certain selected WOA set (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), \(\theta_{H2}\)) = (118.23\({}^{\circ}\), 149.56\({}^{\circ}\), 179.42\({}^{\circ}\), 123.46\({}^{\circ}\)), and find (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), \(\theta_{H2}\)) where \(t_{GTF1}\) is wavelength-tuned. At the above selected WOA set indicated by Set I\({}_{\text{GTF1}}\) and fixed orientation angles of PMF 1 and PMF 2 (\(\theta_{P1}\) = 0\({}^{\circ}\) and \(\theta_{P2}\) = 22.5\({}^{\circ}\)), the \(\text{SOP}_{\text{out}}\) trace of PMF 2, or \(SE_{out,GTF1}\), is obtained over a wavelength range from \(\lambda_{0}\) (=1548.0 nm) to \(\lambda_{0}\) + \(\Delta\lambda\) (=0.8 nm), as shown in Figure 3a. At Set I\({}_{\text{GTF1}}\), \(O(\lambda_{0})\) indicated in Figure 3a is located nearest to LHP on the Poincare sphere, which makes the filter transmittance be maximized at \(\lambda_{0}\). If we put the wavelength at which \(t_{GTF1}\) is maximized as \(\lambda_{peak}\), \(O(\lambda_{0})\) becomes \(O(\lambda_{peak})\), that is, \(\lambda_{peak}\) = \(\lambda_{0}\) = 1548.0 nm at Set I\({}_{\text{GTF1}}\). The \(\text{SOP}_{\text{in}}\) trace of PMF 2, or \(SE_{in,GTF1}\) in Figure 3b, can be derived by utilizing the inverse matrix of \(T_{PMF2}\) (\(T_{PMF2}\)\({}^{-1}\)) and the \(SE_{out,GTF1}\) trace in Figure 3a.