## 2 Phase Modulation Principle Based on Spectral Evolution of State of Polarization (SOP)
no insertion losses are assumed to exist in any optical components of the filter, and all waveplates are regarded as wavelength-independent. \[T= \begin{bmatrix}1&0\\ 0&0\end{bmatrix} T_{PMF2}(\theta_{P2})T_{HWP2}(\theta_{H2})T_{QWP2}\big{(}\theta_{Q 2}\big{)}T_{PMF1}(\theta_{P1})T_{HWP1}(\theta_{H1})T_{QWP1}\big{(}\theta_{Q1} \big{)} \begin{bmatrix}1&0\\ 0&0\end{bmatrix} \] \[+ \begin{bmatrix}0&0\\ 0&1\end{bmatrix} T_{QWP1}\big{(}-\theta_{Q1}\big{)}T_{HWP1}(-\theta_{H1})T_{ PMF1}(-\theta_{P1})\times \tag{1}\] \[T_{QWP2}\big{(}-\theta_{Q2}\big{)}T_{HWP2}(-\theta_{H2})T_{PMF2}( -\theta_{P2}) \begin{bmatrix}0&0\\ 0&1\end{bmatrix} \]  
\[t_{filter}=\frac{1}{4}(P_{0}+P_{1}\cos\Gamma+P_{2}\sin\Gamma)^{2}+\frac{1}{4}(Q _{0}+Q_{1}\cos\Gamma+Q_{2}\sin\Gamma)^{2} \tag{2}\]  
Here, \(T_{QWP1}\), \(T_{HWP1}\), \(T_{PMF1}\), \(T_{QWP2}\), \(T_{HWP2}\), and \(T_{PMF2}\) mean the transfer matrices of QWP 1, HWP 1, PMF 1, QWP 2, HWP 2, and PMF 2, whose slow-axis orientation angles are \(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{P1}\), \(\theta_{Q2}\), \(\theta_{H2}\), and \(\theta_{P2}\) (given by \(22.5^{\circ}\) here) for the \(x\) axis, respectively. \(P_{0}=-\)sin\(\alpha\)sin\(\beta-\sin(\gamma-\pi/4)\)sin\((\delta-2\theta_{P1})\), \(Q_{0}=\cos\))cos\(\beta-\cos(\alpha-\pi/4)\)cos\((\delta-2\theta_{P1})\), \(P_{1}=-\)sin\(\alpha\)sin\(\beta+\sin(\gamma-\pi/4)\)sin\((\delta-2\theta_{P1})\), \(Q_{1}=\cos\))cos\(\beta\)\(+\cos(\alpha-\pi/4)\)cos\((\delta-2\theta_{P1})\), \(P_{2}=\cos(\gamma-\pi/4)\)cos\(\beta+\cos\)\(\alpha\)cos\((\delta-2\theta_{P1})\), and \(Q_{2}=\sin(\alpha-\pi/4)\)sin\(\beta-\sin\)\(\gamma\)sin\((\delta-2\theta_{P1})\), where \(\alpha=2\theta_{H2}-(\theta_{Q1}+\theta_{Q2})\), \(\beta=2\theta_{H1}-(\theta_{Q1}+\theta_{Q2})\), \(\gamma=2\theta_{H2}+(\theta_{Q1}-\theta_{Q2})\), and \(\delta=2\theta_{H1}-(\theta_{Q1}-\theta_{Q2})\). Before we begin to address the principle of the phase modulation of a GTF, let us consider the previous conclusions on the relationship between SOP changes and the continuous wavelength tuning of a narrowband transmittance \(t_{n}\) given by (3) in the comb filter composed of two PMF segments, two OWS's of an HWP and a QWP, and the third HWP. [23, 24]. \[t_{n}=[3-4\cos(\Gamma+\phi)+\cos 2(\Gamma+\phi)]/8 \tag{3}\]  
The additional phase delay difference \(\phi\) in (3) defines the absolute wavelength position of the narrowband transmission spectrum. In the previous studies [23, 24], eight WOA sets (from Set I\({}_{\text{n}}\) to Set VIII\({}_{\text{n}}\)) of four waveplates (except for the third HWP) were utilized to display eight wavelength-tuned spectra calculated from \(t_{n}\) with eight \(\phi\) values from \(0^{\circ}\) to \(315^{\circ}\) (increment: \(45^{\circ}\)). When the wavelength of input light increases from \(\lambda_{0}\) to \(\lambda_{0}+\Delta\lambda\), where \(\Delta\lambda\) implies the FSR, the SOP\({}_{\text{in}}\) of PMF 2 at Set I\({}_{\text{n}}\) (\(\phi=0^{\circ}\)) spectrally evolves on the Poincare sphere, as shown in Figure 2a. This spectral evolution indicated by pink circles is denoted by \(SE_{in}\). With increasing wavelength, the SOP\({}_{\text{in}}\) moves along the \(SE_{in}\) trace rotating CW around the normal vector \(\mathbf{p_{1}}\), when \(\mathbf{p_{1}}\) points the observer. The rotation direction of the SOP\({}_{\text{in}}\) can also be thought using a left-hand rule. If the direction of \(\mathbf{p_{1}}\) is the same as that of the left thumb, the direction of the rest of the fingers is the rotation direction of the SOP\({}_{\text{in}}\) with the increase of the wavelength. In general, the SOP\({}_{\text{out}}\) of the birefringent medium rotates around its slow axis on the Poincare sphere according to the wavelength \(\lambda\) of light passing through the medium.