## 2 Phase Modulation Principle Based on Spectral Evolution of State of Polarization (SOP)  
Figure 1a shows a schematic diagram of the proposed comb filter comprised of a PBS, two PMF segments whose lengths are equal (denoted by PMF 1 and PMF 2), two QWPs (denoted by QWP 1 and QWP 2), and two HWPs (denoted by HWP 1 and HWP 2). As shown in Figure 1a, one OWS of QWP 1 and HWP 1 is placed ahead of PMF 1, and the other OWS of QWP 2 and HWP 2 is located  
between PMF 1 and PMF 2. Every OWS can effectively change the phase delay difference between two principal modes of the individual PMF segment, and the combination of QWP 2 and HWP 2, or the second OWS, has an additional function to effectively modify the relative angular difference between the principal axes of PMF 1 and PMF 2. The slow axis of PMF 2 butt-coupled to port 3 of the PBS is \(22.5^{\circ}\) away from the TM polarization axis (denoted by \(x\) axis) of the PBS. An input beam introduced into port IN of the filter is decomposed into two orthogonal linear polarization components, such as linear horizontal polarization (LHP) and linear vertical polarization (LVP), which circulate through the polarization-diversified loop of the filter in the clockwise (CW) and counterclockwise (CCW) directions, respectively. Figure 1b shows the two propagation paths of light travelling through the filter, designated as CW and CCW paths. When the SOP in of the filter is LHP, input light propagates along the CW path and sequentially passes through a linear horizontal polarizer (\(x\) axis), QWP 1 (with its slow axis oriented at \(\theta_{Q1}\) for the \(x\) axis), HWP 1 (oriented at \(\theta_{H1}\)), PMF 1 (oriented at \(\theta_{P1}\)), QWP 2 (oriented at \(\theta_{Q2}\)), HWP 2 (oriented at \(\theta_{H2}\)), PMF 2 (oriented at \(\theta_{P2}\) = \(22.5^{\circ}\)), and a linear horizontal analyzer (\(x\) axis). In the case of the SOP in of LVP, input light propagates through a linear vertical polarizer (\(y\) axis), PMF 2 (\(-\theta_{P2}\) oriented), HWP 2 (\(-\theta_{H2}\) oriented), QWP 2 (\(-\theta_{Q2}\) oriented), PMF 1 (\(-\theta_{P1}\) oriented), HWP 1 (\(-\theta_{H1}\) oriented), QWP 1 (\(-\theta_{Q1}\) oriented), and a linear vertical analyzer (\(y\) axis) in turn, circulating along the CCW path. F and S displayed on the waveplates and the PMFs indicate their fast and slow axes, respectively. Polarization interference is created by optical birefringence in an optical structure composed of two linear polarizers and BEs inserted between them. In the case of one BE like a PMF segment, the transmittance function of this optical structure becomes a sinusoidally varying function of \(\Gamma\), the phase delay difference between two principal modes of the BE, which is represented by \(2\pi BL/\lambda\), where \(B\), \(L\), and \(\lambda\) are the birefringence of the BE, the length of the BE, and the free space wavelength, respectively. It is possible to tune the wavelength of this polarization interference spectrum by modifying \(\Gamma\), and the modification of \(\Gamma\) can be done by adding a supplementary phase delay difference \(\phi\) to \(\Gamma\) and modulating this \(\phi\)[21]. In the same way, to tune the wavelength position of the first-order comb spectrum generated with two BEs, or two PMF segments [26], \(\phi\), which is added to \(\Gamma\) for individual PMF, should be modulated equally for both PMF 1 and PMF 2. This simultaneous \(\phi\) modulation can be accomplished by changing the SOP in of PMF 1 and PMF 2 with two OWS's. When we increase \(\phi\) from \(0^{\circ}\) to \(360^{\circ}\) by appropriately adjusting the SOP in of each PMF, the comb spectrum makes a redshift by an interference period, i.e., a free spectral range (FSR). The filter transmittance function \(t_{\mathit{filter}}\) can be derived from the Jones transfer matrix [27]\(T\) given by (1) and is represented as (2). In this derivation,  
Figure 1: (**a**) Schematic diagram of the PDLS-based first-order fiber comb filter and (**b**) propagation path of light travelling through the filter.