## 2 Principle of Operation
F and S appearing on the wave retarders and the HBF's denote their fast and slow axes, respectively. Two orthogonally polarized light (i.e., LHP and LVP components) circulates in the polarization-diversity loop along the CW and CCW paths, respectively. In each path, polarization interference occurs owing to \(\Gamma\) of each HBF, leading to an interference spectrum. Because the input optical power of the filter is distributed to ports 2 and 3 of the PBS according to its input SOP, the insertion losses of these two interference spectra (for the CW and CCW paths) also rely on the input SOP. However, there is no difference in other spectral characteristics of the two interference spectra, for example, a free spectral range (FSR) that determines a channel spacing of the comb filter [17]. This is because the Jones transfer matrices of all the wave retarders and HBF's used here are lossless and unitary, which can also be confirmed in the following Jones matrix formulation of the filter transmittance. Owing to the orthogonality of the SOP's of these two interference spectra, the output transmission spectrum of the filter can be obtained by simple arithmetic sum of them. Moreover, as any two orthogonal polarization bases (e.g., LHP and LVP) can generate an arbitrary SOP, the output spectrum is independent of input polarization in our filter [17]. A comb spectrum created by polarization interference has a sinusoidal transmittance function of \(\Gamma\). As a simple example, the transmittance function of a polarization interference spectrum created in an HBF segment sandwiched by two linear polarizers is represented by \(p+q\text{cos}\Gamma\) (where \(p\) and \(q\) are real constants) called the zeroth-order transmittance function. If the number of the HBF segments inserted between linear polarizers becomes the integer \(M\) (\(\geq\)2), this transmittance function is called the (\(M\)-1)-order transmittance function, which includes sinusoidal functions of \(\Gamma\), such as \(\text{cos}^{M}\Gamma\), \(\text{cos}^{M-1}\Gamma\), and \(\text{cos}\Gamma\)[25]. To continuously tune the wavelength of a higher-order transmittance function, an extra phase retardation difference \(\psi\) should be added to the phase retardation difference \(\Gamma\) of each HBF segment. For this, in view of the transmittance function, (\(\Gamma+\psi\)) should be substituted for \(\Gamma\) in the sinusoidal functions (\(\text{cos}^{M}\Gamma\), \(\text{cos}^{M-1}\Gamma\), etc.). As we want to tune a narrowband comb spectrum of the first-order transmittance function (\(M=2\)), the effective phase retardation difference of each HBF segment between linear polarizers needs to become (\(\Gamma+\psi\)) while the effective slow-axis AOA's of HBF 1 and HBF 2 are set as 22.5\({}^{\circ}\) and 67.5\({}^{\circ}\), respectively, as can be found in the CW path of Figure 1c. These two effective AOA's are inferred by analyzing the AOA relationship between birefringent components in a conventional fan Solc filter with a narrowband transmittance [26]. The extra phase difference \(\psi\) and AOA's (\(\theta_{B1}\) and \(\theta_{B2}\)) of HBF can be effectively modified by changing the input polarization of HBF 1 or HBF 2. This input SOP of each HBF can be determined by controlling the AOA's of the four wave retarders (QWR 1, HWR 1, HWR 2, and QWR 2). If we can set \(\psi\) to increase from 0\({}^{\circ}\) to 360\({}^{\circ}\) through the AOA adjustment of the wave retarders, the narrowband comb spectrum can be continuously red-shifted by a wavelength displacement corresponding to one FSR. The general transmittance \(t_{filter}\) of our filter can be derived using the Jones transfer matrix \(T\), which contains the transfer matrices of two QWR's, two HWR's, two HBF's, and two polarizers, obtained along the CW and CCW paths, the first and second terms of the right side of (1), respectively. It is assumed here that there is no insertion loss in all the optical components and the wave retarders are frequency-independent. \[