## 3 Spectral Calculation and Experimental Demonstration  
Figure 3 shows the calculated narrowband transmission spectra of the proposed filter, obtained at the eight selected FAOA sets (sets I to VIII) over the wavelength range from 1548 to 1552 nm. In this spectrum calculation, the length \(L\) and birefringence \(B\) of each HBF (HBF 1 or HBF 2) were set as 7.2 m and 4.166 \(\times\) 10\({}^{-4}\) to make \(\Delta\lambda\) of the narrowband spectrum become \(\sim\)0.8 nm at 1550 nm, respectively. As can be checked from the figure, the narrowband spectrum moves towards a longer wavelength region as the FAOA set changes from set I to set VIII. If one peak wavelength of the comb spectrum at set I is denoted by \(\lambda_{0}\) (=1550.4 nm), as shown in the left-top subplot of Figure 3, \(\lambda_{0}\) increases from 1550.5 to 1551.1 nm by 0.1 nm per set, while the FAOA set switches from set II to set VIII. In particular, it was also confirmed through additional spectral calculations that \(\lambda_{0}\) linearly increased with \(\psi\) for finer values of \(\psi\) (for instance, 360 values of \(\psi\) starting from 0\({}^{\circ}\) with a step of 1\({}^{\circ}\)), although the calculation results were not provided here. This implies that the narrowband comb spectrum can be continuously tuned in its wavelength. As a result, this calculated result corroborates that our comb filter can be continuously frequency-tuned within \(\Delta\lambda\) by properly selecting FAOA sets. For experimental demonstration of the calculated results, an actual implementation of our filter was done by incorporating a PBS (OZ Optics) pigtailed with single-mode fiber (SMF), two SMF-pigtailed QWR's (OZ Optics), two SMF-pigtailed HWR's (OZ Optics), and two equal-length bow-tie HBF segments (Fibercore), as shown in Figure 1. Considering the birefringence (\(\sim\)4.166 \(\times\) 10\({}^{-4}\)) of the HBF, its length was tailored to be \(\sim\)7.12 m so that \(\Delta\lambda\) became \(\sim\)0.8 nm at 1550 nm. The transmission spectra of the constructed filter were measured using a broadband light source (Fiberlabs FL7701) and an optical spectrum analyzer (Yokogawa AQ6370C). Figure 4 shows an actual experimental setup for measurement of the transmission spectra of the filter. As shown in Figure 4, the input and output ports (ports 1 and 4) of the filter were connected to the broadband light source and optical spectrum analyzer, respectively, with FC/PC type fiber patchcords. The sensitivity and resolution bandwidth of the optical spectrum analyzer were set as HIGH1 and 0.02 nm, respectively, to acquire high resolution and high contrast optical spectra. To avoid unwanted displacements of all the optical components of the filter, we taped them up on the optical table so that they were immobilized during the spectrum measurement. Figure 4: Actual experimental setup for measurement of transmission spectra of the constructed filter. Figure 3: Calculated narrowband transmission spectra of the proposed filter, obtained at the eight selected four AOA (FAOA) sets (sets I to VIII) shown in Figure 2 over the wavelength range from 1548 to 1552 nm. Figure 5 shows the narrowband transmission spectra, measured at the eight FAOA sets (sets I to VIII) in Table 1. The resolution bandwidth of the optical spectrum analyzer was 0.05 nm during the measurement. For the eight comb spectra, the average insertion loss was measured to be \(\sim\)5.88 dB and primarily attributed to the insertion losses of the PBS and wave retarders and the fiber splicing losses between different types of fibers (e.g., HBF and SMF). When we sequentially switched the FAOA set from set I to set VIII, the narrowband transmission spectrum moved towards a longer wavelength region by \(\sim\)0.1 nm, leading to an entire wavelength shift of \(\sim\)0.7 nm. Figure 6a shows the variation of the peak wavelength denoted by \(\lambda_{0}\) in Figure 5, measured at the eight FAOA sets. A skyblue solid line indicates a linear fit of the measured data of \(\lambda_{0}\).