## 2 Principle of Operation
Figure 2a shows the AOA sets of four wave retarders (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{H2}\), \(\theta_{Q2}\)), referred hereafter to as four AOA (FAOA) sets, indicated by skyblue squares, blue circles, olive triangles, and green inverted triangles, respectively, for the extra phase difference \(\psi\) (from \(0^{\circ}\) to \(360^{\circ}\) with a step of \(1^{\circ}\)), which are found for the continuous wavelength tuning of \(t_{narrow}\) in (5) at \(\theta_{B1}=0^{\circ}\). As \(\psi\) increases, \(\theta_{Q1}\) and \(\theta_{H1}\) alternate with \(\psi\) at the same frequency, and they are bounded in \(-22.5^{\circ}<\theta_{Q1}<22.5^{\circ}\) and \(28.6^{\circ}<\theta_{H1}<61.4^{\circ}\). \(\theta_{Q1}(\psi)\) and \(\theta_{H1}(\psi)\) seem to have similar alternating trajectories, but they are not sinusoidal functions of \(\psi\). On the other hand, \(\theta_{H2}\) linearly increases with \(\psi\) in an AOA range of \(33.4^{\circ}<\theta_{H2}<123.5^{\circ}\), and \(\theta_{Q2}\) is not a function of \(\psi\) but a constant value of \(67.5^{\circ}\). From Figure 2a, we can clearly see that an FAOA set (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{H2}\), \(\theta_{Q2}\)) corresponds to any \(\psi\) increasing from \(0^{\circ}\) to \(360^{\circ}\) with a step of \(1^{\circ}\). This means that the narrowband transmittance \(t_{narrow}\) can be wavelength-tuned if an FAOA set is picked along these four trajectories of \(\theta_{Q1}(\psi)\), \(\theta_{H1}(\psi)\), \(\theta_{H2}(\psi)\), and \(\theta_{Q2}(\psi)\) so that \(\psi\) increases from \(0^{\circ}\) to \(360^{\circ}\). Moreover, it can effortlessly be confirmed from quantitative calculations that the same locus is obtained for each trajectory shown in Figure 2a, even for \(\psi\) with \(360\) or more steps split at an angle smaller than \(1^{\circ}\), which proves once again the continuous frequency tunability of \(t_{narrow}\) in our filter. In the case of \(\theta_{B1}\
eq 0^{\circ}\), \(\theta_{H1}\) and \(\theta_{H2}\) are simultaneously increased by \(\theta_{B1}/2\) for all \(\psi\)Figure 2b,c show four types of loci of (\(\theta_{Q1}\), \(\theta_{H1}\)) and (\(\theta_{H2}\), \(\theta_{Q2}\)), which are plotted for \(\psi\) (from 0\({}^{\circ}\) to 360\({}^{\circ}\)) at \(\theta_{B1}\) = 0\({}^{\circ}\) in the Cartesian coordinate systems of (\(\theta_{Q1}\), \(\theta_{H1}\)) and (\(\theta_{H2}\), \(\theta_{Q2}\)), respectively. There are four types of FAOA sets that can implement the continuous frequency tuning of the narrowband spectrum, and the AOA sets represented in Figure 2a show one of them. In Figure 2b, four subplots displayed as Types I to IV, that is, the four loci of (\(\theta_{Q1}\), \(\theta_{H1}\)), show Lissajous trajectories plotted using both \(\theta_{Q1}\)(\(\psi\)) and \(\theta_{H1}\)(\(\psi\)) in these four types of FAOA sets. Similarly, in Figure 2c, four subplots denoted by Types I to IV, i.e., the four loci of (\(\theta_{H2}\), \(\theta_{Q2}\)), are obtained using both \(\theta_{H2}\)(\(\psi\)) and \(\theta_{Q2}\)(\(\psi\)) in the four types of FAOA sets. In Figure 2b,c, the locus plots of Types I to IV are indicated by blue circles, green squares, red triangles, and violet inverted triangles, respectively. Among the four types of loci, two Type I loci in Figure 2b,c are drawn out from \(\theta_{Q1}\)(\(\psi\)) and \(\theta_{H1}\)(\(\psi\)) traces and \(\theta_{H2}\)(\(\psi\)) and \(\theta_{Q2}\)(\(\psi\)) traces in Figure 2a, respectively. In terms of Types I and II loci in Figure 2b, the point of (\(\theta_{Q1}\), \(\theta_{H1}\)) on the elliptical locus moves CCW along the locus, starting at a void circle and a void square, respectively, while \(\psi\) increases from 0\({}^{\circ}\) to 360\({}^{\circ}\). As opposed to the former case, in the case of Types III and IV loci in Figure 2b, the point of (\(\theta_{Q1}\), \(\theta_{H1}\)) moves CW along the locus, starting at a void triangle and a void inverted triangle, respectively, with increasing \(\psi\) from 0\({}^{\circ}\) to 360\({}^{\circ}\).