## 2 Principle of Operation
In particular, the elliptical locus of Type I is determined by the following relations (6) and (7), which can be derived by considering the \(\psi\)-driven evolution of the output SOP of HWR 1 on the Poincare sphere [22]. Figure 2: (**a**) Azimuth orientation angle (AOA) sets of four wave retarders (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{H2}\), \(\theta_{Q2}\)), 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 \(t_{narrow}\) at \(\theta_{B1}\) = 0\({}^{\circ}\). Four types of loci of (**b**) (\(\theta_{Q1}\), \(\theta_{H1}\)) and (**c**) (\(\theta_{H2}\), \(\theta_{Q2}\)), 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, and (**d**) four types of loci of (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{H2}\)), which are plotted for \(\psi\) (from 0\({}^{\circ}\) to 360\({}^{\circ}\)) at \(\theta_{B1}\) = 0\({}^{\circ}\) in the Cartesian coordinate system of (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{H2}\)). \[\tan\!\left(4\theta_{H1}-2\theta_{Q1}\right)=\frac{\sin 2\theta_{B1}-\cos 2 \theta_{B1}\sin\psi}{\cos 2\theta_{B1}+\sin 2\theta_{B1}\sin\psi} \tag{6}\]  
and  
\[\sin 2\theta_{Q1}=-\frac{1}{\sqrt{2}}\cos\psi \tag{7}\]  
These simple and deterministic locus patterns pave the way for easy prediction of FAOA sets for the continuous wavelength tuning of narrowband spectra. For Types I and III loci in Figure 2c, the point of AOA's (\(\theta_{H2}\), \(\theta_{Q2}\)) on the linear locus goes from left to right with \(\theta_{Q2}\) maintained as 67.5\({}^{\circ}\), as \(\psi\) increases from 0\({}^{\circ}\) to 360\({}^{\circ}\). On the contrary, for Types II and IV loci in Figure 2c, the point of (\(\theta_{H2}\), \(\theta_{Q2}\)) on the locus shifts from right to left keeping \(\theta_{Q2}=157.5^{\circ}\) with the increase of \(\psi\) from 0\({}^{\circ}\) to 360\({}^{\circ}\). To grasp the total loci at a glance, as shown in Figure 2d, we plotted the four loci of (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{H2}\)) with respect to \(\psi\) (from 0\({}^{\circ}\) to 360\({}^{\circ}\)) at \(\theta_{B1}=0^{\circ}\) in the Cartesian coordinate system of \(\theta_{Q1}\), \(\theta_{H1}\), and \(\theta_{H2}\), using \(\theta_{Q1}(\psi)\), \(\theta_{H1}(\psi)\), and \(\theta_{H2}(\psi)\) shown in Figure 2a. To maintain consistency, the locus plots of Types I to IV are indicated by blue circles, green squares, red triangles, and violet inverted triangles, respectively, as in Figure 2b,c. In terms of Types I and III loci (\(\theta_{Q2}=67.5^{\circ}\)), the point of AOA's (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{H2}\)) on these helical loci rises from the bottom with the increase of \(\psi\) (from bright to dark markers). In contrast, for Types II and IV loci (\(\theta_{Q2}=157.5^{\circ}\)), the point of (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{H2}\)) on the loci comes down from the top with increasing \(\psi\). For each type of locus, eight points indicated by I to VIII show the eight sets of FAOA's selected to spectrally shift \(t_{narrow}\). From set I to set VIII, an additional phase difference \(\psi\) increases from 0\({}^{\circ}\) to 315\({}^{\circ}\) with a step of 45\({}^{\circ}\). In other words, if we put the FSR of the narrowband transmission spectrum as \(\Delta\lambda\), the narrowband transmission spectrum at set I is red-shifted by \(\Delta\lambda\)/8, \(\Delta\lambda\)/4, \(3\Delta\lambda\)/8, \(\Delta\lambda\)/2, \(5\Delta\lambda\)/8, \(3\Delta\lambda\)/4, and 7\(\Delta\lambda\)/8 at sets II to VIII, respectively.