## 4 WOA's for Phase Modulation of GTF and Its Phase-Modulated Spectra
Next, the loci of (\(\theta_{Q1}\), \(\theta_{H1}\)) and (\(\theta_{Q1}\), \(\theta_{Q2}\)), displayed by blueish squares and reddish circles, are plotted with \(\theta_{Q1}\)(\(\phi\)), \(\theta_{H1}\)(\(\phi\)), and \(\theta_{Q2}\)(\(\phi\)) in Figure 4a at \(\phi\) ranges of 0\({}^{\circ}\)-180\({}^{\circ}\) and 0\({}^{\circ}\)-360\({}^{\circ}\), respectively, as shown in Figure 4c. In terms of the locus of (\(\theta_{Q1}\), \(\theta_{H1}\)), the point of (\(\theta_{Q1}\), \(\theta_{H1}\)) makes a round along this elliptical locus from (105\({}^{\circ}\), 154\({}^{\circ}\)) in the CW direction, while \(\phi\) increases from 0\({}^{\circ}\) to 180\({}^{\circ}\). It can be seen that the right side of this elliptical locus is much denser than its left side, which suggests that an equal displacement of the point (\(\theta_{Q1}\), \(\theta_{H1}\)) on the locus results in a relatively greater change in \(\phi\) at the right side. Even if the (\(\theta_{Q1}\), \(\theta_{H1}\)) locus is not represented using analytic mathematical expressions as suggested in the previous work [22], the desirable phase \(\phi\), i.e., the desirable transmission dip (or peak) wavelength, can be set and be continuously changed by selecting and shifting (\(\theta_{Q1}\), \(\theta_{H1}\)) on this locus, respectively, while letting \(\theta_{Q2}\) and \(\theta_{H2}\) follow the WOA curves of \(\theta_{Q2}\)(\(\phi\)) and \(\theta_{H2}\)(\(\phi\)) in Figure 4a, respectively. In terms of the locus of (\(\theta_{Q1}\), \(\theta_{Q2}\)), the point of (\(\theta_{Q1}\), \(\theta_{Q2}\)) makes a closed parabolic trace for \(\phi\) increasing from 0\({}^{\circ}\) to 360\({}^{\circ}\). Like the (\(\theta_{H2}\), \(\theta_{H1}\)) and (\(\theta_{H2}\), \(\theta_{Q1}\)) loci in Figure 4b, this (\(\theta_{Q1}\), \(\theta_{Q2}\)) locus also has a two-fold symmetry with respect to the \(\theta_{Q1}\) axis because \(\theta_{Q2}\)(\(\phi\)) has a period two times longer than \(\theta_{Q1}\)(\(\phi\)). Moreover, the density difference on the locus is analogous to that of the (\(\theta_{Q1}\), \(\theta_{H1}\)) locus. Figure 4d shows the loci of (\(\theta_{Q2}\), \(\theta_{H1}\)) and (\(\theta_{Q2}\), \(\theta_{H2}\)), displayed by blueish squares and reddish circles, respectively. \(\theta_{Q2}\)(\(\phi\)), \(\theta_{H1}\)(\(\phi\)), and \(\theta_{H2}\)(\(\phi\)) in Figure 4a are utilized to plot these loci at a \(\phi\) range of 0\({}^{\circ}\)-360\({}^{\circ}\). (\(\theta_{Q2}\), \(\theta_{H1}\)) forms a locus shaped like inverted [PERSON]'s eyes with increasing \(\phi\) (from 0\({}^{\circ}\) to 360\({}^{\circ}\)), which also has a two-fold symmetry with respect to the \(\theta_{H1}\) axis owing to the period of \(\theta_{Q2}\)(\(\phi\)) two times longer than that of \(\theta_{H1}\)(\(\phi\)). It can also be found that the lower outer part of the (\(\theta_{Q2}\), \(\theta_{H1}\)) locus is sparser than its other parts. The locus of (\(\theta_{Q2}\), \(\theta_{H2}\)) forms a parallelogram-shaped trace, which makes one CW revolution along the locus, starting at (17\({}^{\circ}\), 2\({}^{\circ}\)), while \(\phi\) varies from 0\({}^{\circ}\) to 360\({}^{\circ}\). The left and right sides of the (\(\theta_{Q2}\), \(\theta_{H2}\)) locus are slightly sparser than its other parts. This indicates that an identical displacement of the point (\(\theta_{Q2}\), \(\theta_{H2}\)) along the locus causes a relatively smaller modification of \(\phi\) at these parts of the locus. In sum, it is confirmed from Figure 4 that one can always find a WOA set (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), \(that the phase of \(t_{GTF2}\) can be continuously modulated, also indicating that \(t_{GTF2}\) can be continuously frequency-tuned. As long as an initial WOA set where any GTF is obtained is given, one can always find a WOA set inducing an arbitrary phase in this GTF.