## 4 WOA's for Phase Modulation of GTF and Its Phase-Modulated Spectra  
Here, we explore WOA sets for continuous phase modulation of another GTF \(t_{GTF2}\), which is defined at another different (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), \(\theta_{H2}\)) = (105\({}^{\circ}\), 154\({}^{\circ}\), 17\({}^{\circ}\), 2\({}^{\circ}\)) with \(\theta_{P1}\) = 0\({}^{\circ}\) and \(\theta_{P2}\) = 22.5\({}^{\circ}\), by using our reverse tracing method. Specifically, 360 sets of (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), \(\theta_{H2}\)), where an additional phase shift of 1\({}^{\circ}\)-360\({}^{\circ}\) (increment: 1\({}^{\circ}\)) can be induced in \(t_{GTF2}\), were found with our method. Figure 4a shows the 360 sets of (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), \(\theta_{H2}\)) as functions of \(\phi\) ranging from 0\({}^{\circ}\) to 360\({}^{\circ}\) (step: 1\({}^{\circ}\)), displayed by blue circles (\(\theta_{Q1}\)), green squares (\(\theta_{H1}\)), red diamonds (\(\theta_{Q2}\)), and violet triangles (\(\theta_{H2}\)), respectively, which are found at fixed values of \(\theta_{P1}\) = 0\({}^{\circ}\) and \(\theta_{P2}\) = 22.5\({}^{\circ}\). In contrast with WOA sets for continuous wavelength tuning of an analytic transmittance \(t_{n}\)[23, 24], all four WOA's (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), and \(\theta_{H2}\)) do not take the form of simple mathematical functions of \(\phi\) and have similar patterns either. Whereas \(\theta_{Q1}\)(\(\phi\)) and \(\theta_{H1}\)(\(\phi\)) have a period of 180\({}^{\circ}\), the period of \(\theta_{Q2}\)(\(\phi\)) or \(\theta_{H2}\)(\(\phi\)) is 360\({}^{\circ}\). \(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), and \(\theta_{H2}\) are bounded in the WOA ranges of 63.5\({}^{\circ}\)\(<\)\(\theta_{Q1}\)\(<\) 116.6\({}^{\circ}\), 115.4\({}^{\circ}\)\(<\)\(\theta_{H1}\)\(<\) 154.6\({}^{\circ}\), -37.2\({}^{\circ}\)\(<\)\(\theta_{Q2}\)\(<\) 37.2\({}^{\circ}\), and -18.1\({}^{\circ}\)\(<\)\(\theta_{H2}\)\(<\) 40.6\({}^{\circ}\), respectively. In order to reveal implicit information on the 360 sets of (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{Q2}\), \(\theta_{H2}\)) in Figure 4a, we investigated the relationship between any two of the four WOA's. Figure 4b shows the loci of (\(\theta_{H2}\), \(\theta_{H1}\)) and (\(\theta_{H2}\), \(\theta_{Q1}\)), displayed by blueish squares and reddish circles, respectively. These loci were drawn using \(\theta_{H2}\)(\(\phi\)), \(\theta_{H1}\)(\(\phi\)), and \(\theta_{Q1}\)(\(\phi\)) in Figure 4a in the Cartesian coordinate system at a range of \(\phi\) from 0\({}^{\circ}\) to 360\({}^{\circ}\). As \(\phi\) increases from 0\({}^{\circ}\) to 360\({}^{\circ}\), (\(\theta_{H2}\), \(\theta_{H1}\)) or (\(\theta_{H2}\), \(\theta_{Q1}\)) forms a closed locus, whose shape is similar to a squeezed goggle or an infinity symbol, respectively. Because the period of \(\theta_{H2}\)(\(\phi\)) is two times longer than that of \(\theta_{H1}\)(\(\phi\)) or \(\theta_{Q1}\)(\(\phi\)), both loci have two-fold symmetry. It is readily seen that the density of the point (\(\theta_{H2}\), \(\theta_{H1}\)) or (\(\theta_{H2}\), \(\theta_{Q1}\)), the interval between two locus points, is different along the locus. In terms of the locus of (\(\theta_{H2}\), \(\theta_{H1}\)), the lower inner part of the locus is sparser than its other parts. This implies that the \(\phi\) variation is smaller at the lower inner part of the locus for the same amounts of changes in \(\theta_{H2}\) and \(\theta_{H1}\). Similarly, in terms of the locus of (\(\theta_{H2}\), \(\theta_{Q1}\)), the upper part of the locus is denser than its lower part, which means that the \(\phi\) variation is larger at the upper part of the locus for the same amounts of changes in \(\theta_{H2}\) and \(\theta_{Q1}\).