Arbitrary Phase Modulation of General Transmittance Function of First-Order Optical Comb Filter with Ordered Sets of Quarter- and Half-Wave Plates

: Here we theoretically and experimentally demonstrated the arbitrary phase modulation of a general transmittance function (GTF) of the ﬁrst-order optical comb ﬁlter based on a polarization-diversity loop structure, which employed two ordered waveplate sets (OWS’s) of a quarter-wave plate (QWP) and a half-wave plate (HWP). The proposed comb ﬁlter is composed of a polarization beam splitter (PBS), two equal-length polarization-maintaining ﬁber (PMF) segments, and two OWS’s of a QWP and an HWP with each set located before each PMF segment. The second PMF segment is butt-coupled to one port of the PBS so that its principal axis should be 22.5 ◦ away from the horizontal axis of the PBS. First, we explained a scheme to ﬁnd four waveplate orientation angles (WOA’s) allowing the phase of a GTF to be arbitrarily modulated, using the way each component of the ﬁlter, such as a waveplate or PMF segment, a ﬀ ects its input or output polarization. Then, with the WOA ﬁnding method, we derived WOA sets of the four waveplates, which could give arbitrary phase retardations φ ’s from 0 ◦ to 360 ◦ to a GTF chosen here arbitrarily. Finally, we showed phase-modulated GTF’s calculated at eight selected WOA sets allowing φ ’s to be 0 ◦ , 45 ◦ , 90 ◦ , 135 ◦ , 180 ◦ , 225 ◦ , 270 ◦ , and 315 ◦ , and then the predicted results were veriﬁed by experimentally measured results. It is concluded from the theoretical and experimental demonstrations that the GTF of our ﬁlter based on the OWS of a QWP and an HWP can be arbitrarily phase-modulated by properly controlling the WOA’s of the four waveplates. we showed phase-modulated GTF’s eight 0 ◦ , ◦ , ◦ , ◦ , ◦ , ◦ and 315 ◦ the theoretically predicted results experimentally veriﬁed. is from the theoretical and experimental demonstrations that the GTF of our ﬁlter based on the OWS of a QWP and an HWP can be arbitrarily phase-modulated by appropriately controlling the WOA’s of the four waveplates. Our demonstration is expected to be to ﬁber-optic applications, demand frequency-tunable comb ﬁlters as photonic signal and multiwavelength lasing.


Introduction
To date, optical comb filters have provided versatile spectrum controllability in the fields of interrogation of optical fiber sensors, wavelength routing in optical communications, and photonic processing of microwave signals [1][2][3][4][5]. In these comb filters, the capability of continuous wavelength tuning is a crucial function to pass desired spectral components or reject unwanted ones in optical signal processing. By utilizing a Mach-Zehnder interferometer structure [6,7], Sagnac birefringence loop structure [8,9], Lyot-type birefringence loop structure [10,11], and polarization-diversified loop structure (PDLS) [12][13][14][15][16], much effort has been exerted in implementing this continuous tunability of the absolute wavelength in comb filters. Among these filter configurations, optical comb filters based on a PDLS have superior flexibility in the wavelength switching or tuning of their output spectra [12][13][14][15][16][17][18][19][20]. Since 2017, successive works have been done to realize the continuous wavelength tuning of periodic transmission spectra in PDLS-based fiber comb filters [21][22][23][24][25]. A continuously Figure 1a shows a schematic diagram of the proposed comb filter comprised of a PBS, two PMF segments whose lengths are equal (denoted by PMF 1 and PMF 2), two QWPs (denoted by QWP 1 and QWP 2), and two HWPs (denoted by HWP 1 and HWP 2). As shown in Figure 1a, one OWS of QWP 1 and HWP 1 is placed ahead of PMF 1, and the other OWS of QWP 2 and HWP 2 is located Appl. Sci. 2020, 10, 5434 3 of 15 between PMF 1 and PMF 2. Every OWS can effectively change the phase delay difference between two principal modes of the individual PMF segment, and the combination of QWP 2 and HWP 2, or the second OWS, has an additional function to effectively modify the relative angular difference between the principal axes of PMF 1 and PMF 2. The slow axis of PMF 2 butt-coupled to port 3 of the PBS is 22.5 • away from the TM polarization axis (denoted by x axis) of the PBS. An input beam introduced into port IN of the filter is decomposed into two orthogonal linear polarization components, such as linear horizontal polarization (LHP) and linear vertical polarization (LVP), which circulate through the polarization-diversified loop of the filter in the clockwise (CW) and counterclockwise (CCW) directions, respectively. Figure 1b shows the two propagation paths of light travelling through the filter, designated as CW and CCW paths. When the SOP in of the filter is LHP, input light propagates along the CW path and sequentially passes through a linear horizontal polarizer (x axis), QWP 1 (with its slow axis oriented at θ Q1 for the x axis), HWP 1 (oriented at θ H1 ), PMF 1 (oriented at θ P1 ), QWP 2 (oriented at θ Q2 ), HWP 2 (oriented at θ H2 ), PMF 2 (oriented at θ P2 = 22.5 • ), and a linear horizontal analyzer (x axis). In the case of the SOP in of LVP, input light propagates through a linear vertical polarizer (y axis), PMF 2 (−θ P2 oriented), HWP 2 (−θ H2 oriented), QWP 2 (−θ Q2 oriented), PMF 1 (−θ P1 oriented), HWP 1 (−θ H1 oriented), QWP 1 (−θ Q1 oriented), and a linear vertical analyzer (y axis) in turn, circulating along the CCW path. F and S displayed on the waveplates and the PMFs indicate their fast and slow axes, respectively.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 14 QWP 2), and two HWPs (denoted by HWP 1 and HWP 2). As shown in Figure 1a, one OWS of QWP 1 and HWP 1 is placed ahead of PMF 1, and the other OWS of QWP 2 and HWP 2 is located between PMF 1 and PMF 2. Every OWS can effectively change the phase delay difference between two principal modes of the individual PMF segment, and the combination of QWP 2 and HWP 2, or the second OWS, has an additional function to effectively modify the relative angular difference between the principal axes of PMF 1 and PMF 2. The slow axis of PMF 2 butt-coupled to port 3 of the PBS is 22.5° away from the transverse-magnetic polarization axis (denoted by x axis) of the PBS. An input beam introduced into port IN of the filter is decomposed into two orthogonal linear polarization components, such as linear horizontal polarization (LHP) and linear vertical polarization (LVP), which circulate through the polarization-diversified loop of the filter in the clockwise (CW) and counterclockwise (CCW) directions, respectively. Figure 1b shows the two propagation paths of light travelling through the filter, designated as CW and CCW paths. When the SOPin of the filter is LHP, input light propagates along the CW path and sequentially passes through a linear horizontal polarizer (x axis), QWP 1 (with its slow axis oriented at θQ1 for the x axis), HWP 1 (oriented at θH1), PMF 1 (oriented at θP1), QWP 2 (oriented at θQ2), HWP 2 (oriented at θH2), PMF 2 (oriented at θP2 = 22.5°), and a linear horizontal analyzer (x axis). In the case of the SOPin of LVP, input light propagates through a linear vertical polarizer (y axis), PMF 2 (−θP2 oriented), HWP 2 (−θH2 oriented), QWP 2 (−θQ2 oriented), PMF 1 (−θP1 oriented), HWP 1 (−θH1 oriented), QWP 1 (−θQ1 oriented), and a linear vertical analyzer (y axis) in turn, circulating along the CCW path. F and S displayed on the waveplates and the PMFs indicate their fast and slow axes, respectively. Polarization interference is created by optical birefringence in an optical structure composed of two linear polarizers and BEs inserted between them. In the case of one BE like a PMF segment, the transmittance function of this optical structure becomes a sinusoidally varying function of Γ, the phase delay difference between two principal modes of the BE, which is represented by 2πBL/λ, where B, L, and λ are the birefringence of the BE, the length of the BE, and the free space wavelength, respectively. It is possible to tune the wavelength of this polarization interference spectrum by modifying Γ, and the modification of Γ can be done by adding a supplementary phase delay difference ϕ to Γ and modulating this ϕ [21]. In the same way, to tune the wavelength position of the first-order comb spectrum generated with two BEs, or two PMF segments [26], ϕ, which is added to Γ for individual PMF, should be modulated equally for both PMF 1 and PMF 2. This simultaneous ϕ modulation can be accomplished by changing the SOPin's of PMF 1 and PMF 2 with two OWS's. When we increase ϕ from 0° to 360° by appropriately adjusting the SOPin of each PMF, the comb spectrum makes a red shift by an interference period, i.e., a free spectral range (FSR). The filter transmittance function tfilter can be derived from the Jones transfer matrix [27] T given by (1) and is represented as (2). In this derivation, no insertion losses are assumed to exist in any optical components of the filter, and all waveplates are regarded as wavelength-independent.  P  HWP  H  QWP  Q  PMF  P  HWP  H  QWP  Q   T  T  T  T  T  T  Polarization interference is created by optical birefringence in an optical structure composed of two linear polarizers and BEs inserted between them. In the case of one BE like a PMF segment, the transmittance function of this optical structure becomes a sinusoidally varying function of Γ, the phase delay difference between two principal modes of the BE, which is represented by 2πBL/λ, where B, L, and λ are the birefringence of the BE, the length of the BE, and the free space wavelength, respectively. It is possible to tune the wavelength of this polarization interference spectrum by modifying Γ, and the modification of Γ can be done by adding a supplementary phase delay difference φ to Γ and modulating this φ [21]. In the same way, to tune the wavelength position of the first-order comb spectrum generated with two BEs, or two PMF segments [26], φ, which is added to Γ for individual PMF, should be modulated equally for both PMF 1 and PMF 2. This simultaneous φ modulation can be accomplished by changing the SOP in 's of PMF 1 and PMF 2 with two OWS's. When we increase φ from 0 • to 360 • by appropriately adjusting the SOP in of each PMF, the comb spectrum makes a redshift by an interference period, i.e., a free spectral range (FSR). The filter transmittance function t filter can be derived from the Jones transfer matrix [27] T given by (1) and is represented as (2). In this derivation, no insertion losses are assumed to exist in any optical components of the filter, and all waveplates are regarded as wavelength-independent.
Before we begin to address the principle of the phase modulation of a GTF, let us consider the previous conclusions on the relationship between SOP changes and the continuous wavelength tuning of a narrowband transmittance t n given by (3) in the comb filter composed of two PMF segments, two OWS's of an HWP and a QWP, and the third HWP. [23,24].
The additional phase delay difference φ in (3) defines the absolute wavelength position of the narrowband transmission spectrum. In the previous studies [23,24], eight WOA sets (from Set I n to Set VIII n ) of four waveplates (except for the third HWP) were utilized to display eight wavelength-tuned spectra calculated from t n with eight φ values from 0 • to 315 • (increment: 45 • ). When the wavelength of input light increases from λ 0 to λ 0 + ∆λ, where ∆λ implies the FSR, the SOP in of PMF 2 at Set I n (φ = 0 • ) spectrally evolves on the Poincare sphere, as shown in Figure 2a. This spectral evolution indicated by pink circles is denoted by SE in . With increasing wavelength, the SOP in moves along the SE in trace rotating CW around the normal vector p 1 , when p 1 points the observer. The rotation direction of the SOP in can also be thought using a left-hand rule. If the direction of p 1 is the same as that of the left thumb, the direction of the rest of the fingers is the rotation direction of the SOP in with the increase of the wavelength. In general, the SOP out of the birefringent medium rotates around its slow axis on the Poincare sphere according to the wavelength λ of light passing through the medium. The vector AB connecting A (2ε = 0 • , 2ψ = −135 • ) and B (2ε = 0 • , 2ψ = 45 • ), which lie on the Poincare sphere, points the direction of the slow axis of PMF 2 (θ P2 = 22.5 • ) and is denoted by the vector p 2 . Here, 2ε (−90 • ≤ 2ε ≤ 90 • ) and 2ψ (−180 • ≤ 2ψ ≤ 180 • ) are the latitude and longitude of the Poincare sphere, respectively. For convenience, the plane whose normal vector is p 1 , that is, the plane of the SE in trace, is designated as the SOP in plane, and another plane with its normal vector of p 2 is designated as the SOP out plane. Actually, p 2 is an axis of revolution, around which the SE in trace moves as φ varies. Hence, p 1 and the SOP out plane are parallel, and p 2 and the SOP in plane are parallel, ending up with p 1 and p 2 being perpendicular. At Set II n (φ = 45 • ), the SE in trace shown in Figure 2a rotates CCW by 45 • around both p 1 and p 2 , and thus p 1 shown in Figure 2a rotates 45 • CCW about p 2 as well. Analogously, for the remaining six sets (III n -VIII n ), the SE in trace at Set I n rotates CCW by 90 • -315 • (with an increment of 45 • ) around both p 1 and p 2 . the position of O(λ0) along the SEout trace. On the SEout trace at Set IIn, therefore, OII(λ0 + ∆λ/8) is located at C′, and λ0 + ∆λ/8 becomes λpeak because C′ is nearest to LHP. This implies that the narrowband transmission spectrum moves towards a longer wavelength region by ∆λ/8, that is, ϕ is modulated by 45° making tn redshift by (45°/360°)∆λ = ∆λ/8. Similarly, for remaining Sets IIIn-VIIIn, ϕ is modulated by 90° to 315° with an increment of 45° making tn redshift by ∆λ/4 to 7∆λ/8 with a step of ∆λ/8, and λpeak changes from λ0 + ∆λ/4 to λ0 + 7∆λ/8 with an increment of ∆λ/8. In the case of a GTF, or an arbitrary transmittance function, obtained in the proposed filter structure, the SOPout trace of PMF 2, designated as SEout,GTF trace, is presumed to be a completely different trajectory in comparison with the SEout trace shown in Figure 2b. However, it can be deduced from the aforementioned conclusions of the previous works that a location change of O(λ0) along the SEout,GTF trace, accompanied by a shift in λpeak, is responsible for a phase modulation in the GTF, resulting in a wavelength shift in the transmission spectrum. For a given SEout,GTF trace, the corresponding SOPin trace of PMF 2, designated as SEin,GTF trace, should be a circle on the Poincare sphere as well, since a wavelength-dependent BE through which light has passed is just one, PMF 1. However, in terms of the radius and the normal vector (p1) of the circular trace, this SEin,GTF trace will be quite different from the SEin trace shown in Figure 2a. Because it is assumed that all waveplates used here are independent of wavelength, they do not distort the shape of the SOPin or the SOPout Likewise, the SOP out 's of PMF 2 at Sets I n (φ = 0 • ) and II n (φ = 45 • ) spectrally evolve on the Poincare sphere over the same wavelength range of λ 0 to λ 0 + ∆λ, as shown in Figure 2b. This spectral evolution also indicated by pink circles is designated as SE out . This SE out trace is embodied by making a CW rotation of the wavelength-dependent SOP in of PMF 2 about p 2 by an angle ϕ(λ) given by 360 • (λ − λ 0 )/∆λ. When PMF 2 is passed through by light at λ 0 , the SOP in of PMF 2 at λ 0 , C (2ε = 0 • , 2ψ = 0 • ) shown in Figure 2a, rotates CW around p 2 by ϕ(λ 0 ) = 0 • , and the SOP out of PMF 2 at λ 0 becomes C (2ε = 0 • , 2ψ = 0 • ) shown in Figure 2b without a change in the position on the Poincare sphere. Similarly, the SOP in of PMF 2 at λ 0 + ∆λ/4 rotates CW around p 2 by ϕ(λ 0 + ∆λ/4) = 90 • during its passage through PMF 2. If we put the SOP out of PMF 2 at a certain wavelength λ as O(λ), O(λ 0 ) at Set I n , marked by O I (λ 0 ) in the figure, becomes C . Owing to the wavelength-dependent transformation of the SOP in of PMF 2, O(λ) forms a non-circular trajectory SE out shaped like a droplet as λ increases from λ 0 to λ 0 + ∆λ. During the wavelength increase of ∆λ starting from λ 0 , O(λ) goes round this entire SE out trace starting from C in the CW direction around the S 2 axis. If we assume the wavelength at which t n is maximized as λ peak , λ peak can be regarded as λ 0 at Set I n because t n reaches the maximum when O(λ) is nearest to LHP (2ε = 0 • , 2ψ = 0 • ) on the Poincare sphere. At Set II n , O II (λ 0 ), i.e., the SOP out of PMF 2 at λ 0 , becomes C distant from O I (λ 0 ) by an angular displacement that corresponds to ∆λ/8 and also goes round the entire SE out trace starting from C about the S 2 axis in the CW direction, as the wavelength increases from λ 0 to λ 0 + ∆λ. Here, O I (λ 0 ) or O II (λ 0 ) is the initial point of the spectral evolution of O(λ). The position difference between O I (λ 0 ) and O II (λ 0 ) originates from the fact that, while the WOA set goes from Set I n to Set II n , the SE in trace at Set I n is rotated CCW by 45 • not only about p 1 on the SOP in plane but also about p 2 on the SOP out plane. This means that simultaneously rotating the SE in trace CCW about both p 1 and p 2 can change the position of O(λ 0 ) along the SE out trace. On the SE out trace at Set II n , therefore, O II (λ 0 + ∆λ/8) is located at C , and λ 0 + ∆λ/8 becomes λ peak because C is nearest to LHP. This implies that the narrowband transmission spectrum moves towards a longer wavelength region by ∆λ/8, that is, φ is modulated by 45 • making t n be redshifted by (45 • /360 • )∆λ = ∆λ/8. Similarly, for remaining Sets III n -VIII n , φ is modulated by 90 • to 315 • with an increment of 45 • making t n be redshifted by ∆λ/4 to 7∆λ/8 with a step of ∆λ/8, and λ peak changes from λ 0 + ∆λ/4 to λ 0 + 7∆λ/8 with an increment of ∆λ/8.
In the case of a GTF, or an arbitrary transmittance function, obtained in the proposed filter structure, the SOP out trace of PMF 2, designated as SE out,GTF trace, is presumed to be a completely different trajectory in comparison with the SE out trace shown in Figure 2b. However, it can be deduced from the aforementioned conclusions of the previous works that a location change of O(λ 0 ) along the SE out,GTF trace, accompanied by a shift in λ peak , is responsible for a phase modulation in the GTF, resulting in a wavelength shift in the transmission spectrum. For a given SE out,GTF trace, the corresponding SOP in trace of PMF 2, designated as SE in,GTF trace, should be a circle on the Poincare sphere as well, since a wavelength-dependent BE through which light has passed is just one, PMF 1. However, in terms of the radius and the normal vector (p 1 ) of the circular trace, this SE in,GTF trace will be quite different from the SE in trace shown in Figure 2a. Because it is assumed that all waveplates used here are independent of wavelength, they do not distort the shape of the SOP in or the SOP out trace. Thus, the SOP out trace of PMF 1 also has a circular shape identical to that of the SE in,GTF trace. Like the relationship between the SE in and SE out traces, the increase in λ peak results from the CCW revolution of the SE in,GTF trace about both p 1 on the SOP in plane and p 2 on the SOP out plane. If the SE in,GTF trace is revolved CCW about both p 1 and p 2 by an amount of ξ, the transmission spectrum makes a redshift of (ξ/360 • )∆λ, resulting in an increase of (ξ/360 • )∆λ in λ peak . That is to say, ξ can be regarded as φ, or the additional phase delay difference. Thus, as ξ varies from 0 • to 360 • , the phase of the GTF also changes from 0 • to 360 • , which means that the GTF is able to be phase-modulated continuously by varying ξ. Briefly, one simultaneous CCW rotation of the SE in,GTF trace about both p 1 and p 2 , implemented by controlling four waveplates contained within the proposed filter, enables its transmission spectrum to be redshifted by an FSR (∆λ), in other words, its phase to be modulated by 360 • .

Finding Algorithm of Waveplate Orientation Angles (WOA's) for Phase Modulation of General Transmittance Function (GTF)
Considering the aforementioned polarization conditions required for arbitrarily modulating the phase of a GTF or continuously tuning its wavelength, the WOA's (θ Q1 , θ H1 , θ Q2 , θ H2 ) for this arbitrary phase modulation (i.e., continuous wavelength tuning) are investigated with our unique WOA-finding algorithm. This algorithm is suggested as it is quite sophisticated to draw out an exact mathematical expression of a GTF having an application-specific unique spectrum. Moreover, although the mathematical expression t GTF of a GTF is given, it is cumbersome to solve (θ Q1 , θ H1 , θ Q2 , θ H2 ) from trigonometric simultaneous equations obtained from direct comparison of t GTF with t filter in (2). To explain our angle-finding approach, let us choose a specific GTF t GTF1 , which is procured at a certain selected WOA set (θ Q1 , θ H1 , θ Q2 , θ H2 ) = (118.23 • , 149.56 • , 179.42 • , 123.46 • ), and find (θ Q1 , θ H1 , θ Q2 , θ H2 ) where t GTF1 is wavelength-tuned. At the above selected WOA set indicated by Set I GTF1 and fixed orientation angles of PMF 1 and PMF 2 (θ P1 = 0 • and θ P2 = 22.5 • ), the SOP out trace of PMF 2, or SE out,GTF1 , is obtained over a wavelength range from λ 0 (=1548.0 nm) to λ 0 + ∆λ (=0.8 nm), as shown in Figure 3a. At Set I GTF1 , O(λ 0 ) indicated in Figure 3a is located nearest to LHP on the Poincare sphere, which makes the filter transmittance be maximized at λ 0 . If we put the wavelength at which t GTF1 is maximized as λ peak , O(λ 0 ) becomes O(λ peak ), that is, λ peak = λ 0 = 1548.0 nm at Set I GTF1 . The SOP in trace of PMF 2, or SE in,GTF1 in Figure 3b, can be derived by utilizing the inverse matrix of T PMF2 (T PMF2 −1 ) and the SE out,GTF1 trace in Figure 3a. Based on the foregoing description on the phase modulation of the GTF using the SE in,GTF and SE out,GTF traces, another SE in,GTF1 trace shown in Figure 3c can be obtained by rotating the SE in,GTF1 trace at Set I GTF1 CCW by 45 • around both p 1 and p 2 , for the phase modulation of 45 • (i.e., the wavelength tuning of ∆λ/8 = 0.1 nm) in t GTF1 . It is easily expected that this new SE in,GTF1 trace will be obtained at another WOA set (θ Q1 , θ H1 , θ Q2 , θ H2 ) designated later as Set II GTF1 .
Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 14 coincides with the S1 axis on the Poincare sphere as the orientation angle θP1 of PMF 1 is 0°, as shown in Figure 3e. Then, θH2 and θQ2 can be determined by finding the SOPout trace of PMF 1 to satisfy this constraint for p1. What we should focus on is a way to obtain Set II GTF1 , another WOA set described above. Basically, the SOP out of one component of the filter is equivalent to the SOP in of the following component in any propagation path shown in Figure 1b. As our discussion is confined to the CW path, as an example, the SOP out of PMF 1 is equivalent to the SOP in of QWP 2. The same thing can be said for SOP traces. For example, the SOP in trace of HWP 1 is the same as the SOP out trace of QWP 1. A QWP and an HWP rotate input polarization CCW by 90 • and 180 • about their slow axes on the Poincare sphere, respectively. If an SOP trace, composed of multiple different SOP's distributed on the Poincare sphere, enters a waveplate, therefore, not its shape but its position is altered by the waveplate. For a given SE in,GTF1 trace at Set II GTF1 (in Figure 3c), a certain SOP out trace of QWP 2 can be obtained by using the SE in,GTF1 trace (i.e., the SOP out trace of HWP 2) and the inverse matrix of T HWP2 , or T HWP2 −1 , if θ H2 is fixed as one value (unknown yet). With this SOP out trace of QWP 2, shown in Figure 3d, a corresponding SOP out trace of PMF 1, which is transformed into the SOP out trace of QWP 2 after passing through QWP 2, can also be derived similarly by using the predetermined SOP out trace of QWP 2 and T QWP2 −1 , if θ Q2 is fixed as another value (unknown either), as shown in Figure 3e. Because the SOP rotation due to a QWP or an HWP applies equally to all points comprising an SOP trace, the shape of the SOP trace created by PMF 1 remains the same after passage through HWP 2 or QWP 2, and thus the trajectory shape of the SOP out trace of QWP 2 or PMF 1 is equal to that of the SE in,GTF1 trace. In terms of the SOP out trace of PMF 1, however, p 1 , the normal vector of the trace, coincides with the S 1 axis on the Poincare sphere as the orientation angle θ P1 of PMF 1 is 0 • , as shown in Figure 3e. Then, θ H2 and θ Q2 can be determined by finding the SOP out trace of PMF 1 to satisfy this constraint for p 1 . After both θ H2 and θ Q2 are elucidated, the SOP out of HWP 1, which is displayed as an SOP location on the Poincare sphere, can simply be determined using the SOP out trace of PMF 1 and T PMF1 −1 , as shown in Figure 3f. In particular, the validity of θ H2 and θ Q2 determined above can be verified by checking whether this SOP in of PMF 1 is a single SOP point or not. That is to say, if θ H2 and θ Q2 predetermined abiding by the above constraint for p 1 are suitable for the WOA set at Set II GTF1 , the SOP in of PMF 1 should be unchanged even with variations in the wavelength from λ 0 to λ 0 + ∆λ.
As θ H2 and θ Q2 are found from the SE in,GTF1 trace at Set II GTF1 , this SOP in of PMF 1 (i.e., the SOP out of HWP 1) is the starting point to unveil θ H1 and θ Q1 . By using the SOP out of HWP 1 (in Figure 3f) and T HWP1 −1 , the SOP out of QWP 1 can be obtained, if we set θ H1 as a fixed value (unknown yet). Moreover, for a fixed value (unknown either) of θ Q1 , a corresponding SOP in of QWP 1 can be drawn out from the above SOP out of QWP 1 and T QWP1 −1 . Then, to find θ H1 andθ Q1 , we can harness the fundamental SOP constraint that the SOP in of QWP 1 should be LHP because light emerging from port 2 of the PBS is linear horizontally polarized. Through this reverse tracing algorithm described above, (θ Q1 , θ H1 , θ Q2 , θ H2 ) at Set II GTF1 could be found to be (108.49 • , 132.67 • , 201.92 • , 134.71 • ), and it was also confirmed from additional calculations, although not suggested here, that several degenerate WOA sets could also exist for t GTF1 at Set II GTF1 . It is figured out from the SOP out trace of PMF 2 at Set II GTF1 , the SE out,GTF1 trace in Figure 3g, that O(λ 0 ) at Set II GTF1 differs from O(λ 0 ) at Set I GTF1 , although the shape of the SE out,GTF1 trace at Set II GTF1 remains unchanged in comparison with the shape of the SE out,GTF1 trace at Set I GTF1 . From O(λ 0 ) at Set I GTF1 , O(λ 0 ) at Set II GTF1 is distant by an angular displacement that corresponds to ∆λ/8 along this SE out,GTF1 trace towards decreasing wavelength. This implies that O(λ 0 + ∆λ/8) at Set II GTF1 is the same as O(λ 0 ) at Set I GTF1 and also as O(λ peak ) at Set II GTF1 , and λ peak becomes λ 0 + ∆λ/8 at Set II GTF1 , resulting in a wavelength shift of ∆λ/8 in t GTF1 (i.e., a phase modulation of 45 • ). Figure 3h shows two transmission spectra of t GTF1 , which are calculated at Sets I GTF1 and II GTF1 and displayed by blue and red solid lines, respectively. It is apparent that the transmission spectrum at Set II GTF1 is redshifted by ∆λ/8 (= 0.1 nm) compared with that at Set I GTF1 . In short, for a phase modulation of η, or a wavelength tuning of (η/360 • )∆λ, in t GTF1 , one should find a new SE in,GTF1 trace first, obtained by revolving the SE in,GTF1 trace at Set I GTF1 CCW by η in angle about p 1 and p 2 . Then, one can find (θ Q1 , θ H1 , θ Q2 , θ H2 ) for this new SE in,GTF1 trace through the WOA-searching procedures addressed above. If we pick η within 0 • to 360 • , t GTF1 can be arbitrarily phase-modulated. This arbitrary phase modulation can be utilized directly for the continuous wavelength tuning of t GTF1 .
Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 14 within ∆λ using the WOA sets shown in Figure 4, implying the arbitrary phase modulation capability of tGTF2 within 360°.   To examine if the WOA sets found by our angle-finding method can embody the predicted continuous phase modulation of t GTF2 , the transmission spectra of t GTF2 were calculated at eight WOA sets, which were chosen from the WOA sets found above for the phase modulation of t GTF2 , with θ P1 and θ P2 set at 0 • and 22.5 • , respectively. The chosen WOA sets, denoted by Sets I GTF2 , II GTF2 , III GTF2 , IV GTF2 , V GTF2 , VI GTF2 , VII GTF2 , and VIII GTF2 , enable φ's to be chosen as 0 • , 45 • , 90 • , 135 • , 180 • , 225 • , 270 • , and 315 • , respectively. The transmission spectra of the GTF t GTF2 , shown in Figure 5, were calculated at the eight WOA sets (Sets I GTF2 -VIII GTF2 ) in a wavelength range of 1548-1552 nm. For the calculation of the transmission spectrum, the length L and birefringence B of PMF (PMF 1 or PMF 2) were set as 7.2 m and 4.166 × 10 −4 , respectively, so that ∆λ became~0.8 nm at 1550 nm. It can be seen from the figure that the first-order comb spectrum of deformed narrow passbands makes a redshift as φ is modulated as 0 • to 315 • (increment: 45 • ). If the peak wavelength of one passband is designated as λ peak,GTF at Set I GTF2 (φ = 0 • ), one step transition in the WOA set (e.g., from Set II GTF2 to Set III GTF2 ) leads to an increase of 45 • in φ and of 0.1 nm in λ peak,GTF . Linear movement of λ peak,GTF with respect to φ, displayed as red-dotted arrows, directly tells us that the continuous or the desired arbitrary phase modulation of t GTF2 can be realized. These calculation results clearly manifest that t GTF2 can be continuously wavelength-shifted within ∆λ using the WOA sets shown in Figure 4, implying the arbitrary phase modulation capability of t GTF2 within 360 • . Figure 4. (a) Four WOA's θQ1 (blue circles), θH1 (green squares), θQ2 (red diamonds), and θH2 (violet triangles) as a function of extra phase difference ϕ (from 0° to 360° with a step of 1°) for phase modulation of another GTF (tGTF2) at θP1 = 0° and θP2 = 22.5°. (b) Loci of (θH2, θH1) and (θH2, θQ1), displayed by blueish squares and reddish circles, respectively. (c) Loci of (θQ1, θH1) and (θQ1, θ Q2), indicated as blueish squares (ϕ: 0°-180°) and reddish circles (ϕ: 0°-360°), respectively. (d) Loci of (θQ2, θH1) and (θQ2, θH2), displayed by blueish squares and reddish circles, respectively.

Experimental Verification of Phase-Modulated Spectra
In order to verify the predicted phase modulation capability of t GTF2 through experiments, our filter was manufactured with a fiber-pigtailed PBS (OZ Optics) with four ports, two fiber-pigtailed QWPs (OZ Optics), two fiber-pigtailed HWPs (OZ Optics), and two~7.12 m-long segments of bow-tie PMF (Fibercore) with a birefringence of~4.166 × 10 −4 , as shown in Figure 1a. Figure 6 shows an actual experimental setup for measuring the transmission spectrum of the constructed filter. As shown in Figure 6, we employed a broadband light source (BLS, Fiberlabs FL7701) covering S, C, and L bands and an optical spectrum analyzer (OSA, Yokogawa AQ6370C) to monitor the transmission spectra of the constructed filter. The input and output ports (ports 1 and 4) of the filter were connected to the BLS and OSA, respectively, with optical fiber patchcords (FC/PC type). For acquisition of high resolution and contrast optical spectra, we set the sensitivity and resolution bandwidth of the OSA as HIGH1 and 0.05 nm, respectively. To avoid unwanted position changes of all the optical components comprising the filter, we taped them up on the optical table so that they were immobilized during the spectrum measurement.
The transmission spectra of the GTF t GTF2 , shown in Figure 7, were measured at the eight WOA sets (Sets I GTF2 -VIII GTF2 ) in a wavelength range of 1548-1552 nm. From the measured spectra, the FSR of the constructed filter was evaluated as~0.8 nm around 1550 nm. This FSR is determined by L (~7.12 m) and B (~4.166 × 10 −4 ) of PMF used here and increases with wavelength. As confirmed from the theoretical spectra in Figure 5, while the WOA set is switched from Set I GTF2 (φ = 0 • ) to Set VIII GTF2 (φ = 315 • ), the comb spectrum of deformed narrow passbands, which seems to be analogous to the comb spectrum shown in Figure 5, shifts towards a longer wavelength region step by step bỹ 0.1 nm, resulting in a total wavelength displacement of~0.7 nm. The inset shows the variation of  Sets I GTF2 , II GTF2 , III GTF2 , IV GTF2 , V GTF2 , VI GTF2 , VII GTF2 , and VIII GTF2 , respectively. Blue squares designated as I GTF2 -VIII GTF2 indicate the λ peak,GTF positions of transmission spectra obtained at eight Sets I GTF2 -VIII GTF2 . It can be figured out from the inset that λ peak,GTF very linearly increases with φ showing an adjusted R 2 value of~0.99946 and the tuning of the spectrum wavelength can be mediated by the phase modulation. In particular, through additional experiments, continuous frequency tuning capability was confirmed with respect to numerous φ values arbitrarily selected from 0 • to 360 • except integer multiples of 45 • as well. Thus, it is experimentally verified that the GTF t GTF2 can be arbitrarily phase-modulated by properly controlling (θ Q1 , θ H1 , θ Q2 , θ H2 ), specifically, appropriately selecting (θ Q1 , θ H1 , θ Q2 , θ H2 ) that satisfies the WOA sets suggested in Figure 4. Ultimately, we conclude that any GTF generated in the proposed comb filter can be phase-modulated, or wavelength-tuned, continuously by taking advantage of WOA sets that can always be drawn out with our reverse tracing algorithm.

Experimental Verification of Phase-Modulated Spectra
In order to verify the predicted phase modulation capability of tGTF2 through experiments, our filter was manufactured with a fiber-pigtailed PBS (OZ Optics) with four ports, two fiber-pigtailed QWPs (OZ Optics), two fiber-pigtailed HWPs (OZ Optics), and two ~7.12 m-long segments of bow-tie PMF (Fibercore) with a birefringence of ~4.166 × 10 −4 , as shown in Figure 1a. Figure 6 shows an actual experimental setup for measuring the transmission spectrum of the constructed filter. As shown in Figure 6, we employed a broadband light source (BLS, Fiberlabs FL7701) covering S, C, and L bands and an optical spectrum analyzer (OSA, Yokogawa AQ6370C) to monitor the transmission spectra of the constructed filter. The input and output ports (ports 1 and 4) of the filter were connected to the BLS and OSA, respectively, with optical fiber patchcords (FC/PC type). For acquisition of high resolution and contrast optical spectra, we set the sensitivity and resolution bandwidth of the OSA as HIGH1 and 0.05 nm, respectively. To avoid unwanted position changes of all the optical components comprising the filter, we taped them up on the optical table so that they were immobilized during the spectrum measurement. The transmission spectra of the GTF tGTF2, shown in Figure 7, were measured at the eight WOA sets (Sets IGTF2-VIIIGTF2) in a wavelength range of 1548-1552 nm. From the measured spectra, the FSR of the constructed filter was evaluated as ~0.8 nm around 1550 nm. This FSR is determined by L (~7.12 m) and B (~4.166 × 10 −4 ) of PMF used here and increases with wavelength. As confirmed from the theoretical spectra in Figure 5, while the WOA set is switched from Set IGTF2 (ϕ = 0°) to Set VIIIGTF2 (ϕ = 315°), the comb spectrum of deformed narrow passbands, which seems to be analogous to the comb from 0° to 360° except integer multiples of 45° as well. Thus, it is experimentally verified that the GTF tGTF2 can be arbitrarily phase-modulated by properly controlling (θQ1, θH1, θQ2, θH2), specifically, appropriately selecting (θQ1, θH1, θQ2, θH2) satisfying the WOA sets suggested in Figure 4. Ultimately, we conclude that any GTF generated in the proposed comb filter can be phase-modulated, or wavelength-tuned, continuously by taking advantage of WOA sets that can always be drawn out with our reverse tracing algorithm.

Conclusions
We demonstrated the arbitrary phase modulation of a GTF, which could be obtained in the first-order fiber comb filter based on the PDLS. The proposed filter was composed of a PBS, two OWS's of an HWP and a QWP, and two PMF segments. Each OWS was located before each PMF segment, and the second PMF segment was butt-coupled to one port of the PBS so that its principal axis should be 22.5° away from the transverse-magnetic polarization axis of the PBS. Basically, polarization conditions, which should be satisfied to modify the phase of a GTF as a desired value, were explained using the spectral evolution of the SOPin and SOPout of the second PMF in the narrowband transmittance function (tn), on the basis of the continuous wavelength tuning mechanism of previously reported PDLS-based first-order narrowband comb filters. Then, we explained a systematic scheme to find four WOA's for the arbitrary phase modulation of a GTF Figure 7. Experimental phase-modulated transmission spectra of GTF (t GTF2 ), measured at eight WOA sets (Sets I GTF2 -VIII GTF2 ).

Conclusions
We demonstrated the arbitrary phase modulation of a GTF, which could be obtained in the first-order fiber comb filter based on the PDLS. The proposed filter was composed of a PBS, two OWS's of an HWP and a QWP, and two PMF segments. Each OWS was located before each PMF segment, and the second PMF segment was butt-coupled to one port of the PBS so that its principal axis should be 22.5 • away from the TM polarization axis of the PBS. Basically, polarization conditions, which should be satisfied to modify the phase of a GTF as a desired value, were explained using the spectral evolution of the SOP in and SOP out of the second PMF in the narrowband transmittance function (t n ), on the basis of the continuous wavelength tuning mechanism of previously reported PDLS-based first-order narrowband comb filters. Then, we explained a systematic scheme to find four WOA's for the arbitrary phase modulation of a GTF using the effect of each component of the filter, such as a waveplate or PMF segment, on its SOP in or SOP out . By exploiting this WOA finding method, we derived WOA sets of the four waveplates, which could give arbitrary phase delays φ's from 0 • to 360 • to a GTF chosen here arbitrarily. Finally, we showed phase-modulated GTF's calculated at eight selected WOA sets allowing φ's to be 0 • , 45 • , 90 • , 135 • , 180 • , 225 • , 270 • , and 315 • , and then the theoretically predicted results were experimentally verified. It is concluded from the theoretical and experimental demonstrations that the GTF of our filter based on the OWS of a QWP and an HWP can be arbitrarily phase-modulated by appropriately controlling the WOA's of the four waveplates. Our demonstration is expected to be beneficial to fiber-optic applications, which demand frequency-tunable comb filters with specific transmittances, such as microwave photonic signal processing, optical sensor interrogation, and multiwavelength lasing.