1. Introduction

Up to the present, fiber comb filters have provided much benefit such as simple structure, ease of fabrication, convenient spectrum controllability, and good fiber compatibility in the areas of multiwavelength or waveband switching in wavelength-routed optical networks [1–3], microwave photonic signal processing [4,5], and optical label swapping in subcarrier multiplexed systems [6,7]. Continuous wavelength tunability in fiber comb filters is essential to extract or reject desired spectral components in the optical signal processing. Lots of efforts have been made to realize the continuous frequency tunability of comb filters by using a Sagnac birefringence loop mirror [8,9], a Mach-Zehnder interferometer [10,11], a Lyot-type birefringence interferometer [12,13], and a polarization-diversified loop structure (PDLS) [14–18]. In comparison with other filter structures described above, PDLS-based comb filters offer flexible wavelength switchability and tunability [14–18]. By incorporating some combinations of a half-wave plate (HWP) and a quarter-wave plate (QWP), continuous wavelength tuning was implemented in a PDLS-based zeroth-order comb filter, which employed one polarization-maintaining fiber (PMF) segment as a birefringent element [15,18]. The first-order version of this filter, which can be constructed by combining two PMF segments appropriately selected in their azimuth and length, e.g., concatenating them in Solc- or Lyot-type configuration [19,20], features useful and sophisticated transmittance functions with flat-top or narrow passbands. For these first-order transmittance functions, wavelength switching was achieved by varying the orientation angles of waveplates contained in the filter [19–21].

Recently, continuous wavelength tuning of a flat-top band transmittance function (FBTF) has been realized in the first-order PDLS-based comb filter composed of a polarization beam splitter (PBS), two HWPs, two QWPs, and two PMF segments [22]. Furthermore, continuous frequency tuning of a narrow band transmission function (NBTF) was also embodied in the first-order PDLS-based comb filter [23]. The azimuth angle adjustment of the four or five waveplates could provide synchronous modulation of an additional phase difference of 0 to 360° between two orthogonal modes of each PMF for two PMF segments. The input and output states of polarization (SOPs) of the second PMF segment were rigorously investigated on the basis of the Poincare sphere representation, and the frequency tuning scheme of its transmission spectrum was found by analyzing the spectral evolution of these SOPs. However, continuous wavelength tuning of other transmission spectra, except for passband-flattened and -shrunk spectra, achievable in this filter structure was not accomplished. In particular, the qualitative and quantitative way of continuously tuning the absolute phase of a general transmittance function (GTF), except for flat-top and narrow band transmittance functions, has been disregarded and not been discussed yet. In several specific applications such as the label erasing and spectral flattening, optical filters with unique spectral shapes other than flat-top and narrow band shapes are highly demanded, and the functionality to tune their spectrum to the desired wavelength location can offer great efficiency to their entire optical systems. It is readily expected that the orientation angles of the waveplates for the extra phase shift of this transmittance function from 0 to 360° are totally different from those obtained for the continuous wavelength tuning of the FBTF and the NBTF in the previous works [22,23]. Moreover, the previous approach of finding four or five waveplate orientation angles (WOAs) for the continuous wavelength tuning, where simple analytic equations for the FBTF or NBTF are utilized and compared with the filter transmittance to trace them [22,23], is hard to be equally applied to the case of the GTF. This is because it is fairly cumbersome to derive the exact analytic expression of the GTF and further to find these WOAs through its direct comparison with the filter transmittance with six unknown coefficients [22].

Here we show the continuous wavelength tunability of a GTF mostly obtainable in the PDLS-based first-order fiber comb filter which has the same filter structure as that of the previous work [22]. First, we revisit the previous results on the continuous wavelength tuning of the FBTF, including WOAs for continuous extra phase shifts of 1−360° and wavelength-tuned transmission spectra. Using two selected transmission spectra, whose wavelength locations differ by 0.1 nm, a thorough investigation is made on the relationship between the behavior of the input SOPs (SOP_in’s) and output SOPs (SOP_out’s) of the second PMF on the Poincare sphere and the frequency tuning of the filter transmittance. The continuous wavelength tunability of the GTF as well as FBTF can be inferred from this qualitative investigation on the principle of the wavelength tuning. SOP conditions, which should be satisfied to continuously tune the GTF, are explained based on the tuning mechanism of the flat-top band transmittance. Then, by considering the effect of the four waveplates (i.e., two HWPs and two QWPs) and two PMF segments on the SOP_out of each element within the filter, we suggest a step-by-step approach to seek four WOAs for continuous wavelength tuning of the GTF. Three hundred sixty sets of the four WOAs, which give a phase shift from 1 to 360° (with a step of 1°) to a specific transmittance function as an exemplificative GTF, are obtained with the suggested angle search scheme. In particular, this scheme can be exploited to implement the wavelength tuning of every transmittance function including the FBTF. Finally, wavelength-tuned transmission spectra are displayed at eight WOA sets, which are selected among the above angle sets so that they can give additional phase shifts of 0, 45, 90, 135, 180, 225, 270, and 315° to the original transmittance function. The eight calculated transmission spectra clearly show that the GTF of the filter can be continuously wavelength-tuned by properly choosing WOAs. This qualitative and quantitative prediction on the continuous frequency tuning of the GTF is also experimentally verified.

2. Principle of wavelength tuning of FBTF and GTF

Before we begin to deal with the wavelength tuning of the GTF, we revisit the previous results on the continuous wavelength tuning of the FBTF. Figure 1(a) shows a schematic diagram of the first-order fiber comb filter proposed in [22], which consists of a four-port PBS, two equal-length PMF segments (denoted by PMF 1 and PMF 2), two HWPs (denoted by HWP 1 and HWP 2), and two QWPs (denoted by QWP 1 and QWP 2). A set of an HWP and a QWP (i.e., HWP 1 and QWP 1) is located before PMF 1, and another set of an HWP and a QWP (i.e., HWP 2 and QWP 2) before PMF 2. Each waveplate set controls the effective phase difference between fast and slow axes of each PMF, and the second waveplate set (i.e., HWP 2 and QWP 2) determines in addition the relative angular difference between two PMF segments. The PMF 2 segment is directly connected to port R of the PBS so that its slow-axis orientation angle becomes 22.5° with respect to the horizontal axis of the PBS. Input light entering port IN of the PBS is separated into two linearly polarized beams with orthogonal polarization, i.e., linear horizontal polarization (LHP) and linear vertical polarization (LVP), which propagate along the fiber loop of the filter in clockwise (CW) and counterclockwise (CCW) directions, respectively. As shown in Fig. 1(b), when linear horizontally polarized input light propagates along the CW path, it passes through a linear horizontal polarizer (x axis), HWP 1 (with its slow axis oriented at θ_h1 with respect to the x axis), QWP 1 (oriented at θ_q1), PMF 1 (oriented at θ_p1), HWP 2 (oriented at θ_h2), QWP 2 (oriented at θ_q2), PMF 2 (oriented at θ_p2 = 22.5°), and a linear horizontal analyzer (x axis) in turn, where the x axis indicates the horizontal axis of the PBS. Similarly, in the case of the light propagation along the CCW path, a linear vertically polarized input beam goes through a linear vertical polarizer (y axis), PMF 2 (−θ_p2 oriented), QWP 2 (−θ_q2 oriented), HWP 2 (−θ_h2 oriented), PMF 1 (−θ_p1 oriented), QWP 1 (−θ_q1 oriented), HWP 1 (−θ_h1 oriented), and a linear vertical analyzer (y axis), where the y axis implies the vertical axis of the PBS. Here, F and S mean the fast and slow axes, respectively, of waveplates or PMF.

 

Fig. 1 (a) Schematic diagram of the PDLS-based first-order fiber comb filter, (b) propagation path of light traveling through the filter, (c) four WOAs (θ_h1, θ_q1, θ_h2, θ_q2) with respect to ϕ (from 0 to 360° with a step of 1°), obtained for continuous frequency tuning of the FBTF, and (d) calculated passband-flattened transmission spectra at two WOA sets (Sets I and II). An inset of (d) shows the variation of the dip wavelength λ₀ when ϕ changes from 0 to 360° with a step of 1°. Roman numerals I, II, III, IV, V, VI, VII, and VIII indicate eight WOA sets satisfying ϕ = 0, 45, 90, 135, 180, 225, 270, and 315°, respectively.

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In a polarization-interference-based comb spectrum created in one PMF segment inserted between two polarizers, its transmittance function determining the interference spectrum is described as a sinusoidal function of the phase delay difference Γ (= 2πBL/λ) between two orthogonal modes of PMF, i.e., p + qcosΓ (where p and q are real values), which is referred to as the zeroth-order transmittance one. In terms of Γ, B, L, and λ are the PMF birefringence, PMF length, and wavelength in vacuum, respectively. The zeroth-order comb spectrum can be frequency-shifted by modifying this phase delay difference Γ “effectively” in a figurative sense. This effective modification alludes to giving a supplementary phase difference ϕ to the original phase difference Γ₀ of PMF itself [18]. Likewise, for the spectral tuning of the first-order comb spectrum formed by two PMF segments [24], an extra phase difference ϕ should be added to Γ₀ of each PMF simultaneously. The addition of ϕ can be done by controlling the SOP_in of each PMF, and the SOP_in control can be performed using waveplates located before each PMF. When ϕ increases from 0 to 360°, the first-order comb spectrum shifts toward a longer wavelength region by one free spectral range (FSR), that is, is wavelength-tuned over a full wavelength range. The transmittance t of the filter can be derived using Jones transfer matrices T_CW and T_CCW [25], given by Eqs. (1) and (2), obtained for the CW and CCW paths shown in Fig. 1(b), respectively, on the basis of Jones matrix formulation. It is assumed that there are no insertion losses (ILs) of optical components comprising the filter. On top of the IL, the phase delay difference of each waveplate is also regarded as independent of wavelength.

Figure 1(c) shows the WOAs of the four waveplates, or (θ_h1, θ_q1, θ_h2, θ_q2), which can introduce an extra phase difference ϕ from 0 to 360° into t_flat, which are obtained through direct quantitative comparison between Eqs. (3) and (4) with θ_p1 and θ_p2 fixed at 0 and 22.5°, respectively. With increasing ϕ, θ_h1 and θ_q1 (denoted by black squares and red circles, respectively) show similar alternating patterns, that is, decrease first, then increase, and finally decrease again. θ_h1 and θ_q1 are bounded in −16.4° < θ_h1 < 16.4° and −22.5° ≤ θ_q1 ≤ 22.5°, respectively, but they are not sinusoidal functions of ϕ. θ_h2 (green triangles) linearly increases with ϕ, starting from 33.75° and ending up with 123.75°, and θ_q2 (blue inverted triangles) is a constant, given by 67.5°. Figure 1(d) shows calculated passband-flattened transmission spectra at two WOA sets (Sets I and II), which are denoted by blue circles and red squares, respectively. Here, the eight selected sets of (θ_h1, θ_q1, θ_h2, θ_q2) allowing ϕ in Eq. (4) to be 0, 45, 90, 135, 180, 225, 270, and 315° are designated as Sets I, II, III, IV, V, VI, VII, and VIII, respectively. The length L and birefringence B of each PMF were set as 7.2 m and 4.166 × 10⁻⁴ to achieve an FSR of ~0.8 nm at 1550 nm, respectively. The wavelength of one transmission dip indicated as a sky blue arrow is denoted by λ₀ ( = 1548.0 nm) at Set I. It can be found from the figure that λ₀ increases from 1548.0 to 1548.1 nm while the WOA set switches from Set I to Set II. The inset of Fig. 1(d) shows the variation of the dip wavelength λ₀ when ϕ changes from 0 to 360° with a step of 1°. Blue circles indicated as I, II, III, IV, V, VI, VII, and VIII show λ₀ locations (1548.0, 1548.1, 1548.2, 1548.3, 1548.4, 1548.5, 1548.6, and 1548.7 nm) with respect to Sets I, II, III, IV, V, VI, VII, and VIII, respectively. Closely spaced equi-step red circular symbols in the inset show that the passband-flattened comb spectrum can be continuously wavelength-shifted.

Using the above two WOA sets (Sets I and II), a thorough investigation is made on the relationship between the spectral evolution of the SOPs, especially the SOP_in and SOP_out of PMF 2 in the CW path of Fig. 1(b), and the frequency tuning of the filter transmittance t_flat. Although only the CW path is considered here, there is no loss of generality because exactly the same can be done with the CCW path. Figure 2(a) shows the spectral evolution of the SOP_in of PMF 2 at Set I on the Poincare sphere, which is denoted by C_in, when the wavelength of light propagating through the filter varies from λ₀ to λ₀ + Δλ, where Δλ corresponds to the wavelength spacing between two transmission minima, that is, the FSR. As can be seen from the figure, the SOP_in begins with point P with coordinates (2ε = 0°, 2ψ = 180°) at λ₀, becomes point Q with coordinates (2ε = −45°, 2ψ = 135°) at λ₀ + Δλ/4, and reaches point R with coordinates (2ε = 0°, 2ψ = 90°) at λ₀ + Δλ/2, rotating in a CW direction around the normal vector n with increasing wavelength. 2ε (−90° ≤ 2ε ≤ 90°) and 2ψ (−180° ≤ 2ψ ≤ 180°) are the latitude and longitude of the Poincare sphere, respectively. The sense of rotation follows the convention that an SOP of light getting through a birefringent component is rotated CW around its slow axis with increasing wavelength. Furthermore, the rule of choosing the direction of the normal vector n follows the left-hand rule: If you point the thumb of your left hand in the direction of n, your four fingers point in the direction of the wavelength increase. The vector AB, which connects two points A (2ε = 0°, 2ψ = −135°) and B (2ε = 0°, 2ψ = 45°) on the Poincare sphere, implies an axis of revolution of the C_in trace according to the variation of ϕ. Actually, the vector AB indicates the slow axis of PMF 2, whose orientation angle θ_p2 is 22.5°, on the Poincare sphere. Figures 2(b) and 2(c) display the same SOP_in trace as one shown in Fig. 2(a) on new two-dimensional (2-D) planes whose normal vectors are AB and n, which are referred to as “SOP_out” and “SOP_in” planes in convenience, respectively. It is confirmed from these two figures that AB and n are perpendicular and AB and the plane of the C_in trace are parallel with each other. Figure 2(d) shows the C_in trace at Set II on the Poincare sphere, which is obtained through 45° CCW rotation of the C_in trace at Set I around n and AB, at the wavelength range from λ₀ to λ₀ + Δλ. At Set II, n of the C_in trace at Set I is also rotated CCW by 45° around AB. Similarly, Figs. 2(e) and 2(f) depict the C_in trace at Set II on the SOP_out and SOP_in planes whose normal vectors are AB and n, respectively. It can be found from Figs. 2(d)–2(f) that the SOP_in varies from point S (2ε = 45°, 2ψ = −157.5°) to U (2ε = 45°, 2ψ = 45°) via T (2ε = 0°, 2ψ = 135°) as the wavelength increases from λ₀ + Δλ/8 to λ₀ + 5Δλ/8 via λ₀ + 3Δλ/8. In other words, the wavelength at which the SOP_in at Set II reaches point S or T or U is shifted by Δλ/8, compared with the case of point P or Q or R at Set I.

 

Fig. 2 (a) Poincare sphere representation of spectral evolution of the SOP_in of PMF 2 at Set I, the same SOP_in traces as one shown in (a) viewed on new 2-D planes whose normal vectors are (b) AB and (c) n, (d) Poincare sphere representation of spectral evolution of the SOP_in of PMF 2 at Set II, the C_in traces shown in (d) viewed on new planes whose normal vectors are (e) AB and (f) n, Poincare sphere representations of spectral evolution of the SOP_out of PMF 2 at Sets (g) I and (h) II, and (i) eight O(λ₀) locations at the eight selected WOA sets (Sets I−VIII).

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Figure 2(g) shows the spectral evolution of the SOP_out of PMF 2 at Set I on the Poincare sphere, which is denoted by C_out, over the same wavelength range (from λ₀ to λ₀ + Δλ). This C_out trace is obtained by rotating each SOP_in of PMF 2 at Set I CW around AB (when viewed on the SOP_out plane like Fig. 2(b) or 2(e)) by a wavelength-dependent angle ρ(λ) given by (360°/Δλ)(λ − λ₀). For example, while light at λ₀ passes through PMF 2, the SOP_in of PMF 2, i.e., P shown in Fig. 2(a), is revolved CW around AB by ρ(λ₀) = 0° and results in P′ shown in Fig. 2(g). If the SOP_out of PMF 2 at λ₀ is referred to as O(λ₀) for simplicity, P′ can be regarded as O(λ₀). Similarly, during the passage of light at λ₀ + Δλ/4 or λ₀ + Δλ/2 through PMF 2, the SOP_in of PMF 2 (Q or R) is rotated CW around AB by ρ(λ₀ + Δλ/4) = 90° or ρ(λ₀ + Δλ/2) = 180° and becomes Q′ or R′, shown in Fig. 2(g), respectively. That is, Q′ and R′ are equivalent to O(λ₀ + Δλ/4) and O(λ₀ + Δλ/2), respectively. Due to this wavelength-dependent rotation of the SOP_in of PMF 2, the SOP_out of PMF 2 forms a trajectory C_out of a droplet shape, which makes one CW rotation, starting at P′, around the S₂ axis along the C_out trace as the wavelength increases from λ₀ to λ₀ + Δλ. If we assume the wavelength where t_flat is minimized as λ_d, λ_d is equal to λ₀ at Set I as the transmission minimum occurs at P′.

Likewise, in the case of the C_out trace obtained at Set II over the same wavelength range, shown in Fig. 2(h), this trace is obtained by rotating each SOP_in of PMF 2 at Set II CW around AB on the SOP_out plane by ρ(λ). Hence, points S, T, and U at Set II, which correspond to the SOP_in’s of PMF 2 at λ₀ + Δλ/8, λ₀ + 3Δλ/8, and λ₀ + 5Δλ/8, are moved CW around AB by 45, 135, and 225° during their passage through PMF 2 and end up with S′, T′, and U′ in Fig. 2(h), respectively. Consequently, S′, T′, and U′ on the C_out trace of Fig. 2(h) are identical in location to P′, Q′, and R′ on the C_out trace of Fig. 2(g), respectively. Considering other wavelength components except for those mentioned above, it can readily be seen that the C_out trace at Set II has the same trajectory shape as that at Set I. The only difference between both trajectories in Figs. 2(g) and 2(h) is that the initial point of the spectral evolution beginning with λ₀, i.e., O(λ₀), which is indicated by a purple arrow with a legend of λ₀, is different with each other. Specifically, O(λ₀) in Fig. 2(h) is distant from S′, which is equivalent to O(λ₀ + Δλ/8) at Set II, by an angular displacement corresponding to Δλ/8. For the two C_out traces in Figs. 2(g) and 2(h), the same trajectory shape and different O(λ₀) are attributed to 45° CCW rotation of the SOP_in trace at Set I around AB on the SOP_out plane and around n on the SOP_in plane when the WOA set switches from Set I to Set II. In other words, rotating the C_in trace CCW around both AB and n by a certain angle leads to a change in the position of O(λ₀) along the C_out trace without modification of its trajectory shape. In the case of the C_out trace at Set II, therefore, λ_d becomes λ₀ + Δλ/8, which means that the transmission spectrum red-shifts by Δλ/8, that is, t_flat is tuned by ϕ = 45°. Figure 2(i) shows eight O(λ₀) locations at the eight selected WOA sets (Sets I−VIII). For Sets III−VIII, λ_d changes from λ₀ + Δλ/4 to λ₀ + 7Δλ/8 by a step of Δλ/8, and t_flat is tuned by ϕ increasing from 90 to 315° by a step of 45°.

In terms of the GTF, it is natural that the SOP_out trace of PMF 2, designated as the C_out,arb1 trace, has a totally different trajectory shape compared with the C_out trace obtained in the FBTF. However, it can be inferred that a change in the O(λ₀) position along the C_out,arb1 trace is responsible for a wavelength shift in the transmission spectrum. This is because the location change of O(λ₀) leads to a change in λ_d. For a given C_out,arb1 trace, the SOP_in trace of PMF 2, referred to as the C_in,arb1 trace, also has a circular trajectory because light has gone through a birefringent fiber segment (i.e., PMF 1) once in the CW path, but its radius and normal vector (n) are quite different from those of the C_in trace in Fig. 2. Like the relationship between C_in and C_out traces, the abovementioned increase in λ_d is caused by CCW rotation of the C_in,arb1 trace around AB on the SOP_out plane and around n on the SOP_in plane. If the C_in,arb1 trace is rotated CCW by φ around both AB and n, λ_d increases by (φ/360°)Δλ, resulting in a red-shift of (φ/360°)Δλ in the transmission spectrum. That is, φ is equivalent to ϕ, which can be added to the original transmittance function. Hence, a choice of φ from 0 to 360° can make the absolute phase of the GTF vary by 360°, which implies its continuous wavelength tunability. In brief, one CCW revolution of the C_in,arb1 trace around both AB and n, mediated by the WOA control, allows its transmission spectrum to move towards longer wavelength region by Δλ.

3. WOAs for wavelength tuning of GTF

On the basis of the SOP conditions as described above, which should be satisfied to continuously tune the GTF, WOAs (θ_h1, θ_q1, θ_h2, θ_q2) for this continuous wavelength tuning are investigated using a new angle-finding scheme different from the previous approach [22]. This new scheme is introduced because it is not effortless to derive the analytic expression of the GTF with desired spectral characteristics and, even if it is derived, further troublesome to find (θ_h1, θ_q1, θ_h2, θ_q2) through its direct comparison with the filter transmittance in Eq. (3) with six unknown coefficients [22]. Here, as an example of the GTF, we target a specific transmittance function t_arb1 obtained at an arbitrarily chosen WOA set (θ_h1, θ_q1, θ_h2, θ_q2) = (57.9°, 89.8°, 122.6°, 155.1°) and find (θ_h1, θ_q1, θ_h2, θ_q2) which enables t_arb1 to be frequency-tuned. For this chosen WOA set (denoted as Set I_arb1) with θ_p1 and θ_p2 set at 0° and 22.5°, respectively, the SOP_out trace of PMF 2, i.e., C_out,arb1, can be obtained, as shown in Fig. 3(a). At Set I_arb1, O(λ₀) is located at a point with coordinates (2ε = 0°, 2ψ = 0°), i.e., LHP, resulting in the maximum transmittance at this wavelength. This wavelength where the transmittance is maximized is referred to as λ_p here and is equal to λ₀ ( = 1548.0 nm) at Set I_arb1. By using the inverse matrix of the transfer matrix of PMF 2 (i.e., T_P2⁻¹), the SOP_in trace of PMF 2 (i.e., C_in,arb1) shown in Fig. 3(b) can be obtained from the C_out,arb1 trace in Fig. 3(a). On the basis of previous discussion on the spectral tuning of the filter transmittance using both C_in,arb1 and C_out,arb1 traces, a new C_in,arb1 trace can be determined, as shown in Fig. 3(c), by making 45° CCW rotation of the C_in,arb1 trace at Set I_arb1 (shown in Fig. 3(b)) around both AB and n, for a wavelength tuning of Δλ/8 in the transmission spectrum. It is obvious that this new C_in,arb1 trace originates from another WOA set (θ_h1, θ_q1, θ_h2, θ_q2), which will be denoted as Set II_arb1 later.

 

Fig. 3 Spectral evolution of SOPs in t_arb1 obtained at Set I_arb1: (a) SOP_out and (b) SOP_in traces of PMF 2 at Set I_arb1. Spectral evolution of SOPs in the wavelength-tuned version of t_arb1, obtained at Set II_arb1: (c) SOP_in trace of PMF 2, (d) SOP_out trace of HWP 2, (e) SOP_out trace of PMF 1, (f) SOP_out of QWP 1, (g) SOP_out of HWP 1, and (h) SOP_out trace of PMF 2. (i) Two calculated transmission spectra obtained at Sets I_arb1 and II_arb1, indicated as blue circular and red rectangular symbols, respectively.

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Now we describe the scheme to find the new WOA set (i.e., Set II_arb1). Because only the CW path shown in Fig. 1(b) is considered here, light passes through HWP 1, QWP 1, PMF 1, HWP 2, QWP 2, and PMF 2 in turn. For example, the SOP_in trace of PMF 1 is equivalent to the SOP_out trace of QWP 1, and the same is true for the SOP_in of PMF 2 and the SOP_out of QWP 2. When light propagates through a QWP or an HWP, it rotates the SOP of input light by 90° or 180° CCW around its axis determined by its slow axis orientation angle on the Poincare sphere, respectively. Thus, waveplates do not alter the trajectory shape of an SOP trace. For a given C_in,arb1 trace at Set II_arb1, the SOP_out trace of HWP 2, which is transformed into this C_in,arb1 trace after passing through QWP 2, can be determined if θ_q2 is found. Because QWP 2 does not affect the trace shape, the SOP_out trace of HWP 2 should have the same circular trajectory shape as the C_in,arb1 trace. However, its center should be on the equator of the Poincare sphere, as the SOP_out trace of PMF 1 is always centered on the sphere’s equator, specifically on LHP here (θ_p1 = 0°). If we utilize this C_in,arb1 trace and the inverse matrix of T_Q2 (i.e., T_Q2⁻¹), lots of candidates for the SOP_out trace of HWP 2 can be calculated and have different centers according to θ_q2. Then, one of these candidates, which satisfies the abovementioned condition for the trace center, is the SOP_out trace of HWP 2, shown in Fig. 3(d). Here two different values of θ_q2, which differ by 90° with each other, can be found, resulting in two different SOP_out traces of HWP 2. Similarly, after the SOP_out trace of HWP 2 is sought, the SOP_out trace of PMF 1, which becomes this SOP_out trace of HWP 2 after passing through HWP 2, can be determined if θ_h2 is revealed. The SOP_out trace of PMF 1 should have the same circular trajectory shape as the C_in,arb1 trace, but its normal vector n should be parallel to the S₁ axis of the Poincare sphere because θ_p1 = 0°. By incorporating the previously found SOP_out trace of HWP 2 and T_H2⁻¹, some candidates for the SOP_out trace of PMF 1 can be calculated with respect to θ_h2, and among them one whose center is LHP can be selected as the SOP_out trace of PMF 1, shown in Fig. 3(e). Likewise, through this deduction process, two values of θ_h2, which differ by 90° with each other, can be found, but they result in the same SOP_out trace of PMF 1 unlike the case of θ_q2. This is because 180° rotations of an SOP trace (or an SOP) around the slow axis of the HWP and its 180°-rotated one (e.g., S₁ and –S₁ axes, respectively) lead to the same result.

After both θ_q2 and θ_h2 are found, the SOP_out of QWP 1 can be readily found using the SOP_out trace of PMF 1 and T_P1⁻¹ and should be one point on the Poincare sphere, as shown in Fig. 3(f). Among some candidates for the SOP_out of HWP 1, calculated with respect to θ_q1 using the SOP_out of QWP 1 (in Fig. 3(f)) and T_Q1⁻¹, one that exists on the equator should be the SOP_out of HWP 1, as shown in Fig. 3(g). One can find two different values of θ_q1, which differ by 90° with each other, and these two values of θ_q1 account for two different SOP_out’s of HWP 1. Finally, in order to find θ_h1, one can utilize the constraint that the SOP_in of HWP 1 should be LHP. With this constraint, θ_h1 can be sought with respect to candidates for the SOP_in of HWP 1, obtained by the SOP_out of HWP 1 (in Fig. 3(g)) and T_H1⁻¹. Likewise in the case of θ_h2, two values of θ_h1, which differ by 90° with each other, can be found, but two SOP_in’s of HWP 1 obtained by these two values of θ_h1 are congruent unlike the case of θ_q1, that is, the SOP_in of HWP 1 is uniquely LHP. Through this whole process depicted above, the new WOA set (Set II_arb1) can be found as (θ_h1, θ_q1, θ_h2, θ_q2) = (63.7°, 112.0°, 111.0°, 154.2°), and it can also be seen that 16 ( = 2 × 2 × 2 × 2) degenerate sets exist for Set II_arb1. Figure 3(h) shows the SOP_out trace of PMF 2, i.e., the C_out,arb1 trace, at Set II_arb1 found above. It can be checked from the figure that its trajectory shape is identical to that of the C_out,arb1 trace at Set I_arb1, but O(λ₀) is different from that at Set I_arb1. Along the C_out,arb1 trace, O(λ₀) at Set II_arb1 is moved from O(λ₀) at Set I_arb1, which is equivalent to O(λ₀ + Δλ/8) at Set II_arb1, by an angular displacement corresponding to Δλ/8 in the direction of wavelength decrease. This means that λ_p = λ₀ + Δλ/8, which results in a red-shift of Δλ/8 in the transmission spectrum of t_arb1. Figure 3(i) shows two calculated transmission spectra obtained at Sets I_arb1 and II_arb1, indicated as blue circular and red rectangular symbols, respectively. It can be confirmed from the figure that the transmission spectrum at Set II_arb1 shows a red-shift of Δλ/8 in comparison with that at Set I_arb1. On behalf of a wavelength tuning of a certain value δ in the transmission spectrum, a new C_in,arb1 trace should be arranged by making CCW rotation of the C_in,arb1 trace at Set I_arb1 by an angle of 360° × (δ/Δλ) around both AB and n. The WOA set for this new C_in,arb1 trace can be determined by reiterating the angle-finding processes mentioned before. In consequence, the GTF targeted here, or t_arb1, can be continuously wavelength-tuned by varying δ from 0 to Δλ, and this wavelength tunability can be generalized to any transmittance function created in the first-order comb filter. Moreover, this angle-finding scheme can be exploited to implement the wavelength tuning of every transmittance function including the FBTF and NBTF.

Then, to apply the proposed angle-finding method to another GTF, WOA sets for its continuous wavelength tuning are explored with respect to a new arbitrarily chosen transmittance function t_arb2 obtained at a randomly selected WOA set (θ_h1, θ_q1, θ_h2, θ_q2) = (20.0°, 80.0°, 95.0°, 130.0°). Three hundred sixty WOA sets, which can introduce an extra phase shift from 1 to 360° (with a step of 1°) to t_arb2, are quantitatively examined with our search scheme. Figure 4(a) shows four WOAs (θ_h1, θ_q1, θ_h2, θ_q2) with respect to ϕ (from 0 to 360° with a step of 1°), indicated as black squares, red circles, green triangles, and blue inverted triangles, respectively. These WOAs for wavelength tuning of t_arb2 are obtained at the fixed azimuthal angles of PMF 1 and PMF 2, i.e., θ_p1 = 0° and θ_p2 = 22.5°, because they directly depend on θ_p1 and θ_p2. With increasing ϕ, all the four WOAs show alternating patterns, which are totally different with each other. θ_h1(ϕ) and θ_q1(ϕ) have a period of 180°, but remaining two WOAs (θ_h2 and θ_q2) have a period of 360°. θ_h1, θ_q1, θ_h2, and θ_q2 are bounded in 11.9° < θ_h1 < 78.1°, 49.6° < θ_q1 < 130.4°, 76.1° < θ_h2 < 126.4°, and 79.6° ≤ θ_q2 ≤ 145.4°, respectively. It can be found that these four WOAs are much more complex periodical functions of ϕ than those shown in Fig. 1(c). To get more intuitive information on the WOA sets for the GTF tuning, the relationship between any two of θ_h1, θ_q1, θ_h2, and θ_q2 is investigated. Figure 4(b) shows the locus of θ_h1(ϕ) and θ_q1(ϕ) plotted in the Cartesian coordinate system of θ_h1 and θ_q1 in the ϕ range of 0−180°. The point of (θ_h1, θ_q1) on this parallelogram-shaped locus makes one CCW revolution along the locus, starting at (20.0°, 80.0°), while ϕ increases from 0 to 180°. It can be seen that upper and lower sides of this parallelogram-shaped trace are quite denser than other remaining sides, which suggests that a small change in (θ_h1, θ_q1) along the trace gives rise to a relatively larger modulation of ϕ in these regions (upper and lower sides). Although this locus of (θ_h1, θ_q1) is not governed by simple trigonometric equations like the case of t_flat, suggested in our previous study [22], ϕ can be expected to be gradually and continuously changed by selecting (θ_h1, θ_q1) along this locus with (θ_h2, θ_q2) satisfying constraints given by Fig. 4(a).

 

Fig. 4 (a) Four WOAs θ_h1 (black squares), θ_q1 (red circles), θ_h2 (green triangles), and θ_q2 (blue inverted triangles) as a function of extra phase difference ϕ (from 0 to 360° with a step of 1°) for wavelength tuning of the second GTF (t_arb2) at θ_p1 = 0° and θ_p2 = 22.5°. (b) Locus of (θ_h1, θ_q1) plotted using θ_h1(ϕ) and θ_q1(ϕ) of (a) in Cartesian coordinate system (ϕ: 0−180°). (c) Loci of (θ_h2, θ_h1) and (θ_h2, θ_q2), indicated as pinkish circles and greenish squares, respectively, which are plotted using θ_h2(ϕ), θ_h1(ϕ), and θ_q2(ϕ) of (a) in Cartesian coordinate system (ϕ: 0−360°). (d) Loci of (θ_q2, θ_h1) and (θ_q2, θ_q1), indicated as pinkish circles and greenish squares, respectively, which are plotted using θ_q2(ϕ), θ_h1(ϕ), and θ_q1(ϕ) of (a) in Cartesian coordinate system (ϕ: 0−360°).

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Figure 4(c) shows the loci of (θ_h2, θ_h1) and (θ_h2, θ_q2), indicated as pinkish circles and greenish squares, respectively, which are plotted using θ_h2(ϕ), θ_h1(ϕ), and θ_q2(ϕ) of Fig. 4(a) in the Cartesian coordinate system over the ϕ range of 0−360°. The point of (θ_h2, θ_h1) makes a ribbon-like trace with the increase of ϕ from 0 to 360°, and this trace has two-fold symmetry as the period of θ_h2 is twice as long as that of θ_h1. The difference in the density of the point (θ_h2, θ_h1) can readily be found along the trace. Then, the locus of (θ_h2, θ_q2) shows a ϕ-dependent behavior analogous to that of (θ_h1, θ_q1), which makes one CCW revolution along the locus, starting at (95.0°, 130.0°), while ϕ increases from 0 to 360°. Also in this case, upper and lower sides of the trace are much denser than other remaining sides. Through the simultaneous use of the loci of (θ_h1, θ_q1) and (θ_h2, θ_q2), the tuning of ϕ based on the control of (θ_h1, θ_q1, θ_h2, θ_q2) can be more intuitively understood even though the analytic equations governing these behaviors of (θ_h1, θ_q1) and (θ_h2, θ_q2) are difficult to be provided. Figure 4(d) shows the loci of (θ_q2, θ_h1) and (θ_q2, θ_q1), indicated as pinkish circles and greenish squares, respectively, which are plotted using θ_q2(ϕ), θ_h1(ϕ), and θ_q1(ϕ) of Fig. 4(a) in the Cartesian coordinate system (ϕ: 0−360°). The point of (θ_q2, θ_h1) or (θ_q2, θ_q1) makes a double-helix-like or sea-mew-like trace for ϕ increasing from 0 to 360°, respectively. These two traces also have two-fold symmetry because the period of θ_q2 is twice as long as that of θ_h1 or θ_q1. Similarly, in each trace, there is the density difference along the trace. When compared with the case of Fig. 1(c), where the loci of (θ_q2, θ_h1) and (θ_q2, θ_q1) will be just linear lines with θ_q2 fixed at 67.5°, these loci for the wavelength tuning of t_arb2 are far more complicated than those for t_flat. For another GTF different from t_arb2, new loci of (θ_h1, θ_q1, θ_h2, θ_q2) different from Figs. 4(b)–4(d) will be obtained. In brief, it is clearly seen from Fig. 4 that a WOA set (θ_h1, θ_q1, θ_h2, θ_q2) can always be found for any ϕ that increases from 0 to 360° with a step of 1°, which proves that the spectral location of the second selected GTF (i.e., t_arb2) can be continuously tuned. This finding can be extended to any other GTFs.

4. SOP behavior and transmission spectra

To validate if the SOP_in trace of PMF 2 shows the predicted behavior at the WOA sets found by the proposed approach, the spectral evolution of the SOP_in of PMF 2 was calculated within the FSR Δλ at eight WOA sets selected among the above WOA sets for the wavelength tuning of t_arb2 (θ_p1 = 0° and θ_p2 = 22.5°). These eight selected WOA sets, which are designated as Sets I_arb2, II_arb2, III_arb2, IV_arb2, V_arb2, VI_arb2, VII_arb2, and VIII_arb2 in convenience, allow ϕ to be 0, 45, 90, 135, 180, 225, 270, and 315°, respectively. Figures 5(a), 5(b), 5(c), 5(d), 5(e), 5(f), 5(g), and 5(h) show the spectral evolution of the SOP_in of PMF 2 on the SOP_out plane, calculated at Sets I_arb2, II_arb2, III_arb2, IV_arb2, V_arb2, VI_arb2, VII_arb2, and VIII_arb2, respectively, when the wavelength increases from λ₀ to λ₀ + Δλ. As can be seen from Figs. 5(a) and 5(b), the SOP_in trace of PMF 2, denoted by C_in,arb2, and O(λ₀) in the trace are rotated CCW by 45° around AB and n, respectively, when the WOA set switches from I_arb2 to II_arb2. This 45° CCW rotation of the C_in,arb2 trace and its O(λ₀) around AB and n, respectively, can also be checked when the WOA set proceeds step by step from II_arb2 to VIII_arb2. This spectral behavior of the SOP_in of PMF 2 corroborates the WOA sets obtained in Section 3.

 

Fig. 5 Calculated SOP_in traces of PMF 2 at eight selected sets (a) I_arb2, (b) II_arb2, (c) III_arb2, (d) IV_arb2, (e) V_arb2, (f) VI_arb2, (g) VII_arb2, and (h) VIII_arb2, depicted on the SOP_out plane, and (i) calculated SOP_out trace of PMF 2 and eight O(λ₀) locations at Sets I_arb2−VIII_arb2.

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Figure 5(i) shows the calculated spectral evolution of the SOP_out of PMF 2, which is denoted by C_out,arb2, over the wavelength range of λ₀ to λ₀ + Δλ for the eight Sets I_arb2−VIII_arb2. As is the same with the C_out trace in Fig. 2(i), this C_out,arb2 trace can also be obtained by rotating the C_in,arb2 trace CW around AB by ρ(λ) = (360°/Δλ)(λ − λ₀). For the eight selected WOA sets, the trajectory shape of the C_out,arb2 trace remains unchanged, but O(λ₀) has eight different locations, which are equally spaced in wavelength and marked by I_arb2−VIII_arb2. While the wavelength increases from λ₀ to λ₀ + Δλ at Sets I_arb2, the SOP_out of PMF 2 evolves from I_arb2 and returns to I_arb2 via VIII_arb2, VII_arb2, VI_arb2, V_arb2, IV_arb2, III_arb2, and II_arb2, in turn. This evolution feature that the evolution direction according to the wavelength increase is opposite to the increasing order of the eight selected WOA sets makes the transmission spectrum of the second chosen GTF (t_arb2) red-shift when the WOA set proceeds starting from Set I_arb2. This is because the wavelength, at which the same SOP is obtained on the C_out,arb2 trace, increases with increasing WOA set order. Moreover, as mentioned in Section 2, a change in the position of O(λ₀) along the C_out,arb2 trace results in a change in the spectral location of t_arb2. In terms of two adjacent WOA sets (e.g., Sets I_arb2 and II_arb2), the spectral displacement due to the movement of O(λ₀) (e.g., from I_arb2 to II_arb2) corresponds to Δλ/8 in the transmission spectrum because the angular displacements of both the C_in,arb2 trace and its O(λ₀) are equal to 45° ( = 360°/8). This calculated result on the spectral behavior of the SOP_out of PMF 2 confirms again that the GTF t_arb2 can be continuously wavelength-tuned through the WOA sets obtained in Section 3.

Figure 6 shows the calculated transmission spectra of the GTF t_arb2, obtained at the eight WOA sets (Sets I_arb2−VIII_arb2) in the wavelength range from 1548 to 1552 nm. In this calculation of the transmission spectra, the length L and birefringence B of each PMF were set as 7.2 m and 4.166 × 10⁻⁴ to achieve Δλ = ~0.8 nm at 1550 nm, respectively. It can be found from the figure that the first-order comb spectrum with asymmetrically distorted flat-top passbands moves towards a longer wavelength region while the WOA set switches from Set I_arb2 to Set VIII_arb2. If the wavelength of one transmission dip indicated as a red arrow is designated as λ_d,arb at Set I_arb2, λ_d,arb increases by 0.1 nm per set while the WOA set changes from Set II_arb2 to Set VIII_arb2. The inset shows the variation of λ_d,arb, induced by the WOA sets giving eight equi-step values of ϕ starting from 0° (with a step of 45°). Red circles denoted by I_arb2−VIII_arb2 indicate λ_d,arb locations with respect to the eight sets (Sets I_arb2−VIII_arb2). Equispaced symbols and linear λ_d,arb response to ϕ directly tell us that the continuous and predictable frequency tuning is realizable. This theoretical result obviously manifests that the WOA sets derived in Section 3 endow t_arb2 with a continuous wavelength shift within Δλ, implying the capability of continuous wavelength tuning in the GTF.

 

Fig. 6 Calculated wavelength-tuned transmission spectra of the GTF t_arb2, obtained at eight WOA sets (Sets I_arb2−VIII_arb2).

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In order to experimentally verify the predicted wavelength tunability of the GTF t_arb2, the PDLS-based first-order fiber comb filter was constructed using a four-port PBS (OZ Optics), two HWPs (OZ Optics), two QWPs (OZ Optics), and two bow-tie PMF segments (Fibercore) of the same length (~7.12 m) and birefringence (~4.166 × 10⁻⁴), as shown in Fig. 1(a). The PBS and waveplates were pigtailed with 1-m-long single-mode fiber (SMF). The transmission spectra of the fabricated filter were observed using a nearly unpolarized broadband light source (Fiberlabs FL7701) and an optical spectrum analyzer (Yokogawa AQ6370C). In all the measurements, the resolution bandwidth, sample points, and sensitivity of the optical spectrum analyzer were arranged as 0.02 nm, 4001 points (0.005 nm step), and “HIGH1”, respectively. The FSR of the filter, determined by the length and birefringence of PMF, was measured as ~0.8 nm around 1550 nm. Figure 7 shows transmission spectra of the GTF t_arb2, measured at the eight WOA sets (Sets I_arb2−VIII_arb2) in the wavelength range from 1548 to 1552 nm. As predicted in the calculated spectra, when the WOA set is changed from Set I_arb2 to Set VIII_arb2, the first-order comb spectrum, whose spectral shape is similar to one in Fig. 6, redshifts step by step by ~0.1 nm, resulting in a total wavelength displacement of ~0.7 nm. A red-dashed line indicates the calculated transmission spectrum of t_arb2 at Set I_arb2 for comparison between theoretical and experimental spectra. The IL of the calculated spectrum was adjusted to the level of the measured spectrum for clear comparison. The FSR of the fabricated filter is observed to be slightly greater than 0.8 nm.

 

Fig. 7 Experimental wavelength-tuned transmission spectra of the GTF t_arb2, measured at eight WOA sets (Sets I_arb2−VIII_arb2).

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The inset shows the variation of λ_d,arb (indicated as a red arrow), obtained at the eight WOA sets. Likewise, red circles denoted by I_arb2−VIII_arb2 indicate λ_d,arb locations with respect to the eight Sets I_arb2−VIII_arb2. As can be found from the inset, ϕ and λ_d,arb have highly linear relationship where an adjusted R² value is evaluated as ~0.99931. Moreover, the wavelength tunability of t_arb2 was also examined for arbitrary ϕ values from 0 to 360° besides n × (45°) (n = 0, 1, 2, 3, 4, 5, 6, and 7). That is, it is experimentally confirmed that the GTF t_arb2 can be flexibly and continuously wavelength-tuned by adjusting (θ_h1, θ_q1, θ_h2, θ_q2) according to the WOA sets obtained in Section 3. In consequence, it is concluded that any GTF created in our filter can be continuously frequency-tuned by harnessing the WOA sets which can be derived using the proposed angle-finding method. In addition, for the eight comb spectra, the average IL and extinction ratio (ER) were measured as ~4.37 dB and > 21.45 dB, respectively. This IL mainly comes from the PBS, waveplates, and fusion splicing points between PMF and SMF. The ER difference between theoretical and experimental spectra is presumed to be caused by the slight birefringence of SMF pigtailed to the PBS and waveplates [26]. Besides, the temperature or pressure effect on the SOP evolution and thus the filter performance should not be discounted. A temperature- or pressure-induced birefringence change or microbending of PMF and even SMF can modulate the SOP of light propagating through the optical elements within the filter. For an ambient temperature variation within 1−2 °C, this temperature-induced effect is ignorable, but large temperature changes over 10 °C or more can severely aggravate the wavelength-tuning operation of the filter. Similarly, external pressure changes greater than 0.1 MPa can also disturb the filter operation. Accordingly, on behalf of the operation stability of the filter, the whole filter device except for lead-in and lead-out fibers can be hermetically sealed to protect the filter from external temperature or pressure perturbations, and even a thermoelectric cooler may be employed for more robust operation.

5. Conclusions

In summary, we demonstrated theoretically and experimentally the continuous wavelength tunability of the GTF obtainable in the PDLS-based first-order fiber comb filter, comprised of a PBS, two HWPs, two QWPs, and two PMF segments. First, we revisited the previous results on the continuous wavelength tuning of the FBTF (t_flat) and a thorough investigation was made on the relationship between the behavior of the SOP_in’s and SOP_out’s of the second PMF on the Poincare sphere and the frequency tuning of the filter transmittance. The continuous wavelength tunability of the GTF as well as t_flat could be inferred from this qualitative investigation on the principle of the wavelength tuning. SOP conditions for continuous wavelength tuning of the GTF were explained based on the tuning mechanism of t_flat: simultaneous CCW rotation of the PMF 2 SOP_in trace around the slow axis of PMF 2 (i.e., AB) on the SOP_out plane and the normal vector of this trace (i.e., n) on the SOP_in plane. Then, by using the proposed angle-finding approach to seek four WOAs (θ_h1, θ_q1, θ_h2, θ_q2) for continuous wavelength tuning of the GTF (in reverse order from θ_q2 to θ_h1), three hundred sixty sets of (θ_h1, θ_q1, θ_h2, θ_q2), which gave a phase shift from 1 to 360° (with a step of 1°) to a specific transmittance function as an exemplificative GTF, were predicted theoretically. In particular, this angle-finding scheme can be exploited to implement the wavelength tuning of every transmittance function including t_flat. Finally, wavelength-tuned transmission spectra were calculated at eight WOA sets, which were selected among the above angle sets so that they could give additional phase shifts of 0, 45, 90, 135, 180, 225, 270, and 315° to the selected GTF. The eight calculated transmission spectra clearly show that the selected GTF of the filter can be continuously wavelength-tuned by properly choosing WOAs. This qualitative and quantitative prediction on the continuous frequency tuning of the GTF was also experimentally verified. Therefore, it is concluded that any GTF created in our filter can be continuously frequency-tuned by harnessing the WOA sets which can be derived using the proposed angle-finding method. This result can be beneficially utilized for optical applications that require tunable optical comb filters with unique spectral transmittances.