1 Introduction

Owing to simple structure, fiber compatibility, and ease of use, fiber comb filters have been considered as useful optical signal-processing elements that can be utilized to route transmission signals or block unwanted signals causing crosstalks in wavelength-division-multiplexed optical networks. In addition to signal-processing devices, fiber comb filters can also be applied to fiber lasers [1, 2], microwave transversal filters [3], optical pulse train generation [4], and so on. The ability to precisely adjust the transmission spectrum of a comb filter to a desired wavelength location can greatly enhance its efficiency when the filter is used in wavelength-routing devices or multiwavelength sources. Numerous studies have been conducted to impart tunability to a comb filter using a Sagnac birefringence loop [1, 5, 6], Lyot-type birefringence combination [7, 8], fiber gratings [9, 10], Mach–Zehnder interferometers (MZIs) [11–13], and polarization-diversity loop structure (PDLS) [14–16]. Particularly, comb filters based on the PDLS [17, 18] formed through a polarization beam splitter (PBS) have great advantages in that wavelength switching or tuning can be performed [14, 19], which cannot be realized in fiber-coupler-based Sagnac birefringent filters [20]. Moreover, PDLS-based comb filters are basically more robust to environmental perturbations compared with MZI-based comb filters [11–13] and independent of input polarization unlike Lyot-type comb filters [7, 8] or some Sagnac interferometer-based comb filters [1]. The wavelength tuning of fiber gratings can be obtained by applying tension, pressure, or temperature to them. However, this tuning technique can adversely affect the durability and reliability of the grating, including the performance degradation of spectral properties such as bandwidth, insertion loss (IL), and channel isolation.

The spectral tuning of the PDLS-based comb filters has already been accomplished using the wave retarder combination (WRC) of a half-wave retarder (HWR) and a quarter-wave retarder (QWR) in this sequence before high birefringence fiber (HBF) [14]. This WRC can give a phase change in the range of 0 to 2π to the filter transmittance. Up to now, however, other WRC groups, e.g., two QWRs, two HWRs, and a set of a QWR and an HWR, have not been suggested for the realization of frequency-tunable PDLS-based comb filters. On top of that, another HWR was also used in the previous study [14] to align the effective azimuth angle of the slow axis of HBF to ±45° for the horizontal axis of the PBS. The use of this HWR can degrade the structural simplicity and operational convenience of the comb filter or may worsen filter loss and spectral distortion, caused by its wavelength-dependent phase retardation difference.

In this paper, we propose a continuously tunable polarization-independent zeroth-order fiber comb filter based on the PDLS using some selected optimal WRC groups, one set of which is composed of two wave retarders. The selected WRC groups contain a set of two QWRs, a set of a QWR and an HWR, and a set of an HWR and a QWR. The proposed filter was constructed using a fiber-pigtailed PBS, one of the three selected WRC groups, and HBF. One end of HBF was butt-coupled to one port of the PBS so that the slow axis of HBF should be oriented at 45° with respect to the horizontal axis of the PBS, and the other end was connected to the WRC group. Three kinds of comb filters were fabricated by utilizing the three selected WRC groups. As far as we know, this is the first demonstration of a PDLS-based polarization-independent zeroth-order fiber comb filter, of which spectral tuning can be performed with only two wave retarders. First, we investigated light polarization conditions needed to continuously tune the transmission spectrum of the filter in the viewpoint of input and output states of polarization (SOPs) of HBF. Then, three filter transmittances were theoretically derived through Jones matrix formulation, and eight special azimuth angle sets of two wave retarders, which gave the transmittance function eight kinds of phase shifts of 0 to –7π/4 with a step of –π/4, were found based on the transmittance derived for each WRC group. It was experimentally verified that filter transmission spectra could be wavelength-tuned by ~0.1 nm whenever an azimuth angle set of two wave retarders was switched to an adjacent set for all three fabricated filters. It was also experimentally confirmed that azimuth angles of each selected WRC group, which induced an arbitrary phase shift in the transmittance function, could be always found, that is, the fabricated filter could be continuously tuned within one channel bandwidth by the appropriate control of wave retarder azimuth angles. This continuous tuning feature of our filter may be beneficially applied to optical label switching [21, 22] and waveband switching [23, 24] besides multiwavelength lasing, microwave and optical signal processing, and optical sensor interrogation. Moreover, when the filter is involved in integrated optic devices, its structural simplicity is expected to contribute to enhancing their operation stability and performance.

2 Principles of operation

Figure 1a shows a schematic diagram of the proposed filter composed of a fiber-pigtailed PBS, a WRC group composed of two wave retarders denoted by WR 1 and WR 2, and HBF connected to port 2 of the PBS. The slow axis of HBF has an angle offset of 45° with respect to the horizontal axis of the PBS. WRC groups (or WR 1, WR 2) chosen for filter tuning include (QWR, QWR), (QWR, HWR), and (HWR, QWR), designated as WRC groups A, B, and C, respectively. Figure 1b shows the optical path of an input beam propagating through the filter. An input beam to the PBS is divided into linear horizontally and vertically polarized beams, which emerge from ports 1 and 2 of the PBS, respectively. These two beams with SOPs of linear horizontal polarization (LHP) and linear vertical polarization (LVP) propagate through the polarization-diversity fiber loop in clockwise (CW) and counterclockwise (CCW) directions, respectively. In a 45°-oriented HBF segment sandwiched between two polarizers, shown in Fig. 1b except for wave retarders, polarization interference is created due to phase difference between two linear polarization components aligned along the principal axes of HBF [19]. The wave retarders can spectrally shift the interference spectrum, which is caused by a change in a free spectral range (FSR), by varying the effective birefringence of the entire fiber loop [19].

Fig. 1
figure 1

a Schematic diagram of the proposed filter and b optical path of input beam propagating through filter

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If we assume the horizontal and vertical axes of the PBS as x- and y-axes in Fig. 1b, respectively, an input beam passes through a horizontal polarizer (x-axis), WR 1 (with its slow axis oriented at θ 1 for the x-axis), WR 2 (with its slow axis oriented at θ 2), HBF (with its slow axis oriented at 45°), and horizontal polarizer (x-axis) along the CW path or through a vertical polarizer (y-axis), HBF (−45° oriented), WR 2 (−θ 2 oriented), WR 1 (−θ 1 oriented), and vertical analyzer (y-axis) along the CCW path. Here, F and S indicate the fast and slow axes of birefringent elements, respectively, such as wave retarders and HBF. As described above, an interference spectrum is generated in each path due to polarization interference. These two interference spectra have an equal FSR, but their ILs are different depending on input polarization. Because their polarization states are orthogonal with each other, their optical power superposition gives the output power spectrum of the filter, and the filter output spectrum is input-polarization-independent [19].

The filter transmission spectrum, or a sinusoidal function of a phase difference Γ between fast- and slow-axis components emerging from HBF [19], can be frequency-shifted by varying the effective birefringence of the fiber loop, that is, adding a phase retardation difference φ to Γ as shown in Fig. 2. Here, Γ is given by 2πBL/λ, where B, L, and λ are the HBF birefringence, HBF length, and wavelength in vacuum, respectively. The sinusoidal function of Γ is given by cos(2πBL/λ) in a polarization interference spectrum, of which FSR is determined by B and L of the HBF segment. This sinusoidal transmittance function is defined as the zeroth-order one [25]. When the number of HBF segments used for polarization interference is increased up to an integer N (≥2), the transmittance function contains cosN(2πBL/λ), which is indicated as a (N − 1)th-order one. If the additional phase difference φ varies from 0 to 2π, the comb spectrum moves in the wavelength domain by one FSR, that is, it is fully tuned within a channel spacing. φ can be modulated by altering the SOPin of HBF, which means that the proper choice of the SOPin allows φ to have values from 0 to 2π. For instance, in the case of an SOPin of right circular polarization (RCP), a slow-axis component leads a fast-axis one by π/2, resulting in φ of −π/2. Hence, the spectral tuning of the filter can be achieved by varying the SOPin of HBF with a suitable WRC group.

Fig. 2
figure 2

Spectral evolution of SOPout of HBF in CW path for eight special SOPin sets (from Set I to VIII)

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In order to find a group of the SOPin’s, which enable the filter wavelength tuning, the spectral evolution of the SOPout of HBF was examined for the eight special SOPin’s when input light passes through the filter along the CW path, as shown in Fig. 2. The CCW path is not dealt with here because the same result is obtained. To prevent confusion in the determination of polarization ellipses, it is assumed that an observer at port 1 of the PBS sees light leaving him in the CW path. The eight special SOPin’s include LHP, horizontally oriented right elliptical polarization (HOREP), RCP, vertically oriented right elliptical polarization (VOREP), LVP, vertically oriented left elliptical polarization (VOLEP), LCP, and horizontally oriented left elliptical polarization (HOLEP), displayed in turn from Set I to Set VIII in Fig. 2. These SOPin’s allow light entering HBF to have φ values of 0, −π/4, −π/2, −3π/4, −π, −5π/4, −3π/2, and − 7π/4, in sequence. With the increase of wavelength, the SOPout of HBF evolves with a certain period that determines the FSR Δλ of the filter transmission spectrum. For example, if the SOPin is LHP, the SOPout varies with wavelength, starting from LHP at a certain wavelength λ n and ending up with LHP at another wavelength λ n+1 closest to λ n (λ n  < λ n+1), resulting in Δλ = λ n+1 − λ n . While the wavelength varies from λ n to λ n+1 by Δλ/8, the SOPout changes in the order of LHP, HOREP, RCP, VOREP, LVP, VOLEP, LCP, HOLEP, and LHP, as shown in eight polarization ellipses indicated as 1 to 8 at Set I of Fig. 2, except for the last ellipse (or LHP). This SOPout evolution allows the polarizer output to have a one-period comb spectrum whose maximum and minimum transmittances occur when SOPout’s are LHP and LVP, respectively. For different SOPin’s, modified φ shifts the above SOPout evolution in the wavelength domain, resulting in the spectral tuning of the filter.

In the case of Set I where the SOPin is LHP (φ = 0), if the SOPout is assumed to vary from LHP to HOLEP with the increase of wavelength from λ 1 to λ 8 (=λ 1 + 7Δλ/8) by Δλ/8, maximum and minimum transmittances occur at λ max = λ 1 and λ min = λ 5 (=λ 1 + Δλ/2), respectively. In Set II where the SOPin is HOREP (φ = −π/4), the SOPout changes with increasing wavelength from λ 1 to λ 8 in the order of HOREP, RCP, VOREP, LVP, VOLEP, LCP, HOLEP, and LHP. This SOPout evolution makes the maximum and minimum transmittance wavelengths (λ max and λ min) become λ 8 and λ 4, respectively, resulting in a spectral shift of − Δλ/8 in comparison with Set I, which implies that the transmission spectrum is blue-shifted by Δλ/8 due to an additional phase difference φ of −π/4. Similarly, in Set III where the SOPin is RCP (φ = −π/2), the spectral evolution of the SOPout is obtained as follows: RCP at λ 1, VOREP at λ 2, LVP at λ 3, VOLEP at λ 4, LCP at λ 5, HOLEP at λ 6, LHP at λ 7, and HOREP at λ 8. Additional spectral shift of the SOPout evolution array by Δλ/8 causes λ max and λ min to be λ 7 and λ 3, respectively, and the transmission spectrum to be blue-shifted by Δλ/4, compared with Set I. Further, Sets IV, V, VI, VII, and VIII, at which the SOPin’s are VOREP (φ = −3π/4), LVP (φ =−π), VOLEP (φ = −5π/4), LCP (φ = −3π/2), and HOLEP (φ = −7π/4), allow (λ max, λ min) to be (λ 6, λ 2), (λ 5, λ 1), (λ 4, λ 8), (λ 3, λ 7), and (λ 2, λ 6), resulting in the blue-shifts of 3Δλ/8, Δλ/2, 5Δλ/8, 3Δλ/4, and 7Δλ/8, respectively, compared with Set I. As a result, seven special sets (Sets II–VIII) can cause a comb spectrum obtained at Set I to be sequentially blue-shifted by Δλ/8 to 7Δλ/8 with a step of Δλ/8. Other SOPin’s for continuous frequency tuning can also be effortlessly found considering these special SOPin’s.

The above-mentioned qualitative discussion can be quantitatively supported using Jones matrix formulation [26]. The theoretical filter transmittance can be obtained from the transfer matrix T given by (1). In this formulation, any ILs of optical components contained in the filter are not included, and it is assumed that the wave retarder birefringence is not dispersive. Further detailed formulation can be found in [19].

$$\begin{aligned} T = \left[ {\begin{array}{*{20}c} 1 \quad 0 \\ 0\quad 0 \\ \end{array} } \right]T_{{{\text{HBF}}}} (45^\circ )T_{{{\text{WR2}}}} (\theta _{2} )T_{{{\text{WR1}}}} (\theta _{1} )\left[ {\begin{array}{*{20}c} 1\quad 0 \\ 0 \quad 0 \\ \end{array} } \right] \\ + \left[ {\begin{array}{*{20}c} 0 \quad 0 \\ 0 \quad1 \\ \end{array} } \right]T_{{{\text{WR1}}}} ( - \theta _{1} )T_{{{\text{WR2}}}} ( - \theta _{2} )T_{{{\text{HBF}}}} ( - 45^\circ )\left[ {\begin{array}{*{20}c} 0 \quad 0 \\ 0 \quad 1 \\ \end{array} } \right] \\ \end{aligned},$$
(1)

where T WR1, T WR2, and T HBF are the Jones matrices of the WR 1, the WR 2, and HBF that have slow-axis azimuth angles of θ 1, θ 2, and 45° with respect to the x-axis, respectively. At the WRC group A, the WR 1 and WR 2 are two QWRs with azimuth angles of θ A1 and θ A2, designated as QWR 1 and QWR 2, respectively. At the WRC groups B and C, the WR 1 and WR 2 are a set of a QWR and an HWR with azimuth angles of θ B1 and θ B2 and a set of an HWR and a QWR with θ C1 and θ C2, respectively. Filter transmittances t A, t B, and t C shown in (2), (3), and (4) can be obtained from (1) for the three WRC groups, respectively.

$$\begin{aligned} t_{{\text{A}}} = \frac{1}{2} + \frac{1}{2}\left[ {\cos (2\theta _{{{\text{A}}1}} - 2\theta _{{{\text{A}}2}} )\cos 2\theta _{{{\text{A}}1}} \cos 2\theta _{{{\text{A}}2}} - \sin 2\theta _{{{\text{A}}1}} \sin 2\theta _{{{\text{A}}2}} } \right]\cos \Gamma \\ + \frac{1}{2}\left[ {\sin (2\theta _{{{\text{A}}1}} - 2\theta _{{{\text{A}}2}} )\cos 2\theta _{{{\text{A}}1}} } \right]\sin \Gamma \\ \end{aligned}$$
(2)
$$t_{{\text{B}}} = \frac{1}{2} + \frac{1}{2}\left[ {\cos (2\theta _{{{\text{B}}1}} - 2\theta _{{{\text{B}}2}} )\cos 2\theta _{{{\text{B}}1}} } \right]\cos \Gamma + \frac{1}{2}\sin (2\theta _{{{\text{B}}1}} )\sin \Gamma$$
(3)
$$t_{{\text{C}}} = \frac{1}{2} + \frac{1}{2}\left[ {\cos (4\theta _{{{\text{C}}1}} - 2\theta _{{{\text{C}}2}} )\cos 2\theta _{{{\text{C}}2}} } \right]\cos \Gamma + \frac{1}{2}\left[ {\sin (4\theta _{{{\text{C}}1}} - 2\theta _{{{\text{C}}2}} )} \right]\sin \Gamma$$
(4)

When considering wave retarder angle sets that can induce an arbitrary phase shift from 0° to −360° in the transmittance function for the three WRC groups, the loci of the wave retarder angles are very different depending on WRC groups. At the WRC group A, the loci of (θ A1, θ A2) have a period of π (180°) for both θ A1 and θ A2 and are wave-like curves aligned along a diagonal direction in the Cartesian coordinate system of θ A1 and θ A2. In the case of the WRC group B, the loci of (θ B1, θ B2) are linear curves aligned along a diagonal direction in the Cartesian coordinate system of θ B1 and θ B2 with periods of π (180°) and π/2 (90°) for θ B1 and θ B2, respectively. Finally, in the WRC group C, the loci of (θ C1, θ C2) are linear curves aligned along a horizontal (θ C1 axis) direction in the Cartesian coordinate system of θ C1 and θ C2 with periods of π/2 (90°) and π (180°) for θ C1 and θ C2, respectively. The linear loci of (θ B1, θ B2) make it easy to tune the transmittance function, compared with those of (θ A1, θ A2), because θ B2 can be determined by θ B1. In particular, the WRC group C can provide the easiest tuning scheme of the transmittance function among the three WRC groups, as θ C2 can be fixed at one angle value.

On the basis of the derived transmittances, azimuth angle sets of the wave retarders, at which spectral tuning operation could be realized, were investigated for the three WRC groups. Table 1 shows eight special wave retarder azimuth angle sets (θ 1, θ 2) for the discrete tuning operation and the corresponding filter transmittances for the three WRC groups. Eight special angle sets, which transform LHP (SOP before the WRC group) into eight SOPin’s corresponding to Sets I, II, III, IV, V, VI, VII, and VIII, were determined so that a raised cosine transmittance function, expressed by (1 + cosΓ)/2, had phase delays of 0, –π/4, –π/2, –3π/4, –π, –5π/4, –3π/2, and –7π/4, respectively. Moreover, the QWR and HWR have π and π/2 angular symmetry in the three transmittances, respectively. As can be seen from the table, the filter transmission spectrum is blue-shifted by Δλ/8, Δλ/4, 3Δλ/8, Δλ/2, 5Δλ/8, 3Δλ/4, and 7Δλ/8 at Sets II, III, IV, V, VI, VII, and VIII, respectively, when the transmittance at Set I, that is, (1 + cosΓ)/2, is taken as the reference transmittance. These eight special angle sets for the discrete wavelength tuning can be found in all three WRC groups. On the basis of these special angle sets, other wave retarder angle sets for continuous wavelength tuning can also be found for the three WRC groups. On the other hand, a WRC group composed of two HWRs is not chosen for the tuning because it is impossible for this WRC group to convert LHP into the above eight SOPin’s.

Table 1 Eight special wave retarder azimuth angle sets for discrete tuning operation and corresponding filter transmittances for three WRC groups
Full size table

The filter transmittances predicted in Table 1 are displayed as comb spectra in Fig. 3. Figure 3 shows the calculated transmission spectra of the filter, obtained at the eight special wave retarder angle sets (Sets I–VIII) of Table 1. In this calculation, the length L and birefringence B of HBF were set as 7.2 m and 4.166 × 10−4 so that the filter FSR became ~0.8 nm around 1550 nm, respectively. It can be seen from the figure that the transmission spectrum has one of transmission minima at ~1550.0 nm in the case of Set I and this dip moves toward a shorter wavelength region by 0.1 nm at Set II. When the wave retarder angle set is selected as from Set III to Set VIII, the transmission dip at 1550.0 nm is blue-shifted by 0.2 to 0.7 nm with a step of 0.1 nm. In other words, it is confirmed that the full wavelength tuning within the FSR can be achieved using the special wave retarder angles given in Table 1 for the three WRC groups.

Fig. 3
figure 3

Discrete wavelength tuning of filter transmission spectrum, obtained at eight special wave retarder azimuth angle sets (Sets I–VIII)

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3 Experimental results and discussion

In order to verify the theoretical prediction of the wavelength tunability of the filter, it was fabricated using a fiber-pigtailed four-port PBS (OZ Optics), one of three selected WRC groups, and bow-tie type HBF (Fibercore). One end of HBF was butt-coupled to the PBS with its slow axis being oriented by 45° for the horizontal axis of the PBS, and the other end was connected to the WRC group. Three kinds of comb filters were constructed using the three selected WRC groups such as two QWRs (OZ Optics), a set of a QWR and an HWR (OZ Optics), and a set of an HWR and a QWR. The WRC group was placed in front of HBF whose birefringence and length were ~4.166 × 10−4 and ~7.12 m, respectively. The wavelength spacing between transmission minima was ~0.81 nm around 1550 nm. A broadband light source (Fiberlabs FL7701) and an optical spectrum analyzer (Yokogawa AQ6370C) were utilized to measure the transmission spectrum of the fabricated filter. The resolution bandwidth of the optical spectrum analyzer was set as 0.02 nm in the measurements.

Figure 4a–c shows wavelength-tuned transmission spectra measured at the eight special sets (Sets I–VIII) for WRC groups A, B, and C, shown in Table 2, respectively, and the insets of Fig. 4 show the transmission spectra superimposed over the eight special sets. ILs were measured as ~4.74, ~3.46, and ~3.47 dB for WRC groups A, B, and C, respectively. Main sources of the loss are the PBS (including fiber coupling loss), two wave retarders, and fusion splicing loss between HBF and single-mode fiber (SMF) used for wave retarder pigtails. And the extinction ratios were measured as ~20.33, ~21.76, and ~21.07 dB for WRC groups A, B, and C, respectively. It is observed from the superimposed plot that the IL is nearly uniform and the ER is more than 20 dB over 10 nm. In particular, the spectral flatness between channels was measured as ~0.08 dB for the WRC group A, ~0.32 dB for the WRC group B, and ~0.21 dB for the WRC group C, respectively. As predicted in the theoretical analysis, the filter transmission spectrum shifted toward shorter wavelength by ~0.1 nm whenever the wave retarder angle set was changed to a neighboring set, e.g., from Set I to Set II shown in Table 2, for all three WRC groups. Deviations between the predicted and measured wave retarder angle sets seem to be caused by the weak form birefringence of SMF used for fiber pigtails of the PBS and wave retarders, which can act as another wave retarder [27]. There was no noticeable difference in the wavelength-tuned transmission spectra among the three WRC groups. It was also confirmed that the filter could be continuously tuned within the FSR by specifying appropriate wave retarder angles outside the angle sets in Table 2. In order to check the angle tolerance of the proposed filter, variations of the IL, ER, and dip wavelength were investigated with respect to orientation angle deviations in the wave retarders of ±2° with a step of 0.5° at Set I for the three WRC groups. In all three WRC groups, the IL and ER could be kept as <0.066 and >20.7 dB for angle deviations of ±2°, respectively. For these angle deviations, the maximum deviation of the dip wavelength was ~0.04 nm, resulting in an angle sensitivity of ~0.01 nm/°. Moreover, input polarization independence was also checked using an additional polarization control instrument (Agilent 8169 A), and the maximum polarization sensitivity was measured as <0.5 dB.

Fig. 4
figure 4

Wavelength-tuned transmission spectra measured at eight special sets (Sets I–VIII) for WRC groups a A, b B, and c C. Insets show the transmission spectra superimposed over the eight special sets

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Table 2 Experimentally obtained eight special wave retarder azimuth angle sets for discrete tuning operation for three WRC groups
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4 Conclusions

In summary, we demonstrated three kinds of continuously tunable polarization-independent zeroth-order PDLS-based fiber comb filters by incorporating three WRC groups, that is, a set of two QWRs, a set of a QWR and an HWR, and a set of an HWR and a QWR. Each filter was constructed using a fiber-pigtailed PBS, one of the three selected WRC groups, and HBF. One end of HBF was butt-coupled to one port of the PBS so that the slow axis of HBF should be oriented at 45° with respect to the horizontal axis of the PBS, and the other end was connected to the WRC group. First, we explored light polarization conditions required for the continuous tuning of the filter transmission spectrum in the viewpoint of SOPin’s and SOPout’s of HBF. Next, three filter transmittances were analytically derived by utilizing Jones matrix formulation. Then, eight special azimuth angle sets of two wave retarders, which gave the transmittance function eight kinds of phase shifts of 0 to –7π/4 with a step of –π/4, were found based on the transmittance derived for each WRC group. Theoretical prediction was verified by experimental demonstration. It was found that the filter transmission spectrum shifted toward shorter wavelength by ~0.1 nm whenever the wave retarder angle set was changed to a neighboring set for all three WRC groups. It was also experimentally confirmed that azimuth angles of each selected WRC group, which induced an arbitrary phase shift in the transmittance function, could be always found, that is, the fabricated filter could be continuously tuned within one FSR by the appropriate control of wave retarder azimuth angles.