Optical-fiber comb filters have attracted considerable attention for a wide variety of applications in multiwavelength optical systems, such as dense-wavelength-division-multiplexing (DWDM) systems, multiwavelength laser sources, and nonlinear optical switching devices [1-6]. They serve as essential components, due to their capability of selecting desired signals and isolating adjacent signals.

To improve the flexibility of optical comb filters, various approaches to implement continuous wavelength tunability have been proposed, using Sagnac birefringence filters (SBFs) based on birefringence combination [7-12], a double-loop Mach-Zehnder interferometer (MZI) [4, 13, 14], or a fiber Bragg grating [15, 16]. In particular, extensive investigations on comb filters using the polarization-diversified loop configuration (PDLC) have been carried out, due to their ability to switch or continuously tune the pass bands. The authors have proposed zeroth-order tunable comb filters using a section of polarization-maintaining fiber (PMF) [17, 18] and first-order Solc and Lyot comb filters using [19, 20] two sections of PMF based on PDLC, respectively.

Recently, a continuously tunable polarization-independent pass-band-flattened fiber comb consisting of two sections of PMF was proposed and demonstrated experimentally [21]. The filter transmission and orientation-angle sets of wave plates for wavelength tuning were derived through Jones-matrix formulations. The relationship between wavelength tuning and the final state of polarization (SOP) within the PDLC was also discussed. The arbitrary wavelength tunability in the transmission spectrum was verified by theoretical investigation and experimental results. However, an exact analytic expression for a flat-top transmittance function is required to obtain the wave plate’s orientation angles for its continuous wavelength tuning. Besides, the other optical properties of the filter have not been investigated thoroughly.

In this paper, we investigate the characteristics of the tunable pass-band-flattened fiber comb filter. The operating principles of the filter are described, and the orientation-angle loci of the wave plates for wavelength tuning are obtained by the use of the Poincaré sphere. The algorithms for pass-band-shape design and wavelength tuning are presented in detail. The multiple orientation-angle sets of the wave plates are derived using a geometrical method via the Poincaré sphere. The optical characteristics are also analyzed, including group-velocity dispersion and differential group delay (DGD).

Figure 1 shows the schematic of the tunable filter based on the PDLC, which consists of a PBS and two birefringent groups of an HWP, a QWP, and a section of PMF, designated as HWP1, QWP1, and PMF1 in the first group and HWP2, QWP2, and PMF2 in the second group. We denote the PBS’s horizontal and vertical axes as the x axis and y axis respectively. The orientation angle of each optical element is defined as the angle between its slow axis and the x axis. Arbitrarily polarized light incident from the IN port might have x- and y-axis components, which are separated by the PBS. The x component travels in the clockwise (CW) direction, while the y component moves in the counterclockwise (CCW) direction in the PDLC. PMF2 is butt-coupled to the PBS so that the slow axis of PMF2 is oriented at 22.5° with respect to the x axis.

In fiber comb filters based on the PDLC, the filter spectrum is generated by the wavelength-dependent phase retardation Γ = 2πBL/λ between two orthogonal principal states of polarization of the PMF, where B, L, and λ are the birefringence, length of the PMF, and wavelength of light in vacuum respectively. The birefringence and length of both PMFs are 4.166 × 10⁻⁴ and 7.2 m respectively, leading to a channel spacing of 0.8 nm.

Generally, a birefringent element allows a point-of-input SOP (s₁,s₂,s₃) on the Poincaré sphere to be rotated by its phase retardation CCW (looking from its slow axis to its fast axis) around the axis, passing its two eigenpolarization states in the Poincaré-sphere representation [22]. We concentrate our attention here on the x component of the light’s polarization vector, since the optical characteristics can be investigated simply by analyzing the SOP evolution of light traveling CW within the PDLC [22]. The input SOP is x-polarized and represented by a point (1,0,0) on the Poincaré sphere, and will be converted into a linear polarization with azimuthal angle of 4θ_h1 by HWP1. If we assume the wave plates to be achromatic, their output SOPs of broadband light will be identical. After passing HWP1, light passes through QWP1, and its output SOP can be obtained by rotating the SOP around the 2θ_q1 axis CCW by an angle of π/2. PMF1 moves the SOPs by Γ, which is dependent on wavelength, and the output SOPs become a circle centered on the point with an azimuthal angle of 2θ_PMF1 on the equator, with a radius of the distance between the SOP after QWP1 and the center of the circle. Hence, the broadband light source is dispersed into a circular locus by rotation about the axis lying on the equator with longitude 2θ_PMF1, and the locus is rotated by successive HWP2 and QWP2 before PMF2, while remaining circular in shape over the FSR.

For convenience, we designate 1548.0, 1548.2, 1548.4, and 1548.6 nm as λ₁, λ₂, λ₃, and λ₄, and their corresponding phase retardations of the PMF as Γ₁, Γ₂, Γ₃, and Γ₄, respectively. First let us consider the angle combination that produces a transmission spectrum with dip wavelength λ₁. The SOP loci before PMF2 are shown with transmittance phase shift of 0° (red) and 45° (blue) in Fig. 2(a). The red circular locus represents the SOP for dip wavelength λ₁ before PMF2. The SOP of λ₁ remains unmoved, while that of λ₃ moves toward (1,0,0) after PMF2, which correspond to the center wavelengths of the stop band and pass band respectively. For a 0.1-nm shift (ΔΓ = 45°), the circle is rotated around the 2θ_PMF1 = 45° axis tightly between two green circles, and spins at the same time CCW by 45°, becoming the blue circle shown in Fig. 2(a). Thus the SOP of 1548.1 nm (= (λ₁+λ₂)/2), which is located simultaneously on the blue and green circles, will move to (-1,0,0) through rotation by 45° CW, and will be extinguished at the output port. Two circular loci (red and blue) are further dispersed after PMF2, but the final loci are the same in shape but different in wavelength location, implying wavelength tuning. The wavelength locations are those at phase shift 0° (red) and 45° (blue) in Fig. 2(b).

The corresponding final polarization ellipses after PMF2 at the individual wavelengths of light shown in Fig. 2(b) are shown in Figs. 3(a) and 3(b) for the two cases 0° and 45° respectively.

To see the tunability and flat pass band clearly, the SOP loci before (red circles) and after PMF2 (blue circles) of light traveling CW within the PDLC are replotted as world maps in Figs. 4(a), 4(b), 4(c), and 4(d), where the center wavelength of the stop band corresponds to λ₁, λ₂, λ₃, and λ₄ respectively. The SOP loci after PMF2 are identical in shape but different in wavelength location, leading to wavelength tunability of the filter. From the isocontours of transmission shown in Fig. 4(a), it is obvious that the 2-dB bandwidth is larger than 0.4 nm, since Δλ = λ₄ - λ₂ = 0.4 nm.

To clearly observe continuous wavelength tunability of the filter, the orientation-angle sets of four rotatable wave plates (θ_h1, θ_q1, θ_h2, θ_q2) for individual center wavelength λ_p of the pass band are investigated using the method explained above for θ_p1 = 0°. The loci of (θ_h1, θ_q1) and (θ_h1, θ_h2) are shown in Figs. 5(a) and 5(b) respectively. The tuning over the FSR is implemented for the angle ranges −16.4° ≤ θ_h1 ≤ 16.4°, 67.5° ≤ θ_q1 ≤ 112.5°, and −56.2° ≤ θ_h2 ≤ 34.1° respectively. The projection of (θ_h1, θ_q1, θ_h2, θ_q2) onto the (θ_h1, θ_q1) plane looks like a Lissajous pattern, while that into (θ_h1, θ_h2) becomes a spiral. All trajectories have the periodicity of the FSR (0.8 nm), and θ_q2 is fixed at 67.5°.

To investigate the trend, the loci for various values of θ_p1 are shown in Fig. 6(a) and 6(b). For variation of θ_p1 with a fine wavelength-shift step of 0.005 nm, the loci are always obtained, and thus there is no need to delicately align a PMF when splicing, which becomes favorable for implementing the present filter.

To demonstrate continuous tunability, the filter was constructed using a fiber-pigtailed four-port PBS (OZ Optics^TM), two fiber-pigtailed HWPs (OZ Optics^TM), two fiber-pigtailed QWPs (OZ Optics^TM), and two equal-length bow-tie-type PMF segments (Fibercore^TM), as shown in Fig. 1. A broadband source (BBS) was amplified with a spontaneous emission source (Fiberlabs FL7701^TM), and an optical spectrum analyzer (OSA) (Yokogawa AQ6370C^TM) was employed to measure the transmission spectra of the filter. The calculated and measured transmission spectra are shown in Figs. 7(a)~7(d) for dip wavelengths of 1548.0, 1548.2, 1548.4, and 1548.6 nm respectively. One can see that the calculated and measured transmission spectra are shifted by 0.2 nm. The insertion loss was measured as 5.87 dB, which originates from the PBS, the butt-coupling of PBS and PMF2, inherent insertion losses of HWPs and QWPs, and fusion splicing of PMFs and single-mode fibers for wave-plate pigtails.

It is often desirable that comb filters have both a flattened pass band and small dispersion in each pass band. To validate the optical properties of the filter, further simulations were carried out. The calculated pass band and its corresponding group-delay time are shown in Fig. 8(a). The 3-dB bandwidth is calculated to be 0.5125 nm, which is larger than that of a zeroth-order comb filter by 0.4 nm [18].

The dispersion and differential group delay are also essential parameters for designing wavelength-selective filters in optical networks. The group-velocity dispersion is defined as the derivative of the group-delay time with respect to the wavelength, as follows:

where τ_g is the group-delay time.

The differential group delay (DGD) of a device is defined as the maximum difference in group delay between all polarization states. The calculated group-velocity dispersion and DGD with each maximum value of 15.5 ps/nm and 14.2 ps are shown in Fig. 8(b), which features good characteristics, compared to fiber gratings [23].

We will show in this section that for given orientation angles of the wave plates, the angle combinations that produce the same transmission spectrum can be deduced by the use of the Poincaré sphere. Let us consider an angle locus that covers the whole FSR, and assume that (θ_h1, θ_q1, θ_h2, θ_q2) is a point on it. The HWP1 rotates the incident light by π into a point on the equator making an angle of 4θ_h1 with respect to the S₁ axis, and thus θ_h1 has a periodicity of π/2. Similarly, the HWP2 rotates the SOP circle after PMF1 by π and locates its center at a point that is apart arc distance 4θ_h2, also implying a periodicity of π/2. If we assume a new angle set (θ_h1′, θ_q1′, θ_h2′, θ_q2′), these relations are expressed as follows:

The S₁-S₂ projection planes from the Poincaré sphere for multiple angle sets in a locus that result in the same transmission spectrum through the combination of the respective optical elements are depicted in Fig. 9. Suppose that point A represents the SOP after QWP1, as shown in Fig. 9(a). If θ_q1′ = θ_q1 + π/2, the condition for the new SOP after QWP1 to reach point A is such that 4θ_h1′ is symmetric to 4θ_h1 with respect to 2θ_q1; that is, θ_h1′=−θ_h1 + θ_q1, as shown in Fig. 9(a).

Meanwhile, let us consider another case for the combination of HWP1, QWP1, PMF1, HWP2, and QWP2. In Fig. 9(b), if 4θ_h1′ and 4θ_h2′ are symmetric to 4θ_h1 and 4θ_h2 with respect to 2θ_p1, and θ_q2′ = θ_q2 + π/2, the new angles are as follows:

Suppose that B and C represent the centers of circular SOPs after QWP2, whose azimuthal angles are 4θ_h2 − 2θ_p1 and 4θ_h2′−2θ_p1 respectively. For the same circular SOP after QWP2, B and C must be symmetric with respect to QWP2, and thus the relation for the orientation angles of HWP2 and the other elements is as follows:

The multiple loci obtained as explained earlier are shown in Figs. 10(a) and 10(b) for (θ_h1, θ_q1) and (θ_h1, θ_h2) respectively. Periodic gaps occur and are calculated to be θ_h1, θ_q1, and θ_h2 (Δθ_h1, Δθ_q1, Δθ_h2) as 57.2°, 45°, and 57.2° respectively. It is noteworthy that the locus at the center of the (θ_h1, θ_q1) figure consists of two opposite directional traces that are rotating CW and CCW.

We characterize the optical properties of a tunable pass-band-flattened fiber comb filter based on the PDLC, using the Poincaré-sphere representation. The flatness of the pass band is checked quantitatively in the design process by visualization of the SOP through a world map. The working principle of the filter is described, and tuning performance is verified experimentally. The pass-band-shape-design and wavelength-tuning algorithms are presented in detail. The orientation-angle loci of the wave plates, and their periodicities through the combination of the respective optical elements, are deduced using a geometrical method via the Poincaré sphere. The optical characteristics are investigated, including transmission, group-velocity dispersion, and DGD. It is found that the 3-dB bandwidth, maximum dispersion, and DGD of the filter are respectively 0.5125 nm, 15.5 ps/nm, and 14.2 ps over the FSR of 0.8 nm. The characterization in this work provides a qualitative understanding of the optical properties of the fiber comb filter based on the PDLC, and this filter may find significant applications in multiwavelength optical systems.