1. Introduction

Bridging the microwave and the mid-infrared, terahertz (THz) radiation (typically from 0.1 THz to 10 THz) has sparked an increasingly wide variety of applications, including homeland security, non-destructive material inspection, biology spectroscopy, as well as ultra-fast ultra-broadband wireless communication [1, 2], thanks to the low photon energy, the high transparency and the wideband characteristics of the THz radiation. In order to build integrated functional THz systems [3], essential components handling THz waves are in great demand, such as waveguides, filters, beam splitters/combiners, polarizers, and modulators. Nevertheless, the key obstacle hindering the development of practical, compact and flexible THz devices is the lack of low-absorption materials transporting THz waves. Because dry air is considered the most commonly lossless medium in the THz field, one solution is manipulating THz waves with additional components in free-space. For instance, the spatial modulator holds the ability to control THz wavefront [4], whilst the photo-designed THz device offers frequency modulation [5]. In addition, novel structured devices (e.g., photonic crystals and metamaterials) have been investigated and used as resonator [6], bandpass filter [7], polarizer [8], metamodulator [9] and broadband plasmonic absorber [10] in THz field. In contrast to spatially separated components, THz functional devices integrated on waveguides offer an alternative feasible option, being beneficial to achieving robust, compact THz systems. To lower the propagation loss, nowadays THz waveguide designs generally incorporate an air-core with innovative cladding structures [11–16], or guidance along the surface of a subwavelength waveguide [17–19]. Accelerated by the rapid development of THz waveguides, waveguide-based couplers [20], splitters [21], filters [22], resonators [23], and (de)multiplexers [24] have been demonstrated to manipulate the THz radiation, which have found exciting applications in sensing [25], spectroscopy [26], and wireless communication systems [22]. Waveguide-based THz gratings, featuring passband or stopband at desired frequencies, have attracted substantial attention. In 2012, Zhou et al. reported THz Bragg gratings written on a polymer fiber [27]. Limited by the circularly asymmetric inscription, only one polarized mode can be fully modulated by the notch-type grating, remaining the other polarization state much less modulated. Assisted with an extra paper grating, a plasmonic two-wire waveguide based THz grating has been realized, which filters the wave polarized perpendicular to the paper grating [28]. Recently, 3D printed THz metallic waveguide gratings have been applied for dispersion compensation, which is a basic signal processing element for transmission link [22]. Aforementioned waveguide-based THz gratings filter either only one polarization state, or the degenerate modes.

Manipulation of THz polarization-maintaining radiation is one of the key processes in the context of polarization-sensitive communication [29], sensing [25, 30], imaging [31] and spectroscopy [32] in THz range. Taking THz wireless communication link as an example, the polarization-division-multiplexing technique could double the transmission capacity, where two orthogonally polarized THz beams simultaneously propagate [33]. By deliberately introducing structural anisotropy in waveguide cross-section, massive efforts have been directed at achieving highly birefringent THz waveguides, including photonic crystal fibers [34–36], dielectric-coated metallic hollow fibers [37], metamaterial waveguides [38], pipe waveguides [39] and porous fibers [40]. In the optical communication band, gratings inscribed in polarization-maintaining optical fibers have been successfully implemented in optical fiber based sensor [41], laser [42], and mode converter [43]. However, “on-line” filtering (i.e., filtering electromagnetic wave during propagation along the waveguide) of polarization-maintaining waves remains an unaddressed challenge at THz field, despite the importance of such capability for polarization-sensitive THz systems.

Here, we propose a subwavelength birefringent polymer waveguide for on-line filtering of two orthogonally polarized THz waves operating, i.e. waveguide-based polarization-maintaining filters. We demonstrate THz bandstop and bandpass filters for both polarization states, which are achieved by incorporating Bragg grating (periodic structural perturbations) along the wave propagation direction. In Section 2 of this paper, we present the operation mechanism of the proposed THz grating and in Section 3, the impacts of geometrical parameters on filtering properties, including extinction ratio (ER) and full-width half-maximum (FWHM) of the bandstop filter, are investigated. In Section 4, we manifest a narrow bandpass filter by introducing a π-shift in the middle of the grating, followed by analyzing the fabrication tolerances of the phase-shift region. The simulation results illustrate the proposed uniform grating possesses polarization-maintaining stopband characteristics (ER>20.9 dB and FWHM~21.1 GHz of two polarization states). Additionally, the phase-shifted grating allows narrow transmission peaks with below 1.7 GHz FWHM in the spectrum. We expect the proposed THz filters would be potential candidates for polarization-sensitive THz systems.

2. Operation mechanism of polarization-maintaining subwavelength filter

Geometrical anisotropy of the subwavelength waveguide cross-section (e.g., rectangular shape and elliptical shape) splits two orthogonally polarized fundamental modes, contributing to the polarization-maintaining transmission. The subwavelength waveguide-based grating can be thus constructed by periodically varying the waveguide dimension along the wave propagation direction. Figure 1 shows a schematic sketch of the proposed THz grating, where the THz beam is assumed guiding along $z$ axis. In a single grating unit, we label the large- and small-sized cells as $C_1$ and $C_2$, respectively. All geometrical parameters of the grating are denoted in the insets of Fig. 1. Grating pitch $\text{Λ}$ equals to $L_1+L_2$, $r_g=L_1/\text{Λ}$ represents the fraction of $C_1$ in a grating unit, and grating length is expressed as $N\times \text{Λ}$. Regarding the grating cross-sectional parameters, the ratios between $d_{1y}$ and $d_{1x}$ as well as $d_{2y}$ and $d_{2x}$ are kept the same, denoted as $r_d=d_{1y}/d_{1x}=d_{2y}/d_{2x}$. We choose Zeonex (trade name of cycloolefin polymer, Grade 480R) [44] as the hosting material of the structure due to its low absorption in THz range and we use the frequency-dependent dispersive complex refractive index of Zeonex [45] in the numerical models.

 

Fig. 1 Schematic of the THz polarization-maintaining subwavelength grating. Insets: defined geometrical parameters of grating in $yz$ and $xy$ planes.

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Large and small subwavelength rectangular cells support modes with different refractive indices (i.e., $n_{eff1}$ and $n_{eff2}$ corresponding to $C_1$ and $C_2$, respectively), thus yielding the effective index modulation. Such mode coupling resembles the optical fiber Bragg grating, which can be approximately formulated by Bragg condition [46]:

To find appropriate grating cross-sectional parameters ($d_{1x}$, $d_{1y}$, $d_{2x}$ and $d_{2y}$) that determine $n_{eff1}$ and $n_{eff2}$ in Eq. (1), we first study a conventional subwavelength rectangular waveguide using frequency-domain mode analysis employing wave optics module of COMSOL [47], which is based on finite element method. The side lengths of the waveguide cross-section are $d_x$ and $d_y$, and the ratio between them is $r_d=d_y/d_x$ (i.e., the geometrical anisotropy). In the COMSOL model, to create a subwavelength waveguide with an infinite air cladding, we use a perfectly matched layer (PML) around the box of simulation far away from the structure. Regarding the material parameter used here, the complex refractive index of Zeonex is 1.52 + i0.001 at target 0.3 THz. Figures 2(a), 2(b) and 2(c) respectively present modal $n_{eff}$, loss and birefringence ($\Delta n_{eff}$) of the waveguides as a function of $d_x$ at $f_B=0.3 \text{THz}$, where the $r_d$ values are chosen to be 0.7, 0.8 and 0.9. Two birefringent modes, polarized in $x$ and $y$ directions, are denoted as x-pol. (solid curves) and y-pol. (dashed curves), respectively. As expected, the $n_{eff}$ values vary with $d_x$ (a quasi-proportional relationship) as shown in Fig. 2(a), illustrating the feasibility of periodically modifying the waveguide dimension to realize the Bragg coupling. In general, a large spacing between $n_{eff1}$ and $n_{eff2}$ could enhance the effective index modulation, so the proposed grating requires $C_1$ and $C_2$ with a large dimensional difference between them and minimum losses. As expected and can be seen in Fig. 2(b), the modal losses are very low for small $d_x$, and gradually increase with $d_x$ [18]. For $d_x<0.65 \text{mm}$, the modal losses of waveguides with three $r_d$ values are below 0.53 dB/cm, which offers a wide selection range of $d_x$ to obtain a pair of $n_{eff1}$ and $n_{eff2}$ with large enough separation. Additionally, we observe that the modal losses of x-pol. (solid curves) are higher than those of y-pol. (dashed curves). This is expected as due to wider waveguide cross-section along $x$ direction. The separation between losses of x-pol. and y-pol. increases as the anisotropy of waveguide cross-section becomes larger. Figure 2(c) shows the achievable birefringence with the chosen parameters. For $d_x$ ranging from 0.23 mm to 0.65 mm, the subwavelength rectangular waveguides are capable for polarization-maintaining operation with $\Delta n_{eff}$ ranging from $1\times {10}^{-4}$ to $3.9\times {10}^{-2}$. This identifies the range from which the cross-sectional parameters of $C_1$ and $C_2$ can be chosen. For the waveguides with $r_d=0.8$, Fig. 2(d) shows the normalized electric field distributions of two strictly orthogonally polarized modes when $d_x=0.6 \text{mm}$ and $d_x=0.25 \text{mm}$, in which the blue arrows represent the electric vectors. We observe that, both x-pol. and y-pol. modes propagate well with the $d_x=0.6 \text{mm}$ waveguide (fractional power in the core over 70%), but the guided modes for $d_x=0.25 \text{mm}$ waveguide are very weakly confined (fractional power in the core below 1.3%). Please note that, to determine the final values of parameters, we also need to consider the impact of $r_d$ on filter performances, which will be discussed in Section 3.

 

Fig. 2 Modal (a) $n_{eff}$, (b) loss and (c) $\Delta n_{eff}$ of the subwavelength rectangular waveguides as a function of $d_x$ at 0.3 THz for $r_d$ values of 0.7 (green curves), 0.8 (red curves) and 0.9 (blue curves). (d) Normalized electric field distributions of rectangular waveguides ($r_d=0.8$) with $d_x=0.6 \text{mm}$ and (b) $d_x=0.25 \text{mm}$. The blue arrows represent the electric vectors.

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Next, we need to identify the waveguide dimensions where the proposed grating will be incorporated. Here, we can integrate the grating in a waveguide with $C_1$ cross-section [scenario (a) in Fig. 3, which is mode launching via large waveguide] or a waveguide with $C_2$ cross-section [scenario (b) in Fig. 3, which is mode launching via small waveguide]. To investigate this, we use a finite-difference time-domain (FDTD) model (Lumerical FDTD-solutions) [48], in which a PML boundary is applied to mimic an infinite air background. The THz pulse is launched using the waveguide mode. After propagating through the grating, the THz wave is detected by a field monitor to evaluate the transmittance. The transmission spectra for both scenarios are shown in Fig. 3(c). The simulation results confirm that the periodical geometrical perturbations introduced along propagation direction indeed produce Bragg couplings for the two orthogonally polarized waves, which works as a THz polarization-maintaining bandstop filter. We observe that the position of the stopbands for both scenarios overlap with scenario (b) having slightly wider FWHMs (1.3%) compared to that of scenario (a). We also find that the reflective frequency of y-pol. is roughly 4 GHz higher than that of x-pol. This is expected as the effective index of y-pol. modes are blue-shifted. More importantly, the transmission outside stopband of scenario (a) [red curves in Fig. 3(c)] is significantly higher than that of scenario (b) [blue curves in Fig. 3(c)] for both polarization states. This is due to the fact that the input mode of scenario (b) has a large portion of power distributed outside the waveguide core [fractional power in the air cladding above 98.7%, see field distributions for $d_x=0.25 \text{mm}$ waveguide in Fig. 2(d)], being severely disturbed by the large-sized lossy polymer cells ($C_1$). A subwavelength waveguide with most of the power in the air cladding, is susceptible to any perturbation, i.e., suffering from high radiation losses due to perturbations. In opposite, the mode launched from the scenario (a) [fractional power in the core over 70%, see field distributions for $d_x=0.6 \text{mm}$ waveguide in Fig. 2(d)] is modulated by smaller cross-sections ($C_2$ with low losses), leading into higher transmittance outside the stopband. Hence, for the rest of simulations in this work, we use the large waveguide, i.e., scenario (a), for launching the mode into the THz gratings. In practice, THz radiation is most often coupled into a device from a lineally polarized focused Gaussian or quasi-Gaussian beam. Thanks to the strictly linear polarization of the waveguide modes [see modal profiles in Fig. 2(d)], we believe the proposed grating could be efficiently excited. Furthermore, regarding the selection of dielectric material, we notice that the proposed gratings made of polymers with higher material absorption would degrade the expected performance, in particular the transmission outside filter stopband because of the relatively higher modal propagation losses of the waveguide. Therefore, highly transparent polymers at THz frequency are preferable.

 

Fig. 3 Schematic mode launching from scenario (a): large waveguide and scenario (b): small waveguide. (c) Grating transmission spectra of scenario (a) (red curves) and scenario (b) (blue curves) waveguides. Solid and dashed curves represent x-pol. and y-pol. states, respectively. Grating parameters as $d_{1x}=0.6 \text{mm}$, $d_{2x}=0.25 \text{mm}$, $r_d=0.8$, $r_g=0.7$, $\text{Λ}=0.435 \text{mm}$ and $N=29$, leading to x-pol. stopband at around $f_B$ = 0.3 THz.

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To elaborate the proposed polarization-maintaining filter further, we look into the snapshots of the normalized power distribution of both polarizations at different locations of the transmission spectra, as displayed in Fig. 4. Grating parameters used here can be found in caption of Fig. 4, and the detailed analyses of impact on filter performances will be performed in Section 3. The normalized power distribution shows transmission of THz wave outside the stopband [see Figs. 4(b) and 4(d)], whilst both polarization states are almost totally reflected back in the stopband [see Figs. 4(c) and 4(e)]. The average transmission loss [49] outside the stopbands (0.26-0.34 THz) are estimated 0.11 dB/cm and 0.10 dB/cm (considering 13 cm grating length excluding the input and output waveguides), for x-pol. and y-pol., respectively. In comparison to other waveguide-based gratings reported in THz region, our proposed polarization-maintaining filter has lower transmission losses than the metallic filters, and comparable losses to the subwavelength dielectric filter. Due to the Ohmic loss of metals, the transmission loss of metallic grating could be of order of 1 dB/cm [22, 28], which would be more than 5 times greater than the expected losses of our proposed polymer grating. For the dielectric case, our proposed polarization-maintaining grating would potentially reduce the loss even below the measured value (around 0.2 dB/cm) of the notch-type Topas grating [18], since Topas has slightly higher material absorption than that of Zeonex at around 1 THz. Furthermore, the proposed structure can be potentially used as a narrow band THz polarizer. Figures 4(f) and 4(g) show the normalized power distributions of x-pol. and y-pol. at 0.296 THz (dotted gray line), respectively, where the y-pol. can well propagate (transmittance of −1.16 dB) while the x-pol. is largely reflected (transmittance of −14.5 dB). Similarly, at 0.316 THz, the proposed grating transmits only the x-pol. wave but not the y-pol. wave.

 

Fig. 4 (a) Grating transmission spectra of x-pol. (solid curve) and y-pol. (dashed curve) states. Normalized power distributions of x-pol. (in $yz$ view) at (b) 0.262 THz and (c) 0.304 THz, as well as y-pol. (in $xz$ view) at (d) 0.334 THz and (e) 0.306 THz. The ER and FWHM of x(y)-pol. stopbands are 23.5(20.9) dB and 21.7(20.5) GHz, respectively. Normalized power distributions of (f) x-pol. (in $yz$ view) and (g) y-pol. (in $xz$ view) at 0.296 THz. Grating parameters are $d_{1x}=0.6 \text{mm}$, $d_{2x}=0.25 \text{mm}$, $r_d=0.8$, $r_g=0.7$, $\text{Λ}=0.435 \text{mm}$ and $N=29$.

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3. Design of polarization-maintaining waveguide-based filters

In this section, by exploiting the FDTD model of the proposed THz grating introduced above, we analyze the effects of cross-sectional anisotropy ($r_d$) and grating unit parameters ($r_g$ and $d_{2x}$) on filtering properties (including FWHM and ER) at target reflective frequency around 0.3 THz. Unless otherwise stated, the grating $\text{Λ}$ is calculated from Eq. (1) in this paper.

3.1 Impact of cross-sectional anisotropy

As discussed, the birefringence depends on the waveguide geometrical anisotropy ($r_d$). Here, we vary $r_d$ value to study its influence on filtering property, whilst $d_{1x}$ and $d_{2x}$ are fixed to be 0.6 mm and 0.25 mm, respectively. FDTD results of the gratings with $r_d$ values of 0.7, 0.8 and 0.9 are presented in Fig. 5. We observe that the transmission dips (stopbands) of x-pol. (solid curves) are almost independent of $r_d$. This is due to the fact that the cross-sectional parameters in $x$ direction ($d_{1x}$ and $d_{2x}$) are invariant and the x-pol. mode is designed to operate at $f_B=0.3$ THz. However, the y-pol. stopband (dashed curves), particularly its position and ER, are affected by the cross-sectional anisotropy. As $r_d$ decreases (increasing anisotropy), the central frequencies of y-pol. transmission dips move to higher frequencies. Increasing the cross-sectional anisotropy further separates the effective refractive index of x-pol. and y-pol. modes with y-pol modes having larger shifts to higher frequencies. The ERs for both polarization states decline, in particular there is more decrease for the y-pol. mode. This is owing to the improved birefringence between two polarization states [see Fig. 2(c)] and weaker Bragg coupling [see Fig. 2(a)]. With various $r_d$ values, the fluctuations of filter FWHM are below 1 GHz for both polarization states. In a nutshell, increasing the grating cross-sectional anisotropy enlarges $\Delta n_{eff}$ of the two orthogonally polarized modes, leading into further separation of the stopband position and reduction of the ER.

 

Fig. 5 Grating transmission spectra with cross-sectional anisotropy $r_d$ values of 0.7 (green curves), 0.8 (red curves) and 0.9 (blue curves). Solid and dashed curves represent x-pol. and y-pol. states, respectively. Other grating parameters are $d_{1x}=0.6 \text{mm}$, $d_{2x}=0.25 \text{mm}$, $r_g=0.7$ and $N=29$ .

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3.2 Impact of grating unit structure

Here, we analyze two key parameters of the grating unit ($C_1$ and $C_2$), which are $r_g$ in longitudinal direction and $d_{2x}$ in transverse direction.

First, we vary the relative fraction of $C_1$ in the grating unit ($r_g$) keeping $d_{2x}$ constant. The transmission spectra for x-pol. and y-pol. are shown in Figs. 6(a) and 6(b), respectively. To clearly observe the relative changes for both polarization states, the orthogonal polarization for each case is also shown with dotted line in each figure. When $r_g$ increases, the stopband FWHM and ER for both polarization states decrease, gradually approaching to that of a waveguide without any grating structures (i.e., $r_g=1$). On the other hand, we observe that as $r_g$ declines, the excessive geometrical perturbation results into transmission reduction at high frequencies outside the stopband, which is not a desired effect [27]. Consequently, $r_g$ should be carefully chosen to ensure that the stopband filter has the required ER and FWHM performances, while it maintains high transmission outside the stopband at higher frequencies.

 

Fig. 6 Grating transmission spectra with $r_g$ values of 0.6 (pink solid curves), 0.7 (red solid curves), 0.8 (blue solid curves), 0.9 (green solid curves) and 1 (orange solid curves) for (a) x-pol. and (b) y-pol. states. Other grating parameters are $d_{1x}=0.6 \text{mm}$, $d_{2x}=0.25 \text{mm}$, $r_d=0.8$ and $N=29$.

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Next, we vary the cross-sectional parameter of $C_2$ ($d_{2x}$) in the grating unit meanwhile keeping $r_g$ constant, which is equivalent to varying the $n_{eff2}$ in Eq. (1). Here, the $d_{2x}$ increases from 0.25 mm to 0.35 mm (step of 0.05 mm) Simulated transmission spectra for x-pol. and y-pol. are presented as solid and dashed curves in Fig. 7, respectively. As expected, the grating with low difference between $n_{eff1}$ and $n_{eff2}$ (increasing $d_{2x}$ and keeping $d_{1x}$ invariant) remarkably reduce FWHM and ER of the transmission dip, which is a result of weaker modal coupling. Moreover, we observe the central frequency of the x-pol. stopband moves closer to the target $f_B$ of 0.3 THz when $d_{2x}$ increases. We reveal the deviation of stopband frequency between the theory [Eq. (1)] and numerical results for gratings with very low $d_{2x}$ as follows. When $d_{2x}=0.25 \text{mm}$, the calculated $n_{eff}$ values are 1.0011 and 1.0006 of x-pol. and y-pol. modes, respectively, being quite close to the refractive index of air – in other words, the modes propagate almost in free-space (calculated fractional power in polymer core is less than 1.3% for both polarization states). Although such weakly guided mode may not strictly satisfy the Bragg condition, one could approximately estimate the desired filtering frequency using Eq. (1).

 

Fig. 7 Grating transmission spectra with $d_{2x}$ values of 0.25 mm (red curves), 0.3 mm (blue curves) and 0.35 mm (green curves). Solid and dashed curves represent x-pol. and y-pol. states, respectively. Other grating parameters are $d_{1x}=0.6 \text{mm}$, $r_d=0.8$, $r_g=0.7$, and $N=29$.

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We also numerically investigate the effect of the grating length ($N$) on the filter performance. We observe that, the proposed grating with an increased number of grating cells improves the coupling strength, i.e., the ER. With grating parameters of $d_{1x}=0.6 \text{mm}$, $d_{2x}=0.25 \text{mm}$, $r_d=0.8$, $r_g=0.7$ and $\text{Λ}=0.435 \text{mm}$, the transmittances at reflective frequencies of both polarization states are no higher than 0.01 (i.e., −20 dB) when $N\ge 27$. However, the fabrication costs of a long grating increase accordingly. Thus, we select $N=29$ in all simulations of this work. In addition, it should be noted that the FWHM values and stopband positions are almost invariant with grating length as these parameters are mainly defined by grating unit. Furthermore, varying the grating pitch $(\text{Λ)}$ will shift the center of the stopband (the target $f_B$), as it is expected from Eq. (1). Similar tunability can also be achieved by adjusting grating $r_g$ value with fixed $\text{Λ}$.

4. Exploration of phase-shifted grating

THz ultra-narrow linewidth passband filters, which are capable of selecting specific frequencies, are in great demand for abundant THz applications [50]. These filters eliminate the unwanted information consequently enabling to implement high-sensitive sensors [51], lasers with a narrow beam output [52], and uplink/downlink channels in communication systems [53]. Our proposed on-line waveguide-based bandpass filter can be transformed into an ultra-narrow linewidth passband filters by introducing a phase shift (a Fabry–Perot resonator) in the grating [54]. To achieve this, we modify the length $(L_1)$ of the middle $C_1$ cell to $L_1^{ ,}$ (modified cell labeled as $C_1$) to obtain a π-shift by adjusting the period to $\text{Λ}/2$ [see Fig. 8(a)]. The transmission spectra of the π-shifted grating [Fig. 8(b)] confirms the emergence of transmission at the center of the original stopbands for both orthogonally polarized THz beams with FWHM = 1.6(1.7) GHz and ER = 13.5(12.3) dB for the x(y)-polarization state and $r_d=0.8$. Compared to a metasurface-based narrowband THz bandpass filter recently reported in [50] (note this filter is a spatially isolated device), the FWHM of our proposed waveguide-based filter is over 100 times narrower. In addition, the transmission losses at the center of the transmission peaks are 0.34 dB/cm and 0.28 dB/cm (considering 12.8 cm grating length), for x-pol. and y-pol., respectively. Similar to the bandstop filter, we also investigate the effect of $r_d$ on the waveguide-based narrowband filter. The transmission peaks of x-pol. [solid curves in Fig. 8(b)] are almost invariant with different $r_d$ values, whereas the position of y-pol. passband [dashed curves in Fig. 8(b)] is blue-shifted. Moreover, we observe that the filter ER decreases with a reduced $r_d$ because of the enhanced birefringence [see Fig. 2(c)]. We expect that the linewidth of the proposed THz polarization-maintaining filter could be further improved by including an additional π-shift, similar to optical fiber gratings with two-cavity Fabry–Perot structures [54].

 

Fig. 8 (a) Schematic of the π-shifted grating. (b) π-shifted grating transmission spectra for $r_d$ values of 0.7 (green curves), 0.8 (red curves) and 0.9 (blue curves). Solid and dashed curves represent x-pol. and y-pol. states, respectively. Grating parameters are $d_{1x}=0.6 \text{mm}$, $d_{2x}=0.25 \text{mm}$, $r_g=0.7$, $\text{Λ}=0.435 \text{mm}$ and $N=29$.

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In practice, we anticipate that the proposed polarization-maintaining waveguide grating could be fabricated using the cutting-edge 3D printing technique, which has been widely applied for manufacturing THz devices with many kinds of polymers or filaments [55–57]. Very recently, the Zeonex based devices have been successfully 3D printed with fused deposition modeling process at millimeter-wave frequency [58], which is a promising solution for fabricating the proposed waveguide grating. Transverse and longitudinal resolutions of current 3D printers can be as high as 50 µm and 1 µm, respectively [50]. Here, we analyze the fabrication tolerance of the $C_1^{ ,}$ cell located at the phase-shift point which is the key element to achieve the narrow passband. Because the resolution in longitudinal direction ($z$ axis in our case) is well below the filter dimensions, only fabrication errors of $C_1^{ ,}$ in cross-sectional direction are studied, which include two scenarios: first when $C_1^{ ,}$ has an imperfect cross-section (all the grating unit cells are aligned), and second when $C_1^{ ,}$ is miss-aligned (no cross-section change). Figure 9(a) presents the transmission spectra of a π-shifted filter with ideal and imperfect cross-sections of $C_1^{ ,}$. Such fabrication error would lead into 3% and 8% fluctuations in ER and FWHM for both polarization states, respectively. When $C_1^{ ,}$ is relatively shifted diagonally by 100 µm from the center, Fig. 9(b) shows the transmission peaks have 5% variations in both ER and FWHM. This suggests that the proposed bandpass filter would be tolerant of transverse fabrication errors of the $C_1^{ ,}$ cell. Furthermore, if all cells of the grating are imperfect, for example ± 50 µm lateral variations to every $C_1$ cell, the center of the stopbands or the passbands would be shifted due to the modification of the Eq. (1).

 

Fig. 9 π-shifted grating transmission spectra of ideal (red curves) and imperfect (blue and green curves) gratings, where (a) and (b) show aligned and non-aligned cases, respectively. Ideal grating parameters are $d_{1x}=0.6 \text{mm}$, $d_{2x}=0.25 \text{mm}$, $r_d=0.8$, $r_g=0.7$, $\text{Λ}=0.435 \text{mm}$ and $N=29$.

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5. Conclusion

In summary, we have proposed and investigated the polarization-maintaining filters at THz regime. Grating structures are introduced on highly birefringent subwavelength rectangular polymer waveguides, producing separated coupling responses of two orthogonally polarized modes in spectrum. Unlike spatially individual THz filters working for free-space radiation, the proposed grating devices integrated on waveguides are able to manipulate guided THz modes “on-line”. Using numerical models, effects of structural parameters of the uniform grating on filtering characteristics are comprehensively discussed. Results show that the bandstop filters with ER = 23.5(20.9) dB and FWHM = 21.7(20.5) GHz can be achieved for x(y)-polarized wave. The reflective frequencies of two polarization states, as well as the spacing between them can be tuned by adjusting grating parameters. At designated frequencies, the proposed filter can act as a THz polarizer, reflecting only one polarization state. Furthermore, we demonstrate that the bandstop filter can be transformed into narrow bandpass filter [ER = 13.5(12.3) dB and FWHM = 1.6(1.7) GHz for x(y)-polarizations] by introducing a π-shift unit. The tolerance analyses suggest that we can fabricate these filters using 3D printing technology and the proposed THz bandpass filter is insensitive to fabrication errors at the phase-shift point. As a waveguide-based component, the proposed THz grating may play one of the key roles in many polarization-maintaining THz applications, such as imaging, sensing, and wireless communication.