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We propose square and rectangular pipe waveguides for low-loss THz waveguiding and polarization control. Different from common circular-symmetric THz fibers and waveguides, the proposed rectangular pipe waveguides successfully remove the transmission degeneracy of two orthogonal polarizations and possess polarization sensitivity to the guided THz waves. By measuring the attenuation spectra, we find that the polarization sensitivity depends on the structure of the pipe waveguides. With butt coupling method, it is easy to combine circular pipe waveguides and the rectangular ones.
©2011 Optical Society of America
1. Introduction
Terahertz (THz) fibers and waveguides have been extensively investigated recently. Because the most transparent medium in the THz region is dry air, it is a common approach to guide THz waves in air for low-loss THz waveguiding. There are two primary types of design to guide THz fields in air. The first one is subwavelength structure. The subwavelength fibers minimize the interacting areas between the fibers and THz fields, so the propagation loss due to material absorption and finite metal conductivity can be reduced [1–9]. However, since the guided modes are mostly outside the subwavelength cores, the THz fields experience environmental disturbance and high bending loss. Another common method for low-loss THz waveguiding is air-core structure. For the air-core waveguides, since most THz fields are guided within the air-core region, guiding loss and environmental disturbance can be significantly reduced [10–18]. Hollow cylindrical metallic waveguides are one of common air-core THz waveguides. The two dominant modes of hollow metallic waveguides are TE01 (azimuthally polarized) and TE11 (linearly polarized) [12,17]. With dielectric coating on the inner core of the metallic waveguides, the dominant mode becomes HE11 and an attenuation constant lower than 1 dB/m (≒ 0.002 cm−1) had been achieved [11,12].
Recently, we proposed a dielectric circular air-core pipe for low-loss THz waveguiding [13,14]. The dielectric pipe waveguides are with leaky mode nature and the dominant mode is HE11-like. Without complex fabrication, we used commercial Teflon pipes to demonstrate that the circular pipe waveguides not only possess low attenuation constants (< 0.001 cm−1) and high coupling efficiency (> 80%) [13], but also suffer low bending loss [16]. Besides, a circular pipe waveguide with a high passband width (>1.1 THz) was recently demonstrated [18]. However, the circular structure hides the polarization information of THz waves. For future THz polarization applications, such as polarization controller and filter, it is important to develop low-loss and polarization-sensitive THz fibers and waveguides.
In this paper, we propose dielectric rectangular pipe waveguides for future THz polarization applications. With polyethylene (PE) and polymethylmethacrylate (PMMA) rectangular pipes, we experimentally demonstrate that this kind of waveguide is low-loss and polarization-sensitive. Similar hollow square and rectangular waveguide had been proposed for CO2 laser (10.6-μm wavelength) transmission [19,20]. However, infinite claddings were assumed in [19] and the claddings in [20] were metallic strips coated with multiple dielectric layers. Moreover, the previous works did not put emphasis on the polarization-dependent characteristics. Different from previous researches, the structure we proposed here comprises a rectangular air-core and a thin dielectric cladding layer while this paper focuses on exploring the attenuation spectrum characteristics of fundamental guiding modes and polarization-sensitivity of the rectangular pipe waveguides, especially in the THz regime. Our experimental results indicate that the guiding principle of square and rectangular pipe waveguides is similar to that of circular ones, namely anti-resonant reflection with leaky mode nature. Moreover, the polarization sensitivity of the rectangular pipe waveguides can be controlled by adjusting the structure of the waveguides. At last, we demonstrate that the proposed square and rectangular waveguides can be high-efficiently butt coupled with circular pipe waveguides.
2. Experiment and discussion
We fabricated the square and rectangular pipe waveguides by sticking four PE or PMMA strips. The structure is shown in Fig. 1 . The pipe waveguides consist of a square or rectangular air core region (n1 = 1) and a uniform dielectric cladding with a low refractive index n2. Even though these kinds of air-core waveguides are multi-mode waveguides, the higher-order modes suffer much higher propagation loss [13,14]. In this paper, we focus on the fundamental eigenmode characteristics of the square and rectangular pipe waveguides, so we used long-enough pipe waveguides to eliminate higher-order modes. We conducted our experiments with a CW Gunn oscillator module tunable between 324 GHz and 420 GHz. The emitted CW terahertz waves were directly coupled into the input end of the measured pipe waveguide of 150 cm. We used a cutback method to measure the attenuation constant. Keeping the input end steady and removing 50 cm-long waveguide from the output end, we can obtain attenuation constant by measuring the output powers of the different length pipe waveguides.
Fig. 1 (a) Structure of the square and rectangular pipe waveguide. (b) Cross section of the square pipe waveguide. (c) Cross section of the rectangular pipe waveguide, where n1 = 1 (air). The x-direction and y-direction are defined as the longer axis and the shorter axis of the rectangular pipe, respectively.
Figure 2(a) shows the attenuation constants for two PE square pipes with the same air-core size (S = 8mm) but different cladding thickness (t = 1 and 2mm). It is found that low attenuation on the order of 0.002 cm−1 has been achieved and that the guiding bandwidth becomes approximate one-half as cladding thickness doubles. The attenuation spectra of the square pipe waveguides are similar to those of the circular ones [13,14], indicating that the guiding mechanism of the square pipe waveguides is similar to that of the anti-resonant reflecting optical waveguide (ARROW) [13,14,21]. Because the cladding of pipe waveguides can be viewed as a Fabry-Perot etalon, the resonant frequencies can be theoretically calculated as [14]:
where c is the speed of light in vacuum and m is an integer. The refractive index n2 for PE is about 1.54 [22] in our interested frequency range. If the operating frequency of the guided waves is near the resonant frequencies, cladding becomes transparent and THz waves would penetrate through the cladding outside the pipe waveguides, leading to high guiding loss. On the contrary, near the out-of-resonant frequencies (or anti-resonant frequencies), most THz fields are strongly confined in the air-core region, thus the attenuation constants decrease significantly. For the PE pipe waveguides with t = 2.0 mm, Eq. (1) predicts that there are resonant frequencies near 320 GHz (m = 5) and 384 GHz (m = 6). As shown in Fig. 2(a), the square pipe waveguide with t = 2mm indeed suffer high attenuation near 320 GHz and 384 GHz. We also found from Fig. 2(a) that the resonant frequency of the PE square pipe waveguides with t = 1mm is about 340 GHz, which is slightly different from the theoretical calculation of Eq. (1). The fact that Eq. (1) is well matched with the experimental results of t = 2mm but not with t = 1mm can be attributed to the cladding-thickness variation in the adopted PE strips. We found bigger cladding variations for 1-mm-thick PE strips. Cladding thickness was measured to be between 1.10 mm-1.16 mm for t = 1mm but only between 2.00mm-2.03mm for t = 2mm. Figure 2(b) demonstrates the attenuation constants for two PE square pipes with the same cladding thickness (t = 1mm) but different core sizes (S = 6 and 8mm). It is observed that the attenuation constants decrease as core size increases. This result is also similar to that of the circular pipe waveguides [13,14]. Moreover, we also observed from Fig. 2(b) that the there is an attenuation peak at 384 GHz. Because this phenomenon was not found in circular pipe waveguides [13,14], it can be attributed to the resonant frequency of corner cladding of the square pipe waveguides.Fig. 2 (a) Attenuation spectra of PE square pipe waveguides for t = 1mm (black squares) and t = 2mm (red triangles) with S = 8mm.(b) Attenuation spectra of PE square pipe waveguides for S = 8mm (black squares) and S = 6mm (red triangles) with t = 1mm.
We further investigated the attenuation spectrum characteristics with different cladding materials. The results (Fig. 3(a) ) display that the PE pipe waveguides suffer lower guiding loss than PMMA ones. At anti-resonant frequencies, where pipe waveguides suffer lowest attenuation, the guiding loss of PMMA waveguides is about ten times higher than that of PE waveguides. Since the material absorption loss of PMMA (about 2-3 cm−1) is also approximately ten times higher than that of PE (about 0.2 cm−1) [22–24] within our studied frequency range, it is found that the absorption loss of cladding material dominates the guiding loss of the square pipe waveguides at anti-resonant frequencies. To measure the mode profile of the square pipe waveguides, we put a metallic pinhole of 300 μm in front of a Golay cell, which was mounted on a 2D translational stage, to scan the intensity distribution (near field) at the output endface of the square waveguides. From the measured intensity distribution shown in Fig. 3(b), we found that the guided mode is well-confined in the air core region and resembles the fundamental mode () of a conventional channel waveguides [25].
Fig. 3 (a) Attenuation spectra of PMMA (black squares) and PE (red triangles) square pipe waveguide with S = 8mm, t = 2mm. (b) The measured intensity distribution () at the output end of a 100-cm-long PE square pipe waveguide with S = 6mm and t = 1mm at 376 GHz.
To measure the attenuation spectra of rectangular pipes, we defined the x and y direction as the longer and shorter axis of the rectangular pipe waveguides, respectively. The THz waves emitted from the Gunn oscillator module were linearly polarized. As a result, after measuring x-polarized THz waves, we then rotated the rectangular pipe waveguides by 90 degrees and measured the y-polarized THz waves. Figure 4(a) shows the attenuation spectra of the rectangular pipe waveguide with L = 10mm, W = 6mm, and t = 1mm. We found that the rectangular pipe waveguides successfully remove the transmission degeneracy of two orthogonal polarizations. The attenuation constants are strongly dependent on the polarization of THz waves. As polarization parallels to the longer axis of the rectangular waveguides, the attenuation decreases significantly. From Fig. 4(a), we found that in the anti-resonant frequency regime (364 GHz-412 GHz), the average attenuation of the x-polarized waves (0.004 cm−1) is about 4 times smaller than that of the y-polarized THz waves (0.017 cm−1), and the maximum difference in attenuation constant occurs at 384 GHz, where the attenuation constant of the x-polarized waves is about 11 times smaller than that of the y-polarized waves. It is also found from Fig. 4(a) that x-polarization and y-polarization have the same resonant frequency (348 GHz), which can be also inferred from Eq. (1). We then changed the ratio of L to W and investigated the relation between polarization-sensitivity and structure of waveguides. Figure 4(b) shows the attenuation spectra of a rectangular pipe waveguide with L = 10mm, W = 8mm, and t = 1mm. Compared to Fig. 4(a), Fig. 4(b) shows weak polarization-sensitivity: in the anti-resonant regime, the ratio of average attenuation of the y-polarized waves (0.007 cm−1) to that of the x-polarized THz waves (0.004 cm−1) becomes 1.8, and the maximum difference in attenuation constant reduces to 4.8 times (at 392 GHz). It is found that the rectangular pipe waveguides with larger ratio of L to W possess higher polarization-sensitivity.
Fig. 4 x-polarized (black squares) and y-polarized (red triangles) attenuation spectra of (a) PE rectangular pipe waveguides with L = 10mm, W = 6mm, t = 1mm. (b) PE rectangular pipe waveguides with L = 10mm, W = 8mm, t = 1mm. (c) The measured intensity distribution () at the output end of a PE rectangular pipe waveguide of 100cm long at 400 GHz with L = 10mm, W = 8mm, and t = 1mm. (d)The attenuation spectra of a special rectangular pipe waveguides: the air core region is square. The thickness of top and bottom cladding is 1mm, while the thickness of left and right cladding is 2mm. The x direction and y-direction are defined as the longer axis and shorter axis, respectively.
Plausible explanation can be obtained from ray optics viewpoints: when the polarization is parallel to the longer axis, the guided THz waves are TE waves for the longer cladding pair, but TM waves for the shorter cladding pair. Because of the structure of rectangle, there are more bounces between longer cladding pair than shorter cladding pair. That is to say, as the polarization parallels to the longer axis, there are more bounce numbers at the air-cladding interface for TE waves than TM waves. Since the reflectivity for the TE waves is higher than that of the TM waves, THz waves with polarization parallel to the longer axis lead to better mode confinement and lower guiding loss. On the contrary, if the polarization is parallel to the shorter axis, there will be more reflections for TM waves than TE waves, leading to relatively higher attenuation. We also measured the mode distribution of PE rectangular pipe waveguides. Figure 4(c) indicates that THz fields are well guided in the rectangular air core region with an excellent mode quality ().We further investigated another kind of PE rectangular pipe waveguides (shown in Fig. 4(d)). The air core is square (S = 8mm), but the cladding thickness is different: the top and bottom cladding is 1-mm-thick, while the left and right cladding is 2-mm-thick. By measuring the attenuation spectra (Fig. 4(d)), we found that this kind of rectangular pipe waveguide possesses weak polarization-sensitivity.
We found that it is hard to bend the square and rectangular pipe waveguides, while the circular ones possess magnificent flexibility [16]. To combine these two systems, we used an end-butt coupling method. The setup is shown in Fig. 5(a) . We butt-coupled Teflon (n2≒1.4) circular pipes (core diameter = 8mm, cladding thickness = 1mm) with PE square (S = 8mm, t = 1mm) and rectangular (L = 10mm, W = 6mm, t = 1mm) pipes. The emitted CW THz waves were directly coupled into the circular pipe waveguide. We measured the output power (PI) of the circular waveguide first. After butt coupling the circular pipe waveguide with the square or the rectangular one, we measured the output power (PO) again. The butt coupling efficiency (C) was calculated as:
α is the attenuation constant of the square (Fig. 2(a)) or the rectangular (Fig. 4(a)) pipe waveguide. Z represents lengths (50 cm in this experiment) of the square or rectangular waveguide. Before measuring the butt coupling efficiency, we measured the attenuation spectrum of the Teflon circular pipe waveguide first (shown in Fig. 5(b)). It is found that the resonant frequency of the circular pipe is around 328 GHz. From Fig. 2(a) and Fig. 4(a), we noticed that the resonant frequency of the PE square and rectangular pipes is about 340GHz. Figure 5(c) shows the butt coupling efficiency spectra. It is observed that the coupling efficiency approaches zero as the operating frequency is near the resonant frequencies of circular and square pipe waveguides. However, the coupling efficiency increases significantly as the operating frequency is away from the resonant frequencies. In the anti-resonant regime (364 GHz-412 GHz), the average coupling efficiency between circular and square waveguides is about 89%. We further obtained almost 100% coupling efficiency at 404 GHz. To achieve high coupling efficiency, it is essential to make operating frequency at the anti-resonant regime of both the circular and square pipe waveguide. From Eq. (1), it is found that anti-resonant regime depends on the cladding thickness and the refractive index of cladding. As a result, it is better to couple the circular and square pipe waveguide which have the same cladding material and cladding thickness. In our experiment, the circular and square waveguides are made of different materials, because it is relatively hard to fabricate Teflon square pipe waveguides. Another concern for high coupling efficiency is the mode size. It is expected that high coupling efficiency can be achieved if the circular and square pipe waveguide are with the same air-core sizes.Fig. 5 (a) Experimental setup for butt coupling. (b) The attenuation spectrum of a Teflon circular pipe with a core diameter (D) of 8mm and a cladding thickness (t) of 1mm. (c) Measurement of butt coupling efficiency between circular (D = 8mm, t = 1mm) and square (S = 8mm, t = 1mm) pipe waveguide (black squares), and between circular (D = 8mm, t = 1mm) and rectangular (L = 10mm, W = 6mm, t = 1mm) pipe waveguide (red triangles for x polarization and blue circles for y polarization).
For the coupling between circular and rectangular pipes, as shown in Fig. 5(c), higher coupling efficiency can be obtained if polarization is parallel to the longer axis (x-polarized) of the rectangular waveguide. The coupling efficiency for x polarization is about 84% on average in the anti-resonant regime and the maximum value is 93% occurring at 380 GHz, while the coupling efficiency for y polarization reduced to 69% on average in the anti-resonant regime. The difference in coupling efficiency reveals that the modal profiles of the circular and rectangular pipes are more matched in the x polarization than in the y polarization. This characteristic makes the rectangular pipe waveguides a good candidate for THz polarization controller. If unpolarized THz waves are guided within a circular pipe waveguide and then coupled into a rectangular waveguide, x-polarized waves not only possess higher coupling efficiency but also suffer lower attenuation in the rectangular waveguide. As a result, after propagating in a long-enough rectangular pipe waveguide, x-polarized THz waves would dominate the output power.
3. Conclusion
We propose dielectric polarization-sensitive square and rectangular pipe waveguides for THz waveguiding and polarization-control. For square and rectangular pipe waveguides, our measurement indicates that THz fields are well-confined in the air-core region, and low attenuation constants on the order of 0.002 cm−1 can be achieved. At the anti-resonant frequency regime, the absorption loss of cladding material dominates the guiding loss of the pipe waveguides. For PE rectangular pipe waveguides, they are not only low-loss but also polarization-sensitive. The polarization-sensitivity can be controlled by adjusting the structure of the rectangular pipe waveguide. We further demonstrate that the square and rectangular pipe waveguides can be high-efficiently coupled with the circular ones. It is expected that the proposed waveguides have high potential for polarization–sensitive waveguide devices at the THz regime such as polarization waveguide controllers, waveguide filters and polarization-maintaining fibers.
Acknowledgments
This work was sponsored by the National Science Council of Taiwan (NSC) under grants NSC 100-2120-M-002-009, NSC 100-2221-E-002-183-MY3, and NSC100-2218-E-146-001.
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