We experimentally demonstrate a terahertz (THz) leaky mode directional coupler for future THz applications. The proposed directional coupler comprises two square pipe waveguides. The coupling efficiency is investigated for different frequencies, polarizations, and core sizes. Rectangular pipe-waveguide-based directional couplers and the issue of insertion loss are also discussed. It is found that the THz directional coupler works most efficiently in the minimal-attenuation wavelength regime.
© 2011 OSA
The development of terahertz (THz) fibers and waveguides has progressed significantly during the past decade [1–10]. For future THz communication and imaging applications, it is crucial to develop directional couplers for their capability of power switching and controls. In the THz regime, directional couplers based on dielectric subwavelength fibers [11–15] and quartz circuits  have been experimentally demonstrated. Since THz fields are guided outside the subwavelength fibers, the directional coupler based on subwavelength fibers can efficiently couple THz fields from the input fiber to the coupled fiber with a short overlap length . However, subwavelength fibers are vulnerable to environmental disturbance and suffer high bending loss. The THz directional coupler based on quartz circuits exhibits broadband (330GHz – 500 GHz) functionality but suffer relatively high loss .
Recently, we proposed dielectric pipe waveguides for low-loss THz waveguiding [9,10]. The pipe waveguides consist of a circular air core region and a thin dielectric cladding. THz fields are well-guided in the air core region and suffer low attenuation. It was found that the guiding principle of the pipe waveguides is anti-resonant reflection with a leaky mode nature [9,10,17]. With commercial Teflon pipes, we previously demonstrated that attenuation constants lower than 0.001 cm−1 can be achieved  and that the pipe waveguides possess magnificent flexibility and suffer low bending loss . Furthermore, a recent study indicated that thin-wall silica circular pipe waveguides hold a high passband width and suffer low group velocity dispersion at the anti-resonant regime .
For future THz applications, in this paper we develop a pipe-waveguide-based directional coupler. Since the contact area between two circular pipe waveguides is limited and the THz fields cannot be efficiently transferred between two waveguides, we constructed the directional coupler with two square pipe waveguides  instead of circular ones. Different from previously reported THz directional coupler, the pipe-waveguide-based directional coupler is with a leaky mode nature. Even though leaky mode directional couplers had been discussed in the optical regime [21–23], few of them were proposed in the THz range. Recently, THz directional couplers based on two slab antiresonant reflecting hollow waveguides (ARRHWs) were theoretically discussed , and it was shown that the coupling length can be significantly reduced at the antiresonant frequencies. Since the slab type ARRHWs are the one-dimensional version of the pipe waveguides, the model proposed in  can be treated as a simplified one for the studied directional coupler.
In this paper, the square pipe-waveguide-based THz directional couplers are experimentally investigated for different frequencies, polarizations, and air-core sizes. We further introduce the rectangular directional couplers for comparison. The issue of insertion loss is also discussed for different coupler shapes and frequencies. Because of the guiding mechanism of the pipe waveguides, it is found that the directional coupler works most efficiently in our interested minimal-attenuation wavelength regime.
2. Experimental setup
We fabricated the square pipe waveguides by sticking four polyethylene (PE) strips . The general directional coupler setup is shown in Fig. 1(a) . THz waves emitted from a Gunn oscillator module, which was tunable between 324 GHz and 420 GHz, were directly coupled into a square pipe waveguide. We made two pipe waveguides, the cladding thickness of which was 1mm, touch together to establish the directional coupler. However, we noticed that it is hard to make two pipe waveguides contact perfectly and that there were some inevitable air gaps between two waveguides, which led to inconsistent experimental errors and poorer coupling efficiency. To overcome this problem, we modified the experimental setup (shown in Fig. 1(b)). We replaced the two 1-mm-thick PE strips between two waveguides with a 2-mm-thick strip. We named this 2-mm-thick strip “coupling cladding.” This modified setup worked as if two square pipe waveguides contact perfectly. In the modified setup, THz waves were first coupled into a 100-cm-long square pipe waveguides (input waveguide), which eliminated high order modes and made only the fundamental mode exist . Then we butt-coupled the directional coupler with the input waveguide and measured the output power with a Golay cell. The coupling efficiency (C) was defined as:
3. Results and discussion
Before measuring the performance of the directional coupler, we measured the attenuation spectra of the square pipe waveguides first. Figure 2 shows the attenuation spectra of two PE square pipe waveguides with a cladding thickness (t) of 1mm and 2mm, respectively. The side length (S) of air core was kept the same (6mm). We recently demonstrated that, like the circular ones, the guiding principle of the square pipe waveguides is the anti-resonant reflection mechanism [9,10,20]. As a Fabry-Perot resonator, at the resonant frequencies, the cladding is nearly transparent and THz waves penetrate through the cladding outside the pipe waveguides. On the other hand, at the anti-resonant frequency regime, THz fields are strongly confined in the air-core region and suffer low attenuation. From Fig. 2, we found that 376 GHz is the resonant frequency and the anti-resonant frequency for the pipe waveguide with t = 2mm and t = 1mm, respectively. That is to say, if the operating frequency is 376 GHz, THz waves are well-confined in the core region and suffer low attenuation in a pipe waveguide with t = 1mm, but as we make two square pipe waveguides with t = 1mm touch together, THz waves can be efficiently transferred between two pipe waveguides, because the cladding between them, the thickness of which is 2mm, becomes nearly transparent at 376GHz. As a result, the directional coupler works most efficiently in our interested minimal-attenuation wavelength regime.
Figure 3 demonstrates the coupling characteristics of a pipe-waveguide-based directional coupler with S=6mm and t=1mm. It is found that the directional coupler works most efficiently at the resonant frequency of the coupling cladding, namely 376GHz, while as the operating frequency is away from 376 GHz, it takes longer overlap length to reach high coupling efficiency. This phenomenon is well matched with the simulation results shown in . From Fig. 3(a), it is observed that the coupling efficiency of the directional coupler operated at 376GHz reaches over 90% as the overlap length is 25cm, while the directional coupler operated at 368GHz should be with a longer overlap length (about 47.5 cm) to reach over 90% coupling efficiency. It is expected that the overlap length should be even longer for the directional coupler operated at 360GHz to reach 90% coupling efficiency. This particular coupling phenomenon has been explained by  and is attributed to a three-mode beating behavior. Here, we provide an alternative but intuitive explanation in the following. As stated previously, the coupling cladding is nearly transparent at resonant frequencies, but it becomes less transparent as operating frequencies are away from the resonant frequencies. As a result, at the resonant frequencies of the coupling cladding, THz waves can easily penetrate the coupling cladding, leading to higher coupling efficiency.
Figure 3(a) and Fig. 3(b) demonstrate the coupling performance of the directional coupler for different polarizations: TM- and TE-polarization to the coupling cladding, respectively. It is found that the directional coupler is polarization-sensitive. As THz waves are TM polarized to the coupling cladding (shown in Fig. 3(a)), higher coupling efficiency can be obtained. As we can see from the Fig. 3, at 376GHz, the coupling efficiency reaches 90.5% for TM polarization but only 42.8% for TE polarization as the overlap length is 25cm. Since the transmittivity of TM waves is higher than that of TE waves, THz waves with TM-polarization characteristics would more likely penetrate through the coupling cladding into another waveguide, thus lead to higher coupling efficiency.
We further investigated the dependence of coupling efficiency on air-core sizes. Figure 4(a) shows the coupling efficiency of the directional couplers with S=6mm, S=8mm, and S=10mm. The operating frequency was 376GHz and THz waves were TM-polarized to the coupling cladding. We found that the directional coupler with smaller air-core sizes works more efficiently. This phenomenon can be explained by ray optics: As THz waves are guided in the pipe waveguides, THz waves would bounce back and forth between claddings. The smaller the core size is, the more frequent bounces occur at the coupling cladding, leading to higher coupling efficiency. Furthermore, the pipe waveguides with a smaller core size suffer higher attenuation [9,10], which results from the fact that weaker reflection (higher transmission) occurs at the air-cladding interfaces for the pipe waveguides with a smaller core size, leading to higher coupling efficiency.
If coupling length is defined as the overlap length where coupling efficiency is maximum, it is found from Fig. 4(b) that the coupling length decreases as S decreases. It is expected that the coupling length would be shorter than 25cm if S is smaller than 6mm. However, the square pipe waveguides with a small air core are not recommended for THz delivery due to high guiding loss . Our study indicated that the square pipe waveguides with S ranging from 6mm to 10mm would be ideal for practical THz applications for their low guiding loss and acceptable coupling length. Figure 4(a) also indicates that the measured maximum coupling efficiency ranges from 88% (S=10mm) to 95% (S=8mm).
We then modified the square directional couplers into rectangular ones. The air core size was 10mmx6mm and the cladding thickness was 1mm. We measured the coupling efficiency of the longer-side-coupled (LSC) and shorter-side-coupled (SSC) rectangular directional coupler (shown in Fig. 5 ). The operating frequency was 376GHz and THz waves were TM-polarized to the coupling cladding. It is observed that the LSC directional coupler worked more efficiently than the SSC directional coupler. From the viewpoint of ray optics, this phenomenon is reasonable. As two rectangular waveguides are coupled with the longer cladding, there are more bounces happening at the coupling cladding due to the rectangular structure. Also, it is logical that higher coupling efficiency can be obtained as the contact area between two waveguides increases. The LSC directional coupler reaches about 95% coupling efficiency as the overlap length is 20cm, while the SSC directional coupler should be 50-cm-long to reach such high coupling efficiency.
By comparing the LSC rectangular (with a 10mmx6mm air core) and the square (S=10mm) directional coupler, we found that the ratio of the waveguide longer to shorter sides affects coupling length. The coupling claddings of these two kinds of directional coupler are with the same width, but the coupling length of the rectangular directional coupler (20 cm) is significantly shorter than that of the square directional coupler (47.5 cm). We can infer that if the width of coupling side is fixed, the LSC rectangular directional coupler with a shorter horizontal side works more efficiently. This inference can be also attributed to the fact that there are more bounces happening at the coupling cladding for the LSC directional coupler with shorter horizontal sides. From the ray optics viewpoint, the bounce number happening at the coupling cladding would be inverse proportional to the length of horizontal side of the LSC rectangular directional coupler. However, if the bounce number is the only one factor to affect coupling length, the coupling length of the square directional coupler with S=10mm would be 20cm (coupling length of the LSC directional coupler) ÷ 6mm (the length of the horizontal side of the LSC directional coupler) x 10mm (the length of the horizontal side of the square directional coupler)≒ 33.3cm, which is smaller than the experimental result (47.5 cm). As a result, bounce number is not the only one factor to influence coupling length and other issue should be taken into account. As we discussed in the case of the square directional coupler, stronger reflection (weaker transmission) occurs at the coupling cladding for the directional coupler with a bigger air core size, leading to lower coupling efficiency. Since the square directional coupler with S = 10mm possesses a bigger air core region than the LSC directional coupler, it is reasonable that the experimental coupling length of the square directional coupler is longer than the coupling length predicted by the bounce number method.
As we can see from Fig. 1, in the directional coupler region, we replaced the two 1-mm-thick strips between two waveguides (Fig. 1(a)) with a 2-mm-thick strip (Fig. 1(b)), which makes the directional coupler become an integral and compact device. To measure the loss of the directional coupler, in Fig. 1(b), we treat PO and PA + PB as the input power and the total output power, respectively. The insertion loss (L) of the direction couplers can be calculated as:
The insertion loss includes the guiding loss of the directional coupler and the butt-coupling loss between the input waveguide and the directional coupler. Even though the coupling loss is theoretically 0, the previous experimental results showed that the coupling loss ranges between 0 and 0.46 dB [20,25]. As shown in Table 1 , the insertion loss of the pipe-waveguide-based 3 dB couplers (50% coupling efficiency) is about 1.5 dB, which is smaller than the insertion loss of the directional coupler based on quartz circuits (2.5-4 dB) .
From Table 1, we found that the insertion loss is nearly independent of the coupler shapes. The primary cause is that there is compensation between guiding loss and overlap length. The directional couplers which work more efficiently (namely, reach 50% coupling efficiency with shorter overlap length) are made of the pipe waveguides with higher attenuation. For example, even though the smaller square directional couplers work more efficiently than the bigger one, the pipe waveguides with a smaller air-core also suffer higher guiding loss [9,20]. Therefore, there is an insignificant relation between insertion loss and the size of the square directional couplers (within the butt-coupling loss range). The same phenomena happen in the case of rectangular directional couplers: in order to make THz waves TM-polarized to the coupling plane, for the longer-side-coupled rectangular couplers, the polarization of THz waves is parallel to the shorter axis of the rectangular waveguides, while for the shorter-side-coupled couplers, the polarization is parallel to the longer axis of the rectangular waveguides. At the same time, THz waves suffer lower attenuation if polarization is parallel to the longer axis of rectangular waveguides . Consequently, to obtain the same coupling efficiency, even though the longer-side-coupled coupler works with shorter overlap length than the shorter-side-coupled coupler, there is little difference in the insertion loss between them because of the compensation between guiding loss and overlap length.
We further investigated the relation between insertion loss and operating frequency. From Table 2 , it is observed that the insertion loss increases as the operating frequency is away from 376GHz. Although we can find from Fig. 2 that the square pipe waveguide with S = 6mm and t = 1mm suffers about the same attenuation for 376GHz, 368GHz, and 360GHz, the pipe-waveguide-based directional coupler operated at 376GHz work most efficiently. As a result, to obtain the same coupling efficiency, the overlap length should be longer if the operating frequency is away from 376GHz, leading to higher insertion loss.
We propose a leaky mode THz directional coupler for future THz applications. We construct the directional coupler with two square pipe waveguides so that the contact area between two waveguides can be significantly increased, leading to higher coupling efficiency. Due to the anti-resonant reflecting guiding mechanism, the proposed directional coupler works most efficiently in the minimal-attenuation wavelength regime. The directional coupler is polarization-sensitive and higher coupling efficiency can be obtained if guided waves are TM-polarized to the cladding between two waveguides. Moreover, the directional coupler with smaller air-core sizes works more efficiently. We further introduce rectangular pipe-waveguide-based directional couplers and the longer-side-coupled rectangular directional coupler is preferred. It is also found that the insertion loss is independent of the coupler shapes, while the insertion loss increases as the operating frequency is away from the resonant frequency of the coupling cladding. We believe that the pipe-waveguide-based directional coupler is promising for many THz applications, including THz communications, sensing, and imaging.
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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