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In this paper, a self-supporting polymer pipe is proposed and investigated for THz wave transmission. Utilizing fiber drawing technique for polymer fiber, self-supporting pipes with wall thickness of several tens micrometers can be fabricated using polymethylmethacrylate (PMMA). The guiding mechanism and transmission characteristics of the self-supporting pipes are investigated theoretically, showing that it can support single-mode transmission at THz band. The self-supporting pipe samples with different structure parameters are fabricated and measured experimentally, showing that it can support single HE11 mode transmission. Theoretical analysis and experimental results show that this self-supporting polymer pipe is a promising candidate for low loss THz fibers.
©2013 Optical Society of America
1. Introduction
In recent years, terahertz (THz) technology has shown a rapid development due to its promising applications on security, sensing and communications [1–5]. THz fibers are basic elements in systems for THz applications. Since low loss materials, as fused silica for the near infrared band, are not available for the THz band, various fiber structures with air core or air holes were proposed to realize low loss THz transmission by confining the THz waves in dry air, such as hollow core fiber with metal layer [6] or metal/dielectric layers [7], all-polymer Bragg fibers [8], ferroelectric PVDF cladding waveguides [9], honeycomb band gap fibers [10,11], porous subwavelength fibers [12, 13] and suspended core fibers [14]. Recently, simple dielectric pipes attract much attention as a scheme to realize low loss THz transmission. They guide THz waves by the anti-resonance effect of their thin walls, which act as transverse Fabry-Perot etalons. Previous work shows that the thin wall dielectric pipe supports a series of low loss transmission bands separated by high loss frequencies satisfying the resonance condition of the thin wall [15–17]. The width of the transmission band is determined by the thickness and index of the thin wall [18]. However, the transmission loss of this kind of THz waveguides is still limited [19]. Although enlarging the core size has been proven to be an effective way to reduce transmission loss, it reduces the flexibility of the pipe [15, 18], which is an important required characteristic of THz fibers. On the other hand, large core size will lead to multi-mode transmission, hence, the output field pattern of the large core dielectric pipe is sensitive to the coupling condition, leading to a limited beam quality and stability. Another way to improve the transmission properties of the dielectric pipe is reducing its wall thickness [19]. In our previous work [20], by drawing a PMMA tube to a pipe with a wall thickness of about 200 μm, the potential of the thin wall dielectric pipe on low loss THz transmission has been demonstrated. However, farther reduction of the wall thickness is limited by the mechanical strength of the pipe required in its fabrication processes and applications.
In this paper, a self-supporting polymer pipe is proposed and investigated for low loss THz transmission. Theoretical analysis indicates that low transmission loss and large transmission bandwidth can be achieved in this structure. Furthermore, by analyzing the characteristics of the fundamental mode and higher order modes in the self-supporting pipe, we find this structure supports single-mode transmission at THz band. Then the self-supporting polymer pipes are fabricated and measured utilizing a CO2 laser-pumped THz laser operating at 3.1THz. The mechanism of low loss single mode transmission in the self-supporting pipe is demonstrated experimentally.
2. Self-supporting polymer pipe fabricated by PMMA
In our previous work, thin wall PMMA pipes are fabricated from PMMA tubes utilizing the polymer fiber drawing technique. Their wall thickness and diameter are about 200 μm and 3~4 mm, respectively, controlled by the drawing condition. In this paper, to farther reduce the wall thickness, a new fiber preform is realized by filling seven such PMMA pipes into a large PMMA tube. Then the preform is drawn to a pipe with compound structure shown in Fig. 1(a). In the drawing process, proper pressure is added in the seven pipes to reduce the wall thickness and eliminate the spaces between the pipes and the outer tube. It can be seen that a thin hexagonal inner pipe is fixed in the center of the outer pipe supporting by six thin walls, hence, it is called “self-supporting pipe”. The thickness of the inner pipe and supporting walls is about 20~40 μm, far thinner than the PMMA pipe in our previous work. While, the typical thickness and diameter of the outer pipe are about 200~300 μm and 3~5mm, respectively.
Fig. 1 The self-supporting PMMA pipe and its models for theoretical analysis. (a) The photo of the transverse section, (b) Model with hexagonal inner pipe, (c) Model with round inner pipe without supporting
3. Theoretical analysis
Two kinds of models are proposed to analyze the transmission characteristics of self-supporting PMMA pipe theoretically. The first one is shown in Fig. 1(b), which has a hexagonal inner pipe supported by six thin walls. The distance between the apex and the center of the hexagonal inner pipe is denoted by rh, the thickness of the hexagonal inner pipe is denoted by th. The radius and thickness of the outer pipe are denoted by Rh and Th, respectively. The transmission characteristics of this waveguide model can be calculated by the finite element method (FEM). Considering that the light confinement in this structure is mainly due to the reflection of the inner pipe, the model shown in Fig. 1(b) can be farther simplified to Fig. 1(c), in which the inner pipe is simplified to a round thin wall pipe, while the supporting walls are neglected. Similar simplification has been applied in the analysis on Kagome fiber [21, 22]. In this model, the radius and thickness of the inner pipe are denoted by r and t, while, the radius and thickness of the outer pipe are denoted by R and T. The simplified model shown in Fig. 1(c) can be calculated by the transfer matrix method (TMM).
Firstly, the attenuation spectrum of the HE11 mode from 2THz to 5THz in this structure are calculated and shown in Fig. 2. The green triangles and black squares are the calculation results of the models shown in Fig. 1(b) and 1(c), respectively. In the calculation, typical structure parameters are used, which are shown in Table 1.
Fig. 2 Calculated attenuation spectra of HE11 mode in self-supporting pipes. The green down-triangles are calculated by model of Fig. 1(b), the black squares are calculated by model of Fig. 1(c), the blue up-triangles and red circles are calculated attenuation spectra of HE11 mode in only the inner pipe and only the outer pipe, respectively
Table 1. Structure parameters used in calculation
The index of the polymer material is assumed to 1.53 + 0.003i [23]. It can be seen that the attenuation spectra calculated by the two models agree well. Although the simplified model may underestimate the attenuation at low frequency region, the low loss transmission bands and high loss frequency region between them can be indicated correctly by it. In the following analysis, we use the simplified model [shown in Fig. 1(c)] to investigate the guiding mechanism and transmission characteristics of the self-supporting pipe.
To demonstrate the guiding mechanism of the self-supporting pipe, the attenuation spectra of the inner pipe and the outer pipe in the simplified model are calculated separately, and shown in Fig. 2 as the blue up-triangles and red solid circles, respectively. It can be seen that due to its thin thickness, the transmission band of inner pipe is very broad, almost all the frequency region of the calculation is in a low loss anti-resonance transmission band, which shows that due to the extreme thin wall thickness, the inner pipe alone can act as an effective antiresonant THz waveguide. While, the transmission band of the outer pipe is far narrow than that of the inner pipe. At the frequencies that the outer pipe is in the anti-resonance band, the attenuation of the self-supporting pipe is lower than that of the inner pipe, since the anti-resonance of the outer pipe provides additional light confinement to the field leaking to the area between the inner and the outer pipes. On the other hand, at the frequencies that the outer pipe has high loss due to the resonance condition of its wall, the attenuation of the self-supporting pipe is close to that of the inner pipe, showing that the field leaking from the inner pipe is lost due to the leakage and the material absorption of the outer pipe. At the frequencies that the inner pipe has high loss, it can be expected that the field cannot confined in the inner pipe effectively, much field leaks out of the inner pipe and the light confinement is mainly provide by the outer pipe. It worth to note that the calculated attenuation by the simplified model with inner pipe is always lower than the case of outer pipe only, showing that the introduction of the inner pipe do not introduce additional loss, but improve the transmission performance by combining the guiding effects of inner pipe and outer pipe together.
Figure 3 shows the variations of the attenuation spectra of the HE11 mode in the self-supporting pipe when the structure parameters changes. According to the analysis on Fig. 2, the calculation results can be understood clearly by the effects of the inner pipe and the outer pipe. Figure 3(a) and 3(b) show the results when the radius r and wall thickness t of the inner pipe changes, respectively. Other parameters are unchanged as Table 1 in the calculation. It can be seen that the increasing r doesn’t change the low loss transmission band, while it is extremely helpful in reducing the attenuation coefficient. On the other hand, the transmission band of the self-supporting pipe is sensitive to the variation of t. These results are similar to the case that only the inner pipe is considered. The spectrum variations under different radius R and wall thickness T of the outer pipe are calculated in a relatively narrower band and shown in Fig. 3(c) and 3(d) respectively, in which other structural parameters are also unchanged as Table 1. It can be seen that the resonance condition in the outer pipe wall leads to the ripples in the spectrum, which are determined by T. On the other hand, the attenuation spectrum is almost unchanged under different R. According to Fig. 3, it can be concluded that to realize a THz fiber at a certain frequency, t is the main design parameters to optimize the low loss transmission band. Larger r is preferred to reduce the transmission loss, while R can be designed to a relatively small value, which is helpful to improve the flexibility of the self-supporting pipe as a THz fiber.
Fig. 3 The impacts of structure parameter variations on the attenuation spectra of the self-supporting pipe calculated by the simplified model shown in Fig. 1(c).
Figure 3(a) also shows that the variation of the attenuation spectrum is not continuous with the increasing r. There is a rapid reduction when r changes from 0.8 mm to 1.0 mm. To show this effect clearly, the attenuation of several typical modes in the self-supporting pipe are calculated at 3.1 THz under different r, which is shown in Fig. 4. Other parameters used in the calculation are unchanged as Table 1. The black squares are the results of HE11 mode, showing the rapid reduction of the modal attenuation clearly. The red circles, blue up-triangles and green down-triangles are the results of TE01, TM01, HE21 modes, respectively. All of them show a sharp reduction of the transmission loss as r increases. However, the value of r for the sharp reduction of the HE11 mode is smaller than those of the higher order modes. It provides a possibility to realize single HE11 mode transmission if r is designed properly, which is shown as the grid region in Fig. 4. To show the mechanism of the single mode transmission in the self-supporting pipe, the electrical field distributions under different r are also calculated and shown in Fig. 4. It can be seen that for the pipe with an r larger than the single mode transmission region (grid region), the field confined in the inner pipe by the anti-resonance of its thin wall. This field distribution is kept as the r decreases to the single mode transmission region. However, if r decreases to a value lower than the grid region, the field extends to the area between the inner pipe and the outer pipe. In this case, the light mainly guides by the outer pipe, which leads to a rapid increase of the modal attenuation. For the higher order modes, the value of r for the field extension is higher than that of the HE11 mode, leading to the single mode transmission in the self-supporting pipe. Hence, to realize a single mode transmission THz fiber operating at a certain frequency, r should be designed at a proper value to avoid the field extension of the HE11 mode on one hand, and maintain the high attenuations of the higher order modes on the other hand.
Fig. 4 Attenuation coefficients of HE11, TE01, TM01, and HE21 modes under different r at 3.1THz in the simplified model and field patterns of HE11 mode under r = 0.5mm, 1mm and 1.5mm, respectively
Figure 5 is calculated attenuation spectra of HE11, TE01, TM01, and HE21 modes in the simplified model under r = 1mm, which is in the region supporting single mode transmission shown in Fig. 4. It can be seen that the attenuation of HE11 mode is quite low in the all frequency region of the calculation (2THz~5THz). While, the attenuations of higher order modes are far higher and show high loss regions periodically due to the resonance condition of the wall of the outer pipe, which indicate that these modes are mainly confined by the outer pipe. The calculation of modal field distributions also show that the field of HE11 mode is mainly distributed in the core region and those of the higher order modes are mainly distributed in the area between the inner pipe and the outer pipe. These results show that the single mode transmission band of the self-supporting pipe is broad, not limited to a specific frequency range.
Fig. 5 Attenuation spectra of HE11, TE01, TM01, and HE21 modes in the simplified model under r = 1mm, which is in the region supporting single mode transmission shown in Fig. 4
4. Experiment results
The experimental setup to measure the transmission characteristics of the self-supporting PMMA pipes is shown in Fig. 6. A CO2 laser-pumped THz laser (SIFIR-50 FPL Far-Infrared Laser System) provides a THz wave output at 3.1 THz. A Teflon lens with a focal length of 150 mm is used to focus the wave to a variable diaphragm. Its diameter is the same as that of the inner pipe of the sample. The self-supporting pipe sample places right after the diaphragm. Both ends of the sample, tinfoil is used to cover the area between the inner pipe and outer pipe to eliminate the impacts of the modes guiding in this region, with a hole in the center for light coupling to and out of the modes guiding in the inner pipe. A THz camera (Pyroelectric Array Camera Pyrocamtm Iii Series) is used to detect the field pattern at the output end of the sample.
Three self-supporting pipe samples are fabricated with different structural parameters, rh of which is about 0.5mm, 1mm, 1.5mm respectively, corresponding to the three different conditions in Fig. 4. Figure 7 is the measured output field patterns of different samples under various coupling conditions, which is adjusted by changing the location of the sample input end and the angle between the sample and the output direction of the laser. The sample 1 has the smallest inner pipe size among the three samples. The measured output field is very weak under all the possible coupling conditions, which is shown in Fig. 7(a). It is due to that the fields of HE11 mode and other higher order modes extend to the area between the inner pipe and the outer pipe, which is blocked by the tinfoil at the output end of the pipe. The sample 3 has the largest pipe size among the three samples. Figure 7(c) shows the output field patterns of the sample 3 under different coupling conditions. It can be seen that the field pattern varies with the coupling condition, showing the characteristics of multi-mode transmission clearly. The measured results for the sample 2 are shown in Fig. 7(b). It can be seen that the output field pattern is a single round spot under different coupling condition, showing that in this sample single HE11 mode transmission is realized. Hence, these experiment results demonstrate the mechanism of single mode transmission in the self-supporting pipe analyzed in Fig. 4.
Fig. 7 Typical field patterns at the output end of the pipe samples with the lengths of 0.15m and rh of 0.5mm (a), 1mm (b), 1.5mm (c).
Calculated by the measurement results of the THz wave intensity behind the diaphragm at the input end and the output intensity at the output end of the pipe, the insertion loss of the sample 2 (15cm in length) is 5.34 dB including the coupling loss at the input end.
5. Summary
In this paper, a self-supporting polymer pipe is proposed for THz wave transmission. Utilizing fiber drawing technique for polymer fiber, self-supporting pipes with wall thickness of several tens micrometers can be fabricated using PMMA. Utilizing the simplified model for the pipe, the effects of the inner pipe and outer pipe are analyzed. Theoretical analysis also shows that the pipe can support single HE11 mode transmission if the radius of the inner pipe is designed properly. In experiments, the self-supporting pipe samples with different structure parameters are fabricated and measured, demonstrating the single mode transmission in the pipe sample. The insertion loss of the single HE11 mode transmission of a 15cm sample is 5.34 dB. Theoretical analysis and experimental measurement show that the proposed self-supporting polymer pipe is a promising candidate for high-quality THz fibers.
Acknowledgment
This work is supported in part by 973 Programs of China under Contract No. 2010CB327600, Nature Science Foundation of Beijing under Grant No. 4102016 and Basic Research Foundation of Tsinghua National Laboratory for Information Science and Technology (TNList).
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