## Characteristics of bent terahertz antiresonant reflecting pipe waveguides |

Optics Express, Vol. 22, Issue 7, pp. 8460-8472 (2014)

http://dx.doi.org/10.1364/OE.22.008460

Acrobat PDF (1468 KB)

### Abstract

Bending characteristics of the terahertz (THz) pipe waveguides are numerically investigated. Numerical results reveal that the inherent periodic feature of the loss spectrum, resulting from the antiresonant reflection guiding mechanism, is nearly unaffected under bending. However, attenuation constant of the fundamental (HE_{11}) mode becomes polarization dependent for the bent pipe waveguide, and the polarization perpendicular to the bending plane experiences less bending losses. Moreover, unlike the straight case where a larger air-core diameter leads to a smaller attenuation constant, increasing core diameter of the bent pipe waveguide is unable to reduce attenuation constant effectively if the propagation mode is a whispering gallery mode. Finally, behavior of the bent pipe waveguide connected to a straight one is also examined in this work.

© 2014 Optical Society of America

## 1. Introduction

1. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics **1**(2), 97–105 (2007). [CrossRef]

5. J. Federici and L. Moeller, “Review of terahertz and subterahertz wireless communications,” J. Appl. Phys. **107**(11), 111101 (2010). [CrossRef]

6. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B **17**(5), 851–863 (2000). [CrossRef]

16. M. Rozé, B. Ung, A. Mazhorova, M. Walther, and M. Skorobogatiy, “Suspended core subwavelength fibers: towards practical designs for low-loss terahertz guidance,” Opt. Express **19**(10), 9127–9138 (2011). [CrossRef] [PubMed]

17. C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, “Low-index terahertz pipe waveguides,” Opt. Lett. **34**(21), 3457–3459 (2009). [CrossRef] [PubMed]

18. C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express **18**(1), 309–322 (2010). [CrossRef] [PubMed]

19. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO_{2}–Si multilayer structures,” Appl. Phys. Lett. **49**(1), 13–15 (1986). [CrossRef]

^{−1}, under straight condition [17

17. C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, “Low-index terahertz pipe waveguides,” Opt. Lett. **34**(21), 3457–3459 (2009). [CrossRef] [PubMed]

20. E. Nguema, D. Férachou, G. Humbert, J. L. Auguste, and J. M. Blondy, “Broadband terahertz transmission within the air channel of thin-wall pipe,” Opt. Lett. **36**(10), 1782–1784 (2011). [CrossRef] [PubMed]

21. B. You, J.-Y. Lu, C.-P. Yu, T.-A. Liu, and J.-L. Peng, “Terahertz refractive index sensors using dielectric pipe waveguides,” Opt. Express **20**(6), 5858–5866 (2012). [CrossRef] [PubMed]

22. A. Mazhorova, A. Markov, B. Ung, M. Rozé, S. Gorgutsa, and M. Skorobogatiy, “Thin chalcogenide capillaries as efficient waveguides from mid-infrared to terahertz,” J. Opt. Soc. Am. B **29**(8), 2116–2123 (2012). [CrossRef]

23. M. F. Xiao, J. Liu, W. Zhang, J. L. Shen, and Y. D. Huang, “THz wave transmission in thin-wall PMMA pipes fabricated by fiber drawing technique,” Opt. Commun. **298-299**, 101–105 (2013). [CrossRef]

23. M. F. Xiao, J. Liu, W. Zhang, J. L. Shen, and Y. D. Huang, “THz wave transmission in thin-wall PMMA pipes fabricated by fiber drawing technique,” Opt. Commun. **298-299**, 101–105 (2013). [CrossRef]

24. J.-T. Lu, Y.-C. Hsueh, Y.-R. Huang, Y.-J. Hwang, and C.-K. Sun, “Bending loss of terahertz pipe waveguides,” Opt. Express **18**(25), 26332–26338 (2010). [CrossRef] [PubMed]

25. W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Field deformation in a curved single-mode fibre,” Electron. Lett. **14**(5), 130–132 (1978). [CrossRef]

26. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. **11**(2), 75–83 (1975). [CrossRef]

27. S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. **14**(9), 2085–2092 (1996). [CrossRef]

30. J. Xiao, H. Ni, and X. Sun, “Full-vector mode solver for bending waveguides based on the finite-difference frequency-domain method in cylindrical coordinate systems,” Opt. Lett. **33**(16), 1848–1850 (2008). [CrossRef] [PubMed]

29. K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, “Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends,” Opt. Express **14**(23), 11128–11141 (2006). [CrossRef] [PubMed]

30. J. Xiao, H. Ni, and X. Sun, “Full-vector mode solver for bending waveguides based on the finite-difference frequency-domain method in cylindrical coordinate systems,” Opt. Lett. **33**(16), 1848–1850 (2008). [CrossRef] [PubMed]

31. F. L. Teixeira and W. C. Chew, “PML-FDTD in cylindrical and spherical grids,” IEEE Microwave Guided Wave Lett. **7**(9), 285–287 (1997). [CrossRef]

## 2. Structure for numerical modeling

*x*-

*z*plane and the bending radius

*R*is measured from the bending center to the point where

*x*= 0. Cross-section of the pipe waveguide is shown in Fig. 1(b). It consists of a large air core with refractive index

*n*

_{1}

*=*1 and a uniform dielectric cladding layer with refractive index

*n*

_{2}. The thickness

*t*of the cladding layer is much smaller than the diameter

*D*of the air core.

18. C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express **18**(1), 309–322 (2010). [CrossRef] [PubMed]

19. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO_{2}–Si multilayer structures,” Appl. Phys. Lett. **49**(1), 13–15 (1986). [CrossRef]

18. C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express **18**(1), 309–322 (2010). [CrossRef] [PubMed]

*c*is the speed of light in vacuum and

*m*is an integer.

30. J. Xiao, H. Ni, and X. Sun, “Full-vector mode solver for bending waveguides based on the finite-difference frequency-domain method in cylindrical coordinate systems,” Opt. Lett. **33**(16), 1848–1850 (2008). [CrossRef] [PubMed]

*x*-

*y*plane, with (

*x*,

*y*) = (0, 0) being the waveguide center. Essentially, the FDFD mode solver is to solve an eigenvalue matrix equation in the form:where

*E*and

_{x}*E*are transverse components of the electric field and

_{y}*β*is the complex propagation constant. Once Eq. (2) is solved, bending loss, in terms of attenuation constant, can be obtained as −2Im(

*β*). In subsequent simulations, the following parameters are assumed:

*D*= 9 mm,

*t*= 0.5 mm, and

*n*

_{2}= 1.4. We primarily investigate the fundamental (HE

_{11}) mode in this work.

## 3. Numerical results

### 3.1 General characteristics

*x*-polarized) and the polarization perpendicular to the bending plane (

*y*-polarized) are both shown in Fig. 2. The bending radius is

*R*= 200 cm. Attenuation spectrum of the straight pipe waveguide is also shown for comparison, where the polarization is not defined for the straight case, because attenuation constants of the

*x*- and

*y*-polarized HE

_{11}modes are the same owing to cylindrical symmetry. In Fig. 2, it is noted that two discontinuities (around 300 and 600 GHz, respectively) occur in the attenuation spectrum of the straight pipe waveguide. These discontinuity frequencies are the resonant frequencies predicted by Eq. (1), which can be confirmed by substituting

*m*= 2 and 3 into Eq. (1) and the resultant resonant frequencies are 306 and 612 GHz, respectively. As previously stated, near the resonant frequencies, the cladding is almost transparent and THz waves could hardly be confined in the air-core region. Thus local maximum losses occur. These resonant frequencies make the attenuation spectrum periodic, and such a periodicity is a unique feature of the ARROW-like waveguides. Obviously, the bent pipe waveguide exhibits the same periodic behavior as that of the straight waveguide, in that the two discontinuities in the attenuation spectrum of the bent pipe waveguide also coincide with the resonant frequencies. Hence, simulation results in Fig. 2 reveal that the inherent periodic characteristics of the THz pipe waveguide is nearly unaffected under bending. In addition, as expected, attenuation constants of the bent pipe waveguide are larger than those of the straight one because of the radiationlosses caused by bending. However, it is found that the

*y*-polarized mode suffers less bending losses than the

*x*-polarized one does. The polarization effect on bending will be discussed later.

*y*-polarized mode. Since there are three pass bands in the attenuation spectra shown in Fig. 2, we choose one frequency from each band and the results are calculated at 200, 500, and 800 GHz, respectively. Clearly, owing to the bending, field profile of the THz wave not only shifts toward the outer cladding (toward the +

*x*direction) but also narrows substantially. The shift of the THz wave from the air-core center increases as the bending radius decreases. It also increase with frequency. According to the simulations, the field shift is polarization independent, i.e., it is the same for both

*x*- and

*y*-polarized modes. One simulated example is illustrated in Fig. 3(b), where the intensity distributions and the electric field vector distributions of both polarizations are displayed. They are calculated at 800 GHz and

*R*= 75 cm. If the shift ratio is defined as: (the shift of intensity peak from the core center) / (the core radius) × 100%, for the case shown in Fig. 3(b), the shift ratios are 67% and 66% for the

*x*- and

*y*- polarized modes, respectively. From Fig. (3), it is noted that even though the field profiles suffer considerable deformation, THz waves are still well-confined in the air-core region. This suggests that the THz pipe waveguide may sustain severe bending without suffering excessive bending losses.

### 3.2 Polarization effect

*x*- and

*y*-polarized modes as a function of bending radius are shown in Fig. 4. The results are calculated at 800 GHz. Clearly, attenuation constants for both polarizations are almost the same when the bending radius is large, e.g.,

*R*= 1000 cm. Actually, if the bending radius approaches infinity, which is the case of straight pipe waveguide, there would be no difference between the attenuation constants of

*x*- and

*y*-polarizations because of the symmetrical geometry under straight condition. When the bending radius decreases, as expected, the attenuation constant of either polarization increases. However, it is found that the attenuation constant of the

*y*-polarized mode is less than that of the

*x*-polarized one, and the difference between the two polarizations is getting obvious with decreasing bending radius. This phenomenon is much different from that observed in the total-internal-reflection-based waveguides, such as the conventional step-index fiber. For the step-index fiber, bending losses of the

*x*- and

*y*-polarized HE

_{11}modes are the same [32

32. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. **66**(3), 216–220 (1976). [CrossRef]

33. R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. **43**(10), 899–909 (2007). [CrossRef]

*y*-polarized HE

_{11}mode experiences less bending loss than the

*x*-polarized one does. Similar polarization effect was also observed in the bent hollow-core Bragg fiber [34

34. M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Full-vectorial coupled mode theory for the evaluation of macro-bending loss in multimode fibers. application to the hollow-core photonic bandgap fibers,” Opt. Express **16**(19), 14945–14953 (2008). [CrossRef] [PubMed]

*x*-polarized (parallel to the bending plane), the wave close to the outer cladding is like a transverse magnetic one (TM-like), i.e., the electric field is perpendicular to the outer core-cladding interface, as shown in the left case of Fig. 3(b). On the contrary, for the

*y*-polarized THz wave (perpendicular to the bending plane), the electric field is parallel to the outer core-cladding interface which makes the wave like a transverse electric wave (TE-like), as shown in the right case of Fig. 3(b). It is well known from the electromagnetic theory that reflection is larger for the TE wave than for the TM wave [35]. As a result, the bending loss for the

*y*-polarized HE

_{11}mode is smaller than that of the

*x*-polarized one. In other words, the

*y*-polarization, i.e., the polarization perpendicular to the bending plane, would be more preferred for the THz pipe waveguides when bending is taken into consideration.

### 3.3 Influence of the air-core diameter

*y*-polarized mode. Intensity distributions of the bent pipe waveguides with various core diameters are shown in Fig. 5, where the cladding thickness is still fixed to 0.5 mm. The frequency is 500 GHz and the bending radius is

*R*= 200 cm. Simulated results for the straight case are also shown for comparison. Clearly, under the same bending condition, the larger core diameter the pipe waveguide has, the more severe deformation the field profile suffers. In Fig. 5, shift ratios of the intensity peak are 3%, 13%, 34%, 51%, and 61% for

*D*= 3, 6, 9, 12, and 15 mm, respectively.

**18**(1), 309–322 (2010). [CrossRef] [PubMed]

*D*

^{4}for the THz pipe waveguide under straight condition [17

17. C.-H. Lai, Y.-C. Hsueh, H.-W. Chen, Y.-J. Huang, H.-C. Chang, and C.-K. Sun, “Low-index terahertz pipe waveguides,” Opt. Lett. **34**(21), 3457–3459 (2009). [CrossRef] [PubMed]

*θ*at the outer core-cladding interface, where the incident angle is defined with respect to the interface normal as shown in Figs. 7(a) and 7(b). At the point of transition, the reflected ray from the outer core-cladding interface must graze the inner core-cladding interface tangentially, as shown in Fig. 7(c). Let the incident angle satisfying this condition be denoted as

_{i}*θ*. From Fig. 7(c), one haswhere

_{WGM}*O*is the point of bending center, and

*A*and

*B*are the points the ray touches the outer and inner core-cladding interfaces, respectively. For the bent pipe waveguide, if

*θ*<

_{i}*θ*, the propagation mode is a perturbed mode; otherwise, if

_{WGM}*θ*>

_{i}*θ*, it is a whispering gallery mode. From Eq. (3),

_{WGM}*θ*would be smaller for a smaller

_{WGM}*R*or a larger

*D*. In addition,

*θ*would be larger at higher frequencies [18

_{i}**18**(1), 309–322 (2010). [CrossRef] [PubMed]

26. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. **11**(2), 75–83 (1975). [CrossRef]

*θ*for the bent pipe waveguides operating at 500 GHz, with the bending radii of

_{i}*R*= 75, 200, 500, and 1000 cm, are shown in Figs. 8(a)–8(d), respectively. The critical incident angles

*θ*obtained according to Eq. (3) are also displayed in the figures. When

_{WGM}*θ*=

_{i}*θ*, transition between the perturbed mode and the whispering gallery mode will take place. From Fig. 8, it is found transitions occur at the core diameters

_{WGM}*D*= 4.3, 6.0, 8.1, and 10.3 mm for

*R*= 75, 200, 500, and 1000 cm, respectively. The results indicate that, taking

*R*= 75 cm as an example, if the core diameter is less than 4.3 mm,

*θ*<

_{i}*θ*, thus the propagation mode is a perturbed mode; on the other hand, if the core diameter is larger than 4.3 mm,

_{WGM}*θ*>

_{i}*θ*, and it becomes a whispering gallery mode. The four critical diameters with which transitions occur are shown as colored arrows in Fig. 6(b) for the frequency of 500 GHz, where one color corresponds to one bending radius. From Fig. 6(b), it is apparent that the four critical diameters are consistent with the four points where the attenuation curves of bent pipe waveguides diverge from that of the straight one. That is to say, the critical diameter for some bending radius, e.g.,

_{WGM}*R*= 75cm, agrees with the boundary point between the sharp and flat segments of the attenuation curve for that bending radius. Critical diameters for 200 and 800 GHz are also shown as colored arrows in Figs. 6(a) and 6(c), respectively, and agreement between the critical diameters and the boundary points can be clearly observed as well. Such an agreement exhibited in Fig. 6 provides a strong support of the aforementioned inference that the sharp segment corresponds to the perturbed mode, while the flat segment corresponds to the whispering gallery mode. Therefore, when the bent pipe waveguide is operated in the perturbed mode, increasing the air-core diameter can significantly reduce the attenuation constant; however, if the waveguide is in the whispering gallery mode, increasing the core diameter would be in vain for reduction of the attenuation constant.

### 3.4 Considering the straight-to-bent interface

_{11}mode propagates from the straight one into the bent one, coupling loss will occur due to the mismatch between the field distributions of the straight and bent HE

_{11}modes. Moreover, higher order modes will be excited in the bent section. To study the problem with a straight-to-bent interface, one of the commonly used techniques is the coupled mode theory [34

34. M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Full-vectorial coupled mode theory for the evaluation of macro-bending loss in multimode fibers. application to the hollow-core photonic bandgap fibers,” Opt. Express **16**(19), 14945–14953 (2008). [CrossRef] [PubMed]

**18**(1), 309–322 (2010). [CrossRef] [PubMed]

*P*of the

_{i}*i*-th mode can be calculated according to the following overlap integral:where

**E**

*is the field distribution of the input,*

_{in}**E**

*is the field distribution of the*

_{i}*i*-th mode in the bent pipe waveguide, and * represents the operation of complex conjugate. Since the eigen modes of the pipe waveguide are leaky, cut-off approximation [34

34. M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Full-vectorial coupled mode theory for the evaluation of macro-bending loss in multimode fibers. application to the hollow-core photonic bandgap fibers,” Opt. Express **16**(19), 14945–14953 (2008). [CrossRef] [PubMed]

*N*is the number of modes in the bent pipe waveguide,

*α*is the attenuation constant of the

_{i}*i*-th mode, and

*L*is the length of the bent waveguide. The total power loss (in dB) after propagation through the bent pipe waveguide can be obtained as –10 log(

*P*).

_{out}_{11}mode, the next three higher order modes are the TE

_{01}, HE

_{21}, and TM

_{01}modes, respectively [18

**18**(1), 309–322 (2010). [CrossRef] [PubMed]

*R*= 500 cm. Note that there are two orientations of the HE

_{21}mode shown in Fig. 9. They are denoted as HE

_{21o}and HE

_{21e}depending on whether they are odd or even with respect to the bending plane. In the straight case, the two orientations are identical except a rotation of 45°. They become different in the bent case, because the rotational symmetry is destroyed upon bending. Calculated attenuation constants under the assumed bending condition are 0.0034, 0.0063, 0.0072, and 0.0102 cm

^{–1}for the TE

_{01}, HE

_{21o}, HE

_{21e}, and TM

_{01}modes, respectively.

*y*- and

*x*-polarized HE

_{11}modes in the bent pipe waveguide are denoted as HE

_{11y}and HE

_{11x}. Under the same bending condition, attenuation constants of the HE

_{11y}and HE

_{11x}modes are 0.0018 and 0.0022 cm

^{–1}, respectively.

*y*-polarized HE

_{11}mode of a straight pipe waveguide. Taking

*R*= 500 cm as an example, with the

*y*-polarized input, excited modal powers of the HE

_{11y}, HE

_{11x}, TE

_{01}, HE

_{21o}, HE

_{21e}, and TM

_{01}modes in the bent pipe waveguide calculated according to Eq. (4) are 0.978, 0, 0.018, 0.004, 0, and 0, respectively. Clearly, excited modal powers of the HE

_{11x}, HE

_{21e}, and TM

_{01}modes are all zero. It is because the

*y*-component of the electric field is absent for the HE

_{11x}mode and antisymmetric with respect to the

*x*-axis for the HE

_{21e}and TM

_{01}modes. Therefore, only the HE

_{11y}, TE

_{01}, and HE

_{21o}modes can be excited by the

*y*-polarized HE

_{11}mode of the straight waveguide. Figures 10(a)–10(c) plot the total power losses of the bent pipe waveguide as a function of waveguide length

*L*for

*R*= 500, 200, and 75 cm, respectively. Two conditions are shown for comparison. One (

*N*= 1) considers only the HE

_{11y}mode in Eq. (5), and the other (

*N*= 3) takes the HE

_{11y}, TE

_{01}, and HE

_{21o}modes into account. Note that the power loss at

*L*= 0 represents the coupling loss occurring at the straight-to-bent interface. As shown in Fig. 10, the coupling loss of the HE

_{11y}mode (

*N*= 1) increases with decreasing bending radius. The coupling losses are 0.1, 0.5, and 2.3 dB for

*R*= 500, 200, and 75 cm, respectively, and correspondingly, 98%, 88%, and 59% of the input power are transferred into the HE

_{11y}mode. Hence, when there is a straight-to-bent interface, it should be enough to describe the bending behavior of the pipe waveguide by considering only the HE

_{11y}mode for

*R*= 500 and 200 cm, but insufficient for

*R*= 75 cm. If the two higher order TE

_{01}and HE

_{21o}modes are included into simulation, the total coupling losses (

*N*= 3) reduce to 0, 0, and 0.3 dB for the three bending radii. Thus the simulation can be improved by the inclusion of the higher order modes. It is noted that the coupling loss for

*R*= 75 cm is not zero (0.3 dB) when three modes are considered, indicating that there might be additional higher order modes excited by the missing input power. In general, more higher order modes are necessary to have a more complete analysis, especially for the case of strong bending.

## 4. Conclusion

_{11}mode) is polarization dependent, and the polarization perpendicular to the bending plane experiences a smaller bending loss than that of the polarization parallel to the bending plane. Moreover, when the bent pipe waveguide is operated in the perturbed mode, like the straight case, increasing the air-core diameter can significantly reduce the attenuation constant; nevertheless, if the waveguide is operated in the whispering gallery mode, increasing the core diameter would be ineffective for reduction of the attenuation constant. Finally, bending behavior of the pipe waveguide with a straight-to-bent interface is examined by using the coupled mode theory. It is shown that simulation can be improved by considering the higher order modes, especially for the case of strong bending.

## Appendix

26. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. **11**(2), 75–83 (1975). [CrossRef]

*θ*at the outer core-cladding interface for the bent pipe waveguide. Consider the straight case first, and Fig. 11(a) depicts the longitudinal cross-section of the straight pipe waveguide. It has been shown that the incident angle

_{i}*θ*for the straight pipe waveguide is given by [18

_{i}**18**(1), 309–322 (2010). [CrossRef] [PubMed]

*β*is the complex propagation constant,

*n*

_{1}is the refractive index of the core, and

*k*

_{0}is the free-space wavenumber.

**11**(2), 75–83 (1975). [CrossRef]

*n*(

*x*,

*y*) is the original refractive index profile of the bent pipe waveguide, and

*n'*(

*x*,

*y*) is the transformed index profile of its equivalent straight waveguide. In Eq. (7), the latter approximation holds when

*x*

*R*, i.e., the distance from the waveguide center is far smaller than the bending radius. The transformed index distribution

*n'*is tilted with respect to the original one, as shown in Fig. 11(b), where the corresponding field distribution for the bent pipe waveguide is illustrated as well. The index distribution and the field profile are shown along the

*x*-axis (

*y*= 0). It is noted that refractive index of the core of the equivalent straight waveguide, after conformal transformation, is no longer homogeneous. This prevents the direct usage of Eq. (6) to determine the incident angle, because the core index becomes position dependent. However, since most of the energy is around the peak of the field, it seems reasonable to represent the core index with the equivalent one

*n*, which is the refractive index in the position

_{eq}*x*where the field peak takes place. The equivalent index

_{max}*n*is given byBy replacing

_{eq}*n*

_{1}in Eq. (6) with

*n*, the incident angle

_{eq}*θ*at the outer core-cladding interface of the bent pipe waveguide may be estimated as

_{i}## Acknowledgment

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20. | E. Nguema, D. Férachou, G. Humbert, J. L. Auguste, and J. M. Blondy, “Broadband terahertz transmission within the air channel of thin-wall pipe,” Opt. Lett. |

21. | B. You, J.-Y. Lu, C.-P. Yu, T.-A. Liu, and J.-L. Peng, “Terahertz refractive index sensors using dielectric pipe waveguides,” Opt. Express |

22. | A. Mazhorova, A. Markov, B. Ung, M. Rozé, S. Gorgutsa, and M. Skorobogatiy, “Thin chalcogenide capillaries as efficient waveguides from mid-infrared to terahertz,” J. Opt. Soc. Am. B |

23. | M. F. Xiao, J. Liu, W. Zhang, J. L. Shen, and Y. D. Huang, “THz wave transmission in thin-wall PMMA pipes fabricated by fiber drawing technique,” Opt. Commun. |

24. | J.-T. Lu, Y.-C. Hsueh, Y.-R. Huang, Y.-J. Hwang, and C.-K. Sun, “Bending loss of terahertz pipe waveguides,” Opt. Express |

25. | W. A. Gambling, H. Matsumura, and C. M. Ragdale, “Field deformation in a curved single-mode fibre,” Electron. Lett. |

26. | M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. |

27. | S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol. |

28. | N. N. Feng, G. R. Zhou, C. Xu, and W. P. Huang, “Computation of full-vector modes for bending waveguide using cylindrical perfectly matched layers,” J. Lightwave Technol. |

29. | K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, “Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends,” Opt. Express |

30. | J. Xiao, H. Ni, and X. Sun, “Full-vector mode solver for bending waveguides based on the finite-difference frequency-domain method in cylindrical coordinate systems,” Opt. Lett. |

31. | F. L. Teixeira and W. C. Chew, “PML-FDTD in cylindrical and spherical grids,” IEEE Microwave Guided Wave Lett. |

32. | D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. |

33. | R. T. Schermer and J. H. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. |

34. | M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Full-vectorial coupled mode theory for the evaluation of macro-bending loss in multimode fibers. application to the hollow-core photonic bandgap fibers,” Opt. Express |

35. | H. A. Haus, |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(230.7370) Optical devices : Waveguides

(040.2235) Detectors : Far infrared or terahertz

**ToC Category:**

Terahertz optics

**History**

Original Manuscript: January 2, 2014

Revised Manuscript: February 14, 2014

Manuscript Accepted: March 13, 2014

Published: April 2, 2014

**Citation**

Chih-Hsien Lai, Teng Chang, and Yi-Siang Yeh, "Characteristics of bent terahertz antiresonant reflecting pipe waveguides," Opt. Express **22**, 8460-8472 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8460

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