Modal characteristics of the THz pipe waveguide, which is a thin pipe consisting of a large air core and a thin dielectric layer with uniform but low index, are investigated. Modal indices and attenuation constants are calculated for various core diameters, cladding thicknesses, and cladding refractive indices. Numerical results reveal that the guiding mechanism of the leaky core modes, which transmit most of the power in the air-core region, is that of the antiresonant reflecting guiding. Moreover, modal patterns including modal intensity distributions and electric field vector distributions are shown for the fundamental and higher order modes. Experiments using time-domain spectroscopy with PMMA pipes also confirm the antiresonant reflecting guiding mechanism.
©2010 Optical Society of America
Terahertz (THz) technology has attracted growing research interests in recent years. Applications of THz technology are increasing in a broad variety of areas including biology and medical science, security, imaging, environment monitoring, and information transmission [1,2]. Most of THz systems are large in size because of the usage of free space for THz wave propagation. The major difficulty for THz waveguiding is due to the lack of low-loss materials at the THz range. Various waveguide structures have been proposed according to conventional microwaves [3–5] or photonics [6–8] technologies. However, strong absorption loss in dielectrics and high skin-depth loss of metals prevent these waveguides from long-distance THz transmission applications. To minimize the absorption loss, it is feasible to design waveguides with less power within the lossy material and majority of the power transmitted through the air. Two different types of THz waveguides have been proposed based on this concept. The subwavelength type , where its core dimension is much smaller than the wavelength, has high power fraction outside the core. But severe cross-talk might be encountered if two such waveguides are placed closely . On the other hand, the air-core type, which is essentially an air-core fiber surrounded by a structured cladding, confines most of the power within the air-core region. As a result, outside interference is minimized. Various cladding structures have been applied to the air-core-type waveguides which attempted to provide high reflection at the core-cladding interface. The high reflections could be provided by metallic [11–13], ferroelectric [14,15], or multiple-layered periodic dielectric [16–18] materials. However, owing to the complex cladding structures, previously demonstrated air-core THz waveguides might be difficult to fabricate.
Recently, the low-index pipe waveguide was proposed and experimentally demonstrated for THz waveguiding . Much different from all previously reported air-core THz waveguides which bear high reflection materials or periodic dielectric structures to provide high cladding reflection, structure of the THz pipe waveguide is very simple and it is only a thin pipe consisting of a large air core (with a diameter greater than the wavelength) and a thin dielectric layer (with a thickness much smaller than the core diameter) with uniform but low index. The pipe waveguide is also different from the one utilizing low-index discontinuity , in which a subwavelength air core as well as a small air-core-diameter to cladding-thickness ratio are assumed and the THz waves are guided by the mechanism of total internal reflection (TIR). Guiding mechanism of the pipe waveguide is similar to that of the antiresonant reflecting optical waveguide (ARROW) , but is realized with a single low-index dielectric layer. Using commercially available Teflon air pipes up to 3 meters long as the example pipe waveguides, it was experimentally confirmed that THz waves can be successfully guided in the central air core with excellent mode qualities, high coupling efficiency (as high as 80%), and controllable bandwidth, while the measured attenuation constants can be as low as 0.0008 cm−1 .
Similar pipe structure has ever been seen for infrared (IR) transmission and the leakage loss has been approximately estimated [22,23]. Besides, it has been pointed out that in similar but more complicated photonic bandgap (PBG) optical waveguides, the guiding mechanism could be PBG guiding or antiresonant reflecting guiding [24–26]. However, no detailed modal analysis giving modal patterns, modal indices, and attenuation constants, which are important for waveguide applications, were conducted for the pipe waveguides in the above-mentioned works, especially for THz band.
In this paper, we focus on the theoretical aspect of the THz pipe waveguides, by illustrating the guiding mechanism and providing the modal analysis. Modal characteristics of the pipe waveguides are numerically investigated by the finite-difference frequency-domain (FDFD) mode solver . Modal indices and attenuation constants of the leaky core modes are calculated for various core diameters, cladding thicknesses, and cladding refractive indices. Results are compared with the resonance-frequency formula derived from the Fabry-Perot etalon and the antiresonant reflecting guiding mechanism is theoretically indentified. The effect of material absorption is then examined. Modal patterns of the fundamental mode and several higher-order modes are also presented. At last, through the experiments utilizing time-domain spectroscopy [28,29] with PMMA (polymethylmethacrylate) pipes, the guiding mechanism of the THz pipe waveguides is experimentally confirmed to be that of the antiresonant reflecting guiding. A ray-optics analysis is provided in the appendix, which shows consistency with the FDFD simulations.
2. Guiding mechanism of the core modes
Transverse cross-section of the pipe waveguide is shown in Fig. 1(a) . It has a large air-core (n 1 = 1) with diameter D and a thin ring cladding with thickness t. The medium outside the cladding is also the air. Material of the cladding is assumed to be dielectric with a uniform low refractive index n 2.
The pipe waveguide supports cladding modes for which fields are confined within the cladding region. These modes are guided based on the TIR owing to the higher refractive index of the cladding. However, the cladding modes attenuate rapidly since high material absorption losses are encountered. Alternatively, the pipe waveguide also supports core modes with fields confined in the air-core region. As the refractive index of the core is less than that of the cladding, fields of the core modes will oscillate and radiate through the cladding, making the core modes leaky with the propagation constants being complex. However, as the core modes suffer less material absorption losses than the cladding modes, they are the dominant modes in the pipe waveguide. In this work, only the characteristics of the core modes are investigated.
Guiding mechanism of the core modes in the THz pipe waveguide is similar to that of ARROWs . The ARROW mechanism can be briefly described by viewing the cladding of the pipe waveguide as a Fabry-Perot etalon as shown in Fig. 1(b), where θ 1 and θ 2 are angles of inclined wave vectors with respect to the interface normal in media 1 and 2, respectively. It is well known that a Fabry-Perot etalon behaves like a resonator . Its transmission spectrum is periodic and narrow peaks corresponding to transmission maximums occur at the frequencies at which the etalon resonates. That is to say, at or near the frequencies at which the cladding of the THz pipe waveguide resonates, nearly no reflection takes place at the core-cladding interface and thus fields could hardly exist inside the core region. On the other hand, under the antiresonant conditions of the cladding, i.e., at the frequencies away from the resonant ones, considerable reflections at the core-cladding interface cause the core modes to appear resulting from the fact that waves bounce back and forth inside the core region.
Resonant frequencies of the cladding are given by Eq. (1) becomesEquation (2) can be used to predict the frequencies at or near which the core modes of the pipe waveguide do not exist. Moreover, the period of the resonance frequencies of the cladding is
3. Modal index and attenuation constant of the fundamental mode
In this and the following sections, modal characteristics of the pipe waveguide are numerically examined over the frequency range between 200 and 900 GHz. If not particularly specified, we assume the parameters of the pipe waveguide are: core diameter D = 9 mm, cladding thickness t = 1 mm, and cladding refractive index n 2 = 1.4. Modal index and attenuation constant of the guided core mode are defined as Re(β)/k 0 and −2Im(β), respectively, where β is the complex propagation constant and k 0 is the free-space wavenumber. Numerical results are obtained utilizing the FDFD mode solver . For the calculations, the computational window size is 14 mm by 14 mm, the grid size is Δx = Δy = 0.025 mm, and the perfectly matched layer (PML) is placed at the outmost to surround the computational window .
We first investigate the effect of core diameter by considering D = 7 mm and 9 mm, respectively. Modal indices and attenuation constants of the fundamental (the lowest) core mode are shown in Figs. 2(a) and 2(b), respectively. It is observed that there are discontinuities in Fig. 2 and the discontinuity frequencies repeat periodically. At these frequencies, no core modes can be found by the mode solver, and fields of the obtained guided modes are confined either almost or at least with a significant ratio in the cladding. Substituting m = 2, 3, 4, and 5 into Eq. (2), the obtained resonant frequencies are 306, 459, 612, and 765 GHz, respectively, which coincide with the discontinuity frequencies in Fig. 2. In fact, the discontinuities correspond to the resonance conditions of the cladding under which core modes of the pipe waveguide do not exist . It is obvious that the discontinuity frequencies are independent of the core diameter, which can also be implied from Eq. (2).
Just as in the conventional microwave hollow-core metallic waveguides where Re(β) < k 0, Re(β)’s of the leaky core modes in the pipe waveguide are smaller than k 0, leading to modal index, which is defined to be Re(β)/k 0, being less than 1, as can be seen in Fig. 2(a). The incident angle θ 1 at the core-cladding interface shown in Fig. 1(b) can be derived from the modal index by taking the inverse sine operation. For example, from Fig. 2(a), modal index for the D = 9 mm structure is from 0.9910 to 0.9996, meaning that θ 1 is from 82.3° to 88.4°, which confirms glancing reflections assumed previously in deriving Eq. (2). From the trend of the modal index shown in Fig. 2(a), it can be interpreted that the incident angle θ 1 is larger for higher frequencies and larger core diameters.
From Fig. 2(b), the attenuation constant of the D = 9 mm structure is found to be smaller than that of the D = 7 mm one. That is to say, pipe waveguides with larger core diameter suffer less waveguide losses. This is reasonable and can be explained as follows. For pipe waveguides with larger core diameter, the distance between two consecutive reflections at the core-cladding interface is larger than that of the waveguides with smaller core diameter. Moreover, the more important is, as mentioned above, larger core diameter results in larger θ 1, i.e., the wave propagates more parallel to the core-cladding interface. Therefore, there are not only less number of reflections under the same waveguide length but also higher reflectivity for each reflection for waveguides with larger core diameter and, as a result, the loss coming from imperfect reflections off the core-cladding interface decreases. This phenomenon hints that, to reduce leaky losses for terahertz waves propagating in the pipe waveguide, the structure with larger diameter might be desired. Also note that from Fig. 2(b), the attenuation constant is getting smaller as the frequency increases, again due to larger θ 1 at the core-cladding interface.
Next, we reduce the cladding thickness from t = 1.0 mm to t = 0.5 mm. The results of modal indices and attenuation constants are shown in Figs. 3(a) and 3(b), respectively. As can be seen from Fig. 3(a), the thickness of the cladding has little effect on the modal-index value. However, discontinuity frequencies that are relevant to the cladding resonant frequencies depend significantly on the thickness of the cladding, as shown in Figs. 3(a) and 3(b). As the thickness is reduced to one half, i.e., from 1.0 mm to 0.5 mm, the period of the discontinuity frequency is doubled. This phenomenon complies with what can be deduced from Eq. (3). In addition, the cladding resonant frequencies for t = 0.5 mm derived from Eq. (2) with m = 1 and 2 are 306 GHz and 612 GHz, respectively, which also coincide with the discontinuity frequencies shown in Fig. 3.
From Figs. 2 and 3, it is found that discontinuity frequencies of the pipe waveguide are independent of the core diameter and are inversely proportional to the cladding thickness. These characteristics agree with those of the cladding resonance frequencies as described by Eq. (2). Moreover, values of the discontinuity frequencies coincide with those obtained from Eq. (2). Therefore, we can conclude that the discontinuity frequencies are exactly the same as the resonant frequencies of the cladding, and hence confirm guiding mechanism of the pipe waveguide to be antiresonant reflecting guiding.
In Fig. 3(b), if the available bandwidth is defined as the continuous frequency range over which the core mode can be guided, one can find that the available bandwidth for t = 0.5 mm is about twice than that for t = 1.0 mm. Thus, it suggests that thinner cladding is desirable to provide a broader bandwidth.
Influence of the refractive index of the cladding n 2 is shown in Fig. 4 for n 2 = 1.4 and 1.6, respectively. Like the thickness of the cladding, it is clear that the refractive index has little effect on the modal-index value, but has significant influence on the resonant frequencies of the cladding. From Eq. (2) with m = 2 to 7, the resonant frequencies for n 2 = 1.6 are 240, 360, 480, 600, 721, and 841 GHz, respectively. Again, they coincide with the discontinuity frequencies shown in Fig. 4. From Fig. 4(b), it is also observed that the bandwidth is larger for lower refractive index, meaning that low-index material is preferred for the pipe waveguide.
Previous discussions assume that the refractive index of the cladding is real. In other words, material absorption is not considered. Here, the effect of material absorption is examined by assuming the refractive index of the cladding is n 2 = 1.4 – j0.0015. Modal indices and attenuation constants obtained with and without material absorption are shown in Figs. 5(a) and 5(b), respectively. Figure 5(a) shows that modal indices are practically unaffected under such material absorption. As for the attenuation constants, the effect of material absorption is not obvious in the lower frequency region (e.g., lower than 400 GHz), as can be seen from Fig. 5(b). However, as the frequency increases, the attenuation constant magnitude will be increased and the increment is more significant at higher frequencies.
4. Modal patterns of the fundamental and higher order modes
The first twelve lowest guiding modes of the pipe waveguide are solved at the frequency of 380 GHz. Figure 6 shows the loci of the complex propagation constant β normalized to k 0 of the twelve lowest modes. Mode numbers are labeled according to the ranking of the modal index, i.e., Re(β)/k 0. Note that the imaginary part of the complex propagation constant is negative. Hence the larger the Im(β)/k 0 of the guiding mode is, the smaller the attenuation constant will be. It is clear the lowest mode (mode 1), or the fundamental mode, has the smallest attenuation constant at the simulated frequency and it becomes the dominant mode for the pipe waveguide investigated here.
Modal intensity distributions and electric field vector distributions of the first twelve lowest modes are shown in Figs. 7(a) and 7(b), respectively. Note that x and y components of the calculated electric field, Ex and Ey, are complex. To plot the electric field vector distributions, the x component of the vector is taken as the absolute value of Ex multiplied by the sign (positive or negative) of the real part of Ex, and the y component is taken similarly. Electric field vector distribution of mode 1 (the fundamental mode) resembles that of the HE11 mode of a conventional step-index fiber. Hence the fundamental mode of the pipe waveguide is labeled as the HE11 mode. Similarly, all the remaining higher order modes of the pipe waveguide can find their counterparts from a step-index fiber. Therefore the higher order modes are also labeled according to the designation for a step-index fiber.
In Fig. 6, modes 2, 3, and 4 have almost the same modal indices, thus they can be grouped as the first set of higher order modes. Modal intensity distributions of these three modes look identical and they are distinguished from the electrical field vector distribution as TM01, HE21, and TE01 modes, respectively. Although real parts of the propagation constants are almost the same, it is interesting to note that imaginary parts are very different among the three modes, leading to very different attenuation constants. This could be understood from the electromagnetic theory by knowing that the reflection coefficient of the TE mode is larger than that of the TM mode. Since core modes are guided based on antiresonant reflection, the more the wave reflects at the core-cladding interface, the more proportion of the fields can be guided inside the core region, and the smaller the attenuation constant will be. Consequently, the attenuation constant of mode 4 (TE01) is smaller than that of mode 2 (TM01). Moreover, the attenuation constant of mode 3 (HE21) lies between those of modes 2 and 4 due to its hybrid nature (mixed TE and TM). This phenomenon also appears among modes 10, 11, and 12, in that the sequence of attenuation constant magnitudes is: mode 12 (TE02) > mode 11 (HE22) > mode 10 (TM02).
Finally, modal indices and attenuation constants of the fundamental mode and the first set of the three higher order modes are shown in Figs. 8(a) and 8(b), respectively. It is clear that the basic characteristics of the higher order modes are the same as those of the fundamental mode, including the same discontinuity frequencies. However, as seen in Fig. 8(a), modal indices of the three higher order modes are almost the same and are less than that of the fundamental mode. This implies that these three modes have almost the same incident angles θ 1 at the core-cladding interface and propagate less parallel than the fundamental mode. Due to the smaller θ 1, these higher order modes have larger attenuation constants as can be seen in
Figure 8(b). In the Appendix, we show the attenuation constants can be calculated using a different approach based on a geometric consideration.
5. Experimental verification of the resonant frequencies
In our previous work , experiments using Teflon pipes as the pipe waveguides showed good consistency with the simulated results. However, owing to the nonuniform thicknesses of the commercial Teflon air pipes, with thickness variations of 10%, measured resonant frequencies of the cladding had a slight frequency shift from the theoretical predictions. Since the resonant frequency is an important feature for the pipe waveguides, it is verified here using PMMA pipes with a better cladding uniformity by measuring the transmission spectra through the standard terahertz time-domain spectroscopy [28,29]. Figure 9 shows the experimental setup. THz pulse was generated and detected by the LT-GaAs based photoconductive switches, which were both pumped by a 100-fs Ti:sapphire pulse laser. The generated THz pulse was first focused by a pair of parabolic mirrors with 80-mm effective focal length, then coupled into the pipe waveguide, and finally collimated by a PE lens to the receiving antenna for detection.
The pipe waveguides used for experiment are made of PMMA with the refractive index of 1.6 . They are 15 cm long and have the cladding-thickness variations below 3%. Normalized transmission spectra of the PMMA pipes with two different sizes, one is with D = 8 mm and t = 1.0 mm and the other is with D = 4 mm and t = 2.0 mm, are shown in Figs. 10(a) and 10(b), respectively. Resonant frequencies of the cladding obtained from Eq. (2) are also shown for comparison. It is clear that the transmission minima and the resonant frequencies match very well, indicating that guiding mechanism of the pipe waveguides is indeed that of antiresonant reflecting guiding. At the resonant frequencies, transmissions significantly drop because there are no core modes to deliver the THz powers. Note that the two absorption peaks of water vapor, 558 GHz and 753 GHz , can also be observed in Fig. 10, as pointed out by arrows.
The pipe waveguide is promising for THz waveguiding owning to its low-loss feature and simple structure. In this paper, modal characteristics of the leaky core modes of the pipe waveguide are investigated in the THz range. Utilizing the FDFD mode solver, modal indices and attenuation constants of the core modes which are frequency dependent are calculated for different core diameters, cladding thicknesses, and cladding refractive indices. It is found that at certain frequencies no core modes can exist. Comparing these frequencies with the resonant frequencies of the cladding reveals that the guiding mechanism of the core modes is that of the antiresonant reflecting guiding. Calculated results also suggest that, to have a low-loss and high-bandwidth pipe waveguide, large core diameter, thin cladding thickness, and low refractive index are desired. The effect of material absorption is also examined, which shows that the attenuation constant magnitude will be increased and the increment is more significant at higher frequencies. Moreover, modal patterns are shown for the fundamental mode and the higher order modes, including modal intensity distributions and electric field vector distributions. It is observed that modal patterns of the core modes of the pipe waveguide resemble those of the guided modes of the step-index fiber. From the spectrum of the attenuation constant, it shows that the fundamental mode (HE11-like) has the smallest attenuation constant and is the dominant mode for the pipe waveguide investigated. Using time-domain spectroscopy to measure the transmission spectra of PMMA pipes, guiding mechanism of the pipe waveguides was experimentally confirmed to be that of antiresonant reflecting guiding.
In previous sections, the attenuation constants of the pipe waveguides, defined as −2Im(β), are obtained directly from the FDFD mode solver . In this appendix, we show that the attenuation constants are consistent with those obtained by an alternative approach in which the derivation starts from the modal indices, i.e., Re(β)/k 0.
Consider the longitudinal cross-section of the pipe waveguide as shown in Fig. 11 . In the core region, the propagation constant β and the wave vector n 1 k 0 are related as , where θ 1 is the incident angle with respect to the normal of the core-cladding interface. Since the modal index (also called the effective index) neff is defined as n eff = Re(β)/k 0, we have
From basic electromagnetic theory, it is known that the reflection coefficients (reflected amplitude per incident amplitude) of a plane wave incident at the n 1-n 2 interface are different for TE (electric field in the x-z plane) and TM (electric field in the y-z plane) polarizations, and are given by 
As stated previously, the cladding of the pipe waveguide can be viewed as a Fabry-Perot etalon. Hence the reflectivity (reflected power per incident power) of the “mirror” formed by the core-cladding interface for TE polarization is 
where ω is the angular frequency. The reflectivity for TM polarization RTM is the same as Eq. (7) except rTE is replaced by rTM. Note that the refracted angle θ 2 in Eq. (7) can be replaced by θ 1 via Snell’s law.
In Fig. 11, let P be the power along the propagation direction and α is the attenuation constant of the leaky core mode. We have
where L is the propagation distance, R is the reflectivity, and N is the number of reflections at the core-cladding interface within the distance L. Knowing that, one can obtain from Eq. (8) that . Hence for TE and TM modes, the attenuation constants are respectively
Moreover, the attenuation constants for the hybrid modes, i.e., HE and EH modes, are obtained by taking the arithmetic average of those of TE and TM modes .
Note that Eq. (11) holds when the TE, TM, HE, and EH modes have the same θ 1. In short, after calculating the incident angle θ 1 from the modal index, the attenuation constants of the leaky core modes of the pipe waveguide can be obtained via Eqs. (5)−(7) and (9)−(11). Note that θ 1 has significant influence on the attenuation constant, as can be observed from Eqs. (9) and (10) where not only the tangent but also RTE and RTM terms are strongly dependent on θ 1.
To validate the Fabry-Perot-based procedure mentioned above, the attenuation constants of the four lowest order modes (HE11, TM01, HE21, and TE01) of the pipe waveguide are calculated from the modal indices. The case with core diameter D = 9 mm, cladding thickness t = 1.0 mm, and cladding refractive index n 2 = 1.4 is considered for validation. Starting from
the modal indices shown in Fig. 8(a), the incident angles θ 1 are calculated by applying Eq. (4). Two sets of θ 1 will be obtained. The first set of θ 1 includes larger angles for the HE11 mode and the second set of θ 1 includes smaller angles for the TM01, HE21, and TE01 modes. Then, following the procedure described above, the attenuation constant αHE 11 of the HE11 mode can be obtained utilizing the first set of θ 1, and αTM 01, αHE 21, and αTE 01 can be obtained utilizing the second set of θ 1, with αHE 21 = (αTM 01 + αTE 01)/2. These results obtained according to the procedure (denoted as “FP”) are shown in Fig. 12 , and are compared with the attenuation constants calculated directly from the FDFD mode solver (denoted as “FDFD”). As can be seen in Fig. 12, results obtained by the two approaches almost match each other. This match reveals several facts. First, the practicality of the procedure to derive the attenuation constant (related to the imaginary part of β) from the modal index (related to the real part of β) is confirmed. Second, since the procedure is developed based on the operating principle of a Fabry-Perot etalon, guiding mechanism of the pipe waveguide is again confirmed to be that of antiresonant reflection guiding. Third, the accuracy of the FDFD mode solver  is proven.
As shown in Fig. 2(b), the attenuation constants for D = 7 mm and D = 9 mm are much different. Since the incident angles θ 1 for the two diameters are different too, and since both D and θ 1 will affect the attenuation constants, it is interesting to distinguish which one is more important for the attenuation constants. Using the procedure described in this appendix, three conditions are assumed for the calculation of the attenuation constants: (1) D = 7 mm and θ 1 is assumed to be that of D = 7 mm, i.e., θ 1 is the inverse sine of the modal index of D = 7 mm, (2) D = 7 mm and θ 1 is assumed to be that of D = 9 mm, i.e., θ 1 is the inverse sine of the modal index of D = 9 mm, and (3) D = 9 mm and θ 1 is assumed to be that of D = 9 mm, i.e., θ 1 is the inverse sine of the modal index of D = 9 mm. The difference between the attenuation constants obtained under conditions (1) and (2) represents the contribution from θ 1 only, while the difference between the results obtained under conditions (2) and (3) represents the contribution from D only. From Fig. 13 , it is clear that the incident angle θ 1 has much more contribution to the attenuation constant than the core diameter D does. Hence, the incident angle θ 1 plays an important role in the modal propagation within the pipe waveguide.
This work was supported in part by the National Science Council of the Republic of China under grant NSC 97-2221-E-002-043-MY2, NSC 97-2221-E-002-047-MY3, NSC 96-2628-E-043-MY3, NSC 97-2120-M-002-010, NSC 94-218-M-008-009, in part by the Excellent Research Projects of National Taiwan University under grant 98R0062-07, and in part by the Ministry of Education of the Republic of China under “The Aim of Top University Plan” grant.
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