We report a design of low-loss THz Bragg fibers with a core size on the order of wavelength that operates near the cutoff frequency of its TE01 mode. We also propose a broadband Y-type mode converter based on branched rectangular metallic waveguides to facilitate coupling between the TE01 mode of the Bragg fiber and the TEM mode in free space with 60% efficiency. Our fiber holds strong promise to facilitate beam-wave interaction in gyrotron for high-efficiency THz generation.
© 2015 Optical Society of America
Terahertz (THz) technology has aroused strong interest over the past decade due to its many promising applications in communication , imaging and spectroscopy , biology and medicine , and astronomy . To improve performance, coherent THz sources with high power are sought after. Gyrotrons, which are the waveguide implementation of electron cyclotron maser (ECM), is able to generate high-power coherent THz radiation from energetic gyrating electrons, and has been a subject of intense research over the past few decades [5–7]. To achieve high power, high gain through efficient beam-wave interaction is necessary and a low-loss TE01 mode is generally employed . Under the beam-wave synchronism condition, relativistic gyrating electrons would exhibit either azimuthal bunching, originating from the modulation of electrons’ gyrating frequency by transverse electric field, or axial bunching, originating from the modulation of electrons’ drift velocity by transverse magnetic field . The competition between these two bunching mechanisms results in low gain and thus should be avoided [5, 6]. A small-core cylindrical metallic waveguide with an operating frequency close to the TE01 cutoff frequency is generally employed to suppress axial bunching. The near-cutoff operation, however, results in large ohmic loss where the power attenuation can be as high as 1 cm−1 around one terahertz [8, 9]. A low-loss THz waveguide with small-core dimension is therefore highly desirable for THz gyrotron.
In order to reduce the propagation loss in metallic waveguides, a single layer of dielectric coating has been added on the inner surface of metallic walls to form so-called hybrid-cladded waveguides [10–14]. By carefully choosing the thickness of the dielectric film, destructive interference at the operating frequency reduces the field strength at lossy metallic boundary and hence lowers the propagation loss. This low loss, however, is obtained at the expense of available bandwidth, as a thin-film etalon is typically very narrowband .
On a parallel front, hollow-core THz photonic bandgap (PBG) fibers, including photonic-crystal holey fibers that utilize two-dimensional index modulation [15, 16] or Bragg fibers that utilize radial index modulation [17–19], have become an active research topic recently. Because of the strong photonic bandgap confinement and their all-dielectric nature, these fibers can exhibit much lower loss compared to their hollow-core metallic or hybrid-cladded counterparts [10, 20]. To this end, the majority of efforts focused on PBG fibers with large core dimension (core diameter to wavelength ratio > 15), where the operating THz band is far above the cutoff frequency to render very low-loss and low-dispersion propagation [17–20]. Such PBG fibers can be used to transport broadband THz radiation for communication and sensing [1–3, 10], but are not suitable as the interaction tubes for gyrotron due to extremely serious mode competition and potentially unstable beam-wave synchronism caused by the competition between azimuthal bunching and axial bunching .
Another challenge for operating Bragg fibers in TE01 mode is its low coupling efficiency by traditional sources such as photoconductive-switching antenna  and optical rectification . When the linearly polarized THz wave from these sources impinges on Bragg fibers, HE11 mode will be excited instead of the azimuthally polarized TE01 mode [8, 18]. To the best of our knowledge, only a couple experiments have succeeded in exciting/extracting TE01 mode in the visible/infrared regions [22, 23]. Consequently, efficient excitation of the TE01 mode in THz waveguides is a critical issue for the application as high-power THz sources.
In this work, we present a design of low-loss small-core Bragg fiber operated around one terahertz, which is slightly above the cutoff frequency of the TE01 mode for efficient gyrotron operation. In addition, we propose a novel coupling scheme based on a Y-type mode converter to excite the TE01 mode in the designed fiber, and quantify its coupling efficiency using rigorous eigenmode expansion method. It is well known that a hollow-core Bragg fiber is intrinsically a leaky waveguide whose eigenmodes are either a continuum of radiation modes or a discrete set of leaky modes . While modal expansion utilizing the continuum has been demonstrated in one-dimensional open waveguides , it is very complicated to apply to a two-dimensional Bragg fiber. To effectively model the modal coupling between the mode converter and the Bragg fiber, we modify our fiber design by adding a copper outer cladding to the Bragg fiber such that electric field becomes zero at the dielectric-metal interface. Eigenmodes then become normalizable and discretized, facilitating the eigenmode expansion. The effect of this copper outer cladding upon the modal coupling will be shown to be negligible. As the typical interaction length for THz gyrotron is only several millimeters [6, 7], our Bragg fibers can be considered short and rigid. We therefore do not consider bending loss here.
This article is organized as follows. Section 2 briefly describes the design guideline for the small-core metal-cladded Bragg fibers. Section 3 presents the characteristics of the TE01 mode in the designed Bragg fiber, including its dispersion, field distribution, and ultra-low attenuation (power loss coefficient < 0.1 cm−1). Section 4 shows the characteristics of the additional twenty eigenmodes with azimuthal symmetry that are necessary for the coupling analysis. The design and performance of the Y-type mode converter is presented in Sec. 5, and the coupling efficiency of desired TE01 mode is analyzed and quantified in Sec. 6.
2. Design of a small-core Bragg fiber with metallic cladding operating in TE01 mode
As explained above, in this work we consider a hollow-core metal-cladded Bragg fiber. The index profile is shown in Fig. 1, where () and () represent the refractive index and the thickness of high-index (low-index) layer, followed by a metallic outer cladding. For low-loss operation, the materials used in Bragg layers should possess large refractive index contrast and low dielectric losses [18–20, 25–27]. We therefore choose high-density polyethylene (HDPE) as the high-index material which has a refractive index of 1.530 and low power attenuation constant of 0.098 cm−1 around one terahertz . On the other hand, we choose air as the low-index material in this work. While the true “air layer” is impossible to realize, a reasonably low-index layer with index close to air can be implemented either by cobweb-type configuration  or by using the highly-porous polymer , both of which contain sub-wavelength supporting structures so that their effects on the dispersion and the attenuation of the guided wave are negligible .
Similar to traditional metallic waveguides, the core radius governs the cutoff frequency of guided modes in a Bragg fiber. The cutoff frequency of TE01 mode in a metallic waveguide with a radius equal to 202 μm is 0.905 THz, which is slightly below the operating frequency of one terahertz. We therefore adopt it as the core radius of our small-core Bragg fiber. Due to the near-cutoff operation, the light rays strike the Bragg layers at nearly normal incidence. To ensure its maximum reflection, the thickness of each layer can be obtained following the well-known quarter-wave stack condition: and , where is the matching wavelength in vacuum [18, 30]. As the current stopband is expected broad due to the large index contrast between HDPE and air, we choose a matching frequency = 0.920 THz ( = 326 μm) to ensure good power confinement in the vicinity of one terahertz. Hence, the thickness of the high-index HDPE dielectric is 53 μm and of low-index air layer is 81 μm. The period of the Bragg pair () is therefore 134 μm. The ratio of the core diameter to the operating wavelength is 1.24, which is at least an order of magnitude smaller than conventional THz PBG fibers [20,27]. The total number of Bragg pairs in the present design is chosen to be 10, which should be sufficient to suppress the leaky loss of TE modes, given the large-index-contrast employed in current design [20, 27]. These Bragg layers are finally enclosed by a copper cladding whose conductivity is 5.96 × 107 S/m at 20°C, with a thickness much larger than the skin depth of copper, which is < 1 μm at one terahertz.
3. Characteristics of the TE01 mode in the metal-cladded Bragg fiber
The complex propagation constant of the TE01 mode in the proposed fiber (denoted as ), where is the propagation constant and represents the power loss coefficient, is solved by the transfer matrix method with metallic boundary condition [25, 26]. The dispersion relation of the TE01 mode is shown as the black solid curve in Fig. 2. The projected photonic band structure associated with corresponding planar Bragg mirror is also shown in Fig. 2, where the grey/blank regions represent the pass/stop bands and the solid purple curve represents the light line. For comparison, the dispersion relation of TE01 mode in a cylindrical metallic waveguide with the same core size (denoted as ) is also shown as the blue dots. Even though the current design does not have large enough refractive index contrast to warrant an Omni-directional bandgap [20, 27], Fig. 2 indicates the dispersion of the Bragg fiber closely follows that of the metallic waveguide. In particular, the cutoff frequency of the mode is 0.908 THz, very close to that of the mode in the metallic waveguide (0.905 THz).
The similar behaviors of and dispersions have rooted in their field patterns. Figure 3(a) compares the radial distributions of their transverse electric-field magnitude () at 1 THz. As indicated, these two modes have nearly identical field distributions inside the core. While the electric field of mode (blue dots) reduces to zero at the core’s boundary, the field of mode (black solid) extends into the dielectric layers with oscillating magnitude before it is terminated at the outer metallic wall. This is a characteristic of a Bloch wave with the decaying amplitude in a photonic bandgap . The field magnitude drops to below −20 dB in the last Bragg pair, which effectively reduces the ohmic loss at the dielectric-metal interface. The low ohmic loss, together with the low material loss of the selected dielectrics, renders a low power loss coefficient () less than 0.1 cm−1 (i.e., power is attenuated to 0.43 dB or 90.48% of its original value after propagating 1 cm) over a broad bandwidth from 0.966 to 1.500 THz as shown by the black solid curve in Fig. 3(b). Since at 1 THz (0.076 cm−1) is nearly one order of magnitude smaller than that in the stainless-steel cylindrical waveguide (0.825 cm−1) , our proposed metal-cladded HDPE/air Bragg fiber is very promising for beam-wave interaction or THz transmission, compared to traditional small-core metallic waveguides. For comparison, the red dashed curve in Fig. 3(b) shows of mode in a fiber without the copper outer cladding, in which the leaky field extended beyond the Bragg layers is free to radiate. As demonstrated, there is no significant difference for in the presence and the absence of copper cladding, indicating the negligible effect of outmost metallic boundary on power attenuation.
4. Characteristics of the eigenmodes in the metal-cladded Bragg fiber
In this section we analyze the first twenty eigenmodes in the metal-cladded HDPE/air Bragg fiber that exhibit the same azimuthal symmetry as the TE01 mode. These results will be used later in the modal coupling analysis (Sec. 6). We calculate their dispersion relation, power loss coefficient, and the electric field distribution (i.e., the amplitude of ) using transfer matrix method, as shown in Figs. 4, 5, and 6, respectively. The 11th mode is the TE01 mode as described in details in Sec. 3.
Modes 1-10 are propagating modes in the metal-cladded Bragg fiber other than the TE01 mode (11th mode). They represent the cladding modes, as their dispersions lie in the passband of the Bragg mirror [grey regions in Fig. 4(a)]. Since modes 1-7 lie below the light line (solid purple curve), their fields are evanescent in the core, while modes 9 and 10 have propagating field components in the core. In addition, these cladding modes have energy concentrated in the dielectric cladding and thus exhibit higher loss than the TE01 mode as shown in Fig. 5(a). The increase of loss originates primarily from the HDPE absorption while only a small contribution (0.2-2%) is due to the ohmic loss of metallic walls. Cladding modes with higher order have larger electric field energy density () in the lossy HDPE, leading to the higher modal loss. Modes 12-20 represent the first 9 evanescent modes in the Bragg fiber. These modes have their cutoff frequencies higher than the current operating band (0.950 THz to 1.050 THz), rendering very small propagation constants (exactly zero in the absence of HDPE loss) and very large power loss coefficients. As their dispersion lie above the light line and within the bandgap, these evanescent modes have oscillatory transverse electric field with a decaying amplitude in the cladding.
As mentioned in the introduction, the copper outer cladding in our fiber design is introduced to facilitate modal coupling analysis between the mode converter and the Bragg fiber (Sec. 6). It is therefore important to investigate the effect of this metallic cladding on the eigenmode characteristics, especially their power losses. We found the loss increment is negligible (less than 0.01%) for the mode (i.e. 11th mode), due to the strong confinement of the Bragg layers. The loss increases marginally by 0.2% - 2% for the propagating cladding modes (mode 1-10) as their transverse fields extend beyond the dielectric layers and into the metallic wall. The propagating loss does increase considerably (more than 10%) for the evanescent modes (mode 12-20) because of their extremely small propagation constants such that the fields spend considerable time in the copper. These higher-order modes, however, have negligible contribution to modal coupling, as will be discussed in Sec. 6.
5. Design and characteristics of a Y-type TE01 mode converter
In this section, we propose a new scheme to efficiently couple the traditional THz sources with linearly polarized TEM mode to the mode of the Bragg fiber by employing a Y-type mode converter . The scheme is illustrated in Fig. 7(a) where a mode converter, made of hollow tunnels carved out of a solid copper, is butt coupled to a Bragg fiber. The same scheme can also be used to convert the mode back into linearly polarized TEM mode in free space. The basic idea behind this coupling scheme is to explore the similarity in the TE01 field distribution between the metallic cylindrical waveguide and the fiber, as shown in Fig. 3(a). A typical Y-type mode converter comprises two-stage power-dividing junctions made of branched rectangular metallic waveguides, followed by a mode-converting section made of cylindrical metallic waveguide . The electric field distribution along with its polarization inside the proposed Y-type TE01 mode converter is depicted in Fig. 7(b). The input end of the rectangular metallic waveguide is operated in the fundamental TE10 mode (denoted as ), which can be easily excited by the linearly polarized free-space THz signal . As the injected THz wave () travels to the 1st-stage power-dividing junction, it splits into two signals with equal magnitude and phase. The two divided signals propagate on their own along the waveguide tunnels and are further divided into four sub-signals by 2nd-stage power dividers, which are attached symmetrically on the side wall of the mode-converting cylindrical waveguide. These four linearly-polarized fields of the rectangular waveguides with equal amplitude and phase circumnavigate the cylindrical waveguide to jointly excite its modes with four-fold symmetry (i.e., = 0, 4, 8...etc.). This precludes the excitation of lower-order modes of the metallic cylindrical waveguide (i.e., , , , and ), owing to the mismatched azimuthal symmetry in field distribution and polarization. As the operating band is chosen slightly above the cutoff, only mode can be excited and all other higher-order modes (where = 0, 4, 8...etc.) are evanescent and thus are also precluded. Consequently, only the very high-purity mode will emerge at the output end of the mode converter, as shown in Fig. 7(b).
A simple design of Y-type mode converter that works at around one terahertz is presented as follows. The width () and height () of the input rectangular waveguide are respectively 239 μm and 119 μm, yielding a cutoff frequency of 0.628 THz for mode. As the rectangular waveguide is branched, its width stays the same (239 μm) to ensure a fixed cutoff as well as a fixed propagation constant of the mode. The height of the branched rectangular waveguide, on the other hand, is reduced by half (to 60 μm) at every branching point to divide the power equally. Before the second branch point, it is tapered back to its original value (119 μm). The core radius of the cylindrical waveguide in the mode-converting section is chosen to match that of the Bragg fiber (202 μm). Figure 8 shows the spectral characteristics of the mode converter, calculated by High Frequency Structural Simulator (HFSS, Ansys) using realistic material property of copper with conductivity of 5.80 × 107 S/m. As indicated, the mode converter has very high transmittance of the TE01 mode [> −0.4 dB or 90%], very low transmittances of other lower-order modes [< −40 dB], and a very low reflectance [< −20 dB or 1%], over the operating band from 0.971 THz to 1.024THz. This indicates a mode purity as high as 99.99%. The coupling loss is dominated by the ohmic loss of the conducting wall. The fractional bandwidth of our mode converter, calculated based on a −0.2dB (or 95% of the peak transmission) bandwidth, is nearly 10%, which is one order of magnitude larger than previously reported coaxial-cavity couplers  or serpentine schemes [34, 35]. The broader bandwidth of proposed Y-type mode converter can be attributed to the non-resonance structure adopted for coupling, whereas in the coaxial-cavity couplers  TE011 cavity mode needs to be excited for further coupling of TE01 mode, which significantly limits their bandwidths. On the other hand, the superior performance compared to the serpentine couplers [34, 35] results from the shorter mode-conversion length in present scheme (several wavelengths), while serpentine couplers usually requires much longer length for mode conversion (more than ten wavelengths), suffering from serious ohmic loss and narrow transmission bandwidth in THz region.
6. Excitation of the TE01 mode in Bragg fibers using a Y-type mode converter
In this section, we investigate the excitation efficiency of the mode in the metal-cladded Bragg fiber by a mode generated at the end of the mode converter as depicted in the last section. According to the modal effect [8, 18], eigenmodes with the same azimuthal symmetry as the incoming mode, on both sides of the coupling interface, will be excited in order to meet boundary conditions. At the mode converter side, these modes are essentially , whereas at the Bragg fiber side the first twenty of these modes are calculated in Sec. 4. Denote the electric and magnetic fields of m-th (n-th) eigenmodes in the Bragg fiber (metallic waveguide) by () and (), respectively. The boundary conditions at the coupling interface requireFig. 1. In principle, the larger the numbers of mode is considered in modal analysis ( and ), the more accurate the result is expected at the expense of computation time. In practice we found that and are sufficient to give converged values of and , and is used for the following analysis. Based on the field reflection and field transmission coefficients ( and ), the reflectance and transmittance (conversion efficiency of fiber mode) can be calculated from Poynting vectors according to
Figure 9(a) shows of the first fifteen modes () in the mode converter. Notably, for are all smaller than and are not shown for clarity. It is clear that not only the fundamental mode but also the higher-order modes are excited to meet the transverse boundary condition. These higher-order modes, however, do not carry power as they are evanescent for being under the below-cutoff operation. As shown in Fig. 9(b), the total reflection power is less than 0.05 for the whole operating band, indicating that most of the injected power is coupled into the fiber under the current coupling scheme.
Figure 9(c) and 9(d) respectively show the magnitude of field transmission coefficient and the transmittance of the first thirteen azimuthally symmetric eigenmodes in the Bragg fiber (). Note that for are smaller than and thus are not displayed for clarity, and for are zero as they are evanescent modes and carry no power. A few observations can be made: firstly, most of the injected power is converted into mode (the 11th mode) with an average conversion efficiency ~60% over this frequency range. Secondly, among all the other excited modes, the transmittance increases from mode 1 to mode 8, then decreases for modes 9 and 10. Thirdly, although higher-order evanescent modes in the fiber (modes 12-20) are also expected to be excited to meet the boundary condition, we found their field transmission coefficients () are negligible (on the order of 10−6). These trends can be understood as follows. The first term on the right-hand side of Eq. (4) is related to the mode overlap integral between the incident wave () in the mode converter and the targeted eigenmode of the Bragg fiber. The second term, on the other hand, contains contribution from all eigenmodes of the mode converter, each of which is proportional to the overlap integral between the mode of the mode converter and the targeted fiber mode, multiplied by the field reflection coefficient . Even though of each metallic-waveguide mode has similar order of magnitude as demonstrated in Fig. 9(a), higher-order mode has much larger and hence smaller [see Eq. (9) in the Appendix], leading to diminishing mode overlap integral. The field transmission coefficient can therefore be approximated to becomeEquation (7) can be used to explain the above observations: firstly, as mode (the 11th mode) has similar field distribution as the mode, is the largest among all the excited modes. Secondly, as seen in Fig. 6(a), modes 1-8 have increasing presence in the core, resulting in increasing mode overlap with the mode and therefore larger . Modes 9-10, on the other hand, have decreasing weight in the core and thus smaller . Thirdly, all evanescent modes of the Bragg fiber have very large mode order () with rapid field oscillation in the core. This leads to diminishing mode overlap integral with the mode, which means they have very little contributions in modal coupling. Lastly, as is dominantly related to the overlap integral between the incident wave and the targeted fiber eigenmode, the significant increase of the modal loss of evanescent modes in Bragg fibers () upon adding the metallic wall has little effect on the excitation of propagating modes (), as allured in last paragraph of Sec. 4.
Once excited at the coupling interface, each eigenmode propagates forward with its own complex propagation constant . The total transverse field will evolve along the propagation length due to modal interference and can be calculated according toFig. 5), mode would eventually dominate—so called effective single-mode regime . Figure 10 shows the superimposed transverse electric field magnitude calculated by Eq. (8) for several representative distances along the fiber. For comparison, of is also shown (black dashed). As expected, within the core at the coupling interface ( = 0 mm, red solid) resembles strongly of the cylindrical metallic waveguide (Fig. 3a). Due to the finite numbers of eigenmodes involved in the eigenmode expansion, outside the core, although small (< −20 dB), is not exactly zero as required by the metallic transverse boundary condition. At 0.01 mm (yellow solid), in the cladding near the core rises and exhibits some level of amplitude modulation. At 0.1 mm (green solid), in the cladding has developed periodic pattern with the same periodicity as the Bragg layer, except its value is too high at the core-cladding interface and too low in the cladding compared to the mode. The overshoot at the core-cladding interface is corrected at = 10 mm (blue solid), while in the cladding it is over-corrected. At 1000 mm (orange solid), all cladding modes are nearly died off [ for all ] and has nearly converged to the pure mode.
In this work we present a design of a hollow-core dielectric Bragg fiber with a copper outer cladding, intended to operate at 1 THz near the cutoff of the TE01 mode. The core diameter is on the order of the operating wavelength, which is at least an order of magnitude smaller than conventional THz PBG fibers. By employing a quarter-wave design at the cutoff frequency, our Bragg fiber exhibits a power loss coefficient as low as 0.1 cm−1 – a significant improvement compared to existing small-core stainless-steel THz waveguides. Characteristics of the first twenty azimuthally symmetry eigenmodes, including dispersion relation, power loss coefficient, and field distribution, are studied using transfer matrix method. We also propose a broadband Y-type mode converter based on symmetrically branched rectangular metallic waveguides to facilitate coupling between the TE01 mode of the Bragg fiber and the TEM mode in free space. The mode converter features a mode purity as high as 99.99% and has a fractional bandwidth one order of magnitude larger than previously reported schemes. Using rigorous eigenmode expansion method involving the first twenty eigenmodes in the Bragg fiber and the mode converter, the excitation efficiency of is as high as 60%. Although cladding and evanescent modes in the Bragg fiber are also excited, their propagation loss is significantly higher than that of the , rendering an effective single-mode fiber after a propagation length of 1 meter. Due to its low loss, our proposed Bragg fibers holds strong promise to facilitate beam-wave interaction in gyrotron for high-power THz generation. Our Y-type mode converter is also expected to benefit other research that requires operating at TE01 mode in fibers.
The following modal analysis is applied to analyze the field reflection coefficients and the field transmission coefficients at the junction between the mode converter and the Bragg fiber. The transverse electric and magnetic fields for the mode ( and ) in the metallic cylindrical waveguide can be written analytically as36]. On the other hand, the transverse electric and magnetic fields in the Bragg fiber ( and ) do not have analytic expression but can be solved by the transfer matrix method as demonstrated in Fig. 6. The orthogonality condition in the metal-cladded Bragg fiber requires37]Fig. 1). Equation (4) of Sec. 6 can be obtained by multiplying Eq. (1) with and integrating over the fiber’s cross-section. Following Eq. (11), we haveEq. (2) with , integrating over the cross-section of the metallic cylindrical waveguide, and following Eq. (12), we obtain
For numerical calculation, we must truncate the summation in Eq. (13) to , i.e. the total number of eigenmodes considered in the metallic waveguide. Similarly, the summations in Eq. (14) is truncated to (the total number of eigenmodes considered in the Bragg fiber). In principle the larger and are, the more accurate the final result would be at the expense of computation time. Equation (3) of Sec. 6 can be obtained by substituting Eq. (13)in to Eq. (14), which reduces to a set of linear equations for solving the field reflection coefficient :
From Eqs. (15), (16), and (17), the field reflection coefficients of the metallic-waveguide modes could be calculated. Substituting these results into Eq. (13), we can obtain the field transmission coefficients of the fiber modes.
This work was sponsored by the National Science Council of Taiwan under Contract No. NSC 101-2112-M-007-005-MY3. The authors appreciate technical support from Ansys Inc.
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