We present a high-Q 3D waveguide transmission filter for the THz-domain, based on an inhomogeneous Bragg grating, incorporated into the walls of a metallic slit waveguide. The reasons for the occurring loss mechanisms in the compact component are presented and the losses are minimized by selective mode adaption and by tapering the transitions to the corrugated regions. The performance of the device and the influence of parameter variations are analyzed by detailed numerical simulations. These 3D simulations clearly show the drastic drawback of 2D calculations in designing narrowband 3D metal-dielectric waveguide filters and could even lead to a better performance than known designs in 2D technology.
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High-Q Bragg grating filters offer an extremely high frequency selectivity and are therefore commonly used for example in distributed-Bragg-reflector (DBR) lasers or in fiber filters for the optical frequency range. The necessary high-Q transmission resonance is achieved with a line defect in a one-dimensional photonic crystal (PC) . Previous research lead to a better understanding of the wave propagation in such structures  and refined the frequency characteristic for potential applications . Careful layout of the grating geometry and the use of tapered transitions [4–6] provided detailed insight into the loss mechanisms that are responsible for the limitation of the resonance Q-factor , which is defined as the ratio of resonance frequency to its −3dB bandwidth. With the help of efficient 2D calculations this allowed a significant enhancement of the achievable Q-factor, which strongly depends on the necessary peak transmission . The advantage of micro resonators like this is the combination of small size and high efficiency. So the same technique is also used in other areas, as for example for Surface Plasmon Bragg gratings in metal-insulator-metal waveguides [8,9].
However, most of the proposed designs aim at high-Q filters for the optical regime, especially in the near infrared. Concerning the so called terahertz gap (0.1 – 10 THz), the wide area of interesting applications  is accompanied with a lack of naturally occurring materials for the design of functional components needed for the exploitation of this frequency range. Narrow-band filters in particular are needed for example for wavelength division multiplexing (WDM) in wireless high speed THz communication  or for bio-sensing applications . First designs for this matter, based on parallel plate waveguides (PPWG) [12,13] and photonic crystal slab (PCS) waveguides [14,15] have already been presented and show that high-Q resonators for this frequency range are possible. The disadvantage of PPWGs is the missing lateral limitation, which is necessary for a compact layout that could possibly be integrated. PCS waveguides on the other hand need a relative complex design. In this paper we present a compact, high-Q filter, based on a metallic slit waveguide, which has proved very good properties for THz transmission regarding attenuation, dispersion and mode confinement .
We explain the effects of the lateral limitation on loss and bandwidth and propose a layout that provides even higher Q-factors for a certain peak transmission than in recent 2D publications. For a detailed analysis and optimization of the filter performance a 3D model is used to consider all loss mechanisms in the compact 3D structure affecting the resonance Q-factor. These losses are reduced by selective insertion of a silicon (Si) layer for a better field distribution [17,18] and by tapering the transitions between the waveguide and the corrugation. Selectively inserting silicon into homogeneous slot waveguides has been recently investigated for plasmonic applications in the telecommunication frequency range for improved spatial localization and enhancement of the propagation length  and has also been characterized in the THz range in .
2. Layout and simulated properties
The presented filter structure (Fig. 1a ) consists of two thin rectangular gold sections of thickness t = 500 µm and length l = 1600 µm, arranged in a distance of g = 120 µm. The inside walls of the so created metallic slit waveguide are periodically corrugated with N = 8 rectangular grooves on each side of a center phase-shift region without corrugation (Fig. 1b), leading to the effect of Bragg reflection due to the effective index modulation. The lattice discontinuity in the center results in a cavity, allowing the creation of a narrowband defect state in the stopband of the Bragg reflection . In the center of the air gap there is a thin (s = 30 µm) undoped silicon plate of same thickness and length as the metal parts, which leads to an improved lateral field confinement . The geometry of a standard groove is illustrated in Fig. 1c with a groove width of w = 45 µm and a groove depth of d = 30 µm. The Bragg frequency, which defines the position of the stopband, and the resonance frequency are determined by the two parameters grating period Λ = 2w = 90 µm and phase-shift length p = 125 µm. For the calculation of these parameters the effective index of the wave, which is strongly influenced by the dielectric material has to be considered. The relatively small number of grating lines with strong mode coupling allows the compact size of the filter, but also demands deep grooves, leading to high radiation loss .
The grooves are inserted in a way that the averaged gap distance is kept constant, resulting in the fact that the metal teeth loom into the air gap. Towards the transitions between corrugation and plain waveguide the depth of the respective first groove is reduced to d’ = d/2 = 15 µm (Fig. 1b). These tapering measures reduce the mode mismatch between the mode in the corrugated area and on the waveguide, resulting in better transmission results [4–7].
For the numerical calculations software based on the finite integration technique (FIT) is used . All boundaries are defined as perfectly matched layers (PML) that absorb any field components leaving the simulation range, avoiding disturbing reflections. The metallic as well as the dielectric losses are addressed with different loss models, in which the Drude model (with ωp = 13,7e15 1/s and ωc = 40,84e12 1/s ) for the metallic and the constant-fit-tan δ model (with a loss factor of 6e-4 ) for the dielectric losses are best suited. The waveguide is excited with the fundamental mode, determined by the port mode calculation algorithm included in . To accelerate the highly time-consuming calculation symmetry planes are assumed in the center of the structure, which reduces the number of mesh cells without influencing the simulation results.
For reference purposes a quasi-2D calculation of the filter structure without regarding metallic losses and without dielectric center material was performed, representing a laterally unlimited waveguide. Therefore the boundaries directly at the metal-air interface in y-direction were assumed to be perfect magnetic conductors, so that there is no electric field component in y-direction, leaking out of the air gap. The calculated normalized transmission spectrum of the described filter is illustrated in Fig. 2a . For all simulations the parameters g, w and d were chosen to achieve best fit to the overall design goal of at least 80% transmission at 1 THz. A sharp resonance peak appears near the frequency of 1 THz within a broad stopband. The relatively small frequency shift results from insufficient knowledge of the actual effective index of the wave in the waveguide and can easily be tuned by changing the grating period Λ. The resonance peak shows an excellent characteristic with a Q-factor of Q = 7021 at a transmission of 91% and with sideband suppression ratios of −60 dB and −55 dB.
A lateral limitation leads to a drastic reduction of the filter performance. The same structure with thickness t = 500 µm leads after parameter optimization to the characteristic in Fig. 2b, remaining at a Q-factor of Q = 47 with an amplitude transmission of 82% and sideband suppression ratios of −16 dB and −12 dB. Obviously, the boundary in y-direction has a strong impact on the frequency characteristic. The reason for the degradation is based on transversal loss mechanisms motivating the following analysis: Depending on the structure thickness the accumulated radiated power is compared to the Q-factor of the resonance (Fig. 3 ).
With increasing thickness t radiation is reduced and Q increases until it saturates at about t = 1500 µm, which indicates that radiation loss is the primary restricting factor for the limitation of Q. The origin of this loss mechanism are field components outside of the air gap, which are typically present in metallic slit waveguides , (Fig. 4a ) and which tend to couple to radiation modes. These field components are scaled with the structure thickness and need to be avoided in a low-loss design. They can be drastically reduced by filling the whole air gap with silicon, as used in  for a homogeneous waveguide, which leads to a lateral field confinement in y-direction (Fig. 4b). Consequently, a saturation of Q is already observable at t = 500 µm and a Q-factor of over 6000 at a transmission exceeding 80% is again possible, however, assuming lossless materials.
Taking into account the dielectric loss in Si only, this already almost cancels out the beneficial effect of the additional lateral waveguiding in y-direction. This is shown in Fig. 5a by a comparison of the influences of different material losses on the transmission at resonance. For the completely Si-filled waveguide resonator a calculation without loss is performed, again designed for over 80% transmission at 1 THz. Further simulations with separate dielectric and metallic losses, as well as including both kinds of material loss are added for the same parameter set. The figure clearly shows how dielectric and metal loss contribute to the total material loss, indicating that metal loss has considerably larger impact on Q. The observable frequency shifts of the resonance peak are due to the applied material models.
To reduce the influence of the illustrated loss mechanisms, another modification is necessary: A lateral limitation of the silicon element in x-direction concentrates the field in the air gaps between Si and metal surface  and thus reduces the field components and loss inside of the dielectric, without losing the confinement in y-direction. Figure 4c shows the distribution of the x-component of the electric field in a z-cut through the waveguide with this modification.
Considering the metallic loss with the new design, another benefit of the chosen structure emerges. The modified field distribution prevents high field intensities at the metal surface and thus minimizes the ohmic losses as well, which consequently provides the possibility of designing the low-loss 3D high-Q resonator. So, Si as dielectric material has several advantages: the high refractive index leads to strong field confinement and high effective indexes, and thus to a compact filter size. Furthermore, the losses of undoped Si in the THz frequency range are low compared to other dielectrics .
The calculated transmission spectrum of the optimized filter structure is illustrated in Fig. 5b. A structure thickness of t = 500 µm was chosen due to the observed saturation effect and the length of the phase-shift region is selected to p = 125 µm, which allows a nearly centered transmission peak within the stopband. With a Q-factor of Q = 530 at a peak transmission of 80% and sideband suppression ratios of about −30 dB the performance is still outstanding for a compact 3D resonator device in the THz-domain. The simulated frequency characteristic even exceeds the results of recent filters in 2D PPWG technology [12,13]. So, an accordingly realized 3D filter could provide selectivity on the same level as current 2D structures, but with the advantage of lower lateral geometrical extension. If necessary, the tradeoff between peak transmission and Q-factor can be further tuned to higher values of Q by changing the modulation depth d . Also the usage of high resistivity silicon (HR-Si) with an even lower loss factor might lead to an improvement of the Q-factor.
We presented an efficient, compact 3D bandpass filter for THz frequencies implemented as an inhomogeneous 1D photonic crystal structure with a line defect incorporated into a metallic slit waveguide. The high-Q transmission band was optimized by detailed 3D simulations and the occurring loss mechanisms were presented and reduced by selective insertion of a silicon layer into the waveguide.
As the presented structure is designed for the THz range, it has the advantage of relatively large dimensions, enabling alternative, probably even mechanical fabrication methods. The high Q-factor of the device provides the possibility of operation in channel filtering or sensor applications.
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