Design, Simulation, and Characterization of MEMS-Based Slot Waveguides

This contribution presents a design approach as well as characterization results for dielectric slot waveguides fabricated in highly resistive (HR) silicon (HR-Si) designed for operation in the terahertz (THz) frequency range. The authors discuss the fundamentals of dielectric slot waveguides and the respective figures of merit. Furthermore, analytical solutions as well as numerical results from field simulations of the propagating mode are presented and discussed. Prototypes are fabricated in a process for HR-Si microelectromechanical systems (MEMSs), including a silicon-based metamaterial to mechanically support the waveguide while preserving a strong field confinement. In addition, actuator concepts to implement device functionalities, such as phase shifting capabilities, are demonstrated. The prototypes are measured and characterized using vector network analyzer (VNA)-based two-port measurements in the frequency range from 220 to 330 GHz. The devices are characterized in terms of their field confinement, loss, and effective mode index. For the first time, the authors report on the experimental validation of dielectric slot waveguides in this frequency range and provide values for the obtained losses.


I. INTRODUCTION
T ERAHERTZ (THz) signals and systems are of high interest to a vast range of research topics in communication, instrumentation, and sensor technology. As the so-called THz gap [1] is being more and more addressed by studies from both sides of the spectrum, namely, electronic and photonic systems, concepts from either side are evaluated in this specific frequency range between 100 GHz and 1 THz. While, from the electronic point of view, metallic waveguides are the transmission architecture of choice, photonic fibers, i.e., dielectric waveguides, have already proven advantages for higher frequencies, for example, the absence of metallic losses or a larger single mode bandwidth. In either case, the waveguide topology aims to achieve guidance of THz waves with great field control and confinement [2]. Photonicintegrated circuits (PICs) benefit from low-loss and lowdispersion silicon waveguides [3]. As the wavelength of THz radiation is in the submillimeter range, it can be directly addressed by microelectromechanical systems (MEMSs). Thus, the integration of MEMS in THz systems is a promising consequence.
Combining dielectric waveguides with MEMS is a well-known concept in optical communication, and it becomes an emerging field for systems in the THz range. The key idea is to steer beams [4] by combining silicon-based actuators with silicon-based waveguides into THz-integrated circuits (TIC) and transmissive phase shifting waveguides. Therefore, the selection of the most suitable waveguide concept from the point of view of THz and MEMS technology plays a major role. Rib waveguides are well known for integrated-optical circuits in silicon and are investigated further on, e.g., with comb-drive actuators [5]. Considering the scaling, there are scaled-down hollow metallic waveguides using silicon micromachining and metallization for fabrication of channels. These make use of concepts known from radio frequency (RF) MEMS by downscaling hollow silicon waveguides and producing them by MEMS technology with a thin-film metallization [6]. In [7], such silicon-based hollow waveguides in combination with MEMS actuation, including phase shifters and switches, are introduced, achieving waveguides that exhibit a low insertion loss per unit length (0.02-0.07 dB/mm). However, the fabrication of a closed cavity and the required smooth surface of the metallized inner walls are challenging [8].
Scaled-up dielectric waveguides are known from farinfrared (IR) optics applications and are usually realized in highly resistive (HR) silicon (HR-Si) serving as a dielectric material. HR-Si with a resistivity exceeding 10 k cm −1 exhibits transparency in a frequency range from 250 GHz up to 270 THz showing an almost constant refractive index of 3.4-3.5 (ε r ≈ 12) [9]. Consequently, waveguide concepts known from near IR [10], [11] can be scaled to THz applications [12]. HR-Si has a negligible conductivity and a minimum loss in the THz window at room temperature. Although the conductivity and loss increase with temperature [13], [14], dielectric slot waveguides with a confinement of the This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ electromagnetic field in the slot between the dielectrics are promising to overcome a temperature-dependent absorption.
Another aspect to be considered is the refractive index causing Fresnel reflections at each interface.
However, it is proven for IR applications that these reflections can be controlled by using suitable adiabatic refractive index gradients [15]. The photonic crystal waveguides are among the scaled-up dielectric waveguides showing a very low frequency-dependent loss and achieving high-quality factors; thus, they are most suitable for narrowband applications [16], [17]. Rib waveguides or waveguides with an effective medium cladding achieve a much broader transmission characteristic [18]. Thus, these exhibit a nonresonant behavior and are usually limited by the cutoff frequency of the second mode at increasing frequencies and by decreasing guidance at decreasing frequencies. A slot waveguide originally established for near IR is based on the discontinuity of the electrical field at high-index-contrast interfaces and features a double core structure with a high confinement of the field in the low-index gap [10], [11]. The slot waveguide features two beams with the height h and the width w forming a slot with the width s; see Fig. 1(a). The beams confine the electromagnetic field and, therefore, serve as dielectric frames for the electromagnetic wave in the slot. Slot waveguides with two opposing silicon beams enclosing a gap are predestined for MEMS-based actuation and fabrication.
In our recent conference paper [19], we already demonstrated the MEMS actuators and their ability to manipulate and tune the slot-waveguide architecture, while at the same time ensuring freestanding silicon beams to not degrade the field confinement [20]. In addition, no carrier substrate is needed [21] in our full-silicon process, as the beams are stabilized to their sides. However, the simulated propagation properties of the fabricated slot waveguides were only briefly addressed and had not been measured and validated yet. In this article, we present a complete design, numerical simulation, and characterization of an optimized slot-waveguide structure for submillimeter wave and THz signals starting at 300 GHz. Since the waveguide structure has to be mechanically fixed and can be subject to manipulation in order to provide, e.g., phase shifter functionality, influence of the surrounding material on the propagation properties is investigated. A lowindex material is preferred to alter the behavior of the slot as little as possible. Therefore, we propose a structured silicon (Si) metamaterial to support the slot waveguide mechanically, which can precisely be manufactured using MEMS processes and has significantly lower refractive index than silicon. The effect of different metamaterial structures is, thus, investigated as well.

II. DESIGN AND SIMULATION
As depicted in Fig. 1(a), the dielectric slot waveguide has two main parameters that can be optimized in the design process: the slot width s and the width of the silicon beams w. In addition, the height of the waveguide h may be subject to optimization. However, fabrication can be significantly simplified, if a silicon wafer having a standard thickness can be used. Furthermore, the refractive index of the surrounding material has a strong impact on the waveguide performance, as well. Assuming an air-filled slot and silicon beams that define the slot waveguide itself, the field can be strongly confined in the slot region, if the surrounding refractive index is low, compared with the beam material. If the whole structure is embedded into a silicon wafer, the high refractive index material, filling the x z plane [ Fig. 1(a)], causes the field to leave the waveguide and instead propagate in the dielectric slabs [11]. Therefore, a low-index material is preferred.

A. Electromagnetic Field Discussion
The fundamentals of the slot waveguide are discussed in, e.g., [11]. The fundamental mode is of a quasi-transverse electric (quasi-TE) type and has a dominant electric field component pointing into the horizontal (x-direction) direction. The mode can be discussed assuming parallel dielectric slabs with infinite height h. The discontinuities of the refractive index in the cross section of the slot waveguide cause discontinuities in the dominant x-component of the electric field, as well. Since the normal component of the electric displacement field D has to be continuous at the material interfaces, the electric field strength E must be much stronger in the slot region than in the beam region. The x-component of the dominant mode's electric field E in the different regions is given by [10] Explicitly, the field in these regions can be formulated by cosh(γ slot a)cos(γ si (|x| − a)) Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
The dimensions a = 0.5s and b = 0.5s + w denote the boundaries among the slot E 1 (x), beam E 2 (x), and cladding E 3 (x) region with the respective refractive indices n slot , n si , and n clad . The propagation constants γ slot , γ si , and γ clad in these regions can be calculated from Helmholtz equation Use the free-space wavenumber k 0 and the mode's effective refractive index n eff , which is derived in [22]. The fields E and D according to (1)-(4) are depicted in Fig. 2 for an idealized slot waveguide with s = 100 µm and w = 100 µm. Alternatively to an analytical solution of the Helmholtz equation, n eff can be found by formulating an inverse problem that minimizes the discontinuities in D. In the considered scenario, n eff = 2.02 is identified by both methods.

B. Design Methodology
In order to design the slot waveguide, a suitable figure of merit for its ability to guide a broadband electromagnetic wave must be found. For that purpose, the field confinement factor C is usually used [10], [11]. It is defined as power ratio of the field components within and outside the slot region As seen in (9), it can be calculated by integrating the wave's (time-averaged) Poynting vectorS over the cross region (segments) of the waveguide. Therefore, a high-field confinement corresponds to a mode field that is less influenced by the cladding material. Fig. 3 presents the confinement factor C as a function of the slot width s and the beamwidth w. Its maximum confinement can be considered as an initial optimization point from which to start the design. A maximum is found at 100-µm beamwidth and 60-µm slot width. The refractive index of the cladding material was set to 1.82 (see Section II-C). Subsequently, we used the mode solver from CST Microwave Studio to analyze the waveguide, taking the finite height h of the Si beams into account. Again, the simulation was conducted around 300-GHz center frequency, which was also considered in the mode solver. Within the numerical simulation, the maximum confinement 0.36 is achieved at 100-µm beamwidth and 58-µm slot width. Of course, the absolute confinement results differ from the analytical simulation due to the finite beam height; nevertheless, they qualitatively agree very well.

C. Supporting Metamaterial Structure
Since the silicon beams need a mechanical fixture [20], a supporting material must be used. Furthermore, the material  should be made from a low refractive index material. Since the slot waveguides are fabricated on an HR-Si wafer, we propose to utilize a metamaterial approach by purposefully etching the silicon outside the beams to lower the effective refractive index in this region. Therefore, different unit cell structures are considered: 1) hexagons; 2) rhombs; and 3) circles (see Fig. 4). All of these are much smaller than the considered wavelength at 300 GHz, so that neither photonic crystal effects nor wave scattering occurs at these. Instead, we aim at creating an effectively homogenous material with a significantly lowered average refractive index than the original silicon. Structures 1)-3) were analyzed in terms of volume fraction ratio air/Si and effective refractive index [23]. Table I summarizes the considered design parameters of these three metamaterials. The effective refractive index of these materials was identified using a simulative transmission line study in combination with the Baker-Jarvis method [24] to extract the dielectric properties from the scattering parameters of a meta material filled waveguide.
In order to enhance the simulation performance and optimization of the waveguide, an effective medium is preferred against a full detailed simulation of the etched metamaterial structure. Therefore, we compare the (simulated) slotwaveguides transmission properties of an effective medium to mechanically support the waveguide with a detailed simulation once for each of the unit cells. For this purpose, a 10-mm-long Si-slot waveguide with h = 200 µm, s = 100 µm, and w = 100 µm is considered in CST Microwave Studio. From the broadband (220-330 GHz) simulation results, the transmission phase's frequency dependence is used to determine the effective wavenumber, propagation constant, and refractive index of the waveguide. Table II summarizes these values for the three different unit cells and the different simulation types. Again, a simplified and original model behaves sufficiently similar, having low discrepancies. Therefore, the effective medium simplification can be used with a very good accuracy, if the effective dielectric properties are determined beforehand by separate simulations. As can be seen in Table II, the   TABLE II   STRUCTURED AND EFFECTIVE MEDIUM SIMULATION RESULTS circular unit cell in the supporting structure has a much larger effective dielectric constant than the other unit cells. Thus, the fundamental mode's effective index is also comparably large, causing a degradation of the mode confinement. In Section III, we, therefore, consider only the rhombical and hexagonal structures.

D. Phase Shifters [19]
Since the design parameters of the dielectric slot waveguide directly determine the effective refractive index and, therefore, the wavenumber as well as the phase and group velocity, manipulating these parameters allows to realize functional waveguide components, subsystems, and systems. As a prototyping example, we investigate the capability of the silicon slot waveguide to act as a wideband phase shifter by manipulating the slot width using two MEMS actuators, one on each side of the slot, directly actuating the silicon beams, as shown in Fig. 5. Using the MEMS actuators, the original slot width of 100 µm can be varied highly precise from 46.4 to 182.4 µm. Fig. 6 shows the simulated effective refractive index of the fundamental mode with respect to the slot width, as well as the phase difference per micrometer, compared with the original width. It can be seen that a phase difference of 30 • per millimeter is achieved for the narrowest slot width of 40 µm. Negative phase differences, i.e., higher group velocities, are obtained for slot width larger than 100 µm. Due to the nonlinear behavior of the effective index, the obtained phase differences are significantly smaller in this area.

E. Bend Waveguides
Since a functional waveguide system must be able to realize bends and curves, direction changing waveguides are a crucial point to investigate on [25]. From dielectric rib waveguides, it is well known [26] that too narrow bend radii (with respect to wavelength) may cause radiational losses; i.e., the wave does not follow the curvature. In principle, the same holds for slot waveguides. In order to improve the guidance of the field during the bend, tapering the silicon beams is a suitable approach. Bending losses of dielectric waveguides can, e.g., be described by a coordinate system transformation [27], transforming the bend into a straight waveguide. Using this model, radiation occurs, if the local refractive index of the waveguide and its surrounding is larger than the effective one of the considered mode. This effect can either be reduced    7. Microscopy image of the bend silicon slot waveguide with hexagonal metamaterial structure. In order to improve the field guidance, the outer silicon beam was tapered from w si = 100 µm to w taper = 130 µm. by larger bend radii or by tuning the local refractive index. Therefore, we taper the outer silicon beam from w si = 100 µm to w taper = 130 µm, as can also be seen in Fig. 7. The larger refractive index keeps the field near the beam and improves the guidance.

A. Fabrication Process
The structures are fabricated on HR (exceeding 10 k cm −1 ) silicon substrates with a 100-mm diameter, a 200-± 10-µm thickness, and double-side polished surfaces. The first fabrication step is the deposition of a 1600-µm SiO 2 layer on the front of the substrate and a 200-nm SiO 2 layer on the back of the substrate by plasma-enhanced chemical vapor deposition (PECVD), as schematically shown in Fig. 8(a). During deep reactive ion etching (DRIE), the SiO 2 layer on the front serves as a hard mask, while SiO 2 on the back is used as an etching stop to prevent an etching of the carrier wafer. During lithography, we use an AZ MIR 701 positive photoresist [ Fig. 8(c)]. After lithography, we pattern the SiO 2 layer by reactive ion etching (RIE), as shown in Fig. 8(d). During DRIE [ Fig. 8(e)], the HR substrate is placed on a carrier wafer and fixed with fomblin oil to ensure the cooling during etching. As the HR silicon is highly susceptible to overheating, we use a two minutes break after each 25 cycles of etching. The etch rate is 0.32 µm per cycle. After DRIE, the structures are still placed on the carrier substrate. As the structures are surrounded by an etching frame, they can be removed from the silicon substrate without additional sawing process. The chip is placed on a 3-D print with commercially available glue [ Fig. 8(f)]. Afterward, the areas in front of the waveguides are broken out mechanically. Fig. 8(g) shows the fabricated device showing four straight waveguide sections (two slot and two rib waveguides) and two cosine-shaped bends that realize a waveguide displacement by 4 mm.

B. Experimental Setup
The test structures are characterized using a vector network analyzer (VNA) with frequency conversion modules to address the WR3.4 frequency range (220-330 GHz). The used VNA is of type Rohde & Schwarz ZNA67 and is used alongside two VDI WR3.4 VNAX modules. Therefore, full two-port measurements are done. The experimental setup is calibrated in the VNAX flange plane using the through-reflect-match (TRM) method [28]. A photograph of the setup is presented in Fig. 9, showing the waveguide probe antennas and the six test structures in the center. Numbering the waveguides from 1 (front) to 6 (back): sample 1 is slot waveguide with hexagonally shaped fixture; sample 2 is slot waveguide with rhombical fixture, and samples 3 and 4 are rib waveguides with hexagonal and rhombical fixtures. Samples 5 and 6 are slot-waveguide bends (4-mm displacement) with both having a rhombical fixture, but in sample 6, the Si beams of the waveguide are tapered, as described in Section II-E, while in sample 5, the waveguide profile is not altered during the bend. The signals are coupled into the test structures via free space, using two WR3.4 waveguide probe antennas. In order to estimate the insertion and coupling losses of these, a direct probe-to-probe measurement in varying distances is from (nearly) 0 to 40 mm. The transmission curve for the center frequency of 275 GHz is presented in Fig. 10. It can be seen that the measured path loss fits very well to a free-space path loss model, assuming an antenna gain of 7.5 dB taken from the datasheet. The ringing effect of multiple reflections can be identified to start at approximately 2-cm separation distance, showing that the performed calibration is able to sufficiently suppress them for closer antenna separations. Since the waveguides, i.e., the wafer, are very thin, a significant coupling through free space above and below the waveguide is expected. Thus, the free-space path loss is a measure for the lower boundary of the dynamic range in this setup. Thus, a calibrated dynamic range of ≈30 dB at 2-cm measurement distance is achieved.

C. Characterization Results: Field Pattern
Due to the very fine structures to be characterized, we have employed a scanning measurement setup that allows to raster the coupling planes to both sides of the dielectric waveguides, as depicted in Fig. 9. Fig. 11 shows the transmission magnitude S21 at 300 GHz for the slot waveguide with the hexagonal metamaterial structure. Due to the raster scan using a WR3.4 waveguide, which is much larger than the slot itself, the spatial field shape is strongly smoothed. Nevertheless, it can be seen that the field maximum is well within the slot region, and a very symmetric field profile is achieved. This can be explained by the averaging effect of the WR3.4 probe. The Fig. 10. Measured path loss as a function of antenna separation distance. Measurement matches well with free-space path loss, even at narrow spacings. The measurement is used as a reference, against which the waveguide transmission is compared. Fig. 11. Scanned transmission magnitude at 300 GHz of the slot waveguide with hexagonal structured metamaterial. The focusing of the field at the measured path loss as a function of antenna separation distance. Measurement matches well with free-space path loss, even at narrow spacings. The measurement is used as a reference, against which the waveguide transmission is compared. field polarization is mainly aligned with the horizontal axis in Fig. 11; this, therefore, corresponds to the shorter edge of the WR3.4 waveguide. Thus, the averaging effect is less pronounced on the x-axis than on the y-axis, corresponding to the longer WR3.4 edge, respectively.
In order to provide a more quantitative analysis, Fig. 12 presents the transmission magnitudes of both slot waveguides (hexagonal, rhombical metamaterial) and both rib waveguides in the major Eand H -plane cuts through the maximum coupling point. The E-plane corresponds to the electric field polarization, which lies within the wafer plane, whereas the H -plane is orthogonal to this, i.e., vertical to the wafer. All aforementioned measured quantities are plotted with respect to the left-hand side y-axis (black).
As already stated above, the field magnitudes in Figs. 11 and 12 do not resemble the theoretical curve of Fig. 2 but are a convolution of the dielectric and metallic waveguide modes. As a deconvolution of the WR3.4 pattern is difficult, we simulated the coupling behavior in another full wave simulation assuming a distance of 1 mm between the WR3.4 probe tip and the slot waveguide. To estimate this offset, we removed the sample without changing the setup and measured the transmission pulse delay in time domain. In the simulation, we then introduced a varying horizontal and vertical offset Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. (a) Horizontal (H -plane, wafer plane) cut of the transmission magnitude in Fig. 11. (b) Vertical (E-plane) cut of Fig. 11. In addition to the presented slot waveguide, rib waveguides with different supportings were characterized as well. However, the proposed slot-waveguide/hexagonal metamaterial structure provides the highest transmission and best confinement. For this structure, simulation results are presented by the red dashed curve that validate the mode pattern.
(±0.5 mm) resembling the raster scan in the measurement. The simulated coupling result is shown in Fig. 12 by the red dotted curve, which is plotted with respect to the right-hand side y-axis (red). Since only the coupling at the end of the waveguide was simulated, the absolute transmission values differ significantly in this idealized simulation. The simulated transmission factor pattern with respect to the scan offset is in very good agreement with our measurements. Therefore, this supports our assumption that the slot-waveguide mode in the experiment agrees with the expected one. In addition, it can be seen that the absolute transmission magnitude of the slot-waveguide/hexagonal structure combination is significantly larger than the other samples and far above the 32-dB free-space path loss measured at 12-mm separation distance (see Fig. 10).
Using the same procedure, we investigated on the transmission behavior of the bend waveguides that realize a beam displacement of 4 mm by following a cosine-shaped trace. The result of the raster scan at different frequencies is depicted in Fig. 13. In this case, the feed position is fixed at −2 mm, while the receive port scans along the x-axis. As can be seen from the blue solid curve, a strong coupling is achieved for the measurement at 300 GHz at +2 mm using the tapered bend Fig. 13. Horizontal (H -plane, wafer plane) cut of the transmission magnitude on a slot waveguide with a bend. The waveguide excitation occurs at −2 mm, and the receive signal is expected at +2 mm. Only the tapered bend presents a clear maximum at this position and only at the design frequency of 300 GHz and above. For significantly lower frequencies, the slot does not guide the wave, as is also depicted in the simulation field monitors.
(see Fig. 7). In contrast, the blue dashed curve is measured at the same frequency but in the bend without an additional taper (sample 5). This one shows no distinct maximum, instead the propagation mode of the slot can not follow the bend and propagates as a substrate mode in almost the complete Si plane. The same behavior can be observed for both samples (green curves), at frequencies way below the design frequency of 300 GHz. In this case, the slot does not guide the wave, due to the larger wavelength, i.e., less confined field, and it is, thus, radiated when the bend starts.

D. Characterization Results: Transmission Measurements
Of course, one of the most interesting key performance indicators of the fabricated dielectric waveguides is the transmission loss within the structure. As already indicated above, the transmission between both probe antennas is significantly increased by placing the dielectric waveguides in between. Fig. 14(a) presents the measured S21 of the straight slot waveguide with hexagonal structured fixture. In addition, the free-space coupling without the slot is shown. Furthermore, we have simulated the slot waveguide including the feeding probe antennas to both sides. The simulated S21 is also shown in Fig. 14(a).
In order to estimate the loss per unit length within the waveguide, we used the transmission through free space [ Fig. 14(a) (green)] to measure the pulse shape and deconvolute the hollow waveguide dispersion that occurs within the probe antennas, which are of significant length (40 mm each). Afterward, the transmission level is corrected for the measured (and also filtered/time gated) free-space path loss. Therefore, we first considered the scenario without a dielectric waveguide in between, where there is a separation distance of 12 mm leading to ≈32 dB of path loss. We compared this with the second scenario, including the dielectric waveguide, where 10 mm of the propagation distance is replaced by the waveguide. Subsequently, we compensated twice for the additional path loss of 1 mm (7 dB) on each side of the dielectric waveguide. The results obtained by this correction are presented in Fig. 14 for the slot [Fig. 14(c)] and rib [ Fig. 14(d)] waveguides. It can be seen that the samples with the rhombical unit cell exhibit strong attenuation starting from ≈300 GHz, which may Time delay (x-axis) and magnitude (y-axis) are normalized to the face-to-face measurement of the probe antennas in order to de-embed the propagation within the antennas. From the peak positions, the individual time delay of the pulses traveling through each of the waveguides can be directly measured. Furthermore, it is apparent that none of the dielectric waveguides exhibits significant dispersion. be related to some resonant behavior caused by the longer diagonal of the rhombus (236 µm), which is close to a quarter wavelength in this frequency range. In the end, we obtained the average loss values per millimeter that are summarized in Table III. As can be seen, the slot waveguide having the hexagonal metamaterial support exhibits the lowest loss as well as the lowest effective index. In the literature, dielectric slot waveguides are typically discussed in the 1550-nm band exhibiting losses in between 0.4 and 1.2 dB/mm [29]. To the best of our knowledge, the only contribution to the lower THz region is given by Nagel et al. [12], discussing dielectric tube waveguide that share a low-index region in the center with the concept of slot waveguides. For these tubes, the authors report losses between 0.1 and 0.3 dB/mm alongside an effective index of 1.79, which is comparable to the obtained values in this publication. The cladding in [12] is made from polyimide having an refractive index of 1.87, which is also in the same region as our hexagonal thinned silicon (1.84). In [21], an HR-Si slot waveguide on a quartz substrate is presented at frequencies above 500 GHz exhibiting 0.168-dB/mm loss. In Table III, the measured effective mode indices are compared with the simulated ones using the effective medium approach. As can be seen, the results fit quite well, but exhibit some discrepancies with the exact model simulation (see Table II). We assume that this is due to numerical problems of the used port mode solver, if strongly inhomogenous materials are near the excitation ports.

E. Characterization Results: Effective Indices
In addition to the characterization of the transmission losses, the effective index of the propagating mode in the dielectric waveguide can be well estimated used for validating the design and simulation approach using the effective metamaterial medium, as well. Therefore, we consider the time-domain representation of the measured transmission parameter S21. The transformation to time domain is done using an inversechirp-Z [31] transform in combination with a Hann window in order to provide a well-resolved representation of the transmission pulse. Fig. 15 presents the time-domain pulses of all considered waveguides as well as the transmission between the two probes face-to-face, i.e., without spacing (distance nearly 0 mm) and with 12-mm spacing (as in the waveguide measurements), respectively. The effective index can be calculated by the ratio of the time delays with and without the waveguide in between the antennas.
The results are summarized in Table III. To compare the measurements with the aforementioned simulations (Section II-C), the simulation results for the effective medium approach are listed (in parentheses) as well. The measured effective indices are in very good agreement with the simulated ones, validating the effective medium approach as well as the overall design.

IV. CONCLUSION
This article presents a design approach for dielectric slot waveguides in the THz range fabricated on an HR-Si platform, allowing for co-integration with and manipulation by MEMS actuators. We present measurements from dielectric slot waveguides made from HR-Si in the frequency range below 1 THz, evaluating the effective mode indices as well as losses of these. At the design frequency of 300 GHz (WR3.4), the measured losses are in the range of 0.15 dB/mm and are well in line with the only other reported values (0.1-0.3 dB/mm) in this range for a different, yet comparable, dielectric tube waveguide [12]. Furthermore, they are in line with a silicon slot waveguide on a quartz substrate (0.168 dB/mm), presented in [21].
In addition, measurements on dielectric rib waveguides are conducted and validate the measurements in conjunction with detailed simulations.
Our contribution discusses the fundamentals of dielectric slot-waveguide design and presents results from full wave simulations with a special focus on structured HR-Si used as a low-index metamaterial to fix and support the waveguide. Measurements on 10-mm-long samples are conducted using a VNA with WR3.4 extensions to cover the frequency range. The measured transmission parameters are evaluated to investigate on the mode pattern, the losses, as well as the effective mode index. We demonstrated the ability to guide the fields through bended structures by means of tapered silicon beams that are needed to ensure field confinement.
Moreover, it was proven that the design parameters known from waveguides for optical communication are still valid for dielectric waveguides with an almost three orders of magnitude larger wavelength.