1. Introduction

The generation of THz signals at high power levels and the efficient transmission of THz signals using wave guiding structures are two major challenges researchers are currently facing while trying to satisfy the strong demand for compact, low-cost and robust THz systems. Certain key applications anticipated at microwave or optical frequencies (e.g. integrated circuit devices or fiber-based lasers, respectively) seem to be impossible at THz frequencies due to the lack of sufficiently efficient waveguides.

One serious problem is the residual absorption – even in the so called “THz-transparent” materials – preventing THz signal transmission over longer distances. Dry air is the most transparent broadly available transmission medium for THz waves, but a technically useful waveguide has of course to be materialized in some way. Therefore, waveguide structures that confine only low power fractions within lossy material are highly desirable. Recent approaches in this direction, e.g. with metal-based Sommerfeld wires, demonstrated very low attenuation and dispersion [1]. However, these waveguides suffer from a large field extension into air [2] as well as from large radiation loss at bendings [3]. These severe drawbacks for applications requiring low cross-talk between closely spaced lines (e.g. integrated devices) or flexible handling (e.g. endoscopy) require alternative solutions.

In this work a new concept of THz waveguides is demonstrated, that might have the potential to make some of the THz applications yet impossible become reality. The concept is based on dielectric waveguides which can be divided into two groups according to their guiding mechanism: Total internal reflection (TIR), also called index guiding, is a classic method that is in use in the majority of optical waveguides since many decades. External reflection generated e.g. by interference effects can be found in more recent developments like photonic crystal fibers (PCFs) [4]. TIR-based dielectric THz waveguides reported until now include polyethylene [5] and sapphire fibers [6] or polyethylene ribbons [7]. THz photonic crystal fibers with high-index core wave guiding have been demonstrated with high-density polyethylene [8] or Teflon [9] as dielectric material. Unfortunately, there is a common trade-off that dielectric THz waveguide concepts are suffering from: Attenuation increases with increasing confinement as, for example, demonstrated with TIR-based dielectric waveguides having sub-wavelength diameters [5,6]. In theory, this trade-off can be circumvented using hollow-core PCFs, which have already been demonstrated as optical waveguides [10]. In such a waveguide the most of the light is guided in the hollow core placed in a periodically structured cladding. However, a fabrication technology that is able to build these structures with the required level of perfection for optical application is still missing. Additional losses caused by variations in the fiber structure along the length, scattering at multiple surfaces, deformation at bends, surface roughness and insufficient mode confinement are responsible for the fact that today optical hollow-core PCFs still fall behind the loss performance of conventional bulk fibers.

Recently, a different approach based on index guiding has been proposed by Almeida et al. for integrated optical devices [11]: A high-index dielectric single-mode rectangular waveguide split along the axis of wave propagation. The guided wave is linearly polarized in the normal direction to the high-index-contrast splitting interface. According to Maxwell’s equations the normal component of the electric flux density D _norm has to be continuous at dielectric interfaces. Hence a large index discontinuity causes a large field amplitude discontinuity with increased field amplitude on the low-index side. Therefore, it is possible to guide and confine a substantial fraction of power in the low-index split – even though the guiding mechanism is TIR.

The aim of this work is to transfer this basic principle to the THz range and to evaluate how successfully it can be applied there. Although a large body of electronic and optical waveguide concepts is known to be rather inapplicable for THz transmission, this idea stemming from the field of optics deserves a closer inspection.

 

Fig. 1. Geometries of the considered dielectric waveguide structures: (a) A split rectangular waveguide and (b) a tube waveguide.

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Two dielectric waveguide structures with a low-index discontinuity as depicted in Fig. 1 are compared: First, a split rectangular waveguide (SRW) composed of two dielectric sections, each with a cross-section having width w and height h separated by an air gap of width g; second, a tube waveguide (TW) with an outer radius R and inner air core with radius r. Both structures gain to a different extent from field enhancement effects. To the best of our knowledge these structures have not been considered for THz transmission up till now. The tube waveguide differs in a crucial point from other known optical hollow dielectric waveguides: Having sub-wavelength dimensions in the transverse direction it only supports single-mode transmission. An optical dielectric tube waveguide in the sense of this work would have a transverse size on the order of a nanotube. Sub-wavelength optical nanowire waveguides on the other hand have been reported recently [12]. Hollow dielectric light guiding tubes have indeed been considered for telecommunication applications in the 1960s but with only little success [13]. Much more successfully, hollow multimode waveguides are applied for the transmission of high-power levels of infrared light [14]: At an operating wavelength of 10.6 µm certain glasses show anomalous dispersion with an index of refraction n<1. For example the refractive index of a waveguide cladding made of sapphire is n=0.67. In this case the air-core with n=1 is the high-index core and the waveguide is a classic TIR-based waveguide. However, such dielectric material properties are not common at other wavelengths. Transmission properties of dielectric single-mode flexible rod and tube waveguides made of plastic have been investigated recently at millimeter-wave frequencies between 33 GHz and 50 GHz [15]. Here we investigate loss, dispersion and mode confinement properties of the proposed structures in the THz range and compare the results with state-of-the-art THz waveguides.

2. Numerical field simulations

The geometry of the investigated structures are shown in Fig. 1 within orthogonal coordinates (x,y). The waveguide structures are calculated using numerical electromagnetic field simulation software based on the finite element method with a non-uniform adaptively optimized mesh [16]. The material considered for the SRW is float-zone, high-resistivity silicon. According to THz time-domain spectroscopy measurements this material has a refractive index of n=3.417 and a dielectric dissipation factor tanδ=0.00001 at a frequency f=0.7 THz [17]. The material considered for the TW is fused silica with n=1.95 and tanδ=0.001 at f=0.5 THz as experimentally determined in [18]. For air we set n=1 and tanδ=0. The power absorption coefficient α_m given in the above mentioned literature is converted into the dissipation factor by using the relations k=α_m·c/(4πf) and tanδ=2·n·k/(n²-k²), with c being the speed of light. n and k are the real and imaginary part of the complex refractive index n̂(f)=n(f)-ik(f). Only the fundamental quasi-TE mode, which has no frequency cut-off, is considered for the SRW and the TW. The main electric field component is aligned in y-direction. Two-dimensional field solutions are calculated for the whole waveguide cross-section including the outer free-space region, which spans over a radius of more than three wavelengths from the waveguide center to account for evanescent fields.

 

Fig. 2. Distribution of the normalized scalar z-component of the time-averaged Poynting vector S_z over a linear color scale in the cross section area of the waveguides. (a) Float-zone silicon split rectangular waveguide at f=0.7 THz with w=54 µm, h=90 µm and g=18 µm. (b) Fused silica tube waveguide at f=0.5 THz with R=181.5 µm and r=27 µm.

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We consider first the spatial distribution of the power density of the electromagnetic wave propagating in z-direction along the waveguide which is given by the time-averaged z-component of the Poynting vector S_z. In order to see some of the characteristic differences between both structures, S_z is mapped over the cross-section of a SRW and a TW with typical dimensions in Fig. 2(a) and 2(b), respectively, under high-confinement conditions. In both structures the maximum value of S_z is clearly observed within the inner low-index regions. The geometry of the TW, however, exhibits considerable power density leakage into the dielectric material at the inner regions where the main electric field component is aligned parallel to the dielectric interface. This effect is due to the continuity condition of the tangential electric field component at the dielectric interface. In addition to the geometrical aspect, the overall fractional power in the dielectric region increases for the fused silica TW in comparison to the silicon SRW because of the lower dielectric contrast at the material-air interfaces. A further difference between both structures is given by the additional local maxima of S_z at the inner edges of the SRW.

In order to optimize the waveguides in terms of minimum attenuation, the spatial power distribution is a critical parameter: Attenuation due to dielectric loss increases with the fractional power within the lossy dielectric material region, radiation loss at bendings increases with the weakly guided power fraction within the outer free-space region. Therefore, to achieve lowest possible losses the power confined in the inner air region has to be maximized. We distinguish between a materialized region of the waveguide and an inner air region as sketched in Fig. 3. In case of the TW the inner air region is directly given by the air core. An extended gap region is considered as the inner air region of the SRW to account for the well confined power contributions at the gap openings as displayed in Fig. 2. The surrounding area of both waveguides is air as well.

 

Fig. 3. Geometrical definition of the regions considered as dielectric material (blue), inner (red) and outer (white) air regions.

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The power ratio guided in the dielectric material is given by $r_m=\int {\int }_{A_m}{S}_{z}dA⁄\int {\int }_A{S}_{z}dA$, with A_m being the cross-section area of the dielectric material and A the cross-section area of the total solution range. A is kept constant in all following calculations. The power ratio confined in the core or gap area A_c is given by $r_c=\int {\int }_{A_c}{S}_{z}dA⁄\int {\int }_A{S}_{z}dA$. We focus now on the scaling of the waveguide cross-section under a fixed aspect ratio. Fig. 4 shows r_m and r_c for both waveguide structures as a function of their scaling. In Fig. 4(a) the cross-section of the SRW is scaled with h=1.6̄ w=5g and the TW with R=2r. Additionally, the power ratio in a conventional rectangular waveguide (g=0) and a circular waveguide (r=0) are shown for comparison. In this case the transverse scaling is determined by h=1.6̄ w and R, respectively. For the data shown in Fig. 4(a) and 4(b) the frequency considered for the rectangular waveguides is f=0.7 THz, and for the circular waveguides f=0.5 THz.

We first attend to Fig. 4(a). In case of the rectangular waveguide with g=0 the fractional power within the material switches immediately from zero to 80 % as soon as the waveguide reaches a certain threshold in size. After this initial rise a much slower increase to 100% can be observed as the size of the waveguide is further increased. In the SRW, in contrast, the power ratio within the dielectric material is considerably reduced since a substantial part of the power is confined within the air gap. We can identify an optimal size of the SRW around h=90 µm where almost 55 % of the power is confined within the low-index gap area and less than 25 % of power is guided within the high-index material. For smaller sizes of the SRW, the power ratios in the gap area and in material are reduced, because the overall waveguide has sub-wavelength dimensions. For sizes bigger than the optimum size, the power ratio within the gap area decreases while the power ratio within the dielectric material increases. In this range both parts of the SRW are getting large enough to act as a pair of coupled rectangular waveguides with transverse dimensions greater than the wavelength. The absolute maxima of S_z are then spatially located in the center of the dielectric sections. The qualitative behavior of the scaled TW is very similar. However, the power ratio confined in the air core is only 26 % and the power ratio in the material is still ca. 50 % in the best case.

 

Fig. 4. Power ratio confined in the material and in the low-index gap or channel region for a SRW and TW. (a) Variation of h and R with h=1.6̄ w=5g and R=2r (b) Variation of g and r with h=90 µm, w=54 µm, r=R-90 µm.

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In Fig. 4(b) the gap width and the core radius are varied, whereas the respective size of the rectangular dielectric sections and the tube cladding thickness are kept constant. An absolute maximum of the power ratio within the low-index core or gap region is observed at ca. r=130 µm and g=21 µm. For g→0 the power ratio within the dielectric material of the rectangular waveguide increases continuously. This is expected since the resulting rectangular waveguide for g=0 has a large filling factor. For large gap widths g, however, the transverse dimensions of the separated rectangular waveguides are in the sub-wavelength regime. Therefore, only small power fractions are guided in the material for large gap widths. Considering the TW on the other hand, the volume of the dielectric material increases with the core radius r. For increasing values of r the power is predominantly confined in the tube cladding regions with the main electric field component in tangential direction to the cladding surface. As a result, the power ratio in material stays on a relatively high level around 50 % as the core radius increases.

We now focus on the frequency-dependent transmission properties of the proposed dielectric waveguide structures. We consider the attenuation α (f) and the effective relative permittivity ε _r,eff(f) of the fundamental mode propagating through a straight transmission line. The numerical field simulations account for the dielectric loss of the SRW and TW and for metallic loss of the Sommerfeld wire considered for comparison. Material dispersion of float-zone silicon and fused silica is not included since it is negligibly small in the considered frequency range of 0.3–0.9 THz [17,18]. In Fig. 5(a) and 5(b) the simulated transmission data of circular and rectangular dielectric waveguide structures with (g>0, r>0) and without (g=0, r=0) a low-index discontinuity and of a metallic Sommerfeld wire are compared. The size of the dielectric waveguides has been chosen to support strong low-index confinement in the observed frequency range of 0.3–0.9 THz.

 

Fig. 5. Frequency-dependent (a) attenuation and (b) effective permittivity of tube and rectangular dielectric waveguide structures with and without a low-index gap. For comparison, the transmission properties of a Sommerfeld copper wire with a radius R=1 mm are shown as well.

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The attenuation behavior is shown in Fig. 5(a). We note that there is a scaling difference of one order of magnitude between the left axis for the tube waveguide data and the right axis for the rectangular and Sommerfeld waveguide data. The attenuation of the dielectric waveguides is proportional to r_m·tan δ·f, with rm being the power ratio confined within the lossy material. The dielectric waveguides without low-index discontinuity exhibit three characteristic regimes with different increasing attenuation gradients. The first regime is observed at low frequencies, e.g. in Fig. 5(a) at the rectangular waveguide data for f<0.55 THz: The wavelength of the guided signal is large in comparison to the transverse dimensions of the waveguide. In this regime the attenuation is extremely low. However, mode confinement is extremely low as well. This regime is technically not very attractive since strong radiation losses can be expected at bended structures. As the frequency increases further a rapid increase of α is observed for the circular waveguide with r=0 for f<0.5 THz and for the rectangular waveguide with g=0 for 0.55 THz<f<0.65 THz. In this frequency range the attenuation gradient is dominated by the strong increase of r_m as observed in Fig. 4(a). For higher frequencies r_m saturates. As a consequence, the increase of α (f) is mainly proportional to tan δ·f at higher frequencies. The implementation of the low-index discontinuity to the dielectric waveguide structures generates a distinct reduction of α in accordance to the above mentioned decrease of r_m. We compare the attenuation of the SRW with the attenuation of the fundamental radial surface-wave mode on a metallic Sommerfeld wire. The attenuation of a straight Sommerfeld wire is generated by ohmic loss which decreases for increasing radii R [2]. In our example the attenuation of a Copper-based Sommerfeld wire with a conductivity of 5.8 10⁷ S/m and a radius as large as 1 mm clearly exceeds the loss of the much smaller SRW in the observed frequency range. The dispersion of the Sommerfeld wire in terms of dεr(f)/df, however, can be completely neglected in comparison to the noticeable dispersion of dielectric waveguides, as can be seen in Fig. 5(b). The low-index discontinuity of the dielectric waveguides, nevertheless, reduces the dispersion very effectively in comparison to the conventional dielectric structure by lowering the overall effective permittivity.

The most striking criteria in this comparison, however, is given by the fact that the SRW is able to exhibit lowest loss and highest confinement at the same time: For a SRW with h=1.6 w=5g=90 µm 75% of the propagated power at f=0.7 THz is confined in a cross-section area – including the material and the gap section – of only 0.015 mm². The area surrounding the considered Sommerfeld wire in which 75 % of the power is propagated at this frequency is more than 10⁴ times larger [19]. In view of applications like integrated photonic THz devices or near-field imaging the very efficient confinement provided by the SRW is extremely helpful in order to achieve low cross-talk between closely spaced transmission lines or sub-wavelength resolution, respectively. Considering bended waveguide structures this conclusion becomes even more evident. In a bended waveguide, the modal field becomes radially distorted. This perturbation causes coupling from the core mode to radiation modes [20]. The perturbation – and thus the bending loss – increases as the transverse extension of the guided mode increases. As a result, single-mode dielectric waveguides usually exhibit a bend loss edge at long wavelengths where the mode extension strongly exceeds the waveguide cross-section.

 

Fig. 6. Electric field magnitude distribution of the fundamental transmission mode at (a) a bended Sommerfeld wire (R=140 µm, f=500 GHz), (b) a bended TW (R=181.5 µm and r=27 µm, f=0.5 THz) and (c) a bended SRW (h=90 µm, w=54 µm and g=18 µm, f=700 GHz), the latter both with the E-field polarized in parallel direction to the bending plane. The bending radius of the applied 45° segment of a circle is 2 mm.

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For comparison we simulate the three-dimensional field distribution at three bended THz waveguides: a Sommerfeld wire with R=140 µm, a TW and a SRW with dimensions and dielectric properties as described in Fig. 2. The applied bending radius is 2 mm. For the Sommerfeld wire we consider the fundamental radial TM01 mode. The fundamental quasi-TE mode as depicted in Fig. 2(a) is considered for the TW and the SRW with the main electric field component polarized in parallel direction to the bending plane. In Fig. 6 the electric field magnitude distribution of all waveguides is compared within the bending plane. The field distribution at the bended Sommerfeld wire in Fig. 6(a) exhibits a clear generation of radiated field contributions in a direction outwards of the bend: As given by the effective permittivity data in Fig. 5(b) the surface-waves on metallic Sommerfeld wires propagate almost with the speed of light. Since this velocity can not be exceeded, the phase difference between the inner and the outer side of the bended wire increases along the propagation axis as can be observed very well in Fig. 6(a). As a consequence the guided surface-wave mode is radially distorted. Radiation loss of THz surface-waves on bended wires and curved surfaces has been experimentally investigated recently [3,21]. The TW and the SRW, on the other hand, exhibit much better field confinement and consequently negligible radiation loss in comparison. The overall pure bending loss values excluding dielectric dissipation loss calculated for the Sommerfeld wire, the TW and the SRW in Fig. 6 are 0.33 dB/°, 0.0015 dB/° and 0.0004 dB/°, respectively.

3. Experimental investigations

Since a split rectangular waveguide was not available at the time of writing, the following experimental investigations – which should be considered as a proof-of-principle – focus on the tube waveguide. A very important advantage of the TW is its commercial availability. Tubes made of fused silica are widely used for gas or liquid phase chromatography [22]. However, to the best of our knowledge they have not been applied as THz waveguides until now. Fortunately, the spectrum of available tube sizes corresponds very well with the optimum dimensions required for THz signal transmission. The dielectric waveguides used here are fused silica tubes with an outer protective polyimide coating [23]. The core radius is r=12.5 µm and the outer radius including the 14-µm-thick polyimide film is R=181.5 µm. We choose this geometry with comparably high loss in order to have easier conditions for the quantitative determination of frequency-dependent attenuation using time-domain techniques, which is technically demanding for waveguides with extremely low loss levels at THz frequencies.

3.1 Measurement set-up

The experimental determination of the propagation parameters of the dielectric tube waveguide is based on a standard THz time-domain pump-probe scheme. Femtosecond laser pulses at 780 nm wavelength and 78 MHz repetition rate are used for photoconductive excitation and detection of THz signals. For impulsive THz signal generation and detection two identical photoconductive antenna devices are used. They are fabricated in planar metal-based waveguide technology and consist of a coplanar waveguide fed Yagi-Uda antenna structure embedded in a parallel plate waveguide. The parallel plate waveguide serves as the out- or input port of the emitter/detector device. A photoconductive LT-GaAs switch is integrated into the feeding line of the antenna. This switch is biased at 10 V and optically excited for THz pulse generation. The optical excitation along with the incident THz electric field induces a photocurrent in the detector antenna which is recorded by the data acquisition system. The above components have been described in more detail in an earlier publication [24].

For the transmission measurements the tube waveguide is placed and aligned between two THz antennas at a distance of a few micrometers to the parallel plate waveguide port. A linearly polarized electromagnetic THz wave is coupled from the parallel plate waveguide into the tube waveguide, where it excites a fundamental quasi-HE11 mode. Time resolved measurements of the transmitted THz wave are performed by changing the relative time delay between the excitation and the detection pulse with a mechanical delay stage. The excitation beam is chopped at 12 kHz for lock-in detection of the transmitted THz signals. In order to determine the complex frequency dependent propagation constant γ(f)=α(f)+iβ(f) of the waveguide we perform comparative time-domain transient measurements with waveguides of differing lengths in a range from 30 mm to 150 mm. The time-domain transients are Fourier transformed and the frequency-dependent propagation parameters are derived from these data sets using linear regression fits as described in [25]. The effective relative permittivity of the waveguide ε_r,eff (f) is calculated from β(f) using the relation ε_r,eff(f)=(βc/2πf)².

 

Fig. 7. Time-domain THz signals measured at two tube waveguides of 30 mm and 40 mm length. The data has been time-shifted and normalized to the peak amplitude of the 30-mm-signal to point up dispersion and attenuation effects.

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3.2 Results

As an example in Fig. 7, two THz signals are compared after transmission through tube waveguides of 30 and 40 mm length. For a better illustration of attenuation and dispersion effects, the signals have been shifted in time to the point of origin and normalized to the peak of the 30-mm-signal. The multiple signal oscillations originate from the limited bandwidth property of the impulsively driven Yagi-Uda THz emitter, which has been designed for optimal emission and reception of frequencies around 0.5 THz. Except from a slight pulse broadening, the shape of both signals remains approximately the same. We observe an amplitude reduction of ca. 1.8 % per mm.

The measured frequency-dependent attenuation and effective relative permittivity of the TW are now compared with simulation data. In our simulations we assume for fused silica the above mentioned literature data (ε_r=3.8 and tanδ=0.001) and comparatively ε_r=4.5 and tanδ=0.002. For the polyimide cladding ε_r=3.5 and tanδ=0.008 are chosen. Fig. 8(a) shows the attenuation of the TW at frequencies in the range of 0.3 to 0.7 THz. The simulation data exhibit a monotonic increase of the attenuation with frequency, which is also observed at the measured curve for frequencies around 0.5 THz. Since the emitter and detector devices are optimized for a frequency f=0.5 THz, we obtain the most reliable results nearby this frequency. The strong fluctuations of the measured data at frequencies below 0.4 THz and above 0.6 THz result from a reduced signal-to-noise ratio. The measured data agree very well with the simulation considering the increased loss tangent value of 0.002. The frequency-dependent effective relative permittivity is shown in Fig. 8(b). Here, the measurement agrees well with the simulation where ε_r=4.5. The deviations from the literature data might be a result of impurities within the fused silica material.

4. Discussion and conclusion

 

Fig. 8. Measured and simulated (a) attenuation α and (b) effective permittivity ε_r,eff of the tube waveguide as a function of frequency. The grey marked range exhibits a high level of noise due to the limited amplitude bandwidth of the applied emitter/detector devices. Phase noise is considerably lower than amplitude noise.

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Although the proposed THz SRW has not been available for experiments, we can already point to some of the technical consequences that the transfer from optical to THz frequencies implies for low-index discontinuity waveguides: First, one can expect that losses due to surface-roughness, which are a serious problem at optical wavelengths, will be much lower since surface-roughness on the order of THz wavelengths should be avoidable. In contrast to optical SRWs having transverse dimensions in the nm-range, the THz-range analog is basically large enough to be even mechanically self-supporting. Nevertheless, some kind of supporting low-index substrate or structure – not considered in this study – will certainly have to be added to the SRW in order to build a photonic device.

Our investigations show that low-index discontinuity single-mode dielectric THz waveguides provide an outstanding combination of high mode confinement and low transmission losses currently not realizable with any other metal-based or photonic crystal approach. With these exceptional properties they might provide some breakthroughs for a multitude of novel THz applications. Split rectangular waveguide structures appear very attractive for integrated photonic THz devices, while tube waveguides – due to their robust and flexible mechanical properties – may find their way in endoscopy-like applications or inter-device THz transmission.