## Abstract

We describe the fabrication and characterization of plasmonic waveguides based on a periodic one-dimensional array of symmetric and asymmetric T-shaped structures. The devices are fabricated in a polymer resin using conventional 3D printing and subsequently overcoated with ~500 nm of Au. Using terahertz (THz) time-domain spectroscopy, we systematically measure the guided-wave transmission properties of the devices as a function of the different geometrical parameters. Through these measurements, we find that the resonance frequency associated with the lowest order mode depends primarily on the structure height and the cap width and appears to be independent of its lateral width. We also perform numerical simulations using the same geometrical parameters and find excellent agreement between experiment and simulation. We fabricate a waveguide in which the lateral width of the T-shaped structures is tapered in a linear fashion. While the spectrum of this device is similar to one without tapering, we observe relatively little reduction in the mode size, even as the structure width is reduced by a factor of eight.

© 2014 Optical Society of America

## 1. Introduction

The ability to control the propagation properties of optical radiation is one of the fundamental goals of optics. Waveguides offer a powerful solution to this goal, since they allow for point-to-point transport across potentially complex geometries. In the terahertz (THz) spectral range, defined as extending from 100 GHz to 30 THz [1], the progress in developing such devices has been dramatically slower. The difficulty, in large part, lies in the fact that many conventional dielectrics and semiconductors are highly lossy in this frequency regime. Nevertheless, a number of interesting implementations have been developed [2]. Many of these device geometries have been inspired by existing microwave and optical waveguides. In the former category, for example, a variety of metal-based waveguides, including parallel plate waveguides [3], slot waveguides [4], hollow cylindrical waveguides [5,6] and single metal wires [7] have been shown to allow for low loss and, in some cases, low dispersion [3,4,7] propagation. In the latter category, a number of different dielectric-based waveguide implementations have been described, including solid core dielectric waveguides [8,9] and plastic microstructured waveguides [10].

In recent years, an alternative approach based on guiding surface plasmon-polaritons (SPPs) [11] on structured metal films has been pursued. In the earliest implementations, the metal was etched with rectangular holes to form either one-dimensional [12] or two-dimensional [13] structured surfaces. In general, tailoring the surface structure of a metal can dramatically alter the dispersion properties of propagating bound electromagnetic waves [12,14,15]. In the case of waveguides that are based on holes that either partially [13,16] or completely perforate the metal film [12,17], the longest aperture dimension, usually transverse to the propagation direction, determines the frequency of the lowest order propagating mode. Subsequently, a number of interesting theoretical and experimental studies have examined a variety of other geometries that allow for low loss THz guided-wave propagation [18–24]. Among these, there are a number of proposals that rely on building different metallic shapes on a metal surface. A unique aspect of two of these geometries is the fact that the in-plane transverse dimension does not affect which frequencies are guided [20,22]. Waveguides based on such architectures may allow for focusing of radiation, by simply tapering the transverse dimension of the structures along the propagation axis. Of the two geometries, only the domino structure has been examined experimentally at microwave [25] and THz [23] frequencies. While the domino structures are interesting, waveguides based on inverse-L shaped structures [21], and variations of that geometry, have been predicted to allow for tighter confinement of the propagating wave. This may be useful for tightly focusing the guided mode, as mentioned above, and allow for better performance on curved geometries.

In this submission, we fabricate and characterize a family of waveguides that have been predicted to allow for higher lateral confinement than all other predicted geometries. The devices consist of one-dimensional arrays of symmetric or asymmetric metallic T-shaped structures fabricated on a metalized substrate. Given the significant complexity in fabricating such structures using conventional lithographic techniques, we extend our recent work on 3D printing of plasmonic structures to fabricate these devices [26]. The symmetric structures are T-shaped, while the most extreme asymmetric structures have the shape of an inverted L. Using THz time-domain spectroscopy, we measure the transmission spectrum and other guided-wave properties of the devices as a function of the structure height, lateral width, asymmetry and periodicity. We also perform 3D numerical finite-difference time-domain (FDTD) simulations to validate our observations. Experimentally and numerically, we find that the lowest order resonance is defined only by the height and longitudinal overhang in the individual structures, thus confirming that the resonance frequencies are independent of the lateral width. Such a finding would suggest that laterally tapered waveguides based on these structures would be well suited to strongly focus the guided THz radiation. However, in contrast to theory and numerical simulations, we find that tapered waveguides do not concentrate THz radiation very strongly. We discuss the reasons for this.

## 2. Experimental details

In Fig. 1(a), we show a schematic diagram of a representative T-shaped waveguide. The individual structures are composed of two parts: a lower support that has a height of h_{1} and a width of g and an upper structure that has a height of h_{2} and total width, along the waveguide axis, of w + g, where w = w_{1} + w_{2}. Thus, for a symmetric T-structure, w_{1} = w_{2} = w/2, while for the most asymmetric L-shaped structure, w_{1} = 0 and w_{2} = w or w_{1} = w and w_{2} = 0. Each of the individual structures has a lateral width, L, with the spacing between structures given by p, the periodicity. All of the fabricated devices are 7 cm long. Experimentally, the waveguides were made using a commercially available professional grade 3D printer (Object EDEN 260V), which had a printing resolution of 600 dpi in the x-y plane and 1600 dpi along z-axis. The devices were printed using a polymer resin (Vero White) on a support platform. After the resin solidified, the devices were detached from the support and sputter deposited with Au. The measured surface roughness of the printed device was ~3 µm rms, which was substantially larger that the ~50 nm rms surface roughness associated with metal foils that have previously been used to fabricate other plasmonic waveguides [12,16,17]. Among the various deposition techniques, sputtering is the most omnidirectional and allows for more uniform metal coatings on complex non-planar geometries. To ensure that the bottom surfaces of the T or inverted-L structures were coated with Au, we turned the samples multiple times between deposition runs. On the upper (planar) surfaces of the waveguide, the measured Au film thickness was ~500 nm. We have previously shown that the properties of the propagating SPP are determined by the surface metal layer and are independent of the underlying material once the metal thickness is greater than approximately twice the skin depth (δ~150 nm at 0.3 THz frequency) [27]. We do not know the thickness of the Au on the bottom surfaces. However, based on the transmission properties, we do not believe that the metal thickness is sufficiently thin anywhere that it plays a detrimental role in the guided-wave properties. In Figs. 1(b) and 1(c), we present images of the metalized symmetric T and asymmetric inverted-L structures, respectively.

In Fig. 1(a), we also show a simplified excitation and detection scheme to measure the guided-wave properties. We used an amplified ultrafast Ti: Sapphire laser as the optical source for generation and coherent detection of broadband THz radiation using nonlinear optical crystals [28,29]. The output of the laser was split 80:20 to yield the optical pump and probe beams, respectively. Broadband THz radiation was generated using a 1 mm thick <110> ZnTe crystal. This THz radiation was collected using a parabolic mirror and focused onto one end of the waveguide using a 150 mm focal length TPX (polymethyl-pentene) lens. In contrast to earlier work where we used a common input groove to couple freely propagating THz radiation to SPPs, here the (corrugated) structures acted as the input coupler to the waveguide. Regardless of whether we used a groove fabricated into the substrate of the device or the T-shaped structures that sat above the substrate surface, the structured geometry acted to scatter a fraction of the normally incident radiation into SPPs that propagated along the direction of the waveguide. We used a second 1mm thick <110> ZnTe crystal to measure the properties of the propagating SPPs via electro-optic sampling [28,29]. By moving the ZnTe detection crystal to different points along x, y and z-axes, we could measure the propagation properties of the E_{z} component of the SPP wave anywhere above the surface of the waveguide. While the phase information is important, in the results that follow, we present only the spectral amplitude of E_{z}, |E_{z}|.

We performed 3D numerical finite-difference time-domain (FDTD) simulations using a commercial software program XFDTD (Remcom Inc.) to model the propagation properties of the different waveguide geometries. The metal was modeled as a perfect electrical conductor. While such an approximation was certainly not strictly valid, it appeared to model the resonance frequencies well, since they were determined by the geometrical parameters of the device. However, such an assumption underestimated the resonance linewidths, since metal losses were not taken into account. The surrounding dielectric medium was assumed to be air. We used a spatial grid size of 25 µm, which was sufficient to ensure convergence of the numerical calculations, and perfectly matched layer absorbing boundary conditions for all boundaries. For the input electric field, we used a dipole source placed just above the substrate surface with a temporal profile determined by the derivative of a Gaussian pulse that had the same bandwidth (and similar pulse shape) as was available in the experimental work. All simulated results were obtained by recording the field properties at specific spatial points, typically centered on the waveguide (except for y-axis data) in order to match the experimental measurements.

## 3. Experimental results, simulation and discussion

We begin by first considering the role of symmetry on the guided-wave properties of the devices. In Fig. 2(a), we show the experimentally measured transmission spectra for five separate 7 cm long waveguides, in which the values of w_{1} and w_{2} were varied, but where w = w_{1} + w_{2} = 200 µm was kept constant. For all five waveguides, the other device parameters – g = 200 µm, h_{1} = h_{2} = 150 µm, p = 500 µm, L = 800 µm – were kept constant. In comparing the spectra, the frequencies associated with the resonance peaks appear to shift slightly relative to one another. However, all of the dips on the high frequency side of the resonance occur at the same frequency, 0.126 THz. This is consistent with our earlier finding that the dips (minima) on the high frequency side of each resonance are the relevant parameter, not the frequencies associated with the resonance peaks [30].

In Fig. 2(b), we show the numerically simulated transmission spectra for the exact same waveguide parameters. There are some similarities and differences between these two sets of spectra. First of all, in the numerical simulations, there are no obvious dips on the high frequency side of each resonance. We have previously observed this in simulations related to other guided-wave devices [16]. Nevertheless, the experimentally determined frequencies corresponding to the sharp dips and the frequencies corresponding to extrapolation of the trailing edge of each resonance (0.129 THz) in the simulated data match extremely well. Furthermore, in the simulated data, different waveguide parameters lead to different resonance amplitudes. For example, waveguides with w_{1} = 0 µm and w_{1} = 200 µm not only have the highest amplitude, but also have the same amplitude, as would be expected by symmetry; waveguides with w_{1} = 50 µm and w_{1} = 150 µm have the next lower amplitude, while devices with w_{1} = 100 µm (the symmetric case) have the lowest amplitude. The waveguides with w_{1} = 0 µm and w_{1} = 200 µm may exhibit the highest amplitudes because they have been predicted to have the highest lateral confinement, which may lead to higher overall confinement [22]. Although there is some variation in the resonance amplitudes in the experimental data, there is no equivalent pattern. Finally, based on the periodicity of the structures, p = 500 µm, the Bragg frequency is given by ν_{B} = c/2p = 0.3 THz, where c is the speed of light in vacuum. Any propagating mode with a transverse wave number beyond the first Brillouin zone would exhibit high propagation loss and, therefore, would not be evident in the measured spectral detection window.

Next, we consider the role of structure height in determining the guided-wave properties. In Fig. 3(a), we show the experimentally measured transmission spectra for five separate 7 cm long waveguides, in which the values of h_{1} and h_{2} were varied, but where h = h_{1} + h_{2} = 400 µm was kept constant. As with the earlier devices, the other parameters – g = 200 µm, w_{1} = w_{2} = 100 µm, p = 500 µm, L = 800 µm – were constant across all five devices. We used a somewhat larger value of h here than was used for the devices in Fig. 1. The primary reason for this was to increase the number of variants (values of h_{1}) that we could fabricate. Since the symmetric T-shaped structures used here are larger than those used for Fig. 1, the high frequency side minima occurred at a slightly lower frequency, ~0.105 THz. In Fig. 3(b), we show the numerically simulated transmission spectra for the exact same waveguide parameters. Here, the frequencies corresponding to extrapolation of the trailing edge of each resonance in the simulated data occur at ~0.11 THz, which is in good agreement with the experimental data. Finally, we note that the resonance amplitudes are all approximately similar in the simulated spectra, but show somewhat greater variation in the experimental spectra.

We now consider the role of periodicity in determining the guided-wave transmission properties. In Fig. 4(a), we show the experimentally measured transmission spectra for four separate 7 cm long waveguides, in which the value of p was varied between 500 µm and 800 µm. As with the earlier devices, the other parameters – g = 200 µm, w_{1} = w_{2} = 100 µm, h_{1} = h_{2} = 100 µm and L = 800 µm – were constant across all four devices. In addition, we once again used a different value of h. Aside from the issue of fabricating a wide enough range of devices (i.e. sufficient variation in a geometrical parameter), using different values of h was necessary to develop a model for the resonance frequencies, as discussed below. Since the symmetric T-shaped structures used here are smaller (lower value of h) than for the waveguides used for Figs. 1 and 2, the high frequency side minima occur at a slightly higher frequency, ~0.165 THz. However, there is some spread in the location of the frequency minimum (~0.008 THz between the four devices). In Fig. 3(b), we show the numerically simulated transmission spectra for the exact same waveguide parameters. Here, the frequencies corresponding to extrapolation of the trailing edge of each resonance in the simulated data occur at ~0.169 THz and, although there is also some variation in this frequency, it is in good agreement with the value found from the experimentally measured spectra. These data demonstrate that the dependence upon the periodicity, p, is weak.

Finally, we consider the dependence on L, the structure width. In Fig. 5(a), we show the experimentally measured transmission spectra for four separate 7 cm long waveguides, in which the value of L was varied between 200 µm and 800 µm. As with the earlier devices, the other parameters – g = 200 µm, w_{1} = w_{2} = 100 µm, h_{1} = h_{2} = 100 µm and p = 500 µm – were kept constant across all four devices. Here, the values of h, w, g and p are identical to those for the previous set of spectra. We find that the frequency associated with the high frequency side minima all occur at 0.165 THz, with minimal variation. This is consistent with earlier theoretical predictions with so-called domino [20] and inverted-L [22] based waveguides that showed that the transmission spectrum is essentially independent of the lateral width, L. In Fig. 5(b), we show the numerically simulated transmission spectra for the exact same waveguide parameters. Here, the frequencies corresponding to extrapolation of the trailing edge of each resonance in the simulated data occur at ~0.170 THz. Again, we observe excellent agreement between experimental and simulation results.

Given the wide variation in the parameters used to fabricate the waveguides, it is useful at this point to summarize the basic results. In Table 1, we list the device parameters that were varied along with the corresponding frequencies associated with the resonance minima.

Based on the experimental and simulated resonance minima frequencies, we find that a very simple phenomenological approximation relating the frequency of the transmission minimum on the high frequency side of each resonance with the geometrical parameters of the devices is given by

It is interesting to note that Eq. (1) is approximately similar to that given in [22], where the authors theoretically examined waveguides based on inverted-L structures and found that the resonance frequency was approximately given by ν_{m}≅c / [4 (h

_{1}+ w + g)], using our notation. The sum of the geometrical parameters in this equation is slightly larger than that in Eq. (1) and likely offsets the fact that we use a slightly larger non-integer multiplier in the denominator (4.5 versus 4). Another possible explanation for the use of a non-integer value in Eq. (1) may arise from the fact that the top of the T-shaped structures is open (i.e. there is a gap between structures), while the bottom surface is entirely metalized. We have previously shown that small offsets from integer value indexes were necessary to characterize the resonance properties of waveguides based on blind holes [16,26], which are metalized on the bottom and open on top. The physical basis for this relation arises from the fact that the space between the structures appears to create a groove that has an effective depth given by (h + w), corresponding to a resonance frequency of c/[4.5(h + w)] [22].

In addition to understanding the guided-wave spectral properties of these devices, it is important to know how tightly bound the propagating wave is along the two transverse dimensions. If we first consider the out-of-plane extent of the bound propagating wave, its spatial properties can be measured by simply moving the optical probe beam along the z-axis and recording the magnitude of the E_{z} component of the electric field at the resonance peak frequency. In Fig. 6, we show the measured electric field amplitude as a function of distance above the waveguide surface (i.e. along the z-axis) for the symmetric (T-shaped) and completely asymmetric (inverted-L shaped) structures. The 1/e out-of-plane decay length is ~3 mm in the former case and ~3.4 mm in the latter case. Since a number of different waveguide geometries have been investigated experimentally, with different resonance properties, it is useful to compare these values in terms of the corresponding wavelength. Thus, for example, in earlier work for waveguides fabricated using rectangular holes in a thick metal film [12], we found that the 1/e out-of-plane spatial extent was ~1.69 mm at a resonant wavelength of ~1 mm, corresponding to a out-of-plane decay length of ~1.7 λ. For the two waveguides discussed here (Fig. 6), the resonant wavelength is ~1.75 mm and the corresponding out-of-plane decay length is ~1.7 λ (symmetric case) and ~1.9 λ (asymmetric case). Thus, waveguides, based on depressions (holes) and protrusions (T-shaped structures) appear to behave very much the same in this respect. Next, we consider the propagation properties along the x-axis for the lowest order mode. In Fig. 7, we show the loss properties for propagation along the waveguide for the same two waveguides. An exponential fit to the data for both waveguides yields a loss parameter of ~0.18 cm^{−1}, corresponding to 1/e propagation lengths of ~5.6 (symmetric T) and ~5.5 cm (inverted-L) for the two waveguides. In comparison to the waveguides fabricated in stainless steel [12], where the 1/e propagation length was ~8 cm, the waveguides described here exhibit somewhat greater loss.

While the last two sets of data give information about the mode properties, it is helpful if they can be visualized in a slightly different manner. In Figs. 8(a) and 8(b), we show snapshots of the magnitude of the electric field in the xz plane for the symmetric T and asymmetric inverted L based waveguides. In both cases, the measurement was taken at the center (laterally) of the structures, assuming an input frequency that corresponded to the resonance peak for each device. In Figs. 8(c) and 8(d), we show snapshots of the magnitude of the electric field in the xy plane for both waveguides immediately above the upper surface of the structures. It is interesting to note that the field distributions for the two waveguides are similar in nature and the xy distributions look nearly identical, though they exhibit slightly different magnitudes. It is also worth noting that the field decays very rapidly along the y-axis and beyond the structures, for snapshots in Figs. 8(c) and 8(d). While the lateral width of the structures was L = 800 µm for both sets of simulations, we observed nearly identical mode properties as L decreased. This observation, which is consistent with earlier theoretical calculations [22], is the basis for assuming that a tapered waveguide would allow for concentration of the guided-wave mode. Figure 8(e) shows the color coding applied to all four snapshots.

With these measurements and simulations in mind, we now address whether or not a tapered waveguide can concentrate the guided-wave mode as well as has been predicted [22]. In order to accomplish this, we need to measure the mode profile along the y-axis of a tapered device. We fabricated a single waveguide in which the width L linearly decreased from L = 800 µm on the input side to L = 100 µm on the output side over a length of 7 cm. Since there appeared to be relatively little variation in the guided-wave properties between devices with different structures (symmetric T to inverted L), we used symmetric T-shaped structures with h_{1} = h_{2} = 100 µm, w_{1} = w_{2} = 100 µm, g = 200 µm and p = 500 µm. In Fig. 9(a), we show an image of the laterally tapered device. The corresponding guided-wave transmission spectrum associated with the full device is shown in Fig. 9(b). Although the experimental and simulated resonance peak frequencies do not match, the frequencies associated with the resonance dips are the same and are consistent with the spectra in Fig. 5. In order to measure the guided-wave width, we simply moved the probe beam along the y-axis and recorded the magnitude of the E_{z} component of the THz electric field at the resonance peak frequency at 4 different positions along the waveguide. These positions corresponded to structure widths of L = 700 µm (1 cm from the input), L = 500 µm (3 cm from the input), L = 300 µm (5 cm from the input), and L = 100 µm (end of waveguide). In Fig. 9(c), we show the measured y-dependence for each of the positions along with a Gaussian fit to the data. The width of the lowest order guided-wave mode, Γ (full-width at half maximum (FWHM) of the fit) is: Γ = 4.8 mm for L = 700 μm, Γ = 4.4 mm for L = 500 μm, Γ = 4.0 mm for L = 300 μm, and Γ = 3.7 mm for L = 100 μm. Thus, for the peak frequency of 0.151 THz (λ = 2 mm), the widths are 2.4 λ (L = 700 µm), 2.2 λ (L = 500 µm), 2 λ (L = 300 µm) and ~1.85 λ (L = 100 µm). Interestingly, Γ does not vary appreciably, even as the structure width is reduced by a factor of 8. If we once again compare to waveguides based on rectangular holes in a thick metal film [12], the FWHM beam width was ~2.2 mm at a resonant wavelength of ~1 mm (corresponding to a FWHM beam width of 2.2 λ). Thus, there appears to be little difference between these different types of waveguides in this regard also. These results are in stark contrast to the theoretical [22] and simulation results. We attribute the wider than expected guided-wave mode width to the fact that real metals exhibit loss. Such losses not only increase the resonance linewidth, but also reduce the level of mode confinement possible in a waveguide. This is consistent with earlier set of waveguides, in which higher losses corresponded to a lower level of confinement for the guided-wave mode [16,26]. Thus, the assumption of lossless metals in theoretical models can lead to significant deviations when considering mode confinement.

## 4. Conclusion

In conclusion, we have designed, fabricated and characterized THz waveguides based on a one-dimensional array of symmetric and asymmetric T-shaped structures. The devices were fabricated using conventional 3D printing and then sputter coated with ~500 nm of Au. We have measured the guided-wave transmission properties of these devices as a function of the structure asymmetry, height, width and periodicity and found that the frequency of the lowest order resonance was determined primarily by the sum of the height and the length of the overhang (w_{1} + w_{2}). However, the resonance properties appear to be independent of the level of asymmetry in the T-shaped structure. We also measured the properties of the lowest order mode along both transverse axes of the waveguide. We found that the 1/e out-of-plane decay length was ~1.8 λ and varied slightly with geometrical changes. Similarly, we found that the mode profile along the y-axis appears Gaussian in shape, as is true of many other waveguide geometries, and has a FWHM width ~2 λ. Finally, we fabricated a tapered waveguide and found that the propagating mode does not become tightly focused, as the structure width becomes much smaller than a wavelength. We attribute this to the fact that metal losses lead to reduced confinement of the guided-wave mode. Aside from the details of the experimental results described here, the excellent agreement between the experimental and simulation results among all of the different devices demonstrates the high level of reproducibility in the 3D printing process for THz device fabrication and suggests exciting new possibilities for device development.

## Acknowledgments

This work was supported by the NSF MRSEC program at the University of Utah under grant # DMR 1121252.

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